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Dynamics of a four-well system interacting with a dissipative environment Dubé, Martin 1993

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DYNAMICS OF A FOUR-WELL SYSTEM INTERACTING WITH ADISSIPATIVE ENVIRONMENTByMartin DubeB. Sc. (Physique) Universite Laval, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standard THE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© Martin Dube, 1993Department ofThe University of British olumbiaVancouver, Canadai?3DateIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)DE-6 (2/88)AbstractThe dynamics of sytems that can be mapped to a four-well problem in the presence ofa dissipative environment is examined. The probabilities of finding the systems in oneof the four state as a function of time are found in the limit of overdamped oscillations.Also in this limit, the time evolution of the density matrix of a four-well system is found.The diagonals elements are then rederived by considering an infinite chain of wells.iiTable of ContentsAbstract^ iiTable of Contents^ iiiList of Figures^ viList of Tables^ viiAcknowledgements^ viii1 Introduction and Basic Theory 11.1 Description of the Systems ^ 31.1.1^Two-Well Systems 31.1.2^Two Two-Well Systems ^ 61.1.3^Four-Well Systems 81.2 Introduction of the Environment^ 101.3 Low-Temperature Limit ^ 131.4 Path Integrals and the Density Matrix ^ 152 Two-Level System 212.1 Explicit Expression for P(t) ^ 212.1.1^Merging of the Two Paths 222.1.2^Form of the Influence Functional ^ 272.2 Noninteracting-Blip Approximation 29iii2.32.2.1^Ohmic Dissipation ^2.2.2^Superohmic Dissipation Results in the Different Regimes ^2932332.3.1^c = 0, Ohmic Case 352.3.2^c = 0, Superohmic Case ^ 362.3.3^c^0 383 Interacting Two-Level Systems 423.1 Construction of the Influence Functional ^ 443.1.1^Form of F12 for the Possible Configurations ^ 503.2 Expressions for the Different Probabilities 523.2.1^P++ ^ 553.2.2^p 633.2.3^p+_ and p_.+ ^ 643.3 Analysis of the Results and Limits of Validity ^ 653.3.1^Ohmic Dissipation ^ 673.3.2^Superohmic Dissipation 683.3.3^Subohmic Dissipation^ 694 Four-Well System 734.1 Density Matrix ^ 764.1.1^Diagonal Elements ^ 784.1.2^Off-Diagonal Elements 794.1.3^Laplace Inversion and Analysis ^ 814.2 Infinite Chain of Wells ^ 844.2.1^Mapping to the Four Wells ^ 844.2.2^Calculation of the Generating Function ^ 86iv5 Conclusion^ 94A Calculation of the tunneling matrix element^97B Laplace Inversion of P(\)^ 101B.1 Ohmic Dissipation, c = 0  101B.1.1 Line a = 1/2, Arbitrary T ^  102B.1.2 High Temperature and/or Strong Dissipation^ 103B.1.3 T = 0 ^  104B.2 Superohmic Dissipation, c = 0 ^  105Bibliography^ 109List of Figures1.1 Double-Well Potential ^ 41.2 Schematic Representation of Two Two-Level Systems ^ 71.3 System-Environment Interaction ^ 202.1 Path of the Coordinate q(t) 232.2 Single Path Over the Four States {q(t), q'(t)} ^ 242.3 Blips and Sojourns ^ 292.4 Form of P(t) for Various Values of a at T = 0 ^ 373.1 Type-I and Type-2 Paths^. ........ ^ 473.2 Possible Configurations for n 1 = n2 = 1 .^........ 493.3 Paths for the Overdamped System 543.4 Examples of the Four Class of Paths ^ 563.5 Illustration of the 3 Groups of Blips 573.6 Schematic Summation of the Paths for p++ ^ 603.7 Probabilities for Various Values of the Bias (Bias^—el2kT) ^ 704.1 Mapping to the Four Wells ^ 854.2 Path in the -OM , q'(t)} Space 874.3 Restrained Paths ^ 92viList of Tables3.1 Upper limits of integration in Equation 3.2 ^ 453.2 Charge of the Last Sojourns for the Different Probabilities ^ 484.1 Ending States for the Matrix Elements pig ^ 764.2 Values of the Charges (j for n = 1, m = 2 884.3 Values of the Charges rij for n = 1, m = 2 ^ 88viiAcknowledgementsI would like to thank my supervisor, Philip Stamp, for suggesting this project andguiding me through it, as well as Nicolai Prokof'ev and Rob Kiefl for their very valuablecomments. J'aimerais tout particulierement remercier mes parents et ma famille pour lesupport et l'affection qu'ils m'ont toujours donnes.viiiChapter 1Introduction and Basic TheoryIn many cases, a quantum mechanical object cannot be considered as isolated from itsexternal environment; they are coupled together, and their interaction can have severeconsequences. Generally, the effect of this interaction will be to destroy any possiblequantum behaviour, specially for macroscopic objects. This can be understood by con-sidering the environment as a "measuring apparatus" observing the system at a givenfrequency coc,. The effect of this observation is to project the system in the state in whichit was observed (collapse of the wave-function), thus reducing considerably any quan-tum transitions. Recently, however, the problem of a system tunneling in a potentialcomposed of two wells and interacting with its environment has been thoroughly exam-ined [1, 2] and it was found that, under some conditions, the system would still displayquantum coherence (i.e., it is still in a superposition of quantum states). The analysiswas made using a general formalism designed by Caldeira and Leggett [3] that had beenpreviously utilised to examine problems ranging from the quantum brownian motion ofa particle [4, 5, 6, 7, 8] to the tunneling of a macroscopic coordinate out of a metastablewell [2, 3, 9, 10].The main goal of this thesis is to extend the analysis of the two-well problem to aconfiguration made of four wells or to systems that can be mapped into one. Unfortu-nately, the analysis cannot be as complete. It was found that the only solvable limit isfor the case of high temperature and/or strong dissipation, where the systems are over-damped. If the systems were very weakly coupled to the environment, it would possible1Chapter 1. Introduction and Basic Theory^ 2to use perturbation theory, but this approach shall not be used due to the lack of time.Nevertheless, some insight into the system is gained and some results can be applied tosystems relaxing exponentially.The rest of this chapter will be devoted to the description of the different systemsthat will be looked at, as well as the theory necessary for their study. In Chapter 2, ashort review of the work of Leggett et. al. [2] on the two-well problem is presented. Atthe same time, concepts and techniques used in the next chapters as well as the veryimportant noninteracting-blip approximation are introduced. More importantly maybe,some limitations of the theory and of the approximations are discussed. The bulk ofthe thesis is to be found in Chapters 3 and 4. In Chapter 3, it is shown how a systemcomposed of two two-well subsystems can be mapped to a four-well system. However,the mapping is not perfect. The external environment couples them together, therebyintroducing a bias in the wells that complicates the situation. An explicit expression forthe probability that the system starting in a given position at time t = 0 ends up in oneof its possible states at a later time t is calculated. It shows that the systems will relaxexponentially to one of these states provided that the coupling to the environment is veryweak. If it is not, the system stays localised in its starting well. Furthermore, in the caseof subohmic dissipation (to be defined below), the system is localised independently ofthe strength of the coupling. In chapter 4 the "genuine" four-well problem is studied. Inthis case, the environment doesn't give any bias, and in the limit of high-temperatureand/or strong dissipation, an expression for the reduced density matrix of the systemcan be found, from which the expectation values of various operators can be found. Thediagonal matrix elements are next rederived by considering an infinite chain of wells withperiodic boundary conditions. A better understanding of the approximations used isthen gained.Chapter 1. Introduction and Basic Theory^ 31.1 Description of the SystemsFor definiteness, we shall assume that the quantum system interacting with its environ-ment is a particle of mass M, labeled by a coordinate q and moving inside a potentialwell V(q). Its Lagrangian is then1L = —2 Mgt — V(q).It is convenient to think of it as a particle, but is by no means necessary. The variableq can have a meaning other than spatial. For example, it can very well be the magneticflux through a SQUID [3], the magnetisation vector of a magnet [11] or the phase in-side a superconductor[12], just to give a few examples. In these cases, the form of theLagrangian can change, but the general spirit of the discussion is not affected.All the calculations will be done in the semi-classical approximation, also known asthe WKB approximation. For a particle, this amounts to consider only its classical pathand the small gaussian fluctuations around it when performing a path integration. Italso corresponds, for a potential well, to the case where the tunneling induced splitting ismuch smaller than the separation between the energy levels. This last point will becomeclearer below.1.1.1 Two-Well SystemsThe simplest multiwell configuration is the two-well system shown on Figure 1.1 Thewells can be chosen to be symmetric with respect to q = 0, with their minima beinglocated at +qo/2 and a potential barrier of height Vo between them. In energy, theyneed not to symmetrical; there can be a bias c between them. For definiteness, it willbe assumed that the well with minimum at +q0/2 has a minimum energy of -1-€/2. Thenof course, € can be either positive or negative. The tunneling of the system below the( 0 11 0( 0 — i, ay =i 0(1 002 =Crx =0 —1(1.3)Chapter 1. Introduction and Basic Theory^ 4hw eV(g)4 Cm.2^ 2Figure 1.1: Double-Well Potentialbarrier induces a splitting in the energy levels. In particular, we define the splitting ofthe ground state to be hAo and those two levels are separated from the next energy levelby an energy hwo .Now, at low temperatures (such that kT/h < w0 ) and for A o < wo , only the groundstate is occupied and need to be taken into account In effect, the system has beentruncated to a two-level problem. Then, by introducing the transition amplitude A0 /2and defining the state vectors IR) and IL), corresponding to the system being in the rightand left wells respectively, the following Hamiltonian can be written1H = --2 hAcrz —2 az ,^ (1.2)where ci are the Pauli matricesThis Hamiltonian can easily be diagonalised, yielding the two eigenvaluesh ^E± = ±!EI = ±-20 2 + (0) 2 (1.4)Chapter 1.^Introduction and Basic Theory 5and their corresponding eigenfunctions0± = A± [(E± + OIR) — nAolL)]^(1.5)with the constantA± = 1^i{1/2.^(1.6)(E± + E) 2 + (hAo) 2 ]For symmetric wells (E = 0), this reduces to E± = +hA0 /2, with the odd and eveneigenfunctions 0± = (IR) + 14)/0.Assuming that at time t = 0 the system is in the right well, the probabilities PR andPL that it will successively be found in the right or left well at time t arePR(t) = [1 + (1*) 2^2^+ t  cos 21E1 .0,1^(1.7)2PL (t) =^1— (FEIE ) — (FAA ) 2 cos 2lElt/h] .^(1.8)It then becomes interesting to define the function P(t) = PR (t) — PL(t) which in thesymmetric case has the very simple form^P(t) = cos(Dot) ,^ (1.9)showing explicitly the oscillatory behaviour of the system. The goal of chapter 2 is to findin as much of a quantitative way as possible how this function is influenced by dissipation.Several physical realisations of these systems can be found. The system that originallymotivated this work was a SQUID [3], where the flux through it can oscillate between twostable states, but the analysis can also be applied to a chiral molecule, a classic examplebeing the NH3 molecule [13], to an impurity diffusing incoherently in a solid [14, 15], toa two-level atom inside a radiation bath [16], spin glasses [17] (although the model is notvery adequate in this case [18]) and so forth. Another interesting example is the Kondoproblem, where a spin-1/2 impurity interacts with the Fermi sea. This last exampleChapter 1. Introduction and Basic Theory^ 6is particularly important since it can be shown [2, 19] that the Kondo Hamiltoniancan be mapped to the Hamiltonian of the two-level system interacting with an ohmicenvironment. In particular, the case a = 1/2 (to be defined below) corresponds to alimit where the Kondo problem is exactly solvable (Toulouse limit). Some insight into theKondo problem can then be gained by solving the two-level system and vice-versa. Someother examples of two-state systems are a magnetic domain wall pinned by two impurities[20] and the magnetisation vector of a giant magnetic molecule [20, 21, 22, 23, 24].However, the analysis of their interaction with the environment requires more powerfulmethods than those that will be presented here [22, 25, 26, 27].1.1.2 Two Two-Well SystemsThis class of system is essentially similar to the previous one. We consider two two-wellsystems, with coordinates q i and q2 , and an interaction term of the form Jqi • q2 /2 canalso be included if necessary. By the truncation procedure, only the ground states areused and we assign them the transition matrix elements 0 1 /2 and A 2 /2. The truncatedHamiltonian is then1^1H = --2hA i cr (1) — —hA 2 cr (2) + Ja (1) (r (2)x ^x^z^z •(1.10)This problem can be mapped to a four-well problem by assigning to the total Hilbert space641 )0642 ) the basis {+, +}, {+, —}, {—, —}, {—, +} where the first (second) member of thepair characterises the state of the system 1 (2), the +(—) corresponding to the right (left)well. A schematic representation of the two systems is shown in Figure 1.2. To restrictthe systems to their ground states, we must have A i << co,(31) andA 2 << ce. It is thenpossible to neglect the possibility of having the two systems tunneling simultaneously.The analogue of the two-level Hamiltonian is thenChapter 1. Introduction and Basic Theory^ 7142I Ai-r+-A :Figure 1.2: Schematic Representation of Two Two-Level SystemsJ —h.A 2^0^—hA iH= 1 —h0 2 J —h0 1 02^0 —h0 1 J —h0 2—hA i 0 —hA 2 Jand the splitting due to the tunneling is now fourfold,h ^2 = ±-2(A i + 0 2 ) 2 + (J/h) 2E3,4 = ±2 v(O1 — .6,2) 2 + (J1h) 2 ,with the resulting partition function(1.12)Z = 2 cosh OV(Ai + 02) 2 + (J/h) 2 + 2 cosh OV(A i — 6,2 ) 2 + (J/h) 2 (1.13)In the absence of the environment and with zero coupling between the two systems(J = 0), the probabilities p++ , p+_, p_+ and p__ of the system being initially in {+, +}to be in one of the four states {+, {+, —}, {—, +} or {—, —} at a time t are1—4[1 + cos Alt] [1 + cos A21P++ =Chapter 1. Introduction and Basic Theory^ 81= -4[1 - cos A l t] [1 - cos O tt]1p_+ = -4 sin(A i t) sin(A 2t).(1.14)In Chapter 3, the effect of the environment on these functions will be examined. Forthe rest of the work, it will be assumed that J = 0. The new physics brought in by theenvironment then appears more clearly.1.1.3 Four-Well SystemsThis last system has four potential wells whose minimums in the X-Y plane are locatedat {-+x0 /2, +y0/2}, {+x o/2, -yo /2}, {-x0 /2, -y0/2} and {-x 0 /2, +y0 /2} , all the wellshaving the same minimum energy. Again, the problem may be truncated, and we intro-duce the transition amplitudes A s and Ay as well as the vectors 11), 12),13) and 14), thatcorrespond to the system being in the states {+, +}, -}, {-, -}, {-, +} respectively.Neglecting the diagonal transitions, the Hamiltonian is4 0AyAy00AsA,0H= -h2 0 As 0 Ay(1.15)\ A, 0 A2 0with eigenvalues identical to those in Equation 1.12 but with J = 0. For symmetric wells(As = Ay ), the energies and corresponding eigenvectors are4 10El = -hA^= 2 11E2 = 0^=10(1.16)\01\1 1 1 1 1 1 0 —1 O 11 1 1 1 1 0 0 0 0; P2 =1 1 1 1 `' -1 0 1 01 1 1 1 0 0 0 0 j1= 4 (1.18)Chapter 1. Introduction and Basic Theory^ 91E3 = 0^= 1E4 = +hA " - 2— 11(1.17)— 11and the density matrices pi = 111)i )(zki I obtained from these four states are 4 0 0 0 00 1 0 —10 0 0 00 —1 0 1 /1 —1 1 —1P3121P4 =—1 1 —1 11 —1 1 —1—1 1 —1 1.^(1.19)Assuming an initial density matrix of the pure state form1 0 0 00 0 0 0P(0) =0 0 0 0\ 0 0 0 0its expression at time t is/ P11 P12 P13 P12P12 P22 P23 P22P =P13 P23 P33 P23\PI2 P22 P23 P22 /with1P11 = -4 [cos Dot + 1] 2 , p22 = —4 sin e Dot(1.20)(1.21)Chapter 1. Introduction and Basic Theory^ 101P33 = —4 [cos Dot — 1] 2 , p12 = —4 sin Aot[l + cos Dot]Ll1^ z . A^„P13 = —4 [1 — cos Dot] [1 + cos D ot] , P23 =-  S111D ot [1 — cos Dot] .(1.22)The modification of the density matrix brought about by the environment will be exam-ined in chapter 4.The physical realisation of these kind of systems might be less frequent than that ofa two-well system, but they are still quite important for themselves. Some immediateexamples come from generalisation of two-level systems, e.g., an impurity diffusing insidea crystal with four accessible states, a chiral molecule, a four-level atom in a photon bathand many other systems.1.2 Introduction of the EnvironmentIf a particle in a multiwell system were totally isolated, it would undergo coherent bandmotion. But, inside a crystal, it can interact with electrons, for a metal; phonons, for aninsulator; or even the magnons in the case of a domain wall inside a magnet. A distinctiontherefore has to be made between the particle and the degrees of freedom with whichit interacts. The former will be called the system and the latter will be referred to asthe environment. The "universe" will be the combination of the two. The environmentthen is defined as all the degrees of freedom that are weakly coupled to the system understudy, i.e., they must be weakly perturbed by the motion of the system. If any one ofthese is not weakly perturbed, then it must be included in the system. Of course, thatdoes not mean that the effect of the environment will be weak. Many weak couplings canproduce an enormous effect, but not always sufficient to destroy any quantum behaviour,as will be seen later.At low enough energy, a smooth potential can always be treated in an harmonicChapter 1. Introduction and Basic Theory^ 11approximation. It is then plausible to treat the environment as a bath of harmonicoscillators of mass ma and frequency coa , where a refers to the ath oscillator. In thisthesis, it will be considered that the coupling between the bath and the system is linear,that is, the ath oscillator and the system are coupled through a term of the form Ca xaq(t),Ca being a coupling constant and xa the deviation of the oscillator from its equilibriumposition. This will be valid as long as all the coordinates have a small value. Thus, thebath and interaction Lagrangian can be written as :7nce 2— rncxw 2 x2 q E caxa^(q)^(1.23)1enll ( X^L int (X q) = 2 ^"When coupling the system to the environment, we introduce a shift in the totalpotential energy. The role of the counterterm 0(q) is to cancel this unphysical effect ifnecessary. It is easy to see that the oscillators bring a new effective potentialVeil (q) -= V(q) — q2 E2^m,4 (1.24)and the counterterm is chosen accordingly. The total Lagrangian that will be consideredis then1^. 