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Superconducting qubits : survey and theoretical investigations for solid state quantum computing Gupta, Santosh Kumar 2006

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SUPERCONDUCTING QUBITS: SURVEY AND THEORETICAL INVESTIGATIONS FOR SOLID STATE QUANTUM COMPUTING by SANTOSH K U M A R G U P T A B.Sc, The University of New Brunswick, 2004 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Physics) T H E UNIVERSITY OF BRITISH C O L U M B I A September 2006 © Santosh Kumar Gupta, 2006 A b s t r a c t Superconducting qubits have in recent years become a promising candidate for the implementation of a quantum computer due to their design flexibility, good protection from decohering elementary excitations, and availability of well developed fabrication and measurement techniques. Superconducting flux qubits, for which the effect of offset charge noise is reduced due to the fact that the Josephson energy dominates over the charging energy, corre-spond to one of the proposed means of designing a superconducting qubit. A nonlinear dispersive readout scheme of flux qubits involving a D C SQUID magnetometer that avoids the effects of on-chip dissipation can be readily implemented, yielding high contrast output for single qubit readout. Cou-pling schemes via nonlinear Josephson elements have also been realized. On the other hand, while the means of isolating superconducting qubits from ex-ternal noise sources has been found, the mechanisms by which they undergo relaxation and decoherence due to intrinsic noise sources in the junctions themselves are not very well understood, and the question of how to deal with these noise sources remains unanswered in the general case. Other ii questions deal with the problem of experimentally observing entanglement in an array of coupled superconducting qubits, and finding the means by which the existence of entanglement in a typical laboratory setup may to some extent be verified by measurements on a global scale. Following a brief introductory review, we will first investigate the influence of a Two-Level Fluctuator on a D C SQUID driven by a finite current bias. Then we intro-duce a directly measurable signature of multiqubit entanglement for a large system of qubits and show that it is compatible to a recently introduced measure of global entanglement. iii C o n t e n t s Abstract ii Table of Contents • • • • i v List of Tables vi List of Figures vii Acknowledgements viii 1 Introduction 1 1.1 Macroscopic Quantum Coherence 2 1.2 Superconductivity 7 1.3 Josephson Effect 13 1.4 Macroscopic Quantum Systems 17 1.5 Fluctuations and Dissipation 20 1.6 Two-level State Reduction and the Design of Qubits for Quan-tum Computation 26 1.7 Relaxation and Decoherence . 31 1.8 Readout 36 1.9 Entanglement 38 1.10 Quantification and Measurement of Entanglement 41 1.11 Objectives 45 2 DC SQUID measurement of a Two-Level Fluctuator 47 2.1 Preliminaries 47 2.2 D C SQUID with symmetric junctions 49 2.3 Screening Current Correlations 51 2.4 Effects of Small Asymmetries 55 2.5 Criteria of the observability of the fluctuator state 57 iv 3 Comparison of two methods of quantifying entanglement ap-plied to a quadruple array of flux qubits 61 3.1 Preliminaries 61 3.2 Numerical Results 65 4 Conclusions 71 Bibliography 73 v L i s t of T a b l e s 2.1 Minimum signal-to-noise ratio for different current bias strengths vi L i s t o f F i g u r e s 2.1 Diagram of D C SQUID circuit under the RCSJ model coupled to a two-level dipole 48 3.1 Four qubit array 63 3.2 Nonlinear Susceptibility and Global Entanglement Quantifi-cation Methods compared graphically for lowest eigenstates. . 68 3.3 Nonlinear Susceptibility and Global Entanglement Quantifi-cation Methods compared graphically for higher states. . . . . 69 3.4 Average difference between two entanglement quantities for different eigenstates 70 vii A c k n o w l e d g e m e n t s I would like to thank Alexandre Zagoskin for his supervisory role, for regular revisions of the draft manuscript during its development, and useful infor-mation. I would also like to thank Philip Stamp and Anatoly Smirnov for reading the draft manuscript and for discussions related to the topics covered. Finally I would like to thank the faculty and staff of the U B C Physics and Astronomy department and the Pacific Institute of Theoretical Physics for organizing and hosting the many informative workshops, conferences, and seminars related to the quantum information sciences and other fields in which I have taken interest. viii C h a p t e r 1 I n t r o d u c t i o n Before delving into a discussion of superconducting quantum circuits, which will then lead into a discussion of superconducting qubits, I think it will be ap-propriate to begin by explaining the basic concepts of macroscopic quantum coherence, superconductivity, and the Josephson effect. Subsequently I will give a discussion of superconducting quantum circuits as macroscopic quan-tum systems, including a description of models of the influence of dissipation on quantum fluctuations. Following this I will describe how superconducting qubits may be designed for quantum computing applications, focusing on a special prototype of superconducting qubit-the superconducting flux qubit-which will be the subject of study in the remaining chapters. I will briefly describe relaxation, decoherence, and methods of readout and entanglement of superconducting qubits. At the end I will finish with a discussion of the problem of entanglement quantification and describe some proposed methods 1 of quantifying entanglement. 1.1 Macroscopic Quantum Coherence In 1935, shortly after Einstein, Podolsky, and Rosen (EPR) published their paper on the paradox bearing their initials, Schroedinger, the author of the wave theory of quantum mechanics, wrote a paper describing a hypothetical experiment designed to further illustrate the seemingly contradictory rela-tionship between the postulates of quantum mechanics and classically ob-served phenomena [1]. This hypothetical experiment involved a cat trapped in a steel chamber containing a radioactive Geiger counter. The state of an atom in the counter will be such that when it has not decayed the cat will remain unaffected, but upon decay it will release a trigger which will result in cyanide gas being released into the chamber and the following death of the cat. Thus the state of the atom being either unchanged or decayed is correlated with the states of the cat being either alive or dead respectively. This thought experiment has motivated a series of investigations con-cerning what is known as macroscopic quantum coherence, which will be illustrated as follows. Suppose now that the atom is in a superposition state 72 ( I T ) + |!))> where ||) corresponds to the non-decayed state and ||) cor-responds to the decayed state of the atom. Then, due to the correlations between the atom and the cat, the principles of quantum mechanics as it is applied to closed systems imply that the total system of atom plus cat will 2 evolve into a superposition state ^ ( |f) \alive) + \i) \dead)). (one of these principles tells us that the state of the total system is a tensor product of the state of the individual systems atom and cat respectively, and another one of these principles tells us that the system will evolve in a linear fashion, so that after a sufficient amount of time has passed, there will be a 50% chance that the system will be in a state ||) \alive) where the atom has not decayed and the cat is alive and a 50% chance that the system will be in a state |J.) \dead) where the atom has decayed and the cat is dead. The important thing to note is that the probability of the cat being alive or dead is not simply a matter of chance of being in either state in the classical sense but arises from the cat being in a linear superposition of these states, so that the cat in fact does not exist in either of these states until it is observed by looking through the window of the chamber.) The idea that a macroscopic system such as a cat can exist in a super-position state is inconsistent with our intuition of how an "everyday object" ought to behave. For one thing, that the cat can be neither alive nor dead and that it collapses into one of these states only when it is observed, has its roots in the wave-particle duality of matter and is, to say the least, difficult to grasp (to this day no one has yet "understood" this concept). Nonethe-less, we have a priori no reason to suggest why quantum physics, as it is understood for closed systems, breaks down at the macroscopic level. Hence to provide a rationale for why the cat undergoes a transformation from a coherent linear combination of states to an incoherent mixture, a process 3 known as decoherence, it has been suggested that a macroscopic system or-dinarily wi l l deviate a lot from the behavior of a pure closed system and must be treated as open systems which are strongly coupled to their envi-ronment. This coupling results in dissipation of the quantum system, ie. the irreversible transfer of energy to the environment. Thus, for example, the cat in the original thought experiment irreversibly radiates thermal energy into the environment which is a source of decoherence. There have been a few different schemes for modeling the decoherence process. One model was proposed by Leggett and Caldeira [2, 3], in which the quantum system investigated was a model spin 1/2 system coupled to the environment so that any single mode of the environment is only weakly perturbed by the quantum system, resulting in a harmonic oscillator bath model of the environment. The authors were able to explain how the system-environment coupling led to an exponential decay of the off-diagonal elements of the system density matrix. In the original scheme used to investigate de-coherence mechanisms in the Caldeira-Leggett paper, the system considered was what is known as an R F S Q U I D (radio-frequency superconducting quan-tum interference device). The device consists of a superconducting loop of a metal such as niobium interrupted by a single Josephson junction (a thin in-sulating barrier between two superconducting capacitive plates) wi th a given capacitance, cri t ical current, and geometrical loop self-inductance through which there is an imposed external flux. The Hamil tonian of this system is of the form 4 ($ - $ext)2 I0(p0 COS 2 7 T $ ) (1.1) 2L 2ir 00 where p$ = C $ Here $ is the flux generated by the S Q U I D circulating current loop and <3>ext is the external flux threading the S Q U I D . Io, C, and L are the crit ical current, capacitance, and self-inductance of the Josephson junction respec-tively and 0o is the flux quantum. When the external flux is set to half a flux quantum, the potential has the form of a degenerate double well centered around this flux point. The ground state localized in each well form the two states of the qubit. The eigenstates of the Hamil tonian are antisymmetric and symmetric combinations of the states in either well. The nonlinearity of the Josephson junction makes it possible for the qubit level separation to be much smaller than the excited states of the Hamil tonian so that we can achieve a two-state system. The symmetric flux states of the R F S Q U I D are considered to be macroscopically distinct since they result from the action of millions of Cooper pairs traveling through the circuit. The goal of any investigation concerning macroscopic quantum coher-ence, then is to see if we can observe in the laboratory the superposition of macroscopically distinct flux states [4]. One way of doing this is by seeing if the system can be observed to undergo time dependent Rab i Oscillations, 5 ie. in which the probability of being either in one state or another oscillates sinusoidally. Such a phenomenon involves tunneling back and forth between the two localized states, a purely quantum mechanical effect with no classical counterpart, and is a signature of quantum coherence. Macroscopic Quantum Coherence in systems such as the R F SQUID are now observed experimentally on a routine basis, and their magnetic moment in the first set of experiments that observed coherent oscillations in these systems [5] were about I O ^ V B - Since this is equivalent to the magnetic mo-ment of around 10 1 0 electron spins aligned with each other, it is definitely reasonable to consider this system as being macroscopic. It is useful to note however that this is only one type of macroscopic quantum coherence, be-cause the macroscopically distinct states arise from a macroscopic number of microscopic systems that are indistinguishable from each other (ie. the Cooper pairs). Another type of macroscopic quantum coherence, which is perhaps closer to an actual "Schroedinger's Cat" and has not yet been ob-served, arises when we have a macroscopic array of identical but distinguish-able systems (in the sense of Maxwellian statistics) which exist collectively in a superposition state [6]. We now turn to a discussion of a specific type of phenomenon, related to the R F SQUID already mentioned above, for which macroscopic quantum coherence is a hallmark: superconductivity. 