@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Hürlimann, Martin Dominik"@en ; dcterms:issued "2010-10-11T17:17:15Z"@en, "1989"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """A new type of atomic hydrogen maser that operates in a dilution refrigerator has been developed. In this device, the hydrogen atoms circulate back and forth between a microwave pumped state invertor in high field and the maser cavity in zero field. A prototype maser with a small maser cavity has been built and the results obtained so far are encouraging. Stable maser oscillations were observed for temperatures of the maser bulb between 230 mK and 660 mK and for densities up to 3 x 1012cm⁻³. The short term frequency stability was measured with the help of two high quality quartz crystal oscillators by the three-cornered-hat method. The observed fractional frequency fluctuations for an averaging time of 1 s were 6.3 ± 3.7 x 10⁻¹⁴, which is lower than the results from the best room temperature masers. In conjunction with the stability measurements, the phase noise of the maser electronics was investigated. In particular, the temperature dependence of the phase noise of the cooled preamplifier was measured and it was shown that anomalous high noise levels between 2.2 K and 4.2 K are caused by the boiling of the liquid helium. From the temperature dependence of the maser frequency, the binding energy E[sub B] of H on ⁴He could be determined to a high precision. The result is E[sub B] = 1.011 ± 0.010 K. An extensive computer simulation program has been written that models the operation of the cryogenic hydrogen maser. It has been used to analyze and interpret some of the data. In addition, this simulation program is helpful for the design of an improved second generation cryogenic maser. Based on the present data and the model calculations, a new pumping scheme is proposed that is expected to increase the efficiency of the state invertor significantly."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/29116?expand=metadata"@en ; skos:note "C R Y O G E N I C H Y D R O G E N M A S E R By . Martin Dominik Hiirlimann Dipl. Natw. ETH, Swiss Federal Institute of Technology, Zurich A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1989 © Martin Dominik Hiirlimann, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department DE-6 (2/88) Abstract A new type of atomic hydrogen maser that operates in a dilution refrigerator has been developed. In this device, the hydrogen atoms circulate back and forth between a mi-crowave pumped state invertor in high field and the maser cavity in zero field. A prototype maser with a small maser cavity has been built and the results obtained so far are encouraging. Stable maser oscillations were observed for temperatures of the maser bulb between 230 mK and 660 mK and for densities up to 3 x 1012cm-3. The short term frequency stability was measured with the help of two high quality quartz crystal oscillators by the three-cornered-hat method. The observed fractional frequency fluctuations for an averaging time of 1 s were 6.3 ± 3.7 x 10-14, which is lower than the results from the best room temperature masers. In conjunction with the stability measurements, the phase noise of the maser electronics was investigated. In particular, the temperature dependence of the phase noise of the cooled preamplifier was measured and it was shown that anomalous high noise levels between 2.2 K and 4.2 K are caused by the boiling of the liquid helium. From the temperature dependence of the maser frequency, the binding energy EB of H on 4He could be determined to a high precision. The result is EB = 1.011 ± 0.010 K. An extensive computer simulation program has been written that models the opera-tion of the cryogenic hydrogen maser. It has been used to analyze and interpret some of the data. In addition, this simulation program is helpful for the design of an improved second generation cryogenic maser. Based on the present data and the model calcula-tions, a new pumping scheme is proposed that is expected to increase the efficiency of the state invertor significantly. ii Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgement x 1 Introduction 1 2 Theory of Maser Operation 8 2.1 Hyperfine states of atomic hydrogen 8 2.2 Interaction with rf magnetic field 11 2.3 Relaxation, Bloch equation 12 2.4 Stimulated emission 14 2.5 Power emitted by atoms 17 3 Properties of atomic hydrogen at low temperatures 20 3.1 Interactions with the walls 20 3.2 Recombination 23 3.3 Hyperfine frequency shift 25 3.4 Diffusion 27 3.5 Spin exchange 27 4 Considerations on frequency stability 32 iii 4.1 Characterization of frequency stability 32 4.2 Intrinsic frequency instabilities 33 4.3 Electronic Noise 35 4.4 Frequency shifts 38 4.4.1 Magnetic field shifts 38 4.4.2 Buffer gas and wall shifts 39 4.4.3 Cavity pulling and spin exchange shifts 40 5 The U B C cryogenic hydrogen maser 44 5.1 Principle of closed cycle cryogenic hydrogen maser 45 5.2 General characteristics of the design of the prototype apparatus 48 5.3 Maser cavity 51 5.3.1 Tuning and coupling of maser cavity 51 5.3.2 Maser bulb 54 5.3.3 Cooling bath 55 5.3.4 Thermometry 56 5.4 State selector 57 5.4.1 Microwave cavity 57 5.4.2 Relaxing foil 59 5.4.3 State selector magnet 59 5.5 Magnetic shielding of the maser cavity 61 5.6 Hydrogen discharge source 63 5.7 Electronics for state selector microwave pump 65 5.8 1420 MHz detection system 67 6 Experimental Results 73 6.1 Start up procedure 73 iv 6.1.1 Cool down 73 6.1.2 Adjustment of bias field at the maser cavity 74 6.1.3 Cross relaxation in the maser bulb 75 6.1.4 State selector magnetic field 79 6.2 Power versus density measurements 80 6.2.1 Density calibration 80 6.2.2 Recombination 85 6.2.3 Power output of maser 87 6.3 Temperature dependence of maser frequency 90 6.4 Frequency stability 96 6.4.1 Three-cornered-hat method 96 6.4.2 Details of stability measurements 98 6.4.3 Results of the stability measurements 100 6.5 Flicker phase noise measurements 105 6.5.1 Maser preamplifier 105 6.5.2 Phase noise of receiver 109 7 Computer simulation of the maser 113 7.1 Model of the maser 113 7.1.1 Flow 116 7.1.2 One-body spin relaxation 117 7.1.3 Microwave pumping in the state selector cavity 119 7.1.4 Spin exchange relaxation 120 7.1.5 Maser action 121 7.1.6 Overall rate equations 122 7.2 Results of the simulation 123 v 7.2.1 Choice of parameters 123 7.2.2 Power output of maser and state selector efficiency 125 7.2.3 Simulation results relevant for density calibration 128 8 Conclusions and outlook 131 8.1 The present cryogenic maser 131 8.2 Suggested improvements of the cryogenic maser design 132 8.2.1 Full size maser cavity . 132 8.2.2 Proposed new pumping scheme 133 8.2.3 Optimization of maser 135 8.3 Open questions 137 Appendices 138 A Magnetic moment operators 138 B The Slater equation 140 C Measures of frequency stability 143 D Magnetic field of a solenoid inside a long superconducting shield 147 E Eigenmodes of a sapphire loaded microwave cavity 153 E.l General Solution 153 E.2 Boundary conditions 156 E.3 Application to the cryogenic hydrogen maser 159 Bibliography 163 vi List of Tables 3.1 Binding energies of atomic hydrogen to different substrates 22 3.2 Calculated values of the spin exchange parameters 28 C.l Transformations between power-law spectral density Sy(f) and Allan vari-ance o-y2(r) 145 vii List of Figures 1.1 Basic scheme of the conventional hydrogen maser 2 2.1 Hyperfine energies as a function of magnetic field 10 2.2 Magnetization and effective magnetic field in rotating frame 15 5.1 Principle of UBC cryogenic hydrogen maser 46 5.2 Schematic drawing of relaxing foil 47 5.3 Simplified diagram of the cryogenic hydrogen maser 49 5.4 Design of maser cavity 52 5.5 Variable coupling assembly of maser cavity 53 5.6 Details of the state selector region 58 5.7 Low temperature atomic hydrogen discharge source 64 5.8 Schematics of the state selector pump electronics 66 5.9 Schematic diagram of the spectrometer 68 6.1 Transverse relaxation time versus bias field 76 6.2 Relaxation of longitudinal magnetization in the maser bulb versus time . 78 6.3 Maser resonance line width versus total density in the maser cavity . . . 84 6.4 Decay of density in maser bulb while maser is oscillating 86 6.5 Output power of the maser versus total density of hydrogen atoms in the maser bulb 88 6.6 Observed frequency of the maser versus temperature of maser bulb . . . 91 6.7 Fit to the observed temperature dependence of the maser frequency . . . 94 viii 6.8 Compilation of reported values of the binding energy of H to 4He . . . . 95 6.9 Block diagrams of electronics used for measurement of frequency stability 99 6.10 Fractional frequency fluctuations of the cryogenic hydrogen maser and the two reference quartz crystal oscillators . 101 6.11 Relative frequency fluctuations versus averaging time for cryogenic hydro-gen maser and other high quality frequency sources . 103 6.12 Spectral density of phase fluctuations of maser preamplifier for different temperatures 107 6.13 Temperature dependence of spectral density of phase fluctuations of maser preamplifier near helium A-point . 108 6.14 Measured phase noise of various components of maser receiver 110 7.1 Model of cryogenic maser used in computer simulation 114 7.2 Calculated power output of maser versus density in maser bulb 126 7.3 Calculated population inversion outside the maser bulb 127 7.4 Calculated values of p$ + p^> versus density in maser bulb 128 7.5 Calculated density correction factors for the density calibration 129 B.l Electrical model of microwave cavity coupled to output electronics and containing rf magnetization Mz 142 D.l Solenoid inside superconducting shield 148 D. 2 Solenoid with correction coils inside superconducting shield 151 E. l Schematic drawing of the sapphire loaded microwave cavity 154 E.2 Length of the sapphire loaded cavity as a function of inner diameter . . . 160 E.3 Frequencies of all cavity modes below 2 GHz in the sapphire loaded cavity 162 ix Acknowledgement First of all, I would like to thank my 'Doktorvater' Walter Hardy who supervised this work over the last seven years. He conceived this project and his active participation in all aspects of the experiment were essential for its success. I also profited from the theoretical support from John Berlinsky in the early stage of the project. Rick Cline was of great help in the cryogenic design work. In addition, he built the dilution unit used in this experiment and he also assured that an efficient computer system was always operational in our laboratory. In this experiment, we relied extensively on the excellent service of the departmental electronics shop and machine shop. In particular, I would like to thank Phil Akers and George Babinger for their superb machining of the maser parts. This project would have been impossible without the support from many people out-side this university in form of advice and loan of equipment. I am especially indebted to Stuart Crampton and Fred Walls. I had many helpful discussions with Stuart Crampton about the basic maser equations and the maser computer simulation. He was also very generous in lending us two conventional hydrogen masers. We are very grateful to Fred Walls who shipped us the two best quartz crystal oscillators in possession of the National Bureau of Standards (now NIST). Without this loan, it would not have been possible to perform the reported frequency stability measurements. David Allan and Dick Davis from NBS loaned us a GPS receiver system which enabled us to calibrate our local os-cillators. We also acknowledge the loan of a rubidium standard by David Wineland and the loan of magnetic shielding material by Bob Vessot. I would like to thank Bud Bridges for his assistance in the early design work and x Erwin Klein for his help during a low temperature run. All the members of the atomic hydrogen group at UBC have given me countless assistance and have created a very pleasant working environment. I would like to thank in particular Ichiro Shinkoda, Meritt Reynolds, Mike Hayden, Esa Tjukanov and Chris Lusher. I am grateful to the University of British Columbia and the Killam Foundation for their support in form of postgraduate fellowships. Finally, I thank my wife Elaine for her patience and for accepting the fact that low temperature experiments often take much longer than initially anticipated. xi Chapter 1 Introduction The atomic hydrogen maser is presently the most stable atomic clock for averaging times r < 106 s. In the best devices, fractional frequency fluctuations of less than IO- 1 5 for r 104 s have been achieved [1]. This extraordinary frequency stability has led to several spectacular applications: Vessot and collaborators [2] used a space-borne hydrogen maser to measure the gravitational red shift of the earth to a precision of 7 x 10-5. The success of the very long baseline interferometry (see e.g. [3]) also depends to a large extent on the stability of the hydrogen maser. Other applications of the hydrogen maser include the navigation of space probes in deep space. The atomic hydrogen maser was initially conceived and developed at Harvard Uni-versity in 1960 by H. M. Goldenberg, D. Kleppner and N. F. Ramsey [4]. They realized that collisions of atomic hydrogen with paraffin or Teflon coated walls do not perturb the hyperfine levels of hydrogen atoms significantly. In fact, it is possible to confine a low density sample of hydrogen atoms that radiate at the hyperfine transition for up to 1 s in a Teflon coated bulb, before collisions destroy the phase coherence. Such long interaction times between the hydrogen atoms and the microwave radiation give rise to a very narrow line width. This is an essential prerequisite for any atomic clock. An excellent discussion of the theory and the principle of the atomic hydrogen maser by its inventors can be found in [5] and [6]. The basic principle is illustrated in fig. 1.1. Molecular hydrogen is dissociated in an rf discharge and a beam of atomic hydrogen emerges from the source. The beam is passed through a hexapole magnet. This magnet 1 Chapter 1. Introduction 2 H-FLUX Figure 1.1: Basic scheme of the conventional hydrogen maser. Also shown is the hyperfine diagram and the typical behavior of the output power versus the hydrogen flux. Chapter 1. Introduction 3 acts as a state selector and focuses only atoms which are in the upper two hyperfine states into a Teflon coated bulb. This storage bulb is located inside a microwave cavity with a high quality factor which is tuned to the hyperfine transition at zero magnetic field, the famous '21 cm line'. The hydrogen maser uses the AF — 1, Amj? = 0 transition (| c ) to | a )), because its frequency is to first order independent of magnetic field. A small bias field is applied along the cavity rf magnetic field (Hz) to remove the degeneracy with the Amp = ± 1 transitions. If the flux of state selected hydrogen atoms exceeds a certain threshold value, stable maser oscillations can be observed. Efficient pumps remove all the hydrogen atoms leaving the maser bulb and keep the background pressure very low. It is indicated in fig. 1.1 that the output power of the maser does not increase linearly with the hydrogen flux, but it turns over and the maser stops to oscillate altogether if the flux of hydrogen atoms exceeds a critical level. The maximum power output is typically limited to about 10~12 W. There is only a relatively small range of hydrogen fluxes or hydrogen densities at which maser action is possible. In order to understand this behavior, the effect of collisions between hydrogen atoms has to be taken into account. In particular, collisions between hydrogen atoms can lead to spin exchange which both broadens the atomic line and leads to fast relaxation between the hyperfine levels that reduces the polarization of the incoming hydrogen atoms. As the density is increased, these effects increase proportionally and they are responsible for the observed density dependence of the maser output power. Although the frequency'stability of conventional hydrogen masers is already extremely high, there is demand for clocks with even better performance. Intended applications in-clude better deep space navigation and experiments searching for gravity waves. Another example that illustrates the need for better clocks is the fact that the long term stabil-ity of the period of the millisecond pulsar PSR 1937+21 exceeds the performance of all presently available atomic clocks [7]. Chapter 1. Introduction 4 The stability of state-of-the-art hydrogen masers is limited by electronic and thermal noise [1]. This will be further discussed in chapter 4 of this thesis. It is unlikely that the performance of conventional hydrogen masers, operated at room temperature, can be further improved by a significant amount, except for very long averaging times, where external factors degrade the stability. In 1978, Crampton, Phillips and Kleppner [8] pointed out that hydrogen masers operated at low temperatures might have an improved stability. Vessot et al. [1] came to the same conclusion. The main reason is the much smaller spin exchange broadening at cryogenic temperatures, which implies that the maximum available power output of the maser increases, as does the range of operating densities. In addition, the thermal noise is reduced, a cooled low noise preamplifier can be used, and the structural stability is much higher at low temperatures. However, when a conventional hydrogen maser was cooled to low temperature, it was soon found that Teflon is not a suitable wall coating at low temperature, and no maser oscillation could be observed below about 50 K [9]. The first maser signal at 10 K was observed by Crampton et al. [10] who used frozen neon as the storage surface. Unfortunately, this cryogenic wall coating evaporates slowly and its use is therefore not very suitable in a high performance hydrogen maser. Approximately at the same time, several groups started to develop techniques to study atomic hydrogen at temperatures below 1 K. These efforts have been directed towards the possible observation of the Bose Einstein condensation in spin polarized hydrogen and were initially completely unrelated to the maser research. In the first step, Silvera and Walraven [11] showed that atomic hydrogen can be stored successfully at low temperature in a container lined with a film of superfluid helium. In a series of experiments, Hardy et al. [12] studied the properties of atomic hydrogen at temperatures below 1 K and zero magnetic field by means of pulsed magnetic resonance at the hyperfine transition. They confirmed that the spin exchange broadening is at these temperatures about 1000 times Chapter 1. Introduction 5 smaller than at room temperature. In addition, they found that the hyperfine frequency shift due to the combined effect of the helium buffer gas and interactions with the walls goes through an extremum at around 0.5 K. This is therefore the natural temperature to operate a cryogenic hydrogen maser with 4He coated walls. Berlinsky and Hardy [13] analyzed in detail the theoretical frequency stability of such a device. They concluded that an improvement of up to three orders of magnitude compared to the stability of a conventional hydrogen maser operated at room temperature can be expected. This prediction has been the main motivation for the work described in this thesis. In spring 1986, three groups reported almost simultaneously the successful operation of cryogenic hydrogen masers with helium coated walls. The first report came from the MIT group [14]. In their apparatus, a hydrogen discharge source is placed inside a high magnetic field and cooled to 0.3 K. Atoms that are predominantly in the upper two hyperfine states emerge from this source and reach a microwave cavity in zero field. After the cavity quality factor was enhanced by external feedback, self-sustained maser oscillations were observed. This apparatus was not intended to be a stable frequency source, but it was used as a rather elegant test