1^1L = —2 Mgt — V(q) E 2—77/c,( a 2 c'2 —^co" x 2 — q E ca xa — q2 E 2m (..,d 2 x 2a a^a^a ^a(1.25)It should be noted that this Lagrangian is quite general and can describe many physi-cal cases. Among these are the tunneling of the magnetic flux through a SQUID, diffusionof muons and impurities in solids, configurations of a chiral molecules and many others.However, it is insufficient for a least three cases [11] : motion of domain walls inside amagnet, where the coupling to the bath is non-linear; tunneling of the total magnetisa-tion of a giant magnetic molecule, where one has to deal with a topological phase factor;and a Fermi liquid with singular interactions, for which there is an essential interactionbetween the oscillators forming the bath even as the energy becomes very small. TheseChapter 1. Introduction and Basic Theory^ 12systems can nevertheless be treated by using a more general Lagrangian or a microscopicapproach.The parameters Ca , ma and wa can now be related to the physical situation in twoways. One can start directly from a microscopic analysis of the interaction of the systemand the environment and derive their explicit form or, for a macroscopic object, througha knowledge of its classical equation of motion [28]. Such an equation has to include adissipative term (e.g., of the form 74 for ohmic dissipation) and it is possible to relate itto the parameters of the environment by comparing it to the friction term obtained fromthe model Lagrangian 1.25. To this end, we introduce the spectral function7rJo (w) =2 E^ 8(w— coa )^ (1.26)a mawaand it turns out that only the behaviour of this function is needed to take the dissipativeeffects into account. In particular, for the ohmic case, we must have [3, Appendix C]A(w) = w (1.27)and in general, it will be assumed that the spectral function will be proportional to somepower of the frequency.Jo(w)^ws^(1.28)The case s = 1 corresponds to ohmic dissipation, and the cases .s > 1 and s < 1 will bereferred to as superohmic and subohmic dissipation respectively.The above procedure treats the environment as a bath of bosons, but in many cases,say diffusion in a metal, the system is coupled to a bath of fermions and it is notobvious that this formalism is correct. But, for fermions, the elementary excitationsof the bath are the particle-hole pairs and it can then be shown [29, 30] that at lowenergies and provided that there are no singular interaction between them, fermions givea form of dissipation identical the to boson's one with ohmic dissipation, but, for objectsChapter 1. Introduction and Basic Theory^ 13interacting directly with the environment, where the value of the dissipation parametera E-: q471127rh has to be less than unity. Other cases of ohmic dissipation are 2-phononprocesses and magnon induced dissipation in magnets, however with coefficients 7/ thatare now temperature dependent. Superohmic dissipation is produced mainly by 1-phononprocesses and corresponds to a value s = 3 for 3-dimensional phenomena [19]. Subohmicdissipation, or even non-integer values of s haven't been observed yet, but they couldsoon be useful as the field of nanophysics becomes more and more evolved.1.3 Low-Temperature LimitThe Lagrangian 1.25 describes a system in a very general potential V(x). However, atlow temperatures, only the ground state of the wells will be occupied and we can use thetight-binding approximation. In this limit, we assign a tunneling matrix element A/2 tothe system and describe its motion as a series of jumps between the wells. The amplitudethat a jump takes place between the times t and t + dt is then iLdt/2. The procedureto compute A is briefly described in appendix A, it is simply given byA = Ae-sclih . (1.29)where Sci, is the classical euclidean action of the system and A is a constant describinggaussian fluctuations around the minimum action path. For the case of tunneling out ofa metastable well, the decay rate F is given by a similar formula [31].For a system interacting with the bath, the environmental degrees of freedom mustbe integrated out beforehand [32, 2], yielding an effective actionSeff I,h 1, Y14 2 +vmdr+ 12 lc° dr 113h ds a(T — s)[q(T) — q(s)1 2 , (1.30)- 00^oChapter 1. Introduction and Basic Theory^ 14witha(T — 8) = 127 Jo Jo(w) e-wIT-s1 dw (1.31)and the classical action is the action for the path minimising Seff. Since a(T — s) isalways positive, this indicates that the environment will tend to suppress tunneling, buta careful analysis shows that it is not entirely suppressed.However, the complete coherence process is composed of two events, each takingplace on a quite different time scale tunneling itself is a fairly rapid phenomena, beingaccomplished in a time interval (bounce time, Tb ) given by (c.f., Equation A.7)+4o / 2 dq Tb =^^ (1.32)q0/2 /2V(q)/MBut, the coherence requires a much longer time interval since it is made of several tran-sitions, each of them separated from the others by a time interval in which the systemstays in a given well. Now, by viewing again the environment as a measuring apparatus,we see that there are two important frequency scales. Oscillators with a high frequency(co„ coo ) observe the system very often and will therefore reduce the tunneling matrixelement while low-frequency ones (co, D o ) act directly on the system's coherence andcan destroy it completely. It is also possible to see this by noting that the environment isactually a driving force of frequency co„ on the free oscillating system. If the frequency ofthe force is very far from Do , it barely renormalises it to A, but as the driving frequencyis lowered, resonance occurs and the dissipative effects become quite important. In anycases, this separation of the frequencies can be taken into account by introducing anarbitrary cutoff frequency co, such that A << < coo and by redefining the spectralfunction.Jo(w) = e — w/wcJo (w) + (1 — e —w/wc)Jo (w)J(w)^Ji (w),^ (1.33)Chapter 1. Introduction and Basic Theory^ 15where J(w) and J'(w) are respectively the low and high frequency part of the spectralfunction. Since only the high frequency oscillators are important in the expression for A,Jo(w) must be replaced by J'(w) in Equation 1.30 and all that remains to be done is to takethe low-frequency oscillators into account. These will be integrated out when calculatingthe density matrix of the system. But now, the different environmental parameters arerelated by22^ wmE  ^8(w — w oe ) = J(w) Jo (w)e -wiwc .^ (1.34)a aThis is the form of the spectral function that will be used throughout the remaining ofthis work.With the choice of J'(w) as the relevant spectral function for the tunneling matrixelement, it can be shown [33, 2] that the dependence of A on w e factorises to give anexpression of the form A f(a)g(wo ,^) where f depends only on the shape of thepotential V(q) and the dissipation strength through the parameter a and g in a functionof co, and of the type of dissipation. For ohmic dissipation, g = (wo lwo)a while forsuperohmic dissipation, g 1. A is mostly independent of we in this last case. Now,if this whole truncation procedure is to be correct, we should expect that the arbitrarycutoff frequency we will disappear in the final results. For the two-well and four-wellsystems, this in indeed the case (c.f., Chapter 2 and 4). However, two two-well systemsare coupled together by the bath through a term which is directly dependent on wc , thusin principle allowing a measurement of it, somewhat similarly to a free particle interactingwith its environment [34].1.4 Path Integrals and the Density MatrixIn this thesis, we shall be interested in a particle moving back and forth between two ormore wells. An essential question is then : given that at time t, the particle is in the wellhD[q] exp i S[qf,qi] q(i f)=(1 fq(t1)=qi(1.36)Chapter 1. Introduction and Basic Theory^ 16qi , what is the amplitude that it will be in the well qf at time t f ? That is, we wish toknow the value of the propagatorK[qf ,tf ;qi ,t i] = (qf e- iH(t f -t 2 )1hi qi )^(1.35)where the propagator is expressed as a path integral [36, 38, 39] in the second expression,S[x f , x i] being the action of the particle, equal totfS[q f , qi] = f dt L(q. , q) ,s(1.37)L(q, q) being its Lagrangian. In the path integral formalism, the propagator is a sum overall the possible continuous paths starting from q i at ti and ending at qf at tf . However, toeach path is associated a weighting factor exp(iS/h). Trajectories not having an extremalaction will oscillate with an extremely high frequency and will interfere destructively witheach other. Only paths with an extremal action will contribute effectively to the sum. Inparticular, asking for the path of minimum action gives back the famous Euler-Lagrangeequations, from which the classical path is found. The classical action is then the integralof the Lagrangian for this particular path.We now have the amplitude that we were looking for, but the interesting quantity isthe corresponding probability p(x f, i fj xi, ti ). From the laws of quantum mechanics, it issimply given by the square of the sum of the amplitudes.p(qf ,tf ;qi ,t i ) = 1.1ilqf ,tf ;qi ,td1 2q(t f)=q f^f)=4ff(ti)=9i D[q](ti)=9i D[q'] exp -h S[q(t)] - S[q'(t)] ,where the two paths, q(t) and q'(t) must have the same boundary conditions.(1.38)(1.39)Chapter 1. Introduction and Basic Theory^ 17Squaring the propagator gives a nice representation of the probability to find thesystem in a given state, but more formally, the probabilities of finding a system in oneof its eigenstates are given by the diagonal elements of its density matrix. Assuming aninitial density matrix p(0) at time t = 0, its time evolution is [32]p(t) = c iHt/ hp (o) e -FiHt/h,^ (1.40)In the case of a system interacting with its environment, p(t) represents the densitymatrix of the whole universe, an uninteresting quantity. What will be considered is thereduced density matrix, in which the environmental degrees of freedom have been tracedoutIO(q, q', t) = I dR (qR1 p(t) IV R) •^ (1.41)where R represents the collective coordinates of the bath and q is the system's generalisedcoordinate. By inserting the unity operator1 = I dx I dR Ix R)(x RI ,^ (1.42)it becomesfi (q, q', t) = J dx dy dR dR' dQ' (qRle -dliln IxR') (xklp(0)1Y0 (YVIe'lh lq'R) , (1.43)but the terms formed by the exponential of the Hamiltonian are simply the propagators,so that13(q,q',t) = I dx dy dR dR' dQ' K[q, R,t; x, R' , 0] (xR'Ip(0)1yQ') Klq' , R,t; y, Q' , 0] .(1.44)Next, we assume that at t = 0, the universe density matrix can be decoupled into asystem's and an environment's part asP(0) = Psys(0) p„,,(0)^(1.45)Chapter 1. Introduction and Basic Theory^ 18and, noting that the total action is the sum of the system's and environment's individualactions plus an interaction term SintS[q, Q] = So[q] + Senv[Q] + Smt[q, Q] ,^(1.46)the reduced density matrix can be rewritten asfi(q, q', t) = f dx dy J(q, q', t; x, y, 0) psys (0)^(1.47)with the kernelJ(q, q' ,t; x, y, 0) = f D[x]^ei(S[x]—S[y])1hD[y]^F[x, y]The influence functional F[x, y] is formally [35, 36, 37]F[x, y] = dR dR' dQ' p(0) k[R, t; R', 0; x] klR,t; Q', 0; y]Jwith the "reduced" propagatorU(t)=Rk[R, t; R', 0; x] = ei(Senv[11-FS,nt[U,splhfu (0)=R, D[U](1.48)(1.49)(1.50)The environment has been formally integrated out, and all its effects are embodied in theinfluence functional. Clearly, we have F[x, y] = Fly, x], but also, F[x, x] = 1 for all timet. This is so because if the x and y paths are the same, the propagators differ only bythe path taken by the environment. But then, it is a general property of the propagatorsthatdR K[R, t; R', 0] K* [R, t; Q', 0] = 8(R' — Q') (1.51)and since Tr( au env( 0 )) = 1 by definition, the result follows. The same line of reasoning isused to show that if the coupling between the system and the environment is zero, thenF[x, y] = 1, as should be expected, since in this case, the system and the environment areChapter 1. Introduction and Basic Theory^ 19decoupled at all time: we are then merely taking the trace of the environmental matrixalone !Another important property of the influence functionals is that for a system interact-ing with several external environments all independent of each other, the total influencefunctional is the product of the individual ones:^F[x , y] H Fa[x, y]^ (1.52)with Fa the functional of the a th external system.For the case of a system coupled linearly to an harmonic oscillator by a term of theform f (q), the actions are quadratic and the path integrations can be performed exactly.The most general form of the influence functional is thenFa [x , y] = exp fot dr Jr ds [f (r) — f' (r)] [f (s)^s) — f' (s)^s)]^(1.53)where f (r) and f(7) are evaluated along the path x and y respectively. In particular, if^the environment stays in equilibrium, we get: -ya (r, s)^-ya (r — s).For the bath of harmonic oscillators and the special form f (r) = CAN , this functionis [36, p. 343]Ca ^[ ,^os w(T — s) 170,(7- — s) =^+ 2 c2777,,,wah ephw _ 1(1.54)with /5' = 1/kT, a form that will be extremely useful through the remaining of the text.All this can also be given a diagrammatic interpretation. To this end, consider thetwo paths q(r) and q'(7- ) running parallel in time. The environment brings an interactionbetween these path through the vertices 7(7 — s) and -y* (r — s) as represented in Figure1.3.The first two (a and b) represent the self-energy corrections of a given path while thelast two correspond to an interference between them. It is these last two vertices thatChapter 1. Introduction and Basic Theory^ 20 r (r-sS^T^q(t) q(t)e -7 V-1)q (t)•)q(t)Y (T-s )• q . (t)q(t)(r-s )^ q (t)b) d)Figure 1.3: System-Environment Interactionare responsible for the lost of the system's coherence. The expression for the densitymatrix then correspond to the sum of all the possible paths made up from these fourvertices combined in any distinct ways. Actually, this interpretation is the last stage of amore complex formalism due to Kjeldish [40, 41, 42]. There, one consider two paths, onerunning from t = 0 to t = +oo and the other from t +oo to t 0. The interaction isthen described through a set of four Green's functions corresponding to the four possiblekind of interactions (the four vertices of Figure 1.3). After a proper reordering of thetimes, one gets the previous form of the density matrix.All the tools required for the analysis of a system with its environment are now laiddown and ready to be used.Chapter 2Two-Level System2.1 Explicit Expression for P(t)As was seen in the introduction, a two-well system is conveniently characterised by thefunction P(t) = PR (t) — PO), equal to cos(D ot) for symmetric wells without dissipation.This chapter will be concerned with what happens to this function when the wells arecoupled to their environment. Both symmetric and asymmetric wells will be examined,and the dissipation will be treated using the techniques introduced in Chapter 1. It ismainly a review of some parts of the paper by Leggett et. al. [2] , but provides a verynice introduction to concepts that will be used throughout the remaining of the text.Taking the initial density matrix of the system to bepsys(q, q ' , 0) = (q — go /2) 6 (q' — go/2),^ (2.1)the probability that it will be found again in the right well at a later time t isp(qo/2, q0 /2, t) = f D q(t) f Dq' (t) A[q(t)]^(t)] F[q(t), q' (0]^(2.2)where A[q] is the transition amplitude for the system of going from one well to the otherif the environment was absent and F[q, q'] is the influence functional. However, since weare now dealing with a bath of oscillators, the total influence functional is the productof the functionals for the at h oscillator, as given by Equation 1.54. The structure of thisfunctional will stay the same, the only difference comes in the function 7(7- — .