6 1.2 Superconductivity Superconductivity is a phenomenon that occurs in certain materials when the material is cooled to a sufficiently low temperature, which conventional models predict should be within about 30 K of absolute zero, though in actual experiments superconductivity has been observed in some materials at a temperature as high as around 130 K. There are three main effects that characterize this phenomenon [7]. Firstly, the resistance of the material vanishes below a critical tempera-ture that depends on which material is being analyzed (technically speaking, this is referred to as perfect conductivity). For example, suppose we cooled the material below this critical temperature by placing it in contact with a container of some substance like liquid helium which exists in this phase at the kinds of temperatures we need. (He-4 typically liquefies around 4 K. Nowadays to achieve temperatures as low as about 20 mK we make use of a combination of two different isotopes of liquid Helium, namely He-3 and He-4, which separates into a concentrated phase of He-3 with low He-4 con-centration and a diluted phase of He-3 with high He-4 concentration below a temperature of 700 mK. This is known as a dilution refrigerator). If we now took some voltage source and connected the ends of its leads to an ordinary resistor and measured the resulting current flowing through the circuit, and then inserted a sample block of the superconducting material in series with the resistor and measured the resulting current flowing through the circuit, 7 we would find that in both cases the resulting current has the same value. In addition, suppose we now remove the superconductor and wrap it into a closed ring. Then if we induce a steady current in this ring , we would find that the current would persist in the ring for an extremely long time, perhaps longer even than the age of the universe. Secondly, the superconductor expels magnetic fields from its core-not only wi l l it shield magnetic fields from the core, but it wi l l also expel a magnetic field that is already present in the conductor when it is cooled below the crit ical temperature (technically speaking this is referred to as perfect diamagnetism). There wi l l be a small amount of magnetic penetration at the surface of the conductor known as the London penetration depth, which is typically on the order of 50 to 500 nm when the temperature of the material is near zero. Consequently, there can be no flow of electric current within the core of the conductor (since the current would generate a magnetic field there), and it can only exist in the vicinity of the surface of the conductor. Finally, if we were to monitor the heat capacity of the superconducting material as we varied its temperature, we would find that above Tc the heat capacity varies linearly wi th temperature, as is the way a normal conductor behaves, but as we lowered the temperature below the crit ical temperature we would find a very sharp (essentially discontinuous) increase in the heat capacity followed by a rapid decay (ie. the heat capacity varies as e~? for a constant a) leading to a nearly constant zero heat capacity near zero temperature. This behavior suggests the existence of an energy gap between 8 superconducting and and normal states of electrons in the material. Another interesting effect that arises as a consequence of a material pos-sessing superconductivity, and which wi l l be especially relevant to our sub-sequent discussion on superconducting quantum circuits, is what is known as flux quantization. To explain the origins of this effect I wi l l make use of a description of the supercurrent that does not rely too heavily on the de-tails of the microscopic origins of superconductivity (such as interactions of the electrons wi th the surrounding lattice of ions), but instead describes the behavior of the supercurrent in a global sense. To this end, it wi l l be convenient to describe the supercurrent analogously to an electromagnetic wave. The "photons" making up the wave are pairs of electrons bound to each other, referred to as "Cooper pairs" (more formally each of the Cooper pairs have the quantum statistical properties of bosons). In the present context where we are dealing with a macroscopic number of Cooper pairs in the supercurrent, it is appropriate to treat the supercurrent as a classical wave in the same manner that an electromagnetic wave is treated classically when there are a macroscopic number of photons. More formally, we may describe the supercurrent by a wave function ib(r) which has an amplitude whose square gives us the charge density p(r) of superconducting electrons at a particular location at the surface of the conductor and which has an associated phase </>(r). 9 ^(r) = p(T)12ei^r) (1.2) ( A b r i e f n o t e : t e c h n i c a l l y s p e a k i n g , t h i s w a v e f u n c t i o n i s k n o w n a s a n order parameter i n a p h e n o m e n o l o g i c a l t h e o r y o f s u p e r c o n d u c t i v i t y k n o w n a s t h e G i n z b u r g - L a n d a u t h e o r y ) . T h e p h a s e i n i t s e l f h a s n o p h y s i c a l m e a n i n g . I t d o e s n o t d e s c r i b e t h e m o d -u l a t i o n o f t h e s u p e r c o n d u c t i n g c h a r g e d e n s i t y i n t h e s a m e w a y t h a t t h e p h a s e o f a n e l e c t r o m a g n e t i c w a v e d e s c r i b e s t h e m o d u l a t i o n o f i t s f i e l d s t r e n g t h ( t h i s i s w h y i t i s n e c e s s a r y h e r e t o u s e t h e c o m p l e x n o t a t i o n ) . I t p l a y s t h e s a m e r o l e a s i n t h e q u a n t u m m e c h a n i c s o f a s i n g l e p a r t i c l e f o r e x p l a i n i n g w a v e - l i k e i n t e r f e r e n c e e f f e c t s . O n t h e o t h e r h a n d , p h a s e d i f f e r e n c e s , p a r t i c u l a r l y w i t h r e s p e c t t o s p a t i a l t r a n s l a t i o n s ( i e . t h e g r a d i e n t o f t h e p h a s e ) i s p h y s i c a l l y m e a n i n g f u l : i t i s r e l a t e d t o t h e s u p e r c o n d u c t i n g c u r r e n t d e n s i t y J . M o r e f o r m a l l y w e c a n e x p r e s s t h e r e l a t i o n s h i p a s w h e r e A i s t h e v e c t o r p o t e n t i a l a t t h e p o i n t w e a r e e v a l u a t i n g a n d $ 0 i s N o w t h e n t o d e m o n s t r a t e f l u x q u a n t i z a t i o n s u p p o s e t h a t w e a g a i n w r a p t h e s u p e r c o n d u c t o r i n t o a c l o s e d l o o p . T h e n w e c a n v i s u a l i z e t h i s ' s u p e r c u r -(1.3) t h e flux quantum e q u a l t o . 10 rent wave' at various points around the loop from the functional dependence of the wavefunction on position given above. We can imagine monitoring the phase starting at one point, traversing the loop, until we finally return to the end point. Any discontinuity in the phase between the 'end' of the closed loop and the 'beginning' of the closed loop that is not an integer multiple of 27T will lead to an infinite current or infinite vector potential, hence we must conclude that the phase between the beginning and the end can only differ by an integer multiple of 27r (a more common way of expressing this is that the phase must be single-valued, though we cannot say it is completely continuous because a phase difference of an integer multiple of 27r does not alter the wave function). Another possible way of looking at this is to imag-ine the supercurrent wave traveling around the loop. For most cases when it returns to the starting point it will destructively interfere with itself, and it will only interfere with itself constructively when the phase after it has returned to the beginning point is the same as the phase it had at when it started out up to a term that is an integral multiple of 27r, in the manner of stationary waves (the problem with this picture is that it does not seem necessarily true that the supercurrent wave actually travels around the loop and is time dependent. The phase can be time dependent, in the presence of an electric field which causes the supercurrent density J to change in time according to the London electrodynamic equations [7]) Suppose we examine this supercurrent wave at the center of the supercon-ductor while it is in the closed loop configuration, where it was mentioned 11 previously that the supercurrent is zero. Then since the phase difference around the loop is an integral multiple of 27r, we have the following result which shows that the total flux enclosed by the loop is quantized, ie. it is an integer multiple of the flux quantum. Note that the total applied ex-ternal flux itself can naturally take on a continuous range of values. The superconducting loop behaves like a sort of inductor wi th loop inductance L whose circulating current (again created on the surface of the supercon-ductor) screens the applied external flux so that the total resultant flux is quantized. Bu t this is no ordinary inductor (though it would be if instead of a closed loop there were externals leads connected to the ends of a supercon-ductor that is bent into a loop-like or coil shape like an ordinary solenoid, for then the phase would no longer have to be single-valued). The only time we have zero screening current is when the external flux is an integer multiple of a flux quantum. If the applied external flux is wi thin a half flux quantum of an integer multiple of a flux quantum, then the resultant flux output wi l l be the product of that same integer and the flux quantum. In fact we have for the screening current is: (1.4) 12 is = y (2?rn - $ e x i ) (1.5) where n is determined by $ e I j as mentioned above, indicating the special relationship between the screening current and the external flux which is not linear. We now turn to a discussion of another kind of component with a special type of inductor: the Josephson junction. 