Statistics of work performed on a forced quantum ... - Semantic Scholar
PHYSICAL REVIEW E 78, 011115 共2008兲
Statistics of work performed on a forced quantum oscillator Peter Talkner, P. Sekhar Burada, and Peter Hänggi Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany 共Received 19 March 2008; published 18 July 2008兲 Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator. In this special case two parameters are introduced that are sufficient to completely characterize the force protocol. Explicit results for the characteristic function of work and the corresponding probability distribution are provided and discussed for three different types of initial states of the oscillator: microcanonical, canonical, and coherent states. Depending on the choice of the initial state the probability distributions of the performed work may greatly differ. This result in particular also holds true for identical force protocols. General fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed. DOI: 10.1103/PhysRevE.78.011115
PACS number共s兲: 05.30.⫺d, 05.70.Ln, 05.40.⫺a
I. INTRODUCTION
During the last decade various fluctuation and work theorems 关1,2兴 have been formulated and discussed. They characterize among other things the full nonlinear response of a system under the action of a time-dependent force 关3,4兴. These theorems have been derived and experimentally confirmed primarily for classical systems 关5–7兴. Quantum mechanical generalizations were proposed recently 关8–15兴. Conceptual problems arise, though, in the context of quantum mechanics if one tries to generalize those classical relations that require, for example, the specification of a system’s trajectory extending over some interval of time, or the simultaneous measurement of noncommuting observables. For example, the measurement of work performed by an external force on an otherwise isolated system may be accomplished in the framework of classical physics in principle in two different ways. The first method is based on two measurements of the energy, one at the beginning and the second at the end of the considered process. This method becomes unreliable in practice if the system is large and the work performed on the system is negligibly small compared to the total energy of the system. Such a situation typically arises if the system of interest, on which the force exclusively acts, interacts with its environment. In order to retain an isolated system, the large system made of the open system and its environment must be considered. Again, the work performed on the system results as the difference of the energies of the total system, which may both be very large. For classical systems, this unfortunate situation can be circumvented by a second method, by monitoring the state of the relevant small system during the time when the force is acting. Having this information at hand, one can determine the work by integrating the power supplied to the system at each instant of time. The respective power can be inferred from the registered state of the system and the known force protocol. In a quantum system, a continuous measurement of even a single observable would strongly influence and possibly manifestly distort the system’s dynamics. Clearly, only the first of the two methods of energy measurement is feasible, at least in principle, in the quantum context. An alternative method based on continuous monitoring has recently been suggested by Esposito and Mukamel 关11兴 1539-3755/2008/78共1兲/011115共11兲
for open quantum systems described by Markovian quantum master equations. There the dynamics of the density matrix is mapped onto a classical rate process for which known fluctuation theorems can be applied 关16兴. This provides an interesting formal approach but its physical meaning has remained unclear 关11兴. Moreover, this approach is restricted to open systems that only weakly interact with their respective environments. In the present paper, the distribution of work is discussed for the exactly solvable system of a driven harmonic oscillator 关15,17兴. In this case, the distribution of work is discrete. We provide formal expressions for this distribution and its corresponding characteristic function which are valid for all initial states of the system as well as for all possible kinds of force protocols. In particular, we determine the characteristic functions and distributions of the work for microcanonical, canonical, and coherent initial states which lead to qualitatively different work distributions. The paper is organized as follows. In Sec. II we review the general form of the characteristic function of work performed on a system in terms of a correlation function of the exponentiated Hamiltonians at the initial and final times of the force protocol. We prove that this particular expression indeed always represents a characteristic function, i.e., the Fourier transform of a probability density. Section III presents various fluctuation and work theorems for canonical and microcanonical initial states. In Sec. IV general expressions for the characteristic function and the corresponding probability distribution of work are derived for a driven harmonic oscillator. Moreover, the expressions for the first four cumulants are derived. The dependence of the work distribution on the force protocol for microcanonical, canonical, and coherent initial states as well as its dependence on the specific parameters of these initial states are investigated. II. CHARACTERISTIC FUNCTION OF WORK
The response of a quantum system on a perturbation by a classical, external force can be characterized by the change of energies contained in the total system. The energy as an observable coincides with the Hamiltonian H共t兲 of the total system. It includes the external force and therefore depends
011115-1
©2008 The American Physical Society
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
on time. We will consider the dynamics of the system only within a finite window of time 关t0 , t f 兴 during which the force is acting in a prescribed way, resulting in a protocol of Hamiltonians which is denoted by 兵H共t兲其t f ,t0. Apart from the action of the external force the system is assumed to be closed. Its dynamics is consequently governed by a unitary time evolution Ut,t0, which is the solution of the Schrödinger equation iប Ut,t0/ t = H共t兲Ut,t0,
Ut0,t0 = 1.
共1兲
As explained in the Introduction, the work w is measured as the difference between the energies of the system at the final and initial times t f and t0. In a single measurement the work is given by the difference of two eigenvalues en共t f 兲 and em共t0兲 of the Hamiltonians H共t兲 at the respective times t f and t0, i.e., by w = en共t f 兲 − em共t0兲. The inherent randomness of the outcome of a quantum measurement in general leads to a measured work that is random. A complete description of the statistical properties of the work performed on the system is provided by the characteristic function Gt f ,t0共u兲 which presents the Fourier transform of the probability density of the work pt f ,t0共w兲, i.e., Gt f ,t0共u兲 =
冕
dw eiuw pt f ,t0共w兲.
共2兲
It can be expressed as a quantum correlation function of the exponentiated Hamiltonian at the initial and the final times 关14兴, i.e., Gt f ,t0共u兲 = 具eiuH共t f 兲e−iuH共t0兲典 ⬅ Tr eiuHH共t f 兲e−iuH共t0兲¯共t0兲, 共3兲
n
兺 Gt ,t 共ui − ui⬘兲zⴱi zi⬘ ⱖ 0 i,i⬘
共7兲
f 0
holds. Here, the asterisk zⴱi denotes the complex conjugate of z i. 共iii兲 Gt f ,t0共0兲 = 1. According to a theorem by Bochner 关18兴 the properties 共i兲–共iii兲 are necessary and sufficient conditions for the function Gt f ,t0共u兲 to be the Fourier transform of the probability measure of a random variable. In short, the first condition ensures that, strictly speaking, the function Gt f ,t0共u兲 is the Fourier transform of a measure, the second condition assures that this measure is positive, and the third condition that it is normalized. Hence the correlation expression Eq. 共3兲 always defines a proper characteristic function. A proof of the properties 共i兲–共iii兲 can be found in Appendix A. III. CANONICAL AND MICROCANONICAL INITIAL STATES
In experiments an external force is often applied on a system that initially is found in a thermodynamic equilibrium state. Depending on whether the system was in weak contact with a heat bath or was totally isolated from its environment, the initial state of the system is described by either a canonical or a microcanonical density matrix. For both situations fluctuation and work theorems are known. We will briefly review these relations. A. Work and fluctuation theorems for canonical initial states
where
If the initial density matrix is canonical, i.e., if HH共t f 兲 =
U†t ,t H共t f 兲Ut f ,t0 f 0
共4兲
denotes the Hamiltonian in the Heisenberg picture. The density matrix ¯共t0兲 follows from the initial density matrix 共t0兲 as a result of the measurement of the Hamiltonian H共t0兲. It is given by ¯共t0兲 = 兺 Pn共t0兲共t0兲Pn共t0兲,
共5兲
n
where the operators Pn共t0兲 denote the eigenprojection operators of the Hamiltonian at time t0, which present a partition of unity:
兺n Pn共t0兲 = 1.
共6兲
Before we apply the general expression 共3兲 to a particular system and investigate its dependence on the initial state 共t0兲, we discuss three general properties of the correlation expression 共3兲 which guarantee that the resulting function Gt f ,t0共u兲 indeed always presents a proper characteristic function of a classical random variable w. 共i兲 Gt f ,t0共u兲 is a continuous function of u. 共ii兲 Gt f ,t0共u兲 is a positive definite function of u, i.e., for all integer numbers n, all real sequences u1 , u2 , . . . , un, and all complex numbers zi, i = 1 , 2 , . . . , n,
共t0兲 = Z−1共t0兲exp关− H共t0兲兴,
共8兲
Z共t0兲 = Tr exp关− H共t0兲兴 = e−F共t0兲
共9兲
where
denotes the partition function and F共t0兲 the free energy, then 关H共t0兲 , 共t0兲兴 = 0 and the first measurement of the energy leaves the density matrix unchanged, such that ¯共t0兲 = 共t0兲. With Eq. 共3兲 this leads to the characteristic function of work for a canonical initial state that was derived in Ref. 关12兴. In this case, Gt f ,t0共u兲 can be continued to an analytical function of u for all 0 ⱕ Iu ⱕ  关13兴, where Iu denotes the imaginary part of u. For the particular value u = i the characteristic function yields the mean value of the exponentiated work, 具exp共−w兲典 and the correlation function expression 共3兲 simplifies to the ratio of the partition functions at the times t f and t0, resulting in the Jarzynski work theorem 具e−w典 = Z共t f 兲/Z共t0兲 = exp兵− 关F共t f 兲 − F共t0兲兴其,
共10兲
where Z共t f 兲 = Tr exp关−H共t f 兲兴 = exp关−F共t f 兲兴. Within the domain of analyticity S = 兵兩u兩0 ⱕ Iu ⱕ 其 the characteristic functions for the original and the time-reversed protocol are related to each other by the following formula 共cf. 关13兴兲:
011115-2
STATISTICS OF WORK PERFORMED ON A FORCED …
Gt f ,t0共u兲 =
Z共t f 兲 Gt ,t 共− u + i兲, Z共t0兲 0 f
PHYSICAL REVIEW E 78, 011115 共2008兲
共11兲
where Gt0,t f 共u兲 refers to processes under the time-reversed protocol 兵H共t兲其to,t f starting from the canonical state Z−1共t f 兲exp关−H共t f 兲兴. An inverse Fourier transform leads to the Tasaki-Crooks fluctuation theorem, which relates the probability densities of work pt f ,t0共w兲 for a given protocol to the density of the work pt0,t f 共w兲 for the time-reversed protocol. This theorem explicitly reads 关13兴 pt f ,t0共w兲 pt0,t f 共− w兲
=
Z共t f 兲 w −关F共t 兲−F共t 兲−w兴 f 0 e =e . Z共t0兲
共12兲
B. Fluctuation theorems for microcanonical initial states
A system in a microcanonical state is described by the density matrix
共t0兲 = E−1共t0兲␦„H共t0兲 − E…,
共13兲
E共t0兲 = Tr ␦„H共t0兲 − E… = exp关S共E,t0兲/kB兴
共14兲
entropy can be determined. For further details see Ref. 关14兴. IV. DRIVEN HARMONIC OSCILLATOR
To illustrate these concepts we consider an example that allows the analytical construction of the probability of work. Specifically, we consider a harmonic oscillator on which a time-dependent force acts during a finite interval of time. Its time evolution is governed by the Hamiltonian H共t兲 = បa†a + f ⴱ共t兲a + f共t兲a† ,
where denotes the angular frequency, and a and a creation and annihilation operators, respectively, which obey the usual commutation relation, i.e., 关a , a†兴 = 1. The complex driving force f共t兲 allows for a coupling to position and/or momentum of the oscillator. We assume that f共t兲 vanishes for times t ⱕ t0 = 0. It is our aim to study the influence of the initial state 共t0兲 on the statistics of work performed on the oscillator. The measurement of H共t0兲 = បa†a at time t0 = 0 then yields the result បn with probability pn = 具n兩共t0兲兩n典.
where
denotes the density of states as a function of the energy E of the system. The density of states can be expressed in terms of the entropy of the system SE共t0兲 provided the spectrum of the system Hamiltonian is sufficiently dense such that the density of states becomes a smooth function on a coarsened energy scale. The microcanonical density matrix commutes with the Hamiltonian H共t0兲. Consequently, ¯共t0兲 and 共t0兲 coincide. The microcanonical quantum Crooks theorem assumes the form 关14兴 pt f ,t0共E,w兲 pt0,t f 共E + w,− w兲
=
冉
冊
S共E + w,t f 兲 − S共E,t0兲 E+w共t f 兲 = exp . E共t0兲 kB 共15兲
Analogous to the canonical case, it relates the probability density pt0,t0共E , w兲 of work w, for a system starting in a microcanonical state with energy E, to the respective quantity for the time-reversed process starting at energy E + w. This quantum theorem is formally identical to the corresponding classical theorem 关19兴. From the microcanonical Crooks theorem the probability density of the time-reversed process can be eliminated to yield the so-called entropy-from-work theorem 关14兴, reading E f 共t f 兲 =
冕
dw E f −w共t0兲pt f ,t0共E f − w,w兲.
