Quantum in the Bohmian framework

R. Sampaio,1, ∗ S. Suomela,1 T. Ala-Nissila,1, 2 J. Anders,3 and T. G. Philbin3,† 1COMP Center of Excellence, Department of Applied Physics, Aalto University, P.O. Box 11000, FI-00076 Aalto, Finland. 2Departments of Mathematical Sciences and Physics, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom. 3CEMPS, Physics and , University of Exeter, Exeter, EX4 4QL, United Kingdom. (Dated: January 7, 2018) At non-zero temperature classical systems exhibit statistical fluctuations of thermodynamic quantities arising from the variation of the system’s initial conditions and its interaction with the environment. The fluctuating work, for example, is characterised by the ensemble of system trajectories in phase and, by including the for various trajectories to occur, a work distribution can be constructed. However, without phase space trajectories, the task of constructing a work distribution in the quantum regime has proven elusive. Here we use quantum trajectories in phase space and define fluctuating work as power integrated along the trajectories, in complete analogy to classical statistical physics. The resulting work probability distribution is valid for any quantum evolution, including cases with coherences in the basis. We demonstrate the quantum work probability distribution and its properties with an exactly solvable example of a driven quantum harmonic oscillator. An important feature of the work distribution is its dependence on the initial statistical mixture of pure states, which is reflected in higher moments of the work. The proposed approach introduces a fundamentally different perspective on , allowing full thermodynamic characterisation of the dynamics of quantum systems, including the measurement process.

I. INTRODUCTION detector [17], the full counting statistics (FCS) approach [26– 28], and work definitions based on entropic principles [19– Landau said “All the concepts and quantities of thermody- 21, 55–58]. Work definitions based on trajectories in Hilbert namics follow most naturally, simply and rigorously from the space include the quantum jump approach [46, 59, 60] and concepts of statistical physics” [1]. While the second law of the framework [25]. None of these ap- thermodynamics puts limits on the work drawn from a system, proaches yields a positive work distribution based on trajecto- this work is most naturally viewed as the average over a sta- ries in phase space. tistical work distribution [2,3]. Classically, the work distribu- tion is constructed from the ensemble of system trajectories in In this paper we provide a positive work distribution for phase space. These trajectories specify the energy of the sys- arbitrary statistical mixtures of wave functions in a manner tem at all , thus allowing the work done on the system to fully analogous to the classical work definition. Contrast- be computed. In the quantum case the concept of fluctuating ing with all previous studies of work in the quantum regime, work has proven elusive [4,5] because, in the conventional this work distribution is established using quantum Hamilton- view, trajectories in quantum are considered to be Jacobi theory which we briefly recall in Sec.II. In Sec. III impossible [6]. Trajectories in phase space can, however, be we show how the fluctuating work is defined for initial pure constructed in an alternative formulation of quantum mechan- states, i.e. wave functions, and derive the work distribution ics [6–11] which makes predictions consistent with experi- for closed processes that begin in an initial statistical mixtures mental results. Here we utilise this approach to define work of wave functions. We show that the average work of this dis- and its distribution function for quantum systems. tribution always coincides with the change in the expectation Numerous definitions of quantum work have been pro- value of the , i.e. energy conservation is en- posed [12–32], the most widely used being the two mea- sured. The relation to the TMP work distribution and the use surement protocol (TMP) definition for closed quantum sys- of the Hamilton-Jacobi approach for finite-dimensional sys- tems [14, 15, 33]. The TMP leads to a quantum version of tems are also discussed. In Sec.IV the Hamilton-Jacobi work the Jarzynski equality [2,3, 14, 34] and the Tasaki-Crooks distribution is illustrated with an exactly solvable example, a relation [14, 35, 36], the classical correspondence has been driven quantum harmonic oscillator. We conclude in Sec.V elucidated [37] and experimental implementation is relatively that the work distribution based on Hamilton-Jacobi theory straightforward [16, 38–44]. The TMP has been extended provides a natural characterisation of work and its fluctua- to open systems [45–49], to continuously measured pro- tions in coherent quantum systems while recovering the clas- cesses [50, 51] and to relativistic systems [52]. Other ap- sical definition in the high-temperature limit. We highlight a proaches include using a power or work operator [4, 12, 13, number of open questions that pertain to the Hamilton-Jacobi 22, 53, 54], measuring the system by coupling it weakly to a approach to work in the quantum regime, such as initial en- tangled states and open dynamics. An important observation is that, in general, the work distribution associated with closed ∗ rui.ferreirasampaio@aalto.fi will depend on the preparation of the ini- † [email protected] tial mixture. 2