1.3 Josephson Effect If we now place an ultra thin piece of insulating material such as an alu-minum oxide barrier between two superconductors, we will find that the whole configuration, known as a Josephson Junction [8], will also behave as a superconductor so long as the measured current is below a threshold value known as the critical current which we will denote as 7r> Hence for a straight DC bias current across the junction below the critical current level, we will not observe any voltage difference across it. As we can imagine, this can be verified in the same manner as for an ordinary superconductor, by placing a resistor in series with the junction and applying a voltage across the combi-nation (in which case the current bias is simply the voltage divided by the resistance of the resistor). 13 On the other hand, if we instead take out the resistor and apply a steady DC voltage directly across the junction we will measure an A C current across the junction with a frequency proportional to the voltage. We can verify that associated with this voltage there will be a capacitive charge q on either side of the junction proportional to the voltage by a constant C. In general we can model the Josephson junction as a type of inductor in parallel with a capacitor of capacitance C. We will note the following relationship when we measure the voltage difference across the junction. Let us denote the branch flux $ as being proportional to the time integral of the voltage across the junction, somewhat analogously to an ordinary flux in the expression of Faraday's law. This branch flux may also be described as being proportional to the difference in the phase of the superconducting plates on either side of the junction, which we can express as the gauge invariant phase 8 = Then we may relate the current across the inductance portion of the junction to the voltage indirectly through the branch flux by the relationship / = IQSW. (8)- The current through the capacitance portion of the junction will as usual be equal to Clearly when a small enough DC current is biased across the junction, the current will be proportional to the branch flux by a constant value of 2 ^ denoted as the Josephson inductance Lj. In general for larger bias cur-rents the Josephson inductance is a nonlinear function of the gauge invariant phase: Lj(8) = 2nitlosS- Hence we refer to this inductance as a nonlinear inductance (also referred to as a parametric inductance). The nonlinearity 14 of the Josephson inductance is ultimately related to the periodicity in the relationship between current and phase, and this periodicity ultimately is related to the discreteness of charge that tunnels across the barrier. Even more generally we can consider what happens when a D C current across the junction exceeds the critical current. Then we no longer have only a supercurrent flowing across the junction, but in addition to this there is a normal current It becomes necessary to add an additional shunt resis-tance to the nonlinear inductor-capacitor combination. Then the dynamics of the whole circuit can be modeled by a second order differential equation, which requires as input parameters the initial inductor current (correspond-ing to the initial S) and initial junction voltage V, known as the Resistively Capacitively Shunted Junction (RCSJ) model The differential equation describing the phase has the form of a mechan-ical pendulum driven by an external torque in a restoring gravitational field. In the transition to treating the Josephson junction as a quantum me-chanical entity, as will be described in section IV, it will be useful to describe the (non-dissipative) dynamics of the junction-that is, taking into account only the inductance and capacitance-by its Hamiltonian. For this we may determine the amount of energy required to acquire a given phase difference across the junction is easily found to be — Ej cos(<5), assuming the convention that we have zero energy when maximum supercurrent (at IQ) flows through the junction. Therefore the Hamiltonian will be (reverting to the notation of branch flux in place of phase) 15 H=^-Ejcos6 (1.6) Here the conjugate momentum to the branch flux p$ is C<j>, and the potential counterpart is referred to as having a washboard potential. When a DC current / is biased across the junction low enough not to cause resistive dissipation in the circuit, there is an additional energy proportional to the product so that the Hamiltonian becomes Because the bias current causes the washboard potential to become slanted downwards, this is referred to as having a tilted washboard potential. Suppose we now wrap the superconducting leads on either side of the-junction into a closed loop as we did previously in the absence of a junction. We will recall that when there is no junction, the flux is quantized. With the junction present we will still get a quantization condition, but now the total phase includes the phase difference across the junction so the quantity that is quantized is the total external flux plus the gauge invariant phase difference. In other words, 16 8 + 27T-P - 2im Hence the gauge invariant phase can be effectively replaced by a term proportional to the applied external flux through the loop modulo 27r. Similar results apply when there are more junctions: we simply add or subtract the phase difference of each junction in the quantization condition. When more than a single junction is present we simply add the gauge invariant phases of the additional junctions in the quantization condition. 1.4 Macroscopic Quantum Systems In the following we will discuss superconducting quantum circuits in the context of 'macroscopic quantum systems'. Subsequently in section V we will explain how the influence of dissipation (the irreversible transfer of energy to its environment) on quantum mechanical fluctuations in the circuit may be modeled. A superconducting quantum circuit, then, is a circuit consisting of leads constructed of superconducting material which typically may be interrupted by ordinary inductors and capacitors (which themselves are made of super-conducting materials), as well as elements like Josephson junctions. In the past, the superconductors have primarily been made of low temperature su-17 perconducting materials such as aluminum and niobium (although there has also been work done on the use of materials such as high-Tc superconductors [9]). Circuits are patterned by means of optical or electron beam lithogra-phy in the same manner as semiconductor circuits are patterned for use in integrated circuits, and typically may be on the scale of a few microns. One of the reasons interest in superconducting quantum circuits first arose was because they can behave as macroscopic quantum systems [10], which we have illustrated to some extent in section I. Macroscopic quantum systems have been of much interest lately due to predictions of new kinds of phenom-ena, which arise due to the fact that these systems couple strongly to their environment (as opposed to, for example, individual nuclear spins). Two kinds of macroscopic quantum systems we will consider are the LC circuit and the Josephson junction circuit. In both cases a typical circuit may be patterned on a chip in equilibrium with a dilution refrigerator at 20 mK enclosed in a copper shielding cham-ber. This temperature value is chosen so that thermal fluctuations do not exceed the energy level spacing set by the resonator frequency, so as to avoid incoherent mixing between the levels. We can model the circuit using an artificial diagram in which we have an inductor in parallel with a capacitor (In the case of the LC circuit which has a linear inductance, the actual inductor could be a coil fabricated of niobium). Typically the inductor will have a value of around 10 nH and the capacitor will have a capacitance of around 1 pF, which leads to a resonant 18 frequency of around 1.6 GHz-a microwave frequency oscillation. Under these conditions we may consider the charge and flux as quantum mechanical variables in themselves. In addition we must ensure that quantum mechanical effects persist long enough to be observed, meaning that damping due to the environment must not cause a decay rate that is close to or exceeds the resonant frequency. Hence the quality of the circuit must be high enough. Other considerations, which for sake of brevity will not be discussed in greater depth, more or less set the temperature, size, and frequency of the resonant circuit. If we make the circuit too big, the inductive coupling to the magnetic environment will lead to stronger damping and quantum fluctua-tions will be quickly washed out. If we make the circuit too small, for one thing the effects of damping on quantum fluctuations will be less noticeable (which was one of the key points of studying macroscopic quantum systems in the first place), and even if the damping is undesirable as in the context of quantum information processors the circuit will be still be less accessible to measurement, and will couple less strongly with other macroscopic quantum systems such as Josephson junction circuits, which we will describe shortly. Consider now the LC circuit in the quantum mechanical regime with its inductor and shunt capacitance. As already mentioned, we may characterize the system by the usual variables q and $ corresponding to the charge across the capacitance and flux through the inductor respectively. We have the relationship q = —C^ from Faraday's law , with q conjugate to —$ and capacitance functioning as mass (assuming the charge difference is taken in 19 the same direction as the current through the inductor). Consequently we find the commutation relation [—$, q] = ih [$, q] = —ih. (We will note that the relation $ = also holds true when the circuit has no current bias, and so we may choose the flux to be a momentum coordinate instead, but in general we will not limit ourselves to this specific case and will assume that there may or may not be a current bias across the circuit.) In the case of the Josephson Junction, because there is a nonlinear rela-tionship between the branch flux through the effective inductance and the current through the effective inductance, it follows that the branch flux will not be proportional to the time rate of the charge across the effective ca-pacitance even when no bias current is present. Hence the branch flux can-not serve as a momentum coordinate as was the case for the LC oscillator, and can only serve as a position coordinate from the Josephson relation q = —C^f, and we have the commutation relation [$, q] = —ih. 1.5 Fluctuations and Dissipation Up to now I have been describing non-dissipative quantum circuits-that is, ones which do not contain a dissipative element such as a resistor and are isolated from other environmental degrees of freedom such as stray mag-netic fields. But suppose that we want to consider a more realistic model of a macroscopic system that takes into account dissipative elements or these other degrees of freedom [10]. For example we may be given the circuit con-20 nected to a shunt admittance Y(u), in which Y(ui) has both a real part and an imaginary part, wi th the real part representing the dissipative element. From numerous basic examples of dissipative systems it is clear that the dy-namics of the system is irreversible, which makes it difficult to study the system using the conventional Hamiltonian formalism since it is generally time reversible. One way to deal with the problem is to follow the approach of the Caldeira-Leggett Model. This approach