共16兲
This theorem allows one to determine the unknown density of states of a system with Hamiltonian H共t f 兲 from the known density of states of a reference system H共t0兲 by means of the statistics of the work that is performed on the system in a process that leads from the reference system to the final system with unknown density of states. In the case of systems with a sufficiently smooth density of states the corresponding
共17兲
†
共18兲
Accordingly, the oscillator is found in the state ¯共t0兲 = 兺 pn兩n典具n兩
共19兲
n
immediately after this measurement. Substituting this density matrix in the general expression for the characteristic function, Eq. 共3兲, one obtains Gt f ,t0共u兲 = 兺 pne−iuបn具n兩eiuHH共t f 兲兩n典.
共20兲
n
For the driven harmonic oscillator the diagonal matrix element of the exponentiated Hamiltonian HH共t f 兲 can be determined 关17兴. For details see Appendix B. With the expression 共B14兲 for the matrix element 具n兩exp关iHH共t f 兲兴兩n典 we find Gt f ,t0共u兲 = eiu兩f共t f 兲兩 ⬁
2/共ប兲
n
⫻ 兺 兺 pn n=0 k=0
= eiu兩f共t f 兲兩
冉冊
2/共ប兲
⬁
exp关共eiuប − 1兲兩z兩2兴
冉
n 兩z兩2共n−k兲 −iuប共n−k兲 iuប 共e − 1兲2共n−k兲 e k 共n − k兲!
exp关共eiuប − 1兲兩z兩2兴
⫻ 兺 pnLn 4兩z兩2sin2 n=0
冊
បu , 2
共21兲
where 兩f共t f 兲兩2 / 共ប兲 denotes a uniform shift of the spectrum of the harmonic oscillator due to the presence of the external force 关cf. Eq. 共B9兲兴 and z=
1 ប
冕
tf
ds ˙f 共s兲exp共is兲
共22兲
0
is a dimensionless functional of the driving force f共t兲 关cf. Eq. 共B7兲兴. This dimensionless quantity vanishes in particular for all-quasistatic forcings, i.e., if the force changes only very slowly with f共t兲 = g共t / t f 兲 for t f → ⬁, where g共兲 is a continuously differentiable function of 僆 关0 , 1兴. We hence call z共t兲
011115-3
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
the rapidity parameter of the force protocol. Finally, Ln共x兲 n = 兺k=0 共 nk 兲共−x兲k / k! denotes the Laguerre polynomial of order n 关21兴. Introducing the cumulant generating function K共u兲 = ln G共u兲, one obtains the cumulants of work kn as the nth derivatives of K共u兲 with respect to u taken at u = 0 关22兴, i.e., kn = 共−i兲ndnK共0兲 / dun. The first four cumulants become k1 = 具w典 =
兩f共t f 兲兩2 + ប兩z兩2 , ប
A. Distributions of work for different initial states
As particular examples of initial states we will discuss microcanonical, canonical, and coherent states. 1. Microcanonical initial state
For a microcanonical initial state with energy បn0 the density matrix becomes
共23兲
冉
冊
1 , 2
k2 = 具w2典 − 具w典2 = 2共ប兲2兩z兩2 具a†a典0 +
共24兲
共t0兲 = ¯共t0兲 = 兩n0典具n0兩. The characteristic function then reads iu兩f共t f 兲兩 Gmc t ,t 共n0,u兲 = e f 0
k3 = 具w3典 − 3具w2典具w典 + 2具w典3 = 共ប兲3兩z兩2 ,
冉
共25兲
共26兲 The odd cumulants of the work are independent of the initial preparation. The even cumulants depend on the factorial moments of the number operator a†a with respect to the initial state ¯共t0兲 such as 具a†a典0 = 兺nnpn and 具a†a共a†a − 1兲典0 = 兺nn共n − 1兲pn, where pn is defined in Eq. 共18兲. Moreover, all cumulants apart from the first one vanish for forcings with z = 0. This holds true in particular for all quasistatic force characteristics. The underlying work probability density then shrinks to a ␦ function at w = 兩f共t f 兲兩2 / 共ប兲. In general, the work probability density follows from the characteristic function by means of an inverse Fourier transformation. Rather than the characteristic function itself, we first consider the function G共u兲 ⬅ exp关−iu兩f共t f 兲兩2 / 共ប兲兴 ⫻Gt f ,t0共u兲. Upon expanding exp关兩z兩2exp共iuប兲兴 into a series of powers of 兩z兩2, we obtain for G共u兲 a Laurent series in the variable exp关iuប兴. The inverse Fourier transformation is given by a series of ␦ functions ␦共w − បr兲, with r 僆 Z, with weights ⬁
qr = e
−兩z兩2
=e
2k
冉 冊冉 冊
兩z兩2共k+m兲 n k
兺 兺 兺 共− 1兲2k−lpn m ! k! m,n=0 k=0 l=0 ⬁
−兩z兩2
n
n min共k+r,2k兲
兺兺 兺 l=0 n=0 k=0
共− 1兲2k−l pn
2k l
␦l+m,k+r
冉 冊冉 冊
兩z兩2共2k+r−l兲 n 共k + r − l兲 ! k! k
2k l
.
2/共ប兲
exp关共eiuប − 1兲兩z兩2兴
⫻Ln0 4兩z兩2sin2
k4 = 具w4典 − 4具w3典具w典 − 3具w2典2 + 12具w2典具w典2 − 6具w典4 = 共ប兲4兩z兩2兵1 + 4具a†a典0 + 6关具a†a共a†a − 1兲典0 − 2具a†a典0兴兩z兩2其.
共29兲
បu 2
冊
共30兲
and, accordingly, the probability qrmc共n0兲 to find a change of energy by w = បr + 兩f共t f 兲兩2 / 共ប兲 emerges as qrmc共n0兲 = e−兩z兩
2
n0 min共k+r,2k兲
兺 兺 k=0 l=0
冉 冊冉 冊
n 共− 1兲2k−l 共k + r − l兲 ! k! k
2k l
兩z兩2共2k+r−l兲 . 共31兲
As expected from the behavior of the moments, all probabilities qrmc共n0兲 with r ⫽ 0 vanish for quasistatic forcing, i.e., if z → 0. The dependence of qrmc共n0兲 on the parameter z is displayed in Fig. 1 for n0 = 0 and 3 as well as for the eight lowest values of r. With increasing values of the rapidity parameter z the distribution is becoming broader. For the fixed value of 兩z兩 = 2 the distribution qrmc共n0兲 is compared for the three initial states with n0 = 0, 10, and 30 in Fig. 2. With increasing value of n0 the distributions become broader. They develop a slightly asymmetric shape with higher peaks at negative values of r compared to those at positive r values. Between these dominant peaks the probability still displays pronounced variations. For a harmonic oscillator, the microcanonical Crooks mc 共n + r兲. One can theorem reduces to the relation qrmc共n兲 = q−r prove that this symmetry is satisfied by the probabilities qrmc共n兲 given by Eq. 共31兲. As a consequence, the ratio mc 共n + r兲 is unity independently of the actual values qrmc共n兲 / q−r of the initial energy, the work, and the force protocol as given by n, r, and z, respectively.
共27兲 2. Canonical initial state
The factor exp关−iu兩f共t f 兲兩2 / 共ប兲兴, by which G共u兲 has to be multiplied to yield Gt f ,t0共u兲, gives rise to a constant shift such that the probability density of work performed on a harmonic oscillator becomes
冠 冉
pt f ,t0共w兲 = 兺 qr␦ w − បr + r
兩f共t f 兲兩2 ប
冊冡
.
共28兲
In the next section we will investigate the influence of the initial state on the statistics of the work.
For a canonical density matrix
共t0兲 = 共1 − e−ប兲e−បa
†a
共32兲
the initial states are distributed according to pn = e−បn共1 − e−ប兲. This allows one to write the sum over n in the characteristic function 共21兲 in terms of the generating function of the Laguerre polynomials 共cf. 关21兴兲 yielding the expression
011115-4
STATISTICS OF WORK PERFORMED ON A FORCED …
PHYSICAL REVIEW E 78, 011115 共2008兲 ⬁
1.0
n min共k+r,2k兲
˜ 2 qrc共˜兲 = e−兩z兩 共1 − e−兲 兺 兺
n=0 k=0
qrmc (0)
⫻
0.5
0
1
2
z
(a)
3
4
5
6
4
3
2
1
0
−1
−2
−3
兺 l=0
冉 冊冉 冊
n 兩z兩2共2k+r−l兲 共k + r − l兲 ! k! k
2k l
˜
共− 1兲le−n 共35兲
,
where ˜ = ប denotes the inverse dimensionless temperature of the initial state. The expression for qrc共˜兲 can be further simplified to read
r
˜ ˜ 2 qrc共˜兲 = e−兩z兩 coth共/2兲er/2Ir
冉
冊
兩z兩2 , sinh ˜/2
共36兲
where I共x兲 denotes the modified Bessel function of the first kind of order 关21兴. For details of the derivation see Appendix C. Note that the following detailed-balance-like symmetry relation exists:
1.0
qrmc (3) 0.5
˜ c ˜ 共兲 = e−rqrc共˜兲, q−r
1
2
z
(b)
3
4
5
6
4
3
2
1
0
−2
−3
r
FIG. 1. Probabilities qrmc共n0兲 for two microcanonical initial states with n0 = 0 共a兲 and 3 共b兲 are depicted for r = −3 , . . . , 4, as functions of the rapidity parameter z defined in Eq. 共B7兲. In both cases the distribution collapses at r = 0 for qausi—static forcing, corresponding to 兩z兩 = 0, and broadens with increasing 兩z兩. Obviously, when starting in the ground state the oscillator cannot deliver work, whence the probability for negative r strictly vanishes. “Stimulated emission” becomes possible from an excited state at finite driving rapidity z, leading to nonzero probabilities qrmc共n0兲 at negative values of r in 共b兲.
冉
Gct ,t 共,u兲 = exp f 0
iu兩f共t f 兲兩2 + 共eiuប − 1兲兩z兩2 ប
冊
sin2共បu/2兲 . − 4兩z兩2 ប e −1
0.2
f 0
n0 = 0 n0 = 10 n0 = 30
共33兲 0.15
Putting u = i, one finds that the two terms in the exponent that are proportional to 兩z兩2 cancel each other, such that one obtains 具e−w典 = Gct ,t 共,i兲 = exp关− 兩f共t f 兲兩2/共ប兲兴.
relating the occurrence of positive and negative work. Figure 3 illustrates this relation, which is closely connected to the Tasaki-Crooks theorem 共12兲, as demonstrated at the end of this section. In Fig. 4 the z dependence of qrc共˜兲 for ˜ = ln共4 / 3兲 is compared for a few small values of r. One finds that, because of the average over the canonical initial distribution, the multipeaked structure of the microcanonical distribution as a function of the rapidity parameter 兩z兩 disappears, and only a single peak remains for each value of r. The temperature dependence of the work distribution is illustrated in Fig. 5. Finally, we verify the validity of the Tasaki-Crooks theorem 共12兲 for a driven oscillator. For this purpose we consider the probability density pt0,t f 共−w兲 for the time-reversed protocol. Since the absolute values of the rapidity parameters coincide for the original and the time-reversed protocols, the probability density of work for the reversed protocols becomes
qrmc (n0 )
0
−1
共37兲
共34兲
The free energy difference of two oscillators with Hamiltonians H共t0兲 = បa†a and H共t f 兲 = បa†a + f ⴱ共t f 兲a + f共t兲a†, each one staying in a canonical state at the temperature , is given by ⌬F = F共t f 兲 − F共t0兲 = 兩f共t f 兲兩2 / 共ប兲. Hence, Eq. 共34兲 agrees with Jarzynski’s work theorem. The probability qrc共˜兲 to find the work w = បr + 兩f共t f 兲兩2 / 共ប兲 if the system starts in a canonical state becomes
0.1 0.05 0
−20
−10
0
r
10
20
30
FIG. 2. 共Color online兲 Probabilities qrmc共n0兲 for a microcanonical initial state with n0 = 0 共circles兲, 10 共diamonds兲, and 30 共triangles兲 are compared for a fixed rapidity parameter 兩z兩 = 2 and r = −22, . . . , 30. The lines serve as a guide for the eye.