^) which21Chapter 2. Two-Level System^ 22now has a sum over a included in it :2 cos wa (7 — s)^-y(r — s)^-(7 — s) = E 2nICL [e-iwc' ( r- s )exp(/3hwa ) — 1This expression can be simplified by separating 7(7 — s) into its real and imaginary partand using the spectral function J(w) to get rid of the summation. We thus obtaint^F[q(t), q'(t)]^exp —^dT J T ds {—iL i (T — s)[q(r) — q'(r)][q(s) V(s)]L2 (r — s)[q(r) — q'(r)][q(s) — q'(s)] }^ (2.4)with the functions L 1 and L2 defined as :L i (r — s) f dw J(w) sin w(7 — s)L2 (r — s) J(w) cos w(r — s) coth(iihw/2) . (2.6)The influence functional is made of two parts : a negative real part, which tends to reducethe weight of a given path, and an imaginary part, that adds a phase to the actions ofthe paths.2.1.1 Merging of the Two PathsNow, by the truncation procedure, q(t) has only two possible values : +g o/2. A path isthen formed of a series of sudden jumps from one well to the other, each one followed bya time interval T during which the system stays in a given well, as shown on Figure 2.1.In order to ensure that the system comes back to its starting point at time t, there mustbe an even number of jumps in the path.The probability p(t), however, is a double path integration; that is, for every pathq(t), one has to sum over all the possible paths q'(t). Essentially, this can be reduced to(2.3)00(2.5)t^t^t1 2 3t^t4t6tt. 0Chapter 2. Two-Level System^ 23g ogo2Figure 2.1: Path of the Coordinate q(t)a summation over one single path formed by the merging of the two paths q(t) and q'(t).The states of this path are then the four possible states {+, {+, —}, {—, {—, —}of the combined system {q(t), q'(t)}, where ± stands for ±q0/2. For simplicity, let us callthese states A,B,C, and D respectively. The probability that we are looking for is thenmade by a sum over all the paths starting from and returning to A.To implement this idea, it is useful to define the new functions :(T)^q1:71 [q (7)^ql(T)] — (2.7)x(T) = qc7 1 [q(7 ) + q'(7 ) ]which, according to the different states, take the values=^—1 for state B(2.8)(2.9)=^0 for states A,D (2.10)=^1 for state C (2.11)X(7 )^=^—1 for state D (2.12)=^0 for states B,C (2.13)=^1 for state A . (2.14)Chapter 2. Two-Level System^ 24Path q(t)Path q'(t)= +1 = +1Combined Path x = + 1 z = - X = + = - z = +1ti t2^t3 t^t 5^t 6^t7 t8Figure 2.2: Single Path Over the Four States {q(t), q'(t)}If we neglect the possibility of a diagonal transition from A to D or B to C, (t) and X(t)are never non-zero simultaneously and the path resulting from the merging can also berepresented by a series of jumps. However, to each segment of this path is now associateda value of or x, the value of which changes from one path to the other. An example ofthe two paths q(t) and q'(t) as well as the resulting path {q(t), q'(t)} is shown on Figure2.2Next, the amplitude of having a transition from one state to the other between thetimes t and t + dt is A[q(t)]dt for the path q(t) and Alq/(t)idt for the path q'(t). Asimultaneous jump of the path would correspond to terms of the order (dt) 2 and thispossibility will be neglected from now on. In effect, it means that the transitions A^Dand B^C are forbidden. From the previous chapter, it is known that A[q] = iA/2.Then, the transitions A C and B D, where the value of q(t) is changing possess anamplitude -1-i0/2 while transitions A B and C D have an amplitude —i0/2.For asymmetric wells, an action is also associated to each time interval T spent insidea given well. For definiteness, let the zero of energy be chosen such that the statesChapter 2. Two-Level System^ 25+qo/2 have an energy +€/2. Then, since 4 0 in the well, the corresponding actions areexp(TicT/2h). Remembering that we are dealing with an exponential term of the formiSo [q] — iSo [q], in the four-state path, these actions take the form exp(—i€(t)T/h).With all these considerations in mind, the formal expression of the path integral cannow be constructed. The paths must start from the state A at t = 0 and return to it atime t. Such a path will clearly be made of 2n jumps, which brings along a total jumpamplitude of (-1)n (A/2) 2n. Let the jumps happen at times tk with to = 0 and I-2n+i t.Then, at each time t2k+1 the system has the choice of going to either B or C and at eachtime t2k it can go to A or D except at t = t2n where it must go back to A. FollowingLeggett, the path can be parametrised by the introduction of two charges, ( 3 and ilk suchthat :Ci 0 for t2j—i < t < t23 (3 = +1 for state B (2.15)= —1 for state C--,- +1 for state A0 for t2k < t < t2k-F1= —1 for state D .Then, the functions 4"(r) and x(r) can be rewritten as :nx (T) = E 7/3^— t23) — e(T — t23+1)](2.16)(2.17)(2.18)3=0= E <3 [e(7- — t23_ 1 ) — e(T — t 23 )].Finally, the path integral is made by a sum from n = 0 to n = oo to account forevery numbers of jumps possible, and at each time tj (to which is associated either (3 or773 ) a sum over the possible values +1 of the charges must be made. This summationaccounts for all the possible ways of doing the return path to A in 2n jumps. Of course,the requirement yo 1 has to be imposed to respect the boundary conditions. Also,Chapter 2. Two-Level System^ 26exactly as in the instanton case (c.f. Appendix 1), an integration has to be made overthe transitions centers. With all these provisos, the probability function P(t) = 2p(t) — 1is equal to :^2n^tP(t) =^(-1)n  2n+1^a{t2n} E frn eft,l, {6}, {70) ,Th=0^2A fo {Cpn,}(2.19)where the action coming from the interval spent into the wells has been incorporated intothe initial influence functional.€ ftFn Fn exp^0 di- (T)]^ (2.20)and where, by definition Po = 2 . The contracted notationft D{t2n}^ ft2n° • •dt2n^dt2n_i^0 dt 2^dt 1^(2.21)has been used to denote the integration over the transition times. This integrationaccounts for the fact that the transitions can take place at any times, but that the firsttransition, t i must be between t = 0 and t2 , that the transition at t 2 must be betweent= 0 and t3 and so on up to t.The counterterms do not contribute to the path integration due to the low tempera-ture approximation of the paths which represents them as sharp jumps. For the mergedpath, the counterterms arebut,— i E 2mc?„co2 Jot [q 2 (r) — (q 1 ) 2 (7)] dra aq2 (r) — (q')2(r) = [q(r) — q'(r)][q(r)^(r)]=^("T) X(T) ,(2.22)(2.23)and since 4"(r) and x(r) are never nonzero simultaneously, there are no contributionsarising from this term.exp —^1 Q2( t23 — t2j-1) — ^ L^Cj (k njk=0^ k=1j=k+1njk = Q2(t23 - t2k-1) + Q2(t2j-1 - t2k) - Q2(t23 - t2k) - Q2(t2j-1 - t2k-1)Chapter 2. Two-Level System^ 272.1.2 Form of the Influence FunctionalBy the parametrisation of the trajectories, the variables r and s have been separated intodiscrete intervals. This allows the time integrals of the influence functional to be donewithout difficulties. For example, the L 2 (r — s) part becomesq0^t2jt2kexp [- --i ^^ Sj yk J 2j_1 dT ^2k_1 ds L 2 ( r — S )Performing the time integration inside L2 (r — s) gives:Q 2 (t) - J dw J(w)[ 1 —.cos wt] coth(/9hiw/2) .(2.24)(2.25)(2.26)(2.27)The restriction j > k comes in because the time variable s varies from s = 0 to s = r,not t. The sum over j alone corresponds to the case j = k, where the limits of integrationare t23 _1 < r < t2j and t2j_1 < s < r.In an exactly similar way, the imaginary part of the influence functional is found tobe: 1n,^ q0 n-1 nexp —i^sj (t2j — t2j_1) + i - -^( rlk X,3=0 k=Oj=k+1Xjk = Q1(t2j — t2k+1) + Q1(t2j_1 — t2k) — Q 1 ( t2j — t2k) — Q1(t2j-1 — t2k+1) .J Jw Q 1 (t) -^dw^sinwtw2(2.28)(2.29)(2.30)Chapter 2. Two-Level System^ 28In this case, there is no sum on j alone because 7/3 corresponds to the time intervalt23 < s < t23+1 while (3 corresponds to t 23 _1 < T < t 23 . Since 0 < s < T , such a termcannot occur.After the integrations, P(t) can be rewritten as :oo^LynP(t) = E( 1) n 2271_ 1n=0E D{t2n} Fi(m)F2({t2}, {0){(3,77.,} °xF3({ti}, {6}, {7/3}) (2.31)with, [q/1 nF1 = exp — — EQ2(t2i — t2i_1)rh j=1(2.32)(2.33)(2.34)[q(2) nF2 = exp -- E c.3 (lc Ajk7h, >k[^2 nE mv-, ,. 1^\ _L  q0 V• /-3 Ilk Xjk •F3 = exp —i—h 2_, c3 kt2j — t23 _1 ) i— z..., srh ,,3 =1 3.-,A basic interpretation of these terms can be given, but first, it is convenient tointroduce some definitions: let the time interval t 23 < t < t23+1 ,where the system iseither in state A or D, be called the j th sojourn. Then, the time interval t 2k_ 1 < t < t2kis defined as the k th blip, with the system being in the states B or C (see Figure 2.3). Itis then clear that the term F1 represents the self-energy of the blips as a function of theirlength. Since the exponent is negative, this term actually reduces the weight of a givenpath and therefore controls the length of the blips. The term ((kA3k is an interactionof the j th and the k th blips of charge (3 and (k respectively. It can be viewed as a pointinteraction between the boundaries of the blips. Therefore, F2 represent the interactionsof all the blips between themselves. The term F3 is made up of two terms. The fist onerepresent the additional action to the path brought by the tilting of the wells while theChapter 2. Two-Level System^ 29blips : =± 1 ttl^t2^t3sojourns : n =± 1t2J-3 t21-2^t2j 1^t21 t2.-1Figure 2.3: Blips and Sojournssecond one represents the interaction of the blips with all the preceding sojourns. Thefact that it is imaginary indicates that it is an interference process caused by a returnof the path to the states A or D when coming from B or C. It is a direct result of blipcreation.2.2 Noninteracting-Blip ApproximationNow, Equation 2.31 is incredibly complex and certainly cannot be summed unless someapproximations are made. Fortunately, such an approximation exists : the noninteractingblip approximation, which allows us to keep only the self-energy of the blips. It will bejustified in this section the ohmic (s = 1) and superohmic (s > 1) cases. The subohmiccase (s < 1) doesn't need a special discussion, being quite similar to the ohmic case.2.2.1 Ohmic DissipationIn the ohmic case, the explicit expression of the functions Q 1 (t) and Q 2 (t) are found tobe (GR 3.941.1 [43] , GR 3.943, GR 1.431.1 )Chapter 2. Two-Level System^ 30Q1(t) = 77arctan wct^ (2.35)Q2(t) = - 1 7711).(1 w c2 t 2 )^in—[ 13rhi sinhOh(2.36) The expression for Q2 being exact in the limit w c-1 << /3h, introduced in the truncationprocedure.^The important point to notice is that for t^oo , Q i tends toward a constant valuewhile Q2 diverges logarithmically. This latter point is extremely important. Because ofthe F2 term in Equation 2.31, paths composed of long blips have a very small amplitudeand are effectively suppressed. No such term appears for the contribution of the sojourns,and it can therefore be concluded that the typical length of a sojourn will be much greaterthan a blip's one and that the blips will be far apart from each others. Then, withinlogarithmic corrections, for each j and k, we can set :Q2(t23 — t2k)^Q2(t2i — t2k-1)^ (2.37)Q2(t23 -1 — t2k)^Q2(t2j---1 — t2k-1) • (2.38)This cancels totally the factor Ajk, hence the name of the approximation. Having estab-lished that the blips form a dilute gas, F3 can also be simplified. Since Q i --= 0 for t = 0and is constant for t oo, all the factors X3k can be set to zero for k j — 1. However,for k j — 1 all the terms do not cancel and we are left with :Xj,k = 0 for k j — 1Xj ,j_l^Q1(t2j^t2j_1)^(2.39)which is simply a contribution to the blips self-energy. Within this approximation, P(t)now looks likeoo A2n t^2fAOP(t)^E (-1)n 22n-1^aft 2n 1 exp^Q2(t2 - t2j-1)]n=0^ n jChapter 2. Two-Level System^ 31x E ie n^ 2 nexp^E (t2i — t23_1) +E CJ r7-1—^.(2.40)J =1 J=1The sum over all the (3 can then easily be made. It is simply a product of cosines2q02n H COS —h^'(12 — t2j-1)rh,j-iQi(t2i — t2i-1)j=1(2.41) The sum over the 77 3 can also be made, always keeping in mind the restriction qo1. The resulting form is a product of terms of the form cos(A B) cos(A — B) =2 cos A cos B. The final expression is then2q02271-1 cos {— (t2 t1) —7rh Q1(t2 — t1)]E^ q02 n (4x H cos Ht2j — t2j—i cos —7rh^— t2J-1) •j=2Thus, the complete expression of P(t) reduces to :(2. 42 )(2.43 )P(t) = E (-1)nA2n^D{t2n } [ H g(t2i - t2i_1) + h(t2j - t22-1)^g(t2j - t2J-1)]0n=0^ j=1^ j=2(2.44)with the functions g(t) and h(t) defined asn2^g(t) = cos(ct/h) cos [aQ i (t)] exp [—AQ 2 (t)]( 2.45)n 2^h(t) = sin(Et/h) cos [ 71 ,21,1 exp [422(t)] .^(2.46)In the case of subohmic dissipation, the function Q 2 (t) diverges even more dramati-cally than for ohmic dissipation. The blips are thus far apart and the noninteracting-blipapproximation follows naturally.Chapter 2. Two-Level System^ 322.2.2 Sup erohmic DissipationThe justification of the noninteracting-blip approximation is more delicate in the case ofsuperohmic dissipation, due to the behavior of Q2 (t) for large times. At T = 0 and € 0,the functions Q 1 (t) and Q 2 (t) have the form, (GR 3.944.5, GR 3.944.6)q°27r Q1(t) = Osr(s 1)h[ ^1 ^Iuc2^t2 sin [(s — 1) tan -1 (.),t)] (2.47)n22 7r1° Q2(t)h= #sr(s 1) [1n2hFE As 2 "^ Q3 (t)7r^1 ^2(A.J 2^t2 cos [(s — 1) tan -1 (coct)]]^(2.48)(2.49)where,As = Asr(s — 1) ^ f dw J(w) 27rh^--J(w) Q3 (t)^cico4.02 cos cot(2.50)(2.51)and Os represents the strength of the coupling of the system to the environment.Compared to the ohmic case, the major difference is now that Q 2 (t) approaches aconstant value as t —> oo. This means that long blips are not effectively suppressed andit cannot be assumed that we are dealing with a gas of dilute blips. However, it can beargued that the blip interaction is weak and can thus be ignored altogether.To this end, first notice that without any coupling to the environment Ps = 0), ablip-sojourn pair has a typical duration 1 of A -1 , and that blips and sojourns shouldhave the same duration. Next, when the environment is turned on, this picture shouldremain the same, but now, the typical duration of a pair will be given by a renormalised1 This is due to the fact that without the environment, the summation for p(t) is of the formE(-1)n(At)2r/2n!. The dominant term (path), for a given t has n = At blip-sojourn pairs. Therefore,the average duration of a pair is 0-1.Chapter 2. Two-Level System^ 33inverse splitting^> 0 -1 , by Equation 1.30 . Both Q 1 (t) and Q3 (t) go to zero forlarge times, therefore, if^Aq  Qi(t = 0 -1 ) 7 aQi(t =^b^(2.52)and b << 1 then, the interaction among blips can be ignored. The constant part ofQ 2 (t) cancels due to the structure of Ajk, and the other factors go to zero. After thissimplification, everything follows the same path as before, and the resulting expressionfor P(t) is given by Equation 2.44 with E = 0. Of course, all this justification rests on theassumption that b is indeed small. This has to be checked self-consistently at the end ofthe calculation.At finite temperature, the coth(Ohw/2) brings into Q2 a summation of the form1^s-i[^nOhco, — iwct^ (2.53)For s > 2, the summation is finite. The blips are not dilute, but the argument presentedfor T = 0 still applies. However, for 1 < s < 2, this sum diverges as t oo which forbidspaths where the blips are not dilute. In this case, the argument laid down in the ohmiccase applies directly. In both cases, the use of 2.44 is justified.2.3 Results in the Different RegimesExpression 2.44 is a lot simpler than the formal expression 2.31 but is still relativelycomplicated, due in most part to the fact that it is the difference of the times thatappears in the functions g(t) and h(t). This complication can be resolved by taking theLaplace transform of P(t),P(\) = fo e -AY(t)dt^ (2.54)Chapter 2. Two-Level System^ 34followed by a redefinition of the timesAs a specific example, take the term n = 2 and the part of P(t) which is a functionof g(t) alone. The Laplace transform of this term is :cce^t^f4 dt3 f3 dt2 f2 di:a (t^t a (t^t1) .^(2.55)(-1) 2 ,6, 4^dt _ At^4^^^_ 0 4 — _3)^2 —^0 0 0^0^0Now, by inverting the order of integration from (dt — dt4 — dt3 — dt 2 — dt 1 ) to (dt 1 —dt2 — dt3 — dt4 — dt ) and defining t2j+1 = t2j+1 t2j ; j = 0,4; t 5 = t, it comes :co^ co—At3co([12 e —At2 g(2^e^— At4(- 1) 2 ,6, 4 f^) j°° dt3 dt4 e^g(t4 )^dt C A'A4= ( -1 ) 2 v [g(A)] 2(2.56)where g(A) is the Laplace transform of g(t). Applying this procedure to the entire seriesbrings[^A2h(A)1 ^[6,21n [g(A)P(A) = --- 1 A^E(-1)n^rand performing the summation gives1 — 0 2 h(A)/AP(A) =  A + 02g(A)(2.57)(2.58)A simple expression for the Laplace transform of P(t) is thus obtained. Once thefunctions g(\) and h(A) are known, finding P(t) amounts to taking the inverse Laplacetransformation :P(t) =f^dAeAtP(A)c-i00(2.59)where C is a constant such that the path of integration lies at the right of all the polesof P(\). Of course, such an inversion might not be easy at all, but nevertheless, theexpression for P(A) is a nice step forward.Chapter 2. Two-Level System^ 35A detailed look at all the possible solutions of P(\) is rather lengthy. In the followingsections, a brief overview of the solutions for the different cases will be given. Moredetails can be found in Appendix B.2.3.1 e = 0, Ohmic CaseFor symmetric wells, P(\) takes the simple form1PO) _ + 0290)and the spectral function is(2.60)J(w) = 7/w e-wiwc .^ (2.61)For high temperature and/or strong dissipation, the factor Q 2 (t) in the influence func-tional is extremely efficient in keeping the blips far apart from each others, which meansthat only the region around A = 0 contributes effectively to g(A), and in a first approxi-mation, we can replace g(\) by g(0). The Laplace inversion is then trivial, yieldingP(t) = e -rt^(2.62)with the decay rate, fiF 6. 2^F(a) ^frkT1 2a-1F — A 2g (0) - 2 co, F( cv + 1/2) I hcoc i^(2.63)and a riqV271-h being the dimensionless coupling parameter. Clearly, the initial coher-ent behaviour of the system is totally destroyed by its interaction with the bath.At T = 0, the previous behaviour changes drastically according to the values of a.For a > 1, the system is localised, i.e.,P(t) = 1^ (2.64)for all times [44, 45]. This means that the environment not only destroys all coherence,but also the transitions from one well to the other. As a is lowered, transitions are againChapter 2. Two-Level System^ 36possible, but in a very incoherent way. At a = 1/2, the form of P(t) is again a pureexponential.7r A2[P(t) = e-A ff t = exp — —2 —co,t .