of the efficiency of their hydrogen source. This source has been used in their experiments on the magnetic confinement of atomic hydrogen [15]. About a week after the MIT group, we observed the first maser signal in our apparatus [16]. In our maser, the hydrogen atoms circulate back and forth between a microwave pumped state selector and the maser cavity. The principle of operation and the results obtained with our prototype maser are discussed in detail in this thesis. One month later, the joint group of Harvard University and the Smithsonian Astro-physical Observatory succeeded with their cryogenic maser [17]. The design of this maser is very similar to that of a conventional room temperature maser, except that the maser bulb is coated with a superfluid film. In order to keep the helium background pressure Chapter 1. Introduction 6 low enough and enable the beam of atomic hydrogen to reach the maser bulb, a large sorption pump was installed in their dilution refrigerator. Unfortunately, the apparatus has had problems with low temperature leaks, which has made it impossible to achieve a saturated helium wall coating, and no results of frequency stability have been reported so far. It will be interesting to compare the performance of this type of beam maser with our maser, which is based on a completely different design. This thesis is organized as follows: In chapter 2, the hyperfine structure of hydrogen and the interaction with a classical rf field are discussed. Following a similar approach as taken in the NMR literature when effects of the so-called radiation damping are dis-cussed, we derive the equations relevant for maser action. In chapter 3, the properties of atomic hydrogen at low temperatures that are important for the application of the maser are reviewed. Chapter 4 contains a summary of the possible sources of frequency fluctuations and we discuss the potential stability that can be obtained with a cryo-genic hydrogen maser. The description of the UBC prototype maser and its principle of operation can be found in chapter 5. Chapter 6 contains all the experimental results that have been obtained so far with this apparatus. They include frequency stability measurements, the measurement of the power output, the temperature dependence of the maser frequency, from which we could determine the binding energy of H on 4He, and extensive measurements of phase noise of the maser electronics. In order to be able to interpret some of the results, a computer simulation has been developed that models the operation of the maser. Numerical results and the model on which this simulation is based are presented in chapter 7. In the last chapter, the performance of the present prototype maser is summarized. It also includes suggestions on how the performance of future devices can be further improved. Some of the results of this work have been published previously. After the initial reports of the successful operation of the UBC maser [16], [18], we presented a longer Chapter 1. Introduction 7 paper at the 18th International Conference on Low Temperature Physics [19]. The results of the phase noise measurements on the cooled GaAs preamplifier have been reported in [20]. Chapter 2 Theory of Maser Operation 2.1 Hyperfine states of atomic hydrogen The Hamiltonian, describing the spin degrees of freedom of a hydrogen atom in its elec-tronic ground state in a static magnetic field Bo is H0 = ai-s - 7i(-7eS + 7 Pi) • Bo (2.1) The most recent measurement of the hyperfine frequency ^ is [21] ^ = 1420.405751773(1) MHz and the accepted values for the electronic and proton gyromagnetic ratios are [22] 7e = 1.7608592(12) x l t P s ^ T ' 1 7 P = 2.67522128(81) x l O ^ \" 1 ^ 1 In terms of the bases | ms rrii ), where ms and mt- are the electron and proton spin projections along the applied field Bo, respectively, the eigenstates of (2.1) are in order of increasing energy: , |a> = c o s * | - | , ± ) - Bintf | > (2.2) I M = \\ - \\ ' - \\ ) ( 2 - 3 ) \\c) = Bin*|-i 1) + cos0|± -|) (2.4) \\d) = \\\\,\\), (2.5) Chapter 2. Theory of Maser Operation 9 where tan(20) = ft(7e + 7P)#O a 50.607 mT B0 The corresponding energies are a Hie + 7 p ) £ o a 2 (2.6) (2.7) ED a n . ^(7e + 7p)-^0 a (2.8) (2.9) Figure 2.1 shows the field dependence of the hyperfine energies. At low densities, the natural line width of transitions between the upper and lower hyperfine states is extremely narrow. In free space the lifetime due to spontaneous emission is 3.75 x 1014 s. Experimentally, the line width is determined by other effects such as inhomogeneities in the magnetic field, the limited confinement time of the hydrogen atoms in the bulb or interactions with the walls. By choosing the transition between the | c ) and | a ) state in zero magnetic field to be the maser transition, the contribution to the line width due to magnetic field inhomogeneities can be greatly reduced because this transition frequency depends only in second order on the magnetic field. In the remainder of this chapter, we first calculate the evolution of the magnetization under the influence of an applied rf field. We then make use of a result that relates the oscillating magnetization in a microwave cavity to the induced magnetic rf field in order to take the process of stimulated emission self-consistently into account. We derive results for the population inversion in the cavity, the frequency pulling effect due to cavity mistuning and expressions for the maser output power. Chapter 2. Theory of Maser Operation 10 N o xn 0 ) > CD 5 4 3 2 1 0 - 1 - 2 - 3 - 4 - 5 ' 1 1 1 1 1 -- ^v. 1 0 .1 . 3 B . 4 O U Figure 2.1: Hyperfine energies as a function of magnetic field Chapter 2. Theory of Maser Operation 11 2.2 Interaction with rf magnetic field The interaction of atomic hydrogen with an rf field applied along the static magnetic field is governed by the following Hamiltonian: Note that the electromagnetic field is treated classically and as a consequence, we will not be able to calculate effects due to spontaneous emission with this theory. An rf field along the z-axis can only induce transitions between the | a ) and | c ) state (see Appendix A). We are effectively dealing with a two-level system and it is convenient to rewrite the Hamiltonian (2.10), confined to the 2 dimensional subspace of the | a ) and | c ) states, as the Hamiltonian of a fictitious spin \\ particle. All the quantities within the frame of this fictitious spin \\ particle will be labeled with a The density matrix is written as an expansion of Pauli matrices a H = Ho — %(—fyeSz-+ ipt z)[ioHz cos u>t (2.10) p = \\lW + .<7 (2.11) and the Hamiltonian (2.10) is expressed as (2.12) If we choose the effective gyromagnetic ratio 7 to be 1 = 7e + 7p (2.13) then it is easy to show that in zero applied field, the effective magnetic field B is given by Bx = p.oHx cos cot (2.14) By = 0 (2.15) B, = a = 50.607 mT. (2.16) ft(7c + 7P) Chapter 2. Theory of Maser Operation 12 The expectation values for the components of the effective magnetic moment fl = /17s are related to the elements of the density matrix by (fa) = + lp)(pae + Pea) (2.17) 1 (fa) = -2Hle + lP)(pac- Pea) (2.18) (fa) = U(% + lP)(Paa-Pee) (2.19) Note that both the fictitious longitudinal magnetization and the fictitious transverse magnetization are in real space magnetizations along the z-axis (see Appendix A). With the concept of the fictitious magnetization we can map our problem onto the standard NMR problem and we arrive at the same simple equation of motion for (/x): jt(») = W) * B (2.20) 2.3 Relaxation, Bloch equation In the case of the hydrogen maser, the hydrogen atoms are confined in a bulb for an average duration of Tb, the bulb holding time. In the bulb the atoms collide with each other and with the walls. All these processes affect the evolution of the magnetic moment. In general, it is difficult to calculate the effects for all these mechanisms from first principles. In many cases it has been found that all the relaxation processes can be characterized by only two phenomenological relaxation times, Ti and T2, and that the magnetization M = n#(/2) obeys the so-called Bloch equations (n# is the number density of hydrogen atoms): j M i = 7 ( M (2.21) | * . = 7 ( M x B ) , - + J, ( 2 . 2 2 ) Chapter 2. Theory of Maser Operation 1 3 We have included a source term J s, describing the rate and polarization of hydrogen atoms entering the bulb from the state selector. M 0 is the thermal equilibrium value of Mz and is given by hbJQ Mo = ? f t ( 7 e + 7 P ) » H R=5- ~ iH% + 1p)jUf;nH ( 2 . 2 3 ) It has to be emphasized that the Bloch equations are not always applicable. In the simple case where in addition to relaxation due to the finite bulb holding time there is cross relaxation present between the upper three states, more than one T i is required to describe the correct evolution of M. In chapter 7 , a different approach is discussed where different processes including the nonlinear spin exchange relaxation can be treated exactly. For the rest of this chapter, we will assume that the Bloch equations ( 2 . 2 1 ) and ( 2 . 2 2 ) are valid. We now proceed in the usual way and transform the Bloch equations to rotating coor-dinates [ 2 3 ] . This new coordinate frame is rotating around the direction of quantization (z-axis) with frequency to. The linearly polarized rf magnetic field Bx is decomposed into two counter-rotating circularly polarized components. When u is close to u)0, only one of these components is on resonance and will dominate the dynamics of the magnetization. The presence of the other component, off-resonant by 2u>, causes a slight shift in reso-nance frequency. To lowest order, this so-called Bloch-Siegert frequency shift is given by [ 2 3 ] : A u B.S. Mo(7e + 7 p ) # « 2 ( 2 - 2 4 ) In practical hydrogen masers, this shift is negligibly small and we can ignore the off-resonant component of the rf field. In the rotating frame, the magnetization and the effective magnetic field are then given by (see Fig. 2 . 2 ) : M = M 2z + M r(sinAj - cosAk) ( 2 . 2 5 ) Chapter 2. Theory of Maser Operation 14 H E / / = ^ - ^ Z + l-Hxj (2.26) MoT 2 where we have used the notation too = iBz = (2.27) 2.4 Stimulated emission We have to take into account that in our case of interest, Hx is not an applied field that is controlled by the experimenter. The rf field is generated by the oscillating magnetization in the microwave cavity and it then acts back on the magnetization. This whole process is called stimulated emission of radiation. The Bloch equations (2.21) and (2.22) describe how the rf field acts back on the magnetization. In appendix B the rf field generated by an oscillating magnetization is calculated. This purely electrodynarnic relationship can be easily applied to the quantities Hx and MT in the fictitious spin system. The angle A (see Fig. 2.2) is determined by the detuning of the cavity and is given by (B.10): A = Qi{^--) » ? 9 l { U o - u ) , (2.28) LO COc CO where Qi is the quality factor of the loaded cavity and coc is its resonance frequency. The ratio of the magnitudes of the rf field to the magnetization is to second order in A: In practical devices, the rf magnetic field is not uniform over the whole bulb of the maser. The relations derived in appendix B can still be applied if we interpret Hx as the average z-component of the rf field (in real space) over the maser bulb. The Chapter 2. Theory of Maser Operation 15 z Figure 2.2: Magnetization and effective magnetic field in rotating frame Chapter 2. Theory of Maser Operation 16 rf magnetization is essentially uniform inside the maser bulb due to the fast diffusion present. In this case, the filling factor 77 is [6, 24] \" - vu,rHf/S ^ Vb icavity H2{Y)(Pr where Vb is the volume of the bulb. We now solve the transverse Bloch equation (2.21) in the rotating frame using the self consistent field (2.29). With (2.25), (2.26) and (2.29) we obtain to second order in A: 1 dMT Wr~dT = ^ W ? Q / ( l - A2) M , + 1 (2.31) u-uo = -i^oJ/Q/AM, (2.32) These are just the equations derived by S. Bloom [25] for radiation damping. In the case of stable maser oscillation, MT has to be constant in time and from (2.31) and (2.19) we obtain the population inversion inside the maser bulb 4(1 +A2) 1 nH(Paa - Pec) = - K \\2 n T~ (2-33) From (2.32) and (2.33) then follows immediately a>-u>o = ^ (2.34) Equation (2.34) describes the pulling of the maser frequency by the cavity. As we will discuss in the next chapter, u>o is not exactly given by (2.27); the resonance frequency is slightly affected by the wall shift, buffer gas shift and spin exchange shifts. If we define an atomic quality factor by Qatom = ^ o T 2 , (2.35) then equation (2.34) can be rewritten approximately as cu-cuo w -r^-(w c-w 0). (2.36) tyatom Chapter 2. Theory of Maser Operation 17 For practical masers Qi „„ — pcc)3.s is the population polarization of atoms entering the bulb from the state selector. The electromagnetic energy stored in the cavity can be expressed in terms of rf magnetic field and the filling factor by W = ^-Hl (2.39) The power P radiated by the atoms equals the sum of the power dissipated in the cavity and coupled to the output electronics and it is related to the stored energy W by P = ^ (2.40) Chapter 2. Theory of Maser Operation 18 With equations (2.37) to (2.40) we finally arrive at an expression for the power in terms of hydrogen density and the state selector efficiency: (pec ~ Paa)s.s M > 1 4(1 + A 2) (2.41) Tb \" AkTT! a ft(7. + Tt)Vo'?^r 1r 3. In order to study the dependence of the power on density, we have to make some as-sumptions on the relaxation processes. We make here the simple assumption that the only relevant relaxation processes are due to atoms leaving the bulb and spin exchange relaxation. In addition, we assume that the spin exchange relaxation rates are propor-tional to the total density nn and that the ratio of the transverse to the longitudinal spin exchange relaxation times equals 2 [26]. This is a high temperature approximation and not quite accurate at the operating temperature of a cryogenic hydrogen maser, but qualitatively it is still correct and it is instructive to study this case. Numerical results with the exact treatment of spin exchange relaxation at low temperature are presented in chapter 7. It is convenient to define the following two densities: n o = ,,4<1 + ^> (2.42) rise = * (2.43) <7»e{v)Tb where (v) = (^~) 2 is the thermally averaged collision velocity for hydrogen and crse is the thermally averaged spin exchange cross section. [In the notation used by Verhaar [27] ase = a0(pcc — paa) + t in [6, 13] stem from a different definition of the filling factor T). Chapter 3 Properties of atomic hydrogen at low temperatures 3.1 Interactions with the walls When a conventional hydrogen maser with a Teflon coated bulb is cooled to low tem-peratures, the device stops oscillating below about 50K [28]. The processes responsible for this failure are interactions between the hydrogen atoms and the walls. At room temperature, the precessing hydrogen atoms are almost unaffected by collisions with the Teflon coated walls. Historically, the discovery of such a wall coating was a crucial step in the development of the first hydrogen maser [4]. On closer inspection, the interactions between hydrogen atoms and the walls shift the maser frequency by a small amount. This wall shift is typically —20mHz for a conventional full size maser at room temperature. The imperfect reproducibility of the Teflon coating and the associated uncertainty of the wall shift limits the accuracy of conventional hydrogen masers to about 1 x 10-12. We use a simple model to analyze the shift and broadening of the atomic resonance due to interactions between hydrogen atoms and walls. Attractive van der Waals forces between the substrate and the hydrogen atoms lead to a bound state with binding energy EB- The hyperfine frequency of the weakly polarized adsorbed atoms is shifted from the free space value by us. This frequency shift OJS is expected to be slightly anisotropic with respect to the angle between the surface and the direction of quantization [29], but here we ignore this complication. We further assume that the adsorbed atoms move freely 20 Chapter 3. Properties of atomic hydrogen at low temperatures 21 on the surface and form a two-dimensional gas which is in thermodynamic equilibrium with the three-dimensional gas. We can equate the chemical potentials to obtain the relationship between the number density of adsorbed atoms n„ and the bulk density For low densities, where the interatomic interactions are small, this relationship is given by na = n/yAexpj^j (3.1) where A = (J^ y) 2 = 1-74 n m ( y j 2 is the thermal de Broglie wavelength. The ratio of the average duration of sticking (TS) to the average time between sticking events (TB) is given by the ratio of the total number of atoms in the bound state to the number of atoms in the 3 dimensional gas. In a bulb with surface area Ab and volume Vb we get (T.) Ahna Ab (EB\\ . . In practical cases, this is a small number. Since (rg) is proportional to the inverse of the average velocity of a hydrogen atom and therefore proportional to , the average phase shift per sticking event 0 = U>S(TS) depends on temperature as 0 oc T - 1 exp {•ff-}-With the simple assumption that both TS and TB are distributed according to a Poisson statistics, it is possible to derive the frequency shift and line width of the atomic resonance in the absence of other mechanisms. The results are [23]: SUJW = 1 ^° (3.3) \\TB) 1 + 00 r - (34) At high enough temperatures where ^ <1, the line broadening (3.4) is negligible and there is only a small frequency shift linear in cj>0: 6 U w * = \"stA e X P{ § } (3'5) Chapter 3. Properties of atomic hydrogen at low temperatures 22 At lower temperatures when Q ~ 1, the process of motional narrowing is less effective and the resulting large line broadening can prevent a maser from oscillating. More detailed models for the line shape were discussed by Crampton et al. [30] and by Morrow and Hardy [31] who include the case of a sticking coefficient smaller than 1 and non-Poisson statistics for TB-In order to design a successful low temperature H maser, it becomes obvious from the above simple model that a wall coating with a very small binding energy has to be used. The best candidates are weakly polarizable, low density materials, and table 3.1 lists measured binding energies in units of temperature for a few substrates chosen with these qualities in mind. Substrate Binding Energy Reference Teflon 140 K [28, 32] solid H 2 39.8 K [30] solid Ne 34K [33] liquid He: 4He 1.0K [19] 3He 0.4 K [34] Table 3.1: Binding energies Es/k of atomic hydrogen to different substrates in units of temperature. The value for EB of Teflon was obtained by analyzing the published maser data for o -C 1. The total temperature dependent frequency shift then follows the expression: The buffer gas shift coefficient 8 was measured to be <5(4He) = —1.183 x 10~17Hzcm3 [40] and 6(3He) = -9 ± 1 x 10-18Hzcm3 [12], and the surface frequency shift fa = % was found to be /3(4He) = -49 ± 2 kHz [41] and /s(3He) = -23 ± 2 kHz [34]. It is important to note that 8 and fa have the same sign (negative). This was a key result for the application of the cryogenic H maser, because it implies that the frequency shift has an extremum where the temperature sensitivity of the maser frequency is minimized. A cryogenic hydrogen maser used as a frequency stable source has to be operated close to this frequency maximum, otherwise the requirements on temperature stabilization are unrealistic. The frequency extremum occurs at approximately 0.55 K with 4He walls and at 0.2 K with 3He walls and depends weakly on the area to volume ratio of the bulb. For a bulb with the same size as in a conventional maser (15 cm diameter), the frequency shifts at the optimal temperature are 50 mHz for 4He and 36 mHz for 3He walls, respectively. This is comparable to the wall shift at room temperature in Teflon coated bulbs. We should also mention that at sufficiently low temperatures the sticking coefficient is expected to become exceedingly small [44]. At these low temperatures there is effectively no exchange between surface atoms and bulk atoms any more and the infrequent sticking events do not broaden the hyperfine line significantly. Thus it appears possible to design a maser that is not affected by wall shifts. In such a device, the increasing Kapitza resis-tance at low temperatures forces the density to be kept low, otherwise the recombination heating would be too high. At present, it is not known at what temperature the sticking (3.10) Chapter 3. Properties of atomic hydrogen at low temperatures 27 coefficient becomes small enough for this experiment, but it is safe to predict that such a project will be experimentally very challenging. 