Nonequilibrium quantum thermodynamics of determinantal many-body systems: Application to the Tonks-Girardeau and ideal Fermi gases

We develop a general approach for calculating the characteristic function of the work distribution of quantum many-body systems in a time-varying potential, whose many-body wave function can be cast in the Slater determinant form. Our results are applicable to a wide range of systems including an ideal gas of spinless fermions in one dimension, the Tonks-Girardeau (TG) gas of hard-core bosons, as well as a one-dimensional gas of hard-core anyons. In order to illustrate the utility of our approach, we focus on the TG gas confined to an arbitrary time-dependent trapping potential. In particular, we use the determinant representation of the many-body wave function to characterized the nonequilibrium thermodynamics of the TG gas and obtain exact and computationally tractable expressions— in terms of Fredholm determinants—for the mean work, the work probability distribution function, the nonadiabaticity parameter, and the Loschmidt amplitude. When applied to a harmonically trapped TG gas, our results for the mean work and the nonadiabaticity parameter reduce to those derived previously using an alternative approach. We next propose to use periodic modulation of the trap frequency in order to drive the system to highly non-equilibrium states by taking advantage of the phenomenon of parametric resonance. Under such driving protocol, the nonadiabaticity parameter may reach large values, which indicates a large amount of irreversible work being done on the system as compared to sudden quench protocols considered previously. This scenario is realizable in ultracold atom experiments, aiding fundamental understanding of all thermodynamic properties of the system.

Figure 1

The mean work $\langle W\left(t\right)\rangle$ of a driven Tonks-Girardeau gas in a harmonic trap, normalized by the initial thermal equilibrium energy $\langle \widehat{H}\left(0\right)\rangle$. The driving protocol used here is periodic modulation of the trap strength $V(x,t)=m\omega {\left(t\right)}^2{x}^2/2$, with $\omega {\left(t\right)}^2=\omega {\left(0\right)}^2[1-\alpha sin\left(\mathrm{\Omega }t\right)]$. Panel (a) shows $\langle W\left(t\right)\rangle$ as a function of the dimensionless time $\mathrm{\Omega }t$ and the driving frequency parameter $a\equiv {[2\omega \left(0\right)/\mathrm{\Omega }]}^2$, for the driving amplitude $\alpha =0.9$, whereas panel (c) is for $\alpha =0.5$. This parametrization is the same as the one used in the stability diagram of Fig. 3 of Ref. [42]. Fixing the value of $a$ (at given $\alpha$) corresponds to a specific realization of the driving protocol. Panels (b) and (d) show, respectively, the cuts of $\langle W\left(t\right)\rangle$ at constant $a$ from panels (a) and (c), but for a longer time span. The examples of $\langle W\left(t\right)\rangle$ in (b) at $a=7.5$ and $a=11$ correspond, respectively, to stable (semiperiodic) and unstable (exponentially growing) dynamics invoked by the sinusoidal drive at this value of the modulation amplitude $\alpha =0.9$. Similar examples of unstable behavior can be realized at other values of the parameter $a$ from the high-intensity horizontal bands in panel (a), corresponding to parametric resonances at values of $a$ slightly above $a=j^2$, where $j=1,2,3,...$ [42]. For comparison, in the examples of panel (d), which are for a smaller value of the modulation amplitude ($\alpha =0.5)$, the curve for $a=11$ is no longer unstable, and as an unstable example we show $\langle W\left(t\right)\rangle$ at the primary parametric resonance $a=1$.

Figure 2

The mean reversible work ${\langle W\left(t\right)\rangle }_{\mathrm{rev}}$ for the same driving protocol as in Fig. 1, for two values of the driving amplitude $\alpha$. The difference between the mean work $\langle W\left(t\right)\rangle$ shown in Fig. 1 and the reversible work shown here is the irreversible work.

Figure 3

The work probability distribution function $P\left(W\right)$ versus $W$ for a TG gas in a periodically modulated harmonic trap as in Fig. 1, evaluated at time $\mathrm{\Omega }t=10$. Panels (a) and (b) are, respectively, for the same parameters as the two curves in Fig. 1 (b), representing examples of stable and unstable dynamics. The chemical potential $\mu$ is chosen to result in the average number of particles $\langle N\rangle =20$, and $\langle \widehat{H}\left(0\right)\rangle$ is evaluated at the temperature well below the temperature of quantum degeneracy, $k_{B}T/\langle N\rangle \hslash \omega \left(0\right)=0.1$.

Figure 4

The nonadiabaticity parameter $Q^{*}\left(t\right)$ for the Tonks-Girardeau gas in a periodically modulated harmonic trap. All parameters and representative examples are the same as in Fig. 1.

Figure 5

The base $l\left(t\right)$ of the Loschmidt echo $L\left(t\right)=l{\left(t\right)}^{N^2}$ for the Tonks-Girardeau gas in a periodically modulated harmonic trap. All parameters and representative examples are the same as in Figs. 1 and 4. The gray (dashed) line in panel (b) is the exponentially decaying prediction of $l\left(t\right)=e^{\pi \gamma }{e}^{-i\gamma \tau }$ (where $\tau =\mathrm{\Omega }t$), with the dimensionless decay rate $\gamma =\left|\mathrm{Im}\right(\nu \left)\right|/2$, where $\nu$ is the Floquet exponent [42] equal to $\nu =1-0.182531i$ in this example. For the respective Poincaré maps, explaining the typical features seen in these Loschmidt echo curves, see Fig. 6 and text.

Figure 6

The Poincaré maps for stable (a) and unstable (b) dynamics. The parameter values are the same as in Fig. 5 (a), i.e., $\alpha =0.9$ for both panels, whereas $a=7.5$ for panel (a), and $a=11$ for panel (b). See text for further details.