Page 1

Quantum Crooks fluctuation theorem and quantum Jarzynski equality in the presence of a reservoir

H. T. Quan1and H. Dong2

1Theoretical Division, MS B213, Los Alamos National Laboratory, Los Alamos, NM, 87545, U.S.A.

2Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, P.R. China

We consider the quantum mechanical generalization of Crooks Fluctuation Theorem and Jarzynski Equality

for an open quantum system. The explicit expression for microscopic work for an arbitrary prescribed protocol

is obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum Jarzynski Equality and

their classical counterparts are clarified. Numerical simulations based on a two-level toy model are used to

demonstrate the validity of the quantum version of the two theorems beyond linear response theory regime.

PACS numbers: 05.70.Ln, 05.40.-a

I.INTRODUCTION:

Nonequilibrium thermodynamics has been an intriguing re-

search subject for more than one hundred years [1].

our understanding about nonequilibrium thermodynamic phe-

nomena, especially about those far-from-equilibrium regime

(beyond the linear response regime), remains very limited.

In the past fifteen years, there are several significant break-

throughs in this field, such as Evans-Searls Fluctuation Theo-

rem [2], Jarzynski Equality (JE) [3], and Crooks Fluctuation

Theorem (Crooks FT) [4]. These new theorems not only have

important applications in nanotechnology and biophysics,

such as extracting equilibrium information from nonequilib-

rium measurements, but also shed new light on some fun-

damental problems, such as improving our understanding of

how the thermodynamic reversibility arise from the underly-

ing time reversible dynamics.

Since the seminal work by Jarzynski and Crooks a dozen of

years ago, the studies of nonequilibrium thermodynamics in

small system attract numerous attention [5], and the validity

anduniversalityofthesetwotheoremsinclassicalsystemshas

been extensively studied not only by numerical studies [6], but

alsobyexperimentalexploration[7]insingleRNAmolecules,

and for both deterministic and stochastic processes. For quan-

tum systems, possible quantum extension of Crooks FT and

JE have also been reported [8]. Nevertheless, we notice that

almost all of these reports about quantum extension of Crooks

FT focus on isolated quantum systems [9], and the explicit

expression of microscopic work, and their distributions in the

presence of a heat bath are not extensively studied. In addi-

tion, the relationship between classical and quantum Crooks

FT is not addressed adequately so far. As a result, the experi-

mental studies of quantum Crooks FT and JE are not explored

(an exception is the experimental scheme of quantum JE of

isolated system based on trapped ions [10]).

In this paper, we will give a detailed proof of the validity of

quantumCrooksFTandquantumJEforanopenquantumsys-

tem based on the explicit expression of microscopic work and

their corresponding probability distributions for an arbitrary

prescribed controlling protocol. We also clarify the relation

between quantum Crooks FT, quantum JE and their classical

counterparts. In the last part of the paper, the studies based on

a two-level system are given as an illustration to demonstrate

our central idea.

Yet

FIG. 1: (Color Online) Trajectories of a quantum system in a

nonequilibrium process. Similar to Ref. [4] each step (from tn to

tn+1) is divided into two substeps: the controlling substep of time

τn

change with time, and the relaxation substep of time τn

the energy spectrum (black dashed line) remains unchanged. In the

controlling substep (solid line) work is done, but there is no heat

exchange; While in the relaxation substep, there is heat exchange

between the system and the heat bath, but there is no work done.

Bluetrajectorycorrespondstofastcontrollingprotocol, duringwhich

there are usually interstate excitations in the controlling substep. Red

trajectory corresponds to slow (quantum adiabatic) controlling proto-

col, and the system remains in its instantaneous eigenstate in the con-

trolling substep. Red trajectory is the counterpart of classical case.

Q, in which the energy spectrum (black solid line) of the system

Rin which

II. NOTATIONS AND ASSUMPTIONS:

Crooks FT [4] is firstly derived in classical systems in a mi-

croscopically reversible Markovian stochastic process. In the

proof of a classical Crooks FT, a key technique is to separate

work steps from heat steps. In the following discussion of

quantum extension of Crooks FT and JE, we will employ the

same technique as that used in Ref. [4] to separate the control-

ling process into two substeps: controlling substep and relax-

ation substep (see Fig. 1). The controlling substep proceeds

so quickly in comparison with the thermalization process of

the system that we can ignore the influence of the heat bath

during the controlling substep. So there is only work done in

arXiv:0812.4955v1 [cond-mat.stat-mech] 29 Dec 2008

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2

the controlling substep. In the relaxation substep, on the other

hand, there is only heat exchange.

