Quantum-trajectory thermodynamics with discrete feedback control

Zongping Gong, Yuto Ashida, and Masahito Ueda

Phys. Rev. A 94, 012107 – Published 8 July 2016

Abstract

We employ the quantum-jump-trajectory approach to construct a systematic framework to study the thermodynamics at the trajectory level in a nonequilibrium open quantum system under discrete feedback control. Within this framework, we derive quantum versions of the generalized Jarzynski equalities, which are demonstrated in an isolated pseudospin system and a coherently driven two-level open quantum system. Due to quantum coherence and measurement backaction, a fundamental distinction from the classical generalized Jarzynski equalities emerges in the quantum versions, which is characterized by a large negative information gain reflecting genuinely quantum rare events. A possible experimental scheme to test our findings in superconducting qubits is discussed.

Received 15 February 2016
Revised 12 June 2016

DOI:https://doi.org/10.1103/PhysRevA.94.012107

Physics Subject Headings (PhySH)

Open quantum systems Quantum control Quantum thermodynamics

Statistical Physics & ThermodynamicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Zongping Gong¹, Yuto Ashida¹, and Masahito Ueda^1,2

¹Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
²RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

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Vol. 94, Iss. 1 — July 2016

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Images

Figure 1
A system is weakly coupled to an ideal heat bath with inverse temperature $β$ and simultaneously driven out of equilibrium by a time-dependent inclusive Hamiltonian $H (λ_{t})$ and an exclusive one $h_{t}$ . At the trajectory level, the system is projectively measured twice in the eigenbasis of the instantaneous Hamiltonian at the initial and final times, while the heat bath is subjected to continuous projective monitoring during the whole process.
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Figure 2
A forward QJT (top) and the corresponding time-reversed QJT (bottom) starting from the initial energy eigenstates $| a^{λ_{0}} 〉$ and $Θ | b^{λ_{τ}^{α}} 〉$ , respectively, where $Θ$ is the time-reversal operator. Here the $j_{1} th (j_{1}^{'} th)$ quantum jump occurs at $t_{1} < t_{m} ({\bar{t}}_{1} < {\bar{t}}_{m})$ , the measurement $M_{A} (M_{B_{α}})$ is performed at $t_{m} ({\bar{t}}_{m})$ with the outcome being $α$ , the $j_{2} th (j_{2}^{'} th)$ quantum jump occurs at $t_{2} > t_{m} ({\bar{t}}_{2} > {\bar{t}}_{m})$ , and the forward (backward) trajectory ends at the final energy eigenstate $| b^{λ_{τ}^{α}} 〉 (Θ | a^{λ_{0}} 〉)$ due to the projective measurement $Π^{λ_{τ}^{α}} ({\tilde{Π}}^{λ_{0}} \equiv Θ Π^{λ_{0}} Θ^{†})$ . In general, $| {\bar{ψ}}_{τ - t} 〉 \neq Θ | ψ_{t} 〉$ .
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Figure 3
(a) A quantum rare event in a spinless (time-reversal symmetric, i.e., $Θ = I$ ) two-level system with $t_{m} = 0$ and $M_{A}$ being nothing but the initial PM. Here the outcome of the initial PM at time $t = 0$ is assumed to be the excited state $e$ . Nevertheless, an unexpected quantum jump $g \to e$ occurs at $t_{1}$ , indicated by a tiny jump in the excited state fidelity (blue). A large negative $I_{QJT}$ is implied by the small excited state fidelity at the final state of the time-reversed QJT (green). (b) In the classical limit, the first jump at $t_{1}$ is always $e \to g$ if the system is initially in $e$ because of the absence of quantum superposition, and the time-reversed QJT is given by $| {\bar{ψ}}_{t} 〉 = | ψ_{τ - t} 〉$ .
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Figure 4
$〈 I_{QJT} 〉 / ln 2$ (left) and $(〈 I_{QJT} 〉 + β {〈 W_{diss} 〉}_{\min} / ln 2)$ (right) in the $θ_{0} - θ_{1}$ parameter space. In the left figure, the green and blue planes respectively correspond to the QC-mutual information $I_{QC}$ and 0, where $p_{↑}^{eq}$ is fixed to be $0.8$ . In the right figure, the green curved surface refers to $I_{QC}$ (overestimation from the exact $- β {〈 W_{diss} 〉}_{\min}$ ), while the remaining yellow ones show $〈 I_{QJT} 〉$ for different equilibrium initial states ( $p_{↑}^{eq} = 0.5, 0.8, 0.9, 0.999$ from the lowest to the highest). Note that $〈 I_{QJT} 〉 + β {〈 W_{diss} 〉}_{\min}$ oscillates in the $θ_{0}$ direction, while $I_{QC} + β {〈 W_{diss} 〉}_{\min}$ oscillates in the $θ_{0} = - θ_{1}$ direction.
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Figure 5
Feedback control of a dissipative two-level system based on the initial projective measurement. The strength of the coherent driving $ε$ is tuned to be smaller (larger) if the outcome is the ground state (excited state). The inclusive driving protocol $ω_{t} = ω_{0} + Δ ω t / τ$ and the coupling strength to the heat bath are the same.
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Figure 6
Numerical verification of the generalized quantum Jarzynski equalities in a dissipative two-level system under coherent driving (22). Work distributions (a) without and (b) with feedback control. The distributions of (c) $I_{QJT}$ (17), and (d) the composite variable $W + β^{- 1} I_{QJT}$ for the feedback control process. Insets show the probabilities of the divergent $δ$ -type peaks.
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