3.4 Diffusion At the operating temperature of the cryogenic H maser, the helium vapor density is high enough that it affects the transport properties of hydrogen. The diffusion constant D of hydrogen in a 4He coated cell was measured by Hardy et al. [40] and they obtained an effective hard core diffusion cross section QD = = 20 ± 1 A 2 at T « 1 K (u is the thermally averaged relative speed of a H-He pair). Jochemsen et al. [45] analyzed this problem theoretically and predicted a fairly strong temperature dependence of Qo at temperatures below 1 K. However, the increase of the mean free path at low temperature, defined by Ac = 1 n , is mainly due to the temperature dependence of n#e. For the case of 4He surfaces at a temperature T = 0.5 K, Qp is expected to be « 12 A 2 [45], resulting in Ap fa 2 cm, which is comparable to typical dimensions of our apparatus. By decreasing the temperature to 0.4 K, the mean free path increases to about 1 m and collisions with helium atoms do not appreciably impede the motion of the hydrogen atoms. 3.5 Spin exchange So far we have only considered collisions between hydrogen and helium atoms, which cause a frequency shift without an appreciable line broadening. In comparison, collisions between two hydrogen atoms occur much less frequently at the operating temperature of a cryogenic hydrogen maser, but cause both a frequency shift and a line broadening. This line broadening limits the power output of a maser. The dominant process is electron spin exchange, effects due to dipolar interactions being much smaller. Chapter 3. Properties of atomic hydrogen at low temperatures 28 T 0.5K 300 K X0(v) -1.72 x 10\" -li -3.2 x 10\" -li -2.57 x 10\" -14 1.3 x 10\" -13 A3(t>) -1.67 x 10\" -14 -2.4 x 10\" -13 0o{v) 5.93 x 10\" -14 -9.2 x 10\" -14 7.59 x 10\" -13 -2.1 x 10-11 &2 (v) 1.08 x 10\" -15 4.3 x 10\" -10 Table 3.2: Calculated values [46] of the spin exchange frequency shift parameters A,-(u) and broadening parameters in units of cm3/s. The most extensive theoretical analysis of this problem has been recently completed by Prof. B. J. Verhaar and his collaborators [27, 46], In their fully quantum mechanical calculation, they showed that the frequency shift 6uc and the line broadening Tc due to hydrogen-hydrogen collisions are of the form: (3.11) (3.12) 8u>c = [X0(pcc - Paa) + Ai(/?cc + paa) + A2j (v)nH Tc = [Voipcc - Paa) + ) are listed in table 3.2 for T = 0.5 K and for room temperature [46]. For a given density , the magnitude of both the frequency shift and the line broadening are smaller at 0.5 K than at room temperature. This decrease is especially dramatic for the line width, where for equal densities, pcc + paa = 0.5 and pcc — paa ~ 0 r c ( r = o.5K) T C(T = 300K) 0.9 x 10\" Chapter 3. Properties of atomic hydrogen at low temperatures 29 This implies that for the same line width Tc, a cryogenic maser, compared to a con-ventional maser, can be operated at a more than thousand times higher density. At the optimal line width for frequency stability and all other parameters being equal, it is shown in the next chapter that the power output of the cryogenic maser is also roughly 1000 times higher, which results in an increased signal-to-noise ratio and a much higher . potential frequency stability for the cryogenic maser. This predicted improvement of frequency stability, which is a consequence of the anomalously small spin exchange cross section at low temperature, has been one of the main motivations for the development of cryogenic hydrogen masers [8]. The spin exchange frequency shift parameters A,(u) do not decrease at low tempera-tures by the same amount as the parameters ). As the densities are increased for the low temperature operation, the magnitude of the spin exchange frequency shift becomes in fact larger than at room temperature. The first term in (3.11) does not pose any difficulty. The dynamics of the maser keeps the population inversion nH(pcc — paa) close to zero (see equation 2.33) and proportional to T2~x. This term can then be cancelled by proper mistuning of the cavity, which also results in a frequency shift proportional to T2-1 (see equation 2.34). This is known as the 'spin exchange tuning', introduced by S. Crampton [47] and now routinely used in the operation of conventional masers. Earlier calculations of the frequency shifts induced by spin exchange [26, 48] treated the problem in the degenerate-internal-states approximation which ignores hyperfine in-teractions relative to electron-exchange interactions during collisions. With these calcu-lations only the first term in equation (3.11) was obtained. However, Verhaar et al. [27] showed that the inclusion of the hyperfine interaction during collisions gives rise to the troublesome second and third term in (3.11). These terms cannot be completely cancelled by cavity tuning, so that fluctuations in the density or in the relative populations cause directly a frequency fluctuation. These frequency fluctuations might limit the potential Chapter 3. Properties of atomic hydrogen at low temperatures 30 performance of a cryogenic maser significantly. This is discussed more quantitatively in the next chapter. Prof. Verhaar and his collaborators also calculated the effect of spin exchange on the diagonal elements of the density matrix [49]. They found that in zero field, the following three relaxation processes are most efficient: b c + + d c Gtd-,oa Gcc-.bd a b + + a d c + c Gcc— a^a a + a The backward rate constants are related to the forward rate constants by a Boltzmann factor: Gij-ki = G W i j e x p j — 1 (3.13) At higher temperatures, a longitudinal relaxation time constant Tit3e for spin-exchange can be obtained from these rate constants. The rate equation for n#(/?aa — Pec) = na — nc due to the three considered spin exchange processes is — (na — nc) = —2 (2 Gaa-+cc + G00—M) na2 + 2 (2 Gcc->aa 4- Gcc^bd) nc2 + 2 (Gbd->aa - Gbd^cc) nbnd (3.14) At higher temperatures, Stoof et al. [49] show that Gu-^aa — Gbd-*Cc, in addition, the Boltzmann factors (3.13) can be replaced by unity, and close to equilibrium na 4- nc ~ \\njj-With these simplifications, the rate equation (3.14) becomes: ^ ( n 0 -nc) ~ - [(2Gcc_aa + Gbd^aa) nH] (n„ - nc) (3.15) from which we identify T{^e T\\^e = (2GCc-+aa + Gbd^aa) Tiff (3.16) Chapter 3. Properties of atomic hydrogen at low temperatures 31 Within the degenerate-internal-states approximation and at high temperatures where the Boltzmann factor is unimportant, the ratio of the longitudinal relaxation time constant T\\iSe to the transverse relaxation time constant T 2,se = T c - 1 for H-H collisions is simply [26] ^ = \\ (3-17) Desaintfuscien et al. [32] verified this relationship with a H-maser operated between 77 K and 363 K. At low temperatures however, it is not possible to define a unique Ti | S e, as can be seen from expression (3.14). Morrow and Berlinsky [50] also considered spin exchange relaxation between adsorbed atoms on the helium surface. At the densities and temperatures relevant to the maser, the surface densities are so small that this mechanism is not important. Chapter 4 Considerations on frequency stability 4.1 Characterization of frequency stability The quality of frequency and time standards can be specified by two quantities: accuracy and stability. A clock of a certain type is accurate if its frequency is known exactly and reproducible from one device to the next. Atomic clocks are much more accurate than cavity oscillators and crystal oscillators, where the output frequency depends on some macroscopic length. At present, the most accurate established frequency standards are atomic cesium beam devices and have in fact been chosen to define the unit of second. The accuracy of the best Cs clocks, i.e. the degree to which their frequencies agree, is currently about 10~13 or better; this is about ten times better than the accuracy of state-of-the-art hydrogen masers. Cryogenic hydrogen masers are not expected to be much more accurate. The limiting wall shift at low temperature is comparable in magnitude to that for room temperature masers, and additional problems due to the spin exchange shifts may also play a role in determining the accuracy. Stability is a measure of how constant the frequency of a single device is over a given averaging time. For averaging times r between Is and 106s, conventional hydrogen masers are presently the most stable atomic clocks. Frequency fluctuations of less than 10-15 for r = 1 hour are measured with the best devices [1]. In this chapter we discuss the physical origin of these fluctuations and show that a low temperature operation of a hydrogen maser promises to result in substantially smaller frequency fluctuations and 32 Chapter 4. Considerations on frequency stability 33 therefore improved stability. The specification of frequency stability requires some care. Many stable frequency standards, such as high quality quartz crystal oscillators, exhibit a component of 1// flicker frequency noise, in which case the classical variance of the fractional frequency averaged over a time r diverges. For this reason, the so-called 'Allan variance' o-J/2(r) is used instead, which is easy to measure and it converges for all the major noise types observed in precision oscillators. In the frequency domain, the spectral density of phase fluctuations S^if) is most often used to characterize the frequency stability. The defini-tions of both quantities and their relationship is discussed in appendix C. In this thesis, all frequency stabilities mentioned should be interpreted as ay(r) in the time domain or as Sv(f) in the frequency domain. 4.2 Intrinsic frequency instabilities The dominant source of intrinsic frequency instabilities for a hydrogen maser operated at T « 0.6 K is the background of thermal photons that are present in the maser microwave cavity. The associated electromagnetic field is at random phase relative to the radiation produced by the maser process. In the high temperature limit (T >> ^ = 68.17mK), the resulting frequency instability (square root of Allan variance) for r ^ > <^ ' °m is given by [51]: Sf f i int \\2QltomPT) The atomic quality factor Qatom is related to the transverse relaxation time Ti by Qatom = 2wo^2- P is the power emitted by the atoms. In this expression, T is the temperature of the background photons which depends on the temperature of the maser cavity (and of the isolator termination at the output of the cavity, if present). In our apparatus, the hydrogen atoms are not in \"direct contact with the walls of the microwave Chapter 4. Considerations on frequency stability 34 cavity. This permits, at least in principle, to reduce the intrinsic frequency instabilities by lowering the temperature of the cavity and isolator below the temperature of the hydrogen atoms and the helium film, which has to be kept at around 0.6 K in order to minimize the temperature dependence of the frequency shift. However, the intrinsic frequency instabilities cannot be decreased to arbitrary small values by this method. At lower temperatures (T « ^f2-), spontaneous emission starts to limit the intrinsic stabil-ity, and in this case, the thermal energy kT in equation (4.1) has to be replaced by the photon energy hu>0 [52]. In order to gain some more physical insight, we can rewrite (4.1) approximately as Sf f i n t Watom W / where j-f is the signal to noise ratio of the output amplitude (as opposed to the output power). At low temperatures, the noise term is obviously smaller, but at the same time the signal is larger due to the reduced spin exchange cross section. These two factors result in a large reduction of the intrinsic frequency instabilities. In order to discuss this more quantitatively, we will make use of the expressions for the power output P and the line width derived in chapter 2. The power output P as well as the atomic quality factor Qatom depend on the number density of hydrogen n#. The intrinsic frequency stability can be optimized by operating the maser at the density nopt which maximizes the product Q2atomP- By using expressions (2.45) and (2.46), we obtain for nopt: KHcc faa)s.s T ° 4kT ~ nie In order to simply the expressions, we made in the last step the assumptions that ^ < (pcc - Paa)s.s and that < (pcc - paa)s.s- At the operating temperature of a cryogenic hydrogen maser (T w 0.6 K), this is usually fulfilled since ~ and Chapter 4. Considerations on frequency stability 35 jgp = 2.84 x 10 2. With the same simplifications, the power at nopt becomes 1 Vb Popt « Ti^OTrripcc ~ Paa)s.snSe (4.4) Z lb and we arrive at 61 f * ( y r - i (4.5) int \\hVoVb(pcc- Paa)s.sJ 4.3 Electronic Noise In all hydrogen masers it is found that the frequency fluctuations for short averaging times T are much larger than those predicted by (4.1). This is a result of the fact that the hydrogen maser is intrinsically a low power device, and the microwave signal has to be first amplified before it can be further processed. This amplification process adds noise to the maser signal, which in fact dominates the short term stability. For most state-of-the-art low noise microwave amplifiers, the dominant noise at the frequency and power levels of interest is white phase noise. Its magnitude can be char-acterized by the noise temperature T/v of the amplifier, so that the spectral density of phase fluctuations is given by SM) = ^ (4.6) -* amp where Pamp is the power level at the input of the amplifier. In terms of the coupling coefficient /3 introduced in appendix B (see B.5), Pamp f ° r the first amplifier is related to the power emitted by the atoms P by Pamp = ^ _^ pP (4-7) The power input at subsequent amplification stages is increased by the gain of the pre-ceding stages, so that in practice, only the white phase noise of the first stage contributes significantly to the total phase noise. Chapter 4. Considerations on frequency stability 36 Following Vessot et al. [1], the electronic noise contribution to the fractional frequency fluctuations (square root of Allan variance) is given by 81 f ™ * 1 (4.8) elec \\ PamV J U0T where B is the noise bandwidth of the receiver system. In the derivation of (4.8), two assumptions were made: It was assumed that the Allan variance is measured with some finite dead time and therefore equation (C.12) instead of the expression in table C l was used for the translation from Sv(f) to ay2{r). In the absence of dead time, equation (4.8) has to be multiplied by In addition, it was assumed that a heterodyne re-ceiver system is used with a simple double balanced mixer. The resulting signal at the intermediate frequency /t-/ ( 50.6 mT or alternatively, to use the transition between the | d ) and | a ) levels in zero field as the maser transition. For the latter approach, the sensitivity is given by 6f = 6B0 = 1.399 x 1010 ^ SB0 This way, sensitivities much higher than those obtained with SQUIDS can be achieved at least in principle, provided that a high enough intrinsic frequency stability can be retained. The main experimental difficulty lies in the severe requirement on the homo-geneity of the applied magnetic field. Without a high homogeneity, the atomic line width becomes too broad for maser action. 4.4.2 Buffer gas and wall shifts The buffer gas and wall shifts were already discussed in chapter 3. They are given by f-fo = *n*, + /.^Aexp{||} (4.13) This combined frequency shift has an extremum as a function of temperature which occurs for 4He coated walls at around Text ~ 0.55 K. We are forced to operate the maser very close to this temperature Text in order to minimize the sensitivity to temperature fluctuations. When the maser is kept at Texi, the temperature has to be kept stable to at least 10 [iK if a frequency stability of ^ = 1.4 x 10~18 should be reached. This level^of temperature stability is well within the capability of modern low temperature technology. There are other mechanisms related to the helium buffer gas that set similar require-ments on the temperature stability. The strong temperature dependence of the helium Chapter 4. Considerations on frequency stability 40 cavity Votom vapor density implies that temperature fluctuations cause fluctuations in the dielectric constant e of the buffer gas and therefore fluctuations in the resonance frequency of the maser cavity. According to (2.36), the resulting fluctuations of the maser frequency are attenuated by Q^t'om • If we make the pessimistic but simplifying assumption that the helium buffer gas permeates the whole cavity, we can easily estimate the importance of this effect: S£ = Qi Sf f maser Qatom f For 4He, the dependence of the dielectric constant on the number density nue is accurately described by e - 1 = 2.60 x IO\"24 cm3 nHe(T) (4.15) Together with (3.6) this allows us to relate a temperature fluctuation to a change of the maser frequency. For Q ^ — 10~5 and Text — 0.55 K, a temperature fluctuation of ST — 3 fiK gives rise to ^ = 1.4 x 10-18. In actual devices, the effect will be considerably smaller, because the helium buffer gas is confined to the bulb in the center of the cavity where the electric fields are weak. The size of the effect can be determined with a perturbation calculation. 4.4.3 Cavity pulling and spin exchange shifts As we have seen, frequency detuning of the maser cavity causes frequency shifts in the maser which are attenuated by Q f : V atom r e 1 A Ql , . j. \\ J - JO = ITTrT - n w° - JO) lit ±2 diatom For a potential stability ^ = 1.4 x 10~18 and a ratio Q^tlgm = IO-5, this implies that the resonance frequency of the microwave cavity has to be stable to at least 0.2mHz. Thermal effects on the size of the cavity are not serious at low temperatures, because Chapter 4: Considerations on frequency stability 41 the coefficients of thermal expansion become exceedingly small at temperatures below 1 K. However, great care is still required to assure that the tuning assembly, the output coupling and the possibly varying input impedance of the preamplifier do not cause excessive fluctuations in the resonance frequency fc or the quality factor Q\\. In the operation of conventional masers, the maser cavity is deliberately detuned in order to compensate for the spin exchange shifts to a high degree. According to (2.34) and (3.11), the combined shift is given by f ~ f° = 2~7rT^ + 2~7r [*°(Pcc-P™)+ *i(Pcc +P*a)(v)nH (4.16) It was shown in (2.33) that for stable maser oscillation, the population inversion (pcc—paa) is proportional to T2_1: (pcc - Paa)(v)nH = a(l + A 2 ) ^ (4.17) where we have used the same notation as in [27] by defining a as a - - • Hie + lp) 2PoVQl In conventional masers with atomic hydrogen beams, it has proved difficult to keep the beam intensity and therefore the density and relaxation time T2 in the maser bulb constant to a high degree. Prior to the calculations of Verhaar et al. [27, 46], it was thought that Ax = A2 = 0, 1 in which case the so-called 'spin exchange tuning' [47] of the maser cavity A,e = -aXQ (4.18) eliminates the possible deterioration of the frequency stability due to beam fluctuations. However, Verhaar et al. [27] showed that the shifts proportional to Ax and A2 are not 1A semiclassical treatment of Crampton et al. [55] predicted Ai = 0, A2 0. Chapter 4. Considerations on frequency stability 42 negligible at low temperatures and they studied the implications on the frequency sta-bility. For this analysis, they assumed that the only density dependent part of the Une width is the spin exchange contribution: 77T = 77V• + [°o{pcc ~ Paa) + °\\{pcc + paa) + 2TT The density dependence of the second term can be quite complicated, since in general y- { [ + aA 0(l + A2)] x[cjx(pcc + Paa) + <72] + {\\\\(pcc + paa) + A2] } (v)nH (Pcc + Paa) depends on njj. In the case that this dependence can be neglected, a cavity detuning of A = -alo - Al^cc + H + A 2 (4-21) 1) to almost zero coupling (ft ~ 0) without affecting the tuning of the cavity significantly. At low temperatures, the unloaded quality factor Q0 of the electropolished resonator was initially 2850. In a later run, an electronic tuning assembly was added that lowered Q0 to about 2200. 