II. QUANTUM HAMILTON-JACOBI THEORY.

Quantum mechanics can be formulated as a theory of tra- jectories in phase space as shown by Bohm and others [7– 10, 61, 62]. Writing the for a single particle in polar form, ψ(x,t) = R(x,t) exp(iS(x,t)/h¯), where R ≥ 0 is the amplitude and S ∈ R is the phase of the wave function, and x is the particle position, the Schrodinger¨ equation reduces to two coupled equations for R and S. The imaginary part gives the continuity equation for the probability density R2. The real part has the form of the Hamilton-Jacobi equation FIG. 1. Bohmian trajectories for a driven 1D harmonic oscillator, ∂S(ψ)(x,t) see main text for details. The orange lines show three possible trajec- + E(ψ)(x,t) = 0, (1) ∂t tories of the particle all belonging to the ensemble for the wave func- tion ψ(x,t) that starts in the lowest energy eigenstate. The trajecto- where ries are specified by the particle’s initial position, x0. Also shown is |ψ(x,t)|2 (grey surface); this represents the probability density for p(ψ)(x,t)2 finding the particle on each trajectory of the ensemble. The energy E(ψ)(x,t) = +V(x,p(ψ)(x,t),t) +V (ψ)(x,t). (ψ) 2m Q of the particle at any point of the trajectory is E (xt ,t). (2) Here m is the mass of the particle, V(x,p(ψ)(x,t),t) is the (ψ) (ψ) Under unitary evolution Eq. (3) correctly reduces to the external potential and p (x,t) ≡ ∇S |(x,t). The above is exactly the classical equation for the same problem but energy difference between the end points of the trajectory (ψ) (ψ) (ψ) (ψ) W [xt ] = E (x ,τ) − E (x ,0). For open dynamics, with an additional term, the quantum potential VQ ≡ τ 0 a full integration over the trajectory would have to be per- −h¯ 2(∇2R(ψ))/2mR(ψ). According to Hamilton-Jacobi the- formed, using the total wave function, ψ , of the system and ory, E(ψ)(x,t) and p(ψ)(x,t) are the energy and canonical tot degrees of freedom to which it couples. If a system’s open momentum at point x and t, respectively. To obtain dynamics arises from it coupling to a large reservoir that has the trajectories we integrate Hamilton’s equation of (ψ) (ψ) a well defined temperature, then the difference between the x˙t = ∂pH | (ψ) where xt ≡ x(t) and H (x,p,t) ≡ (ψtot) (ψtot) (xt ,p (xt ,t),t) energy change, E (xτ ,τ) − E (x0,0), and the fluctuat- (ψ) 2 (ψtot) p /2m +V(x,p,t) +V Q (x,t). ing work, W [xt ], can be identified with “fluctuating heat” For each wave function ψ(x,t), this gives an ensemble of absorbed by the system from the reservoir. trajectories, namely one trajectory for each initial position Given definition (3), the probability distribution for quan- x0 of the particle, see Fig.1 for an example. Experimen- tum work when the system starts in an initial pure state ψ(x,0) tally these particle trajectories have been reconstructed using is simply weak measurements [63–65]. The probability for being on a Z 2 (ψ) trajectory with initial position x0 inside an infinitesimal vol- P(W;ψ) = dx0 |ψ(x0,0)| δ(W −W [xt ]). (4) 2 2 ume dx0 is given by R (x0,0)dx0 = |ψ(x0,0)| dx0, i.e. the [10, 66]. The continuity equation ensures that the This is the work distribution assuming that the system is ini- Born rule distribution of the trajectories is preserved in time as tially described by a single wave function and undergoes ei- 2 2 |ψ(xt ,t)| dxt = |ψ(x0,0)| dx0, where dxt is the time evolved ther closed or open dynamics. We now discuss the case where initial infinitesimal volume dx0. Note that whenever the quan- the system starts in a statistical mixture of pure states, {ψ( j)}, tum potential can be neglected from the total energy, the clas- with the index j taken to be discrete for simplicity. We refer sical Hamiltonian and the corresponding equations of motion to statistical mixtures as those that have been prepared in a ( j) are recovered. statistical manner - i.e. with probability p j the pure state ψ was prepared. The probability distribution of work for this statistical mixture is then given by III. WORK IN QUANTUM HAMILTON-JACOBI THEORY. ( j) P{ψ( j)}(W) = ∑ p j P(W;ψ ). (5) A. Fluctuating work definition. j