011115-5
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
(1)
0.2
(2) (3)
5 0
(1) β˜ = 2.0 (2) β˜ = 1.0
−5
−10
−5
0
5
r
冠 冉
⬁
兺
qrc共兲␦ − w − បr −
r=−⬁
兩f共t f 兲兩2 ប
冊冡
, 共38兲
where we took into account the overall shift of the spectrum by the reversed protocol as −兩f共t f 兲兩 / 共ប兲. Multiplying both sides of Eq. 共38兲 with exp关−共⌬F − w兲兴 = exp兵−关兩f共t f 兲兩2 / 共ប兲 − w兴其 and using the symmetry 共37兲, one obtains e−共⌬F−w兲 pt0,t f 共− w兲 = 兺 e−关兩f共t f 兲兩 r
0
10
FIG. 3. 共Color online兲 Natural logarithm of the probability ratio of positive and negative transferred work resulting from Eq. 共36兲 as a function of r for different dimensionless temperatures ˜ =  / ប = 0.5, 1 , 2 共circles, squares, lozenges兲 and different rapidity parameters z = 1 , 2 共large open, small filled symbols兲. The symmetry relation 共37兲 implies a linear law with inclination ˜ 共thin straight broken lines兲 which is independent of the rapidity parameter. The agreement is perfect.
pt0,t f 共− w兲 =
0.1 0.05
(3) β˜ = 0.5
−10
β˜ = 0.5 β˜ = 1.0 β˜ = 5.0
0.15
˜ qrc (β)
˜ c (β)) ˜ ln (qrc (β)/q −r
10
−10
−5
0
10
15
= pt f ,t0共w兲,
20
共39兲
in accordance with the Tasaki-Crooks theorem 共12兲. Conversely, if the Tasaki-Crooks theorem holds for a discrete work distribution of the form 共28兲, then the detailed-balancelike relation 共37兲 follows for the probabilities qrc共˜兲. 3. Coherent initial state
An oscillator prepared in a coherent state 兩␣典 is described by the density matrix
共t0兲 = 兩␣典具␣兩,
共40兲
where
冠 冉
qr 共˜兲 兩f共t f 兲兩2 ប
冠 冉
兩 ␣ 典 = e ␣a
冊冡
兩f共t f 兲兩2 ˜ = 兺 erqrc共˜兲␦ w + បr − ប r
†+␣*a
兩0典,
共41兲
and 兩0典 is the normalized ground state of the oscillator satisfying a兩0典 = 0. Note that the coherent state density matrix does not commute with the Hamiltonian H共t0兲. The measurement of H共t0兲 modifies the coherent state 共40兲 by projecting it onto the eigenstates 兩n典 = 共a†兲n / 冑n!兩0典 of this Hamiltonian, leading to
冊冡
¯共t0兲 = e−兩␣兩
1.0
qrc ( ln(4/3))
2
兺n
兩␣兩2n 兩n典具n兩. n!
共42兲
This implies a Poissonian distribution of the respective energy eigenvalues បn,
0.5
pcs n =
1
5
FIG. 5. 共Color online兲 Probabilities qrc共˜兲 for a canonical initial state as functions of r for 兩z兩 = 2 and different values of the dimensionless inverse temperature ˜ = 0.5 共circles兲, 1 共diamonds兲, and 5 共triangles兲. The lines serve as a guide for the eye.
2/共ប兲−w兴 c
⫻␦ − w − បr −
0
r
2
z
3
4
5
6
4
3
2
1
0
−1
−2
−3
r
FIG. 4. Probabilities qrc关˜ = ln共4 / 3兲兴 for a canonical initial state for r = −3 , . . . , 4 as functions of the parameter z. For the sake of comparability, the dimensionless inverse temperature is chosen such that the average energy in the initial state coincides with the energy 3ប of the microcanonical state in Fig. 1共b兲.
兩␣兩2n −兩␣兩2 e , n!
共43兲
which yields for the characteristic function of work 共21兲 a closed expression of the form
冉 冉冏
Gcs t ,t 共␣,u兲 = exp f 0
iu兩f共t f 兲兩2 + 兩z兩2共eiបu − 1兲 ប
⫻J0 4 ␣z sin
បu 2
冏冊
,
冊
共44兲
where J0共x兲 is the Bessel function of order zero 共cf. Ref
011115-6
STATISTICS OF WORK PERFORMED ON A FORCED …
PHYSICAL REVIEW E 78, 011115 共2008兲 0.2
1.0
0.15 0.5
qrcs (α)
qrcs (3)
|α|2 = 0.1 |α|2 = 1 |α|2 = 10
0.1 0.05
−10 0
0 1
2
z
10
0
r
20
3
(a)
FIG. 6. Probabilities qrcs共␣兲 for a coherent state with parameter 兩␣兩 = 3 for r = −10, . . . , 20 as functions of z.
qrcs共␣兲 = e−兩z兩
2
兺
m=0
冉
兩z兩2m兩␣z兩2兩m−r兩 1 F2 兩m − r兩 + ;兩m − r兩 2 m ! 共兩m − r兩!兲2 1
冊
+ 1,2兩m − r兩 + 1;− 4兩␣z兩2 ,
V. CONCLUSIONS
In this work we studied the statistics of work performed on an externally driven quantum mechanical oscillator by
10
20
|z| = 1 |z| = 2 |z| = 4
0.2 0.15 0.1 0.05
共45兲
where 1F2共a ; b , x ; x兲 denotes a generalized hypergeometric function 关21兴. For details of the derivation, see Appendix D. The dependence of the probabilities qrcs共␣兲 on the rapidity parameter 兩z兩 is illustrated in Fig. 6 for r values ranging from −10 to 20. Increasing values of z lead to a broadening of the distribution and also to a shift toward larger values of r; see also Fig. 7共b兲. This broadening and shift is in accordance with Eq. 共23兲 and 共24兲 for the first two cumulants of the work, which both increase with 兩z兩2. Figure 7共a兲 shows the dependence of the probabilities qrcs共␣兲 on the parameter ␣. Increasing ␣ also leads to a broadening of the work distribution without influencing its mean value 关cf. also Eq. 共23兲兴. In Fig. 8 the probabilities qr are depicted for different initial states. In one case the oscillator is initially prepared in a canonical state at inverse dimensionless temperature ˜ = ប = 0.1. In the other case, the oscillator stays in a coherent state 兩␣典, where the absolute value of ␣ is chosen such that the mean excitation number is the same for both states, i.e., 兩␣兩2 = exp共−ប兲 / 关1 − exp共−ប兲兴. For ប = 0.1 one finds 兩␣兩2 ⬇ 9.51. The two oscillators then are subjected to protocols with the same rapidity parameter 兩z兩 = 2. According to Eqs. 共23兲 and 共24兲 the first two moments of the work performed on the oscillators coincide. Yet the distribution of weight factors qrc共˜兲 and qrcs共␣兲 distinctly differ. Whereas the distribution is pronouncedly bimodal in the case of the coherent state, it is unimodal for the canonical state. The weight factors qrc共˜兲 almost perfectly fall onto a Gaussian probability density which has the same first two moments as the discrete distribution given by qr.
r
0
0
(b)
10
r
20
30
FIG. 7. 共Color online兲 Distribution of work performed on an oscillator which initially is prepared in a coherent state 兩␣典 for different values of ␣ in 共a兲 and of the rapidity parameter z in 共b兲. In 共a兲 the rapidity parameter has the value 兩z兩 = 2. In 共b兲 the coherent state parameter has the value 兩␣兩2 = 1.
means of a correlation function expression for the characteristic function of the work. We demonstrated that this particular expression indeed always represents a proper characteristic function of a random variable, which is the performed 0.06
canonical ensemble coherent state
0.04
qr
⬁
0
0.25
qrcs (α)
关21兴兲. For the probability qrcs共␣兲 of work one obtains with Eq. 共27兲
−10
0.02
0
−20
−10
0
10
r
20
30
40
FIG. 8. 共Color online兲 Distribution of work compared for a canonical and a coherent initial state subject to the same force protocol with rapidity parameter z = 2. With ប = 0.1 and 兩␣兩2 ⬇ 9.51, the expectation values of the energies agree in the two initial states such that according to Eqs. 共23兲 and 共24兲 the first and second moments of the work also coincide. Still the distributions of work greatly differ from each other.
011115-7
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
work in the present context. The proof given here is based on Bochner’s theorem. Note that it holds for general quantum mechanical systems, not only for harmonic oscillators. The considered force linearly couples to the position and momentum of the oscillator. It may describe the influence of an electric field on charged particles in a parabolic trap or the external forcing of a single electromagnetic cavity mode. For this type of additive forcing, the frequency of the oscillator remains unchanged and therefore the level spacing of the eigenvalues of the Hamiltonian is not influenced by the force. The spectrum is only shifted as a whole. As a consequence the work performed on the oscillator is, as a positive or negative integer multiple of the level spacing, a discrete random variable. We determined the first few cumulants of the work for arbitrary force protocols and initial states. A complementary study for a parametrically forced oscillator was recently performed by Deffner and Lutz 关15兴. It turns out that for the harmonic oscillator the statistics of work depends on the force protocol 兵f共t兲其t f ,t0 only through two real parameters, which are 共i兲 the shift of the spectrum, given by L共t f 兲 = 兩f共t f 兲兩2 / 共ប兲, and 共ii兲 the absolute value of the dimensionless quantity z = 兰tt f ˙f 共s兲exp共is兲. This param0 eter vanishes for all quasistatic processes and therefore presents a measure of the rapidity of the force protocol. While the presence of L共t f 兲 only causes an overall shift of the possible values of the work, the rapidity parameter 兩z兩 also influences its distribution. Typically, the distributions move toward larger values of work w and become broader with increasing rapidity 兩z兩, indicating a more violent impact on the oscillator. We also demonstrated that different initial states of the system such as microcanonical, canonical, or coherent states have a large influence on the work statistics. We further note that two different initial density matrices with the same diagonal elements with respect to the energy eigenbasis of the Hamiltonian H共t0兲 lead to identical work distributions even though the two density matrices may be very different in other respects. For example, the coherent pure state 兩␣典具␣兩 and the mixed state exp共−兩␣兩2n兲兺n兩␣兩2 / n ! 兩n典具n兩 cannot be distinguished by means of their respective work statistics. This statistics is also insensitive to the phase of a coherent state.
