Uncertainty Principle Consequences at Thermal Equilibrium

Leonardo A. Pachon,1 Johan F. Triana,1 David Zueco,2, 3 and Paul Brumer4

1Grupo de Fısica Atomica y Molecular, Instituto de Fısica, Facultad de Ciencias Exactas y Naturales,Universidad de Antioquia UdeA; Calle 70 No. 52-21, Medellın, Colombia.

2Instituto de Ciencia de Materiales de Aragon y Departamento de Fısica de la Materia Condensada,CSIC-Universidad de Zaragoza, Zaragoza E-50012, Spain.

3Fundacion ARAID, Paseo Marıa Agustın 36, E-50004 Zaragoza, Spain.4Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum Control,

University of Toronto, Toronto, Canada M5S 3H6(Dated: January 8, 2014)

Contrary to the conventional wisdom that deviations from standard thermodynamics originatefrom the strong coupling to the bath, it is shown that these deviations are intimately linked to thepower spectrum of the thermal bath. Specifically, it is shown that the lower bound of the dispersionof the total energy of the system, imposed by the uncertainty principle, is dominated by the bathpower spectrum and therefore, quantum mechanics inhibits the system thermal-equilibrium-statefrom being described by the canonical Boltzmann’s distribution. This is in sharp contrast to theclassical case, for which the thermal equilibrium distribution of a system interacting via centralforces with pairwise-self-interacting environment, irrespective of the interaction strength, is shown tobe exactly characterized by the canonical Boltzmann distribution. As a consequence of this analysis,we define an effective coupling to the environment that depends on all energy scales in the systemand reservoir interaction. Sample computations in regimes predicted by this effective coupling aredemonstrated. For example, for the case of strong effective coupling, deviations from standardthermodynamics are present and, for the case of weak effective coupling, quantum features such asstationary entanglement are possible at high temperatures.

PACS numbers: 03.65.Yz, 05.70.Ln, 37.10.Jk

Introduction.—Thermodynamics was developed beforethe atomistic description of Nature was formulated. Sta-tistical mechanics was then introduced to understand thelaws of thermodynamics in terms of a microscopic de-scription, thus closing the gap between macroscopic andmicroscopic description. Due to the interest in quantumtechnologies, there is a major ongoing effort to develop aconsistent and well defined extension of thermodynamicsto the quantum regime [1–3]. However, the majority ofthese theories are primarily based on Boltzmann’s origi-nal ideas and are therefore plagued by issues concerningirreversibility, the origin of the second law, the relation be-tween physics and information, the meaning of ergodicity,etc. (see, e.g., Ref. [4]).

Despite these issues, it is now well known that, e.g.,Onsager’s regression hypothesis fails in the quantum realm[5, 6] and that non-Markovian dynamics are relevant in avariety of fields and applications, from foundations [2, 7],to nuclear physics [8], quantum metrology [9, 10] andbiological systems (see, e.g., [11] an references therein).It is also known that the thermal equilibrium state ofa quantum system strongly coupled to a thermal bathdeviates from the canonical Boltzmann distribution [1, 12–14], this is also expected to occur in the classical case [15].Since both are incoherent thermal stationary situations,one would expect that the quantum system is devoidof any coherence and hence, based on the decoherenceprogram [16], that both distributions should coincide.However, one might also suggest that the entanglement

between the system and the bath, which has no classicalcounterpart, could introduce quantum-classical deviations[1]. Furthermore, the fact that extra deviations could bepresent even if the entanglement between the systemand the bath is zero [17] makes the situation even moreintriguing.

Hence, it seems appropriate to find a situation wherethe classical and the quantum contributions to the de-viation from the Boltzmann distribution can be clearlyisolated and examined. Here we show that irrespectiveof the interaction strength, there are no deviations fromBoltzmann’s distribution when a classical system interactsvia central forces with a pairwise-self-interacting environ-ment. Thus, if after quantum-mechanically treating thesame case, deviations from the canonical Boltzmann’sdistribution are present, then they are purely quantumin nature. As shown below, deviations do appear and,based on completely general arguments, are shown to relyon the uncertainty principle characteristic of quantummechanics.