Having clarified the main strategy (separating work substep

from heat substep), let us come to the details of the notations

and assumptions. We employ the same notations and assump-

tions as that in Ref. [4] to prove the quantum Crooks FT. In

Ref. [4] the author assumes discrete time and discrete phase

space. Here, the discrete energy spectrum in a quantum sys-

tem in place of the discrete phase space of a classical system

arises naturally. We also assume discrete time t0, t1, t2, t3,

···, tNfor the quantum system (see Fig. 1). The parameter

λ(t) is controlled according to an arbitrary prescribed proto-

col λ(t0) = λA, λ(t1) = λ1, λ(t2) = λ2, ···, λ(tN) = λB,

where A and B depict the initial and final points of the pro-

cess. Every step tn → tn+1 is seperated into controlling

substep of time τi

ti+1 = ti+ τi

and E(in,λm) to depict the in-th instantaneous eigenstate

and eigenenergy of the system Hamiltonian H(λm), we can

rewrite the trajectory A → B of Ref. [4] in the following way

|i0,λ0? → |i0,λ1?λ1

→ ··· → |iN−1,λN−1? → |iN−1,λN?λN

Qand relaxation substep of time time τi

Q+ τi

1).

R,

R(see Fig.If we use |in,λm?

− →|i1,λ1? → |i1,λ2?λ2

− →|i2,λ2?

− →|iN,λN?.

(1)

In the classical case, the system remains in its in-th state of

the discrete phase space during the controlling substep. Anal-

ogously, in quantum systems, this process corresponds to the

quantum adiabatic regime, i.e., the system remains in its in-

th eigenstate of the instantaneous Hamiltonian when we con-

trol the parameter λ(t) of the Hamiltonian H[λ(t)] so slowly

that the quantum adiabatic conditions are satisfied, and the

above trajectories (1) can be achieved (red trajectory of Fig.

1). However, if we control the parameter of the Hamiltonian

very quickly in the controlling substep, and then the quantum

adiabatic conditions are not satisfied, the trajectory A → B in

general should be written as (see blue trajectory of Fig. 1)

|i0,λ0? → |i?

→ ··· → |iN−1,λN? →??i?

The main difference of the above two kinds of trajectories (1)

and (2) is that after the controlling substep the system may

not be in the same eigenstate as that before the controlling,

i.e., in?= i?

due to randomness caused by quantum non-adiabatic transi-

tion and has no classical counterpart. Actually this difference

of trajectories (1) and (2) highlights the main difference be-

tween the quantum and classical Crooks FT. For a quantum

system, the microscopic work done in every controlling sub-

step is equal to the difference of the energy before and after

the controlling substep: Wn= E(i?

the heat exchanged with the heat bath is equal to the difference

of the energy of the system before and after the relaxation sub-

step Qn= E(in,λn)−E(i?

a whole, we must make 2N times quantum measurements to

confirm the microscopic work done and heat exchanged with

the heat bath. Similar to the classical case, the total work W

0,λ1?λ1

− →|i1,λ1? → |i?

1,λ2?λ2

?λN

− →|i2,λ2?

− →|iN,λN?.

N−1,λN

(2)

n. The internal excitation |in,λn? → |i?

n,λn+1? is

n,λn+1)−E(in,λn), and

n−1,λn). For the trajectory (2) as

performed on the system, and the total heat Q exchanged with

the heat bath are given by the summation of work and heat

in every step, W =?N−1

energy is ∆E = Q + W = E(iN,λN) − E(i0,λ0). Note

that the work and heat depend on the trajectory, but the en-

ergy change depends only on the initial and final energy, and

does not depend on the trajectory.