^ (2.65)It is important to notice that this expression is valid for any temperatures and is com-pletely independent of it.Always at T = 0, when a is lowered below 1/2, the incoherent behaviour is stillpresent, but a coherent damped oscillation is now superimposed on it, their exact ex-pressions being given by Equations B.16 and B.17. The shematic behaviour of P(t) atT = 0 is represented on Figure 2.4. It is only there to give a rough idea of the functions.The real behaviour will be much more complicated. In particular, for a < 1, the incoher-ent part of P(t) should be present. Nevertheless, the general trend is well depicted in thefigure. The previous analysis can also be extended to finite temperature, and for a < 1/2,an upper limit for the temperature below which coherent oscillations are possible can befound [46]. It indicates that the oscillations are more likely to be observed for very smallvalues of a, but at the same time, it holds for a wide range of temperatures.2.3.2 e = 0, Superohmic CaseThe spectral function for the superohmic case isn 2YO 27h^)j(w = )(3s ws 6.-) 1.—s e -w/wc1(2.66)where ci) is the characteristic frequency of the environment. It is introduced solely todefine 13, as a dimensionless coupling constant. The fundamental quantity is of coursethe combination A, co, i-s. For dissipation produced by phonons, Co can be the Debyefrequency, WD .It was shown before that the interblip interaction is weak enough to be neglected.This mean that the blips self-energy will not be too strong itself since it is induced by10 2 4^6time (arbitrary units)8 10alpha = 0 —alpha < 1/2 --- -^alpha = 1/2 ^alpha > 1 ^1.5-10.50-0.5Chapter 2. Two-Level System^ 37Figure 2.4: Form of P(t) for Various Values of a at T = 0Chapter 2. Two-Level System^ 38the environment, and we can expect the behaviour of the system to be preserved in mostpart, which is indeed the case. At T = 0, for all s > 1, the system performs dampedoscillationsP(t) =^cos(At)^ (2.67)where F s = q4J(A)I4h and A is a renormalised tunneling splittingex [ 4  r dco j(w) •2rh o w2 (2.68)Then, it can be proved that the blip's interaction is indeed small for any coupling strength,thus showing the self-consistency of the argument. Again, more details are to be foundin the appendix.For temperatures different than zero and s > 2, the damped coherent oscillations arestill present, but with temperature-dependent parameters2(#) = Aexp [— y717,4 ro dw j(w2 ) cothPh0/211qg^-rs(i3)^—,,j(A)coth[i3h3./2} .(2.69)(2.70)Finally, for 1 < s < 2, the blips are dilute and the form of P(t) is a pure decayingexponential with a decay rate rate given by[ (2 — s) sin 7r3/2^1/(2-s) r r3 —^ si^(A2 inc-311/(2-s)F —^ 2.71)20,F(s — 1) sin 7r(s — 1)^[2 — si ry LkTiAs the temperature is lowered, there will be a crossover from the decaying behaviour tothe damped oscillations observed at low-T.2.3.3^E 0In the biased case, the complete form of P(\) must be used; however, as we shall see,the noninteracting-blip approximation gives an incorrect long term behaviour of P(t) forChapter 2. Two-Level System^ 39many cases. First, let us assume that both g(A) and h(A) can be expanded in a powerseries of ). by taking the expansion of the exponential,g(A) = go + gi A^ (2.72)h(A) = ho + h 1 \^ (2.73)where go (ho ) = limA.o.g(A) (h(A))and gi and h 1 differ from their A = 0 value by thepresence of a t in the integrand :n2g(0) = f dt cos(Et/h) cos [aQ i (t)1n 2exp,_AQ2 ( t ) , ( 2.74 )h(0) = f,c° dtt cos(ct/h) cos [a WO exp[— °,_Q 2 (t)] (2.75)]^72nNow, it can be shown [2, appendix E] that the ratio ho /go is equal to tanh(c/2kT)and is totally independent of the form of the spectral function J(w). Then, if we canassume that both gi and h 1 are much smaller than one, P(t) takes the formP(t) = — tanh(f/2kT) + [1 + tanh(E/2kT)]e -rt (2.76)where F = A 2go . Now, as t -- oo and T --+ 0, P(t) is equal to either +1 or —1 which, inany case indicates a localisation of the particle in one of the wells, even in the presenceof an infinitesimal bias. However, as was just seen, the long time behaviour of P(t) inthe superohmic and ohmic with a < 1 is totally the opposite and there are no reasons tobelieve that it should be modified for a small bias. Thus, at T = 0, the noninteracting-blip approximation cannot be considered accurate. The blips do not form a dilute gasand their interaction do not cancel. In the superohmic case, where the approximationwas justified even though the blips were not dilute, working in first order in b in thepresence of a bias gives totally nontrivial terms, impossible to solve.Chapter 2. Two-Level System^ 40There is however one regime in which the approximation can be justified and theresult 2.76 trusted : the overdamped case, where the system relaxes to its long timevalue before performing any oscillations, exactly the case described by 2.76. This is sobecause the basic requirement that g 1 be much smaller than one exactly corresponds tothe condition for the blips to form a dilute gas. To see this, notice that the typical widthof a blip-sojourn pair is governed by the exponential term (see footnote page 32), i.e.,it is of the order of go-1 , while an estimate of the blip length is given by Igi l/go (it isgiven by the first moment of F1 , Equation 2.32 setting aside the cosine factors). Thus,the condition to have a dilute gas of blips is :'21 < AV or, gi < 1^ (2.77)goand since h(A) is of the same order as g(A), we also get h 1 << 1. If these conditions aresatisfied, the blips form a dilute gas and the noninteracting-blip approximation is valid.The interaction between the blips cancel and only their self-energy will contribute to thesum. However, it is important to keep in mind that this is valid only as long as 2.77holds.For ohmic dissipation, with a > 1, gl is of the order of (0/wc ) 2 , much less than oneand so, 2.76 is valid at all temperatures and bias, which simply makes the connectionwith the T = 0 case already analysed. For a < 1, the conditions under which gi < 1 arehAr or T >> a-1 0r . In both cases, the decaying factor is given by [47, 48, 49]F = A2 127kTl2c'-1 cosh(c/2kt)IF(a if/21- kT)1 2^(2.78)2w, hco,^F(2a)For f^0, the temperature must be high and this reduces to the high temperature limitof Equation 2.63 found previously.Next, in the superohmic case, the requirement is E^hA where 0 is the renormalisedChapter 2. Two-Level System^ 41tunneling splitting, and the decay rate isJF = [—E (2.79)In all other cases, the exact form of P(t) is not accurately given by Equation 2.76 andthe problem must be considered as unsolved at the moment.Chapter 3Interacting Two-Level SystemsThis chapter is concerned with the interaction of two two-level systems coupled to theenvironmental bath. The two systems do not interact directly with each others, but thebath provides an indirect interaction which can have severe consequences.The systems, labeled by 1 and 2 are coupled linearly to the bath by two constants6(1) and Cr. A priori, these constants need not to be equal unless the two systems areidentical 1 . The counterterm must be such as to renormalise the potential of system 1and 2 separately. It simply corresponds to the introduction of two separate counterterms.The total Lagrangian is thusL = 2 11114^vi(g1) 2^v2 (q2)^2 ^— m aw2x2[0,92^4,4^[C(2)12E[qi c(1) q20e2)]xce (41.^ (3.1)L--• 27.naw2x2^27/26/W2X2a In the tight-binding limit, we only consider the ground states as before, and theresulting Hilbert space is now the tensor product of each separate truncated Hilbert space.The problem is thus mapped to a four-well problem whose states, labeled by the pair{ql , q2} are {+, +}, —}, {—, +}, {—, —}, + representing g01 /2 or 9,02 /2, the equilibriumpositions of systems 1 or 2. Also in this limit, two tunneling matrix elements A i and A2are introduced, referring to the systems 1 and 2 respectively. A l describes transitions{+, +} {—, +} while A2 is associated with {+, +} ,= {+, —}. For simplicity, it will'However, even if they are identical, the constants can differ by a phase factor. That is, we haveCa^Ca exp i0a . For example, a system at a given position R might bring a phase of the form= ka R.42Chapter 3. Interacting Two-Level Systems^ 43be assumed that no simultaneous transitions of systems 1 and 2 take place.Assuming that the system is initially in the state {+, +}, the different probabilitiesp++ , p+ _, p_+ and p__ of finding the system in one of its four possible states at time t willbe calculated. However, the environment introduces an affective bias (0 of the wells andthe noninteracting-blip approximation is then valid only for overdamped systems. It willbe seen that unless one of the two systems in very weakly coupled to the environment,the system is localised in its starting well for all practical times. If the weak couplingrequirement is satisfied, the system relaxes exponentially into one of the two states {+, +}and { —} with a very strong probability.The problem of two two-level systems interacting with the environment has alreadybeen examined by a different approach. Using a renormalisation technique first utilisedby Anderson et. al. [50] for the Kondo problem and generalised by Cardy [51] to multiwellconfigurations, a somewhat detailed analysis of the partition function of the system wasmade [52, 53]. The results indicate that a phase transition will take place betweencorrelated and uncorrelated motion of the systems. In the correlated phase, they movetogether, while the motion of one system is independent of the motion of the other inthe uncorrelated phase.However, these results are not directly relevant to the present work. For one thing, theanalysis by the renormalisation technique includes from the start the diagonal transitions{+, +} {—, —} and { —, +} {+, —}. Since this in not considered here, the domainof correlated motion is reduced the facto. Also, the phase transitions were found bylooking at the behaviour of the partition function under a change in the cutoff frequencyco, while in this chapter, the probabilities of having the system in a certain state arecalculated, which is certainly quite different. However, this calculation might providesome information about which phase is present for the range of parameters used.Chapter 3. Interacting Two-Level Systems^ 443.1 Construction of the Influence FunctionalThe procedure leading to the reduced density matrix of the two systems is essentiallysimilar to that of the single two-level system. The only modification is an integral overthe second system's coordinates as well as an integration over the different paths takenby it. Therefore, the probability that the combined system 1 ® 2 starting in the state{+, -F} ends up in one of the four states at later time t is^13±±(±goi1 2 ,±q02 1 2,t) = 1 Dq i I Dq2 .f^I Dq2A[qi] B[q2]A*[qi]B*[q2 ]x F[qi, qi, q2, qi] (3.2)where A[qi] = iA i , B[q2] = iA 2 and F[q i ,q2 , VI , q2] is the influence functional comingfrom the integration of the environment. Its form is essentially the same as in the two-well problem, the function f (t) in Equation 1.53 being now given by q i (t)0,1 )+ q2 (t)0,2 ).However, this function enters the influence functional in the form f(r) x f(s). This willgive rise to crossed terms which are a function of both q i and q2 at different times. Ineffect, the systems are coupled through these terms.The lower limits of integration in 3.2 are all the same, the two systems being in thestates qi = +gm /2 and q2 = +q02 /2 at t = 0, but the upper limits will change accordingto the probability being calculated. These upper limits are depicted in table 3.1, where± represents ±q01/2 or ±q01 /2. Of course, the limit of q i must be equal to the limit ofqi and similarly for the second system.As before, the parameters of the bath are related to the physical situation throughthe spectral functions Ji (w) and J2 (w) defined as7r^[C(12Ji. (w) = — E  a 8(w — wa )^ (3.3)2 ^m oiwc,J1(w ) = -721 E [°'2"2a(w — wc,)^ (3.4)Chapter 3. Interacting Two-Level Systems^ 45qi qi q2 q'2p++ + + +p+- + + — —P-+ — — + +P-- — — — —Table 3.1: Upper limits of integration in Equation 3.2but a third spectral function, J12 (W) , has to be defined in order to account for the crossedterms :7rJ12(w) = E c,V')02)2^, 6(c.,)— wa ).a rilawce (3.5)The spectral functions Ji (w) and J2 (w) are of the form Oscose -w/w° which implies thatJ12 (W ) will have the same form but with a coefficient \//31802s. However, since a phasecan be present in q), the spectral function J12(w) can now be complex. Furthermore,since the two systems interact with the same bath, all these spectral functions have thesame cutoff frequency co,.Next, by introducing the following integralsLV ) (7- — s)^dw .1:Au)) sin w(T — s)147) (7- — s) =^dw Ja(c.o) cos w(r — s) coth(3h42)L3 (T — s) I dw J12 (w) sin w(7- — s)0L4 (7- — s) fo c.° dw J12 (w) cos w(r — s) coth( i3h42)with j = 1,2, as well as the following functions6(7 )^— q;(7- )]00(3.6)(3.7)(3.8)(3.9)(3.10)6(7 ) = q021q2 (7) - q2(7 )]^(3.11)Chapter 3. Interacting Two-Level Systems^ 46xi(T) = q0111q1(7) gar)]^ (3.12)X2 (T) = q021 {q2(7 ) + q'2 (r)], (3.13)the influence functional can be written as:F [qh q2, qi, q'2] = Fl[qh^F2[q2, 412] F12{q1, q2, qi , q21^(3.14)with F1 and F2 being the influence functional for systems 1 and 2 as if they where aloneand F12 is the part representing their interaction through the bath. The expressions forF1 and F2 are given by Equation 2.4 provided that one uses the functions relating to agiven system, while F12 isF12 = exp gorigho2  fc: dr LT ds {—iL3 (r — s)[6(r)x 2 (s) 6(r)x i (s)]-EL4 (r — s) [6 (7)6(8) + 2(T ) 1(S)] }^(3.15)The coupling of the two systems through the bath is now clearly apparent. It has adissipatinve and a polarising terms.Now, the probability is made up of a quadruple path integration over two states.Similarly to what has been done for the two-level system, it could be reduced to a singlepath integration, but this would be over sixteen states. It is simpler to consider a doublepath integration, each of these paths being carried over four states. From the structureof the influence functional, we can take one path to be made of a series of 2n 1 transitionsbetween the combined states {q i , } with action i0 1 /2 and jumps taking place at timesj = 1 • • 2n 1 . The transitions along this path will be called type-1 transitions. Then,the second path is a series of 2n 2 jumps over the states {q 2 , q'2 }, with an amplitude i0 2 /2.These type-2 transitions take place at times uk , k = 1 • • • 2n2 . An example of this typeof path is shown in Figure 3.1This separation of the paths allows the use of a parametrisation similar to the oneused in Chapter 2. We introduce the charges^6.3 , 772k and (2k and rewrite the positionChapter 3. Interacting Two-Level Systems^ 47Path 1Path 2 t=0ti^t2^t3^t4 ts^t6tu 1^u 2^u3^u4 u 5 u 6 u7 u 8Figure 3.1: Type-1 and Type-2 Pathsfunctions as:nl^X1(T) = E 713 [0(7 — t23) — 0(T — t23+1)]^(3.16)2=0nle1(7 )^E^[0(7 — t2j_1) — e(T — t23)}^(3.17)i=on2^X2 ( T ) = E 7,2k [8(7 — u2k) — 64(7 — t2,1)]^(3.18)k=0n24.2(7 ) = E c2k [0(T — u2k_1) — 0(7- — u2k)] •^(3.19)k=0The states {qi ,Vi l and {Q2 , q2} as well as the different charges follow the same defi-nitions already given for the two-level problem. The two paths can be viewed as a seriesof parallel blips and sojourns, the boundary constraints being introduced by the valuesof the charges of the first and last sojourns 710, 920/ 711n1 and 772n2 • The values of qlo and720 are fixed to +1 from the definition of the problem, but the values of the two otherscharges are dependent on which probability is being calculated.To obtain the probability p++ , the two paths must end up in the states {+, +} afterperforming 2n 1 and 2n 2 jumps. The total factor coming from the action of the jumps isChapter 3. Interacting Two-Level Systems^ 48Last Type-1 Sojourn Last Type-2 SojournP++ +1 +1P+- +1 —1P-+ —1 +1P-- —1 —1Table 3.2: Charge of the Last Sojourns for the Different Probabilitiesthen2ni A 2 2n2(_i)ni +n22^2(3.20)and the paths are subject to the constraints n 712n2 = +1.However, for p__, the paths must end in the state {—, —}. That means that at leastone blip must be present in the paths considered. They will thus be composed of n 1 + 1and n2 + 2 blips, both n 1 and n2 ranging from zero to the infinity, with the constraints712n2+1^—1 and a total amplitude factor ofA 2ni +2 A 2n2+2^(_ )ni +n2+2 —1^^2 ^2(3.21)Finally, of course, the paths for p+_ and p__F will be formed by a mix of these two cases.For p+_, the constraints are q ini = +1 and 712n2+1 = — 1 with a total action ofAl 2n102 2n2+2(-1)nl +n2 +12^2(3.22)while for p_+ , the constraints are ih ni+i = —1 and 712 n2 = +1 and the action associatedwith the jumps areA 27/1+1 A 2 2n2(_ WI +n2+12^2(3.23)The restriction on the paths for the different probabilities are shown on Table 3.2However, nothing specifies the order in which the transitions happen. A priori, allthe different configurations are possible and must be taken into account. For example,Chapter 3. Interacting Two-Level Systems^ 49t,^ t,dl■■•■••IC,^ cC..)I 1■••■•..,1.■•■■•t.^ tf )Figure 3.2: Possible Configurations for n 1 = n 2 = 1consider a path relevant to the calculation of p++ with n 1 = n 2 = 1; there are fourtransition times t i , t 2 , u 1 and u 2 which can be arranged in 6 different configurations, asrepresented in 3.2, always with the restriction t 1 < t2 < t and u 1 < u 2 < t. In general,there will be2n1 + 2n 2( = (2n 1 + 2n2)! (3.24)2n1 2n1! 