Therefore, the uncertainty principle not only inhibitsthe system’s thermal-equilibrium-state from being de-scribed by the canonical Boltzmann distribution, but foreach system-bath interaction it also selects which thermalequilibrium states are physically accessible. This latter re-mark, formulated here for the first time in the frameworkof quantum thermodynamics, constitutes the cornerstoneof the theory of pointer states (the states which are ro-bust against the presence of the environment) [16] and

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could have deep consequences for an understanding of thethermalization of quantum systems.

Classical Thermal-Equilibrium-State.—To provide awell-defined situation, consider a classical particle ofmass m with potential energy US(q) and HamiltonianHS(p, q) = 1

2mp2 + US(q). Now consider a collection of N

classical particles interacting via the central force poten-tial U i,j

B (qi−qj) and Hamiltonian HB(p, q) =∑N

j [ 12mj

p2j +∑i,j U

i,jB (qi − qj)], with which the classical system inter-

acts via the central force potential energy Vj(qj − q), sothat the total Hamiltonian is given by

H = HS(p, q) +HB(p, q) +

N∑j

Vj(qj − q). (1)

In classical statistical mechanics, the thermal equilibriumdistribution of the system S is defined by

ρS(p, q) =1

Z

∫ N∏j

dpjdqj exp [−H(p, q, pj, qj)β] , (2)

where Z =∫ ∏N

j dpjdqj∫

dpdq exp(−Hβ) denotes thepartition function of the total system, with β = 1/kBTand T being the temperature of the environment. Theintegral over pj in Eq. (2) trivially cancels out with thecorresponding contribution in Z. Due to the particulardependence of Vj and U i,j

B on qi, qj and q, the integralsover qj can be appropriately manipulated, with thenet result that they cancel out with the correspondingcontribution in Z. Thus,

ρS(p, q) = ZS−1 exp [−HS(p, q)β] , (3)

where ZS =∫ ∏N

j dpjdqj exp(−HSβ). Hence, the ther-mal equilibrium distribution of a bounded particle incontact with a pairwise-self-interacting thermal bath viacentral forces, irrespective of the coupling strength, is ex-actly given by the canonical Boltzmann distribution.

This result is surprising because, in the strong couplingregime, there is no apparent physical reason why theequilibrium thermodynamic properties of a system are in-dependent, for a wide class of systems, of both the natureof the bath to which it is coupled and of the functionalform of the observables that mediate the interaction. Thephysical picture that emerges from this result is that inthe long time regime, any dissipative mechanism is equallyeffective in taking the system to thermal equilibrium. Inother words, dissipative dynamics can contract the classi-cal phase-space volume with no fundamental restrictionand therefore, the resultant equilibrium state is indepen-dent of the dissipative coupling and the rate at whichequilibrium [Eq. (3)] is reached. This suggests that theconcept of intrinsic and extensive thermodynamic vari-ables [4] can be extended, in some cases, to the strongcoupling regime. By contrast, and as shown below, in the

quantum realm, dissipative mechanisms are accompaniedby decoherence effects and are bath-nature and coupling-particularities sensitive [16, 18] and are then capable ofinducing a variety different thermal states.

Quantum Thermal-Equilibrium-State—The quantumdescription follows from Eq. (1) by the standard quantiza-tion procedure. Based on the general description given in[3, 13, 14], one can easily extend the classical definitionin Eq. (2) to the quantum regime, namely,

ρS =1

ZtrB exp

−[H(p, q, pj, qj)

]β. (4)

The operator character of the various terms in Eq. (4) andtheir commutativity relations prevent us from proceedingas we did in the classical case. However, these very samecommutativity relations allow the immediate formulationof the following set of inequalities,

[HS, V ] 6= 0⇒ ∆HS∆V ≥ 1

2|〈[HS, V ]〉|, (5a)

[V , HB] 6= 0⇒ ∆V∆HB ≥1

2|〈[V , HB]〉|, (5b)

where V =∑N

j Vj(qj − q) and ∆O =

√〈O2〉 − 〈O〉2

denotes the standard deviation of O, with 〈O〉 = tr(Oρ),ρ being the thermal equilibrium state of the system S andthe bath B.