Similar to the classical case [4] we assume the trajec-

tory (2) to be Markovian, and the forward process starts

from the thermal equilibrium distribution P(|i0,λ0?)

e−βE(i0,λ0)/(?

?

× PF(|i?

n=0[E(i?

n−1,λn)?, and the total change in

n,λn+1) − E(in,λn)],Q =

?N

n=0

?E(in,λn) − E(i?

=

ie−βE(i,λ0)). The joint probability for a

given trajectory (2) can be expressed as

PF(A → B) =P(|i0,λ0?)

N−1

n=0

PF(|in,λn? → |i?

n,λn+1?)

n,λn+1? → |in+1,λn+1?).

(3)

It can be seen that the above probability (3) of a trajectory

for a quantum case is different from the classical case [4]

by the extra term P(|in,λn? → |i?

randomness due to quantum non-adiabatic transition. When

the quantum adiabatic conditions are satisfied, P(|in,λn? →

|i?

in classical systems [4]. We will see later that the quantum

Crooks FT and quantum JE in the quantum adiabatic regime

are the counterpart of classical Crooks FT and classical JE.

To prove the quantum Crooks FT, we also need to con-

sider the time-reversed trajectory [11] of the original trajec-

tory (2). The time-reversed trajectory corresponding to the

forward time trajectory A ← B in Eq. (2) can be written as

Θ|i0,λ0? ← Θ|i?

··· ← Θ|iN−1,λN? ← Θ??i?

where Θ|in,λn? = |in,λn?∗is the microscopic state in the

time-reversed trajectory [12]. The sequence in which states

are visited is reversed, as is the order in which λ is changed.

The work doneW, the heat exchange Qwith the heat bath, the

change of the internal energy ∆E, and the change of free en-

ergy ∆F for the reversed time direction are the negative value

of that of the forward time trajectory. The joint probability for

time reversed trajectory A ← B can be expressed as

n,λn+1?) arising from

n,λn+1?) = δin,i?

n, we regain the probability of a trajectory

0,λ1?λ1

← −Θ|i1,λ1? ← Θ|i?

N−1,λN

1,λ2?λ2

← −

?λN

← −Θ|iN,λN?

(4)

PR(A ← B) =

N−1

?

× PR(Θ|i?

× P(Θ|iN,λN?),

n=0

PR(Θ|in,λn? ← Θ|i?

n,λn+1?)

n,λn+1? ← Θ|in+1,λn+1?)

(5)

whereP(Θ|iN,λN?) = e−βE(iN,λN)/?

Also there is en extra term PR(Θ|in,λn? ← Θ|i?

arising due to the randomness caused by quantum non-

adiabatic transition in comparison with the classical case.

ie−βE(i,λN))isthe

initial thermal distribution for the time-reversed trajectory.

n,λn+1?)

Page 3

3

III.PROOF OF QUANTUM CROOKS FT AND QUANTUM

JE

As we have mentioned before, in a trajectory every step

consists of two substeps, the controlling substep (not neces-

sarily to be quantum adiabatic) and the relaxation substep.

The relaxation substeps are assumed to be microscopically re-

versible, and therefore obey the detailed balance [4, 13] for all

fixed value of the external control parameter λ

PF(??i?

To compare the ratio of the probabilities of forward (3) and

time-reversed (5) trajectories, we also need to know the ratio

of the probabilities in the controlling substep. In the follow-

ing we will focus on the study of controlling substep and its

time reversal. As we mentioned before, during the controlling

substep, the system can be regarded as an isolated quantum

system and the evolution is completely determined by a time-

dependent Hamiltonian H[λ(t)]. For example, when the con-

trolling parameter λ is changed from λnto λn+1, the prob-

ability of the transition from a microscopic state |in,λn? to

another microscopic state |i?

PF(|in,λn? → |i?

where U = Texp{−i?t1

controlling substep, and T is the time-ordered operator. Sim-

ilarly, in the time-reversed trajectory the excitation probabil-

ity from the microscopic state Θ|i?

scopic state Θ|in,λn? in the time reversed trajectory can be

expressed as [14]

n−1,λn

n−1,λn

?→ |in,λn?)

PR(Θ??i?

?← Θ|in,λn?)=

e−βE(in,λn)

e−βE(i?

n−1,λn).