2n 2 !possible configurations [54] and this results in an influence functional whose form dependson a given configuration. Since Fi[qi, qi] and F2{q2, qZ1 represent the interaction betweenblips of the same type, they give the same result independently of the configurations,but Fi2[qi, qi , q2, q] will depend on them, since it corresponds to the interaction betweenblips of different types.Chapter 3. Interacting Two-Level Systems^ 503.1.1 Form of F12 for the Possible ConfigurationsWith the above parametrisation, the homogeneous parts of the influence functional, F1and F2, are identical to the two-well case provided that the correct spectral functions areused. However, F3 is highly dependent on a given configuration.The L4 part, the interaction between blips of different types, always has the sameform, irrespective of the overlap of the blips. For two blips of charge Clj and (2k thisfactor is:exp —63 (2k [(Mitzi — u2k-1 I) Q4(1t2i-i — u2kI) — Q4(It2i-i — u2k--11) Q4(It23 — u2kI)](3.25)where It 2i — u2k_1 I means that one has to take the absolute value of the time difference.Compared to F1 or F2, there are no factors of Q4 alone. This part of the influencefunctional doesn't contribute to the self-energy of the blips. Q4 is similar to Q2 butwith the spectral function J1 2 (u.;) instead of Ji (w) or J2 (w). From now on, the differentfactors q01 ,702 /7rh, qgi lrh, and g02 17rh will be incorporated inside the different Q-functionsto clarify the notation. That is,Q4(t)^qmq°2 1 d, ..112(w) rh Jo w w2[1 cos cot] coth(Aw/2)^(3.26)2Q2')^gOi o d J.1(w)2^—7rh^u.)  w2 [1 — cos wt] coth(Aw/2) . (3.27)The L3 part of the interaction is slightly more complicated and has to be divided intothree cases.First, if the blip and sojourn are not overlapping, i.e., the times corresponding to thelimits of a sojourn are smaller than the blip's limit, the interaction is identical to whatwas found before. An example of this case could be the interaction of the type-2 blipwith the first type one sojourn in Figure 3.2a. The result is:exp 2 77 1 0 (21 [Q3(ui — t0) + Q3(u2 — t1) — Q3(u2 — to) — Q3( 1/1 tl)^(3.28)Chapter 3. Interacting Two-Level Systems^ 51where again, Q3 is identical to Q i with the appropriate replacements :Ji2(w) .Q3(t) = gorigho 2^, sin wtdw w2 (3.29)Next, if a blip takes place entirely within the time interval of a sojourn, complicationsarise. As an example, consider the interaction of the first type-1 blip with the first type-2sojourn on Figure 3.2a. The time integration is:frds sin wer — s) 1 (t2 — t 1 ) — 12- [ sin w(t 2 — to ) — sin w(t i — to )^(3.30)ft dr,^towhich gives the following resultexp i Cii 7/20 [ Q3(ti — to) + Q3(t 2^to))^(t2^t 1 ) I •^(3.31)The effect of this sojourn is to introduce a bias term in the self energy of the blip. Thevalue of the bias is:E^dch., ^q01q02^Ji2 (W)7r^oFor ohmic dissipation, this isgm q02 =^V1/11/24,-/c = 2Val a2h4.0c (ohmic)7while for superohmic dissipation, we getE —2F(s)^s Vi302shw, (superohmic).(3.32)(3.33)(3.34)However, in the subohmic case, s < 1 and the integral diverges badly. As a result-e^—oo (subohmic).^ (3.35)This will have profound implications later on.The minus sign in 3.32 is introduced only to make the connection with the formulasused in Chapter 2 and is mostly a convenient notation. It is not possible to make a definiteChapter 3. Interacting Two-Level Systems^ 52statement regarding which well has the highest energy and which one is the lowest sincethis depends on the state of the combined system {ql , q2 } as a whole. Interestingly, theeffective bias is a function of the cutoff frequency of the bath. It explicitly appears inthe probability function and could therefore be measured, somewhat similarly to the freeparticle case [34]All other cases are results from an overlap of the blips, and the effective bias will alsoappear in there. If the blips are overlapped as in Figure 3.2b, it is convenient to considertogether the interaction of the CH blip with the 920 sojourn and the C21 - 7  interaction.The corresponding factor in the influence functional is thenexp i^7 20 [ Q3(t2 — ui) Q 3 (t 1 — to ) — Q3 (t2 — to) ] x exp i (21 X11 Q3(u2 — t2).x exp —z— [ C11 7/20 (ui - t1) + (21 911 (u2 - t2) • (3.36)And finally, if a blip completely overlaps another blip, as in Figure 3.2c, the result ofthe interaction with the two neighbouring sojourns is. eexp —z— [ C11 7720 (ui - t1) + S11 7721 (t2 - u2)exp i Cu 7720[ Q3 (t i — to) + Q3(t2 — uz.) — Q3(t2 — to)] — i Cil 7721Q3(t2 — u2)1 3 . 37 )All other cases in fig 3.3 are similar to the one above provided that one interchanges thetimes t3 and uk•This summarises the construction of the influence functional for any configuration. Itis interesting to remark that the effective bias is solely caused by a sojourn. In particular,even if one of the system doesn't tunnel, its presence causes the other well to be tilted.3.2 Expressions for the Different ProbabilitiesPerforming the different summations is hopeless. Now, one has to take into accountnot only the interaction between the blips, but also all the possible configurations. TheChapter 3. Interacting Two-Level Systems^ 53noninteracting-blip approximation can be helpful in treating the interaction among blipsof the same type, but is mostly inefficient for blips of different types. This is not onlydue to the fact that the environment introduces a bias in the wells, as was seen, butalso to the fact that two overlapping blips certainly cannot be considered as far apartand the approximation cannot be justified in this case. However, it was seen in section2.3 that even in the presence of a bias, the blips will form a dilute gas if the system isoverdamped. Then, if the length of a blip is much smaller than the length of a sojourn,the probability of having two blips of different types overlapping is very small and can beeffectively neglected. In this regime, the total number of possible configurations is thusreduced to all the possible permutations of complete blips. For example, in Figure 3.2,only the paths a and f will be considered. In general, there are nown l + n2 (3.38)n2possible configurations to sum over. An example of this type of path is shown on Figure3.3. The interaction between blips of different types mediated through the environment isnow only of the type 3.31, with a sojourn overlapping completely a blip. This argument issomewhat similar to an argument made by Fisher and Zwerger [7] who considered "first-order blips" in a calculation relating to the quantum brownian motion. This statementwill be made more precise in Chapter 4.Within this approximation, not only do all the interactions between blips of the sametype cancel, but also between blips of different types. The analogue of Ask, Equation3.25 is zero and the two Q3 terms in Equation 3.31 also cancel each others. All that isleft is the self-interaction of a given blip. However, a blip still overlaps with a sojournand this brings a bias term whose sign depends on the charge of the sojourn, contraryto the two-level system where the bias was constant. For example, the self-energy of aChapter 3. Interacting Two-Level Systems^ 54n1-- n2 -- 3. There are 20 possible permutationsFigure 3.3: Paths for the Overdamped Systemtype-1 blip isexp [-i Cij I477 2 k - t2.j^t2j-1)^Culj^ri1)kb2; - t2i--1)1 , (3.39)where 6 3 is the charge of the blip and 7/ 2k is the charge of the type-2 sojourn overlappingthe blip.Blips of a given type are well ordered amongst themselves. What changes from oneconfiguration to the other is the ordering between blips of different type. Therefore,the summation over the {( ii } and {(2k} is independent of the configuration and can beperformed immediately. Terms similar to 2.41 are then obtained :ni2n' II cos [qi-10 ) (t2i - t2j-1) - 7 2P(t2j t2j-1)]j=1n2fI(2)^ ex 2n2 11 cos n,k-i Lei (u2k - u2k-1) - 91P — (U2k U2k-1)k=1(3.40)where rhp and 712P are the charges of the overlapping sojournsThe fact that the blips are not overlapping also allows the use of the Laplace transformto perform the summation. For simplicity, let us definef± (A) - 2AJoy01 °3 dt Cate-Q (21) (t) cos[Q (11) (t) f -ht] (3.41)Chapter 3. Interacting Two-Level Systems^ 55g± (A) = f2A o dt e -At e -c42)(0 cos[Q? ) (t)^1-t] . (3.42)The contribution of the blips to a given path will now be expressed as a function of thef± (A) and g± PO. All that remains to be done is the sum over the configurations and theset of the {ll ij} and {7720. This must be performed simultaneously since the summationover the n gives a different result from one configuration to the other.3.2.1 p++To find p++ (t), it is convenient to separate the total probability into seven different termsp(t) = 1 + 7)1 (0 p2 (t)+1412) (t)^Pi2 ) (t)^Pi32) (t) + 1442) (t)^(3.43)The factor 1 represents the path without any blips at all, pi (t) and p2 (t) are formed bypaths where only one of the systems tunnel, the other staying stationary, and the fourremaining terms account for the paths composed of the two types of blips. Since thepath with no blips at all is already counted separately, the summations in both p i (t) andp2 (t) must start at n i = 1 and n 2 = 1. To include the paths made of the two types ofblips, let us define four classes of path according to their first and last blips. Paths ofclass-1 start and end with a type-1 blip while paths of class-2 start and end with a blip oftype-2. A path starting with a type-1 blip and ending with a type-2 blip will be referredto as a class-3 path, while a class-4 path begins with a type-2 blip and ends with a type-1blips. Then, The terms A22), pit ) and A42) are formed of paths of class-1,-2,-3 and -4respectively. An example of each of these paths is represented in Figure 3.4To find the contributions to a path from the self-energy of the blips, it is convenientto separate them into three groups1) decoupledChapter 3. Interacting Two-Level Systems^ 56Class 1Class 2Class 3Class 4 n^n^Figure 3.4: Examples of the Four Class of PathsChapter 3. Interacting Two-Level Systems^ 57C 11^C12^ 4-13^ C1417 10^17 11^ 11 12^I^I 71 13^1^1^17 14'21^'22^C23^'2417 20^ri 21^n 17 22 23^f^17 24^to t1 t2^t3 t4^t o t 6^ t7 t 8U 1 U2^U3 U4^U5 U 6^ U7 U 8Figure 3.5: Illustration of the 3 Groups of Blips2) partially coupled3) coupledThese three groups are present on Figure 3.5. We define a decoupled blip to beindependent of the values of the charges 77 appearing in the expression of its self-energy.The blip 1, of charge CH is clearly the only blip to fall in this category.Next, a coupled blip is defined as a blip whose following sojourn causes the bias toappear in a different type blip. The blips C21-C 12 -(22-(13-C23 and C14 correspond to thisdefinition. The cosine parts of their self-energy is:^r^n (2) f^E ,^r^n(i)t ^t3 )^e ,cosL 7 20t ( U2 — U1) — r/iihku2 — ILO] X COS0711Wi ku,^44 — 1,  — 7121 hkt4 — t3 )}^r^n (2) / ^r^n (1) 14^4 \X COS 1 7122 k.d1 0.16 — u5) — 7112 —l i6 — u5 )1 X COS17112W1 Y'6 — 1'5 ) — 7/23 —h (t6 — 4)] (3.44)hkts —^r^,-,(2),^ ,^ r^r) (1) (4^e iX E0S17/23Vi U8 — U7) — 7113 — Y/28 — ti7)] X COS17/13W1 kb8 — t7) — 7/24 —^4)] )h hwhich shows the coupling between these blips.Finally, a partially coupled blip is one for which the bias term depends on the chargeof the overlapping sojourn, but whose following sojourn doesn't causes any bias. TheChapter 3. Interacting Two-Level Systems^ 58blip (24 corresponds to this definition. The cosine part of its self-energy is(cos[7]21^2) (.14 — u3) — 7112^— u3)] (3.45)and clearly, from the figure, the sojourn 77 22 following this blip doesn't cause any biassince it immediately precedes another type-2 blip.After summation over the 17's, the contribution of a decoupled blip is clearly f_ or g_,depending on its type. For coupled blips, however, things are a little more complicated.Let us define IQ' for Q 1 (t) f Et/hi, the argument of the cosine part of the self-energy.Then, for a chain of blips, whose arguments are Q i • • qty , the summation over the77, gives2 {cos[QT] cos[QM • • • {cos[Q N] cos[Qk]}2{cos[Q 1 ] — cos[QM • • • {cos[QA1 — cos[Q0 (3.46)An explicit derivation of this result has not been found, but it can be proved by induction.Finally, for the partially coupled blips, no matter what is the sign of the effective bias,the summation over the remaining 77 brings a term of the form {cos[Q - ] cos[Q11. Theblips are effectively taken out of the coupled chain.In summary, for each paths, the first blip gives a factor f_ for paths of the class-1and -3 or g_ for class-2 and -4 paths. Then, the chains of coupled blips contribute via aterm2[ (^f +) 3 g^g + )k (f f+ )3 g^g+ )k (3.47)where j = k for paths of class 1 and 2, k = j 1 for class 3 and j = k^1 for class 4.Finally, each partially coupled blip gives a factor (g_ g+ ) or (f_ f+ ) depending onits type.The chains of coupled blips will be used as the skeleton of the complete summation.Since each chain is characterised by a number j, a sum over all the possible chains isChapter 3. Interacting Two-Level Systems^ 59done by a sum over all the possible values of j. Then, for each chain, the partiallycoupled blips are summed from 0 to oo for each type of blip. Of course, there are manyways to place all the partially coupled blips, and since their contribution to the pathis independent of their positions, all their possible locations can be accounted for byintroducing combinatorial factors in the summation. Since all the chains of coupled blipsare independent from each other, this completes the summation over the configurations.To use the language of field theory, it could be said that the chains of coupled blipsare 'dressed' by adding to them all the partially coupled blips. The total probability isthen obtained by summing the contributions of all these dressed chains. This process isrepresented schematically in Figure 3.6.For paths of class 1, a given value of j implies that there are j + 1 type-1 coupledblips and j type-2 coupled blips. Therefore, the 1 added type-1 blips and the k addedtype-2 ones can be distributed in(1+j)(k+j-1)1^kpossible ways. Similarly, the combinatorial factor for paths of class-2 will be(1+j-1k+j)1^k(3.48)(3.49)where 1 and k refer to type-1 and 2 blips respectively. In these two cases, the summationgoes from j = 1 to j = oo.Next, for the paths of class-3 and -4, the case j = 0 also has to be included. Thiscase corresponds to the absence of coupled blips, i.e., all type-1 and all the type-2 blipsare grouped together. The other values of j correspond to j + 1 type-1 blips and j + 1Chapter 3. Interacting Two-Level Systems^ 60a)•••••1217=1:SIMM= illmimmortimm=^I•MlinIMMee•■= P =++^.....s.....■..= 1•1111111■11111•11rEL•■■•i++ +^•^•b)=^n^n^++ +^• ^•^n n- +. n^n n^++ n^n n^+Figure 3.6: Schematic Summation of the Paths for p++ctz /+z-1z ^1a 1=0^1(3.54)(1 — a)Chapter 3. Interacting Two-Level Systems^ 61type-2 coupled blips. The combinatorial factor is then(1-1(k+j) .1(3.50)Now, the summation can be performed. The summations for both p 1 and p2 are clearly asum over partially coupled blips, except for the first one, exactly similar to what happenedin Chapter 2. After taking the Laplace transform, we getpi (A) = E(f- + f+) 3j=0f_A 1 — (f_ + f+ ) •(3.51)And clearly, p 2 will be similar to it but with f replaced by g1^g_p2(A)A 1 — (g_ + g+ ) •Next, the explicit summation for A l2) is(3.52)p12) (A) =^f- E [(f- + f+)3 (g- + g+)3 + (f- - f+)3 (g- - g+)33=1x EE i + j^k+i -1 (f- + f-F) 1 (g- + g+) k -^(3'53)1=0 k=0^1The dressing of the coupled chains is brought by the summations over 1 and k. Knowingthat [54]the summation is easily performed to givep112) (A) = 1 ^f-2A 1 — (f_ + f+) [P1 (A) P2 (A) — 2 ]^(3.55)withF1 — [1 — (f_ + f+ )][1 — (g_ + g+ )]1 — (f_ + f+ ) — (g_ + g+ )(3.56)Chapter 3. Interacting Two-Level Systems^ 62[1 — (f_ + f+ )] [1 — (g_ + g+ )][1 — (f_ + f+ )][1 — (g_ + g+ )] - (f_ - f+ )(g_ - g+ ) .Clearly again, the expression of p1 2) is equal to the previous one by interchanging f andg.P12) (A) = 21A 1 — (g_ + g+) [P1 (A)H-P2 (A) — 2]^(3.58)In the same spirit, the expression for 13 (132) isn(3) =^[ f_ f + y (g^g )3 + 111.23=v(f- f + )3 (g^g _03 + 1cc co (1+j)(k+jx^)E E^(f^g_ok1=0 k=0 1which is equal to(3.59)Pi32) (A) = f- [1 — (f_ f+^[Pi(A)(g_ + g+ ) + P2 (A)(g_ — g+ )] . (3.60)2A^)][11 — (g- + g+ )]Again, the expression for p142) is given by interchanging the f± and the g± . Finally,collecting all the terms and simplifying brings1^1^f_+g_P++( A) = jt  2A 1 — (f- f+) (g- g+ )1^(1_ + g_)/2 — f_g+ — f+g_(3.61)A [1 — (f_ + f+ )][1 — (g_ + g+ )] - (g- - g+)(f- - g+)At this point, we can revert to the notation of Chapter 2 and defineP2 = (3.57)gi (A) = ZN4 f dt c-At-Q (2i) (t) cos[Q (11) (0] cos[d /h.]g2(\) = OZ2)dt e —At—Q2(^(t) COS [0 ) (t)] COS [et/h]0h i (A) = D1 o dt e -At-Q2(1) (t) sin[ICA1) (t)] sin[et/h]h 2 (A) = 02 f dt e- At-Q2(2) (t) sin[Q (12) (t)] sin[d/h] .