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Jarzynski Equality for Driven Quantum Field Theories

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Jarzynski Equality for Driven Quantum Field Theories CALT-TH-2017-052 Jarzynski Equality for Driven Quantum Field Theories Anthony Bartolotta1, ∗ and Sebastian Deffner2, y 1Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 USA 2Department of Physics, University of Maryland Baltimore County, Baltimore, MD 21250 USA (Dated: October 4, 2017) The fluctuation theorems, and in particular, the Jarzynski equality, are the most important pillars of modern non-equilibrium statistical mechanics. We extend the quantum Jarzynski equality to- gether with the Two-Time Measurement Formalism to their ultimate range of validity { to quantum field theories. To this end, we focus on a time-dependent version of scalar phi-four. We find closed form expressions for the resulting work distribution function, and we find that they are proper phys- ical observables of the quantum field theory. Also, we show explicitly that the Jarzynski equality and Crooks fluctuation theorems hold at one-loop order independent of the renormalization scale. As a numerical case study, we compute the work distributions for an infinitely smooth protocol in the ultra-relativistic regime. In this case, it is found that work done through processes with pair creation is the dominant contribution. PACS numbers: 05.70.Ln, 05.30.-d, 11.10.-z, 05.40.-a I. INTRODUCTION The most prominent (integral) fluctuation theorem [11] is the Jarzynski equality [12], which holds for all systems In physics there are two kinds of theories to describe initially prepared in equilibrium and undergoing isother- motion: microscopic theories whose range of validity is mal processes, determined by a length scale and the amount of kinetic hexp (−βW )i = exp (−β∆F ) ; (2) energy, such as classical mechanics or quantum mechan- ics; and phenomenological theories, such as thermody- where β is the inverse temperature, W is the thermody- namics, which are valid as long as external observables namic work, and ∆F is the free energy difference between remain close to some equilibrium value. the instantaneous equilibrium states at the initial and fi- Over the last two centuries, microscopic theories have nal times. In its original inception the Jarzynski equality undergone a rapid development from classical mechanics (2) was formulated for classical systems with Hamiltonian over special relativity and quantum mechanics to quan- [12] and Langevin dynamics [13]. Thus, W is essentially tum field theory. While quantum field theories were a notion from classical mechanics, where work is given by originally developed for particle physics and cosmology, a force along a trajectory. The advent of modern fluctu- this apporach has also been shown to be powerful in the ation theorems for classical systems [5{8, 10, 12{14] has description of condensed matter systems. Examples in- spurred the development of a new field, which has been clude quasiparticle excitations in graphene, cavity quan- dubbed stochastic thermodynamics [15{20]. tum electrodynamics, topological insulators, and many In the study of nanoscale systems out of thermal equi- more [1{4]. librium, it is natural to ask in what regimes quantum In contrast to the evolution of microscopic theories, the effects become significant and how fluctuation theorems development of thermodynamics has been rather stag- apply to quantum systems [21{34]. Nevertheless, it took nant { until only two decades ago when the first fluctua- another decade before it was clearly stated that in quan- tion theorems were discovered [5{8]. Conventional ther- tum mechanical systems W is not a quantum observ- modynamics can only fully describe infinitely slow, equi- able in the usual sense [35]. This means that there is no librium processes. About all real, finite-time processes hermitian operator, whose eigenvalues are the classically the second law of thermodynamics only asserts that some observable values of W . This is the case because thermo- amount of entropy is dissipated into the environment, dynamic work is a path dependent quantity { a non-exact which can be expressed with the average, irreversible en- differential. Hence, thermodynamic work is rather given tropy production as hΣi ≥ 0 [9]. The (detailed) fluc- by a time-ordered correlation function [24, 25, 35]. tuation theorem makes this statement more precise by arXiv:1710.00829v1 [cond-mat.stat-mech] 2 Oct 2017 To gain more insight into the underlying statistics of expressing that negative fluctuations of the entropy pro- quantum work the Two-Time Measurement Formalism duction are exponentially unlikely [5{8, 10], [36, 37] has proven powerful: In this formulation, a quan- tum system is prepared in contact with a heat bath of in- P(−Σ) = exp (−Σ) P(Σ) : (1) verse temperature β. The system is then decoupled from the environment and a projective measurement onto the initial energy eigenbasis is performed. Then, the system is let to evolve before another projective measurement ∗ [email