theorem of Stone 关20兴, each of the exponential operators exp关−iuH共t0兲兴 and exp关iuHH共t f 兲兴 forms a strongly continuous one-parameter group of unitary operators with parameter u. As the trace of a product of two strongly continuous operator valued functions of u with the density operator ¯共t0兲, which is a trace class operator and independent of u, the characteristic function 共3兲 is a continuous function of u. Proof of property (ii): Gt f ,t0共u兲 is a positive definite function of u. Using the cyclic invariance of the trace and the fact that H共t0兲 and ¯共t0兲 commute with each other, we can rewrite the left-hand side of the inequality 共7兲 as n
Gt ,t 共ui − 兺 i,j f 0
n
u j兲zⴱi z j
= 兺 Tr ei共ui−u j兲HH共t f 兲e−i共ui−u j兲H共t0兲¯共t0兲zⴱi z j i,j
= Tr A†A¯共t0兲 ⱖ 0,
共A1兲
A = 兺 zni e−iuiHH共t f 兲eiuiH共t0兲
共A2兲
where
i
is a bounded operator and A† its adjoint. The last inequality in 共A1兲 immediately follows from the positivity of A†A and of the density matrix ¯共t0兲. Proof of property (iii): Gt f ,t0共0兲 = 1. For u = 0 the exponential operators exp关−iuH共t0兲兴 and exp关iuHH共t f 兲兴 become unity. By means of Eqs. 共5兲 and 共6兲 the trace over the density matrix ¯共t0兲 reduces to the trace of the initial density matrix 共t0兲, which is 1. APPENDIX B: THE MATRIX ELEMENT Šn円exp[iuHH(tf)]円n‹
The total time rate of change of the Hamiltonian HH共t兲 coincides with its partial derivative with respect to time. Hence, we obtain with Eq. 共17兲 for the driven oscillator dHH共t兲 ˙ ⴱ † 共t兲, = f 共t兲aH共t兲 − ˙f 共t兲aH dt
† 共t兲 denote annihilation and creation opwhere aH共t兲 and aH erators, respectively, in the Heisenberg picture, which are given by
i ប
aH共t兲 = e−ita −
ACKNOWLEDGMENTS
This work has been supported by the Deutsche Forschungsgemeinschaft via the Collaborative Research Centre SFB-486, Project No. A10. Financial support of the German Excellence Initiative via “Nanosystems Initiative Munich” 共NIM兲 is gratefully acknowledged as well. APPENDIX A: PROOF OF THE PROPERTIES OF Gtf,t0(u , v)
We prove that the conditions of Bochner’s theorem are satisfied, and consequently Gt f ,t0共u兲 is a proper characteristic function. Proof of property (i): Gt f ,t0共u兲 is a continuous function of u. The Hamiltonian operators at the two times of measurement t0 and t f are self-adjoint operators. According to the
共B1兲
冕
t
ds e−i共t−s兲 f共s兲
共B2兲
ds ei共t−s兲 f ⴱ共s兲.
共B3兲
0
and † 共t兲 = eita† + aH
i ប
冕
t
0
This yields for HH共t f 兲 HH共t f 兲 = បa†a + Bⴱ共t f 兲a + B共t f 兲a† + C共t兲, where
011115-8
B共t f 兲 =
冕
tf
0
ds ˙f 共s兲eis ,
共B4兲
STATISTICS OF WORK PERFORMED ON A FORCED …
C共t f 兲 =
i ប
冕 冕 tf
s
ds
0
PHYSICAL REVIEW E 78, 011115 共2008兲
ds⬘关f˙ 共s兲f ⴱ共s⬘兲ei共s−s⬘兲
0
− ˙f ⴱ共s兲f共s⬘兲e−i共s−s⬘兲兴.
共B5兲
The unitary operator
which allow us to represent the nth powers of shifted creation and annihilation operators by derivatives of the correⴱ sponding order. The scalar function e−i共xz+yz 兲 can be taken out of the scalar product and the remaining operator can be brought into normal order. It then becomes 关23兴 †
V = eza
†−zⴱa
with B共t f 兲 z= ប
共B7兲
VHH共t f 兲V† = បa†a + L共t f 兲,
共B8兲
transforms HH共t兲 into
+ eiuប共xa + ya† + xy兲兴其,
具n兩eiuHH共t f 兲兩n典 =
共B9兲 =
Va†V† = a† − zⴱ ,
1 iuL共t 兲 2n e f exp关共eiuប − 1兲兩z兩2兴 n n n! x y ⫻exp兩关共eiuប − 1兲共xz + yzⴱ兲 + eiuបxy兴兩x=y=0
Note that V induces a shift of the creation and annihilation operators, VaV† = a − z,
1 iuL共t 兲 n e f exp关共eiuប − 1兲兩z兩2兴 n n! y ⫻ 关共eiuប − 1兲z + eiuy兴nexp兩关共eiuប − 1兲
共B10兲
⫻共兩z兩2 + yzⴱ兲兴兩y=0
and further note that, when acting on the ground state 兩0典 with a兩0典 = 0, the operator V yields the coherent state 兩z典, i.e.,
= eiu兩f共t f 兲兩
V兩0典 = 兩z典.
⫻兺
n
共B11兲
k=0
One finds with these properties 具n兩e
iuHH共t f 兲
e
e
兩z典兩x=y=0 . 共B12兲
Here we have introduced the auxiliary variables x and y
⬁
qrc共˜兲
=e
−兩z兩2
n 兩z兩2共n−k兲 iuបk iuប 共e − 1兲2共n−k兲 . e k 共n − k兲!
To determine the expression 共36兲 for the work distribution qc共˜兲, we start from the general expression given in the first equality of Eq. 共27兲. Interchanging the summation over the indices n and k, we obtain
冉 冊
2k
⬁
˜
冉冊
兩z兩2共k+m兲 2k e −n n 共− 1兲 ␦l+m,k+r 兺 兺 兺 ˜ m ! k! l n=k 1 − e− k m,k=0 l=0
共1兲
= e−兩z兩
l
⬁
2
兺
2k
兺 共− 1兲l
m,k=0 l=0
共2兲
冉冊
exp关共eiuប − 1兲兩z兩2兴
APPENDIX C: WORK DISTRIBUTION FOR A CANONICAL INITIAL STATE
2n 1 = eiuL共t f 兲 n n n! x y ⫻兩具z兩e
2/共ប兲
共B14兲
1 † 兩n典 = 具z兩共a − z兲neiuបa a+iuL共t f 兲共a† − zⴱ兲n兩z典 n!
x共a−z兲 iuបa+a y共a+−zⴱ兲
⬁
r −兩z兩2
= 共− 1兲 e
兺
m=0
共B13兲
where under the normal ordering operator N all creation operators stand left of the annihilation operators. The matrix element with respect to the coherent state 兩z典 can be read off, yielding
where 兩B共t f 兲兩2 兩f共t f 兲兩2 = . L共t f 兲 = C共t f 兲 − ប ប
†
exaeiuបa aeya = N兵exp关共eiuប − 1兲a†a
共B6兲
冉 冊冉 冊
兩z兩2共k+m兲 2k m ! k! l
e − 1
⬁
˜
1
˜
k
␦l+m,k+r
冉
2k 共− 兩z兩2兲m 关− 兩z兩2/共e − 1兲兴k 兺 m! k=兩m−r兩 k! k+r−m 011115-9
冊
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
冉
⬁
共3兲
= 共− 1兲re−兩z兩
共4兲
= e−兩z兩
2
共− 兩z兩2兲m −2兩z兩2/共e˜−1兲 2兩z兩2 e I兩m−r兩 − ˜ m! e − 1
兺 m=0
2coth共˜ /2兲
⬁
兺
m=0
共5兲
= e
−兩z兩2coth共˜/2兲
= e−兩z兩
再
⬁
兺
m=0
2coth共˜ /2兲 ˜r/2
e
冉 冊
兩z兩2m 2兩z兩2 I兩m−r兩 ˜ m! e − 1
Ir
冉 冊 冉 冊 兩z兩2 . sinh共˜/2兲
共1兲
xk
兺 n=k 1 − x
冉冊 冉 冊 n x = k 1−x
k
共C2兲 共2兲
共cf. Ref. 关24兴, formula 5.2.11.3兲. In the second step 共 = 兲 the Kronecker ␦ is used to perform the sum over k. The third
共C1兲
APPENDIX D: WORK DISTRIBUTION FOR A COHERENT INITIAL STATE EQ. (45)
Starting from Eq. 共27兲,we may proceed in an analogous way as in the case of a canonical initial state; cf. Appendix C. According to Eq. 共43兲 a Poissonian average over the binomial 共 nk 兲 has to be performed instead of the geometric average in the first step of Eq. 共C1兲. This yields ⬁
共3兲
兺 n=k
step 共 = 兲 is based on the relation ⬁
xk
兺 k!
k=兩l兩
冉 冊 2k
k+l
= e2xI兩l兩共2x兲
共C3兲
valid for integer l. Here I共x兲 denotes the modified Bessel function of the first kind of order . With I共−x兲 = 共−1兲I共x兲, where is an integer, we come to the right-hand 共4兲
tk
I−k共x兲 = 兺 k=0 k!
冉 冊 2t +1 x
/2
I共冑x2 + 2tx兲
⬁
兺
k=兩m−r兩
=
共C4兲
共cf. 关24兴 5.8.3.1兲. This leads to the final result given in Eq. 共36兲.
关1兴 G. N. Bochkov and Yu. E. Kuzovlev, Sov. Phys. JETP 45, 125 共1977兲. 关2兴 D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 共1993兲. 关3兴 C. Jarzynski, Phys. Rev. Lett. 78, 2690 共1997兲. 关4兴 C. Jarzynski, C. R. Phys. 8, 495 共2007兲. 关5兴 F. Douarche, S. Ciliberto, A. Petrosyan, and I. Rabbiosi, Europhys. Lett. 70, 593 共2005兲.
冉冊
兩␣兩2n −兩␣兩2 n 兩␣兩2k e . = n! k k!
共D1兲
Next the Kronnecker ␦ is used to perform the sum over l, leaving one with two sums of which the inner one over k can be expressed in terms of a generalized hypergeometric function 关21兴, to become
side of the equality 共 = 兲. In the next step the sum on m is rewritten. The term in the square brackets vanishes because I共x兲 is an even function of order . The remaining sum can be performed by means of the identity ⬁
冋 冉 冊 冉 冊册冎
⬁
兩z兩2 兩z兩2 2兩z兩2 2兩z兩2 2兩z兩2 Ir−m ˜ + 兺 Im−r ˜ − Ir−m ˜ m! m=r+1 m! e − 1 e − 1 e − 1
In the first step 共 = 兲 we performed the sum on n according to ⬁
冊
冉
2k 共− 兩␣z兩2兲k 2 共k!兲 k+r−m
冉
冊
共− 兩␣z兩2兲兩m−r兩 F2 兩m − r兩 共兩m − r兩!兲2 1
冊
1 + ;兩m − r兩 + 1,2兩m − r兩 + 1;− 4兩␣z兩2 . 2
共D2兲
This immediately leads to the expression in Eq. 共45兲.
关6兴 C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today 58 共7兲, 43 共2005兲. 关7兴 V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger, Phys. Rev. Lett. 96, 070603 共2006兲. 关8兴 H. Tasaki, e-print arXiv:cond-mat/0009244. 关9兴 S. Mukamel, Phys. Rev. Lett. 90, 170604 共2003兲. 关10兴 W. De Roeck and C. Maes, Phys. Rev. E 69, 026115 共2004兲. 关11兴 M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129
011115-10
STATISTICS OF WORK PERFORMED ON A FORCED …
PHYSICAL REVIEW E 78, 011115 共2008兲
共2006兲. 关12兴 P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75, 050102共R兲 共2007兲. 关13兴 P. Talkner and P. Hänggi, J. Phys. A 40, F569 共2007兲. 关14兴 P. Talkner, P. Hänggi, and M. Morillo, Phys. Rev. E 77, 051131 共2008兲. 关15兴 S. Deffner and E. Lutz, Phys. Rev. E 77, 021128 共2008兲. 关16兴 U. Seifert, J. Phys. A 37, L517 共2004兲. 关17兴 K. Husimi, Prog. Theor. Phys. 9, 381 共1953兲. 关18兴 E. Lukacs, Characteristic Functions 共Griffin, London, 1970兲. 关19兴 B. Cleuren, C. Van den Broeck, and R. Kawai, Phys. Rev. Lett.
96, 050601 共2006兲. 关20兴 K. Yosida, Functional Analysis 共Springer-Verlag, Berlin, 1971兲. 关21兴 L. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products 共Academic, San Diego, 2000兲. 关22兴 N. G. van Kampen, Stochastic Processes in Physics and Chemistry 共North-Holland, Amsterdam, 1992兲. 关23兴 R. M. Wilcox, J. Math. Phys. 8, 962 共1967兲. 关24兴 A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series 共Gordon and Breach, New York, 1986兲, Vol. 1.