This is a positive probability distribution, i.e. P{ψ( j)}(W) ≥ 0 For each wave function ψ(x,t) one can now define work R for all W and dWP ( j) (W) = 1. It is important to note that as power integrated along any trajectory x just as in classical {ψ } t P (W) depends explicitly on the mixture of wave func- mechanics, i.e. the work done on the system between time 0 {ψ( j)} ( j) and τ is [67] tions {ψ }. Our definition of P{ψ( j)}(W) above means that two differ- Z τ H(ψ) ent statistical mixtures will produce, in general, two different (ψ) ∂ W [xt ] ≡ dt . (3) probability distributions of work even when they correspond 0 ∂t xt 3 to the same density operator, ρˆ. We give an explicit example C. Measurements and the TMP. of the mixture dependence of the work probability distribution below. If the system is initially entangled with other degrees A measurement implies a coupling of the system to a mea- of freedom the reduced state of the system will be mixed but suring device leading to open dynamics of the system during it does not correspond to a proper physical mixture of pure the time interval of the measurement. In the Bohmian formu- states. The work distribution is then not given by (5) as there lation measurements can be explicitly included as part of the is no unique set of mixing probabilities. In this case, the full continuous and deterministic dynamics of the system and the wave function ψtot describing the system and correlated de- measuring device [10, 61]. The system still follows a trajec- grees of freedom is required and the work distribution is given tory in phase space and so the work can again be computed as by P(W;ψtot), see Eq. (4). For a discussion of the distinction integrated power along this trajectory. A general discussion of between statistical mixtures and density operators see, for ex- the thermodynamics of measurements lies outside the scope ample, [4, 68–70]. In this paper, we focus on the case where of this paper, but it is pertinent to understand how the work the initial state of the system is a proper statistical mixture of definition (3) relates to the TMP definition. The protocol in pure states. TMP includes two energy measurements of the system [33]. We now describe the TMP in the Hamilton-Jacobi picture and then discuss circumstances in which the TMP work distribu- tion will coincide with that of Eq. (3). B. Average work. The first energy measurement on an initial density oper- ator ρˆ dynamically selects one of the eigenstates φ (n)(x,0) (n) ( j) ˆ For a statistical mixture ρˆ of pure states, {ψ } with of the Hamiltonian H(0) with eigenvalue E0 . This happens probabilities {p j}, the average work under unitary evolu- in a dynamical process [10, 61] in which the measuring de- tion can be obtained from the distribution (5) as hWi ≡ vice moves to give an outcome E(n) which is associated with R 0 dWP{ψ( j)}(W)W, explicitly, a system trajectory belonging to the ensemble of trajectories of the wave function φ (n)(x,0). The probability for obtain- (n) (n) (n) ing outcome E in this measurement is qn = hφ |ρˆ |φ i. hWi ( j) 0 {ψ } The negative time derivative of the phase of the wave func- Z ( j) 2 ( j) ( j) (n) (n) = ∑ p j dx0 |ψ (x0,0)| (E (xτ ,τ) − E (x0,0)) tion φ (x,t) at time t = 0 is E0 . Hence the particle energy (n) j E(φ )(x,0) in Eq. (1) after the first measurement is just the Z ( j) 2 ( j) (n) = ∑ p j dx0 |ψ (x0,0)| E (xτ ,τ) − hE(x,0)i{ψ( j)} energy eigenvalue E0 , for any position of the particle. The j system now evolves according to the Hamiltonian Hˆ (t) such Z (n) ( j) 2 ( j) that at time τ the wave function of the system is Uˆτ φ (x,0). = ∑ p j dxτ |ψ (xτ ,τ)| E (xτ ,τ) − hE(x,0)i{ψ( j)} j At this time another measurement of the