Some implications follow from Eqs. (5). Specifically,since |〈[HS, V ]〉| is a measure of the quantum correlationsbetween the system and the bath, it dictates the lowerbound of ∆HS∆V . This lower bound is different foreach interaction since each particular form of V imposesa different commutation relation. This last statementis precisely what allows, for example, for a connectionbetween the theory of pointer states and quantum thermo-dynamics. The general bounds in Eqs. (5) predict differentthermal equilibrium states for each type of interaction, apurely quantum effect.

For example, since [HS, V ] = 0 implies a pure de-cohering interaction, which can be treated here in theframework of fluctuations without dissipation [19], theequilibrium state is an incoherent mixture of system’seigenstates and is expected to be well characterized bythe canonical Boltzmann distribution [20]. In this case[HS, V ] = 0, so that ∆HS∆V ≥ 0, meaning that thecommutativity relation here results in the minimum lowerbound on ∆HS∆V . Note that the same lowest limit is ob-tained if, as in the classical case, the thermal equilibriumstate of the system ρS is formally the canonical Boltz-mann distribution ρcanS . Specifically, if ρ = ρcanS ⊗ trSρ,

then |〈[HS, V ]〉| = tr([HS, V ]ρ) = tr([ρ, HS]V ) = 0 since[ρcanS , HS] = 0, giving ∆HS∆V ≥ 0. This is just a con-sequence of the fact that the Boltzmann distribution isthe zero-order-in-the-coupling thermal equilibrium stateand therefore, disregards quantum correlations between

3

the system and the bath. Below, it shown that the bathspectrum is also related to the lower bound and thus,Eq. (5) will ALSO allow for a clear connection to otherfundamental features such as the failure of the Onsager’sregression hypothesis in the quantum regime [5, 6].

Influence of the Spectrum of the Bath: Non-MarkovianCharacter at Thermal Equilibrium—Although the set ofinequalities (5) are fairly general, it is not possible to in-fer the role that standard quantities such as the spectraldensity or the spectrum of the bath play in establishingthe thermodynamic bounds above. To provide a concreteexpression for the lower bound in Eq. (5a), the interaction

between the bath particles is set to zero, i.e., U i,jB = 0

and consider the second order picture of the system-bathcentral force interaction, i.e., V ≈

∑Nj

12mjω

2j (qj − q)2,

which yields to the well-known Ullersma-Caldeira-Leggettmodel [21, 22]. After expanding V , it is possible to rede-fine H in Eq. (1) as H = H ′S+H ′B+VSB with VSB = B⊗S.

Here B =∑N

j mjω2j qj and S = q, which act in the Hilbert

space of the bath and the system, respectively. The com-mutator [HS, V ], calculated to second order in VSBβ, isthen given by

|〈[HS, V ]〉| ∝ trS

[HS, S]e−HSβ

~β∫0

dσS(−iσ)K(σ)

,

(6)where ~K(σ) = 〈B(−iσ)B(0)〉B denotes the two-timecorrelation of the bath operators given by [23] K(σ) =π−1

∫dωJ(ω) cosh

(12~βω − iσ

)/ sinh

(12~βω

), J(ω) =

π∑∞

j12mjω

3j δ(ω − ωj) being the spectral density of the

bath. Note that as long as second order perturba-tion theory is valid, Eq. (6) holds for any S and Band can be straightforwardly generalized to the case ofVSB =

∑α Bα ⊗ Sα.