(6)

n,λn+1? can be expressed as

n,λn+1?) = |?i?

t0H[λ(t)]dt} is the unitary matrix

describing the evolution of the isolated quantum system in the

n,λn+1|U |in,λn?|2(7)

n,λn+1? to another micro-

PR(Θ|in,λn? ← Θ|i?

= |

where ΘU← −

(U†)∗= UTis the time-reversed unitary matrix. Because of

the property of the time-reversed transformation Θ|in,λn? =

|in,λn?∗, and the property of the Hermitian conjugate matrix,

(?in,λn|)∗UT(|i?

it is not difficult to prove that

PF(|in,λn? → |i?

PR(Θ|in,λn? ← Θ|i?n,λn+1?)≡ 1.

Based on the above two results (6), (10) and Eqs. (3) and

(5), we reproduce the Crooks FT for a quantum mechanical

system

PF(A → B)

PR(A ← B)= eβ(W−∆F).

From Eq. (11) we group all those trajectories with the same

amount of microscopic work, and obtain

PF(W|a)

PR(−W|−a)= eβ(a−∆F).

n,λn+1?)

Θ

?

?in,λn|← −

?

ΘU← −

Θ (Θ|i?

n,λn+1?)|2,

(8)

Θ = Texp{−i?t1

t0H[λ(t0 + t1 − t)]dt} =

n,λn+1?)∗≡ ?i?

n,λn+1|U |in,λn?

(9)

n,λn+1?)

(10)

(11)

(12)

Eq. (12) is the Crooks FT. Similar to the derivation in Ref. [4],

we obtain the JE for a quantum open system straightforwardly

?e−βW?

of Crooks FT and JE have been reported in some previous

work, the explicit consideration of the influence of the heat

bath, i.e., the explicit expression of microscopic work in the

presence of a heat bath has not been reported before. Also

the relation between quantum and classical trajectories are not

addressed clearly. Hence our quantum mechanical extensions

of Crooks FT and JE are highly nontrivial.

= e−β∆Ffrom?PR(−W|−a)da = 1. Here, we

would like to emphasize that though quantum generalization

IV. ILLUSTRATION OF QUANTUM CROOKS FT AND

QUANTUM JE IN A TWO-LEVEL SYSTEM

0 0.2 0.40.60.81

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

W[ln2kBT]

PF[W],PR[−W]

N = 5Forward

N = 5Backward

N = 10Forward

N = 10Backward

N = 15Forward

N = 15Backward

N = 20Forward

N = 20Backward

0.263

FIG. 2: (Color Online) Microscopic work distribution PF(W) of

forward trajectories (solid lines), and the negative reverse work dis-

tribution PR(−W) of their corresponding time-reversed trajectories

(dashed lines). The probabilities have been normalized. Here we fix

∆(t0)and∆(tN). Differentdistributionsrepresentdifferentcontrol-

ling time (the more steps, the longer control time). The controlling

steps are chosen to be N = 5 (red •), N = 10 (blue ?), N = 15

(green ?), and N = 20 (black ?). It can be seen that the work dis-

tributionsforbothforwardandreversedtrajectoriesarenotGaussian.

Moreover, with the decrease of the controlling speed, the fluctuation

of the distributions decreases, and the difference between the work

distribution of the forward and time-reversed trajectories becomes

less obvious. The corresponding forward and negative reverse work

distribution cross at W = ∆F, and this is a direct consequence of

the quantum Crooks FT. The free energy difference ∆F ia marked

by the red vertical dash-dotted line.

Having generalized the Crooks FT and JE to quantum sys-

tems in the presence of a heat bath. In the following, we

use the studies based on a two-level system [15] as an illus-

tration to demonstrate our main idea. The Hamiltonian of

the two-level system is H = ∆(t)(σz+ 1)/2, where ∆(t)

is the parameter of the Hamiltonian, and σzis Pauli matrix.