energy is per- formed. As for the first measurement, this second mea- = hE(x,τ)i ( j) − hE(x,0)i ( j) {ψ } {ψ } surement dynamically evolves the system to an eigenstate = Tr[Hˆ (τ)ρˆ(τ)] − Tr[Hˆ (0)ρˆ(0)]. (m) (m) χ (x,τ) of the Hamiltonian Hˆ (τ) with eigenvalue Eτ . (6) (m) The probability of obtaining this outcome Eτ , given the Here we have used the short-hand notation hE(x,t)i ( j) ≡ {ψ } first outcome E(n), is the conditional probability p = p R dx |ψ( j)(x ,t)|2 E( j)(x ,t) in lines 2-4, and the 0 m|n ∑ j j t t t (m) ˆ (n) 2 ( j) 2 ( j) 2 |hχ |Uτ |φ i| . The distribution Pρˆ (∆E) for the change in equivariance property dx0 |ψ (x0,0)| = dx |ψ (x ,τ)| τ τ energy outcomes is given by P (∆E) = q P(∆E;φ (n)) with in line 3. We have used the identity E( j)(x,t) ≡ ρˆ ∑n n (m) (n) ( j) ( j) (n) ℜ{(Hψ )(x,t)/ψ (x,t)} in line 5 [10] and thus for P(∆E;φ ) = ∑m pm|n δ(∆E − (Eτ − E0 )). Since in TMP ( j) ( j) the energy change is identified as work this is the TMP work ρˆ(t) = ∑ p j|ψ (t)ihψ (t)| one obtains hE(x,t)i ( j) = j {ψ } distribution. R ( j) 2 ( j) ˆ ∑ j p j dx|ψ (x,t)| E (x,t) = Tr[ρˆ(t)H(t)] at time t. If we now consider the same protocol, but with work de- Thus, for a closed system one always finds that hWi is equal fined as in Eq. (3), then the integrated power must be com- to the change in the expectation value of the Hamiltonian op- puted along the entire system trajectory, including the time † erator of the system, Hˆ (t), i.e. hWi = Tr[Hˆ (τ)Uˆτ ρˆ Uˆτ ] − intervals where the two measurements take place. Any work Tr[Hˆ (0)ρˆ] where ρˆ is the density operator for the initial sta- done on the system during the two measurements will thus tistical mixture and Uˆτ is the evolution operator for the time- be included, in contrast to TMP. Thus the work distribution interval τ. Note that because of the linearity of the averages based on (3) will in general differ from the TMP work distri- the choice of how the initial density operator ρˆ was mixed bution Pρˆ (∆E). This difference arises due to the fact that dur- plays no role, i.e. for any set of pure states {ψ( j)} that cor- ing the measurements the dynamics of the system is actually responds to the ρˆ the same average work is open, as has been pointed out previously by Kammerlander et obtained. We highlight that, in general, this will not be the al., Solinas et al., and Elouard et al. [20, 60, 71]. There are case for higher moments of work, such as the average expo- circumstances, however, where the TMP work definition will nentiated work (see below). agree with the trajectory definition (3). For example, if the 4 system is in a mixture of energy eigenstates before each en- where we must solve for the time evolution of the wave func- ergy measurement, and if in addition the measurements make tion to obtain the phase S(x,t). The energy eigenstates are a negligible contribution to the work compared to the rest of most easily found by rewriting Eq. (7) with Eq. (8) as the protocol, then the work (3) will be equal to the difference   2 in the energy outcomes, in agreement with the TMP work def- 1 A Hˆ (t) = h¯ω bˆ†bˆ + − , (11) inition. 2 2mω2 √ with bˆ = aˆ − αe−iωt and α = −iA/ 2hm¯ ω3. In this form D. Finite-dimensional systems. it is easy to show that the eigenstates of Hˆ (t) at fixed t are displaced number states, i.e. states obtained by acting with While we have here considered particles with continuous the displacement operator on the energy eigenstates of the un- variables, discrete systems, such as qubits, can also be treated driven oscillator (for details and literature on displaced num- in the Hamilton-Jacobi approach [10, 61, 72]. For example, ber states, see [74] for example). In our case the t = 0 en- if a qubit is physically realised by a spin-1/2 particle then one ergy eigenstates are displaced number states |n˜α i ≡ Dˆ (α)|ni, sets up a spinor wave function and then extracts one over- where |ni is the n-th energy eigenstate of the undriven os- all phase from it, which determines the ensemble of parti- cillator and Dˆ (α) is the displacement operator. The energy ˆ 1 2 cle trajectories. If a qubit is taken as the lowest two energy eigenvalues of H(t) are h¯ω n + 2 − |α| at all times. levels of a spin-less particle, then the formulation described here, using a scalar wave function, is sufficient. As an ex- ample, suppose the qubit is realised as the two lowest lev- A. Initial thermal mixture of energy eigenstates els of a particle in a one-dimensional infinite-well potential of width L [72]. The wave function of any given pure state We consider first the case where the system is in an en- of the qubit is given by ψ(x,t) = c0(t)φ0(x) + c1(t)φ1(x), ergy eigenstate |n˜α i at t = 0. Because of the explicit time where c0(t) and c1(t) are time-dependent complex valued dependence in Eq. (7), the system does not remain in an en- p functions and φ0(x) = 2/Lsin(πx/L) is the ergy eigenstate. The time evolution operator (9) contains p the displacement operator, and evolves the initial state |n˜α i and φ1(x) = 2/Lsin(2πx/L) is the exited state. The work 2 2 associated with any operation can then be calculated from to eiA t/(2hm¯ ω )Dˆ (α + iωtα)|ni, which is another displaced Eq. (3). For quantitative details of a spin-1/2 particle in the number state but with t-dependent complex amplitude. The Hamilton-Jacobi formulation, we refer the reader to the text- evolved state is not an energy eigenstate (for t > 0). The phase 2 2 books [10, 61]. S(x,t) of the wave function for eiA t/(2hm¯ ω )Dˆ (α + iωtα)|ni is straightforwardly obtained from the displaced number state results [74] and takes the form IV. EXAMPLE: THE DRIVEN QUANTUM OSCILLATOR.  1 A2t S(x,t) = − h¯ω n + t + To illustrate the work distributions we here consider a quan- 2 2mω2 tum harmonic oscillator of mass m and frequency ω with ex- A − x[cos(ωt) + ωt sin(ωt)] ternal driving, described by the Hamiltonian ω 2 2 A  2 2  pˆ 1 2 2 + 3 2ωt cos(2ωt) + (ω t − 1)sin(2ωt) . Hˆ (xˆ, pˆ,t) = + mω xˆ − xˆ f1(t) − pˆ f2(t) (7) 4mω 2m 2 (12) where the last two terms in Eq. (7) represent the external driv- The quantum trajectories of the particle are then found from ing, given by, Eq. (10) to be A f (t) = −Asin(ωt), f (t) = − cos(ωt), (8) A 1 2 xn(t) = x + [ωt cos(ωt) − sin(ωt)], (13) mω 0 mω2 where A is a real constant. The external driving is chosen with x0 the initial position. Note that the set of trajectories is such that an exact solution for the time-evolution operator can the same for each initial energy eigenstate |n˜α i, i.e. Eq. (13) be found [73], which is given by is independent of n. The work along a trajectory calculated   from Eqs. (3), (12) and (13) is given by At ˆ Tˆ(t,0) = exp √ −aeˆ iωt + aˆ†e−iωt  e−iH0t/h¯ , (9) 2hm¯ ω Aτ[Aτ + 2mx0ω cos(ωτ)] W[x (t)] = . (14) n 2m ˆ † 1  where H0 = h¯ω aˆ aˆ + 2 is the free-oscillator Hamiltonian. The particle trajectories x(t) are given by the solution of We obtain the work distribution by taking into account the probability of the particle following a trajectory (13) with ini- 1 ∂ tial position x0. The probability distribution for the initial po- x˙(t) = S(x,t) − f (t), (10) 2 m ∂x 2 sition is given by |ψn(x0)| , where ψn(x) is the initial wave 5 function for the t = 0 energy eigenstate |n˜α i. The initial posi- where the Gaussian distribution has been shifted by α since tion distribution is the initial Hamiltonian Hˆ (xˆ, pˆ,0) at time t = 0, see Eq. (7), is displaced by Dˆ (α) with energy eigenstates |n˜ i. The different r  2   r 2 α 2 1 mω mωx0 mω initial mixture in Eq. (20) will give different quantum trajecto- |ψn(x0)| = exp − Hn x0 , 2n n! πh¯ h¯ h¯ ries for the particle compared to the mixture in Eq. (17), even (15) though both mixtures correspond to the same density opera- tor. To obtain the work distribution and work averages for which is the same as that for the energy eigenstates of the the coherent-state mixture we must find the trajectories for an undriven oscillator (at t = 0 the probability density is not dis- initial . placed relative to the number state). The probability distri- For the oscillator initially in a coherent state |ηi with com- bution of work for the initial eigenstate (15) is then given by plex amplitude η, with ηR (ηI) the real (imaginary) part of η, Eq. (4).   