Thermalization processes induced by quantum monitoring in multilevel systems

S. Gherardini, G. Giachetti, S. Ruffo, and A. Trombettoni

Phys. Rev. E 104, 034114 – Published 10 September 2021

Abstract

We study the heat statistics of a multilevel $N$ -dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number $M$ of measurements $(M \to \infty)$ . In this context, the conditions allowing for an infinite-temperature thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the nonequilibrium evolution of the system and its initial state. Exceptions to ITT, which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasicommuting) with the Hamiltonian of the quantum system or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces ( $N \to \infty$ ), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits $M \to \infty$ and $N \to \infty$ matters: When $N$ is fixed and $M$ diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A nontrivial result is obtained fixing $M / N^{2}$ where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.

Received 23 January 2021
Revised 2 June 2021
Accepted 17 August 2021

DOI:https://doi.org/10.1103/PhysRevE.104.034114

Physics Subject Headings (PhySH)

Nonequilibrium & irreversible thermodynamics Quantum fluctuations & noise Quantum measurements Quantum statistical mechanics Quantum stochastic processes Quantum thermodynamics

Nonequilibrium systems

Quantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

S. Gherardini ^1,2,3,*, G. Giachetti^1,†, S. Ruffo ^1,4,‡, and A. Trombettoni^5,1,3,§

¹SISSA and INFN, I-34136 Trieste, Italy
²Department of Physics and Astronomy and LENS, University of Florence, I-50019 Sesto Fiorentino, Italy
³CNR-IOM DEMOCRITOS Simulation Center, I-34136 Trieste, Italy
⁴Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, I-50019 Sesto Fiorentino, Italy
⁵Department of Physics, University of Trieste, I-34151 Trieste, Italy

^*gherardini@lens.unifi.it
^†ggiachet@sissa.it
^‡ruffo@sissa.it
^§andreatr@sissa.it

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Issue

Vol. 104, Iss. 3 — September 2021

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Images

Figure 1
Comparison between the initial (dashed-line histogram with blue area) and final (solid-line histogram with red area) heat statistics for a 5-level (a) and 15-level system (b). The Hamiltonian of the system and the initial density operator $ρ_{0}$ are randomly chosen on a basis in which $O$ is diagonal. The number of realizations of the nonequilibrium process is $5 \times 10^{6}$ in $(a)$ and $15 \times 10^{6}$ in (b). In both cases, in the thermodynamic limit of $M$ large (in our numerical simulations $M = 20$ ), each final energy value is equiprobable and such effect can be explained as the thermalization of the system toward a thermal state with $β = 0$ (infinite temperature). In this figure and in the following ones, the parameter $τ_{j} = 1$ is chosen.
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Figure 2
Comparison between the expression (19) of the characteristic function $G (ε) = 〈 e^{- ε Q} 〉$ (blue solid lines), plotted in semilogarithmic scale, and the numerical values (red dotted lines) computed for the 15-level system simulated in Fig. 1.
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Figure 3
Comparison between the theoretical estimate (red solid line) of the occurrence numbers to measure the heat outcomes $ω ℓ$ , as provided by Eq. (39), and the corresponding histogram (blue areas). The latter has been obtained by numerically repeating the nonequilibrium dynamics of sequential measurements over $10^{6}$ realizations on a spin $s = \frac{7}{2}$ . In the three panels, the initial state $ρ_{0}$ has been thermal with inverse temperature, respectively equal to $β = 0, β = 0.5$ , and $β = 1$ .
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Figure 4
Comparison between the spectra of the stochastic matrices $L (τ)$ with $s = 300$ and different values of $τ$ , expressed in terms of the rescaled variable $\frac{k τ}{2 s}$ . The plotted values of $τ$ belongs to the set ${1, 2, ..., 7}$ and are respectively identified by the blue diamond, orange circle, yellow $X$ , and purple solid lines and green, cyan circle, and red $X$ dotted lines. We can observe that, for $\frac{k τ}{2 s}$ smaller than a critical value (numerically determined to be $\approx 0.934$ ), all the data collapse on the same curve as predicted by Eq. (47). In turn, for $\frac{k τ}{2 s} ≪ 1$ we observe the quadratic behavior provided by Eq. (51), namely $f (\frac{τ k}{2 s}) \approx 1 - {(\frac{τ k}{2 s})}^{2}$ .
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Figure 5
Color plot of the matrix of eigenvectors of $L (τ)$ for different values of $τ$ (i.e., $τ = 1, 2, 3, 4, 6, 7$ from the top to the bottom and from left to right), with $s = 300$ . For visualization purposes, we plot on the $y$ axis of each panel the logarithm of each matrix element, while on the $x$ axis there is the index $k$ that labels each eigenvalue (the larger $k$ , the larger the eigenvalue). We can observe that, in spite of the structures developed for the larger values of $τ$ , the eigenstates on the right of each panel, corresponding to the higher part of the spectrum, are practically the same as long as $τ k / 2 s ≲ 1$ .
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