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arXiv:1701.07603 · Physical review. E

Quantum work fluctuations in connection with the Jarzynski equality

Juan D. Jaramillo, Jiawen Deng, Jiangbin Gong

Published Oct 9, 2017• 13 citations
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Abstract

A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change $\mathrm{\ensuremath{\Delta}}F$ of a system at inverse temperature $\ensuremath{\beta}$ from an ensemble average of nonequilibrium exponential work, i.e., $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$. The number of experimental work values needed to reach a given accuracy of $\mathrm{\ensuremath{\Delta}}F$ is determined by the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, denoted $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$. We discover in this work that $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.

DOI
10.1103/physreve.96.042119

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A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change \ensuremathΔF\mathrm{\ensuremath{\Delta}}F\ensuremathΔF of a system at inverse temperature \ensuremathβ\ensuremath{\beta}\ensuremathβ from an ensemble average of nonequilibrium exponential work, i.e., \ensuremath⟨e\ensuremath−\ensuremathβW\ensuremath⟩=e\ensuremath−\ensuremathβ\ensuremathΔF\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}\ensuremath⟨e\ensuremath−\ensuremathβW\ensuremath⟩=e\ensuremath−\ensuremathβ\ensuremathΔF. The number of experimental work values needed to reach a given accuracy of \ensuremathΔF\mathrm{\ensuremath{\Delta}}F\ensuremathΔF is determined by the variance of e\ensuremath−\ensuremathβW{e}^{\ensuremath{-}\ensuremath{\beta}W}e\ensuremath−\ensuremathβW, denoted var(e\ensuremath−\ensuremathβW)\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})var(e\ensuremath−\ensuremathβW). We discover in this work that var(e\ensuremath−\ensuremathβW)\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})var(e\ensuremath−\ensuremathβW) in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging var(e\ensuremath−\ensuremathβW)\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})var(e\ensuremath−\ensuremathβW) at sufficiently low temperatures, markedly different from the classical behavior. The divergence of var(e\ensuremath−\ensuremathβW)\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})var(e\ensuremath−\ensuremathβW) indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.
The central quantity in the celebrated quantum Jarzynski equality is ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where WWW is work and $\ensuremath{\beta}$ is the inverse temperature. The impact of quantum randomness on the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and hence on the predictive power of the Jarzynski estimator is an important problem. Working on a single nitrogen-vacancy center in diamond and riding on an implementation of the two-point measurement of nonequilibrium work with single-shot readouts, we have conducted a direct experimental investigation of the relationship between the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and adiabaticity of nonequilibrium work protocols. It is observed that adiabatic processes minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, thus verifying an early theoretical concept, the so-called principle of minimal work fluctuations. Furthermore, it is experimentally demonstrated that shortcuts-to-adiabaticity control can be exploited to minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ in fast work protocols. Our work should stimulate further experimental studies of quantum effects on the bias and error in the estimates of free-energy differences based on the Jarzynski equality.
Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, in considering the Jarzynski equality $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$, a change in the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ may impact how rapidly the statistical average of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ converges towards the theoretical value ${e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$, where WWW is the work, $\ensuremath{\beta}$ is the inverse temperature, and $\mathrm{\ensuremath{\Delta}}F$ is the free energy difference between two equilibrium states. Motivated by our previous study aiming at the suppression of work fluctuations, here we obtain a principle of minimal work fluctuations. In brief, adiabatic processes as treated in quantum and classical adiabatic theorems yield the minimal fluctuations in ${e}^{\ensuremath{-}\ensuremath{\beta}W}$. In the quantum domain, if a system initially prepared at thermal equilibrium is subjected to a work protocol but isolated from a bath during the time evolution, then a quantum adiabatic process without energy level crossing (or an assisted adiabatic process reaching the same final states as in a conventional adiabatic process) yields the minimal fluctuations in ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where WWW is the quantum work defined by two energy measurements at the beginning and at the end of the process. In the classical domain where the classical work protocol is realizable by an adiabatic process, then the classical adiabatic process also yields the minimal fluctuations in ${e}^{\ensuremath{-}\ensuremath{\beta}W}$. Numerical experiments based on a Landau-Zener process confirm our theory in the quantum domain, and our theory in the classical domain explains our previous numerical findings regarding the suppression of classical work fluctuations [G. Y. Xiao and J. B. Gong, Phys. Rev. E 90, 052132 (2014)].
We first derive the Jarzynski relation [1] between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. We then comment on the information-theoretic underpinning of Jarzynski’s reasoning which helps explain why the Jarzynski relation holds identically both quantumly and classically [2]. We then present a scheme to verify the quantum non-equilibrium fluctuation relations as encapsulated by Jarzynski. We show that the characteristic function of the work distribution of a quantum system (which is basically equal to the Wick rotated average exponential of the thermodynamical work) can be extracted from Ramsey interferometry of a single probe qubit (which need not itself be pure, though it must not be fully depolarized) [3,4].We first derive the Jarzynski relation [1] between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. We then comment on the information-theoretic underpinning of Jarzynski’s reasoning which helps explain why the Jarzynski relation holds identically both quantumly and classically [2]. We then present a scheme to verify the quantum non-equilibrium fluctuation relations as encapsulated by Jarzynski. We show that the characteristic function of the work distribution of a quantum system (which is basically equal to the Wick rotated average exponential of the thermodynamical work) can be extracted from Ramsey interferometry of a single probe qubit (which need not itself be pure, though it must not be fully depolarized) [3,4]. An interesting fact is that while the quantum version of the Jarzynski equality remains satisfied even in the presence of quantum correlations, the individual thermodynamical work moments in the expansion of the free energy are, in fact, sensitive to the genuine quantum correlations [5]. Whether this is a fortuitous coincidence remains to be seen, but it certainly goes towards explaining why the laws of thermodynamics happen to be so robust as to be independent of the underlying micro-physics.We intend to elucidate the subtle connection between Jarzynski’s relation and the “quantum arrow of time”. Finally we comment on the fact that our scheme for measuring the quantum work characteristic function belongs to the computational class known as DCQ1 [6], namely all computations that can be performed with only one pure qubit (and N maximally mixed ones).
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.
Thermodynamics constrains changes to the energy of a system, both deliberate and random, via its first and second laws. When the system is not in equilibrium, fluctuation theorems such as the Jarzynski equality further restrict the distributions of deliberate work done. Such fluctuation theorems have been experimentally verified in small, nonequilibrium quantum systems undergoing unitary or decohering dynamics. Yet, their validity in systems governed by a non-Hermitian Hamiltonian has long been contentious due to the false premise of the Hamiltonian's dual and equivalent roles in dynamics and energetics. Here we show that work fluctuations in a non-Hermitian qubit obey the Jarzynski equality even if its Hamiltonian has complex or purely imaginary eigenvalues. With postselection on a dissipative superconducting circuit undergoing a cyclic parameter sweep, we experimentally quantify the work distribution using projective energy measurements and show that the fate of the Jarzynski equality is determined by the parity-time symmetry of, and the energetics that result from, the corresponding non-Hermitian, Floquet Hamiltonian. By distinguishing the energetics from non-Hermitian dynamics, our results provide the recipe for investigating the nonequilibrium quantum thermodynamics of such open systems. Published by the American Physical Society 2024