Quantum work for sudden quenches in Gaussian random Hamiltonians

In the context of nonequilibrium quantum thermodynamics, variables like work behave stochastically. A particular definition of the work probability density function (pdf) for coherent quantum processes allows the verification of the quantum version of the celebrated fluctuation theorems, due to Jarzynski and Crooks, that apply when the system is driven away from an initial equilibrium thermal state. Such a particular pdf depends basically on the details of the initial and final Hamiltonians, on the temperature of the initial thermal state, and on how some external parameter is changed during the coherent process. Using random matrix theory we derive a simple analytic expression that describes the general behavior of the work characteristic function $G\left(u\right)$, associated with this particular work pdf for sudden quenches, valid for all the traditional Gaussian ensembles of Hamiltonians matrices. This formula well describes the general behavior of $G\left(u\right)$ calculated from single draws of the initial and final Hamiltonians in all ranges of temperatures.

Figure 1

In panels (a) and (b) we plot the real and imaginary part of the work characteristic functions given in Eq. (17) (solid line) and calculated from Eq. (6) using single draws of the initial, $H$, and final, $\tilde{H}$, Hamiltonians of two GOEs (dashed lines). The parameters that define the ensembles are $N=300$, $〈s〉=0.1283$, and $〈\tilde{s}〉=0.1283/2$, and we chose $〈E〉=\langle \tilde{E}\rangle \approx 24$ (arbitrary units of energy) such with this choice we displace both spectra in order to have $E_1=0$. The inverse temperature is $\beta =0.01$ so $N_{\mathrm{eff}}/N\approx 2.6$. In panel (c) we plot the work probability density, as an histogram function (dashed line) calculated from the definition in Eq. (1) and from the Fourier transform of Eq. (17) (solid line). See text for details.

Figure 2

The same as Fig. 1 but with $\beta =0.1$. Therefore $N_{\mathrm{eff}}/N\approx 0.26$ (see text for details).

Figure 3

The same as Fig. 1 but with $\beta =1$ corresponding to $N_{\mathrm{eff}}/N\approx 0.026$ in panels (a) and (b) and the dashed curve in panel (c). The thin solid line in panel (c) corresponds to $〈P\left(w\right)〉$ calculated from two GOEs with $N=5000$ but keeping the values $\beta =1$ and $N_{\mathrm{eff}}/N\approx 0.026$ (see text for details).