The main feature of the quantum thermodynamicbound in Eq. (6) is the presence of the power spectrumof the bath I(ω, T ) = ~J(ω) coth( 1

2~βω), which for bareOhmic dissipation, J(ω) = mγω, at high temperatures,~β → 0, is the flat I(ω, T ) ≈ 2mγkBT . Note that in thishigh temperature limit, the upper limit of the integral inEq. (6) vanishes, leading to the vanishing of the commu-tator, even if [HS, S] 6= 0. A similar series-expansion anal-ysis leads to the conclusion that the thermal equilibriumstate ρS formally approaches the canonical Boltzmanndistribution only when ~β → 0. In other words, in thehigh temperature limit the quantum correlations betweenthe bath and the system disappear and the thermal equi-librium state is described by the canonical distribution,irrespective of the coupling strength or the functionalform of the spectral density J(ω). It is clear that theseresults cast doubt on arguments in favor of canonicaltypicality in the quantum regime [24, 25] at other thathigh temperatures.

For out-of-equilibrium quantum dynamics, the low tem-perature condition, finite ~β, is associated with non-

Markovian dynamics [23, 26]. Since at fixed T , this non-Markovian character can be modified by the functionalform of the spectral density [27], Eq. (6) makes clearthat the equilibrium system properties depend on thenon-Markovian character. This means that the quantumequilibrium statistical properties of a system experienc-ing Markovian dynamics (flat spectrum) are expected todiffer from those of the same system experiencing non-Markovian dynamics (non-flat spectrum), which is insharp contrast to the classical case (see Eq. 3). This canbe clearly understood in terms of the different thermody-namic lower bounds resulting from either Markovian ornon-Markovian interactions [see Eq. (6)].

To make a connection with previous studies, the failureOnsager’s regression hypothesis in quantum mechanics[5, 6] is discussed next. In doing so, note that the hypoth-esis that knowing all mean values suffices to determine thequantum dynamics of the correlation functions is validonly under Markovian dynamics [5, 6] and when corre-lations between the bath and the system are negligibleat equilibrium (in general, at any time) [28]. Based onthe fact that formal Markovian dynamics can only beachieved for flat spectra (bare Ohmic spectral densitywith ~β → 0 [27]), these two conditions can be seen as asingle one when formulated in terms of Eq. (6). Specif-ically, Markovian dynamics imply |〈[HS, V ]〉| → 0 and,hence, the system-bath correlations vanish. This impliesthat Onsager’s regression hypothesis, as well as the Boltz-mann distribution, pertains exclusively to the classicalrealm.

To provide some insight into the magnitude and con-sequences of the fundamental limit derived above and,in particular, of the role of the spectral density, an effec-tive coupling to the bath is introduced below and sam-ple aspects of two complementary regimes are analyzed.Specifically, (i) effective strong coupling characterized bydeviations from standard thermodynamics, and (ii) aneffective weak coupling that is shown below to allow forthe survival of entanglement between two oscillators inthermal equilibrium at high temperatures.

Effective Coupling to the Bath.—For the Ullersma-Caldeira-Leggett model, a standard calculation [23], afterremoving a local contribution in the correlation func-tion, yields K(τ) = 2m

~βddτ

∑∞l=1 γ(|νl|) sin(|νl|τ), where

νl = lΩ, with Ω = 2π/~β, are the Matsubara frequenciesand γ(z) defines an effective coupling to the bath. Notethat γ(|νl|) contains all the information about the cor-relations of the bath operators and therefore defines theinfluence of the bath on the system at thermal equlibrium.

For the subsequent discussion we adopt the mostcommonly used spectral density, the regularized Drudemodel with a high frequency cutoff ωD, J(ω) =m0γω ω

2D/(ω2 + ω2

D

), where γ is the standard strength

coupling constant to the thermal bath and ωD dictatesthe degree of non-Markovian dynamics. For this par-ticular case, the effective coupling is given by γ(|νl|) =

4

FIG. 1. log(Z/Zcan) for a harmonic oscillator as a function of the ratios kBT/~ω0 and ωD/ω0. We compare the partition functionfor γ = 0.1ω0 (a), γ = 0.05ω0 (b), γ = 0.01ω0 (c) and γ = 0.005ω0 (d). We observe that (i) the stronger the damping rate γ, thestronger the deviation from the canonical partition function, (ii) the larger the cutoff frequency, the larger the deviation fromthe canonical partition function and (iii) in the high temperature regime Ω/ωD 1, regardless the damping rate, no deviationsare obtained.