the state at t > 0 is found from Eq. (9) to be exp i √AtηI |υi, We now consider an initial statistical mixture of energy 2hm¯ ω eigenstates at t = 0 with thermal probabilities, where |υi is a coherent state with time-dependent complex amplitude υ = η + √ At . The corresponding wave-function hm  −βh¯ω  −nβh¯ω 2¯ ω pn = 1 − e e , (16) has the phase

1 hAt¯ ηI for the eigenstate labelled by n with eigenvalue S(x,t) = − h¯ωt + √ h¯ω n + 1 − |α|2. The associated density operator is 2 2hm¯ ω 2 h √  √ i the thermal state at inverse temperature β = 1/(kBT) − x At + ηR 2hm¯ ω sin(ωt) − ηI 2hm¯ ω cos(ωt) ∞   ˆ At ρβ = ∑ pn |n˜α ihn˜α |. (17) − h¯ ηR + √ ηI cos(2ωt) n=0 2hm¯ ω " 2 # The work distribution for unitary evolution starting from this h¯ At 2 + ηR + √ − [ηI] sin(2ωt). (22) statistical mixture is given by Eq. (5) using the weights in (16). 2 2hm¯ ω Using Eqs. (14), (15) and (16) we can now compute work averages. For example, the average work is This phase gives the particle trajectories r r ! (Aτ)2 2h¯ At 2h¯ hWi = , (18) xη (t) =x0 − ηR + + ηR cos(ωt) {n} 2m mω mω mω r which is the difference in the energy expectation values at t = 2h¯ t = + ηI sin(ωt). (23) τ and 0, as expected. The average exponentiated work is mω given by For this initial wave function the work done on the oscillator  2 2 −βW A τ β between t = 0 and t = τ is given by he i{n} =exp − 2m " r #    Aτ 2h¯ω h¯ωβ h¯ωβ W[xη (t)] =Aτ + ηR + {A[ωτ cos(ωτ) + sin(ωτ)] × 1 − cos2(ωτ)coth . (19) 2m m 2 2 p o + hm 3 [ ( ( ) − ) + ( )] Before discussing this result in more detail in SectionIVC, 2¯ ω ηR cos ωτ 1 ηI sin ωτ we will first analyse the case of an initial thermal state when " r # 2h¯ it is prepared as a statistical mixture of coherent states. × x − ηR . (24) 0 mω

The probability distribution for the initial position x0 is given B. Initial thermal mixture of coherent states 2 by |ψη (x0)| , where ψη (x0) is the wave function for the co- herent state |ηi at t = 0: Above we constructed the initial thermal state as a mixture   of energy eigenstates, Eq. (17), and obtained the average ex- r " r #2 2 mω  mω 2h¯  ponentiated work Eq. (19). However, the same initial density |ψ (x )| = exp − x − ηR . (25) η 0 πh¯ h¯ 0 mω operator can be written as a mixture of coherent states |ηi, i.e.   Z 2 The work distribution for the initial coherent state |ηi is now ρˆβ = d η pα (η)|ηihη|, (20) determined by Eq. (4) with ψη (x0) as the initial state. The work distribution arising for an initial thermal state ρˆβ where pα (η) is the P-representation [75, 76]. For the thermal when it was prepared as a mixture of coherent states, (20), is state of the harmonic oscillator one finds [75, 76] given by βh¯ω  Z e − 1 n 2 h βh¯ω io 2 p (η) = exp −|η − α| e − 1 , (21) P(W) = d η pα (η)P(W;ψη ) (26) α π {η} 6

( j) 2 ψ( j) with pα (η) given by Eq. (21). Work averages for this mixture distribution ρ(x,p,t) = ∑ j p j|ψ (x,t)| δ(p − ∇S (x,t)) can now be evaluated with Eqs. (21), (24) and (25). is thus in general not of Boltzmann form, even if the mixture ( j) of {ψ } corresponds to a thermal state ρˆβ .

C. Comparison of work averages for two thermal state mixtures. V. CONCLUSION AND DISCUSSION.