5.3.2 Maser bulb A Pyrex bulb confines the hydrogen atoms to a region in the center of the resonator. A commercial glass to metal seal joins the Pyrex bulb to a short section of copper tube which is epoxied (Stycast 1266) into the copper plate covering the resonator. The tip of the bulb extends into a copper tube that acts as a waveguide beyond cutoff. This tip contains a small puddle of liquid helium which ensures that a saturated 4He film covers all the surfaces in contact with hydrogen. The placement of the pool of 4He outside the rf field minimizes possible cavity tuning instabilities. The bulb holding time Tb is determined by the orifice located at the bottom of the brass cone (see fig. 5.4). The diameter of the orifice is 0.65 mm. This results in Tb = ^ = 0.6 s at T = 0.6 K, where A is the cross section of the orifice, Vb = 5.15 cm3 is Av ' ' \" the volume of the bulb and v is the average velocity of a hydrogen atom. We cannot use indium or lead O-ring seals between the brass cone and the copper pieces below and above it, because these superconducting rings would trap magnetic flux and lead to field inhomogeneities. We have successfully used 0.25 mm thick annealed and etched copper washers instead. The flanges of the brass cone and of the atom tube have machined knife edges. The forces acting on these joints are reduced by three additional supports Chapter 5. The UBC cryogenic hydrogen maser 55 between the maser cavity and a platform bolted to the state selector. They consist of a lower section of bakelite and an upper section of copper, where their relative lengths are chosen in such a way that their total thermal contraction matches that of the combined atom tube - brass cone. The bakelite has a low thermal conductivity which ensures that the thermal anchoring of the maser cavity is not affected by these additional supports. These seals with copper washers have withstood more than a dozen cooldowns without developing a leak. 5.3.3 Cooling bath The volume between the inner and outer Pyrex bulb is filled with liquid 4He. The high thermal conductivity of the superfluid ensures that the temperature of the inner Pyrex bulb is uniform. Silver sinter (0.1 cm3, 700 A particle size) reduces the thermal resistance between the cooling bath and the resonator top plate. This way, the temperature of the hydrogen atoms in the maser cavity can be controlled by regulating the temperature of this top plate. The outer bulb has to be completely filled with helium, otherwise the sinter is not immersed in the liquid helium. To ensure this, an additional buffer volume is mounted on top of the resonator and connected to the cooling bath by a short capillary. It can contain up to 4.4 cm3 of liquid helium, which is much larger than the uncertainty in the volume of the cooling bath (Vc.b — 9.7 cm3). In order to protect the apparatus in case of a blocked filling capillary, the buffer volume is equipped with a bursting seal. The top of the volume consists of a 0.025 mm thin brass foil which is punctured by a tungsten needle in case of a large overpressure in the helium bath. The position of the needle is adjusted at room temperature to break the seal at a pressure difference of about 1.5 atm. Chapter 5. The UBC cryogenic hydrogen maser 56 5.3.4 Thermometry The temperature of the maser cavity is regulated by a heater and three different ther-mometers. We use the resistive element of a Beckman 89PR500 Trimpot 2 for the con-struction of a nonmagnetic heater (~ 500 fi). Brass leads are connected with silver paint and then the heater is coated with a thin layer of black epoxy (Stycast 2850FT) to elec-trically insulate it from the thick copper foil that is wrapped around it. The heater is screwed into the top plate of the resonator. Three different types of thermometers are simultaneously used to monitor the tem-perature of the maser cavity: a 3He melting curve thermometer, a germanium resistance thermometer and a carbon resistance thermometer. The 3He melting curve thermometer is used to calibrate the resistance thermometers. It was built by R. Cline following the design by Grey wall and Busch [60]. The germanium resistance thermometer is a Lake Shore model GR 200A-50. It is mounted into a gold plated copper holder and the elec-trical leads are well heatsunk to the body. A 51 fi 1/8 W Matsushita resistor is used for the carbon resistor thermometer. The resistor is ground down to a thickness of about 0.15 mm and then installed in a slotted gold plated copper body. All these thermometers contain small amounts of ferromagnetic or superconducting materials and therefore must not be mounted directly onto the maser cavity. For this reason, a gold plated copper post with a cross section of 90.7 mm2 and a length of 9 cm is screwed into the gold plated copper piece containing the silver sinter. The three thermometers are then screwed into the top end of this post. This results in a low thermal resistance between the hydrogen gas in the maser cavity and the thermometers, but keeps the thermometers far enough away to avoid a degradation of the magnetic field 2 T h i s trimpot does not contain any ferromagnetic or superconducting parts according to information obtained from the manufacturer. The resistor consists of a film of 'Cervit ' deposited on a slab of alumina. The conducting pads are formed from a Ag-Pd alloy. Chapter 5. The UBC cryogenic hydrogen maser 57 homogeneity at the maser bulb. 5.4 State selector The details of the state selector or state invertor are,shown in fig. 5.6. The majority of the high field seeking atoms emerging from the low field region enter first the space with the layers of magnetic foil. From there they can reach the microwave cavity through four small holes of 0.35 mm diameter. Atoms that are pumped up into the | c ) state leave the cavity predominantly through the 2.5 mm wide extension at the bottom of the cavity. This prevents the | c ) atoms from coming in contact with the magnetic foil which would cause transitions to the | d ) state and lower the population inversion nn(pcc — paa)s.s outside the maser bulb. 5.4.1 Microwave cavity The microwave cavity is placed in the center of the superconducting magnet. We use the TMoio mode, which is for this copper cavity resonant at 39.46 GHz and has a quality factor Qi ~ 2600 at low temperatures. The lid of the cavity consists of a 0.25 mm thick copper disk which is glued to the rest of the cavity with conducting silver epoxy (epo tek 410-E). A copper adapter piece is hard soldered to the rectangular coin silver wave guide. The orientation of the waveguide with respect to the coupling iris is adjusted at room temperature so that the coupling coefficient is ft ~ 0.35. It is then joined to the cavity with silver epoxy and some additional layers of clear epoxy (Stycast 1266). At low temperature, the resulting coupling to the state selector cavity is then close to critical coupling. Hydrogen atoms are dissociated in a low temperature source which is connected to the waveguide above the state selector. The source is described later in this chapter. The Chapter 5. The UBC cryogenic hydrogen maser 58 Figure 5.6: Details of the state selector region. A: waveguide for 40 GHz pump mi-crowaves, B: 1.4 T superconducting magnet with end corrections, C: gradient coil, D: cou-pling iris between waveguide and microwave cavity, covered with mylar window with a pinhole which allows H atoms to enter cavity from discharge source (not shown), E: 40 GHz microwave cavity, F: region with 10 layers of relaxing foil, G: indium O-ring seal, H: atom transport tube. Chapter 5. The UBC cryogenic hydrogen maser 59 atoms enter the state selector and the rest of the maser through a pinhole in the 0.04mm thick mylar window that covers the coupling iris. The characteristic time for atoms to leave the state selector through this pinhole was measured to be of the order of 40 s in the present apparatus. 5.4.2 Relaxing foil The magnetic foil is produced from 13 fim thin aluminum foil 3 according to the recipe of Abe et al. [56]. Ten layers of this foil are inserted into the state selector. An indium O-ring joint between the state selector and the atom transport tube makes it possible to disassemble this part of the apparatus nondestructively and replace the foil later, if necessary. 5.4.3 State selector magnet The superconducting magnet is wound on an aluminum former that fits snuggly around the state selector. The main solenoid consists of 19 layers of 0.19 mm diameter NbTi wire with Cu cladding with a total of 4497 turns. A 0.025 mm thick mylar sheet is inserted after every second layer and the whole magnet is potted with GE 7031 varnish. The persistence switch and the superconducting joint are heatsunk to the 1 K pot of the dilution refrigerator. The superconducting joint has a critical current in excess of 18 A and is constructed by sandwiching bare NbTi wire between Nb foils, following the procedure described by Blair et al. [61]. The magnet was originally designed to be placed inside a 15.4 cm long hollow NbTi cylinder with 3.35 cm inner diameter. The magnet was to be charged up only after this cylinder had become superconducting, the intended purpose of the shield being to 3 T h i s thin aluminum foil was obtained from the U B C Pit burger bar where it was intended to be used as hamburger wrapper. Chapter 5. The UBC cryogenic hydrogen maser 60 attenuate the magnetic field outside the magnet exponentially and eliminate possible field gradients at the maser cavity due to the state selector magnet. With this scheme, the homogeneity inside the magnet is also predicted to improve. In appendix D, the magnetic field produced by a solenoid inside an infinitely long superconducting shield is calculated. It is also outlined how to determine the optimal end corrections in this case in order to achieve the highest homogeneity close to the center of the solenoid. However, after an unsuccessful trial we discovered that the superconducting cylinder is not able to shield the 1.43 T strong magnetic field, which is required to bring the | b ) to | c ) transition into resonance with the 39.46 GHz microwave radiation. The critical current of the NbTi cylinder is much smaller than that of work hardened NbTi wire because of a lack of enough strong pinning sites in the annealed material. In fact, the shielding properties of a similar NbTi shield had previously been studied by Muething et al. [62] and they came to the same conclusion, that only small fields can be confined this way. For higher fields, magnetic flux starts to leak out of the superconducting cylinder. This has forced us to remove the NbTi shield around the state selector magnet and modify the end correction coils. When the magnet is charged up for the first time, the current has to be ramped up very slowly. It appears that flux suddenly jumps to an energetically more favorable position. In these events, large amounts of heat are released. At the operating temperature of the magnet, the specific heat is low and the thermal anchoring is also weak. Therefore, already with a small disturbance the condition for a flux jump can be reached. These processes are difficult to analyze exactly because they not only depend on the properties of the magnet, but also on the power supply used. After the first charging, this behavior is not observed again. The magnet can then be completely de-energized and re-energized without any anomalous heating. Flux trapping in the windings might also be responsible for the homogeneity of the Chapter 5. The UBC cryogenic hydrogen maser 61 magnetic field over the volume of the microwave cavity being only 9 x 10-4. The the-oretical prediction is 1.1 x IO-5. A first order gradient along the z-direction can be applied with the gradient coil, however it improves the homogeneity only by a small amount. Although the homogeneity achieved is rather disappointing, it is nevertheless sufficient for this experiment. 5.5 Magnetic shielding of the maser cavity At the maser cavity, a small uniform bias field has to be applied in order to remove the threefold degeneracy of the upper hyperfine states. In our experiments, a typical value of the bias field is 8 fiT (= 80 mG). This is rather large compared to the fields used in other hydrogen masers, but it is necessary to suppress cross relaxation between the upper three states. The relaxation is presumably caused by some magnetic contamination in or close to the maser bulb. It will be essential to reduce such relaxation in future versions that seek to achieve very high stabilities. The bias field has to be kept very stable and in addition, the maser cavity has to be shielded from external magnetic fields. Both requirements can be met by using a superconducting shield around the maser cavity. The whole assembly, consisting of four concentric cylinders, is bolted to the bottom plate of the vacuum can. The outermost layer is a shield made from high permeability foil (co-netic). A ten turn demagnetization coil allows to demagnetize this shield in situ and to reduce the ambient field to a small value. The high-/* metal shield is wrapped around the solenoid that provides the bias field. This one-layer solenoid has 151 turns over a length of 22.8 cm and is wound on an aluminum former of 12.7 cm diameter. The superconducting shield is situated inside the solenoid. It consists of a 30.7 cm long brass cylinder with inner diameter of 9.8 cm and wall thickness of 1.8 mm, and has Chapter 5. The UBC cryogenic hydrogen maser 62 copper flanges soldered on at both ends. A 15 /^m thick layer of lead is electrodeposited on the inside of the cylinder. The lead is coated with a thin layer of polystyrene that prevents it from oxidizing. Allen Bradley carbon resistor thermometers are located on both flanges. In addition, a nonmagnetic heater is wrapped around the top flange. The superconducting shield is separated by a 1.1 mm thick and 8.5 mm wide greased bakelite ring from a copper support which is well heatsunk to 4.2 K. The support positions the shield such that the maser cavity is exactly at its center. For adjusting the bias field, the entire superconducting shield is first driven normal by supplying enough power to the heater on the top flange. Then the desired bias field is applied with the solenoid. Following this, the power to the heater is slowly reduced. As the shield is cooled from the bottom side with a temperature gradient sustained, the interface between the superconducting and normal phase of Pb moves slowly upwards. This not only leads to trapping of the applied bias field but in principle should also make the magnetic field inside the shield more homogeneous. For a shield heated to above the critical temperature of lead (7.19 K), the direct heat load onto the maser cavity due to the blackbody radiation would be of the order of 10 fiW. This is too large for the weak thermal link connecting the cavity to the refrigerator and for this reason a copper heat shield, heatsunk to 4.2 K, was inserted between the lead shield and the maser cavity. We have observed that the bias field trapped in by the lead shield is indeed very stable. In addition, frequency shifts induced by a magnetic field from a solenoid wrapped around the dewar have also been measured. The deduced attenuation by the shield is AABa'T ~ 1-25 x 10-4. On the other hand, our process of trapping in the bias field does not appear to improve the homogeneity of the externally applied bias field noticeably. In particular, the measured relaxation time constants T2 of the free induction decay with the shield in the normal or superconducting state are equal. This indicates that the lead Chapter 5. The UBC cryogenic hydrogen maser 63 shield freezes in all flux lines penetrating the superconducting film and the lead film ends up in a mixed state. 5.6 Hydrogen discharge source Atomic hydrogen is produced by rf dissociation at low temperature. The source, designed and constructed by W. Hardy, is identical to the source used by B. Statt et al. in the ESR experiment and is described in detail in [63]. The top and bottom of the source, which is shown in fig. 5.7, is heatsunk to a special heat exchanger in the refrigerator that is typically operated at 0.4 K. Hydrogen molecules, condensed onto the inner surfaces are dissociated by rf pulses at the resonance frequency of the resonator, which is 45 MHz. Baffles at the bottom of the source thermalize the produced hydrogen atoms. The bottom tube of the source is connected to the waveguide above the state selector via thin nickel bellows and stainless steel tubes. The hydrogen atoms enter the waveguide below a mylar window vacuum seal that is epoxied across the waveguide. From there, the atoms can reach the state selector cavity and the rest of the cryogenic maser as described before. The hydrogen molecules are originally deposited in the source through the fill tube capillary which can be heated above the boiling temperature of H2 (~21 K) while the source is kept cold. After the heat on the capillary is removed, the same fill tube is used to admit enough helium until all the surfaces are covered with a saturated 4He film and a small puddle forms at the tip of the inner maser bulb. A small radioactive source of 60Co (~20 /xC) provides free charges and helps to start the discharge. Only a few pulses of typically 10 /xs duration and 0.4 W peak power are required to produce a high enough atomic hydrogen density in the maser for stable maser oscillations. Depending on the circumstances, the maser could continue to oscillate for Chapter 5. The UBC cryogenic hydrogen maser 64 THERMOMETER HEATER H2,He FILL TUBE COIL TEFLON SPACER CAPACITOR PLATE BAFFLES HEAT SINK TO BASEPLATE Figure 5.7: Low temperature atomic hydrogen discharge source. This figure was first published in [63]. Chapter 5. The UBC cryogenic hydrogen maser 65 longer than an hour before the discharge has to be pulsed again. 