incorporates the dissipative admit-tance into the Hamil tonian by splitting up the admittance into an infinite array of parallel admittances, each individually corresponding to an LC os-cillator (in the following we wi l l assume a linear dissipative element). This circuit can be incorporated into the Hamiltonian as a term ?m + ti IC, (1.8) Where m corresponds to each individual oscillator charge qm, capacitance Cm, flux $ m , and inductance Lm. The time reversibility problem is dealt wi th by considering these infinite degrees of freedom and making the frequency slightly complex u —* ui + irj, where we take the l imit that n is negligibly small. This is the generalized admittance. The Fourier transform of Ym then obtains a real part 21 Sft {Ym(uj)} - y m - u m [6(u - um) + 6(u + um)] (1.9) Here um is the resonant frequency of the circuit and ym = IT V C " The imaginary part of Ym(u) corresponds to the admittance of the non-generalized LC oscillator m. Hence we see that the contribution to the damping is given by the inverse of the inductance. The resonant frequencies u>m are chosen to be equally spaced in a suitably chosen interval A a ; (as a matter of fact, later on it wi l l be useful to assume that this frequency interval spacing should be chosen to be negligibly small so that discrete sums over the parallel admittances can be converted to a continuous integral). W i t h the model of the dissipative admittance so defined, we can de-termine the current correlations in a classical dissipative circuit element at equilibrium temperature T to be Now we wi l l consider a quantum mechanical LC circuit whose correspond-ing admittance is generalized in the same manner to complex frequencies. For notational convenience we wi l l use instead the impedance, which has a similar generalized form as the admittance (1.10) 22 & {ZLCH} = Z0 [5{u - to0) + S(UJ + W o)]| (1.11) Here o>0 is the resonant frequency of the LC circuit and Z 0 = Be-cause of the form of the real part of the dissipative impedance involving delta functions, we can readily find the spectral density C(u>) of the inductor flux $ as a simple product of the spectral density of the non-dissipative circuit and the dissipative part. C M = -U) coth + 1 » { Z ( W ) } (1.12) This is a version of the quantum fluctuation-dissipation theorem. A t this point, i n order to study the effects of dissipation on the quantum LC circuit, it is legitimate to replace the real part of the impedance by the corresponding impedance of a linear resistor wi th resistance R, so that the total impedance of the LC circuit is just Z{u)'= iRLui R(l - ulu2) + iLuj (1.13) Then from the quantum fluctuation-dissipation theorem we can determine the fluctuations in the flux ( $ 2 ) . For brevity we wi l l simply state the low 23 temperature l imit: ) (1.14) 2 TT^/K2 — 1 where K = (2RCcu0) is the damping parameter. Similar results are obtained for the charge fluctuations (Q2). It is interesting to consider now the effects that damping have on quantum fluctuations in the current-biased nonlinear Josephson junction which has the form of a t i l ted washboard potential as described in section III. More precisely we wi l l be concerned wi th how quantum tunneling effects that arise due to these fluctuations are influenced by damping. Even when the bias current is below the crit ical current of the junction it is possible for the junction to make the transition to higher quantized states of the branch flux by quantum tunneling. Here an expression for the tunneling rate T out of one of the wells of the t i l ted washboard potential current is A is the tunneling rate in the l imit of infinitesimally small barrier height, and when the current bias is at close proximity to the crit ical current the exponent B is r = Aexp(-B) (1.15) 24 B = 7.2 AU (1.16) We can also determine an expression that demonstrates how dissipation affects the tunneling rate (also referred to as the Macroscopic Quantum Tun-neling ( M Q T ) rate) if we assume that the effective admittance of the dissipa-tive noise fluctuations in the current bias is extremely small in comparison to the effective admittance of the Josephson junction itself. The corresponding M Q T rate has the same form as for the non-dissipative case but with the exponent taking on the form Here u'Q is the plasma resonant frequency modified by the dissipative admittance. One other interesting effect occurs due to damping when is of order unity. When the dissipative admittance is of the form of an ordinary resistor in parallel wi th the Josephson junction at temperature T = 0, the effective conductance of the junction itself becomes superconducting even when the resistance of the shunt resistor is very low (more precisely when the resistance R < RQ, where RQ = is the resistance quantum). (1.17) 25 1.6 Two-level State Reduct ion and the De-sign of Qubits for Quantum Computat ion Superconducting quantum circuits have also been of interest in recent years in the attempt to design qubits as elements for a quantum computer which would be relatively easy to manipulate and control, though they do suffer from a few drawbacks [11, 12, 13]. Qubits based on photons, atoms, and nuclear spins have the advantage that they are not nearly as strongly cou-pled to their environment and hence do not suffer as much from noise and decoherence. Superconducting qubits have the tendency to be more strongly coupled to environmental degrees of freedom. E x t r a care must be taken in the solid state to decouple the qubit from all sources of noise and decoherence i n its environment. However, the advantage of superconducting qubits is that, being artificially manufactured systems, they have a lot of flexibility in the parameters that characterize them. These parameters can be manipulated and controlled to override the disadvantages due to decoherence. A n addi-tional advantage is that superconducting qubits have large electromagnetic cross-sections and may be coupled together in complex ways (for example we may embed one circuit inside the circuit loop of another, which is not feasible wi th nuclear spins or trapped ions), and more readily scaled to large integrated circuits (this is a consequence of the fact that the fabrication of su-perconducting quantum circuits is achieved wi th lithography techniques that are relatively easy to implement since they have matured through frequent 26 use in the fabrication of conventional semiconductor integrated circuits). In order to make a qubit out of a superconducting quantum circuit, one additional requirement we must add to the requirements of a macroscopic quantum system is nonlinearity. This is essential if we want to achieve re-duction to a two level system, because the energy level spacing between the ground state and excited state must be sufficiently smaller in comparison to the energy level spacing of higher states. Hence an LC circuit may function as a macroscopic quantum system but not as an effective qubit. On the other hand, under suitable conditions, circuits based on Josephson junctions may function as effective two-level systems because of the nonlinear form of its Josephson energy. As an example for illustration, we will consider the R F SQUID system which was originally mentioned earlier in the context of Macroscopic Quan-tum Coherence. To construct an R F SQUID we simply take a single Joseph-son junction and wrap its superconducting leads into a closed loop. As described earlier in section III, the gauge invariant phase 5 becomes linearly dependent on the total flux through the loop. We now have, associated with the closed loop configuration, a (linear) self-inductance L to which an exter-nal flux bias $ e x t is applied normal to the loop (usually with an auxiliary linear inductor coil which also must be superconducting in order to generate fluxes small enough to have good precision within the range of a flux quan-tum. If this auxiliary coil were not superconducting then the fluctuations in the external flux may override the flux itself). The energy corresponding 27 to the self-inductance is simply the usual form of an inductive energy, ie. ^~2L^ • S° w e obtain the Hamiltonian 2C ~J $ 0 ' 2 L The third term in the Hamiltonian is important to provide a potential well in which the flux will be confined and will be prevented from tunneling to states of higher flux. One of the difficulties with the R F SQUID, which typically are fabricated with niobium, is that the need for a finite self-inductance means that the qubit will have the tendency to couple inductively to stray magnetic fields that can be detrimental to the coherence of the qubit. To allow for use of a superconducting qubit with negligible self-inductance while retaining the qualitative two-level spin dynamics we may add two additional junctions to the circuit. This three Josephson junction (3JJ) qubit has one junction that is smaller in size than the other two (which each have a Josephson energy Ej and therefore has a smaller Josephson energy by a factor a. The circuit may be described by the the following potential: U — = 2 + a — cos 4>i — cos (p2 — a cos (27r/ + (pi — 02) Here (pi and 0 2 are the phase differences across the larger junctions and / = ^ is the total external flux through the qubit loop normalized to the 28 flux quantum, which is also referred to as the magnetic frustration. In describing the superconducting flux qubit we implicitly assumed that the Josephson energy Ej was significantly larger than the charging energy Ec = &£. In this case the tunneling barrier is relatively high and the system exists in a superposition state of localized flux states as already mentioned. In other words, it is the flux that is a good quantum number. However we could consider a Josephson junction in the opposite limit where the charging energy will be significantly larger than the Josephson energy. In this case it will be the charge that is a good quantum number and the system will exist in a superposition of two localized charge states where each correspond to a discrete number of Cooper pairs. Furthermore, the charge may be biased by a charge Ng (normalized by the Cooper pair charge to give a dimensionless value) simply by placing the junction in series with a gate capacitor. Hence its Hamiltonian will have the form H = EC(N - Ng)2 - EjcosO (1.18) where N is the measured variable corresponding to the number of Cooper pairs making up the charge on the capacitor, 9 is the gauge invariant phase limited to values between 0 and 2ir , and the charging energy Ec2{c+c ) now includes the gate capacitance Cg. This setup is an alternative way of designing a qubit out of Josephson junctions and is referred to as the Cooper 29 Pair Box (or alternatively the Superconducting Charge Qubit). Final ly there is in addition to the previous two main classes of super-conducting qubits-the flux qubit and the charge qubit-the Current-biased Junction. A s the name suggests, this is simply a Josephson junction wi th a bias current across it. Therefore the Hamiltonian of the system is simply the tilted washboard potential described in section III. The bias current is used to control the phase of the junction, and if set to be near the critical current of the junction then the potential has the cubic form (reverting to the notation of gauge invariant phase S) U(S) = * 0 ( / o - I)(6 - |) - *<f>(6 - \f (1.19) The two-level system corresponds to the lowest two energy levels of this potential. The transition frequency between the levels is very close to the plasma oscillation frequency o»p, which in this case depends on the bias cur-rent: ""'The-A s one might guess, the use of Josephson junctions for qubit systems gives another reason for taking interest in the effects of dissipation on quantum mechanical Josephson junction circuits. Now we make ask how dissipation 30 -)1 (1.20) places limitations on the use of Josephson junction circuits for the purposes of designing a quantum computer. This wi l l now be dealt wi th in more detail in the following section. 