The average work for the initial coherent-state mixture is 2 is routinely taught in a way that hWi{η} = (Aτ) /(2m), which agrees with Eq. (18), as ex- stresses subjectivity and , while abolishing tra- pected. However, when calculating the average exponentiated −βW jectories in phase space. As Bell pointed out, this is a “delib- work he i{η} one finds that it differs from the one obtained erate theoretical choice” that is “not forced on us by experi- for an initial energy-state mixture in (19). In fact, the quantity −βW mental facts” [6]. Using quantum Hamilton-Jacobi theory we he i{η} can diverge for certain values of the parameters, have here generalised the classical definition of work to the −βW in contrast to the result for he i{n}. At high temperature quantum case in a straightforward manner. We have provided −βW he i{η} always converges and to leading order in β it is a positive definite probability distribution characterising the given by work fluctuations in complete generality, including for coher- ent quantum systems. The distribution gives the average work −βW 2 ωτ  2 he i{η} = 1 + βh¯ω sin + O(β ), (27) as the change of energy expectation values for closed systems 2 while recovering the classical result in the presence of clas- whereas the high temperature limit of Eq. (19) is sical dynamics. While this may appear in contradiction with a recently proven “no-go” theorem [77], note that the work β(Aτ)2 distribution (5) is not linear in the density operator of the sys- he−βW i = 1 − sin2(ωτ) + O(β 2). (28) {n} 2m tem. Thus, it is not of the form considered in Ref. [77] and the theorem does not exclude this case. −βW Note that to first order in β, he i{η} is independent of the In contrast to proposals based on trajectories in Hilbert −βW driving (A) whereas he i{n} is not. The dependence on space, such as the quantum jump approach [16, 46, 47, 59, 60], −βW the driving in the high-temperature expansion of he i{η} consistent histories framework [25] or continuous measure- only appears at third order in β. The fact that moments of the ments [17, 23], the Hamilton-Jacobi approach is based on tra- work for a coherent-state mixture can be non-zero even with- jectories in phase space where the system has a well defined out driving follows from the properties of the coherent state energy at all times. These trajectories have been reconstructed in the Hamilton-Jacobi formulation [10]. Although the aver- using weak measurements [63–65], from which the distribu- age over all trajectories of the energy change of the particle tion (5) can be inferred. in a pure (undriven) coherent state is zero, the energy change We here focussed on initial statistical mixtures undergoing along individual trajectories is not zero in general, due to the closed dynamics - but the work distribution defined in (5) ex- quantum potential [10]. This means that higher moments of tends naturally to correlated and open systems. Instead of us- the energy change do not have to vanish. ing the wave function of the system alone, the open case re- For the time-dependent Hamiltonian (11) the partition func- quires the use of the wave function ψtot of the system and any tion associated with the thermal states is the same at any time correlated or interacting degrees of freedom. The dynamics and thus the classical Jarzynski equality is he−βW i = 1. This of the full system would have to be solved and the power in- is correctly obtained for both statistical mixtures, Eq. (28) and tegrated along the phase-space trajectory of the subsystem of Eq. (27), in the high-temperature limit of β → 0, which is the interest. relevant classical limit here. We note that there is no a pri- Measurements are a particular example of open dynamics, ori reason why the mathematical limit h¯ → 0 should give the where the full system consists of the measured system plus the classical physical result here. Indeed, assuming no contribu- measuring device. The Hamilton-Jacobi approach thus has the tions arising from higher orders of β, for the coherent-state ability to explicitly quantify the work done during a measure- mixture, (27), the limit h¯ → 0 happens to coincide with the ment. Furthermore, it can be