(3.62)with( 1 )^1 ^f- [P12 'I= 2A 1 - (f_ + f+ ) 1)1(A) - P2(A)](3.66)Chapter 3. Interacting Two-Level Systems^ 63With these functions, the probability can be rewritten as1 _ 1 i1 i i ,1- 121 ) g2 (1 i 101 1 ^1 4 gi 1- g2 ) i A A + gi + g2= A^1gi ,1lul,,,- 1 [gi (1 + 11' 1 ) + g2 (1 + —h2 ^ (3.63)4^gi^g2 ^+ A(gi + g2 ) + gig2 (1 -^)gi..g21 gig2 11^1/ 1 11 2 1 1 ^1 2^{^god A A 2 + A(gi + g2) + go2(1 - Vt) •3.2.2 p__The calculation of p__ is essentially similar to what was just done. However, a majordifference comes in the contribution of the coupled blips. Since the last sojourns of thepaths now have a charge -1, their contribution is changed tor-2 [ (1- + f+)3 (g- + g+) k - (f- - f+)3 (g- - g-F) kwhere again j = k for paths of class-1 and -2, k = j +1 for paths of class-3 and j = k -I- 1for class-4 paths. Also, since the two paths must at least have one blip in order to finishin the state {-, -}, the factors 1 +pi + p2 are now absent. Apart from this, everything issimilar. The contribution of the decoupled and partially coupled blips is the same, as arethe various combinatorial factors. The principles involved in summing the configurationsare also similar, and we getP - - (A) — Pi.12) + P12) + P(132) +P12) (3.65)(3.64)andP132) (A) = f- ^12A [l - (f- + f+)] [ 1 - (g- + g+)] [Pi(A)(g- + g+) - P2 (g_ - g+ )] . (3.67)Chapter 3. Interacting Two-Level Systems^ 64(2) and (4)p i2 an  p i2 are then obtained by interchanging f and g. After collecting these terms,we obtain1^1^f_ + g_p__ (A) =- A 2A 1 — (f_ + f+ ) — (g_ + g+ )1 ^(f + g_)/2 — f_g+ — f+g_A [1 — (f_ + f+ )][1 — (g_ + g+ )[— (g_ — g+ )(f_ — g+ )and with the use of the definitions 3.63 this becomes(3.68)1 I (, . hLi_. )^+ g2 ( 1 + I)1 1 ^1 i igi V 1-^I^gi / g2 ) i A A + gi + g2+^14 l_gli ( 1 i-r hi ) +gi^.,,,_.2(1 1+ _h211LI A 2^A(gi^Li22i^(3.69)g2^+^+ g2) + glg2(1 — 9192111^1111121 1 ^1 29192 I. 9192 i A A 2 + A(gi + g2 ) + g1g2 (1 — V-2-. ig, ) •Notice that p__ is similar to p++ but that the two last terms are now positive instead ofnegative.3.2.3 p+_ and P-+The paths of^and p_+ are mixed; one of the path must finish in a state {+, +} whilethe other must terminate in {—, —}. Starting with p+_, it is easy to realise that thefactors 1 and /3 1 are absent but that p 2 now has to be included. Furthermore, since thereis now a factor (-1)n 1 +n2+ 1 coming from the jumps, a minus sign will appear in all thesummations. Also, the contributions coming from the coupled blips will be different fromone class to the other. For class-1 and -3 paths, the last blip must be of type-1. Thislast blip thus "sees" a type-2 sojourn of charge —1 and their contribution will be similarto those for p__.1class-1^—2 { (f_ + f+ )j (g_ + g+ — (f_ — f_0 3 (g_ — g+ )3 I^(3.70)Chapter 3. Interacting Two-Level Systems^ 65r, class-3^[ (1_^f + )j (g^g + )3 + 1 ( f ^f+)' (g^g )j +1 I (3.71)However, paths of class-2 and -4 finish with a type-2 blips that is overlapped by a type-1sojourn of charge +1, exactly the same as for p++ .1class-2 -> 2-1class-4 --* - [ (f_2 L[ (f_^f+ ) 3 (g_ + g+ ) 3 + (f- - f+) 3 (g- - 9+) 3If+ ) 3 + 1 (g_^g+ ) 3 - (f_ - f+ )j+1 (g_ - g+ ) 3 ](3.72)(3.73)The complete summation is then readily performed and givesP+-(A)^=^i[gi (1 + -) + g2 (1 +91 g2(3.74)A A +^+ g21 h l[gi (1 +^- 92 (1 + —h24^ .92 )11A 2 +A(gi+ 92) +9'02 (1 - 9192 )Then, in the same manner,p-+(A)^[gi (1 +) + g2 (1 + h1)1111(3.75)92^AA+gl+g2_1 rgi ( 1^g2^+ )14 [^91 )^g2 J A 2 + A(gi + 92) + 9192 (1 -^)9192It can be checked that the sum of the four probabilities is equal to 1/A, that is, 1 aftertaking the inverse Laplace transform as should be expected !3.3 Analysis of the Results and Limits of ValidityThe various expressions for the probabilities are valid only in the overdamped limit.Otherwise, the blips cannot be considered to be dilute and the noninteracting-blip ap-proximation fails. The domains of validity as well as some limiting results will now beexamined for the three different spectral densities. However, first of all, it is importantto keep in mind that one of the basic assumption of the whole analysis is that only theh2 1Chapter 3. Interacting Two-Level Systems^ 66lowest four levels are occupied, which result in the condition hw, >> kT. This pointwill be important below. Now, in the overdamped limit, all the functions g and h areindependent of Agi(A) = gi(0) h 1 ()) = h 1 (0)g2(A) = 92(0) h2 (A) = h 2 (0)and the explicit dependence on h 1 and h 2 can be eliminated by usingh 1 (0)^h 1 (0) = tanh (-)Th. (0)^gi(0)^2kT(3.76)(3.77)The Laplace inversion is then trivial, resulting in a sum of decaying exponentialsp++ (t) = Ci + C2e-^+ Coe -R2t^(3.78)(ri+r2)t + Ge_RitP--(t) = Cl + C2 e-^(ri-Fr2)t - Ge-Rit - a4e-R2t^(3.79)^p+- (t) = C2 [ 1 - e-(ri +r2)t] + C5 [e -Rit + e -R2t ]^(3.80)G[e -Rit - 6 -R2 t] , (3.81)p-+(t) = C2[1 — e -(r l+r 2)1 —where, 1' 1 = gi (0), 1' 2 = g2 (0),1^1R1 = -2 (P 1 + r 2 ) + -2\FR1^1R2 = -2 (1' 1 + r 2 ) - -2/TIR = il + 11 - 2r 1 r 2 0 - 2 tanh 2 (e/2kT)(3.82)(3.83)(3.84)1C1 = -4[1 - tanh(e/2kT)]^ (3.85)1C2 = -4[1 + tanh(e/2kT)] (3.86)Chapter 3. Interacting Two-Level Systems^ 67^1  R2^1 Fl + F2 C3 =^+ [1 + tanh(q2kT)]^(3.87)2 VT? 4 -VT?^., 1 R 1^1 ri + F2 C4^ [1 + tanh(E/2kT)]^(3.88)2 A 4 Arl?1 1'1V—R- r2 ^C5 = 4^[1 + tanh(q2kT)] ,^ (3.89)with gi (0) and g2 (0) given in section 2.3 for the ohmic and superohmic cases.3.3.1 Ohmic DissipationIn the ohmic case, the decay rates are given byF(r(2a 1 )= 2wA 2 r &si7rkT1 2°4-1 cosh(q2kT) ir(ai + iE/271- kT)1 2^, ^c F2 = 2A2 [2rIcT1 2a2-1 cosPh((2ect/22k)T) IF(a2 iq27rkT)I2^2w,^hu.),and the bias term is(3.90)(3.91)^tanh e/2kT = — tanh Vaia2 hwc/kT^ (3.92)If al , a2 > 1 both together, then the noninteracting-blip approximation is valid forall temperature and bias. However, it is clear that the bias term is equal totanh E/2kT = —1^ (3.93)since hw,^kT and this results in the system being totally localised in its starting wellfor all times of interest :p++ (t) = +1p+_(t) = p_ + (t) = p__(t) = 0.^ (3.94)Clearly, this behaviour will persist unless the bias is considerably reduced. The onlyway to do this is to have one or both of the coefficients a much smaller than one. InChapter 3. Interacting Two-Level Systems^ 68this case, the complete decaying behaviour is observed and the system relaxes to theequilibrium positions1p++ (t -> oo) p__ (t oo) = -4[1 + tanh Jaia2hwc/kT] (3.95)p+-(t^oo) p_+ (t^oo) = 4-1[1 - tanh Vai cr2 hcoc /kT].^(3.96)As the value of the bias tends toward zero, the probabilities of occupying the differentstates becomes equal, but for an intermediate value of e/2kT , the states +} and{-, -} are much more likely to be populated. The four probabilities are plotted inFigure 3.7 for the (totally arbitrary) values F 1 = 0.5, F2 = 0.7 and various values of thebias.The domains of validity of the approximation in the ohmic case with a < 1 areA>> hA^— < Va ia2 at low-Twc-1^-1T al^, a2 A2 at low bias(3.97)(3.98)Since the bias is itself a function of the a's, these conditions requires some quite smallvalues for the tunneling matrix element A 1 and 02. However, as the temperature isincreased, the first condition is not as critical and allows the use of a wider range for thematrix elements.3.3.2 Sup erohmic DissipationIn the superohmic case, the decay rates are given byFi = [hA i 2q(11 J(7h)2h(3.99)r2 LA2 1 qg2 j(0)2h(3.100)Chapter 3. Interacting Two-Level Systems^ 69and the bias term istanh e/2kT = — tanh [ji3is#2s ( 4,7c )s-i kT^(3.101)where 3. 1 and 02 are the renormalised tunneling splitting as given in 2.68 and the form ofJ(w) is given in 2.66. The noninteracting-blip approximation is valid at all temperaturesas long as(3.102)Again, unless the coupling are quite small, the system will be localised. If they are, itrelaxes exponentially and the analysis for the ohmic case goes through as is.3.3.3 Sub ohmic DissipationFor subohmic dissipation, the bias diverges to —oo no matter what is the strength of thecoupling from the system to then environment. Therefore, the system is always localised.The effect of the environment is much more drastic in this case. In a sense, this couldhave been expected, since already for the two-well problem, all the coherent behaviouris destroy in a subohmic environment [2].In any case, for all types of spectral densities, the coherent behaviour that the systemwas displaying when it wasn't coupled to the environment is completely wiped out. Evenin the ohmic case with a < 1/2 or with superohmic dissipation for s > 2, where atwo-level system displays coherence, it is gone..........................0.80.60.90.2Chapter 3. Interacting Two-Level Systems^ 70bias^3.005^10^15^20^25^30^35^40^45^50time (arbitrary units)bias = 1.500.60.90.2-0.200.805^10^15^20^25^30^35time (arbitrary units)4 0^45^50Figure 3.7: Probabilities for Various Values of the Bias (Bias a --el2kT)Chapter 3. Interacting Two-Level Systems^ 71Figure 3.7 (continued)bias = 0.75 10.80.60.40.20'0^5^10^15^20^25^30time (arbitrary units)bias^0.5010.80.60.90.200^2^4^6^8^10^12^14^16^18^20time (arbitrary units)108 973^4^5^6time (arbitrary units)218 9 100^1^2^3^4^5^6^7time (arbitrary units)Chapter 3. Interacting Two-Level SystemsFigure 3.7 (end)bias = 0.25bias = 0.0572-0.2010. 80.60.40.2010.80.60.40.20P++ —P-+-----------^■ ..•• ...................................^............^.......^T.........................................^........^.......... 7:77 .... 1:77,:vrrr .........................................Chapter 4Four-Well SystemThis chapter will be concerned with the time evolution of a system composed of fourpotential wells interacting with its environment. The spatial configuration of the wellsis such that they are symmetrical with respect to the x- and y-axis. Their minima arelocated at {+xo/2, -1-y0 /2}, {-1-x0 /2, —yo/2}, {—x 0/2, —yo /2} and {—x 0/2, +yo /2} andthey all have the same minimum energy. The system part of the total Lagrangian is thus1Lys = 2M(x 2 + 2 ) — V(; y) (4.1)The environment is still represented by a bath of bosons and is coupled linearly to thex- and y-coordinate by the constant C,,,. Everything is thus quite similar to the previouschapter and will in fact be treated in the same way. However, a crucial difference appearsin the counterterms. In the case of the two two-level systems, each potential had to berenormalized separately, bringing two different counterterms, but in the present case, thepotential has to be renormalised as a whole, that is, we must useCa2(I) = — [X (T ) + Y(7 )1 2 E 2m co 2 .aThe square in this term gives rise to crossed terms that will cancel the coupling betweenthe x- and y-trajectories produced by the environment.The reduced density matrix of the system is:XXI., Y1) x2, Y2) t) = f dx dy dx' dy' po(x,y , x', y') J(xi, y i , x 2 , y2 , t; x, y, x', V, 0)^(4.3)(4.2)73Chapter 4. Four-Well System^ 74wheres(t)=x,Xi, Yi, X^iv(t)=yi^jxi(-0=^iyi(t).y12:7 Y i, °)J(xi, yi, x2, Y2, t;^,^Dx^Dy^Dx'^Dy'./x(o), i^.., (0=x:^y, (0)=y:Y(0)=Yix exp[—hi (S„,[x, y] — Ssys [x', OAF [x, y, x', y'] . (4.4)Assuming that the system is initially in the pure statepo = 8(x — x0/2)8(y — yo/2)8(x' — 4/2)8(y 1 — y'0 /2)^(4.5)then, the density matrix is simply given by the quadruple path integralp(x i , yl , x 2 , y2, = J(x i , yi , x 2 , y2 , t; x0 /2, yo /2,4/2, yo72)^(4.6)and is actually fairly easy to obtain.In the tight-binding limit, we assign the tunneling matrix element A to transitionsbetween adjacent wells and use the site representation denoting the state of the systemby the well in which it is. That is, we introduce the vectors 11), 12), 13) and 14) to de-scribe the states {+x o/2, +yo/2}, 1+x0/2, —y o /2}, {—x0/2, —yo /2} and {—x o/2, +y0 /2}respectively. For simplicity, let us take four completely symmetric wells. We thus setxo = yo and Ax =- Ay.The influence functional can now be found for this particular system. Of course , itwill be totally similar to the one found in Chapter 3 for the two two-level systems. Weconsider a double path integration, labeled by {x, x'} and {y, y'}, each of these pathsbeing carried over the four possible combinations {+, +}, {+, —1, {—, +} and —1,called A 1 (2), Bi ( 2), Ci ( 2 ) and D 1 (2 ) respectively, the subscript 1 (2) referring to the {x, x'}({y, y'}) path.The parametrisations 3.10 to 3.13 can be used directly provided that q l and q2 arereplaced by x and y respectively. For clarity, the functions ,2, Xi and X 2 can beChapter 4. Four-Well System^ 75renamed to m , y, x and xv . Furthermore, since there is only one coupling constant, wehave11 ' ) = LV ) , 1, 1) = 42)^(4.7)and L3 and L4 differ from these by the presence of xoyo instead of x02 or yo2 . Of course,this apply also to the functions Q.With these modifications, the results of Section 3.1 for the possible blip configurationscan be used and it seems that we are again in the exact same case of having a bias createdby the other path. This however, would be neglecting the counterterms, whose crossedterms are not zero. This particular part isexp i E 2m^ 0 dr [ 2x(r)y(r) — 2x'(r)y'(r)]acvaE oit dr [G(T)xv(T) + Gh^ (T)xs(r)]=^J (4.8)where e is again given by Equation 3.32. Now, this term is nonzero when a sojourn and ablip of different type overlap and exactly cancel the bias term introduced by the influencefunctional. There is no bias introduced by the environment, as it should be.The elimination of the bias is a major simplification but we are still left with theinteraction among the different blips. The noninteracting-blip approximation allows toneglect it for blips of the same type, but it cannot account for the overlapping blips.The different summations can again only be performed in the limit of high-temperatureand/or strong dissipation. In this regime, we only keep the self energy of the blips, andagain the number of possible arrangements of blips is given by Equation 3.38, but sincethere is no bias for this case, the combinatorial factor can be directly implemented in thesummation; there is no need to resort to the chains of coupled blips.Due to the absence of bias, the blips will be dilute enough only for ohmic and subohmicdissipation. The superohmic case cannot be treated accurately. Again, only the ohmicChapter 4. Four-Well System^ 76i j end states i j end states1 1 A 1 ,A2 3 1 C1 ,C21 2 A1,B2 3 2 C1,D21 3 B1,B2 3 3 D1,D21 4 B1 ,A2 3 4 D 1 ,C22 1 A1 ,C2 4 1 C1 ,A22 2 A1 ,D2 4 2 C1 ,B22 3 B1 ,D 2 4 3 D 1 ,B22 4 B1 ,C2 4 4 D i ,A2Table 4.1: Ending States for the Matrix Elements p iicase will be analysed in detail.4.1 Density MatrixThe elements of the density matrix are given by Equation 4.3. Each path starts at timet = 0 from states A l and A2 and must be in the state specified by x 1 ,x 2 and y 1 ,y2 at timet. For each matrix element pi3 , the ending states are shown on table 4.1.It is already known that a path made of n blips starting from and returning to A is,after summation over the charges,TransitionsA,A -21 f_, rA2n H e_Q2(t2,—t2,-) cos Q1(t2i - i2j-1).j.=1(4.9)For a path starting in A and ending at D in n blips, the summation over the charges issimilar, but the factor (-1) will be different. It comes_(_i)n-F1A2n H e _ch(t2,—t2,_ ) cos Q 1 (t2i - t2i_ 1 ) .1TransitionsA,D^ 1-2 j=1(4.10)Things are somewhat different for paths ending in a state B or C. First, there will be anodd number of transitions, so that a factor i will be left in front. Then, the path is endingon a blip, not a sojourn as previously and the last blip has t as one of its boundary. TheChapter 4. Four-Well System^ 77system can also jump from A to B or C in only one transition. In this case, there is nosummation on the charges since the charge 770 is fixed to 1 and the charge C i must beeither +1 or —1 according to whether the end state is B or C. If the path must end inB, the contribution is :Single TransitionA,B = —i—A exp[—Q2(t — t1) iQi(t — t1)] (4.11)while the other possibility givesSingle Transition = +J.—A2 exp[—Q 2 (t — t1) — iQi(t — ti)] • (4.12)This particularity disappears as soon as there is more than one transitions because thecharge of the preceding sojourn is now allowed to have the values +1. Defining a "com-plete blip " as a blip sandwiched between two sojourns, then the contribution of a pathending to B or C and built of n such blips is :n-I-1TransitionsA,B,c = +i-2^)nA2n H e - (22 (t2 -t2,7 -1 cos Q1(t2j t2j-1)j=1with t2n+2 = t and the — (+) sign being for a path ending in B (C).(4.13)Now that the different contributions are known, all the summations can be done.Once again, these are done by taking the Laplace transforms of the various terms. Wewill define f (A) as g(\) in Equation 2.45 with € = 0and u(\) asf (A) = J dt e -At-c)2(0 cos Q1(t)0 (4.14)U( A ) =^dt e- A t -Q2 (t)+iQi(t )0^(4.15)Chapter 4. Four-Well System^ 784.1.1 Diagonal ElementsThe paths leading to the expression for the diagonal matrix elements either stop at A orD. The summation for pll is then :P11(A) = j + 1-- E ( - 1 r 1 ° 2711ni=1^[f( AA) ] ni + 2 1A cE9.„1[A2 ;A: n1d-n2A 1 )n2 A2n2 [f (A) i1 n2.0+ 1 Ea° t° (__ 1 ri+n2nj =1 n2=1( n i + n 2 )n2(4.16)The term 1/A comes from the path without any blip, the second and third terms representthe contribution of the paths where only one of the systems tunnel and the last term ismade of paths composed of the two types of blip. Knowing that [54]00 00 n i + nz )^1 a 1 1 b^1^E 2_,^ani bn2 = 1 1^+ (4.17)—^— 1—(a+ b) 'ni =1 n2=1 n2the summation is straightforward and givesP11 = 141 { 1  + A + :2f(A) + A -I- 2A 2 f (A)] •^(4.18)It is to be noticed that P11 is exactly equal to Equation 3.64 in the limit e = 0 andAi =^as  should be expectedFor p33 , the two paths arrive in a state D. Since there must be a least one blip to getin such a state, the first three terms in Equation 4.16 are not present. The summationis given by the fourth term alone, with (-1)nl+n2 replaced by (-1)Thl+n 2 +2 . The result isthen1 [1 ^2^1P33 =^ (4.19)4 A A + A2f(A) + A + 2A2 f(A) .For the two other diagonal elements, there is a mixed situation. One of the path endsup in A while the other goes in D. Therefore, the summation isP22 = yjt1 t° (_on2A2n2 r f ( (AT—^" [  n2=1(n 1 + n2 1A2 f(A)rn21.^Anz(4.20)1 00 coE (_1)ni i-n2 +14A En1=1 n2=1Chapter 4. Four-Well System^ 79From table 4.1, clearly p44 is equal to p22 . The only modification needed is interchangingn 1 and n 2 . Thus1 [1 ^1 P22 = P44 = 4 A A + 2A 2f(A)](4.21)This completes the calculation of the diagonal elements of the density matrix. It canreadily be checked that their sum is equal to 1, and it is quite natural that p 22 = p44since the wells 2 and 4 are symmetric with respect to the well 1, the starting well.4.1.2 Off-Diagonal ElementsThe diagonal matrix elements are useful to determine the mean position of the systemor the average value of every diagonal operators but, for non-diagonal operators, the fulldensity matrix is needed. The calculations leading to the off-diagonal elements are similarin spirit to those for the diagonal ones, the only difference being that some of the end-states are now B or C. In particular, from table 4.1, it is seen that four of these elementshave their paths ending in a blip state simultaneously. Since such a configuration iseliminated by assumption, we readily get^P13 = P31 = P24 = p42 = 0 .