THERMAL EQUILIBRIUM FROM THE

arXiv:hep-th/9210034v1 7 Oct 1992THERMAL EQUILIBRIUM FROM THEHU-PAZ-ZHANG MASTER EQUATIONJ.R. Anglin†Physics Department, McGill University3600 University StreetMontr´eal, Qu´ebec CANADA H3A 2T8ABSTRACT: The exact master equation for a harmonic oscillator coupled to aheat bath, derived recently by Hu, Paz and Zhang, is simplified by taking the weak-coupling, late-time limit.

The unique time-independent solution to this simplifiedmaster equation is the canonical ensemble at the temperature of the bath.Thefrequency of the oscillator is effectively lowered by the interaction with the bath.† anglin@hep.physics.mcgill.ca

INTRODUCTIONThe evolution of the density matrix of a quantum harmonic oscillator linearlycoupled to a heat bath is a fundamental problem, but it has only recently beensolved exactly[1]. Physical quantum mechanical systems have often been idealizedas isolated, with the sole effect of the environment being the maintenance of a finitetemperature.

The non-trivial role of the environment in decohering the excitationsof a weakly-coupled system is currently still being explored; as a complement to thisresearch, this letter examines the ground state of a harmonic oscillator weakly coupledto a heat bath.If a system is in equilibrium with a thermal environment, to which it is weaklycoupled, it has long been assumed that the reduced density matrix for the systemis given by the canonical ensemble at the environmental temperature.Using thedensity matrix evolution equation (‘master equation’) derived in Reference [1], thisletter confirms this assumption in the case of the harmonic oscillator. The discussionassumes familiarity with the results of Reference [1] (hereafter denoted HPZ), whichwill not be derived here.THE HPZ MASTER EQUATION ATWEAK COUPLING AND LATE TIMEIn HPZ, the following master equation is derived, for the time evolution of thedensity matrix ρ(x, x′) of a simple harmonic oscillator, with the position variable x2

coupled linearly to a heat bath:i ∂∂tρ = −12h ∂2∂x2 −∂2∂x′2 −Ω20(x2 −x′2)iρ+ (x −x′)hA(t)( ∂∂x + ∂∂x′ ) + B(t)(x + x′)iρ−i(x −x′)hC(t)( ∂∂x −∂∂x′ ) + D(t)(x −x′)iρ . (1)Here ¯h and the oscillator mass have been set equal to 1.

The time-dependent realco-efficients defined in HPZ have been renamed A, B, C, and D. The terms in (1)proportional to A and D are responsible for diffusive effects in the evolution of ρ.B could be considered a time-dependent addition to the effective frequency of theoscillator, while C has the effect of a time-dependent dissipation constant.Equation (1) is derived using an initial quantum state that is a direct product ofthe initial oscillator and (thermal) environment states; the authors of HPZ suggestthat some features of their master equation may be artifacts of this actually ratherimplausible initial condition. In order to avoid the spurious time dependences intro-duced by the artificial assumption that the oscillator and heat bath are uncorrelatedat time t = 0, the co-efficients will be replaced by their asymptotic forms at latetimes: A(t) →A∞≡A(∞), etc.

Setting the RHS of (1) equal to zero, one obtainsan equation for the time-independent state into which the open harmonic oscillatormight be expected to settle down at late time. Equivalently, this equation can beconsidered to describe the ground state of a simple harmonic oscillator in the presenceof a thermal environment:0 = −12h ∂2∂x2 −∂2∂x′2 −Ω20(x2 −x′2)iρ+ (x −x′)hA∞( ∂∂x + ∂∂x′ ) + B∞(x + x′)iρ−i(x −x′)hC∞( ∂∂x −∂∂x′ ) + D∞(x −x′)iρ .

(2)3

The time-independent co-efficients have simple forms in the weak-coupling limit,where only terms of up to second order in the bath-oscillator coupling constant areretained[HPZ, eq. 2.46].

In the notation of the present letter,A∞=g2 1Ω0limt→∞Z t0dsZ Γ0dω I(ω) coth βω2 cos ωs sin Ω0sB∞= −g2 limt→∞Z t0dsZ Γ0dω I(ω) sin ωs cos Ω0sC∞=g2 1Ω0limt→∞Z t0dsZ Γ0dω I(ω) sin ωs sin Ω0sD∞=g2 limt→∞Z t0dsZ Γ0dω I(ω) coth βω2 cos ωs cos Ω0s . (3)I(ω) is the spectral density of the environmental heat bath, and g is the bath-oscillatorcoupling constant.

