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
Nonequilibrium thermodynamics has been an intriguing re-
search subject for more than one hundred years .
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 , Jarzynski Equality (JE) , and Crooks Fluctuation
Theorem (Crooks FT) . 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 , and the validity
been extensively studied not only by numerical studies , but
and for both deterministic and stochastic processes. For quan-
tum systems, possible quantum extension of Crooks FT and
JE have also been reported . Nevertheless, we notice that
almost all of these reports about quantum extension of Crooks
FT focus on isolated quantum systems , 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 ).
In this paper, we will give a detailed proof of the validity of
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.
FIG. 1: (Color Online) Trajectories of a quantum system in a
nonequilibrium process. Similar to Ref.  each step (from tn to
tn+1) is divided into two substeps: the controlling substep of time
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.
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
II. NOTATIONS AND ASSUMPTIONS:
Crooks FT  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.  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
43(2005); M.Haw, Phys.World, 20, (11)25, (2007); C.Jarzyn-
ski, Eur. Phys. J. B. 64, 331 (2008) and reference therein.
 D. J. Evans, E.G.D. Cohen, and G.P. Morriss, Phys. Rev. Lett.
71, 2401 (1993); C. Jarzynski, Phys. Rev. E 56, 5018 (1997).
 G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J.
Evans, Phys. Rev. Lett. 89, 050601 (2002); D. M. Carberry, J.
C. Reid, G. M. Wang, E. M. Sevick, D. J. Searles, and Denis J.
Evans, Phys. Rev. Lett. 92, 140601 (2004); J. Liphardt, S. Du-
mont, S.B. Smith, I. Tinoco Jr., C. Bustamante, Science, 296,
1832 (2002); D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I.
Tinoco Jr., C. Bustamante, Nature 437, 231 (2005); N. C. Har-
ris, Y. Song, Ching-Hwa Kiang, Phys. Rev. Lett. 99, 068101
 S. Yukawa, J. Phys. Soc. Jpn 69, 2367 (2000); J. Kur-
chan, arXiv:cond-mat/0007360v2; H. Tasaki, arXiv:cond-
mat/0009244v2; V. Chernyak, S. Mukamel, Phys. Rev. Lett.
93, 048302 (2004); M. Esposito, and S. Mukamel, Phys. Rev.
E. 73, 046129 (2006); P. Talkner, P. H¨ anggi, M. Morillo,
arXiv:0707.2307v1; J. Teifel, G. Mahler, Phys. Rev. E 76,
051126 (2007); H. Schroder, J. Teifel, G. Mahler, Eur. Phys.
J. Special Topics, 151, 181 (2007); P. Talkner, M. Campisi, and
P. H¨ anggi, arXiv:0811.0973v1;
 P. Talkner, P. H¨ anggi, J. Phys. A.: Math. Theor. 40, F569
(2007); S. Deffner, and E. Lutz, Phys. Rev. E 77, 021128
(2008); P. Talkner, P. H¨ anggi, and M. Morillo, Phys. Rev. E
77, 051131 (2008).
 G. Huber, F. Schmidt-Kaler, S. Deffner, E. Lutz, Phys. Rev.
Lett. 101, 070403 (2008).
 For classical systems, if the forward process is described by a
trajectory in the phase space (? p0,? q0) → (? p1,? q1) as the Hamil-
tonian is changed from H(λ0) to H(λ1). The time-reversed
trajectory is (−? p1,? q1) → (−? p0,? q0) as the Hamiltonian is
changed from H(λ1) to H(λ0). For quantum systems, if the
forward trajectory is |ψ(t0)? → |ψ(t1)? as the Hamiltonian is
changed from H(λ0) to H(λ1), the time-reversed trajectory is
Θ|ψ(t1)? → Θ|ψ(t0)? when the Hamiltonian is changed from
H(λ1) to H(λ0) .
 J. J. Sakurai, Modern Quantum Mechanics (Revised Edition),
(Reading, Addison-Wesley, 1994).
 D. Chandler, Introduction to Modern Statistical Mechanics,
(Oxford University Press, New York, 1987).
 C. Jarzynski, and D. K. Wojcik, Phys. Rev. Lett. 92, 230602
(2004); W. De Roeck, C. Maes, Phys. Rev. E 69, 026115
(2004); T. Monnai, Phys. Rev. E 72, 027102 (2005); G. E.
Crooks, Phys. Rev. A 77, 034101 (2008); D. Andrieux and P.
Gaspard, Phys. Rev. Lett. 100, 230404 (2008).
 H. T. Quan. S. Yang, and C. P. Sun, Phys. Rev. E. 78, 021116
 S. Mukamel, Phys. Rev. Lett. 90, 170604 (2003).
 H. T. Quan, arXiv: 0811.2756.
 P. Talkner, E. Lutz, and P. H¨ anggi, Phys. Rev. E 75, 050102(R)
 J. Q. You, and F. Nori, Phys. Today 58, No. 11, 42 (2005).