011115-11
Statistics of work performed on a forced quantum oscillator Peter Talkner, P. Sekhar Burada, and Peter Hänggi Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany 共Received 19 March 2008; published 18 July 2008兲 Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator. In this special case two parameters are introduced that are sufficient to completely characterize the force protocol. Explicit results for the characteristic function of work and the corresponding probability distribution are provided and discussed for three different types of initial states of the oscillator: microcanonical, canonical, and coherent states. Depending on the choice of the initial state the probability distributions of the performed work may greatly differ. This result in particular also holds true for identical force protocols. General fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed. DOI: 10.1103/PhysRevE.78.011115
PACS number共s兲: 05.30.⫺d, 05.70.Ln, 05.40.⫺a
I. INTRODUCTION
During the last decade various fluctuation and work theorems 关1,2兴 have been formulated and discussed. They characterize among other things the full nonlinear response of a system under the action of a time-dependent force 关3,4兴. These theorems have been derived and experimentally confirmed primarily for classical systems 关5–7兴. Quantum mechanical generalizations were proposed recently 关8–15兴. Conceptual problems arise, though, in the context of quantum mechanics if one tries to generalize those classical relations that require, for example, the specification of a system’s trajectory extending over some interval of time, or the simultaneous measurement of noncommuting observables. For example, the measurement of work performed by an external force on an otherwise isolated system may be accomplished in the framework of classical physics in principle in two different ways. The first method is based on two measurements of the energy, one at the beginning and the second at the end of the considered process. This method becomes unreliable in practice if the system is large and the work performed on the system is negligibly small compared to the total energy of the system. Such a situation typically arises if the system of interest, on which the force exclusively acts, interacts with its environment. In order to retain an isolated system, the large system made of the open system and its environment must be considered. Again, the work performed on the system results as the difference of the energies of the total system, which may both be very large. For classical systems, this unfortunate situation can be circumvented by a second method, by monitoring the state of the relevant small system during the time when the force is acting. Having this information at hand, one can determine the work by integrating the power supplied to the system at each instant of time. The respective power can be inferred from the registered state of the system and the known force protocol. In a quantum system, a continuous measurement of even a single observable would strongly influence and possibly manifestly distort the system’s dynamics. Clearly, only the first of the two methods of energy measurement is feasible, at least in principle, in the quantum context. An alternative method based on continuous monitoring has recently been suggested by Esposito and Mukamel 关11兴 1539-3755/2008/78共1兲/011115共11兲
for open quantum systems described by Markovian quantum master equations. There the dynamics of the density matrix is mapped onto a classical rate process for which known fluctuation theorems can be applied 关16兴. This provides an interesting formal approach but its physical meaning has remained unclear 关11兴. Moreover, this approach is restricted to open systems that only weakly interact with their respective environments. In the present paper, the distribution of work is discussed for the exactly solvable system of a driven harmonic oscillator 关15,17兴. In this case, the distribution of work is discrete. We provide formal expressions for this distribution and its corresponding characteristic function which are valid for all initial states of the system as well as for all possible kinds of force protocols. In particular, we determine the characteristic functions and distributions of the work for microcanonical, canonical, and coherent initial states which lead to qualitatively different work distributions. The paper is organized as follows. In Sec. II we review the general form of the characteristic function of work performed on a system in terms of a correlation function of the exponentiated Hamiltonians at the initial and final times of the force protocol. We prove that this particular expression indeed always represents a characteristic function, i.e., the Fourier transform of a probability density. Section III presents various fluctuation and work theorems for canonical and microcanonical initial states. In Sec. IV general expressions for the characteristic function and the corresponding probability distribution of work are derived for a driven harmonic oscillator. Moreover, the expressions for the first four cumulants are derived. The dependence of the work distribution on the force protocol for microcanonical, canonical, and coherent initial states as well as its dependence on the specific parameters of these initial states are investigated. II. CHARACTERISTIC FUNCTION OF WORK
The response of a quantum system on a perturbation by a classical, external force can be characterized by the change of energies contained in the total system. The energy as an observable coincides with the Hamiltonian H共t兲 of the total system. It includes the external force and therefore depends
011115-1
©2008 The American Physical Society
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
on time. We will consider the dynamics of the system only within a finite window of time 关t0 , t f 兴 during which the force is acting in a prescribed way, resulting in a protocol of Hamiltonians which is denoted by 兵H共t兲其t f ,t0. Apart from the action of the external force the system is assumed to be closed. Its dynamics is consequently governed by a unitary time evolution Ut,t0, which is the solution of the Schrödinger equation iប Ut,t0/ t = H共t兲Ut,t0,
Ut0,t0 = 1.
共1兲
As explained in the Introduction, the work w is measured as the difference between the energies of the system at the final and initial times t f and t0. In a single measurement the work is given by the difference of two eigenvalues en共t f 兲 and em共t0兲 of the Hamiltonians H共t兲 at the respective times t f and t0, i.e., by w = en共t f 兲 − em共t0兲. The inherent randomness of the outcome of a quantum measurement in general leads to a measured work that is random. A complete description of the statistical properties of the work performed on the system is provided by the characteristic function Gt f ,t0共u兲 which presents the Fourier transform of the probability density of the work pt f ,t0共w兲, i.e., Gt f ,t0共u兲 =
冕
dw eiuw pt f ,t0共w兲.
共2兲
It can be expressed as a quantum correlation function of the exponentiated Hamiltonian at the initial and the final times 关14兴, i.e., Gt f ,t0共u兲 = 具eiuH共t f 兲e−iuH共t0兲典 ⬅ Tr eiuHH共t f 兲e−iuH共t0兲¯共t0兲, 共3兲
n
兺 Gt ,t 共ui − ui⬘兲zⴱi zi⬘ ⱖ 0 i,i⬘
共7兲
f 0
holds. Here, the asterisk zⴱi denotes the complex conjugate of z i. 共iii兲 Gt f ,t0共0兲 = 1. According to a theorem by Bochner 关18兴 the properties 共i兲–共iii兲 are necessary and sufficient conditions for the function Gt f ,t0共u兲 to be the Fourier transform of the probability measure of a random variable. In short, the first condition ensures that, strictly speaking, the function Gt f ,t0共u兲 is the Fourier transform of a measure, the second condition assures that this measure is positive, and the third condition that it is normalized. Hence the correlation expression Eq. 共3兲 always defines a proper characteristic function. A proof of the properties 共i兲–共iii兲 can be found in Appendix A. III. CANONICAL AND MICROCANONICAL INITIAL STATES
In experiments an external force is often applied on a system that initially is found in a thermodynamic equilibrium state. Depending on whether the system was in weak contact with a heat bath or was totally isolated from its environment, the initial state of the system is described by either a canonical or a microcanonical density matrix. For both situations fluctuation and work theorems are known. We will briefly review these relations. A. Work and fluctuation theorems for canonical initial states
where
If the initial density matrix is canonical, i.e., if HH共t f 兲 =
U†t ,t H共t f 兲Ut f ,t0 f 0
共4兲
denotes the Hamiltonian in the Heisenberg picture. The density matrix ¯共t0兲 follows from the initial density matrix 共t0兲 as a result of the measurement of the Hamiltonian H共t0兲. It is given by ¯共t0兲 = 兺 Pn共t0兲共t0兲Pn共t0兲,
共5兲
n
where the operators Pn共t0兲 denote the eigenprojection operators of the Hamiltonian at time t0, which present a partition of unity:
兺n Pn共t0兲 = 1.
共6兲
Before we apply the general expression 共3兲 to a particular system and investigate its dependence on the initial state 共t0兲, we discuss three general properties of the correlation expression 共3兲 which guarantee that the resulting function Gt f ,t0共u兲 indeed always presents a proper characteristic function of a classical random variable w. 共i兲 Gt f ,t0共u兲 is a continuous function of u. 共ii兲 Gt f ,t0共u兲 is a positive definite function of u, i.e., for all integer numbers n, all real sequences u1 , u2 , . . . , un, and all complex numbers zi, i = 1 , 2 , . . . , n,
共t0兲 = Z−1共t0兲exp关− H共t0兲兴,
共8兲
Z共t0兲 = Tr exp关− H共t0兲兴 = e−F共t0兲
共9兲
where
denotes the partition function and F共t0兲 the free energy, then 关H共t0兲 , 共t0兲兴 = 0 and the first measurement of the energy leaves the density matrix unchanged, such that ¯共t0兲 = 共t0兲. With Eq. 共3兲 this leads to the characteristic function of work for a canonical initial state that was derived in Ref. 关12兴. In this case, Gt f ,t0共u兲 can be continued to an analytical function of u for all 0 ⱕ Iu ⱕ  关13兴, where Iu denotes the imaginary part of u. For the particular value u = i the characteristic function yields the mean value of the exponentiated work, 具exp共−w兲典 and the correlation function expression 共3兲 simplifies to the ratio of the partition functions at the times t f and t0, resulting in the Jarzynski work theorem 具e−w典 = Z共t f 兲/Z共t0兲 = exp兵− 关F共t f 兲 − F共t0兲兴其,
共10兲
where Z共t f 兲 = Tr exp关−H共t f 兲兴 = exp关−F共t f 兲兴. Within the domain of analyticity S = 兵兩u兩0 ⱕ Iu ⱕ 其 the characteristic functions for the original and the time-reversed protocol are related to each other by the following formula 共cf. 关13兴兲:
011115-2
STATISTICS OF WORK PERFORMED ON A FORCED …
Gt f ,t0共u兲 =
Z共t f 兲 Gt ,t 共− u + i兲, Z共t0兲 0 f
PHYSICAL REVIEW E 78, 011115 共2008兲
共11兲
where Gt0,t f 共u兲 refers to processes under the time-reversed protocol 兵H共t兲其to,t f starting from the canonical state Z−1共t f 兲exp关−H共t f 兲兴. An inverse Fourier transform leads to the Tasaki-Crooks fluctuation theorem, which relates the probability densities of work pt f ,t0共w兲 for a given protocol to the density of the work pt0,t f 共w兲 for the time-reversed protocol. This theorem explicitly reads 关13兴 pt f ,t0共w兲 pt0,t f 共− w兲
=
Z共t f 兲 w −关F共t 兲−F共t 兲−w兴 f 0 e =e . Z共t0兲
共12兲
B. Fluctuation theorems for microcanonical initial states
A system in a microcanonical state is described by the density matrix
共t0兲 = E−1共t0兲␦„H共t0兲 − E…,
共13兲
E共t0兲 = Tr ␦„H共t0兲 − E… = exp关S共E,t0兲/kB兴
共14兲
entropy can be determined. For further details see Ref. 关14兴. IV. DRIVEN HARMONIC OSCILLATOR
To illustrate these concepts we consider an example that allows the analytical construction of the probability of work. Specifically, we consider a harmonic oscillator on which a time-dependent force acts during a finite interval of time. Its time evolution is governed by the Hamiltonian H共t兲 = បa†a + f ⴱ共t兲a + f共t兲a† ,
where denotes the angular frequency, and a and a creation and annihilation operators, respectively, which obey the usual commutation relation, i.e., 关a , a†兴 = 1. The complex driving force f共t兲 allows for a coupling to position and/or momentum of the oscillator. We assume that f共t兲 vanishes for times t ⱕ t0 = 0. It is our aim to study the influence of the initial state 共t0兲 on the statistics of work performed on the oscillator. The measurement of H共t0兲 = បa†a at time t0 = 0 then yields the result បn with probability pn = 具n兩共t0兲兩n典.
where
denotes the density of states as a function of the energy E of the system. The density of states can be expressed in terms of the entropy of the system SE共t0兲 provided the spectrum of the system Hamiltonian is sufficiently dense such that the density of states becomes a smooth function on a coarsened energy scale. The microcanonical density matrix commutes with the Hamiltonian H共t0兲. Consequently, ¯共t0兲 and 共t0兲 coincide. The microcanonical quantum Crooks theorem assumes the form 关14兴 pt f ,t0共E,w兲 pt0,t f 共E + w,− w兲
=
冉
冊
S共E + w,t f 兲 − S共E,t0兲 E+w共t f 兲 = exp . E共t0兲 kB 共15兲
Analogous to the canonical case, it relates the probability density pt0,t0共E , w兲 of work w, for a system starting in a microcanonical state with energy E, to the respective quantity for the time-reversed process starting at energy E + w. This quantum theorem is formally identical to the corresponding classical theorem 关19兴. From the microcanonical Crooks theorem the probability density of the time-reversed process can be eliminated to yield the so-called entropy-from-work theorem 关14兴, reading E f 共t f 兲 =
冕
dw E f −w共t0兲pt f ,t0共E f − w,w兲.