The initial and final value of the parameter are ∆A= ∆(t0)

and ∆B = ∆(tN) respectively. The controlling scheme is

the same as that in Ref. [15]: We divide the whole pro-

Page 4

4

cess into N even steps.

step is ∆(tn) = ∆(t0) + n∆, n = 1, 2, ···, N, where

∆ = (∆B− ∆A)/N is the change of the parameter in ev-

ery step. Every step consists of two substeps: the control-

ling substep, in which we change the parameter from ∆(tn)

to ∆n+1= ∆(tn) + ∆, and the relaxation substep. For sim-

plicity, we consider the case where the system reaches ther-

mal equilibrium with the heat bath in every relaxation substep.

Hence, the probability for the forward and reverse relaxation

substep can be expressed as PF(??i?

Θ|in,λn?) = e−βE(i?

sume the quantum adiabatic conditions are satisfied in every

controlling substep. That is PF(|in,λn? → |i?

δin,i?

on these assumptions, the microscopic work distribution for

the forward trajectories can be obtained [15]

Hence the parameter in the nth

n−1,λn

and PR(Θ??i?

?

→ |in,λn?) =

n−1,λn

e−βE(in,λn)/(?

ie−βE(i,λn)),

n−1,λn)/(?

?

←

ie−βE(i,λn)). Also we as-

n,λn+1?) =

n. Based

n, and PR(Θ|in,λn? ← Θ|i?

n,λn+1?) = δin,i?

PF(W|k∆) = PF

e

N−k−1

?

l=0

eβ∆B− eβ(∆A+l∆)

eβ(l+1)∆− 1

,

(13)

where

PF

e=

N

?

j=1

e−β[∆A+(j−1)∆]

1 + e−β[∆A+(j−1)∆],k = 0,1,2,··· ,N. (14)

Similarly, the microscopic work distribution for the time-

reversed trajectory can be expressed as

PR(−W|−k∆) = PR

e

N−k−1

?

l=0

eβ∆[eβ∆B− eβ(∆A+l∆)]

eβ(l+1)∆− 1

,

(15)

where

PR

e=

N

?

j=1

e−β[∆B−(j−1)∆]

1 + e−β[∆B−(j−1)∆],k = 0,1,2,··· ,N. (16)

We plot the above distributions (13) and (14) of microscopic

work in Fig. 2. Here the probability distribution in the ex-

cited state are Pe(∆A) = e−β∆A/(1 + e−β∆A) = 1/3, and

Pe(∆B) = e−β∆B/(1 + e−β∆B) = 1/5. The free energy

difference is ∆FAB = [ln(1 + 1/2) − ln(1 + 1/4)]kBT ≈

0.263ln2kBT. It can be seen (see Fig. 2) that the correspond-

ing forward and negative reverse work distributions cross at

W = ∆F, no matter what the controlling protocol is, and

this result is a direct consequence of Crooks FT. It should be

pointed out that the work distributions (13) and (15) are non-

Gaussian [15]. Hence, the processes discussed here are be-

yond the linear response regime. Yet we will see both Crooks

FT and JE holds. We also plot the logarithm of the ratio of the

forwardandnegativereverseworkdistribution(SeeFig. 3(a)).

It can be seen that all data collapse onto the same straight line.

In addition, the slope of the line is equal to unit, and the line

cross the horizontal axis at W = 0.263ln2kBT = ∆FAB.

Thus our numerical simulation confirms the validity of quan-

tum Crooks FT when the process is beyond the linear response

regime. We also plot the logarithm of the exponent averaged

work ln?e−βW?and averaged work ?W? of the forward pro-

be seen that the averaged work is greater than the free energy

difference ?W? ? ∆F, while the logarithm of the exponent

averaged work is identical to the difference of the free energy

ln?e−βW?≡ ∆F ≈ 0.1823kBT no matter what the control-

the process is beyond the linear response regime.

cess (see Fig. 3(b)) to test the validity of quantum JE. It can

ling protocol is. Hence, Fig. 3(b) verifies quantum JE when

V.CONCLUSION AND REMARKS

In this paper, we explicitly consider the quantum Crooks

FT and quantum JE in the presence of an external heat bath.

