Nonequilibrium quantum-heat statistics under stochastic projective measurements

Stefano Gherardini, Lorenzo Buffoni, Matthias M. Müller, Filippo Caruso, Michele Campisi, Andrea Trombettoni, and Stefano Ruffo

Phys. Rev. E 98, 032108 – Published 7 September 2018

Abstract

In this paper we aim at characterizing the effect of stochastic fluctuations on the distribution of the energy exchanged by a quantum system with the external environment under sequences of quantum measurements performed at random times. Both quenched and annealed averages are considered. The information about fluctuations is encoded in the quantum-heat probability density function, or equivalently in its characteristic function, whose general expression for a quantum system with arbitrary Hamiltonian is derived. We prove that, when a stochastic protocol of measurements is applied, the quantum Jarzynski equality is obeyed. Therefore, the fluctuation relation is robust against the presence of randomness in the times intervals between measurements. Then, for the paradigmatic case of a two-level system, we analytically characterize the quantum-heat transfer. Particular attention is devoted to the limit of large number of measurements and to the effects caused by the stochastic fluctuations. The relation with the stochastic Zeno regime is also discussed.

Received 4 May 2018

DOI:https://doi.org/10.1103/PhysRevE.98.032108

Physics Subject Headings (PhySH)

Fluctuation theorems Nonequilibrium statistical mechanics Quantum thermodynamics

Nonequilibrium systems

Statistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Stefano Gherardini^1,2,3,*, Lorenzo Buffoni^1,4, Matthias M. Müller^1,2, Filippo Caruso^1,2, Michele Campisi^1,3, Andrea Trombettoni^5,6, and Stefano Ruffo^6,7

¹Department of Physics and Astronomy, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
²CNR-INO, QSTAR and LENS, via N. Carrara 1, I-50019 Sesto Fiorentino, Italy
³INFN, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
⁴Department of Information Engineering, University of Florence, via S. Marta 3, I-50139 Florence, Italy
⁵CNR-IOM DEMOCRITOS Simulation Center, via Bonomea 265, I-34136 Trieste, Italy
⁶SISSA, via Bonomea 265, I-34136 Trieste, Italy & INFN, Sezione di Trieste, I-34151 Trieste, Italy
⁷Istituto dei Sistemi Complessi, CNR, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy

^*gherardini@lens.unifi.it

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Vol. 98, Iss. 3 — September 2018

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Figure 1
Analytical result for $G (i β) = 〈 e^{- β Q_{q}} 〉$ for a fixed waiting times sequence as a function of $c_{1}$ for three real values of $a$ (respectively, $a = 0, 0.1, 0.5$ , solid, dotted, and dashed lines). The analytical predictions are compared with the numerical simulations (respectively, crosses, $x$ marks, and circles). The simulations have been performed by applying protocols of $M = 5$ projective measurements, averaged over 1000 realizations in order to numerically derive the mean of the exponential of work, with $E_{\pm} = \pm 1$ . The point in which all the analytical lines are crossing corresponds to take the initial density matrix of the system equal to the thermal state $ρ_{th} = e^{- β H} / Z$ with $β = 1$ .
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Figure 2
$G (i β) = \bar{〈 e^{- β Q_{q}} 〉}$ (solid, dotted, and dashed lines) as a function of $c_{1}$ for three real values of $a$ ( $a = 0, 0.1, 0.5$ , respectively). The stochasticity in the time intervals between measurements is considered to be distributed as an annealed disorder. The analytical prediction is compared to the numerical simulations (crosses, $x$ marks, and circles) for the three values of $a$ . Also in this case, the point in which all the lines are crossing corresponds to the thermal state. Inset: Partial derivative $\partial_{c_{1}} G (i β)$ as a function of the measurement operator parameter ${| a |}^{2}$ , with resolution of $| a | = 0.05$ , for three different types of noise (fixed, quenched, and annealed). The curves have been performed by applying protocols of $M = 5$ projective measurements, averaged over 1000 realizations, with $E_{\pm} = \pm 1$ and $β = 1$ . For $p (τ)$ we have chosen a bimodal probability density function, with values $τ^{(1)} = 0.01, τ^{(2)} = 3$ , and $p_{1} = 0.3$ .
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Figure 3
$Δ λ$ as a function of $〈 τ 〉 = \bar{τ}$ for different values of $T = 1.5, 2, 2.5, 5, 10, 15, 20, 50$ , given respectively by point, circle, $x$ mark, and diamond solid lines and point, circle, $x$ mark dotted lines. By analyzing $Δ λ$ , we are able to compare the maximum value of the mean quantum heat which is transferred by a two-level quantum system subject to different protocols of projective measurements. In the numerics, we have chosen $Δ E = 1, τ^{(1)} = 0.1, τ^{(2)} = 1.5$ , and ${| a |}^{2} = 0.2$ .
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Figure 4
Maximum mean quantum heat ${\bar{〈 Q_{q} 〉}}_{max}^{(an)}$ (in natural units with $ℏ = 1$ ) as a function of $Δ E 〈 τ 〉 \in [0, 4 π]$ when a stochastic sequence of measurements (annealed disorder) is applied to a two-level system. The curves have been obtained by varying $p_{1}$ among the set of values (0,0.25,0.5,0.75,1) (respectively, point-solid, dotted, dashed, dash-dotted, and solid lines) and by choosing $T = 5, τ^{(1)} = 0.1, τ^{(2)} = 0.5$ , and ${| a |}^{2} = 0.2$ .
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Figure 5
Convergence of $\partial_{c_{1}} G (i β)$ [slope of $G (i β)$ w.r.t. $c_{1}$ ] as a function of ${| a |}^{2}$ ( ${| a |}^{2} \in [0, 0.5]$ ) to the asymptotic limit of $M \to \infty$ when the time intervals between measurements are distributed as quenched disorder. We have considered $M = 2$ (solid line), $M = 10$ (dashed line), and $M = 100$ (dash-dotted line), with $E_{\pm} = \pm 1, τ^{(1)} = 0.01, τ^{(2)} = 3$ , and $p_{1} = 0.3$ .
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