共16兲
This theorem allows one to determine the unknown density of states of a system with Hamiltonian H共t f 兲 from the known density of states of a reference system H共t0兲 by means of the statistics of the work that is performed on the system in a process that leads from the reference system to the final system with unknown density of states. In the case of systems with a sufficiently smooth density of states the corresponding
共17兲
†
共18兲
Accordingly, the oscillator is found in the state ¯共t0兲 = 兺 pn兩n典具n兩
共19兲
n
immediately after this measurement. Substituting this density matrix in the general expression for the characteristic function, Eq. 共3兲, one obtains Gt f ,t0共u兲 = 兺 pne−iuបn具n兩eiuHH共t f 兲兩n典.
共20兲
n
For the driven harmonic oscillator the diagonal matrix element of the exponentiated Hamiltonian HH共t f 兲 can be determined 关17兴. For details see Appendix B. With the expression 共B14兲 for the matrix element 具n兩exp关iHH共t f 兲兴兩n典 we find Gt f ,t0共u兲 = eiu兩f共t f 兲兩 ⬁
2/共ប兲
n
⫻ 兺 兺 pn n=0 k=0
= eiu兩f共t f 兲兩
冉冊
2/共ប兲
⬁
exp关共eiuប − 1兲兩z兩2兴
冉
n 兩z兩2共n−k兲 −iuប共n−k兲 iuប 共e − 1兲2共n−k兲 e k 共n − k兲!
exp关共eiuប − 1兲兩z兩2兴
⫻ 兺 pnLn 4兩z兩2sin2 n=0
冊
បu , 2
共21兲
where 兩f共t f 兲兩2 / 共ប兲 denotes a uniform shift of the spectrum of the harmonic oscillator due to the presence of the external force 关cf. Eq. 共B9兲兴 and z=
1 ប
冕
tf
ds ˙f 共s兲exp共is兲
共22兲
0
is a dimensionless functional of the driving force f共t兲 关cf. Eq. 共B7兲兴. This dimensionless quantity vanishes in particular for all-quasistatic forcings, i.e., if the force changes only very slowly with f共t兲 = g共t / t f 兲 for t f → ⬁, where g共兲 is a continuously differentiable function of 僆 关0 , 1兴. We hence call z共t兲
011115-3
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
the rapidity parameter of the force protocol. Finally, Ln共x兲 n = 兺k=0 共 nk 兲共−x兲k / k! denotes the Laguerre polynomial of order n 关21兴. Introducing the cumulant generating function K共u兲 = ln G共u兲, one obtains the cumulants of work kn as the nth derivatives of K共u兲 with respect to u taken at u = 0 关22兴, i.e., kn = 共−i兲ndnK共0兲 / dun. The first four cumulants become k1 = 具w典 =
兩f共t f 兲兩2 + ប兩z兩2 , ប
A. Distributions of work for different initial states
As particular examples of initial states we will discuss microcanonical, canonical, and coherent states. 1. Microcanonical initial state
For a microcanonical initial state with energy បn0 the density matrix becomes
共23兲
冉
冊
1 , 2
k2 = 具w2典 − 具w典2 = 2共ប兲2兩z兩2 具a†a典0 +
共24兲
共t0兲 = ¯共t0兲 = 兩n0典具n0兩. The characteristic function then reads iu兩f共t f 兲兩 Gmc t ,t 共n0,u兲 = e f 0
k3 = 具w3典 − 3具w2典具w典 + 2具w典3 = 共ប兲3兩z兩2 ,
冉
共25兲
共26兲 The odd cumulants of the work are independent of the initial preparation. The even cumulants depend on the factorial moments of the number operator a†a with respect to the initial state ¯共t0兲 such as 具a†a典0 = 兺nnpn and 具a†a共a†a − 1兲典0 = 兺nn共n − 1兲pn, where pn is defined in Eq. 共18兲. Moreover, all cumulants apart from the first one vanish for forcings with z = 0. This holds true in particular for all quasistatic force characteristics. The underlying work probability density then shrinks to a ␦ function at w = 兩f共t f 兲兩2 / 共ប兲. In general, the work probability density follows from the characteristic function by means of an inverse Fourier transformation. Rather than the characteristic function itself, we first consider the function G共u兲 ⬅ exp关−iu兩f共t f 兲兩2 / 共ប兲兴 ⫻Gt f ,t0共u兲. Upon expanding exp关兩z兩2exp共iuប兲兴 into a series of powers of 兩z兩2, we obtain for G共u兲 a Laurent series in the variable exp关iuប兴. The inverse Fourier transformation is given by a series of ␦ functions ␦共w − បr兲, with r 僆 Z, with weights ⬁
qr = e
−兩z兩2
=e
2k
冉 冊冉 冊
兩z兩2共k+m兲 n k
兺 兺 兺 共− 1兲2k−lpn m ! k! m,n=0 k=0 l=0 ⬁
−兩z兩2
n
n min共k+r,2k兲
兺兺 兺 l=0 n=0 k=0
共− 1兲2k−l pn
2k l
␦l+m,k+r
冉 冊冉 冊
兩z兩2共2k+r−l兲 n 共k + r − l兲 ! k! k
2k l
.
2/共ប兲
exp关共eiuប − 1兲兩z兩2兴
⫻Ln0 4兩z兩2sin2
k4 = 具w4典 − 4具w3典具w典 − 3具w2典2 + 12具w2典具w典2 − 6具w典4 = 共ប兲4兩z兩2兵1 + 4具a†a典0 + 6关具a†a共a†a − 1兲典0 − 2具a†a典0兴兩z兩2其.
共29兲
បu 2
冊
共30兲
and, accordingly, the probability qrmc共n0兲 to find a change of energy by w = បr + 兩f共t f 兲兩2 / 共ប兲 emerges as qrmc共n0兲 = e−兩z兩
2
n0 min共k+r,2k兲
兺 兺 k=0 l=0
冉 冊冉 冊
n 共− 1兲2k−l 共k + r − l兲 ! k! k
2k l
兩z兩2共2k+r−l兲 . 共31兲
As expected from the behavior of the moments, all probabilities qrmc共n0兲 with r ⫽ 0 vanish for quasistatic forcing, i.e., if z → 0. The dependence of qrmc共n0兲 on the parameter z is displayed in Fig. 1 for n0 = 0 and 3 as well as for the eight lowest values of r. With increasing values of the rapidity parameter z the distribution is becoming broader. For the fixed value of 兩z兩 = 2 the distribution qrmc共n0兲 is compared for the three initial states with n0 = 0, 10, and 30 in Fig. 2. With increasing value of n0 the distributions become broader. They develop a slightly asymmetric shape with higher peaks at negative values of r compared to those at positive r values. Between these dominant peaks the probability still displays pronounced variations. For a harmonic oscillator, the microcanonical Crooks mc 共n + r兲. One can theorem reduces to the relation qrmc共n兲 = q−r prove that this symmetry is satisfied by the probabilities qrmc共n兲 given by Eq. 共31兲. As a consequence, the ratio mc 共n + r兲 is unity independently of the actual values qrmc共n兲 / q−r of the initial energy, the work, and the force protocol as given by n, r, and z, respectively.
共27兲 2. Canonical initial state
The factor exp关−iu兩f共t f 兲兩2 / 共ប兲兴, by which G共u兲 has to be multiplied to yield Gt f ,t0共u兲, gives rise to a constant shift such that the probability density of work performed on a harmonic oscillator becomes
冠 冉
pt f ,t0共w兲 = 兺 qr␦ w − បr + r
兩f共t f 兲兩2 ប
冊冡
.
共28兲
In the next section we will investigate the influence of the initial state on the statistics of the work.
For a canonical density matrix
共t0兲 = 共1 − e−ប兲e−បa
†a
共32兲
the initial states are distributed according to pn = e−បn共1 − e−ប兲. This allows one to write the sum over n in the characteristic function 共21兲 in terms of the generating function of the Laguerre polynomials 共cf. 关21兴兲 yielding the expression
011115-4
STATISTICS OF WORK PERFORMED ON A FORCED …
PHYSICAL REVIEW E 78, 011115 共2008兲 ⬁
1.0
n min共k+r,2k兲
˜ 2 qrc共˜兲 = e−兩z兩 共1 − e−兲 兺 兺
n=0 k=0
qrmc (0)
⫻
0.5
0
1
2
z
(a)
3
4
5
6
4
3
2
1
0
−1
−2
−3
兺 l=0
冉 冊冉 冊
n 兩z兩2共2k+r−l兲 共k + r − l兲 ! k! k
2k l
˜
共− 1兲le−n 共35兲
,
where ˜ = ប denotes the inverse dimensionless temperature of the initial state. The expression for qrc共˜兲 can be further simplified to read
r
˜ ˜ 2 qrc共˜兲 = e−兩z兩 coth共/2兲er/2Ir
冉
冊
兩z兩2 , sinh ˜/2
共36兲
where I共x兲 denotes the modified Bessel function of the first kind of order 关21兴. For details of the derivation see Appendix C. Note that the following detailed-balance-like symmetry relation exists:
1.0
qrmc (3) 0.5
˜ c ˜ 共兲 = e−rqrc共˜兲, q−r
1
2
z
(b)
3
4
5
6
4
3
2
1
0
−2
−3
r
FIG. 1. Probabilities qrmc共n0兲 for two microcanonical initial states with n0 = 0 共a兲 and 3 共b兲 are depicted for r = −3 , . . . , 4, as functions of the rapidity parameter z defined in Eq. 共B7兲. In both cases the distribution collapses at r = 0 for qausi—static forcing, corresponding to 兩z兩 = 0, and broadens with increasing 兩z兩. Obviously, when starting in the ground state the oscillator cannot deliver work, whence the probability for negative r strictly vanishes. “Stimulated emission” becomes possible from an excited state at finite driving rapidity z, leading to nonzero probabilities qrmc共n0兲 at negative values of r in 共b兲.
冉
Gct ,t 共,u兲 = exp f 0
iu兩f共t f 兲兩2 + 共eiuប − 1兲兩z兩2 ប
冊
sin2共បu/2兲 . − 4兩z兩2 ប e −1
0.2
f 0
n0 = 0 n0 = 10 n0 = 30
共33兲 0.15
Putting u = i, one finds that the two terms in the exponent that are proportional to 兩z兩2 cancel each other, such that one obtains 具e−w典 = Gct ,t 共,i兲 = exp关− 兩f共t f 兲兩2/共ប兲兴.
relating the occurrence of positive and negative work. Figure 3 illustrates this relation, which is closely connected to the Tasaki-Crooks theorem 共12兲, as demonstrated at the end of this section. In Fig. 4 the z dependence of qrc共˜兲 for ˜ = ln共4 / 3兲 is compared for a few small values of r. One finds that, because of the average over the canonical initial distribution, the multipeaked structure of the microcanonical distribution as a function of the rapidity parameter 兩z兩 disappears, and only a single peak remains for each value of r. The temperature dependence of the work distribution is illustrated in Fig. 5. Finally, we verify the validity of the Tasaki-Crooks theorem 共12兲 for a driven oscillator. For this purpose we consider the probability density pt0,t f 共−w兲 for the time-reversed protocol. Since the absolute values of the rapidity parameters coincide for the original and the time-reversed protocols, the probability density of work for the reversed protocols becomes
qrmc (n0 )
0
−1
共37兲
共34兲
The free energy difference of two oscillators with Hamiltonians H共t0兲 = បa†a and H共t f 兲 = បa†a + f ⴱ共t f 兲a + f共t兲a†, each one staying in a canonical state at the temperature , is given by ⌬F = F共t f 兲 − F共t0兲 = 兩f共t f 兲兩2 / 共ប兲. Hence, Eq. 共34兲 agrees with Jarzynski’s work theorem. The probability qrc共˜兲 to find the work w = បr + 兩f共t f 兲兩2 / 共ប兲 if the system starts in a canonical state becomes
0.1 0.05 0
−20
−10
0
r
10
20
30
FIG. 2. 共Color online兲 Probabilities qrmc共n0兲 for a microcanonical initial state with n0 = 0 共circles兲, 10 共diamonds兲, and 30 共triangles兲 are compared for a fixed rapidity parameter 兩z兩 = 2 and r = −22, . . . , 30. The lines serve as a guide for the eye.