Our proof includes the proof of classical Crooks FT as a spe-

cial case. When the quantum adiabatic conditions are satis-

fied, we reproduce the result of Crooks FT and JE for clas-

sical systems. Our work indicates that in quantum systems,

the probabilities (Eqs. (3) and (5)) comes from the quantum

non-adiabatic transition and statistical mechanical random-

ness, while in classical system, the randomness only comes

from the later case. We use the two-level system as an illus-

tration to demonstrate the validity of quantum Crooks FT and

quantum JE beyond the linear response regime.

Before concluding the paper, we would like to mention the

following points. First, though the quantum non-adiabatic

transition is introduced into the controlling substep, this sub-

step is time reversal symmetric. I. e., all the time asymme-

try is due the relaxation substep (statistical mechanical ran-

domness), rather than the controlling substep (quantum non-

adiabatic transition). This is the same as the classical case.

Second, when we change the Hamiltonian slowly, we repro-

duce the proof of Crooks for classical systems. In this sense,

we say that our proof includes the classical Crooks FT and

classical JE as a special case. Third, for classical system,

the Crooks FT and JE have been experimentally verified [7].

However, for a quantum mechanical system, the experimen-

tal exploration on Crooks FT and JE has not been reported

(an exception is [10]). This perhaps is mainly due to the

fact that microscopic work in a quantum mechanical system

is not a well defined observable [18]. There is no well defined

pressure or force for a quantum system [17]. Hence, we can-

not follow the way that we do in classical system to measure

the force and make the integral of the force by the extension.

On the contrary, we will have to introduce quantum measure-

ment processes to confirm the initial and final energy of the

system and then calculate the microscopic work done from

the difference of the initial and final energy difference [16].

Fourth, though the numerical simulations consider only the

special cases: 1) the system reach thermal equilibrium with

the heat bath in every relaxation substep, and 2) the quantum

adiabatic conditions are satisfied in every controlling substep,

thequantumCrooksFTand quantumJEarenotconstrainedin

these special cases. Finally, our numerical simulations based

on a two-level system can possibly be testified by employing

Josephson junction charge qubit [19]. Discussion about em-

ploying Josephson Junction qubit to test the quantum Crooks

FT and quantum JE will be given later.

Page 5

5

00.2 0.40.6 0.81

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

W[ln2kBT]

ln[PF[W]

PR[−W]]

N = 5

N = 10

N = 15

N = 20

0.263

(a)

05 1015 20

0.18

0.182

0.184

0.186

0.188

0.19

0.192

0.194

N

< βW >,ln < eβW>

< βW >

ln < eβW>

ΔF = 0.1832kBT

(b)

FIG. 3: (Color Online) (a) The logarithm of the probabilities of for-

ward and time-reversed trajectories as a function of work. It can

be seen that all data of different work and different control proto-

cols (N = 5 (red •), N = 10 (blue ?), N = 15 (green ?), and

N = 20 (black ?) ) collapse onto the same straight line. The

slop of the line is equal to unity, and the line cross the horizontal

axes at W = ∆F. Thus the numerical result verifies the quan-

tum Crooks FT ln[PF(W|a)/PR(−W|−a)] = β(a − ∆F). (b)

The averaged work VS. the logarithm of averaged exponent work for

different control protocols. It can be seen that the averaged work

?W? (red ?) is always greater than the difference of free energy

∆FAB and differ from one control protocol to another, while the

logarithm of the exponentially averaged work ln?exp[−βW]? (blue

?) is always equivalent to the difference of free energy irrespective

of the control protocols. Thus the numerical result verifies the JE

ln?exp[−βW]? ≡ ∆F.

VI.ACKNOWLEDGMENTS

HTQ thanks Wojciech H. Zurek, G. Crooks and Rishi

Sharma for stimulating discussions and gratefully acknowl-

edges the support of the U.S. Department of Energy through

the LANL/LDRD Program for this work.

[1] S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics,

(North-Holland, Amsterdam, 1962).

[2] D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994);

D. J. Evans and D. J. Searles, Advances in Physics, 51, 1529

(2002).

[3] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).

[4] Crooks, J. Stat. Phys. 90, 1481 (1998); G. E. Crooks, Phys. Rev.

E 60, 2721 (1999); Gavin E. Crooks, Phys. Rev. E 61, 2361

(2000).

[5] C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today, 54, (7)