1.7 Relaxation and Decoherence The effects of dissipation on a qubit system can be characterized by two phenomena: relaxation and decoherence [11]. Relaxation corresponds to the tendency of a quantum state to fall toward either one of the two classical states of the qubit generally due to the transfer of energy. Decoherence refers to alterations of the phase of the qubit state, and hence is also referred to as dephasing. We may visualize these two phenomena more easily if we consider repre-senting the state of the qubit as a point on the surface of a sphere of unit radius called the Block Sphere. In the Bloch sphere picture, we represent the qubit state \ib) as M = c o s ^ | 0 ) + e*8m£|l) (1.21) and let 8 represent the angle from the Nor th pole of the Bloch sphere (ranging from 0 to | ) and the phase 0 represent the azimuthal angle. The arrow pointing from the origin of the sphere to the (8, 0) coordinate corre-31 sponding to the state is referred to as the Block vector. Then relaxation can be visualized as changes in the coordinate 0 so that the Bloch vector diffuses in the latitude direction, and decoherence can be visualized as changes in the coordinate <j) so that the Bloch vector diffuses in the longitude direction. W i t h this three-dimensional way of visualizing the qubit state, it is some-times more convenient to use the language of Cartesian coordinates to de-scribe the Bloch vector. Then the state with Bloch vector pointing along the z-axis is |0), the state with Bloch vector pointing along the x-axis is ^= (|0) + |1)), and the state with Bloch vector pointing along the ?/-axis as ^ ( | 0 > + i | l » We may take as an example the current-biased Josephson junction, which as we have already discussed reduces to a two-level system under appropriate conditions. The effects of relaxation and decoherence in the current-biased Josephson junction due to noise fluctuations in the bias current have been previously studied. [14] If we drive the junction wi th a D C current bias modulated by a low frequency signal / / / plus a microwave signal I(t) = Idc + Iif{t) + I^wdt) cos(o;oiO + I»ws{t) sm(uj0lt) (1-22) hf, lymci a n - d I^ws are varied in time slowly enough that we may consider them effectively to be constant (the relevant parameter determining whether the modulation is 'slow' or 'fast' is the inverse of the beat frequency of the 32 lowest two transitions.) It can be shown that, given the noise current In(t) that modulates the current bias, we can determine the fluctuations in the phase 0 of the qubit due to the bias current noise, (4>n{t)) (&)) = ( ^ J l ^ d f S ^ W ^ (1-23) where Sj(f) is the spectral density of current noise and WW) = (1-24) is a weight function which clearly gives significant weight to low frequency noise due to the -p dependence at high frequencies. If we assume that the spectrum of the current noise is a constant S® at low frequencies we obtain ( « ( t ) ) - ( ^ ) ' s J « / 2 (1.25) The corresponding decoherence rate 7^ is 33 70 = (1.26) So far we have considered the effects of noise on rotations about the z-axis. Rotations in the 9X and 0y directions are produced by the microwave frequency control currents. If we assume the spectral density of the current noise around the transition frequency UJ\Q is constant, then we wi l l find that the microwave contribution to the spectral density is a constant S V ( T ^ ) and we wi l l obtain the relaxation rate 7S, This corresponds to the stimulated absorption rate, and the correspond-ing rate for stimulated emission (|1) —• |0) transition) is similar. So far we have described relaxation and decoherence from current noise fluctuations and implic i t ly assumed that the circuit is otherwise isolated, or effectively in parallel wi th an infinite impedance. Bu t if we now allow dissipation from the Josephson junction we wi l l have to deal wi th a finite impedance. A s discussed previously in section V we can describe the fluctu-ations from a damped LC impedance source by its spectral density making use of the quantum-fluctuation dissipation theorem. The negative frequencies correspond to a Planck energy being removed from the dissipative element, l s = l / 2 ^ 1 0 C 5 / ( - ^ ) (1.27) 34 and hence is an emission process causing the transitions |0) —• |1) in the qubit. The opposite holds true for positive frequencies. The total noise may be represented as resulting from individual contributions of thermal fluctu-ations and dissipation from the dissipative element. Concentrating on the contribution due to dissipation, we wi l l find that the relaxation rate 71 for the |1) —• |0) transition wi l l be Note this agrees wi th the formula for the inverse of the time constant for a classical dissipative circuit. Noise sources intrinsic to Josephson junctions leading to 1/f noise are typically assumed to arise from two-level fluctuators which may be individ-ually represented as random telegraph noise [15]. The assumption that the noise source is Gaussian distributed holds true when there are a large num-ber of fluctuators. However it may also be important to consider the case where there are only a small number of fluctuators present, and in general the analysis is somewhat more complicated [16, 17, 18]. 71 = n(Y(uw)yc 35 1.8 Readout Now I wi l l t ry to explain how one can determine what state the qubit is in, which is referred to as the '"readout process'". A t any given time a superconducting flux qubit may be in a linear superposition of opposite-going states, and we wi l l want to determine the coefficients of this superposition. In general this means we wi l l want to be able to determine the probability at which the qubit is in a given state \L) or \R). Natural ly we wi l l want to determine these probabilities as accurately as possible, but in reality there are a number of difficulties to surmount due to the fact that noise and the measurement back-action have the tendency to disturb the quantum system, causing the qubit to relax and undergo decoherence in the finite time it takes to register the state of the qubit. A n ideal readout process wi l l have the readout variables completely de-coupled from the qubit variables during the time in which the readout is not occurring. This is referred to as the ' " O F F " ' phase. Then when it becomes necessary to read out the state of the qubit the readout variables must be coupled strongly to the qubit variables so as to make a fast projection mea-surement possible, while at the same time the back-action on the qubit must be minimized as much as possible. This is referred to as the ' " O N " ' phase. Readout schemes generally involve ones which lead to on-chip dissipation and others which are referred to as dispersive. The ones which involve on-chip dissipation have the disadvantage that it takes a significant amount of 36 time for the dissipated energy to be removed so that the qubit can be reset to its ini t ial state after the measurement for further operations. A s an example, on-chip dissipation arises when the critical current of a D C S Q U I D readout device coupled to a flux qubit is monitored to determine the flux qubit state [19, 20]. Dispersive readout schemes are designed to address the problem of on-chip dissipation by avoiding the use of states of the readout that lead to these on-chip dissipative effects. For example, in the D C S Q U I D readout of a flux qubit, instead of using the critical current to identify the state of the qubit we may make use of the measurement of the Josephson inductance directly by the response of an A C signal passing through the S Q U I D and the corresponding phase shift [21, 22]. The phase shift is dependent on the Josephson inductance, which in turn is sensitive to the qubit flux due to its nonlinearity. Two kinds of dispersive readout schemes are known to be possible, one of which makes use of low-amplitude resonant A C driving and the other which makes use of high-ampltiude driving. In the former case, we make treat the D C S Q U I D approximatively as a harmonic oscillator. In general, the low-amplitude driving scheme results in a weak signal response and requires the use of a low-noise cryogenic amplifier. Furthermore, recent studies have suggested that the qubit relaxation times are comparable to the time required to apply an impedance measurement, which limits the detection efficiency of the resonator. Experimental studies suggest that use of the D C S Q U I D readout in the nonlinear regime with relatively high-amplitude driving can 37 help circumvent this problem [23]. 1.9 Entanglement We wi l l explain firstly the concept of the entanglement of an array of qubits [24], and then discuss some proposed methods of experimentally achieving entanglement in a controlled manner. First of all when we talk about the entanglement of some system we must keep in mind that there are at least two quantum mechanical subsystems involved, and possibly more (eg. we could have two or more flux qubits). For the time being we wi l l imagine that there are only two subsystems in-volved, just to make things simpler. Somewhat vaguely speaking, if there is entanglement in a system that exists in a pure state, it means that if we measure the state of one subsystem then we wi l l have acquired information about the other system. For example, if we have two flux qubits in one of the so-called " B e l l states" ^ (| |) | j ) — | | ) |f)), then if we measure the flux of one of the qubits, then we can predict with certainty that the other qubit wi l l have the oppositely oriented flux. The two qubits cannot be considered as individual independent subsystems-and what is even more remarkable is that this interdependency between the qubits persists to hold even if the entangled qubits are taken arbitrarily far apart from each other so that they are no longer interacting (in principle, assuming nothing else interferes with the state of the qubits). 