011115-5
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
(1)
0.2
(2) (3)
5 0
(1) β˜ = 2.0 (2) β˜ = 1.0
−5
−10
−5
0
5
r
冠 冉
⬁
兺
qrc共兲␦ − w − បr −
r=−⬁
兩f共t f 兲兩2 ប
冊冡
, 共38兲
where we took into account the overall shift of the spectrum by the reversed protocol as −兩f共t f 兲兩 / 共ប兲. Multiplying both sides of Eq. 共38兲 with exp关−共⌬F − w兲兴 = exp兵−关兩f共t f 兲兩2 / 共ប兲 − w兴其 and using the symmetry 共37兲, one obtains e−共⌬F−w兲 pt0,t f 共− w兲 = 兺 e−关兩f共t f 兲兩 r
0
10
FIG. 3. 共Color online兲 Natural logarithm of the probability ratio of positive and negative transferred work resulting from Eq. 共36兲 as a function of r for different dimensionless temperatures ˜ =  / ប = 0.5, 1 , 2 共circles, squares, lozenges兲 and different rapidity parameters z = 1 , 2 共large open, small filled symbols兲. The symmetry relation 共37兲 implies a linear law with inclination ˜ 共thin straight broken lines兲 which is independent of the rapidity parameter. The agreement is perfect.
pt0,t f 共− w兲 =
0.1 0.05
(3) β˜ = 0.5
−10
β˜ = 0.5 β˜ = 1.0 β˜ = 5.0
0.15
˜ qrc (β)
˜ c (β)) ˜ ln (qrc (β)/q −r
10
−10
−5
0
10
15
= pt f ,t0共w兲,
20
共39兲
in accordance with the Tasaki-Crooks theorem 共12兲. Conversely, if the Tasaki-Crooks theorem holds for a discrete work distribution of the form 共28兲, then the detailed-balancelike relation 共37兲 follows for the probabilities qrc共˜兲. 3. Coherent initial state
An oscillator prepared in a coherent state 兩␣典 is described by the density matrix
共t0兲 = 兩␣典具␣兩,
共40兲
where
冠 冉
qr 共˜兲 兩f共t f 兲兩2 ប
冠 冉
兩 ␣ 典 = e ␣a
冊冡
兩f共t f 兲兩2 ˜ = 兺 erqrc共˜兲␦ w + បr − ប r
†+␣*a
兩0典,
共41兲
and 兩0典 is the normalized ground state of the oscillator satisfying a兩0典 = 0. Note that the coherent state density matrix does not commute with the Hamiltonian H共t0兲. The measurement of H共t0兲 modifies the coherent state 共40兲 by projecting it onto the eigenstates 兩n典 = 共a†兲n / 冑n!兩0典 of this Hamiltonian, leading to
冊冡
¯共t0兲 = e−兩␣兩
1.0
qrc ( ln(4/3))
2
兺n
兩␣兩2n 兩n典具n兩. n!
共42兲
This implies a Poissonian distribution of the respective energy eigenvalues បn,
0.5
pcs n =
1
5
FIG. 5. 共Color online兲 Probabilities qrc共˜兲 for a canonical initial state as functions of r for 兩z兩 = 2 and different values of the dimensionless inverse temperature ˜ = 0.5 共circles兲, 1 共diamonds兲, and 5 共triangles兲. The lines serve as a guide for the eye.
2/共ប兲−w兴 c
⫻␦ − w − បr −
0
r
2
z
3
4
5
6
4
3
2
1
0
−1
−2
−3
r
FIG. 4. Probabilities qrc关˜ = ln共4 / 3兲兴 for a canonical initial state for r = −3 , . . . , 4 as functions of the parameter z. For the sake of comparability, the dimensionless inverse temperature is chosen such that the average energy in the initial state coincides with the energy 3ប of the microcanonical state in Fig. 1共b兲.
兩␣兩2n −兩␣兩2 e , n!
共43兲
which yields for the characteristic function of work 共21兲 a closed expression of the form
冉 冉冏
Gcs t ,t 共␣,u兲 = exp f 0
iu兩f共t f 兲兩2 + 兩z兩2共eiបu − 1兲 ប
⫻J0 4 ␣z sin
បu 2
冏冊
,
冊
共44兲
where J0共x兲 is the Bessel function of order zero 共cf. Ref
011115-6
STATISTICS OF WORK PERFORMED ON A FORCED …
PHYSICAL REVIEW E 78, 011115 共2008兲 0.2
1.0
0.15 0.5
qrcs (α)
qrcs (3)
|α|2 = 0.1 |α|2 = 1 |α|2 = 10
0.1 0.05
−10 0
0 1
2
z
10
0
r
20
3
(a)
FIG. 6. Probabilities qrcs共␣兲 for a coherent state with parameter 兩␣兩 = 3 for r = −10, . . . , 20 as functions of z.
qrcs共␣兲 = e−兩z兩
2
兺
m=0
冉
兩z兩2m兩␣z兩2兩m−r兩 1 F2 兩m − r兩 + ;兩m − r兩 2 m ! 共兩m − r兩!兲2 1
冊
+ 1,2兩m − r兩 + 1;− 4兩␣z兩2 ,
V. CONCLUSIONS
In this work we studied the statistics of work performed on an externally driven quantum mechanical oscillator by
10
20
|z| = 1 |z| = 2 |z| = 4
0.2 0.15 0.1 0.05
共45兲
where 1F2共a ; b , x ; x兲 denotes a generalized hypergeometric function 关21兴. For details of the derivation, see Appendix D. The dependence of the probabilities qrcs共␣兲 on the rapidity parameter 兩z兩 is illustrated in Fig. 6 for r values ranging from −10 to 20. Increasing values of z lead to a broadening of the distribution and also to a shift toward larger values of r; see also Fig. 7共b兲. This broadening and shift is in accordance with Eq. 共23兲 and 共24兲 for the first two cumulants of the work, which both increase with 兩z兩2. Figure 7共a兲 shows the dependence of the probabilities qrcs共␣兲 on the parameter ␣. Increasing ␣ also leads to a broadening of the work distribution without influencing its mean value 关cf. also Eq. 共23兲兴. In Fig. 8 the probabilities qr are depicted for different initial states. In one case the oscillator is initially prepared in a canonical state at inverse dimensionless temperature ˜ = ប = 0.1. In the other case, the oscillator stays in a coherent state 兩␣典, where the absolute value of ␣ is chosen such that the mean excitation number is the same for both states, i.e., 兩␣兩2 = exp共−ប兲 / 关1 − exp共−ប兲兴. For ប = 0.1 one finds 兩␣兩2 ⬇ 9.51. The two oscillators then are subjected to protocols with the same rapidity parameter 兩z兩 = 2. According to Eqs. 共23兲 and 共24兲 the first two moments of the work performed on the oscillators coincide. Yet the distribution of weight factors qrc共˜兲 and qrcs共␣兲 distinctly differ. Whereas the distribution is pronouncedly bimodal in the case of the coherent state, it is unimodal for the canonical state. The weight factors qrc共˜兲 almost perfectly fall onto a Gaussian probability density which has the same first two moments as the discrete distribution given by qr.
r
0
0
(b)
10
r
20
30
FIG. 7. 共Color online兲 Distribution of work performed on an oscillator which initially is prepared in a coherent state 兩␣典 for different values of ␣ in 共a兲 and of the rapidity parameter z in 共b兲. In 共a兲 the rapidity parameter has the value 兩z兩 = 2. In 共b兲 the coherent state parameter has the value 兩␣兩2 = 1.
means of a correlation function expression for the characteristic function of the work. We demonstrated that this particular expression indeed always represents a proper characteristic function of a random variable, which is the performed 0.06
canonical ensemble coherent state
0.04
qr
⬁
0
0.25
qrcs (α)
关21兴兲. For the probability qrcs共␣兲 of work one obtains with Eq. 共27兲
−10
0.02
0
−20
−10
0
10
r
20
30
40
FIG. 8. 共Color online兲 Distribution of work compared for a canonical and a coherent initial state subject to the same force protocol with rapidity parameter z = 2. With ប = 0.1 and 兩␣兩2 ⬇ 9.51, the expectation values of the energies agree in the two initial states such that according to Eqs. 共23兲 and 共24兲 the first and second moments of the work also coincide. Still the distributions of work greatly differ from each other.
011115-7
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
work in the present context. The proof given here is based on Bochner’s theorem. Note that it holds for general quantum mechanical systems, not only for harmonic oscillators. The considered force linearly couples to the position and momentum of the oscillator. It may describe the influence of an electric field on charged particles in a parabolic trap or the external forcing of a single electromagnetic cavity mode. For this type of additive forcing, the frequency of the oscillator remains unchanged and therefore the level spacing of the eigenvalues of the Hamiltonian is not influenced by the force. The spectrum is only shifted as a whole. As a consequence the work performed on the oscillator is, as a positive or negative integer multiple of the level spacing, a discrete random variable. We determined the first few cumulants of the work for arbitrary force protocols and initial states. A complementary study for a parametrically forced oscillator was recently performed by Deffner and Lutz 关15兴. It turns out that for the harmonic oscillator the statistics of work depends on the force protocol 兵f共t兲其t f ,t0 only through two real parameters, which are 共i兲 the shift of the spectrum, given by L共t f 兲 = 兩f共t f 兲兩2 / 共ប兲, and 共ii兲 the absolute value of the dimensionless quantity z = 兰tt f ˙f 共s兲exp共is兲. This param0 eter vanishes for all quasistatic processes and therefore presents a measure of the rapidity of the force protocol. While the presence of L共t f 兲 only causes an overall shift of the possible values of the work, the rapidity parameter 兩z兩 also influences its distribution. Typically, the distributions move toward larger values of work w and become broader with increasing rapidity 兩z兩, indicating a more violent impact on the oscillator. We also demonstrated that different initial states of the system such as microcanonical, canonical, or coherent states have a large influence on the work statistics. We further note that two different initial density matrices with the same diagonal elements with respect to the energy eigenbasis of the Hamiltonian H共t0兲 lead to identical work distributions even though the two density matrices may be very different in other respects. For example, the coherent pure state 兩␣典具␣兩 and the mixed state exp共−兩␣兩2n兲兺n兩␣兩2 / n ! 兩n典具n兩 cannot be distinguished by means of their respective work statistics. This statistics is also insensitive to the phase of a coherent state.
