Anomalous Thermalization in Quantum Collective Models

We show that apparently thermalized states still store relevant amounts of information about their past, information that can be tracked by experiments involving nonequilibrium processes. We provide a condition for the microcanonical quantum Crook’s theorem, and we test it by means of numerical experiments. In the Lipkin-Meshkov-Glick model, two different procedures leading to the same equilibrium states give rise to different statistics of work in nonequilibrium processes. In the Dicke model, two different trajectories for the same nonequilibrium protocol produce different statistics of work. Microcanonical averages provide the correct results for the expectation values of physical observables in all the cases; the microcanonical quantum Crook’s theorem fails in some of them. We conclude that testing quantum fluctuation theorems is mandatory to verify if a system is properly thermalized.

Figure 1

(a) Expected values for $n_t/N$ in the LMG model, for $\alpha =0.5$, $E/N=-0.24$, and $N=1.6\times {10}^4$. State prepared following procedure (i) (left, green online) and with procedure (ii) (right, red online). (Black dotted line) Analytical expected values $⟨\mathcal{O}(E,\alpha )⟩$. (b) Scaling as a function of the system size. Solid circles represent the distance between long-time averages and microcanonical expected values $\mathrm{\Delta }=|\overline{\mathcal{O}}-{\mathcal{O}}_{\mathrm{mic}}(E,\alpha )|$. (Solid squares) The variance of the fluctuations around the equilibrium state ${\sigma }^2=(1/T){\int }_0^{T}dt{[\mathcal{O}(t)-\overline{\mathcal{O}}]}^2$. Results obtained following procedure (i) (light symbols, green online) and following procedure (ii) (dark symbols, red online). Results are double averaged: over the four selected observables, and over ten different set of points ($\alpha =0.2$ and $E/N=-0.6$ together with nine different cases for $\alpha =0.5$: from $E/N=-0.26$ to $-0.22$, with steps $\mathrm{\Delta }E/N=0.005$). Dotted and dashed lines represent fits to power-law behaviors $N^{-\gamma }$. (See main text for details.)

Figure 2

(a) Statistics of work resulting from forward ${\alpha }_i=0.2\to {\alpha }_f=0.5$ and backward ${\alpha }_i=0.5\to {\alpha }_f=0.2$ processes, in the LMG model with $N=1.6\times {10}^4$. (Filled histogram, green online) Initial equilibrium states prepared following procedure (i). (Empty histogram, red online) States prepared following procedure (ii). (Dotted black line) Results of Eq. (1). (Vertical line) Particular case studied in Fig. 4. (b) Scaling of the distance $D=(1/N)\sum _i|h(w_i)-h_{\mathrm{QFR}}(w_i)|/h_{\mathrm{QFR}}(w_i)$, where $h(w_i)$ is the height of the actual histogram at work $w_i$, $h_{\mathrm{QFR}}$ is the theoretical value, Eq. (1), (QFR denotes Quantum Fluctuation Relation) and $N$ is the number of bins. (Light circles, green online) Results following procedure (i). (Dark circles, red online) Results following procedure (ii). (Dashed black line) Power-law fit $N^{-\gamma }$.

Figure 3

Statistics of work of the procedures $\alpha =1.2\to \alpha =2.0\to \alpha =0.6$ (empty histograms, red online) and $\alpha =1.2\to \alpha =0.0\to \alpha =0.6$ (full histograms, green online) in the Dicke model. (a) Fock initial states. (b) Microcanonical initial ensembles. (Dotted black line) Represents the theoretical result from Eq. (1). (Vertical dotted line, blue online) Indicates the cases displayed in Fig. 4.

Figure 4

(a) Energy distributions resulting from procedures (i) (filled curve, green online) and (ii) (empty curve, red online); both are obtained with the LMG model with $\alpha =0.5$, $E/N=-0.24$, and $N=1.6\times {10}^4$. Dotted black line displays the corresponding energies. (b) Energy distribution for the Dicke model with $\alpha =0.6$ and $E/j=3.48$: empty curve (red online) corresponds to the initial Fock state and the filled curve (green online) corresponds to the microcanonical ensemble. Again, the dotted black line displays the corresponding energies. (c) Scaling of the variance of the energy distribution ${\sigma }^2$ as a function of the number of atoms $N$ (solid circles) and scaling of the effective dimension of the Hilbert space $d_{\mathrm{eff}}$ (solid squares). Both are obtained with the LMG model. Results from procedure (i) (light symbols, green online) and from procedure (ii) (dark symbols, red online). (Dashed black line) Power-law fit $N^{-\gamma }$. (Dotted black line) Power-law fit $N^{\beta }$.