38 In a simple example of a Fermi electron 'gas' where we have two free elec-trons confined in a well-isolated box, entanglement occurs naturally even in the absence of interactions [25] (that is, of the ordinary kind such as Coulomb repulsion and magnetic dipole interaction). In this case the ground state of the system involves the Bel l state mentioned above. The entanglement arises as a consequence of the fundamental fermionic indistinguishability of the electrons. Bu t in an array of distinguishable systems, such as an array of su-perconducting flux qubits, the only way entanglement can be produced from a disentangled state is if interactions exist between the subsystems. This is clear from the fact that in the absence of interactions the unitary evolution of the two subsystems wi l l decouple, producing another disentangled state. In general for two qubits any state-including entangled states-can be generated by a combination of independent unitary operations on each in-dividual qubit plus a CNOT gate, which is of the form of a two-qubit gate operation that does nothing when the first qubit is in the "down" state and flips the second qubit when the first qubit is in the "up" state. In the lan-guage of computational basis states |0) and where |0) represents 'down' and |1) represents 'up', this gate operation has the form UCNOT = |0)<0|<8>1 + | 1 } ( 1 | ® < 7 * (1.29) For example we can generate the Bel l state given above starting from the 39 disentangled state 7^5 (IT) — |j))<8>|T) a n d then followed by an implementation of a C N O T gate on the two qubits. The C N O T gate can also be written in a form involving solely Paul i matrices and the identity operator UCNOT = ^ (1 + oz <8> 1 + 1 <8> o-x - az <8> az) (1.30) The C N O T gate requires interactions between the subsystems. One use-ful two-qubit Hamil tonian for generating entanglement involves independent unitary gate operations (with suitably chosen bias and tunneling parameters) plus a flux-coupling interaction term Hint = Jno-z <8> oz (1.31) Here J i 2 represents the coupling energy between qubits 1 and 2. W i t h superconducting flux qubits, the interaction part of this Hamil tonian can be implemented from the inductive coupling of the fluxes of the qubits. The mutual inductance coupling readily gives the term proportional to az ® oz. To see how this can lead to a C N O T gate implementation, consider the following argument. Due to the mutual inductive coupling, the flux state of one of the qubits changes the external flux of the second qubit. A change in the external flux means a change in the energy splitt ing between the ground 40 a n d f i r s t e x c i t e d s t a t e s o f t h e q u b i t , a n d h e n c e t h e a b i l i t y t o f l i p t h e s t a t e o f t h e s e c o n d q u b i t ( w i t h a p p l i c a t i o n o f a r e s o n a n t p u l s e ) d e p e n d s o n t h e f l u x s t a t e o f t h e o t h e r q u b i t . S t u d i e s i n v o l v n g t h e i n d u c t i v e c o u p l i n g o f t w o f l u x q u b i t s h a v e b e e n m a d e p r e v i o u s l y [26]. O n e o f t h e p r o b l e m s o f i m p l e m e n t i n g t h e a b o v e - m e n t i o n e d s c h e m e f o r c o u p l i n g 3 J J f l u x q u b i t s i s t h a t t h e i r l o o p s e l f - i n d u c t a n c e a r e u s u a l l y q u i t e s m a l l , a n d s i n c e t h e s e d e t e r m i n e t h e c o u p l i n g s t r e n g t h s o f t h e i n t e r a c t i o n b e t w e e n t h e q u b i t s t h e s e c a n l e a d t o v e r y s l o w C N O T g a t e i m p l e m e n t a t i o n s . W e u s u a l l y r e q u i r e C N O T g a t e i m p l e m e n t a t i o n s t o b e f a s t e n o u g h s o t h a t i n f o r m a t i o n c a n b e p r o c e s s e d b e f o r e d e c o h e r e n c e r e d u c e s e v e r y t h i n g t o a c l a s s i c a l j u m b l e . H e n c e a n a l t e r n a t i v e s c h e m e p r o p o s e d f o r i m p l e m e n t i n g t h e c o u p l i n g b e t w e e n t h e q u b i t s i n v o l v e s t h e u s e o f a s i n g l e J o s e p h s o n j u n c t i o n a c t i n g a s a m e d i a t o r . I t h a s b e e n s h o w n t h a t t h e c o u p l i n g s t r e n g t h b e t w e e n t h e t w o 3 J J q u b i t s c a n b e t u n a b l e b y m a n i p u l a t i n g t h e s i z e o f t h e j u n c t i o n , w h i c h i n t u r n w o u l d a l t e r i t s c r i t i c a l c u r r e n t [27]. S u c h a s c h e m e h a s b e e n r e c e n t l y u s e d t o d e t e c t e n t a n g l e m e n t i n a n a r r a y o f f o u r f l u x q u b i t d e v i c e s [28]. 1.10 Quantification and Measurement of E n -tanglement R e c e n t l y t h e r e b e e n a g o o d d e a l o f i n t e r e s t i n w h a t a r e r e f e r r e d t o a s ' m e a -s u r e s o f e n t a n g l e m e n t ' o r t h e ' q u a n t i f i c a t i o n o f e n t a n g l e m e n t ' . T o i l l u s t r a t e 41 this concept of an entanglement measure[29, 30] let us assume that we have a system made up of a given number N of quantum mechanical subsystems which may be interacting wi th one another. Suppose that the state of the whole system is a pure state \ip). What we mean by a measure of entan-glement is essentially some kind of function which maps the given state \ip) of the system we are interested in to a scalar quantity R(\ip)) which gives a value of 0 for the case where the system is completely disentangled and a finite value when there is entanglement present in the system. Furthermore, this finite value should somehow characterize how 'strongly' the subsystems are entangled wi th each other, though in general the notion of strong and weak entanglement are vague and ill-defined. Intuitively the entanglement measure is meant to characterize how much more powerful the system is in processing information (due to the fact that it can represent and process quantum information) above its classical counterpart. However, again this notion of quantum information processing power in terms of entanglement is sti l l vague and ill-defined, and as one might guess is part of the problem of defining a good measure of entanglement. It should be intuitively clear, from a general consideration of how quan-tum algorithms on a quantum information processor work [24] that the infor-mation processing capability of a quantum computer (which essentially is an array of qubits coupled to external controls) depends on its capability of ex-isting in pure, entangled states. The 'more' entanglement present in an array of qubits (for example, the more qubits involved in a single non-decomposable 42 state), the greater is its capability of carrying out more demanding quantum algorithms. Of course, a pure entangled state only exists in the idealized limit. As we described in section VII, qubits are subject to dissipation which causes the system to decohere into a mixture. In general, it would be useful to be able, in as few laboratory measurements as possible, to gain a sense of the information processing capability of an array of qubits by a means of quantifying, in a global sense, the amount of entanglement present in the system. An additional requirement for a good measure of entanglement is that if any one of the individual subsystems undergoes a unitary operation, the value of the entanglement measure for the whole system should not change. This is referred to as local invariance. There are a few other requirements for a good measure of entanglement, which I will not discuss in further detail here. However, for sake of completeness some examples of entanglement measures will be discussed. One example of an entanglement measure that is a good measure for pure states is the reduced Von Neumann entropy. For a given system density matrix p the Von Neumann entropy is S = tr(phxp). The reduced Von Neumann entropy is obtained by determining the reduced density matrix of one of the subsystems and determining the entropy of this reduced density matrix. Another example of an entanglement measure is the entanglement of formation, which is related to the reduced Von Neumann entropy but is a good measure for mixed states as well as pure states. In general, however, 43 for multipartite systems (ie. wi th more than 2 subsystems) which may be mixed there is no unique way of satisfying the criteria of a good entanglement measure. One other proposed measure of entanglement for multipartite systems is referred to as the MW measure and is expressed as where Pk is the reduced density matrix over each qubit k. This measure is capable of distinguishing a state which is completely disentangled (R(\ip)) = 0) from a state which has some entanglement present in it R(\ip)) > 0. However it is incapable of distinguishing between a state in which all the qubits are entangled from states in which subgroups of qubits of the whole system are independently entangled, which we may refer to as entanglement clusters. A n example of a quantity directly observable by experiment from which signatures of entanglement can be obtained is the magnetic susceptibility [31], which is described below. The magnetic susceptibility of a material measures the tendency of an array of dipoles or dipoles within a material to align with an applied field. More precisely, if the magnetization M (that is, the total dipole moment per unit volume of a sample) is found to be directly proportional to the applied external field H , then the dimensionless proportionality factor is referred to 44 as the linear magnetic susceptibility. In nature we observe certain materials which have a net magnetization only when a magnetic field is applied. Such materials are adequately de-scribed by the linear susceptibility. O n a microscopic level the dipoles within the material do not interact wi th one another but simply line up wi th the field independently of each other due to their individual magnetic moments. But for systems in which the dipoles interact with each other strongly enough, we wi l l no longer find a linear relationship between its net magnetization in response to an applied field. In general the magnetization wi l l be a nonlinear function of the applied field, and may even depend on the direction of the applied field (for the present we wi l l ignore dependence on the direction). In this case we can often expand the magnetization in terms of higher or-ders of the applied field relative to some suitably chosen fixed value. Each higher order of the expansion corresponds to a higher order of the magnetic susceptibility (quadratic order, cubic order, etc.). 1.11 Objectives In Chapter 2 the dynamics of a D C S Q U I D circuit in response to the sto-chastic oscillations of a two-level magnetic dipole, a classical model of the D C S Q U I D readout of a flux qubit, wi l l be investigated under the condition that no external flux bias is added to the D C S Q U I D ring. This response wi l l be characterized by determining the screening current correlations of the 45 D C S Q U I D . These results wi l l be used in formulating a criterion for reliably distinguishing two-level transitions of the dipole. In Chapter 3 we wi l l com-pare a signature of multiqubit entanglement, the Nonlinear Susceptibility (irreducible component), to a recently introduced measure of global entan-glement. This wi l l be done in the context of an array of four flux qubits, and the differences between the two methods of entanglement quantification wi l l be determined within a pre-defined range of values of the variable input parameters. 46 Chapter 2 DC SQUID measurement of a Two-Level Fluctuator 2.1 Preliminaries Consider the system shown in Fig. 2.1 below. A D C SQUID [8] is connected to the external circuit with the impedance Z. It is inductively coupled to the fluctuating magnetic dipole M(t). This dipole is modeled as a 'Two-level Fluctuator', which for our analysis is described as undergoing a two-level stochastic random telegraph process [15] (this Two-Level Fluctuator can also serve as a semi-classical model of a flux qubit) . We assume that this Fluctuator can take only two positions, M(t) = ±Mz, which translate into two values of the magnetic flux sent by it through the SQUID loop, $(£) = ± $ x - The resulting screening current in the SQUID, is{t), is detected and used to determine the sign of $(£) , and thus the direction of the dipole M. We will use the resistively and capacitively shunted model of the Joseph-47 Bias current I Screening Figure 2.1: Diagram of DC SQUID circuit under the RCSJ model coupled to a two-level dipole. A magnetic flux due to the dipole, stochastically alternating between two levels of opposite sign, penetrates normal to the loop. The screening current is monitored by an amplifier inductively coupled to the loop. From the screening current one may deduce the flux transitions and distinguish them from the background noise in the bias current arising from the external circuit impedance. 