^ (4.22)Next, for p 12 , the paths must end in A l and B2. The structure of the summation is thenP12 =^iA u(A) z.A  Af(A) ct° ( 1r2 I A2f(A) 1 n22 ^2 ^A in2 =1i u(A) "^[A2(A)1 ni^A 7,1=1 (A^f(A) o0 "nl + nz^rA2f(A)1711-En2^(4.23)4 A E E ( 1)ni+n2^{ An1=1 n2=1^n2Chapter 4. Four-Well System^ 80The first term comes from a single transition along the x-path (i.e., involving the x-coordinate) but no transitions on the y-path. The second term is similar, but representspaths with more than one transition along the x-path. The third term involves only onetransition from the x-path but many from the y-path and the last term is due to pathswhere both systems tunnel After summing, we get:P12^=_ : AZru(A) — f(A) i A u(A)4 I^A 4 A A l f (A)i A ^f(t) 4 A + 2A 2f(A)(4.24)Furthermore, since p 14 corresponds to the end-states B 1 and A2, it is exactly equal to p 12with n 1 and n2 interchanged. Also, taking into account that for a density matrix p = p+,we haveP12 = P21 P14 — P4*1 • (4.25)Next, we can calculate p 23 , whose paths finish in the states B 1 and D2. The structure ofthe summation is similar to 4.23 but it has to be taken into account that the y-path nowends at D, not at A. The final result is :ru(A) — f(A) O u(A) — 2f(A)P23^= 4 I_^A^_I 4^A+ A l f (A)o^f (A)(4.26)+2 4 A + 2A 2 f(A)The matrix element p 34 has its paths ending at the states D 1 and C2 and will thus bethe complex conjugate of p23 . Therefore,P32^n*P23 = v  = P34 = P43 • (4.27)Chapter 4. Four-Well System^ 81This ends the calculation of the density matrix. It can be written as:P11^P12^0^P12p(\) =Pit^P22^P23^0 (4.28)P23^P33^P23PI2^0^P23^P22with pil , P22 P33 p12 and p23 given by Equations 4.18, 4.21, 4.19, 4.24 and 4.26 respec-tively.4.1.3 Laplace Inversion and AnalysisThe Laplace inversions can now be performed. Since the summations were done in thelimit or high temperature and/or strong dissipation, we can use the results of section 2.2and put f(A) and u(.\) equal to their value at A = 0.e+iirar 7r^2a-1^r(a)u(A)^u(0) =  ^ (4.29)cos Ira 2wc [Awc ^r(1/2 + a)with f (0) = Re u(0) as is clear from Equation 4.14.Using these functions, the matrix elements are composed of sums of decaying expo-nentials. We obtain1P11^74 [1^2e-rt^e-21 (4.30)1P22^4 — e-21 (4.31)P33^4 [1 — 2e-rt^e-21for the diagonal elements , and(4.32)\/p 2a 1^r(a) tan^i(e-rt e-21) (4.33)' 12^8^we ,ahc,o,[^IF(1/2 + a) [(1^e -rt )^Ira —Y = 9'22X = go2+1 0 0 00 -1 0 0.^(4.36)0 0 -1 00 0 0 +1+ 1^0^0^00^+1^0^00^0^-1^00^0^0^-1Chapter 4. Four-Well System^ 82.v7r A I 7r 1 2a-1^r(a)6-2n)]P12 -^ )^ (1 + e -rt ) tan erai(e -rt8 co, 113hcoc i^F(1/2 +^a)(4.34)for the off-diagonal ones, with the decay rate r = 0 2f(0).A few remarks can be made about these expressions. First, notice that the dependenceof these functions on A and co, is always of the form A/co,. It can then be eliminated byusing the renormalised tunneling splitting O r defined in B.4 :(4.35)The cutoff frequency is thus totally absent from the results. Then, the diagonal matrixelements are perfectly well behaved: they sum to one and at t = 0, pu. = 1 with all theother pig being zero, corresponding to the initial condition. As t —> cc, all the diagonalelements go to a value of 1/4 exponentially, indicating that there is an equal probabilityof finding the system in any well. This is exemplified by a calculation of the expectationvalue of the positions (x) and (y) whose operator matrix representation isSince the average value of an operator is (A(t))^Trp(t)A and both X and Y arediagonal, we readily get(x ) = (y ) = 2e-r, (4.37)The system relaxes exponentially to the value (x)^(y) = 0 at infinite time, whichcorresponds to the 1/4 probabilities found above.Next, it can be checked that the expressions for p++ ,p+-,p__ and p_+ found inChapter 3 for the two two-level systems reduce to the diagonal elements in the limitChapter 4. Four-Well System^ 830, r i F2 . This could have been expected since it is only the bias that differentiatesthese two systems. As well as providing a confirmation of the results, it also gives a veryeasy way to find the diagonal elements of an asymmetric four-well system just by letting—4 0 and keeping F 1 F2.Now, if the diagonal elements are well behaved, the same cannot be said of the off-diagonal ones. First, the basic structure of their Laplace expression makes it impossibleto make an expansion in powers of A, since terms of the form Ay with v > 1, whoseLaplace inversion is not defined, would appear. Also, both p 12 and p23 are not equal tozero at t = 0. This particularity however is due to the approximation of replacing u(\)and f(A) by their value at A = 0. The exact Laplace inversion is well behaved as thetime goes to zero and the expressions should give the correct behaviour of the systemprovided that one is interested in the long time limit (such that t >>A lot more worrying is the fact that for a = 1/2, 3/2, 5/2•, the real part of both p 12and p23 diverges due to the tan 7ra. This unphysical result comes from the path with onlyone transition, which brings the factor u(\). However, some work by Ovchinnikov [55, 56]might provide a clue about what is going on. By using a perturbation treatment of thetwo well problem, he found that at a = 1/2, the ground state of the system shifts suddenlybut that the two energy levels E+ and E_ also follow this shift so that A = E+ — E_ isstill regular. Since it is the splitting A that determines the values of the probability forfinding the system in a given well (i.e., the diagonal elements), it is normal that nothingshows up there. However, the average energy of the system is given by (E) = TrpHand is in terms of the off-diagonal matrix elements. It can then be expected that theshift in the ground state is reflected in the off-diagonal elements. It should be noticedthat the same kind of behaviour also appears for a two level-system interacting with anexternal field [57]. Also, it is possible that at a = 1/2, the environmental oscillatorsbehave in such a way that it is inconsistent to use the limit wct kT. Again, the workChapter 4. Four-Well System^ 84of Ovchinnikov might give some support to this idea.In any cases, the appearance of the divergence is quite serious and indicates thata detailed analysis of the off-diagonal elements for both the two-well and the four-wellproblem is required. This is however outside the scope of this master thesis and for now,the case a = 1/2 must be considered as problematic.4.2 Infinite Chain of WellsThe results for the diagonal elements of the density matrix can be rederived by consideringan infinite chain of wells with periodic boundary conditions. Recently, Weiss et. al. [8, 58]have solved the problem of a particle tunneling in this kind of system but without anyboundary. This section is thus a slight generalisation of their results.Let us consider an infinite chain of wells, with spacing a between their minima locatedat the positions q = na, —oo < n < oo, all at the same energy set to zero. Weiss et.al. were interested in the time evolution of the expectation value for the position of aparticle put into the system inside the well q = 0 at t = 0. To this end, it is convenientto calculate the generating function00z(s, 0 . (esq) , E esna pon(t)^ (4.38)n--cowhere Pon (t) is the probability that the particle will be in the n th well at time t. Themoments of q are then obtained simply by differentiating this function with respect to s.°N 0 1s-0 .(qN(t ) ) = osN z(s, (4.39)4.2.1 Mapping to the Four WellsNow, the mapping to the four-well system without any diagonal transitions can be done.The well at position q = 0 is clearly equivalent to the well {+, +} of the four-well system.Chapter 4. Four-Well System^ 85Infinite ChainNotation -4 -3 -2 -1 0 1 2 3 4Four-WellNotation 1 2 3 4 1 2 3 4 1Figure 4.1: Mapping to the Four WellsBut then, all the wells at positions q = 4ma with —oo < m < oo are also identical to it,since going from q = 0 to q = 4m is like making round trips around the ring made upof the four wells. In the same spirit, all the positions q = (4m + 1)a are equivalent to{+, —}, q = (4m + 2)a correspond to the well {—, —} and all the positions q = (4m + 3)aare identical to {—, +}, as is shown on Figure 4.1.Then, the probability that the particle will be in a given well j of the four-well systemat time t is equal to the sum of the probabilities of finding it in an equivalent well of theinfinite chain, that is:00^00pjj = E E Pon (5n,4m -i-j —1m=—oo n=—oo(4.40)where m is a winding number indicating how many turns around the ring formed by thefour wells are made by the particle. For example, for j = 1, we getP11 = • • • P0-12 + P0-8 + P0-4 + Poo + PO3 + PO8 + P012 • • •^(4.41)At this point, the discrete representation of a Kronecker delta function for the four-wellsystem can be used :bnm = \-"" c if) (n—m)4 {0,}(4.42)Chapter 4. Four-Well System^ 86with {60 = 0,7r/2,71,37r/2. p33 then becomes00^ 00pii =4^E eiBj(1-J^E eiejn Pon (t)m=-00^ n=—oo(4.43)The summation over m will bring in the delta function 8(40 j ) but since it is summedover 0j , it gives one. The summation over n then corresponds exactly to the definitionof the generating function. As a result, p3j can be rewritten as=1^eie2(1—i)Z(i0jia,t).^(4.44)fti).If an expression for the generating function can be found, the mapping to the correspond-ing four-well probabilities is then very easily accomplished.4.2.2 Calculation of the Generating FunctionThe Lagrangian used for the infinite chain of wells is the same as the one described inChapter 1 with the difference that the potential is periodic over the complete range ofpositions : V(q) = V(q na). Again, the counterterm has to be included to cancel theshift in the potential brought in by the coupling to the environment. Since the infinitechain of wells is a 1-dimensional system, the probability Pon of finding the particle in thenth well at time t is given by the double path integralqt ^)—naPon(t) --= f^Dq^Dq A[q]Alq1F[q,q1q ( ( ) ) —na=0^qi((0)=0(4.45)with A[q] being the action of a single jump and F[q, q'] the influence functional, whoseexpression is given by Equation 2.4. The setting is thus quite similar to that for thetwo-well system but the paths are quite different. Here, they start at q(0) = q'(0) = 0and end at q(t) = q'(t) = na after performing 2m transitions, each transition takingplace at a time t3 , as is shown on fig 4.3. At t = 0, the system is at q = q' = 0 andstays these until t = t i where a transition takes it to a neighbouring site. It then staysChapter 4. Four-Well System^ 87q'rn=18n=4tiet ti 5I t,isqFigure 4.2: Path in the {q(t), q'(t)} Spacethere until the next jump at t 2 and so on until the last transition at t = t 2, brings itto q = q' = na. In order to reduce the double path integration into a single one, it isconvenient to introduce the "relative" and "centre of mass" coordinates of the paths,y = q — q' and x = (q q')/2. x(r) and y(T) are then the analogue of the functions 4 . (T)and x(r) introduced in the two-level problem. With respect to these new variables, theboundary conditions are now x(0) = y(0) = 0, y(t) = 0 and x(t) na. To parametrisethe path, we introduce the two charges (3 and 7/3 , of value ±1 and write2max(r) ^E 8(T — ti )2 j=12my(T) = a E^— to .k=1(4.46)(4.47)Furthermore, to respect the boundary conditions, the following constraints on the chargeshave to be introduced2m^2mE7/; 2n , E ci o (4.48)j=1^j=1The constraint on (3 insures that q(t) = q'(t) while the one on 7/3 makes sure that after2m jumps, q(t) = na. All the different ways to get to q = na in 2m jumps are takenChapter 4. Four-Well System^ 88(1 (2 (3 (4+ 1 + 1 - 1 - 1+1 -1 +1 -1+ 1 - 1 + 1- 1 + 1 + 1 - 1- 1 + 1 - 1 + 1-1 -1 +1 +1Table 4.2: Values of the Charges (j for n 1, m = 27)1 712 7/3 714+ 1 + 1 + 1 —1+1 +1 —1 +1+1 —1 +1 +1—1 +1 +1 +1Table 4.3: Values of the Charges qj for n 1, m = 2into account by summing over all the possible set of values for ( 2 and 773 respecting theconstraints. For example, if n 1 and m = 2, the possible values of the charges arelisted in table 4.2 and 4.3, showing that there is a total of 6 x 4 = 24 different trajectoriesfor these values of m and n.In a general case, there will be2m^2mrn n(4.49)possible trajectories.Now, using the parametrisations 4.46 and 4.47 as such in the influence functionalwould be somewhat cumbersome since the values of x and y at a time tj are given bythe sum of the charges of all the preceding transitions, not by a single value ( 2 or yk asin the two-level case. This renders the formulas quite heavy to manipulate and it willChapter 4. Four-Well System^ 89actually be simpler to integrate by parts the exponent of the influence functional to haveit in terms of 4 and q'. Then, the parametrised trajectories are expressed as a sum ofdelta functions and the time integrations become trivial.Let us first consider the L2 part of F[q, q'] expressed as— f: dc o J (w) fot dr fo r ds [q(r)q(s) — q(r)qi (s) — q' (r)q(s) + q' (r)q'(s)](^1cos W \ T—S j .(4.50)As an example, the integration by parts of the term q(r)q' (s) givesq" ) f t dr x(r) sin cor — —1--q(t) 1 ds q' (s) cos co(t — s) -I- 2 0ft ds q' (s)co 0^ W2^0^ Wt^7.—ji^dr./ dS 4(7)4 1 (8) [1 — COS W(T — .5)].^ (4.51)L4.)^13^0The last term in the equation above will obviously bring in a term Q 2 (r — s) multipliedby derivatives of the trajectories. Next, since q(0) = q'(0) and q(t) = q' (t), the threeother terms will cancel when all the terms in 4.50 are integrated by parts. All that is leftis—a12 fo t dT fo 1- ds [q(r) — Cr)] [4(s) — 4'(s)] Q2(7- — s) .^(4.52)The same procedure is then applied to the L 1 part of the influence functional. Afterintegrating and summing the four different terms, it becomesJ0' dr for ds [4(r) — 4 1 ( 7 )] Ws) + 4'(s)] Q 1 (7 — s)+ 7TZ., 10°° d(A) j 4.0(W) jot dr {q2 (r) — (q) 2 (T)] .^(4.53)The last term, which resembles a bias term is then cancelled by the counterterms of theoriginal Lagrangian, whose contribution to the action islo t dr { q2 (r) — (0 2 (r)] E 2rriC coo, = rih Joy du) J(w) fo l dr [q 2 (r) — (q / ) 2 (r)](4.54)Am (s) = 227. E (- 1)m—n esna E Kmn=—m(4.58)Chapter 4. Four-Well System^ 90i.e. , just the opposite of the preceding term ! Once again, the importance of thecounterterms is shown. So, finally, the complete influence functional can be written asF[q,^= exp jot dr fo 'r ds {-F2 i (r) (s) Qi(r — s) + ("7") (s) Q2(7- — s)}/a2^(4.55)and the factor A[q] is simply i0/2. Everything after is then similar to what was donefor the two-level system. The probabilities Pon can be expressed asexpexp(-)2m of aft 2702m j-1[E E^Q2(t.; — tolJ=2i=12m-1^2mZ E^E ck j Qi(tkj=1 k=j+1E Hm Km^(4.56){C.i,7u}t j)]^ (4.57)whereCOPon(t) Em=1.1HmKmand where contracted notation to denote the integration over the transition centres indefined in Equation 2.21. Notice that the summation starts at m = due to thefact that the paths needs at least n jumps to get at the position q = na, which alsoexplains the factor (-1) 7'. This expression looks simpler than what was obtained forthe two-level system but the summation over the charges is not as straightforward asthen because of the constraints imposed on the paths. Nevertheless, it is still possible towork out some type of approximationsNow, the expression for the probabilities can be placed inside the definition of thegenerating function to givez(s,^E A 2m I D{t2m } E Hm Am (s)m=o^{C,}n=mChapter 4. Four-Well System^ 91where the summations over n and m have been inverted to obtain the last result :00^00^oo n=m(4.59)n=-00 m=lnl m=0 n=—mThe summation over the /I's can now be performed exactly. Again, an explicit derivationhas not been found, but the answer can be found by induction. After summation, A mbecomes2m-1Am (s)^sinh(sa/2) 11 sinh(sa/2 iRi , 2m )^(4.60)j=1with R given by2mRj,2m = i ckQi(tk —tJ ).^ (4.61)k=j+1The formal expression for the generating functional has been found, but again, it istoo complicated to be summed exactly. Instead, a modified version of the noninteracting-blip approximation can be put forward. As before, the (negative) L2 part of the influencefunctional, proportional to y(r)y(s), controls the weight of a given path. If a path stayson the main diagonal, where y(T) = 0, the term L2 do not contribute to the path. Next,if it is on one of the main off diagonal lines (to be referred to as tridiagonal from nowon), the L2 is multiplied by a term of the order 0(a 2 ). However, if it goes outside thesediagonals, the factor is of the order O(p 2 a2 ) where p in an integer greater than 1. Inthe regime of high-temperature and/or strong dissipation, the weight of this type ofpaths should be reduced enough to allow their neglect. The only paths that need to beconsidered are those staying inside the two tridiagonals, as shown in Figure 4.3.During each interval of time t 2i — t2i_ 1 , j running from 1 to m, the path has to bein one of the tridiagonal state while it must be on the main diagonal at the other times.Let us then define a "dipole" as the time interval for which the path is off of the maindiagonal and the value of y(r) is ±1. Clearly, this is the analogue of the blips as definedin Chapter 2 and the times when the path is on the main diagonal are equivalent to theFinal positionq(t).(v(t)IfTridiagonals---11. qChapter 4. Four-Well System^ 92q 'Figure 4.3: Restrained Pathssojourns. Now, in order to restrain the paths, the charges ( 2; and Gi_i of a given dipolemust be opposite (i.e., (+, —) or (—, +)) or else the path will wander off the tridiagonals.Furthermore, since y(r) = 0 on the main diagonal, the length of a sojourn will be muchgreater than the length of a blip and the same argument that allowed us to keep only theself-energy of the blips in Chapter 2 can be used. Under this approximation, we obtainmi mAm = [2 sinh 2 (sa/2)] ri cos Q i (t2i — t2j_. 1 )j=1mHm^exp(—Ch(t2Jj=1and the generating function becomes(4.62)(4.63)z(s,i) = E [20 2 sinh 2 (sa/2)] m rt  D{t2m }^f(t2; - t23_1)0nt=0^ j=1with f(t) being the self-energy of a dipole.f (t) = e-Q2(t) cos Q1(t) .