5.7 Electronics for state selector microwave pump The 40 GHz electronics for the selector pump sits on top of the cryostat and is shown schematically in fig. 5.8. The microwave radiation to drive the ESR transition in the state selector cavity is generated by a backward wave oscillator, 4 whose frequency can be swept in the interval between 26.5 GHz and 40 GHz. The microwave power is attenuated and transmitted to the state selector cavity via Ku-band waveguide. Typically, about 50 fiW of power is dissipated in the state selector cavity. A fraction of the reflected signal is coupled to a crystal detector with a 10 dB directional coupler. This is initially used to identify the cavity resonance, and its resonance frequency can be determined approximately with the wavemeter preceding the detector. In order to monitor the frequency of the microwave radiation accurately, a 3 dB directional coupler diverts power into a harmonic mixer, where it is mixed with the 16 th harmonic of the output from a HP8663A frequency synthesizer, set at 2468.77 MHz. The resulting if signal at around 40 MHz is amplified and fed into a counter. After the initial warm up period of the backward wave oscillator, we have found that the frequency of the 39.46 GHz signal is stable to better than 0.4 MHz over the period of about an hour. This fractional frequency instability of 10-5 is considerably smaller than both the fractional inhomogeneity of the state selector magnet (9 x 10-4) and the inverse quality factor of the state selector cavity (4 x 10-4). Therefore, it is not necessary to further stabilize the frequency of the microwave generator. It would be straight-forward to phase lock a 40 GHz source to a 10 MHz quartz crystal oscillator, or even better to the maser signal. The waveguide used to pipe the microwave signal from the generator down to the 4Weinschel model 221 microwave sweep oscillator with a model H 2640 rf oscillator plugin. Chapter 5. The UBC cryogenic hydrogen maser Backward Wave Oscillator, Sweeper Counter A L.P. Filter Harmonic Mixer HP 8663A Synthesizer Wave Meter Detector Oscilloscope to Cryostat and State Selector Cavity Figure 5.8: Schematics of the state selector pump electronics at 39.46 GHz. Chapter 5. The UBC cryogenic hydrogen maser 67 cavity has to be constructed from several sections with different thermal properties. Starting from the bottom, a 9 cm long section of coin silver waveguide is joined to the state selector cavity. Via a hardsoldered phosphor bronze 90° bend, the waveguide is continued in a piece of copper waveguide. At the upper end of this piece, a 0.05 mm thick Mylar window is epoxied (Stycast 1266) across the waveguide and forms a vacuum seal. It confines the superfluid helium film and the hydrogen atoms, which enter the waveguide below the window from the discharge source. The temperature gradient between the state selector (0.4 K) and the top of the vacuum can (4.2 K) is sustained with a 34 cm long section of thin wall stainless steel waveguide. In order to decrease the microwave attenuation without increasing the thermal conductivity appreciably, the inside of the waveguide is tinned with a thin layer of Pb-Sn solder which becomes superconducting at these temperatures. The 90 cm long section of stainless steel waveguide between the vacuum can and the top of the cryostat has a 2 fj,m thick layer of electrodeposited copper on the inside. A second Mylar window is glued across the waveguide on top of the cryostat. The waveguide section between the two Mylar windows can be evacuated through a small hole to the vacuum can above the lower seal. At room temperature, the microwave loss of this section is 3.4 dB at 40 GHz. 5.8 1420 MHz detection system We use a two stage heterodyne system to detect the maser signal. Also incorporated in this spectrometer is a pulse generator that allows us to apply a simple pulse sequence and study the response of the hydrogen atoms at densities below the maser threshold. The spectrometer was built by W. N. Hardy for the previous experiments [12] on hydrogen at the hyperfine transition and it is only slightly modified for our application. A schematic diagram of the spectrometer is shown in fig. 5.9. Chapter 5. The UBC cryogenic hydrogen maser 68 Pulse Generators Multi- Phase plexer Shifter Maser Cavity Figure 5.9: Schematic diagram of the spectrometer. F: Filter, SSB Generator: Sin-gle-Side-Band Generator. Chapter 5. The UBC cryogenic hydrogen maser 69 Briefly, the maser signal is amplified with two low-noise amplifiers and then mixed with a local oscillator signal at 1420 MHz in an image rejection mixer. The local oscillator signal is generated by direct frequency multiplication (xl42) of the 10 MHz reference signal, which might come from a rubidium standard or a high quality quartz crystal oscillator, depending on the experiment. By using an image rejection mixer, the resulting signal at the intermediate frequency of 405 kHz does not contain any contributions from noise components of the maser signal at 1419.595 MHz. The signal at 405 kHz is then filtered and amplified, before it is mixed with the second local oscillator signal, derived from a frequency synthesizer. This low frequency synthesizer is a HP 3330A and we use the same 10 MHz reference source as before. The exact output frequency of the synthesizer is adjusted to give the desired beat frequency. When we observe stable maser oscillation, we choose the beat frequency to be about 10 Hz; for measurements of the free induction decay, the beat frequency is set higher, typically several times the line width of the hyperfine transition. We can apply a simple pulse sequence at the maser cavity, such as a single 7r/2 pulse or a 7r — 7r/2 pulse sequence. The pulses are generated from the two local oscillator signals at 1420 MHz and 405 kHz with a single-sideband generator. The phase shifter, used to produce the two phases of the 405 kHz signal for the SSB generator, is gated. This avoids any interference problems at the hyperfine frequency, because the generated signal at the resonance frequency of the hydrogen atoms is only present for the duration of the pulses. A 20 dB directional coupler 5 mounted on top of the vacuum can at 4.2 K is used to couple the pulses into the coaxial cable leading to the maser cavity. In order to set the tuning and coupling of the maser cavity, we can inject a swept frequency signal through the same directional coupler. The reflected signal is monitored with a crystal detector after the second amplifier. 5Technical Research and Manufacturing Inc., model DBS-205 Chapter 5. The UBC cryogenic hydrogen maser 70 For absolute power measurements, we have to calibrate the sensitivity of the hetero-dyne detection system. For this purpose, a second directional coupler 6 with opposite directivity is installed between the maser cavity and the preamplifier. We use a HP 8663A frequency synthesizer with well calibrated output power levels to generate the calibration signal at 1420.405 MHz. Special care is required for all the components between the maser cavity and the preamplifier. Any loss will increase the effective noise temperature and degrade the short term stability of the maser. The first 35 cm section of semi-rigid coaxial cable in the immediate vicinity of the maser cavity has copper inner and outer conductor with an outer diameter of 0.085 inch. In order to keep the heatload low, we use stainless steel coaxial cable with outer diameter of 0.141 inch for the next section up to the vacuum flange. The inner conductor is replaced by a stainless steel wire that is tinned with soft solder, which becomes superconducting and reduces the microwave losses. At the vacuum can, a commercial ceramic seal provides a superleak tight microwave connection. The two directional couplers, the preamplifier and the two isolators are located di-rectly on top of the vacuum can and are kept immersed in liquid helium at 4.2 K. The GaAs MESFET preamplifier was built by W. N. Hardy for a previous experiment. It is based on a design by Williams et al. [64] and uses a Mitsubishi MGF 1412 and a Mit-subishi MGF1402 FET in the first and second stage, respectively. At 4.2 K, the gain is 21.0 dB with a 3dB bandwidth of 395 MHz and the noise temperature is T/v = 20 K. The results of flicker phase noise measurements on this device are presented in the next chapter. The isolators 7 at the input and output of the preamplifier are surrounded by superconducting lead jackets that shield the isolators from changing magnetic and electric fields, especially at the line frequency. At the operating temperature of 4.2 K, 6Midwest Microwave, model 5010-2011 with 20 dB coupling 7Pamtech, model LTE1102 Chapter 5. The UBC cryogenic hydrogen maser 71 the isolators have a bandwidth of about 300 MHz, and at 1.42 GHz an insertion loss of 0.25 dB and an isolation of 29.4 dB and 21.1 dB, respectively. All three coaxial cables leading from the vacuum flange at 4.2 K to the top of the cryostat at room temperature have 0.141 inch diameter stainless steel outer conductor and silver-plated copper-clad steel inner conductor. The amplifier 8 preceding the image rejection mixer has a noise temperature of 440 K. Since the gain of the preamplifier is only 21 dB, this results in a 3.5 K contribution to the overall noise temperature, which is referred to the input of the preamplifier. In later measurements, we have reduced the contribution of the second and further stages to less than 1 K by adding a lower noise amplifier in front of the Watkins Johnson amplifier. This additional amplifier is identical to the GaAs preamplifier, however it is operated at room temperature, where its noise temperature is about 90 K. The final filter at the output determines the overall noise bandwidth B of the receiver system and therefore affects directly the signal to noise ratio. It is this noise bandwidth B that enters in the expression (4.8) for the maser frequency instabilities due to electronic white phase noise. When we observe free induction decays, the output of the final mixer is usually directly connected to the input of a Tektronix oscilloscope 9. In this case, the variable input filter of the oscilloscope is the final filter indicated in fig. 5.9. However, when the maser is oscillating and we want to make a precise measurement of the frequency fluctuations, we use a separate narrow-band-pass filter at the output of the final mixer. This active filter is centered at 10 Hz and has a 3dB bandwidth of 3 Hz. Its noise bandwidth B is 5.1Hz. In addition to a direct buffered output, the filter has a second output that delivers an approximate square wave. This output signal is derived from the filtered 8Watkins Johnson, model WJ 737-303S 9Tektronix 5440 oscilloscope with a 5A22N differential amplifier. Chapter 5. The UBC cryogenic hydrogen maser 72 signal by a simple circuit that amplifies the signal and subsequently clips the amplitude. The timings of the zero crossings of both filter outputs are identical, however the slopes at the zero crossings are much larger for the 'square wave' output. As will be discussed in greater detail later, the relative frequency stability of the maser is measured by recording the time intervals between a set number of zero crossings of the beat signal. These measurements are affected by fluctuations in the dc offset of the signal and in the trigger level of the counter. These effects are minimized by using the 'square wave' output of the filter, which illustrates its practical advantage over the 'sine wave output'. It is important to note, that the effective noise bandwidth B is the same for both outputs. Chapter 6 Experimental Results The first results on the cryogenic hydrogen maser were obtained on April 14, 1986, when we observed the first maser oscillations in our apparatus. In this chapter, we first describe the experimental procedures used to start up the maser, before we concentrate on the results, which include the frequency stability and the measurement of the binding energy. 6.1 Start up procedure 6.1.1 Cool down Several frustrating months of leak checking preceded the breakthrough of the observation of the first maser signal. Only a few months after delivery, the Oxford Instruments refrigerator started to develop low temperature leaks on the top flange of the vacuum space, presumably in one or many of the numerous stainless steel welds. For several months, we tried to localize these leaks, however without any noticeable success. In our desperation, we turned to alcohol, and to our surprise, it solved some of the problems. We used the following procedure:1 We first cool the refrigerator with liquid nitrogen to 77 K, then we quickly lower the dewar and spray anhydrous ethanol onto the top flange of the vacuum flange. The alcohol freezes immediately on the cold surface and forms a several millimeter thick smooth layer that becomes our new low temperature seal. With this method, we can usually reduce the leak rate sufficiently to operate the dilution 1 T h i s method, to our knowledge first used at Cornell, was brought to our attention by Prof. T . J . Greytak. 73 Chapter 6. Experimental Results 74 refrigerator. The residual helium leak however limits the base temperature of the mixing chamber to 100 mK or higher. In later runs, we have installed three charcoal sorption pumps in the vacuum can, two of them mounted on the underside of the top flange and the third on the bottom plate of the vacuum can. After the initial cool down, we load the discharge source with molecular hydrogen. The source is kept at a temperature of around 0.5 K, while 5 x 10-3 moles of H 2 are deposited onto the inner walls of the discharge source through a heated capillary, as already described. Then the capillary heater is turned off and about 3 x 10~3 moles of 4He are added to the maser through the same capillary. We are now ready to fire the discharge a few times and detect the atoms in the maser cavity by pulsed magnetic resonance. The maser cavity is tuned to the hyperfine transition frequency and the coupling is adjusted to reach critical coupling. Typically, we apply ir/2 pulses and observe the free induction decay with the spectrometer. 6.1.2 Adjustment of bias field at the maser cavity Before the state selector magnet is charged up, we adjust the bias field at the maser cavity to the desired value. The high fj, metal shield has been demagnetized at room temperature and also at 77 K, so that the initially trapped field in the Pb shield is already quite uniform and usually of the order of 1 fiT in amplitude. The bias field can be changed after driving the superconducting shield normal, as was described in the previous chapter. The bias coil is calibrated by measuring the frequency of the free induction decay as a function of the solenoid current and fitting it to the parabola described by equation (4.12). At low density, radiation damping and spin exchange relaxation are unimportant and it is adequate to fit the signal to an exponentially damped sine wave. When the z-component of the bias field is exactly zero, the direction of quantization is orthogonal to the rf field and we can observe the transverse transitions between the Chapter 6. Experimental Results 75 | d ) and | a ) states and between the | b ) and | a ) states (see appendix A). From the beat frequency of these two signals, the transverse component iJj. of the residual static field can be determined. A typical value for H± is 0.2 //T. If we assume that the bias coil is perfectly aligned with the resonator and does not produce a net transverse field at the maser cavity, we can determine the total bias field that the hydrogen atoms are exposed to in the resonator by combining this measurement with the calibration of the bias coil. In most conventional hydrogen masers, the magnetic field is measured by determination of the rf frequency that induces transitions between the | c ) and | b ) state and between the | c ) and | d ) state. The copper walls of our resonator are much thicker than the skin depth at the appropriate frequency (usually a few kHz), which rules out this method in our case. 6.1.3 Cross relaxation in the maser bulb The observed relaxation times T2 are shorter than the bulb holding time Tb (~0.6s) and are field dependent as shown in fig. 6.1. This indicates that a fairly strong relaxation mechanism between the upper three states is present. When the applied magnetic field is increased, it splits the degeneracy of the three states and thus suppresses the cross relax-ation process for energetical reasons. This interpretation can be confirmed by measuring the relaxation of the longitudinal magnetization with a TT-TT/2 pulse sequence.2 In our analysis, we assume that there are three important processes that determine the recovery of the longitudinal magnetization: (i) Flow in and out of the maser bulb. The rate for the atoms to leave the bulb is rb = Tb-1. 2We are using here the language of the fictitious spin |, which was introduced in chapter 2. Chapter 6. Experimental Results 76 . 3 C\\2 2 A A .1 A A A A A * 4 A A * A A 0 0 1 0 1 5 B i a s f i e l d (/xT) Figure 6.1: Transverse relaxation time T2 versus bias field. The temperature of the hydrogen gas is 500 mK and the densities are low enough to avoid large radiation damping and spin exchange effects. Chapter 6. Experimental Results 77 (ii) Cross relaxation between | c ) and | b ) and between | c ) and | d ). We assume that the relaxation rates are equal and we denote them with r c . b c ^ d (iii) We also include longitudinal relaxation between the upper three states and the | a ) state with relaxation rate T;: a b, a ^ c, a ^ d At low densities, spin exchange relaxation can be neglected and it is then easy to find the solution of the set of linear equations describing the longitudinal relaxation with the above assumptions. In thermal equilibrium, the population difference (n0 — nc)eq is given by t \\ t \\ I huo (na - nc)eq = nH(paa - pcc)eq. - ^ nH~gf After a IT pulse at the maser transition, the population difference na — nc gets inverted and it relaxes back to the equilibrium value according to 6(na - nc)(t) = -\\nH^L {| e \" ^ ' ) ' + \\ e - ^ ^ ) (6.1) If we apply a 7r/2 pulse at a time t after the IT pulse, the amplitude of the free induction decay is proportional to (na — nc)(t) and we can verify expression (6.1) experimentally. A typical example from an early run is shown in fig. 6.2, where the normalized deviation of the longitudinal magnetization, proportional to 6(na — nc)(t), is plotted versus the time t between the 7r and TT/2 pulse. The data is well described by the two exponentials of (6.1) and it is obvious that a single relaxation time 2\\ is not sufficient to characterize the longitudinal relaxation. The dotted line in fig. 6.2 indicates the long time behavior and it extrapolates back to a normalized magnitude of | at t = 0. This confirms our prediction that r c IV Chapter 6. Experimental Results 78 T i m e ( s ) Figure 6.2: Relaxation of the longitudinal magnetization in the maser bulb versus time. The solid line is a fit to (6.1) and the dotted line indicates the asymptotic behavior. The temperature of the maser bulb is 460 mK and the bias field is 13 /iT. Chapter 6. Experimental Results 79 This cross relaxation is most likely caused by some magnetic contamination on or near some part of the surface of the maser storage region. It is not caused by magnetic field gradients across the resonator. The evidence is that at higher temperatures where the vapor pressure of 4He is so high as to effectively immobilize the hydrogen atoms, the relaxation time T2* becomes much longer. If there were large field gradients across the bulb present, we would expect T2* to become shorter, because there is no motional averaging possible any more. On the other hand, if magnetic particles on the surface were responsible for the relaxation, at higher temperatures only a few spins could come in contact with them and be relaxed. This results in a longer effective relaxation time, which is in qualitative agreement with the observations. The choice of bias field is determined by a compromise between a low field for minimal sensitivity of the frequency to field fluctuations and a high field for maximal relaxation time T2. In the last run, we have trapped in a field of 8 //T. When the Pb shield becomes superconducting, we do not observe any effect on the frequency or Tj* of the free induction decay. 