48 son junction (RCSJ ) to describe the S Q U I D dynamics. Neglecting the self-inductance of the S Q U I D loop, we obtain: 0 a - 0 f e = 2eV/h I = ^ ( T T + "Br) + V(C* + °b) + Jo,a sin cf>a + J0ib sin 4>b (2.1) (2.2) •a Kb is = {Io,a sin 0 a - Io,b sin 0 o ) / 2 (2-3) 0 a — 4>b + 2-KV = 27rn. (2.4) Here 0 a i j , are the (gauge invariant) phase differences across the Josephson junctions, io,ai6, Ra,b, Ca,b are their respectively crit ical currents, normal re-sistances, and capacitances. The voltage on the S Q U I D , the bias current and the screening current are respectively V, I, and is- The external flux through the S Q U I D loop, created by the magnetic dipole, is $(£) = z/(£)<&0- Since the external flux bias is 0, and the output of the flux qubit is typically less than a flux quantum, it follows that the flux output of the D C S Q U I D wi l l also be less than a flux quantum, ie. n wi l l simply be zero. 2.2 DC SQUID with symmetric junctions We wi l l start from the case of the symmetric S Q U I D (with identical junc-tions), wi th I0, R, and C being the critical current, resistance, and capaci-49 tance of each of the junctions respectively. Then introduce the combinations V=(<f>b + <P*)/2\ £ = ( & - 0 a ) A (2.5) and the dimensionless parameters Is = islh\ a = //(2/ 0); (2.6) T = [2eIQ/hC]l,2t =ujt\ (2.7) U = [2eC/hI0]1/2 V = (ujTj)V/Vj; (2.8) 0=[h/IoR2Ce]1/2 = 1/UJTJ, (2.9) where Tj = RC, and Vj = IQR characterize the quasiparticle properties of the junctions. Denoting the derivative with respect to r by prime, we then reduce the equations (2.1-2.4) to rj" = -fir]' - sin 77 cos £ + a; (2.10) u = v'; (2-11) Is = — cos 77 sin £; (2.12) e = 7ri/. . (2.i3) We assumed that the flux induced by the fluctuator takes only two sym-metric values, V(T) = ± | $ x | / $ 0 - Note this implies that cos£ is a constant due to the fact that the cosine is an even function (in more physical terms, 50 this can be rephrased by saying that the nonlinear Josephson inductance, Lj{v) = 27rj0cos27ri/ ' l s a constant). In the symmetric case above this de-couples the dynamics of 77 from the fluctuations of £ so that the S Q U I D effectively (up to a current noise factor) synchronizes wi th the fluctuator. This provides a simple starting point for further investigation. We can first find the solution for 77(7-), in the presence of the bias current noise Sa(r). Then the behaviour of £ and the screening current can be determined. Cor-rections due to small deviations from the above symmetries can be taken care of perturbatively. In terms of the D C S Q U I D voltage, which depends on the time derivative of the variable 77, we see that the influence of the dipole fluctuations on the D C S Q U I D have been supressed, and consequently the Hamil tonian of the D C S Q U I D readout device is dependent only on the variable 77 and its time derivative, ie. the voltage, and is independent of the dipole fluctuations. The potential component of the Hamiltonian clearly has the form of a tilted washboard potential. 2.3 Screening Current Correlations We can determine the response of the screening current to the fluctuator from (2.12), once we know 77(r). The fluctuations of the latter, without loss of generality, can be ascribed to the effect of the external circuit with impedance Z, and are statistically independent on the fluctuations of v(r) 51 and, in our symmetric case, of £(T) as well (we do not make any mention here what are the possible consequences of taking into account the back action of the S Q U I D on the fluctuator). The fluctuations in 77 w i l l only depend on £ in the presence of asymmetries. The ability to determine the sign of v from Is is restricted by the noisy COST? term in (2.12). More specifically, to emphasize the statistical independence of 77 and £, the autocorrelation function is KIs{T) = (Is(T)Is(O)) = (cos(r ? (T))cos(7 ? (0)))(sin(£(T))sin(£(0))) First we determine the correlator Ksin^(T). We assumed the random telegraph process. Therefore Here A is the mean rate of transitions of the random telegraph two-level system. In order to find the correlator KC0Sr](T), we expand 77 — fj + Sn, where sin rj = a j cosnu. Hence we obtain Kcos v(T)Ksin^(T) sin£| 2 e- A l T l (2-14) Kcosr](T) = cos 2 77 + s in 2 rjK&r](T) (2.15) Here Ksv(T) is the correlator for the fluctuation Sn. 52 The correlator for 5n is obtained from the spectral density using the Wiener-Khinchine theorem KSV(T) = ^ J SSV(LO) cos{uT)dw (2.16) Here we have transformed to dimensionless variables UJ — > — and T —> upT. The expression for the spectral density is obtained as follows. In the standard way, we write in (2.10) a = a + 5a(r), where the random source 5a(r) is separated from the average bias 57, and search the solution to the stochastic equation linearized in 6TJ: 5n" + B5rj[ + cos 777577 = 6a(r) (2.17) We define 7 = COS 7W K = VCOS 777 This yields for the spectral density of <5?7(T) the expression SSa{oj) = ^SSi(u) (2.19) The fluctuations of the bias current, described by the spectral density 53 Ssi{u>), are in our classical scheme (assuming suffienctly high temperature T and bias current) due to equilibrium thermal noise and shot noise, and these contributions can be treated independently. Hence Consequently, the correlator for the fluctuation 8r) can then be written as a sum of these contributions K&TI(T) = Kthermai(T) + KSHOT(T) Using the methods of residue calculus, we can determine each of the contributions of this sum: K m - k Q ^ P [(i + P\T\)(i-p2r2) + 2p2r2} Kthermal{l)-e ^ P \ l - P2T2) + kQup 3 e ^ l / n 2 / 0 2 i t . 1 T L T 1 4 / ? 4 - 2p2r2 + l for K2 = p2 K ' (T\- -f>m k @ u P - 2 / ? 2 + 2/3q"> + % C Q S g T - 2singT] «tkermal{l ) ^ ^ + _ ^ + k&ujp e - l T l / T l 2/02JR1 r ^ 4 + T ! 2 (2K 2 - AP2) + 1 54 for K2 ^ p2 Here q = \/K2 — P2 Kshot = -yea lo e-^T\(l + p\T\) A U + 47T P4 for K2 = 0 Kshot = -pea [gcosir[ 1Q K pq a* -/?sing|T|] + 4 7 r -K for K2 ^ p2 2.4 Effects of Small Asymmetries If we now take asymmetries in resistance, capacitance, and critical current into account, we start with the original equation I = V(Ga + Gb) + V(Ca + Cb) + io,a sin </>a + h,b sin <f>h\ h = (/o, asin0 a - Jo,6sin0h)/2; and obtain the following modified equations (using the same transforma-tion of variables as in the symmetric case) Is = — cos 77 sin £ + e sin 77 cos £ —Prf — sin 77 cos £ — e cos 77 sin £ + a 55 (2.20) (2.21) We may use the same definitions for the parameters we defined for the symmetric case if we simply replace C by Ca+Ct>, I0 by Io>a+Ia<b and G by 2(Ga + Gb). Then we define e = j ° , a ~ ^ << 1 as our perturbation parameter. We write the time evolution of r? as n = cos fj-\-5rf\T)-\-er]1 (r) = cos r)+8r}. Making use of this expression and ignoring terms of order eSrj and e2 we obtain rf" = - / fy 0 ' - sin 77°7 + a (2.22) r]1" = —fir)1' — 7 C O S 7 7 7 7 1 — cos Tj sin £ (2.23) If we Fourier transform this and solve for the spectral component 77^  we obtain 77* = - 0 0 5 7 7 sing „ _ ' 2 1 2 ^ ,2 2.24 [ 7 C O S 7 7 — u;2J2 + (pu)2 (ignoring terms of order e) The autocorrelation function of the fluctuation Sv is then KSv(T) = KSvo(T) + e[5r ?°(r)r /i(r + T) + ^ ( ^ ( r + T)] (2.25) where the overline as usual denotes time average. The term of order e is Ksve (T) = Re[J 8rf(u)(rry{u)eiuiTdLA\ 56 v R e [ J [(u2 - K2)2 + (52u2)2 M (from Parseval's theorem) It may be useful to note that the first order effect of the asymmetry becomes negligible for relatively strong (cos fj 0) bias currents. However our perturbation analysis is not valid for strong bias currents due to large shot noise fluctuations. Nonetheless, the results are suggestive. 2.5 Cr i ter ia of the observability of the fluc-tuator state Let us first consider the case of a classical detector, which directly measures the screening current in the SQUID. We assume a detector of the form of a linear two-port system undergoing equilibrium thermal fluctuations, that is, the signal amplitude is proportional to the screening current: Idet(r) = Ms(r) + SIdet(r), (2.27) where A is the coupling factor (which can include the amplification coeffi-cient), and r5/d e t(r) is the noise of the detector. The following is a criterion which provides a rule for being able to accu-rately detect the dipole transitions amidst the background noise from mea-surements of the screening current. The screening current is a product of 57 a factor cos (rj + 5r}) multiplied by the factor sin £ which fluctuates in the same manner as the dipole. Hence in order for transitions to be accurately detected we have the condition AlsmaxCosrj > 5Idet =>• A5i™x > (cos r / ) - 1 . Where SIdet is the noise current of the detector and Ismax = ^ o| s in£ | . We set 77 = rj + Sr]rms = rj+ \JK&T)(T = 0), where Ksr,(T = 0) = Kthermal(T = 0) + Kshot(T = 0) ^ thermal \ J- 07 2 / 2 /2 v 1 \T*(K?-2p+20q)+l -I * T 1 4 f c 4 + r 1 2 ( 2 K 2 - 4 / 3 2 ) + l [ re2/3 1 J Furthermore we set the detector noise to be of the form of thermal noise 5Idet = y/*kBTdGdAfd Here Td is the effective temperature of the detector, Gd is the equiva-lent thermal conductance, and Afd is the detector noise spectral bandwidth. Hence we have the criterion Ahmax > / ^ c o g _ _ s i n ^ ^ ( T = 0 ^ 1 = T (2.28) SI, det 58 The right hand side of the inequality denoted by T , which is in dimen-sionless units contains parameters for which the numerical values are known and are given below. The left hand side is proportional to the ratio of the noise current to the amplified screening current, and hence T is an upper bound design constraint for this ratio. To get some numerical values for T for different values of the bias current, we make use of the following values [19, 12] R = 29 fl C= l.2pF v = 1 (T 3 Ri = 50 Q Ci = lpF 6 = o0mK J 0 = 11.7 nA a = 0.25 or 0.025 From these we determine TJ = 3.48 x 1 0 ~ u s Lj = 2.81 x lO~nH up = 1.72 x 1 0 n s _ 1 7 « 1 rj = 0.254 or 0.025 K = 0.967 or K » 1 /? = 0.167 k = 1.381 x 10~ 2 3 J/K r i = 5 x 1 0 _ 1 1 s q = 0.952 ISmax = 3.68 x 10~ 8 59 a Stir,™ Y 0.25 0.947 1.3699 0.025 0.0910 1.0026 Table 2.1: Minimum signal-to-noise ratio for different current bias strengths We make a note that in the first case where the bias current is of interme-diate strength the value of 5r)rms is too large (because in the classical regime the shot noise is large) for our perturbative analysis to be valid. 60 Chapter 3 Comparison of two methods of quantifying entanglement applied to a quadruple array of flux qubits 3.1 Preliminaries In the following we wi l l study numerically two different methods of quanti-fying entanglement in the context of an array of four qubits coupled to each other. The first method of entanglement quantification is a recently intro-duced measure of global entanglement related to the M W measure [30]. The second method of entanglement quantification is an observable signature of entanglement related to the magnetic susceptibility [31]. I wi l l then consider the following setup, exactly the same as used in a recent study on four-qubit entanglement [28], which we wi l l study theoret-ically using these two entanglement quantification methods. A n electron 61 micrograph of the four-qubit setup is provided below in F ig . 