^ (4.65)Again, the summation can be performed by taking first the Laplace transform of all(4.64)Chapter 4. Four-Well System^ 93the terms. The summation is then straightforward and gives.C{Z(s, t)} =1 (4.66)A — 20 2 sinh(sa/2)f(A) •Finally, replacing s by i02 /a and performing the summation leading to the four-wellprobabilities p3i, one finds exactly the same expressions 4.18, 4.21 and 4.19 that werefound before. The Laplace inversion is then totally identical and the final results will begiven by sums of decaying exponentials.Not only does this confirm of the previous results, but another interpretation of theapproximation used can also be gained. In Section 4.1, the basic assumption was that noblips of different types would overlap. If this were to be the case, the value of the functionG would have values of +2 or 0. But now, is the equivalent of y(r) definedin 4.47. Therefore, restraining the path to lies between the two tridiagonals is exactlyanalogous to neglecting the paths where the blips will overlap. Once again, however, thisis valid only if the dissipative environment has a strong effect on the dynamics of thesystem.Chapter 5ConclusionSome effects coming from the environment on the dynamics of a four-well system havebeen examined. Within the tight-binding limit, only the ground state of each well isconsidered and the problem is truncated to a four-level one. Two types of system wereanalysed: two two-level systems coupled together through the environmental bath and asymmetric ring of four potential wells. In both cases, only transitions between nearest-neighbours wells were kept, diagonal transitions having a much smaller amplitude thannon-diagonal ones.The calculations were done using path integrations and, within this technique, it wasfound that the only solvable regime is for high temperature and/or strong dissipation,where the blips of different paths do not overlap with each others.For the two coupled two-level systems, it was found that the environment introducesa time-dependent bias in each well. If the dissipation is subohmic, the effect of thiseffective bias is such as to localise the system in its starting well no matter how weak thecoupling to the environment is. In the cases of ohmic and superohmic dissipation, thesystem relaxes exponentially to an equilibrium state. For the solvable regimes mentionnedabove, the probabilities of occupying a given well are then :P++ = P-- = Cl (5.1)P+- = P-+ = C2 (5. 2)with C1 and C2 given by Equations 3.85 and 3.86. Since C 1 > C2, the equality being94Chapter 5. Conclusion^ 95realised as e 0, the environment tends to align the two separate systems together,which is reminiscent of the correlated phase mentioned in Chapter 3. However, the rateat which it relaxes depends crucially on the bias e which is itself a function of al and a2 ,the dissipation parameters. Unless these are very small (Va ia2 kT/hc.,),), the systemwill remain in its starting well for all practical times. Even there, the system will be morelikely to be found in the {+, +} or { —} wells. The environment tends to correlate thetwo separate two-level systems.The problem of the bias does not appear for the ring of four wells. The analysisis then enormously simplified and it becomes possible to obtain an expression for thecomplete density matrix. It is then found that the diagonal elements are exactly equalto the probabilities of the two two-level systems in the limit e —> 0, as they should be.There is an equal probability of 1/4 of occupying any well. Furthermore, these resultscan be rederived by considering an infinite chain of periodic wells to which we imposeboundary conditions. This latter technique is extremely useful since it can readily begeneralised to any number of symmetric wells just by taking the appropriate form ofthe Kronecker 8-function. The drawback of this method, however, is that it cannot giveany information about the off-diagonal elements of the density matrix. These can onlybe obtained by looking at the original four-level problem. This has been done, but anunexpected divergence at a = 1/2, 3/2, 5/2 occurs. Some tentative explanations havebeen proposed but a much more detailed analysis is clearly required. It should be noticedthat this divergence is also present in the two-level problem; remarkably, the off-diagonalelements have never previously been thoroughly examined, even for this problem.In connection with this problem, an analysis of the four-level system with very weakcoupling by perturbation theory should be done. It seems that this is the only wayby which some information regarding the underdamped coherence of the system will begained. This was not necessary for the two-level problem since the noninteracting-blipChapter 5. Conclusion^ 96approximation was able to handle this case. But it cannot here due to the overlappingof different blip types. By perturbation theory, it might also be possible to obtain someunderstanding of the divergence in the off-diagonal matrix elements, although a = 1/2cannot really be considered as weak coupling. In any cases, lots remain to be done onthis type of systems interacting with a dissipative environment.Appendix ACalculation of the tunneling matrix elementWe are here interested in calculating the tunneling matrix element D o . In the case ofa double well, it is given by the energy splitting E+ — E_. For simplicity, the simpleLagrangian of a particle of mass M moving in a potential well V(x) will be used, but theprocedure is similar for more complex systems.To find A, it is most convenient to switch to imaginary time, ie, make the formalsubstitution it T where for this appendix, T will be referred to as the imaginary time.For a transition between the positions q 0/2 and —q0/2 between the (imaginary) times—T/2 and +T/2, the propagator takes the formK[go12,7112;.--go12, —T12] _ (2 le —HT/h, — 2o )=^Dq e -s-E [g] in^(A.1)where SE [q] is the euclidean action,77272SE [q] =^dr [Z,,q` + V (q)] .^ (A.2)—7 It differs from the real time action by the presence of the plus sign in front of thepotential. The motion described by this action is thus that of a particle moving in areversed potential U(x) = —V(x).Now, assuming a set of wavefunctions such as H1n) = En In) to exists and introducingit into the propagator, we getK[qo/ 2 , T/2; —q0 /2, —T/2] = E e —EnT in 0:(qo 12) On ( —q0/2).^(A.3)n97Appendix A. Calculation of the tunneling matrix element^ 98with On (q) being the corresponding wavefunction. By letting the time interval T goto infinity, only the ground state of the system contributes effectively to the sum. Fora transition between two wells, however the ground state of each well is split by thetunneling and these two levels must be included in the sum. The result is:K[qo/2, T/2; —q0 /2, —T/2] = C sinh^ (A.4)with C a constant. Therefore, if the path integral can be performed explicitly, a compar-ison gives the value of the matrix element [31].In the WKB-limit, only the least-action path is considered. It is given by the equation:d2 q^01/(0M dr2 = +Oq(A.5)with boundary conditions q(+T /2) = +qo/2 when T oo. Integrating A.5 with respectto time gives an equation for 41—2 Mq2 — V(q) = E (A.6)E , the energy, being the constant of integration. However, since the tunneling processis translationally time invariant, E is a constant and must be equal to zero to keep theaction finite in the limit T oo. Thus,q2V (q) M •(A.7)The motion described by this equation is called an instanton [60, 61]. At T = —T/2, theparticle is at —q0/2 and slowly starts to pick-up momentum. It crosses q = 0 at timeT = 71 and the momentum decreases until it finally come to rest at +q o /2 at T = +T/2. Inthe semiclassical approximation, the particle stays at —q 0/2 for a very long time, switchesvery rapidly to +qo /2 and stays there for a very long time also. Of course, a similar typeof path can start at +q0/2 and end at —q0/2. It will be called an anti-instanton.Appendix A. Calculation of the tunneling matrix element^ 99The instantons and anti-instantons are the building blocks of the propagator. A pathwill be a string a successive instantons and anti-instantons. It is made of 2n + 1 jumps,n 1 of them being instantons and the remaining n being anti-instantons. These twotypes of jump both have the same classical action, given by4°1 2^I^ScL = - V2MV(q)dqq0/2(A.8)and it will be assumed that the action of a group of n jumps is nSci, which is valid as longas the jumps form a dilute gas. Also, small gaussian fluctuations around the classicaltrajectory must be accounted for. This is done by the inclusion of a factor An in theweight of the path. Then, the amplitude of having a jump between T and T dT isA exp[—ScL/Mdr and an integration has to be done over all the instanton centres. Therequirement that the jumps have to follow each other is then included in the limits ofintegrations. The summation for the propagator is thusT/2K(+qo /2, T/2, —q0 /2, —T/2) = limT—oo n=o LT/2D{t2n+i [Ae—Sci 2n+1 (A.9)where a symbolic notation for the integration over the centers has been introduced,TI2^TI2^ t22D{tn } =^dtn^dtn_i • • • I^dt i .^(A.10)J-T/ -T/2^-T/2^-T/2Since the weight of the path is independent of time in this particular example, the timeintegration is trivial, being equal to T 2n+ 1 /(2n +I)! and the propagator is then the seriesof an hyperbolic sine.K(+qo/2, T/2, —q0 /2, —T/2) = C sinh A ie-saih .^(A.11)A direct comparison with A.4 gives immediatelyA =^ (A.12)Appendix A. Calculation of the tunneling matrix element^ 100where A' is the ration of two determinants and is of the order of the oscillation frequency ofthe particle in a well (the attempt frequency). Its exact form isn't particularly interestingsince the dissipation effect is mostly seen through the exponential of the classical action[3].Appendix BLaplace Inversion of P(A)The Laplace inversion of P(\) is essential if one wants to know the behavior of thesystem. However, even the relatively simple form 2.58 proves to be intractable withoutusing some approximation. In particular, it is not very interesting to know how thesystem evolves at times near t 0. The times t >> coc-1 are much more important. Interms of the Laplace transform, this means that an expansion around .A/w, << 1 can betaken. Furthermore, some approximations are already built into the Laplace transform.These are also of much help.The Laplace inversion will be taken for the ohmic and the superohmic cases for bothzero and non-zero temperature. Since the biased system has already been examined, thisappendix will be concentrating on symmetric wells.B.1 Ohmic Dissipation, € = 0For symmetric wells, P(\) takes the simple form :1P(A) = A + ,2g(A) (B.1)By using the definitions 2.35 and 2.36 for Q 1 (t) and Q 2 (t), and writing the time inde-pendent parts in complex notation, the function g(A) is :^I ^2a^1^1 2ag(A)^dA eAt 27 '[2 sinh yt] Re [1 — iwc t ](B.2)101Appendix B. Laplace Inversion of P(\)^ 102where the dissipation parameter a ----- riq0/27rh has been brought in and y 71- kT /h. Next,the transformation z = At + iA/co, is introduced and the expansion of g(A) is taken. Thefirst order term is, placing A 2 explicitly inside g(A) (GR 3.541.1)A2 ( 27 ) 2a-1^F(1 — 2a)F(a + A/27) g(A) = — ^cos(ar)We we F(1 — a + A/2-y)^(B.3)In anticipation of further results, it is convenient to redefine the tunneling matrix elementby grouping all the temperature-independent terms togetherso that g(A) becomesA, -=- A (A/coc)Aeff  -= [F(1 — 2a) cos(ra)]2(1 1-.) Pr2-y ) 2a-1  F(a + A/27) g(A) = Aeff (Aeff^F(1 — a + A/27)(B.4)(B.5)(B.6)Note that g(\) now depends explicitly on only three parameters, the tunneling element,the temperature and the dissipation. In particular, the dependence on co c , the arbitrarycutoff frequency has been absorbed into A,. Remembering that A is itself a functionof the cutoff frequency (c.f. section 1.3), being proportional to w c', it is clear that therenormalised tunneling splitting is independent of we as it should be.B.1.1 Line a = 1/2, Arbitrary TIn the limit a = 1/2, g(A) = Aeff and is completely independent of .A . The inversion isthen trivial, yielding2P(t) = e—Aefft = exp —^t—coc7r co[where the last form results from the explicit expression for A eff in the limit a -4 1/2(B.7)Appendix B. Laplace Inversion of P(\)^ 103B.1.2 High Temperature and/or Strong DissipationNext, for high temperature, a Taylor expansion of the gamma functions can be taken,resulting in :h g(A) = T -1 [1 2kTA cote7ral^ (B.8)withT_i __ OF A' OrKT) 2o( -1 F(a) (B.9)2 co,^hco, i^F(a + 1/2)The Laplace inversion then follows directly :P(t) =h  1[1 2kTrexp [—t/[7- — hcot(ira)/2kT]] .^(B.10)The appearance of the cot(7ra) might seem strange at first, but it is a result of theTaylor expansion F(x) = F(0)(1 + x0(0) + • • •) where 0(x) , the digamma function is thederivative of the gamma function and obeys the identity 0(1 — z)— 0(z) = 7r cot(7rz). sim-ilarly, the term cos(ira) disappear due to the identity F(1/2+ x)F(1/2 — x) = 7r/ cos(7rx).Of course, this expression for P(t) is valid only as long as the first term of the expansionis much smaller than one. This imposes the restriction on the temperature1 kT^A i ^F(ah)^^] 2(1-a)OT Ltan(7ra)F(a + 1/2)^. (B.11)In particular, for a = 1/2, the tangent goes to infinity, this restriction is satisfied for alltemperature and the resulting expression is equal to the previous one, as expected.Appendix B. Laplace Inversion of P(A)^ 104B.1.3 T = 0At zero temperature, the gamma functions must be expanded for large y by the Stirlingformula, yieldingF(a A/27)^( ) 2°-1(B.12)F(1 — a + A/27)^2y )g(\)^Ae2(flf-a)A2a-1^ (B.13)1 PO) = (B.14)A + A2 ( 1 —a)ef fThis Laplace inversion is not tabulated, as the others are and must be done byconsidering the complex plane.First, for a < 1/2, 2a — 1 < 0 and P(A) must be rewritten asA1-2ceP(A) = )2(1—a)^Ae.(flf—a) • (B.15)This function has poles at Ap = A eff exp[+ir/2(1 — a)]. They are located in the secondand third quadrant and symmetric with respect to the real axis. this function also hasa branch cut, which can be taken to be along the negative real axis. The contour ofintegration will go around the branch cut and the pole in order to have no singularitiesinside it.The segments just above and below the branch cut, where A = re i' and A = re -27respectively, don't cancel each other completely . They contribute an incoherent part toP(t) :Piny (t)^sin(2ra) fcc dz ^e-zyz1-2a7r^o^z4(1 -a)^2z2 ( 1 -a) cos(27ra) + 1(B.16)where y Aefft and z^A/Aeff . This integral is not very interesting since it doesnot have any coherent part, and its explicit expression isn't very informative.Appendix B. Laplace Inversion of P(A)^ 105Next, the four segments joining the cut and the poles cancel entirely each other.Therefore, the only other contribution comes from the contour around the two poles,which gives a coherent part to P(t). Adding them gives :ii- aPcoh (t) =1[^y sin2exp1^a 1 — aIx cosr^71"y cosa1 — a 1. (B.17)2Next, for a > 1/2, P(a) still has the two poles and the branch cut, but now, the polesare past the branch cut and not anymore on the principal sheet. Therefore, the contourof integration does not go around them. As a result, only the part along the cut gives acontribution, which will be exactly equivalent to the incoherent part above.sin(27ra) f.°^e-zy z2a-1P(t) =  ^dz ^(B.18)ir^Jo^z2 + 2z 2" cos(21-a) + z4a-2 .Finally, for a > 1, the first term of the expansion of g(\), proportional to A 2 °' -1 is nolonger the dominant term. When going to the next term, it is found that it is of order0(A(0/Lo c ) 2 ) but due to the condition A/co, << 1, this term has to be neglected, yieldingg(A) = 0. In this case, then, clearlyP(t) = 1^ (B.19)and the system is localised in its starting well.B.2 Superohmic Dissipation, c = 0For the superohmic case, it is convenient to work only with Q 1 (t) and Q3 (t) by absorbingthe time-independent part of Q 2 (t) into a renormalised tunneling amplitudeAppendix B. Laplace Inversion of P(A)^ 106A =-_ A exp { q°  j c° dw Jw(w2 ) 127h o^I. (B.20)In the remaining part of g(A), both Q 1 (t) and Q3 (t) go to zero as t goes to infinity.This indicates the presence of a pole at A = 0, which can formally be taken out in frontg(A) = —1 + f(A)oo^a20)f(A) = I dte-At [cos (-Ch, Qi (t) ea ch (i) — 1]rand the complete function P(\) takes the formA PO) = A2 + 3,- 2 [AAA) + 1 ] .The poles of this expression are given by:(B.21)(B.22)(B.23)A, = +iA[1 + of (x`)]1/2^ (B.24)Now, in the absence of any dissipation, f (A) = 0 and P(A) has two poles, located at+i0. Introducing dissipation should give to these poles a negative real value, such thatP(t) acquires a decaying part.Remembering that we are working in first order in b, the expansion of eq (2.81) around±iA results in the following expression for the polesA / = ±A^ (B.25)3, 2AR -7= —7Re f(±i3,) —I',^ (B.26)and P(t) has now the simple formP(t) = C I's' cos(At) + 1.=‘A sin(3,‘ t) ,^ (B.27)but,Fs^A ) (s-1)— As e-(8-1)As ( 1='wc (B.34)Appendix B. Laplace Inversion of P(\)^ 107where, from the definition of f(A),3,2 fooF s = —2 o dt cos(3,t)f(t) . (B.28)In first order in b, f (A) can be expanded to give q4Q3 (t) / 7rh, and then, using thedefinition of Q3, inverting the order of integration and remembering that10 °° dt cos(At) cos(wt) = 2[8(3, + w) + S(A — co)]^(B.29)we obtain4h J(A) •^ (B.30)Finally, the requirement that b is small has to be checked. First, notice that fort 0 -1 large, both1 sin wt^1 cos wt^ and ^R^CO^T CObehave as a 6-function, thus yielding2ach(t=3,-1),10_2Q3 (t=3._1 ) _, 4 JO') —brh^h 3, —Using the previous result, this all comes down to(B.31)(B.32)(B.33)Since s > 1 and A/co, << 1, this is satisfied for all values of A s , the coupling to theenvironment. Furthermore, by Equation B.33, P(t) can be rewritten as:P(t)^cos(At).^ (B.35)Appendix B. Laplace Inversion of P(\)^ 108At non-zero temperatures, for s > 2, only minor modifications are needed. P(t) isstill given by Equation B.35 but the renormalised tunneling element as well as the decayrate are now temperature dependent. r du.,A(0^ql,), A exp [27h^--J(w) coth(Ohw/ 2)1o wII', = Ih J[A(0)] coth[f3h0(0)/2]4(B.36)(B.37)For 1 < s < 2, the complete form B.1 has to be used. However, if an expansion ofg()) in powers of A such that9(A) = .go + A.gi + • • • ; 91 «1^ (B.38)is possible, the system will relax exponentially to zero with a rate g o . 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