γ/(1 + |l|Ω/ωD). Below we analyze the effective weakcoupling, Ω/ωD 1, and the effective strong coupling,Ω/ωD 1, regimes.

Strong Effective Coupling Regime—To quantify theconsequences of non-flat spectra in this regime, consider asthe system a harmonic oscillator of mass m0 and frequencyω0 coupled to a thermal bath [12, 13]. In particular, we areinterested in quantifying: (i) the generation of squeezingin the thermal equilibrium state, (ii) the deviation from

the canonical partition function Zcan = trSeHSβ and (iii)the deviation from the canonical von-Neumann entropyScan = trS [ρcanln(ρcan)].

For this case the momentum and position variancesare given by [12, 13] 〈p2〉 = m2

0ω20〈q2〉 + ∆, where

〈q2〉 = 〈q2cl〉 + 2m0β

∑∞n=1

[ω20 + ν2n + γ(|νn|)|νn|

]−1and

the squeezing parameter ∆ = −2m0γβ−1∂ lnZ ′/∂γ with

Z ′ = 1~βω

∏∞n=1 |νn|

[ω20 + ν2n + γ(|νn|)|νn|

]−1. We recall

that for this model, the classical theory predicts 〈p2cl〉 =m2

0ω20〈q2cl〉 and 〈q2cl〉 = kBT/m0ω

20 , so that ∆cl = 0.

For the effective weak coupling regime Ω/ωD 1,disgarding terms of the order ω0/ωD and γ/ωD givesπ~γmωD/6Ω [12]. Thus ∆ vanishes at high tempera-tures, and the classical unsqueezed state is recovered.However, for the strong coupling regime Ω/ωD 1,∆ ≈ ~γm ln (2πωD/Ω) [12], meaning that the deviationfrom the canonical state translates into squeezing of theequilibrium state. This feature may be of relevance to-ward the generation of non-classical states, e.g., in nano-mechanical resonators.

Deviations from the canonical result are also evident inthe partition function Z. Figure 1 shows the logarithmicof the ratio of Z to the canonical partition function Zcan

as a function of the dimensionless parameters kBT/~ω0

and ωD/ω0 for (from left to right) γ = 0.1ω0, γ = 0.05ω0,γ = 0.01ω0 and γ = 0.005ω0. Deviations are observed atlow temperatures and for high cutoff frequencies (i.e., inthe effective strong coupling regime). In the opposite limit,regardless of the coupling parameter γ, both calculatedpartition functions show the same behavior, as expected

from the discussion above. For the von Neumann entropyS = trS [ρSln(ρS)] , the behavior of the ratio log(S/Scan) isessentially the same as the one described for the partitionfunction ratio in Fig. 1, and is not shown here.

Since γ(z) = 1m0

∫∞0

dωπJ(ω)ω

2zω2+z2 , the use of different

spectral densities changes the functional form of the ef-fective coupling and therefore, of the thermal equilibriumproperties. Hence, as long as ~β remains finite, differentspectral densities lead to different thermal equilibriumstates.

Weak Effective Coupling Regime—As an examplein the weak effective coupling regime, consider thesurvival of entanglement at thermal equilibrium be-tween two identical harmonic oscillators with massesm0 and frequencies ω0 linearly coupled with couplingconstant c0. The Hamiltonian is given by H = HS +∑N,2

j,α

[p2j,α2mj

+mjω

2j

2 (qj,α − qα)2], with α = 1, 2 and

HS = 12m0

(p21 + p22) + 12m0ω

20(q21 + q22)− c0q1q2. The intro-

duction of independent baths for each oscillator ensuresthat no deviations from Boltzmann’s distribution arepresent in the classical case. This can be verified directlyfrom the multi-particle-system generalization of Eq. (1).