3.1(a). The setup is described as follows. Four aluminum loops deposited on an appropriate substrate (using convential lithography techniques) form the four-qubit array. Each one of the qubits has three Josephson junctions as shown in the circuit diagram (Fig. 3.1(b)). Out of the three junctions in each qubit, one of them has a smaller cross-sectional area as for a typical 3J J flux qubit, and hence has a smaller Josepson energy Ej by the usual factor a as described in Chapter I. The qubit array is surrounded by an inductive superconducting coil coupled to an external capacitance forming a parallel tank circuit wi th inductance LT and capacitance CV- The coupling between the qubits is done by shared junctions. The inductive coil applies an external flux to the whole qubit array proportional to the tank current IT. This tank current is comprised of two components: a D C component IbT and an A C component Idr- The latter component simply functions as a means of resonantly driving the tank circuit for readout purposes (by monitoring the resulting phase shift in the tank), and is much smaller than hr- Hence it wi l l be ignored in our subsequent discussion since in our case we wi l l not be discussing the actual measurement anyway but the entanglement of the qubit array itself. In addition to the 'global' external flux applied through Ibr there are auxialliary coils which apply an external flux to each of the qubits individually. Their respective bias currents which apply these external fluxes are Ibx - J M , where the number 1 corresponds to the qubit in the lower left corner and the remaining qubits are counted in a counter-clockwise manner. 62 Figure 3.1: Four qubit array In (a) we have an electron micrograph of the four qubit array showing how the qubits, auxiliary coils, and shared junctions are arranged. In (b) we have a circuit diagram of the setup. 63 The Hamiltonian of the four-qubit array is l<i<7<4 Here the indices i, j refer to each of the individual qubits in the array, and Ci, Aj refer to the bias and tunneling amplitude respectively. The constants Jij refer to the coupling energies between the qubits. Method I, which we will refer to as the 'Global Entanglement', is deter-mined as follows (for brevity I will denote the state \ip) simply as ib). ( ' \ n vs(i/>) \1< |S|< |S| / 2 ( " - l ) - l (3-2) Here S corresponds to a bipartition of the array of n qubits: that is, it corresponds to a set of m qubits out of the total number of qubits, while S corresponds to the remaining n — m qubits (the complement), so that the qubits are divided into two sets S and S. Here the absolute value delimiters are chosen so that |5| = ra. The variable r]s(ip) is chosen so that it ranges in value from 0 to 1 and is equal to zero for a particular bipartition S when the state of the array of qubits cannot be decomposed into pure states of S and S individually. The product is taken over all possible bipartitions of the array, and the overall exponent in the product is chosen to give a geometrically 64 averaged quantity. The prime is used to indicate that we avoid any possible repeated counting of bipartitions. Hence this overall entanglement measure is non-zero only when the state of the qubit array is fully inseparable, ie. it is globally entangled, and zero otherwise. Method II, which we will refer to as the 'Nonlinear Susceptibility', is precisely determined as follows. RW « E L M ( F _ p s ( F _FyF F \ (3-3) ' P i P2<PiP3<P2 V - ^ P i - C /nA- C /P2 JZln)\J-Jp3 J-'n) This signature of entanglement corresponds to the cubic order suscepti-bility (irreducible component). The numerical algorithms these entanglement quantification methods were borrowed from pre-written M A T L A B algorithms using standard M A T L A B packages and libraries [33]. These algorithms determined the eigenstates of the Hamiltonian and calculated the appropriate entanglement quantity for the given eigenstate according to the definitions of the methods given. 3.2 Numerica l Results In order to make a comparison between the two methods of entanglement quantification, the following procedure was chosen. The value of each entan-glement quantity was numerically calculated in the case where Ibi, ib3> and 65 IM were chosen to have fixed values while IB2 and Ibr are flexible in their range of values. The current bias lb2 ranges in value from -100 pA to 100 p,A while IBT ranges in value from -20 \iA to -18 /J,A. In the case of either method a value of IB2 within the designated range is chosen, and subsequently the value of Ibr which corresponds to the maximum value of the entanglement quantity over the given range is determined. This maximum value is denoted as i £ T . The relationship between Jb*T and IB2 is plotted for both methods in the cases where the qubit array exists in a particular eigenstate \4>m), where m ranges in value from 1 to 16 with m = 1 denoting the ground state of the configuration. The results are depicted in Fig. 3.2 amd Fig. 3.3 below. The plots illustrate the degree of similarity between the two methods for different eigenstates of the qubit array. The differences between the results of the two methods is quantified by p,, where A ^ E K - P i ' l (3-4) J V i=i Here i is used to denote the particular value of IB2 sampled in the given range, p\ and pf1 correspond to the value of IB\ evaluated at the point i for methods I and II respectively, and N denotes the total number of values of Ib2 sampled (in this case TV = 50). In other words, this is the average of the absolute value of the difference in the peak location for the two different 66 methods. The value of n is determined for the first ten eigenstates, and the results are plotted below in Fig. 3.4. 67 -17.5 m = 3 -18 -18.5 < I--Q •19 -19.5 -20 -20.5 •17.5 < * i— -18r -18.5 19 -19.5 -20 -20.5 -100 M Nonlinear Susceptibility Global Entanglement •100 -50 0 50 Ib2 (u A) m = 8 ,'V V \ Nonlinear Susceptibility Global Entanglement -50 0 Ib2 ( L I A) 50 100 100 Figure 3.3: Nonlinear Susceptibility and Global Entanglement Quantification Methods compared graphically for higher states. The results depict the location of the peaks 7b*T for m = 3 and 8. 69 Figure 3.4: Average difference between two entanglement quantities for dif-ferent eigenstates The value of / J is determined for Ib2 ranging in value from -100 fiA to 100 u.A and for different eigenstates corresponding to m ranging in value from 1 to 10. 70 Chapter 4 Conclusions In Chapter 2, the response of a D C SQUID to the transitions of a two-level magnetic dipole fluctuator was investigated. We showed how, in the case of a symmetric D C SQUID, these transitions are mirrored by transitions in the screening current. The noise in the current bias sets limitations on the capability to detect transitions in the screening current within the setup of our classical model. The inequality in (2.32) gives the relationship between the detector noise current and the amplified screening current within these limitations. Given suitable values of the noise temperature and bandwidth of the detector, this inequality tells us what is the minimum value of am-plification A needed to reliably detect transitions in the screening current. For example, with a detector noise temperature — 4.2 K and spectral bandwidth [21] A / d = 1 GHz, and with a — 0.025, we require a minimum amplification of A = 13.1 to detect two-level transitions. Generalizations to 71 the quantum mechanical case is foreseeable in future work. In Chapter 3, two methods of quantifying entanglement were compared. The differences between the two were illustrated graphically by depicting the average difference in the peak location for different eigenstates. The typical value of fi is on the order of 0.15 p,A, within an examined range of 3 /JA, with the magnitude of IBT on a scale of 19 fiA. The results indicate these two methods of quantifying entanglement are compatible even for the higher excited states such as the case m — 8 for which the plot of I£T vs 1^2 has been included in Fig. 3.3. 72 Bibliography [1] Wheeler, J. H. & Zurek, W. H., ed., Quantum Theory and Mea-surement (Princeton University Press, Princeton, 1999) [2] Caldeira, A. O. & Leggett, A. J. , Quantum Tunneling in a Dis-sipative System, Ann. Phys. 149, 374, 1983 [3] Leggett, A. J. et al, Dynamics of the Dissipative Two-State Sys-tem, Rev. Mod. Phys 59, 1, 1987 [4] Leggett, A. J. , Testing the Limits of Quantum Mechanics: Mo-tivation, State of Play, Prospects, J. Phys.: Condens. Matter 14, 415, 2002 [5] Friedman, J. , et al Quantum superposition of distinct macroscopic states, Nature 406, 43, 2000. [6] Takagi, S. , Macroscopic Quantum Tunneling (Cambridge Univer-sity Press, Cambridge, 2002) [7] Tinkham, M . , Introduction to Superconductivity (McGraw-Hill, New York, 1975) [8] Barone, A. & Paterno, G., Physics and Applications of the Joseph-son Effect (John Wiley & Sons, Inc., 1982) [9] Zagoskin, A. M . , d-Wave Superconductors and Quantum Com-puters, Physica C 368, 305, 2002 [10] Devoret, M . H., Quantum Fluctuations in Electrical Circuits, Session LXIII on Quantum Fluctuations, Les Houches, 1995 73 [11] Devoret, M . H., Walraff, A., & Martinis, J. M . , Superconducting Qubits: A Short Review, cond-mat/0411174, 2004 [12] Orlando, T. P. et al, Superconducting Persistent-Current Qubit, Phys. Rev. B 60, 398, 1999 [13] Mooij, J. E . et al, Josephson persistent-current qubit, Science 285, 1036, 1999. [14] Martinis, J. M . , et al, Decoherence of a superconducting qubit due to bias noise, Physical Review B, 67, 2003 [15] Buckingham, M . J. , Noise in Electronic Devices and Systems (John Wiley & Sons, Inc., 1982) [16] Gutman, H., et al, Compensation of decoherence from telegraph noise by means of bang-bang control, cond-mat/0308107, 2003 [17] Falci, G., et al, Dynamical means of suppression of telegraph and 1/f noise due to quantum bistable fluctuators, cond-mat/0312442, 2004 [18] Ithier, G., et al, Decoherence in a superconducting quantum bit circuit, cond-mat/0508588, 2005 [19] Chiorescu, I. et al, Coherent dynamics of a flux qubit coupled to a harmonic oscillator, Nature 431, 159, 2004. [20] Tanaka, H. et al, Single-Shot Readout of Macroscopic Quan-tum Superposition State in a Superconducting Flux Qubit, cond-mat/0407299 [21] Lupascu, A., Verwij, C. J. M . , Schouten, R. N., Harmans, C. J. P. M . &; Mooij, J. E , Nondestructive readout for a superconducting flux qubit, Phys. Rev. Lett. 93, 2004. [22] Lupa§cu, A., et al, Quantum, state detection of a superconducting flux qubit using a dc SQUID in the inductive mode, Phys. Rev. B, 71, 2005 74 [23] Lupascu, A., et al, High-contrast dispersive readout of a supercon-ducting flux qubit using a nonlinear resonator, Phys. Rev. Lett., 96, 2006 [24] Nielsen, M . A.. & Chuang, I. L. , Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000) [25] Griffiths, D. J. , Introduction to Quantum Mechanics (Prentice Hall, New Jersey, 1995) [26] Izmalkov, A., et al, Evidence for entangled states of two coupled flux qubits, Phys. Rev. Lett. 93, 2004. [27] Grajcar, M . et al., Direct Josephson Coupling between supercon-ducting flux qubits, cond-mat/0501085, 2005 [28] Grajcar, M . et al, Four-Qubit Device with mixed couplings, Phys. Rev. Lett. 96, 2006 [29] Boumeester, ed., The Physics of Quantum Information: Quan-tum Cryptography, Quantum Information, Quantum Computa-tion, (Springer, 2000) [30] Love, P. J. , A characterization of global entanglement, quant-ph/0602143, 2006 [31] Zagoskin, A. M . , Nonlinear response and observable signatures of equilibrium entanglement, quant-ph/0510154, 2005 [32] Griffiths, D. J . , Introduction to Electrodynamics, 3rd ed., (Pren-tice Hall, New Jersey, 1999) [33] Algorithms for computing entanglement measures are due to the courtesy of Ilya Slobodov 75 

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