It is assumed that I(ω) is cut offat some high frequency Γ andvanishes at least linearly at zero frequency, so that, as long as the limit t →∞istaken last, the order of the s and ω integrals is arbitrary. The temperature of thebath is kT = β−1, where k is the Boltzman constant.In solving (2) to first order in g2, the co-efficient B∞will be split into two terms:B∞≡Ω0 tanh βΩ02 A∞+ Ω0δΩ.

(4)The first term is chosen to provide a cancellation shown below, while the second termis a renormalizing correction to the effective frequency of the oscillator, proportional tog2. (δΩwill be determined in the next section of this letter.) Define the renormalizedfrequency to be Ω≡Ω0 + δΩ.A standard representation of the delta function allows one to writeC∞= g2π2Ω0I(Ω0) ≃g2π2ΩI(Ω)D∞= g2π2coth βΩ02I(Ω0) ≃g2π2 coth βΩ2 I(Ω) .

(5)4

One can therefore re-write (2) in the final form0 = −12h ∂2∂x2 −∂2∂x′2 −Ω2(x2 −x′2)iρ+ (x −x′)A∞h( ∂∂x + ∂∂x′ ) + Ωtanh βΩ2 (x + x′)iρ−i(x −x′)g2π2ΩI(Ω)h( ∂∂x −∂∂x′ ) + Ωcoth βΩ2 (x −x′)iρ ,(6)keeping only terms of up to first order in g2. This is the weak-coupling, infinite-timelimit of the exact time-independent master equation of HPZ.Define the thermal density matrixρβ(x, x′) = (1 −e−βΩ)∞Xn=0e−nβΩψn(x)ψn(x′) ,(7)where ψn(x) is the wave function for the nth excited state of the harmonic oscillatorwith frequency Ω.It will now be verified that ρ = ρβ is a solution to (6).Theverification is straightforward, and proceeds line by line.The first line of (6) is simply [ ˆHx −ˆHx′]ρβ, for ˆH the harmonic oscillator Hamil-tonian in the position representation, and so equals zero.The second line is proportional to∞Xn=0e−nβΩhΩ1/2 sinh βΩ2 (x + x′) + Ω−1/2 cosh βΩ2 ( ∂∂x + ∂∂x′ )iψn(x)ψn(x′)= 1√2∞Xn=0e−nβΩheβΩ2 (ˆa + ˆa′) −e−βΩ2 (ˆa† + ˆa′†)iψn(x)ψn(x′)=eβΩ2√2∞Xn=0e−nβΩh√n ψn−1(x)ψn(x′) −e−βΩ√n + 1 ψn+1(x)ψn(x′) + [x ↔x′]i=e−βΩ2√2∞Xn=0e−nβΩ√n + 1hψn(x)ψn+1(x′) −ψn+1(x)ψn(x′) + [x ↔x′]i= 0 .

(8)5

Here ˆa =1√2Ω(Ωx + ∂∂x), ˆa† =1√2Ω(Ωx −∂∂x), the primes imply that x is replaced byx′, and ψn satisfiesˆaψn = √n ψn−1ˆa†ψn =√n + 1 ψn+1ψ−1 ≡0 .Similarly, the third line of (6) is proportional to∞Xn=0e−nβΩhΩ1/2 cosh βΩ2 (x −x′) + Ω−1/2 sinh βΩ2 ( ∂∂x −∂∂x′ )iψn(x)ψn(x′)= 1√2∞Xn=0e−nβΩheβΩ2 (ˆa −ˆa′) + e−βΩ2 (ˆa† −ˆa′†)iψn(x)ψn(x′)=eβΩ2√2∞Xn=0e−nβΩh√n ψn−1(x)ψn(x′) + e−βΩ√n + 1 ψn+1(x)ψn(x′) −[x ↔x′]i=e−βΩ2√2∞Xn=0e−nβΩ√n + 1hψn(x)ψn+1(x′) + ψn+1(x)ψn(x′) −[x ↔x′]i= 0 . (9)The canonical ensemble at temperature kT = β−1 is therefore a solution tothe late-time, weak coupling, time-independent master equation.

This equation isa hyperbolic partial differential equation in the two independent variables x and x′.Normalizability of ρ requires that ρ and its derivatives decay to zero for large |x| or|x′|. Normalizability therefore imposes Cauchy initial, final, and boundary conditionson the master equation, which in general over-determine the solution[2].

The solutionwhich has been found is therefore unique. (The terms “initial” and “final” are usedby analogy, since either x or x′ may be considered to play the role of the timelikevariable usually involved in hyperbolic equations.