At equilibrium, the entanglement between the two har-monic oscillators can survive only when kBT/~ω0 1(see, e.g., Ref. [29]). However, this limit only applies inthe Markovian regime and γ → 0. Thus, based on thediscussion above, and supported by the recent observa-tion that non-Markovian dynamics assists entanglementin the longtime limit [9], we expect that this limit needsto be refined in order to account for the non-Markoviancharacter of the interaction and the finite value of γ.

Figure 2 shows the logarithmic negativity for differentvalues of the damping constant γ as a function of thedimensionless ratios kBT/~ω0 and ωD/ω0. As expected,(i) the more coupled the oscillators are, the higher thetemperatures and the damping rate at which entangle-ment can survive at equilibrium (not shown), and (ii) thesmaller the damping rate (the more isolated the systemis), the higher the temperature at which entanglement

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FIG. 2. Logarithmic Negativity in the presence of non-Markovian interactions for c0 = 0.1m0ω20 with γ = 0.1ω0 (a), γ = 0.05ω0

(b), γ = 0.01ω0 (c) and γ = 0.005ω0 (d) as a function of the dimensionless parameters kBT/~ω0 and ωD/ω0. We observethat (i) the stronger the coupling between the oscillators, c0, the higher the temperature at which the system can supportentanglement at equilibrium (not shown); (ii) the higher the damping rate γ, the stronger the influence of the non-Markovianinteraction depicted by the ratio ωD/ω0; and (iii) the smaller the ratio ωD/ω0 (a more non-Markovian interaction), the higherthe temperature at which the system can support entanglement at equilibrium.

can be maintained. The new feature here is that the morenon-Markovian the interaction, the higher the tempera-ture and the damping rate at which entanglement can bemaintained at equilibrium.

Discussion.—It has been shown that the fundamentalbound imposed by the Heisenberg uncertainty principle[Eq. (6)] prevents quantum systems to relax the Gibbsstate dictated by the system Hamiltonian only. TheGibbs state is only recovered in the classical-high T limit(~β → 0). The implications at low-T for quantum ther-modynamics are so crucial as the failure for the Onsagerhypothesis or the difficulty on the definitions of the specificheat [30] and the temperature definition [31]. Specifically,the high temperature regime defined by ~β → 0, modifiedby an appropriate effective coupling, emerges as the maincondition for the vanishing of those deviations. Althoughthe concrete examples presented here are particular forthe second order approximation of the interaction po-tential, the general picture provided here remains valid,albeit more involved, in non-linear cases and with non-Gaussian statistics. The results presented here clarifythe role of non-Markovian dynamics and its relevance atthermal equilibrium, and provide physical insights intohow non-Markovian interactions contribute to protectingquantum features such as entanglement. Moreover, theymay shed light on the role of non-Markovian dynamicsand their effects in the derivation of fundamental limitsin areas such as quantum metrology [10] and quantumspeed limits [7].

The result also serves to evaluate other measures ofquantum correlation. For example, the logarithmic nega-tivity used in Ref. [17] to quantify the system-bath entan-glement is found wanting. That is, it vanishes althoughthere are still deviations from the canonical state. Assuch this measure is unable to quantify the quantumcorrelations and that other kind of quantum correlationwitnesses, such as discord, are required.

Discussions with Profs. S. Mukamel and J. L. GracıaPalacios and Dr. M. Campisi are acknowledged with

pleasure. This work was supported by the Comite parael Desarrollo de la Investigacion –CODI– of Universidadde Antioquia, Colombia under contract number E01651and under the Estrategia de Sostenibilidad 2013-2014, bythe Departamento Administrativo de Ciencia, Tecnologıae Innovacion –COLCIENCIAS– of Colombia under thegrant number 111556934912, by NSERC and by the USAir Force Office of Scientific Research under contract num-ber FA9550-13-1-0005, by the Spanish MINECO projectFIS2011-25167, the Gobierno de Aragon (FENOL group)and the EU (PROMISCE).

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