6.1.4 State selector magnetic field The state selector magnet has to be charged up very slowly for the first time; it takes typically about an hour to reach the operating field of 1.433 T. The frequency of the Ku-band microwave radiation is adjusted to the center of the state selector cavity resonance. At the same time 7r/2 pulses are applied to the maser cavity. As soon as the resonant magnetic field is reached, the free induction decay signal inverts and, if the density is high enough, maser oscillations are observed. After fine adjustments of the microwave frequency and the magnetic field for maximal population inversion, the magnet is put into persistent mode. Chapter 6. Experimental Results 80 6.2 Power versus density measurements 6.2.1 Density calibration An important parameter for the operation of the maser is the total density of hydrogen atoms in the maser bulb, n#. In the present apparatus, it is not possible to measure n# directly, as would be possible, for instance, with a calibrated capacitive pressure gauge. However, the technique of magnetic resonance can be used to infer n#. In thermal equilibrium, the static magnetization Mz is proportional to n# and is given by (2.23): 1 Ti Mz = 2M7e + 7 p ) ^ n H (6.2) With a 7r/2 pulse this static magnetization can be converted into a precessing transverse magnetization MT with the same magnitude as Mz. In this case, the power emitted by the atoms immediately after the pulse is P = fJ-O^oVbTjQl ~ 2 HoLOoVbTlQl Mr r ft 2 (7e + 7 P ) ^ O 1 2 8kT nu1 (6.3) 2 For critical coupling, half of the power P reaches the preamplifier. By measuring the power P absolutely, it is thus possible to determine njj. We have used this method to measure n# in the maser bulb. We first turn off the state selector by either slightly mistuning the microwave radiation applied to the state selector cavity or by changing the magnetic field at the state selector by a small amount. Both methods bring the microwave radiation off resonance with the ESR transition in the state selector cavity, but the second method is preferred for the following reason. Mistuning the microwave frequency changes the power dissipation in the cavity and can lead to small changes in the state selector temperature, which should be avoided. After the transients have decayed, we apply a 7r/2 pulse at the maser cavity and measure the initial power of the free induction decay. Chapter 6. Experimental Results 81 However, it is difficult to obtain accurate results for njj with this method for several reasons. First of all, it is difficult to determine the filling factor rj for the split ring resonator used in the maser. The exact rf field distribution is not known and the filling factor cannot be calculated from the known geometry, but it has to be determined exper-imentally. M. Morrow [38] measured the filling factor for the same split ring resonator, but with a larger outer resonator can, and obtained rj = 0.42. An additional difference to the situation of M. Morrow is that the maser glass bulb extends further out to weaker rf fields compared with the bulb used in the earlier experiment. Both differences tend to decrease the filling factor. There are more fundamental problems with the density determination. Ideally, a low temperature valve at the orifice of the maser bulb, that could be closed in a time short compared with Tb, would confine the atoms in the maser bulb. After the atoms reach their thermal equilibrium, the density n# could be determined from the measured power P with equation (6.3). In the derivation of (6.3), it is assumed that the spin system is in thermal equilibrium described by the temperature T. However, the hydrogen atoms in the maser bulb are also in contact with the state selector, whose temperature is in general lower. During the bulb holding time TJ,, hydrogen atoms encounter many collisions with the walls. Only a few of these collisions are necessary to thermalize the translational degrees of freedom, which therefore reach their equilibrium in a time much shorter than Tb. The spin degrees of freedom however are only thermalized by hydrogen-hydrogen collisions or by interactions with magnetic field gradients, produced for example by magnetic impurities on the walls. At low densities, the collision rate between hydrogen atoms is small and the thermal spin equilibrium in not completely reached within the bulb holding time. Therefore, the static magnetization Mz is larger (for T s s < Tmaaer) than what would be expected based on (6.2). Equation (6.3) leads then to an overestimation of the density in the maser bulb. Chapter 6. Experimental Results 82 So far we have implicitly assumed that the total density in the maser bulb with the state invertor on and off are equal. This is not exactly fulfilled. In the absence of magnetic field gradients between different parts of the apparatus, the ratio of the total densities in different regions of the maser is completely determined by the temperatures of the regions; it does not depend on the population distribution. This is no longer true when different parts of the maser are in different magnetic fields, as is the case for the cryogenic maser. When the microwave radiation in the state invertor is on resonance, it converts high field seeking atoms into low field seeking atoms. As a consequence, the total density nn in the maser bulb is higher when the state invertor is on compared to when it is off. This effect and the effect of incomplete thermalization discussed before compensate each other to a certain degree. The size of the combined effects has been estimated with a. numerical simulation of the maser and is discussed in detail in chapter 7. The results of the simulation indicate that for the present apparatus, operated at TmaaeT = 509 mK and Ts.s = 465 mK, the real densities are approximately 40% higher than the densities deduced from the initial power of the free induction decay with equation (6.3). The size of the correction depends on the relaxation rates and temperatures of the different parts of the maser and is predicted to be much larger at lower temperatures. This makes it desirable to find an additional method to determine the density in the maser bulb. To a first approximation, the only density dependent contribution to the line width V (= T2-1) °f the maser while it is oscillating is due to spin exchange. r = TQ + [a0(pcc-paa) +ax(pcc +paa) +<72](v)nH (6.4) where To does not depend on density and includes contributions from the flow out of the maser bulb and from the cross relaxation mechanism. At the operating conditions of the maser and based on the calculations of Koelman et al. [46], the spin exchange Chapter 6. Experimental Results 83 contribution is dominated by the term proportional to a\\(pcc + paa). For the present situation, pcc + paa can be estimated based on the following observations. The strong cross relaxation in the present maser tends to equalize the upper three populations in the maser bulb, p^ ~ pcc ~ p^. In addition, according to equation (2.33) the population difference pcc—paa is strongly reduced by the maser action. We therefore expect pcc+Paa — 0.5, a result which is confirmed by the numerical simulations presented in chapter 7. Making use of the calculations for ai by Koelman et al. [46], the line width is then expected to depend on the density n# as r ~ T0 + 0 is not much smaller than 1 any more. Above 520 mK, the frequency shifts / — / 0 become very small and spin exchange shifts and their temperature dependence start to play a role. In addition, we use the value for 8 = -1.183 x 10 _ 1 7Hzcm 3 that was measured by Hardy et al. [40] at T ~ 1 K, which is not fully justified since 8 is expected to be weakly temperature dependent [43]. For these reasons, we have excluded the data points above 520 mK. In our least-square fitting routine, we fit ln{(/o — f)VT} and not f0 — / versus T_ 1. The factor of \\fT is included to cancel the temperature dependence of the de Broglie wavelength A. If we assume fs to be temperature independent, the low temperature be-havior of ln{(/0 — f)VT] is a linear function in the inverse temperature T_ 1 with slope ^f- and intercept ln{/ s^A\\/T}. In this way, all data points included are given about equal weight in our fitting procedure. This is desirable, since the main experimental un-certainties are associated with the temperature determination and not with the frequency measurements. The best fit to the data is shown in fig. 6.7, and yields for the three adjustable, pa-rameters: (6.9) /„ = 1420405 769.89 ±0.02 Hz Chapter 6. Experimental Results 93 -r- = 1.011 ± 0.010 K k f, = -71.5 ± 3.0 kHz We have used & = 4.87 cm 1, which was calculated from the known geometry of the bulb. The frequency /o corresponds to the unperturbed hyperfine frequency in the applied bias field and the value obtained in the fit is consistent with the bias field trapped in by the lead shield. The value for the binding energy of hydrogen to 4He is the most precise determination of EB reported so far. In fig. 6.8, we compare our value for EB with values previously published in the literature. There is a surprisingly large spread in the published values for EB- The first deter-mination of EB (a in fig. 6.8) was obtained in a way similar to the present method: the temperature dependence of the frequency shift and recombination rate at zero field was measured by pulsed magnetic resonance on a sealed sample [41]. Both frequency and recombination data indicated EB — 1.15 K. However, the frequency data was analyzed under the assumption that the sticking coefficient over the temperature range of inter-est is only 0.046. Subsequent measurements of the sticking coefficient s by Berkhout et al. [72] have revealed that 5 = 0.33 for temperatures between 80 mK and 0.5 K and that Morrow et al. have therefore overestimated the corrections to the simple expres-sion (6.9), which have to be applied when the phase shift per sticking event <^ 0 becomes comparable to 1. The point indicated by b in fig. 6.8 was obtained from a reanalysis of their frequency data. The corresponding frequency shift fa becomes —61kHz, instead of -49 kHz. With the exception of measurement i in fig. 6.8, all other values of EB were obtained by the analysis of the temperature dependence of recombination rates in high field and rely on the assumption that the intrinsic recombination cross section has no temperature Chapter 6. Experimental Results 94 CM \\ H 1/T (K\"1) Figure 6.7: Observed and fitted temperature dependence of the maser frequency shift from the unperturbed value in the measured bias field. The frequency was measured in Hz, the temperature in K. The solid line is a fit to equation (6.9) as described in text and yields EB = 1.011 ± 0.010 K and f. = -71.5 ± 3.0 kHz. Chapter 6. Experimental Results 95 Figure 6.8: Compilation to the measured values of the binding energy of H to liquid 4He plotted in temporal order. The circles represent measurements from the University of British Columbia [(a) and (b); (h) and (i) from [41], [59, 63] and [65] respectively and (k) from present work]; the diamonds from the University of Amsterdam [(c) and (e) from [66] and [67]]; the squares from MIT [(d) and (g) from [68] and [69]]; the triangle (f) from Cornell [70]; and the cross (j) from Harvard [71]. Point (b) is obtained by a reanalysis of the data of (a) as explained in the text. Chapter 6. Experimental Results 96 dependence. We believe that the surface frequency shift is likely to be less temperature dependent than the recombination cross section. A temperature dependence as weak as leads to an error of the order of 5% or greater in the binding energy and could explain some of the scatter in the published values of EB- Errors in thermometry might also have affected some of the reported values. There is a discrepancy between our value for fs and the value obtained by Pollack et al. [29], which has not yet been resolved. In a field of 6.8 T, Pollack et al. succeeded in observing the | a ) to | b ) NMR free induction decay signal of atoms adsorbed on the surface and measured directly the surface shift. Scaled to zero field, their value corresponds to fs = —43.0 ± 0.8 kHz. Our value for fs depends directly on the value of The quoted uncertainty in fs (3 kHz) does not contain a contribution from the uncertainty in but even with a pessimistic error bar of 10%, our value becomes fs = —71.5 ± 7.8 kHz and is still clearly in disagreement with [29]. It is conceivable that the frequency of adsorbed atoms is not only shifted by Van der Waals effects, but also by interactions with magnetic impurities. As long as an atom samples many impurities within a time interval -^j-, motional averaging will prevent a large line broadening. In this case, it will be difficult to distinguish this shift from the Van der Waals shift. 6.4 Frequency stability 6.4.1 Three-cornered-hat method In the time domain, the frequency stability of an oscillator is usually characterized by the so-called Allan variance' O-!/2(T), which is defined in appendix C. It can be deter-mined by measuring the fluctuations of the output frequency, averaged over a time r. Experimentally, the maser signal at 1420.405 MHz is translated down in two stages to a Chapter 6. Experimental Results 97 beat signal with frequency of approximately 10 Hz. The frequency fluctuations in this low frequency signal can now easily be measured with an electronic counter. The counter registers the time intervals between zero crossings of the beat signal and averages over a given number n of periods (n = 10,100,1000...). These measurements of period are then converted into frequency fluctuations, averaged over r = ^ J J , from which the Allan variances oy2(mT) can be calculated (m = 1,2,3 ...). The dead time between subsequent measurements is one period, i.e. approximately 0.1s in our case. The Allan variance measured this way describes the frequency fluctuations of the maser only if the reference oscillator has a superior performance and does not contribute significantly to the measured frequency fluctuations. In the case of the cryogenic hydrogen maser, there are no such reference oscillators available (at least for short averaging times). Nevertheless, with the so-called three-cornered-hat method it is still possible to determine the contribution of the cryogenic maser alone. This method requires two high quality reference oscillators, whose frequency fluctuations are uncorrelated to each other and to the maser. The measured Allan variance is then just the sum of the intrinsic Allan variance of the two frequency sources involved, e.g. 3 ) + 3T^ + T\\4). In this analysis, we assumed that Tn = T;0 and we neglected the difference between the up- and down-rates. With the same approximations, it can be shown [77] that in the low density limit T2 is related to the relaxation rates by lim T2\": = r<4^3) + If) + 2rj4) (7.9) n-X> From the measurement of the longitudinal and transverse relaxation in the limit of zero density, it is therefore possible to determine T^4-*3), i f and T\\ 4\\ For the one-body relaxation rates in the atom transport tube, we do not have any direct information and we are forced to treat them as adjustable parameters. 7.1.3 Microwave pumping in the state selector cavity In the state selector cavity, the applied microwave radiation induces resonant transitions between the | b ) and | c ) states: d t n \" dt c -«*c (n[ a ) - n^) (7.10) The transition rate Wbc depends on the power level Pmw of microwaves delivered to the cavity, the loaded quality factor Q of the cavity and the magnetic field inhomogeneity inside the cavity: V ^ o l e Q fABy 1 VMB V B J P m w (7'U) w6c If we decrease in our maser the magnetic field around the state selector from 1.433 T to 1.382 T, we bring the | a ) to | d ) ESR transition instead of the | 6) to | c) transition into resonance with the microwave radiation at 39.46 GHz. The transition probability wad is practically equal to Wbc, except that we have to adjust B in (7.11) to the lower value. Chapter 7. Computer simulation of the maser 120 We have included the possibility of driving this other ESR transition in our simulation as an option. In this case, (7.10) has to be replaced of course by: —n(2) dt Ua - - - n( 2 ) \" dt nd (7.12) 7.1.4 Spin exchange relaxation For the effects of collisions between hydrogen atoms, we make use of the theoretical results from the group of Prof. Verhaar. In high field (volume 1 and 2), the dominant spin exchange process is b + d ^ a + c with a calculated rate G M_ o e = 4.3 x 10~13 ^ at a field of 1.4 T [49]. At this field, Stoof et al. [49] found that the relaxation rates of all other spin exchange and dipole relaxation processes are more than two orders of magnitude smaller. Therefore we neglect these additional processes in our simulation and we obtain the following spin exchange contribution for i = 1, 2: i n ( 0 dt a 7-n («) dt r-n (0 dt (7.13) The forward and backward rates Gbd-*ac and Gac-*bd are related by Boltzmann factors '.Ea + E c — E b — Ed\\ Gac->bd = Gbd-tac exp kT (7.14) For zero field (volume 3 and 4) we use the results of Koelman et al. [46]. For 0.2 K < T < 0.6 K, the rate constants for the three dominant processes can be approximated by T b + d ^ a + a c + c ^ b + d c + c a + a GW— = 4-5 x 10-13 B£ (—) G _ w = 4 . 1 x l 0 - - i ( ^ ) Gc^aa = 2.2 x 10-13 of ( 0.08 ,0.6K 0.19 0.07 Chapter 7. Computer simulation of the maser 121 The inverse reaction rates are related to the rates above by the appropriate Boltzmann factor. In volume 3 and 4, we have then the following contribution to the rate equation: d_ dt (») n (0 — 2 G a a . >bd 2Ga Gaa-*bd 2G aa—icc Gaa—tbd 2Gbd—*aa ^Gcc—taa Cbd—*aa Gbd—tcc Gcc—>6d 2Gbd—*cc 2G' cc—•aa 2Gcc—*bd Gbd—taa ^6d—»cc ^cc—*bd ( n «2 \\ (»') («') n «2 (7.15) 7.1.5 Maser action The rate of change of and of due to maser action can be obtained from eq. (2.20): d — (n[^ - nj4)) = t(7e + %)pLQHz cosujt(pac - pca)n {$ (7.16) It depends on the rf magnetic field Hz in the maser cavity and the off-diagonal element pac of the density matrix. These quantities have to be determined in a self-consistent way as was already outlined in chapter 2. Equation (7.16) can therefore not be applied directly in our simulation without including additional equations for Hz and pac. A more direct approach would be based on the relationship 1 P dt Ua - -n<4> dt c V4hu> (7.17) between the rate of change of the two densities and the power emitted by the atoms. However, the expression (2.41) for the power P which was derived in chapter 2, is only valid if a well defined T\\ exists. In the previous chapter is was shown that in the present maser we need at least two time constants to describe the longitudinal relaxation at low densities, and spin exchange relaxation complicates the situation at higher densities even more. Nevertheless, it is possible to take maser action self-consistently into account. Since we are only interested in the stationary solution of the rate equations, we can use directly Chapter 7. Computer simulation of the maser 122 expression (2.33).1 In steady state, the population inversion inside the maser bulb for stable maser oscillations has to be „W _ nW = (nW - nW) + ^(v) ( n « + „W) + ) The density independent line width is identical to (7.9) and it was assumed that inho-mogeneities in the bias field which only dephase the precessing magnetization without causing transitions between the hyperfine states are unimportant. The spin exchange broadening was discussed earlier (see eq. (3.12) and table3.2). 7.1.6 Overall rate equations For the overall rate equations that simulate the maser, it is best to perform a transforma-tion of variables from the variables {n^\\n^\\n^\\ r$} to the new variables { n£) + nW — nW,n|1' + n^,n^ — n^}. From the previous discussion, it is straightforward to derive the set of 16 coupled nonlinear equations that describe the steady state of the 1This approach was first suggested to us by Prof. S. Crampton. Chapter 7. Computer simulation of the maser 123 maser in the case of no maser action (z = 1,2,3,4) It n n («') («') £ It + at flow \\ j + ~dl + di (7.20) To this set of equations, we add the constraint that the total density in the maser bulb n$ or in other cases that the total number of hydrogen atoms in the whole maser is fixed at some predetermined value. We then solve these equations numerically with the Newton-Raphson method [78]. If the population inversion in the maser bulb nSf> — n£4) exceeds the critical value given by the right hand side of equation (7.18), the maser will start to oscillate. Note that jjipffl — n^) is the only rate equation which is affected by maser oscillation. We can therefore simply replace the equation in (7.20) for |(n^-nW) by equation (7.18) and obtain so the steady state density distribution of the oscillating maser. 