)6

DETERMINATION OF δΩFrom equations (3) and (4) one can determine the frequency correction δΩ. Com-paring the definitions of C∞and D∞, one sees thatδΩ= −g2Ωlimt→∞Z Γ0dω I(ω) coth βω2 R(ω, Ω, t) ,(10)where terms of order g4 are neglected, and R is defined byR(ω, Ω, t) ≡tanh βω2Z t0ds sin ωs cos Ωs + tanh βΩ2Z t0ds cos ωs sin Ωs .

(11)R can be evaluated by expanding the hyperbolic tangents in Taylor series, and notingthat, for integer n > 0,Rn(ω, Ω, t) ≡ω2n+1Z t0ds sin ωs cos Ωs + Ω2n+1Z t0ds cos ωs sin Ωs=1ω2 −Ω2h(ω2(n+1) −Ω2(n+1))(1 −cos ωt cos Ω)−ωΩ(ω2n −Ω2n) sin ωt sin Ωti= ω2(n+1) −Ω2(n+1)ω2 −Ω2−nXm=0ω2mΩ2(n−m) cos ωt cos Ωt−ωΩn−1Xm=0ω2mΩ2(n−1−m) sin ωt sin Ωt . (12)One then hasδΩ= −g2Ωlimt→∞∞Xn=0Anβ22n+1 Z Γ0dω I(ω) coth βω2 Rn(ω, Ω, t) ,(13)where An are the co-efficients in the Taylor series for the hyperbolic tangent.The oscillating functions of t in the last two lines of (12) may in fact be ignoredin the limit t →∞: since I(ω) vanishes (at least) linearly at ω = 0, I(ω) coth βω27

may be expanded in a series of non-negative powers of ω, and explicit calculation willshow thatlimt→∞Z Γ0dx xmeitx = 0(14)for m ≥0.Ignoring the t-dependent part of Rn(ω, Ω, t), one therefore has, to leading orderin g2,δΩ= −g2ΩZ Γ0dω I(ω) coth βω2"ω tanh βω2 −Ωtanh βΩ2ω2 −Ω2#. (15)In the high temperature limit, this becomesδΩ|β→zero = −g2ΩZ Γ0dω I(ω)ω,(16)while in the low temperature limit it approachesδΩ|β→∞= −g2ΩZ Γ0dω I(ω)ω + Ω.

(17)CONCLUSIONIn recent years it has been argued that open quantum systems are generically de-cohered by the environment to which they are coupled, in such a way that the stateof the system is rapidly driven towards a mixture of eigenstates of the interactionHamiltonian[1,3,4,5]. This phenomenon is considered to occur on a short time scale;open quantum systems in equilibrium after a long period of time have received com-paratively little attention.

It is nevertheless of fundamental importance to confirmthat the equilibrium state of a system whose position variable is weakly coupled to a8

heat bath is indeed the canonical ensemble, and not a mixture of position eigenstates.This is true at all temperatures, and is independent of the spectral density of the heatbath, to leading order in the coupling g2.At higher order in g2, this result probably no longer holds. Heuristic argumentsjustifying the canonical ensemble typically assume weak coupling between the systemand its environment.

Presumably the equilibrium solution for ρ depends at higherorders in g2 on the specific form of the spectral density I(ω).The infinite-time limit of the master equation derived using uncorrelated initialstates at t = 0 may be conjectured to be equivalent to the master equation onewould obtain at finite times using more realistic initial conditions. The canonicalensemble is an explicit example of a state which does not (at leading order in thecoupling constant) suffer wave-function collapse onto the pointer basis determined bythe coupling to the environment.

This supports the suggestions by several previousresearchers that the role of the initial conditions in decoherence and environmentally-induced superselection needs further investigation.ACKNOWLEDGEMENTSThe author gratefully acknowledges valuable conversations with R.C. Myers.

Thisresearch was supported by NSERC of Canada, and by the Fonds pour la Formationde Chercheurs et l’Aide `a la Recherche du Qu´ebec.9

REFERENCES1.B.L. Hu, Juan Pablo Paz, and Yuhong Zhang, Phys.

Rev. D45, 2843 (1992).2.P.M.

Morse and H. Feshbach, Mathematical Methods of Theoretical Physics(McGraw-Hill; New York, 1953), pp. 683ff.3.W.H.

Zurek, Phys. Rev.

D26, 1862 (1982).4.A.O. Caldeira and A.J.

Leggett, Phys. Rev.

A31, 1059 (1985).5.W.G. Unruh and W.H.

Zurek, Phys. Rev.

D40, 1071 (1989).10

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