applied to initial states with co- classical value of 1. But for the energy-state mixture, (28), herences in the energy basis, whether or not measurements are the h¯ → 0 limit does not give 1, unless one substitutes the performed. The removal of initial coherences through certain 2 2 2 coupling constant A as A = 2mαI (h¯ω)ω where αI is the open processes has been found to allow work extraction [20]. imaginary part of the displacement parameter α. This extracted work could be compared to the average work The expressions show that in general there is no relation obtained from the work distribution (5). More generally, it between the average exponentiated work he−βW i and equilib- will be interesting to investigate how much work is done on rium quantities, such as ∆F, in the quantum case. This arises the system when energy or other observables are measured. because each wave function already contributes an ensemble We note that, as a general feature of the Hamilton-Jacobi of trajectories to the work distribution in Eq. (4) and such en- formulation [10, 61], the predictions for measurement out- sembles are not related to Boltzmann distributions in any di- comes derived from the Hamiltonian-Jacobi approach will rect manner. Only the mixing over different wave functions agree with those of standard quantum mechanics. This in- can introduce thermal weights, and the resulting phase space cludes standard tools of quantum mechanics, such as positive 7 valued operator measurements (POVMs). Our work distribu- sponding trajectories are then recovered. In some cases the tion is not associated with a POVM. This explains why here high temperature limit may also give classical results; and we statistical mixtures rather than density operators determine the showed that for the specific harmonic oscillator example the work distribution, in contrast to previous work definitions. classical Jarzynski equality is recovered in this limit. An open The broader perspective taken here offers new insights into problem is to find the general conditions under which high current challenges in quantum thermodynamics by indicating temperature gives classical statistical results. that the statistical mixture may play a bigger role than usu- Many questions remain open in the field of quantum ther- ally anticipated. It will be interesting to explore implications modynamics. The approach introduced here shows that they of this feature on interpretational aspects of quantum theory are firmly located in the realm of the “speakable” [6]. [78, 79]. As demonstrated in the harmonic oscillator example, the average exponentiated work in the quantum case is not in gen- Acknowledgements. We would like to thank M. Camp- eral related to equilibrium properties such as the free energy isi, I. Starshinov, H. Miller, P. Muratore-Ginanneschi and B. change of the system. This difference from classical statistical Donvil for inspiring discussions and M. Campisi, J. Garra- physics occurs because the quantum thermal phase-space dis- han, C. Jarzynski, P. Solinas and P. Talkner for critical reading tribution is not of Boltzmann form and there is an explicit de- of the manuscript. This work has been in part supported by pendence on the statistical mixture. Future investigations can the Academy of Finland through its CoE grants 284621 and explore how fluctuation relations can be generalized within 287750. R.S. acknowledges support from the Magnus Ehrn- the quantum Hamilton-Jacobi theory. rooth Foundation. J.A. acknowledges support from EPSRC In the Hamilton-Jacobi approach the classical limit is trans- (grant EP/M009165/1) and the Royal Society. This research parent as it occurs whenever the quantum potential can be ne- was supported by the COST network MP1209 “Thermody- glected in (2). The classical equations of motion and corre- namics in the quantum regime”.

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