7.2 Results of the simulation 7.2.1 Choice of parameters It was already mentioned that several parameters needed in the simulation cannot be measured directly in the present apparatus, in particular the one-body rates outside the maser bulb. We can adjust these rates so that the calculated density dependence of the output power agrees with the measured dependence. However, this method does not yield a unique set of parameters and we have not tried to minimize the discrepancies between measurement and simulation in the examples shown below. To illustrate the dependence on the relaxation rate of the magnetic foil in the state selector, we show some numerical results where this rate is varied between 0.15 s_ 1 and Chapter 7. Computer simulation of the maser 124 1.0s-1. For this simulation, the temperature of the maser bulb is chosen to be 509 mK and the temperature of the state selector to be 465 mK. This corresponds to the condition of the second run shown in fig. 6.5 for the power output and of the data in fig. 6.3 which shows the dependence of the maser resonance line width on the density riff and which has been used to calibrate the density determination. The relevant flow rates can be calculated from the known volumes of the different regions and from the cross sections of the orifices connecting them. For the maser bulb holding time we obtain T\\, — 0.6 s (= l/I^4-''3)). This is consistent with the analysis of the relaxation rates of the transverse and longitudinal magnetization MT and Mz as discussed before. From this analysis we also obtain if) = 1.94 s- 1. The rates rj4) are quite small and cannot be reliably extracted from the experimental data. For the results shown, we set — ~ 0.33s 1. In the atom tube, we have to choose some more or less arbitrary values for if^ and if). The rather large value of if) is presumably due to some magnetic dirt on the walls. The copper tube and the brass cone forming the region 3 have been coated carefully with a layer of non-magnetic Stycast 1266 and for this reason, we do not expect to have one-body relaxation rates as large as those encountered in the maser bulb. In the simulation shown, we set = T$ = 0.01 s - 1 and If) = 0.1s-1. The power of the microwave signal injected into the state selector cavity is approximately 50 /iW, resulting in Wbc ~ 2 x 102 s - 1. This is two to three orders of magnitude larger than the relaxation rates of the magnetic foil used in the calculation and therefore the results of the simulation are not sensitive to variations in the power level Pmw This is in agreement with experimental results. Chapter 7. Computer simulation of the maser 125 7.2.2 Power output of maser and state selector efficiency The power emitted by the atoms is given by P = huV*(nP-nW) (7.21) 2 dt V /maser In steady state, the rate of change of n'4'-n[4' due to maser action has to be compensated by all the other contributions. The power output (7.21) is thus equal to p = -^huV4 which can be easily calculated by the expressions given above if the density distribution is known. In fig. 7.2, the calculated power output of the maser is shown as a function of the total density n$ in the maser bulb for three different relaxation rates = of the magnetic foil. It can be seen that the model describes the experimental data of fig. 6.5 at least qualitatively quite well. For the parameters chosen, it is clear that the magnetic foil relaxation forms a bottleneck. With a more efficient foil, the output power and the range of densities that result in oscillations can be greatly increased. The curvature in the power versus density plots is predominantly caused by spin exchange relaxation in the maser bulb. This was already shown in the simplified theory presented in chapter 2. However, some additional curvature is due to the density dependence of the population inversion p^j) — p^} outside the maser bulb. This is shown in fig. 7.3. The population inversion in the atom tube is rather low and decreasing rapidly with density. The main mechanism for this decrease is again spin exchange relaxation, but this time in the atom tube. The increased population inversion for the more efficient magnetic foil is the main reason that the power output of the maser is in that case so much larger. dt V*« U c Jflow + dt V ° c h + dt \\ a C ii (7.22) Chapter 7. Computer simulation of the maser 126 0 1.0x10 2.0x10 3.0x10 4.0x10 Density in Maser Bulb (cm ) Figure 7.2: Calculated power emitted by atoms versus total density in maser bulb for three different values of the relaxation rate of the magnetic foil as indicated. The values of all the other parameters are discussed in the text. Chapter 7. Computer simulation of the maser 127 .5 3 o • i — i ui u CD > M o • r H i — t o .3 2 .1 1 1 1 1 1 1 1 — \\ . \\ 1 0.15 s\"1 -i , i 0 l.OxlO12 2.0xl012 3.0xl012 4.0xl0'2 —3 Density in Maser Bulb (cm ) Figure 7.3: Calculated population inversion p^> — pa3J outside the maser bulb versus total density inside the maser bulb riff for three different values of the relaxation rate of the magnetic foil as indicated. All the other parameters used are discussed in the text. Chapter 7. Computer simulation of the maser .52 128 Density in Maser Bulb (cm ) Figure 7.4: Calculated values of pi 4} + pi 4} versus total density in maser bulb for three different values of the relaxation rate of the magnetic foil as indicated. 7.2.3 Simulation results relevant for density calibration Figure 7.4 shows the calculated values for pi 4} + pi 4), the fraction of atoms in the states | a ) and | c ) in the maser bulb. In chapter 6, we argued that pi 4} + pi 4) has to be close to 0.5 because of the strong cross relaxation present in the maser bulb. This was used in the theoretical prediction of the density dependence of the maser line width which then allowed us to obtain values for the density in the maser bulb. From fig. 7.4 it can be seen that our assumption pa4] + pi£ = 0.5 is quite well fulfilled and is certainly not the main source of any error in the density determination. The discontinuities in the slope of the curves of fig. 7.4 occur at the lower and upper threshold densities for maser oscillations, where pi 4} — pi 4} also changes discontinuously. With the computer simulation, we can also study how well the density in the maser Chapter 7. Computer simulation of the maser 1.8 129 ** o -t-i o 03 «t-i O o OJ u u o CJ CD P 0 l.OxlO12 2.0xl0'2 3.0xl012 4.0X1012 Density in Maser Bulb (cm 3) Figure 7.5: Calculated values for the density correction factors versus the total density in the maser bulb for three different values of the relaxation rate of the magnetic foil. bulb can be measured with the pulsed magnetic resonance technique described in the pre-vious chapter. Our method is based on the assumption that the total density in the maser bulb n$ while it is oscillating is proportional to the population difference (n^ — n^) while the state selector is turned off. The population difference (n^ — n^4^ can be easily measured with a | pulse. If the system is in thermal equilibrium, the proportion-ality constant is given (in the high temperature limit kT4 ^> %u) by therm.eq. In general, eq. (7.23) does not apply exactly to the oscillating maser and the right hand side has to be multiplied by a density correction factor. The calculated values for the correction factor are shown in fig. 7.5. For all three relaxation rates of the magnetic foil, the correction factors are roughly equal and independent of density. This implies that our density calibration which is based on the measured line width of the maser transition n ( 4 ) ni4> - ni4> Chapter 7. Computer simulation of the maser 130 should be essentially correct, because it only relies on a constant correction factor. The upturn at low densities in fig. 7.5 however indicates that we systematically underestimate densities below about 5 x 1011 cm-3. With the numerical simulation program, we have found that the density correction factor is unfortunately rather sensitive to the temperature of the state selector. Therefore, for every temperature of the state selector, a new density calibration has to be performed by measuring the line width versus density. It would be desirable to find a more direct and satisfactory method of density determination. Chapter 8 Conclusions and outlook 8.1 The present cryogenic maser The apparatus described in this thesis is a prototype hydrogen maser that has been primarily designed to study the potential of a new pumping scheme: hydrogen atoms are cycled many times back and forth between the maser cavity and the state selector by means of magnetic field gradients in combination with microwave pumping. In this essentially closed system, no large external pumps are required, in contrast to masers with the traditional beam geometry. The results obtained with our prototype maser are very promising and they indicate that cryogenic hydrogen masers have indeed the potential to improve the state-of-the-art in frequency stability significantly. We have been able to observe stable maser oscillations over the range of temperature between 230 mK and 660 mK and for densities up to 3 X 1012cm-3. The analysis of the temperature dependence of the maser frequency has resulted in a precision determination of the binding energy EB of atomic hydrogen on 4He. Although the maser was not built for optimal frequency stability, the short term stability of our prototype device already exceeds the performance of the best conventional hydrogen masers. The frequency stability of our maser for averaging times r larger than 10 s could not be measured, since reference oscillators with high enough stability for these averaging times were not available to us. In conjunction with the experimental work, an extensive computer simulation program was developed that helps us to understand the 131 Chapter 8. Conclusions and outlook 132 operation of the present maser. In addition, it is useful to guide the development of the second generation maser design. 8.2 Suggested improvements of the cryogenic maser design 8.2.1 Full size maser cavity For the prototype maser, we chose a small size maser cavity in order to keep the design and construction as simple as possible. However, both the frequency instabilities due to thermal noise (4.1) and due to white phase noise in the electronics (4.8) decrease with increasing power. To first order, the maser output power is expected to scale linearly with the volume of the maser bulb. Therefore, a cryogenic hydrogen maser with the highest possible frequency stability has to have a full size microwave cavity. High performance room temperature masers use the TEon mode of a silver plated high Q cavity. The hydrogen atoms are confined by a quartz bulb to a region in the center of the cavity where the phase of the rf magnetic field has the same sign. This design is not very well suited for a cryogenic maser because it would be difficult to control the temperature of the maser bulb to a high degree. An attractive alternative is a dielectrically loaded cavity, where the dielectric medium confines the hydrogen atoms and at the same time reduces the overall size of the maser cavity. The dielectric medium must exhibit low microwave losses, high thermal conductivity and a reasonably high dielectric constant. The best choice is single crystal sapphire with a dielectric constant e « 12 and a loss tangent tan 6 of less than 10~7 at low temperatures. A sapphire loaded cavity in the TEon mode has been used successfully by the Harvard-Smithsonian group [17] in their cryogenic maser. The desired TEon mode is not the fundamental mode in such a cavity. Since the cost of a large single crystal sapphire is high and the cutting and polishing of the surfaces Chapter 8. Conclusions and outlook 133 is demanding and time consuming, the design of the cavity has to be chosen carefully in order to avoid an accidental degeneracy with one of the many other modes. For this reason, the frequencies of all the modes are determined in appendix E. The calculation is complicated by the fact that sapphire is anisotropic and has two dielectric constants e2 and ex.1 8.2.2 Proposed new pumping scheme If the split-ring resonator were to be replaced with a much larger volume sapphire loaded cavity, it is not clear whether the state selector in the present state would be efficient enough for maser oscillations and whether a larger power output would result. Based on the analysis of the experimental results and the simulation calculations, it is obvious that the present apparatus is not optimized. Two of the main problems that have been identified are the low efficiency of the magnetic foil in the high field region and the strong cross relaxation in the maser bulb. However, even if these problems could be solved, the performance of the maser would not improve as drastically as one might naively expect. The present pumping scheme has some intrinsic limitations. The spin exchange relaxation in low field leads to a large background of atoms in the | d ) state in the maser bulb. Two atoms in the | a ) state or two atoms in the | c ) state can be relaxed to a | b ) and a | d ) atom in a collision. a + a b + d c + c ^ b -f d In the absence of strong cross relaxation, these spin exchange processes determine the 1A similar calculation was performed by Folen et al. [79], but their results are not available to non-US citizens. However, we were informed (A. J. Berlinsky, private communications) that these authors did not take the anisotropy of the sapphire correctly into account and obtained erroneous results. Chapter 8. Conclusions and outlook 134 | d ) density in the maser bulb and the atom transport tube. The | b ) atoms are sucked back into the high field region of the state selector, resulting in a depletion of | b ) atoms in low field. The backward rates for the two spin exchange processes are therefore low and a large | d ) density builds up in the low field regions until the backward rate balances the forward rate. This leads to a low population inversion. In order to avoid this problem, a new pumping scheme is proposed which is a variation on the present one. The maser consists again of a region in zero field with the maser cavity, and a microwave cavity in a high magnetic field. In this new scheme, the | a ) to | d ) ESR transition is pumped instead of the | b ) to | c ) transition. In the atom transport tube, a small magnetic field is applied and an rf field, tuned to the | c ) to | d ) transition, equalizes the densities of | d ) and | c ) atoms. This way, a population inversion close to 50 % is obtained outside the maser bulb. With this pumping scheme, no large buildup of | d ) atoms in the maser bulb is possible any more. The | 6 ) atoms that are attracted to the high field region do not get trapped in the ESR cavity. They relax with | d ) atoms, which are always present in the ESR microwave cavity, into an | a ) and | c ) atom by spin exchange. We have modified the simulation computer program and confirmed that, at least theoretically, this pumping scheme results in much higher population inversions and power outputs than the scheme used in the present maser. The additional rf field in the new scheme adds some complexity. However, it gives the experimenter more control and circumvents all the difficulties associated with the magnetic foil. One of the intrinsic problems with the foil that we have not mentioned so far is that its efficiency necessarily decreases with time due to the buildup of a layer of molecular hydrogen on top of it. Statt et al. [59] observed in their ESR experiment, that after atomic hydrogen recombined to H2 and froze out on top of the magnetic impurities in the wall of their container, the one-body relaxation rate decreased significantly. The same process will affect the long term stability of the present maser with the magnetic Chapter 8. Conclusions and outlook 135 foil. In the new scheme, there are three regions with three different magnetic fields. It is therefore important to shield each of them carefully. This can be achieved with super-conducting shields that have high critical current densities, as discussed in appendix D. Alternatively, the shield can be replaced by a counterwound solenoid. This technique was successfully used by Israelsson and Gould [80] in shielding a superconducting magnet. The applied magnetic field in which the | c ) and | d ) densities are equalized by an rf field should be low enough in order to avoid the degradation of the magnetic field homogeneity at the maser cavity. On the other hand, it should be high enough to remove sufficiently the degeneracy between the | d ) to | c ) transition and the | c ) to | b ) transition. A reasonable choice seems to be a field of a few mT, which corresponds to a transition frequency of a few tens of MHz. At these fields, the | c ) to | d ) transition is essentially an electronic spin flip (see appendix A). 8.2.3 Optimization of maser There are many parameters in the cryogenic maser that have to be optimized for the best performance, especially the volumes of the different regions and the sizes of the orifices connecting them. In the absence of any relaxation mechanisms, the inverse holding time for an atom in state | k ) and in volume i is given by 4) = E (8-1) where is given by (7.2) and depends on the volume of region i and the cross sections of the orifices. Ideally, all the desired relaxation and pumping rates in volume i affecting | k ) atoms are large compared with the rate (8.1), and the undesired relaxation rates, such as cross relaxation or possibly spin exchange relaxation rates, are much smaller than (8.1). This design rule helps to determine the optimal values for the different volumes and Chapter 8. Conclusions and outlook 136 the orifice sizes. It is of course difficult to anticipate correctly the one-body relaxation rates in the different volumes. A holding time of special importance is the maser bulb holding time Tb. Its proper choice is to a large extent determined by the intended application of the maser. Within the approximations used in chapter 4, the optimal intrinsic frequency stability (4.5) does not depend on the value of Tb. However, the best operating density nopt nse is proportional to the inverse bulb holding time (see 2.43). A low operating density reduces effects due to spin exchange shifts and recombination. Therefore, for the best long-term stability of a cryogenic maser, the maser bulb holding time should be chosen as large as the cross relaxation rates permit. If the user of the hydrogen maser is primarily interested in its excellent short-term stability, the maser bulb holding time should be adjusted to a smaller value, as can be seen from eq. (4.9). A small bulb holding time results in a larger flux of incoming state selected atoms and in a larger power output, but at the cost of a broader atomic line. For averaging times r where the electronic noise contribution dominates, a reduction of Tb therefore improves the frequency stability. It is not obvious what value should be chosen for the magnetic field B at the microwave cavity. An increase in B also increases the resonant frequency of the ESR transition which implies that the volume of the microwave cavity decreases. This is offset by a larger Boltzmann factor and therefore a larger density of high field seeking atoms in the cavity. It has to be checked with a computer model calculation whether the faster spin exchange and recombination processes due to larger densities cause any difficulties. We also point out that with a large field B, small temperature fluctuations in the state selector can cause significant density fluctuations in the low field regions. This can then lead to frequency fluctuations via the spin exchange shift. Chapter 8. Conclusions and outlook 137 8 . 3 Open questions With the present work, we have not addressed several important questions, whose an-swers will affect the ultimate frequency stability that can be obtained with a cryogenic hydrogen maser. One of them deals with the frequency instabilities induced by the den-sity dependent spin exchange shifts. It will be important to verify experimentally the theoretical predictions for the spin exchange shift cross sections, which were calculated by Koelman et al.