protected] of the energy is performed. As the system is isolated, y deff[email protected] the work performed on the system is identical to the 2 change in energy. Despite its success, this formalism has tudes are detailed in AppendixA. In Sec.V we specialize several limitations [38] including the lack of thermody- to a time-dependent version of λφ4 and discuss its renor- namic accounting for the measurement process [39] and malization. Then, Sec.VI discusses how closed form ex- its inapplicability to coherently controlled quantum sys- pressions for the work distribution function can be calcu- tems [40]. Nevertheless, it is important to remark that lated at leading order using a graph theoretic technique. in complete analogy to how classical mechanics is con- The details of the derivation can be found in AppendixB tained in quantum mechanics (in the appropriate limits) while the closed form expressions for the work distribu- the Two-Time Measurement Formalism produces work tion function are in AppendixC. We discuss the analytic distribution functions which correspond to those of clas- properties of the work distribution function in Sec. VII sical systems in semiclassical approximations [41{45]. and analytically verify both the Crooks fluctuation theo- To date another decade has gone by, yet quantum rem and quantum Jarzynski equality at leading order for stochastic thermodynamics is still rather incomplete. time-dependent λφ4. In Sec. VIII we numerically evalu- How to describe thermodynamic work and entropy pro- ate the work distribution function for a relativistic bath duction in open quantum systems is still hotly debated and a particular driving protocol, and verify the fluctua- [46{56], and with a few exceptions [57{61] most of the tion theorems. Interestingly, we find that the dominant literature is restricted to standard Schr¨odingerquantum process in the work distribution function is particle pair- mechanics. The purpose of the present analysis is to production through a loop diagram, an effect only found significantly broaden the scope of stochastic thermody- in a quantum field theory. We conclude in Sec.IX with namics, and to take the next, important step { extend a few remarks. quantum stochastic thermodynamics to quantum field theories. In the following, we demonstrate that the Two-Time II. PRELIMINARIES: TWO-TIME Measurement Formalism can be systematically used to MEASUREMENT FORMALISM investigate the work distribution functions of a restricted class of quantum field theories, focusing on a time- We begin by reviewing the Two-Time Measurement 4 dependent version of λφ . Closed form expressions for Formalism to establish notions and notation [25]: A these work distributions are found at leading order, in- quantum system is initially, at t = t1, in thermal equilib- cluding loop corrections, through the use of a new dia- rium with a classical heat bath of inverse temperature β grammatic technique and a mapping between finite-time + [62]. At t = t1 + 0 the system is disconnected from the transition amplitudes and infinite-time scattering ampli- heat bath and the energy of the system is projectively tudes. It is found that to the perturbative order con- measured to be E1. The system then evolves according sidered, the work distribution function does not run with to a time dependent protocol until time t = t2. At this the renormalization scale indicating that the distribution time, the energy of the system is measured to be E2. is an observable of the quantum field theory. We verify Let H^ (t) be the Hamiltonian at time t and let U(t2; t1) that the quantum Jarzynski and Crooks fluctuation the- be the time evolution operator from t to t , and Π^ orems hold exactly and are independent of the renormal- 1 2 E is the energy projection operator onto the (potentially ization scale. Due to the form of the work distributions, degenerate) subspace of eigenstates with energy E. This it is straightforward to show that the fluctuation theo- projection operator is time-dependent due to the time- rems hold if one removes the loop corrections (as would dependent Hamiltonian, but for compactness of notation, be the case for a classical field theory) and also in the this dependence will be implicit. non-relativistic limit. As the system starts in equilibrium, the initial state of These results demonstrate that quantum fluctuation the system is given by the thermal density matrix theorems and stochastic thermodynamics can be ex- tended to include quantum field theories, our most fun- ^ damental theory of nature. Thus, our results open the exp −βH(t1) door for future application of fluctuation theorems to the ρ^0 = n o: (3) tr exp −βH^ (t ) study of problems at the forefronts of physics { in con- 1 densed matter physics, particle physics, and cosmology. This paper is organized as follows: In Sec.II we review The probability of measuring energy E1 at time t1 is then the Two-Time Measurement Formalism and the quantum given by Jarzynski equality. We define a restricted class of quan- n^ o tum field theories in Sec.
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