theorem of Stone 关20兴, each of the exponential operators exp关−iuH共t0兲兴 and exp关iuHH共t f 兲兴 forms a strongly continuous one-parameter group of unitary operators with parameter u. As the trace of a product of two strongly continuous operator valued functions of u with the density operator ¯共t0兲, which is a trace class operator and independent of u, the characteristic function 共3兲 is a continuous function of u. Proof of property (ii): Gt f ,t0共u兲 is a positive definite function of u. Using the cyclic invariance of the trace and the fact that H共t0兲 and ¯共t0兲 commute with each other, we can rewrite the left-hand side of the inequality 共7兲 as n
Gt ,t 共ui − 兺 i,j f 0
n
u j兲zⴱi z j
= 兺 Tr ei共ui−u j兲HH共t f 兲e−i共ui−u j兲H共t0兲¯共t0兲zⴱi z j i,j
= Tr A†A¯共t0兲 ⱖ 0,
共A1兲
A = 兺 zni e−iuiHH共t f 兲eiuiH共t0兲
共A2兲
where
i
is a bounded operator and A† its adjoint. The last inequality in 共A1兲 immediately follows from the positivity of A†A and of the density matrix ¯共t0兲. Proof of property (iii): Gt f ,t0共0兲 = 1. For u = 0 the exponential operators exp关−iuH共t0兲兴 and exp关iuHH共t f 兲兴 become unity. By means of Eqs. 共5兲 and 共6兲 the trace over the density matrix ¯共t0兲 reduces to the trace of the initial density matrix 共t0兲, which is 1. APPENDIX B: THE MATRIX ELEMENT Šn円exp[iuHH(tf)]円n‹
The total time rate of change of the Hamiltonian HH共t兲 coincides with its partial derivative with respect to time. Hence, we obtain with Eq. 共17兲 for the driven oscillator dHH共t兲 ˙ ⴱ † 共t兲, = f 共t兲aH共t兲 − ˙f 共t兲aH dt
† 共t兲 denote annihilation and creation opwhere aH共t兲 and aH erators, respectively, in the Heisenberg picture, which are given by
i ប
aH共t兲 = e−ita −
ACKNOWLEDGMENTS
This work has been supported by the Deutsche Forschungsgemeinschaft via the Collaborative Research Centre SFB-486, Project No. A10. Financial support of the German Excellence Initiative via “Nanosystems Initiative Munich” 共NIM兲 is gratefully acknowledged as well. APPENDIX A: PROOF OF THE PROPERTIES OF Gtf,t0(u , v)
We prove that the conditions of Bochner’s theorem are satisfied, and consequently Gt f ,t0共u兲 is a proper characteristic function. Proof of property (i): Gt f ,t0共u兲 is a continuous function of u. The Hamiltonian operators at the two times of measurement t0 and t f are self-adjoint operators. According to the
共B1兲
冕
t
ds e−i共t−s兲 f共s兲
共B2兲
ds ei共t−s兲 f ⴱ共s兲.
共B3兲
0
and † 共t兲 = eita† + aH
i ប
冕
t
0
This yields for HH共t f 兲 HH共t f 兲 = បa†a + Bⴱ共t f 兲a + B共t f 兲a† + C共t兲, where
011115-8
B共t f 兲 =
冕
tf
0
ds ˙f 共s兲eis ,
共B4兲
STATISTICS OF WORK PERFORMED ON A FORCED …
C共t f 兲 =
i ប
冕 冕 tf
s
ds
0
PHYSICAL REVIEW E 78, 011115 共2008兲
ds⬘关f˙ 共s兲f ⴱ共s⬘兲ei共s−s⬘兲
0
− ˙f ⴱ共s兲f共s⬘兲e−i共s−s⬘兲兴.
共B5兲
The unitary operator
which allow us to represent the nth powers of shifted creation and annihilation operators by derivatives of the correⴱ sponding order. The scalar function e−i共xz+yz 兲 can be taken out of the scalar product and the remaining operator can be brought into normal order. It then becomes 关23兴 †
V = eza
†−zⴱa
with B共t f 兲 z= ប
共B7兲
VHH共t f 兲V† = បa†a + L共t f 兲,
共B8兲
transforms HH共t兲 into
+ eiuប共xa + ya† + xy兲兴其,
具n兩eiuHH共t f 兲兩n典 =
共B9兲 =
Va†V† = a† − zⴱ ,
1 iuL共t 兲 2n e f exp关共eiuប − 1兲兩z兩2兴 n n n! x y ⫻exp兩关共eiuប − 1兲共xz + yzⴱ兲 + eiuបxy兴兩x=y=0
Note that V induces a shift of the creation and annihilation operators, VaV† = a − z,
1 iuL共t 兲 n e f exp关共eiuប − 1兲兩z兩2兴 n n! y ⫻ 关共eiuប − 1兲z + eiuy兴nexp兩关共eiuប − 1兲
共B10兲
⫻共兩z兩2 + yzⴱ兲兴兩y=0
and further note that, when acting on the ground state 兩0典 with a兩0典 = 0, the operator V yields the coherent state 兩z典, i.e.,
= eiu兩f共t f 兲兩
V兩0典 = 兩z典.
⫻兺
n
共B11兲
k=0
One finds with these properties 具n兩e
iuHH共t f 兲
e
e
兩z典兩x=y=0 . 共B12兲
Here we have introduced the auxiliary variables x and y
⬁
qrc共˜兲
=e
−兩z兩2
n 兩z兩2共n−k兲 iuបk iuប 共e − 1兲2共n−k兲 . e k 共n − k兲!
To determine the expression 共36兲 for the work distribution qc共˜兲, we start from the general expression given in the first equality of Eq. 共27兲. Interchanging the summation over the indices n and k, we obtain
冉 冊
2k
⬁
˜
冉冊
兩z兩2共k+m兲 2k e −n n 共− 1兲 ␦l+m,k+r 兺 兺 兺 ˜ m ! k! l n=k 1 − e− k m,k=0 l=0
共1兲
= e−兩z兩
l
⬁
2
兺
2k
兺 共− 1兲l
m,k=0 l=0
共2兲
冉冊
exp关共eiuប − 1兲兩z兩2兴
APPENDIX C: WORK DISTRIBUTION FOR A CANONICAL INITIAL STATE
2n 1 = eiuL共t f 兲 n n n! x y ⫻兩具z兩e
2/共ប兲
共B14兲
1 † 兩n典 = 具z兩共a − z兲neiuបa a+iuL共t f 兲共a† − zⴱ兲n兩z典 n!
x共a−z兲 iuបa+a y共a+−zⴱ兲
⬁
r −兩z兩2
= 共− 1兲 e
兺
m=0
共B13兲
where under the normal ordering operator N all creation operators stand left of the annihilation operators. The matrix element with respect to the coherent state 兩z典 can be read off, yielding
where 兩B共t f 兲兩2 兩f共t f 兲兩2 = . L共t f 兲 = C共t f 兲 − ប ប
†
exaeiuបa aeya = N兵exp关共eiuប − 1兲a†a
共B6兲
冉 冊冉 冊
兩z兩2共k+m兲 2k m ! k! l
e − 1
⬁
˜
1
˜
k
␦l+m,k+r
冉
2k 共− 兩z兩2兲m 关− 兩z兩2/共e − 1兲兴k 兺 m! k=兩m−r兩 k! k+r−m 011115-9
冊
PHYSICAL REVIEW E 78, 011115 共2008兲
TALKNER, BURADA, AND HÄNGGI
冉
⬁
共3兲
= 共− 1兲re−兩z兩
共4兲
= e−兩z兩
2
共− 兩z兩2兲m −2兩z兩2/共e˜−1兲 2兩z兩2 e I兩m−r兩 − ˜ m! e − 1
兺 m=0
2coth共˜ /2兲
⬁
兺
m=0
共5兲
= e
−兩z兩2coth共˜/2兲
= e−兩z兩
再
⬁
兺
m=0
2coth共˜ /2兲 ˜r/2
e
冉 冊
兩z兩2m 2兩z兩2 I兩m−r兩 ˜ m! e − 1
Ir
冉 冊 冉 冊 兩z兩2 . sinh共˜/2兲
共1兲
xk
兺 n=k 1 − x
冉冊 冉 冊 n x = k 1−x
k
共C2兲 共2兲
共cf. Ref. 关24兴, formula 5.2.11.3兲. In the second step 共 = 兲 the Kronecker ␦ is used to perform the sum over k. The third
共C1兲
APPENDIX D: WORK DISTRIBUTION FOR A COHERENT INITIAL STATE EQ. (45)
Starting from Eq. 共27兲,we may proceed in an analogous way as in the case of a canonical initial state; cf. Appendix C. According to Eq. 共43兲 a Poissonian average over the binomial 共 nk 兲 has to be performed instead of the geometric average in the first step of Eq. 共C1兲. This yields ⬁
共3兲
兺 n=k
step 共 = 兲 is based on the relation ⬁
xk
兺 k!
k=兩l兩
冉 冊 2k
k+l
= e2xI兩l兩共2x兲
共C3兲
valid for integer l. Here I共x兲 denotes the modified Bessel function of the first kind of order . With I共−x兲 = 共−1兲I共x兲, where is an integer, we come to the right-hand 共4兲
tk
I−k共x兲 = 兺 k=0 k!
冉 冊 2t +1 x
/2
I共冑x2 + 2tx兲
⬁
兺
k=兩m−r兩
=
共C4兲
共cf. 关24兴 5.8.3.1兲. This leads to the final result given in Eq. 共36兲.
关1兴 G. N. Bochkov and Yu. E. Kuzovlev, Sov. Phys. JETP 45, 125 共1977兲. 关2兴 D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 共1993兲. 关3兴 C. Jarzynski, Phys. Rev. Lett. 78, 2690 共1997兲. 关4兴 C. Jarzynski, C. R. Phys. 8, 495 共2007兲. 关5兴 F. Douarche, S. Ciliberto, A. Petrosyan, and I. Rabbiosi, Europhys. Lett. 70, 593 共2005兲.
冉冊
兩␣兩2n −兩␣兩2 n 兩␣兩2k e . = n! k k!
共D1兲
Next the Kronnecker ␦ is used to perform the sum over l, leaving one with two sums of which the inner one over k can be expressed in terms of a generalized hypergeometric function 关21兴, to become
side of the equality 共 = 兲. In the next step the sum on m is rewritten. The term in the square brackets vanishes because I共x兲 is an even function of order . The remaining sum can be performed by means of the identity ⬁
冋 冉 冊 冉 冊册冎
⬁
兩z兩2 兩z兩2 2兩z兩2 2兩z兩2 2兩z兩2 Ir−m ˜ + 兺 Im−r ˜ − Ir−m ˜ m! m=r+1 m! e − 1 e − 1 e − 1
In the first step 共 = 兲 we performed the sum on n according to ⬁
冊
冉
2k 共− 兩␣z兩2兲k 2 共k!兲 k+r−m
冉
冊
共− 兩␣z兩2兲兩m−r兩 F2 兩m − r兩 共兩m − r兩!兲2 1
冊
1 + ;兩m − r兩 + 1,2兩m − r兩 + 1;− 4兩␣z兩2 . 2
共D2兲
This immediately leads to the expression in Eq. 共45兲.
关6兴 C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today 58 共7兲, 43 共2005兲. 关7兴 V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger, Phys. Rev. Lett. 96, 070603 共2006兲. 关8兴 H. Tasaki, e-print arXiv:cond-mat/0009244. 关9兴 S. Mukamel, Phys. Rev. Lett. 90, 170604 共2003兲. 关10兴 W. De Roeck and C. Maes, Phys. Rev. E 69, 026115 共2004兲. 关11兴 M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129
011115-10
STATISTICS OF WORK PERFORMED ON A FORCED …
PHYSICAL REVIEW E 78, 011115 共2008兲
共2006兲. 关12兴 P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75, 050102共R兲 共2007兲. 关13兴 P. Talkner and P. Hänggi, J. Phys. A 40, F569 共2007兲. 关14兴 P. Talkner, P. Hänggi, and M. Morillo, Phys. Rev. E 77, 051131 共2008兲. 关15兴 S. Deffner and E. Lutz, Phys. Rev. E 77, 021128 共2008兲. 关16兴 U. Seifert, J. Phys. A 37, L517 共2004兲. 关17兴 K. Husimi, Prog. Theor. Phys. 9, 381 共1953兲. 关18兴 E. Lukacs, Characteristic Functions 共Griffin, London, 1970兲. 关19兴 B. Cleuren, C. Van den Broeck, and R. Kawai, Phys. Rev. Lett.
96, 050601 共2006兲. 关20兴 K. Yosida, Functional Analysis 共Springer-Verlag, Berlin, 1971兲. 关21兴 L. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products 共Academic, San Diego, 2000兲. 关22兴 N. G. van Kampen, Stochastic Processes in Physics and Chemistry 共North-Holland, Amsterdam, 1992兲. 关23兴 R. M. Wilcox, J. Math. Phys. 8, 962 共1967兲. 关24兴 A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series 共Gordon and Breach, New York, 1986兲, Vol. 1.
011115-11