[46]. In the prototype maser, no attempt has been made to stabilize the density and the populations. It is therefore not known at the present, to which degree these quantities can be held constant and at what level these frequency fluctuations will limit the performance of a cryogenic hydrogen maser. At least for short averaging times, the cryogenic hydrogen maser promises to improve the state-of-the-art in frequency stability by at least an order of magnitude (see fig. 6.11). However, in order to take advantage of this high frequency stability, close attention has to be paid to all electronic components. Meeting all the stringent requirements on phase noise will be a major challenge. Appendix A Magnetic moment operators The interaction of atomic hydrogen with a classical rf magnetic field is determined by the Hamiltonian Hrl = -/2 -Brf{t) (A.l) where the magnetic moment operator fl has a contributions from the electron spin and the proton spin: /I = - % j e s + %~fvi (A.2) For the calculation of the transition probabilities induced by the rf magnetic field in first order perturbation theory, the matrix elements of fi between initial and final state (f\\fi \\i) have to be evaluated. We are interested in transitions between the eigenstates | a ), | b ), | c ) and | d ) of the Hamiltonian Ho (2.1). Below we list the matrix elements of jlx, fiy and jj,z between all these states, or what is equivalent, we list the three operators in basis of the eigenstates of Ho. These expressions are valid for an arbitrary applied dc magnetic field BQ. 0 7esin0+7pcos0 7e sin 0+7P cos 6 0 0 7ecos0+7Psin0 -7e cos 9—7P sin 6 0 0 — 7e cos 0+7P sin# 0 — 7e sin #+7P cos 0 ^—7ecos0 —7Psin0 0 — 7esin0+7pcos0 0 j 138 Appendix A. Magnetic moment operators 139 ^ 0 7 esin0+7 Pcos0 0 7 e cos 0+ 7 P sinf^ —7 esin0—7 Pcos^ 0 7 e cos 9—7P sin# 0 0 —7e cos 0+7 P sin 0 0 7 e sin 6—7P cos 0 ^—7e cos 0 — 7 P sin 6 0 —7 e sin 6+7P cos 0 0 (7e + 7P) C O S 2^ 0 ( 7 e + 7p)sin20 0 0 (7e + 7 P)sin20 7e - 7 P 0 0 - ( 7 e + 7p)cos20 0 0 0 0 0 \"7e + 7p ) The mixing angle 6 is given by tan(20) ^(7e + 7P)-So For zero magnetic field 6 = |, whereas for high fields f?0 ^> % l l a + 1 ^ = 50.6 mT, 6 can be approximated by a 25.304 mT Appendix B The Slater equation We want to calculate the rf magnetic field that is produced by an oscillating magnetization inside a microwave cavity. This problem was first treated by Slater [81], but we follow here the simpler derivation of Wineland et al. [82]. We assume that there is only one cavity mode coupled to the magnetization and we take the magnetization to be oscillating along the z-axis. The microwave cavity, coupled to the output electronics which has impedance Zo, and containing the rf magnetization Mz(t) can then be modeled by the electrical circuit shown in fig. B.l. The magnetic field Hz in the coil with inductance L is proportional to the current I flowing through the coil: Hz = kl (B.l) The voltage across the inductance is then given by where rj is a filling factor. The voltage across the whole circuit has to vanish. 1 o = Vtot = -k Ljt [Hz(t) + vMz(t)} +(R + ^) Hz{t) + I J H,(t) dt (B.3) This equation can be rewritten with the following notations for the resonance frequency of the cavity u>c, the coupling coefficient /?, and the unloaded and loaded cavity quality factors Q0 and Q\\\\ = Tc . c z\\ + 1 where A is given by A = QJ^-OL) (B.10) To second order in A and with the approximation ^ « 1 — ^ » 1, equation (B.9) becomes: Hz{t) = r,Q, ( l - ^ e*'(-*+A)M,(0 (B.ll) The ratio of the amplitudes Hz and Mz is therefore given by rjQi (l — ^r) and A is the deviation from phase quadrature. Figure B.l: Electrical model of microwave cavity coupled to output electronics and con-taining rf magnetization Mz. Appendix C Measures of frequency stability In this appendix we summarize different measures of frequency stability and the relation-ships between them. For more extensive treatment of this subject, the reader is referred to [83] and references therein. We assume that the output signal of the frequency standard, which might be a voltage, is of the form V(t) = [Vo + e(t)} 8m[2irv0t + - ^ = ^ ( C - 3 ) In the time domain, the preferred measure of frequency stability is the so-called 'Allan variance' or 'two-sample variance': = (±(yk+1 - ykf) (C.4) where yk is the fractional frequency y(t) averaged over the time interval tk to tk + r. _ k = 1 „{t) - vo d t = y( r 143 Appendix C. Measures of frequency stability 144 do not affect the Allan variance 0: /+°° •„ , ( 1 +2 f hf2 for f 1 Table C.l: Transformations between power-law spectral densities Sy(f) and Allan Vari-ance or3/2(r). (Euler's constant 7 = 0.577215 ...) Appendix C. Measures of frequency stability 146 r. The two processes can be separated by analyzing the different dependence of cr3/2(r) on the bandwidth fn, which can be decreased in the software. The so-called 'modified Allan variance' is based on this idea and is further discussed in [83]. In many experimental situations, dead time between subsequent measurements of yk cannot be completely avoided. Barnes et al. [84] discuss the effect of dead time on the obtained value for o~y2(r). In general, it gives rise to only small corrections as long as the dead time is short compared to r. However, in the case of white phase noise (a = 2 ) , a dead time comparable to the inverse bandwidth /„ has a significant effect on the measured value of u2/2(r). This can be understood by rewriting the Allan variance <7J,2(T) in terms of the phase: °* 00 = — ( a n ) For white phase noise, the correlation time for ip is determined by the bandwidth fn. For a bandwidth large compared to the inverse dead time t + — t~, it follows that * t + ) = 4(^(0) 2 6(^ (0) 3 In the case of short but finite dead time, the expression in table C.l for zQ, the magnetic field can be expanded into a series of cylindrical waveguide modes beyond cutoff, in our case at zero frequency. The coefficients of expansion can be determined with the techniques developed in section 8.11 of the textbook on electrodynamics by Jackson [85]. The result is (D.l) (D.2) where the coefficients Cj are given by: ^ i ( X j f ) XjbJQ2(Xj) (D.3) 147 Appendix D. Magnetic Held of a solenoid inside a long superconducting shield 148 superconducting shield T solenoid \"A : 2a V. 2b y t__ ___ -z 0 z 0 Figure D.l: Single layer solenoid inside superconducting shield. Appendix D. Magnetic field of a solenoid inside a long superconducting shield 149 and Xj is the j-th zero of the Bessel function J\\(x) (Ji(xj) = 0)-For z < zo, the magnetic field can be calculated by considering the superposition of the following two situations. We take the magnetic field of an infinitely long solenoid with the same radius and current density as before inside the superconducting shield. This field is uniform and is given by From this field, we now subtract the field produced by a semi-infinite solenoid extending from ZQ to oo with radius a. The magnetic field in and around the semi-infinite solenoid (—00 to z0) is therefore With (D.l) to (D.3) and (D.5) to (D.7) we have the complete solution of the semi-infinite solenoid. From this solution, it is now easy to obtain the field of a finite solenoid of length 2z0 (see fig. D.l). The field is the difference between the fields produced by a semi-infinite solenoid extending from — zQ to 00 and another one extending from z0 to 00. In the region —ZQ < z < ZQ, the magnetic field is then given by for p < a for a < p < b (D.4) Hz(a z0), we obtain H2(p,\\z\\>z0) = 2 / n £ C j s i n h ( x i y ) J o ( X i ^ ) e x p | - X j ^ | (D.ll) H,(p,\\z\\>*o) = S / n ^ f c . s i n h t e ^ ^ t e ^ e x p j - x ^ } (D.12) where the coefficients Cj are given by (D.3). Far away from the solenoid (\\z\\ ^ > z0), only the first mode contributes significantly and the magnetic field decays like exp {—3.832^}. The generalization to a multi-layer solenoid is straightforward. Each layer can be approximated by a current sheet of different radius and the contribution to the magnetic field of each layer can be determined by the expressions (D.8) to (D.12). In order to improve the homogeneity inside the solenoid, correction coils are often added at the ends of the solenoid. We illustrate this on a simple practical example. The main solenoid is taken to consist of N layers with radii a,- (i = 1,... N) and extending from — z0 to ZQ. On both ends of the main solenoid, Nc layers with radii <4 (k = 1,... JVC) and width w form the correction coils (see figure D.2). For simplicity, we assume that the current density is identical in all N layers of the main solenoid and Nc layers of the corrections. Inside the main solenoid, but outside the correction coils (\\z\\ < ZQ — w), the contri-bution to the magnetic field from the correction coils is: NC oo HZ(p,z) = 2 / n £ E c M ) e-*& (e** - l) J 0 (xif) cosh(X4) (D.13) i=lj'=l 0 0 NC oo Hfa,z) = -2InJ2ECM)E~X^{EXLF-1) MXJT) sinh(XiT) (D.14) »=ij=i 0 0 where Cj(di) is given by (D.3) with a replaced by d{. The major contribution to the inhomogeneities close to the center of the solenoid are due to the low modes. If the width w of the correction coils is chosen to be b 1 w = — In Xi 1 + S S W i ( x i ? ) (D.15) Appendix D. Magnetic field of a solenoid inside a long superconducting shield 151 superconducting shield solenoid 2a ; 2d, 1 * Z ° : + Z 0 2b w w Figure D.2: Multi-layer solenoid with correction coils inside superconducting shield. Appendix D. Magnetic field of a solenoid inside a long superconducting shield 152 (where xi = 3.8317...), then the total contributions of the lowest mode (j = 1) to the magnetic field inside the solenoid vanishes exactly. With more complicated correction coils, it is also possible to cancel higher modes. Appendix E Eigenmodes of a sapphire loaded microwave cavity In this appendix, the eigenmodes of a cylindrical metallic cavity, partially filled with a ring of anisotropic dielectric material are calculated. In particular, we have in mind the geometry of fig. E.l, where the dielectric ring consists of a single crystal sapphire, oriented along the 2-axis. Sapphire has two dielectric constants, ez along the z-axis and 6 i perpendicular to it. E . l General Solution We treat here the slightly more general case where both the permeability and the di-electric properties are anisotropic. Inside the cavity, the solutions have to satisfy the charge-free and current-free Maxwell equations subject to the appropriate boundary con-ditions: where D VD V xH V x E V -J3 0 dp dt dB dt ( eL 0 0 ^ 0 e x 0 0 0 ez E (E.l) (E.2) (E.3) (E.4) (E.5) 153 Appendix E. Eigenmodes of a sapphire loaded microwave cavity 154 region 1 region 2 Figure E.l: Schematic drawing of the sapphire loaded microwave cavity. ( \\xL 0 0 ^ B = HQ \\ H (E.6) 0 nx. 0 0 0 fiz j The tensor character of e and fj, gives rise to additional terms in the wave equations. For instance, the divergence of E does in general not vanish, but it is given by ex. dz and similarly the divergence of H is fli. oz The symmetries of the problem imply that the solutions are of the form: E(p, i < /> < ^ ) = b2Jp(p%>p) + hYp(^p) (E.18) (E.19) (E.20) In the case that fa1' or ft' is imaginary, the Bessel functions in the corresponding region E.2 Boundary conditions The possible values of the longitudinal propagation constant kz, the frequency LO and the radial wavenumber /3 are determined by the boundary conditions. We assume perfect metallic walls with infinitely high conductivity at z = 0, z = L and p = p2. The components of the electric field parallel to the walls must therefore vanish exactly at the walls. The requirements Ev(z = 0) = Ev(z = L) — 0 and Ep(z = 0) = Ep(z = L) = 0 lead immediately to the quantization of the propagation constant kz: It is more difficult to calculate the quantized values of the radial wavenumbers (3, but as we can see from the expressions (E.ll), (E.13) and (E.15) this is equivalent to finding the eigenfrequencies LO. The solutions of the eigenmodes have to meet the following conditions: kz = n- (n = 0,1,2,...) (E.21) • Ez {p = p2) = 0 • Ev(p = p2) = 0 • Eg is continuous at p = p\\ Appendix E. Eigenmodes of a sapphire loaded microwave cavity 157 • Ev is continuous at p = pi » Hz is continuous at p = pi 9 Hep is continuous at p = p\\ If the four components Ez, Ev, Hz and Hv are continuous at the interface between the two dielectric media, it follows that Dp and Bp are also continuous. The six condi-tions mentioned above lead to a set of linear equations for the six coefficients a,- and (»' = 1,2,3), defined in eq. (E.17) to (E.20): / \\ ai 0-2 03 h [A] \\ \"3 I In the case that ^ and fy' 1 are both real, the matrix A is given by: ?(2) (E.22) [A] 0 0 -UP® PI) - ^ ( /4V) JM ]P2) Yp(/3^p2) 0 0 0 0 0 -c($J'M}P^ ^J'M'px) -JM}PI) Ufflpi) Ytfgpi) {-cgJ'Mpi) 4 2 )WiV) 42>WgV) -^-ufflpi) ^UPVPI) f w k V ) 0 $4 (&P2) cgYlfflpt) 0 (2), 0 0 ,(2) (2). „(2)i (E.23) Appendix E. Eigenmodes of a sapphire loaded microwave cavity 158 where the following abbreviations have been used: (E.24) (E.25) (E.26) and the derivatives of the Bessel functions are indicated by a '. In the case that or jffl is imaginary, the matrix A has to be modified as follows: the imaginary values of /? have to be replaced by their absolute values and the appropriate Bessel functions have to be replaced by the modified Bessel functions. Eigenmodes are characterized by non-trivial solutions of eq. (E.22). Therefore, the determinant of the matrix A has to vanish for all eigenfrequencies: This equation determines the eigenfrequencies. The left hand side of (E.27) depends in numerically. In a simple cylindrical cavity with metallic walls (but without different dielectric media), all the eigenmodes can be categorized either as T M p m n mode or as T E p m n mode. In a TM mode, the magnetic field has only transverse components and no 2-component, and in a T E mode, it is the electric field that has this property. For both types of mode, the eigenmode can be identified with three quantum numbers: the angular quantum number p, the radial quantum number m and the longitudinal quantum number n. The radial quantum number m (m = 1, 2,3,...) labels the modes with identical p and n in order of increasing frequency. det(A) = 0 (E.27) a rather complicated way on u>, namely through the parameters c0l\\, and e$ and through the radial wavenumbers (3Q \\ ft^ and (3$, In general, (E.27) can only be solved Appendix E. Eigenmodes of a sapphire loaded microwave cavity 159 In general, the eigenmodes of a dielectrically loaded cavity do not fall in any of the two categories: both the electric field and the magnetic field of most eigenmodes have a non-vanishing ^ -component and the modes are therefore neither pure T E or T M modes but hybrid modes. The only exceptions are the T E 0 m n , the T M 0 m n and the TM p m 0 modes (TE p mo modes do not exist). In these cases, vanishes and the matrix A can be split into two independent 3 x 3 matrices. For all other combinations of quantum numbers it is not possible to satisfy all the continuity conditions at p = pi simultaneously if Ez = 0 or Hz = 0. E.3 Application to the cryogenic hydrogen maser In this section, the cavity mode calculations are applied to find the best configuration of a sapphire loaded cavity that is to be used in a cryogenic hydrogen maser. The frequency of the operating mode has to coincide with the hyperfine frequency of hydrogen, which is 1420.405 MHz. The choice of the operating cavity mode is determined by the requirements of the hydrogen maser. The radiating hydrogen atoms couple only to the z-component of the magnetic rf field and therefore, all TM modes are excluded. In addition, the rf field Hz should be as uniform as possible in the entire region that is filled with hydrogen atoms in order to suppress the first order Doppler broadening. This applies not only to the magnitude of Hz but also to its phase. In particular, if the spatial average of Hz is zero, the first order Doppler broadening is not suppressed at all and it results in an unacceptable broadening of the maser line. If we assume that the hydrogen atoms fill the entire region (1) and no diaphragms are used to divide up this space, the requirements on the angular and longitudinal quantum numbers are p — 0 and n < 1. Only the T E 0 m i modes satisfy all the requirements, of which the TEon mode is by far the best choice. Appendix E. Eigenmodes of a sapphire loaded microwave cavity 160 20 I—<—i—1—i—1—i—1—r—1—i—1—i—•—r 0 L , i i i , i — i — i — i — i — • — i — i — i — 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 Inner diameter (cm) Figure E.2: Required length of the sapphire loaded cavity that brings the TE 0n frequency in resonance with the hyperfine frequency of hydrogen. The outer diameter of the sap-phire piece is fixed at 11.5 cm. The dielectric constants for sapphire are £j_ = 9.264 and ez = 11.341. As an illustration, we calculate here the size of a sapphire cavity whose TE 0n mode is on resonance with the maser transition, with the additional restriction that the outside diameter of the sapphire ring (2/92) is 11.5 cm. As the inner diameter of the sapphire piece (2pi) is varied, the length has to be adjusted to keep the frequency of the TE 0n mode at its desired value. Equation (E.27) can be used to calculate the required length. For the TEon mode, the matrix A is only a 3 x 3 matrix (the last three columns of row 2, 4 and 5 in eq. (E.23)) and we solve numerically eq. (E.27) for L while we fix u> at the hyperfine frequency of hydrogen. Figure E.2 shows the resulting length of the sapphire loaded cavity as a function of the inner diameter. In this calculation we assume that in the inner region e = fi, = 1. The temperature dependence of the dielectric constants of sapphire has been measured by Loewenstein et al. [86] and by Shelby et al. [87]. Extrapolated to T = OK, Loewenstein et al. obtained e x = 9.290±0.012 and ez = 11.264±0.012, and Shelby et al. obtained e± = 9.264±0.006 Appendix E. Eigenmodes of a sapphire loaded microwave cavity 161 and ez = 11.341 ± 0.006. Note that the two measurements are in slight disagreement. In the present calculations, we have used the values quoted by Shelby et al. [87] and we also set fij_ = fiz = 1. In the application of the hydrogen maser, it is important that the TE 0n mode is well separated in frequency from all the other modes. We have calculated the eigenfrequencies of all modes below 2 GHz as a function of the inner diameter of the sapphire piece, where its length is varied according to the results of fig. E.2. The results are shown in fig. E.3. It is clear from fig. E.3 that the exact dimensions of the sapphire cavity have to be chosen carefully in order to avoid an accidental degeneracy of the TE 0n mode with any of the numerous other modes. We note that it is possible to shift the frequency of an undesired mode sufficiently by a slight modification of the cavity geometry, such as a groove or a ridge. The resulting shifts with the modified geometry can then be calculated from the solutions of the present geometry by the method of cavity perturbation theory. The final design of a sapphire loaded cavity should not only take the results of fig. E.3 into account. For the application of the cryogenic hydrogen maser, it is also important to optimize the volume of the inner region and the filling factor rj. The filling factor 77 (see eq. 2.30) can be easily calculated from the present solution of the TE 0n eigenmode. Appendix E. Eigenmodes of a sapphire loaded microwave cavity 162 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 Inner diameter (cm) Figure E.3: Frequencies of all cavity modes below 2 GHz in the sapphire loaded cavity as a function of the inner diameter of the sapphire ring. The outer diameter is fixed at 11.5 cm and the length is adjusted as shown in fig. E.2 in order to keep the frequency of the TEon mode at 1420.405 MHz. The hybrid modes are only labeled with their angular and longitudinal quantum numbers p and n. Bibliography [1] R. F. C. Vessot, M. W. Levine, and E. M. Mattison. Comparison of theoretical and observed hydrogen maser stability limitation due to thermal noise and the prospect for improvement by low-temperature operation. 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