Work distributions for random sudden quantum quenches

The statistics of work performed on a system by a sudden random quench is investigated. Considering systems with finite dimensional Hilbert spaces we model a sudden random quench by randomly choosing elements from a Gaussian unitary ensemble (GUE) consisting of Hermitian matrices with identically, Gaussian distributed matrix elements. A probability density function (pdf) of work in terms of initial and final energy distributions is derived and evaluated for a two-level system. Explicit results are obtained for quenches with a sharply given initial Hamiltonian, while the work pdfs for quenches between Hamiltonians from two independent GUEs can only be determined in explicit form in the limits of zero and infinite temperature. The same work distribution as for a sudden random quench is obtained for an adiabatic, i.e., infinitely slow, protocol connecting the same initial and final Hamiltonians.

Figure 1

The scaled averaged work distribution ${\sigma }_f{\langle p\left(w\right)\rangle }_{\text{GUE}}^f$ given by Eq. (34) is displayed as a function of $w/\epsilon$ for different inverse temperatures $\beta =0.1/{\sigma }_f$ (red), $\beta =1/{\sigma }_f$ (green), and $\beta =10/{\sigma }_f$ (blue) and different ratios of the initial level spacing and the width of the GUE: $\epsilon /{\sigma }_f=10$ in (a), $\epsilon /{\sigma }_f=1$ in (b), $\epsilon /{\sigma }_f=0.1$ in (c). With increasing width of the GUE the temperature dependence is less pronounced.

Figure 2

The thermally weighted spectral density $q_i\left(e\right){\sigma }_i$ is displayed as a function of the scaled energy $e/{\sigma }_i$ for different inverse temperatures $\beta =0.1/{\sigma }_i$ (red), $\beta =1/{\sigma }_i$ (green), and $\beta =10/{\sigma }_i$ (blue). The distribution for the lowest temperature is already indistinguishable from the ground-state distribution (39) while at the highest temperature corresponding to $\beta {\sigma }_i=0.1$ the limiting case of the spectral density $D_i\left(e\right)$, which is fully symmetric with respect to $e=0$, is apparently not yet reached.

Figure 3

The scaled averaged work distribution ${\sigma }_i{\langle p\left(w\right)\rangle }_{i,f}$ given by Eq. (27) is displayed as a function of $w/{\sigma }_i$ for different values of parameters ${\sigma }_f/{\sigma }_i=0.25$ (red), ${\sigma }_f/{\sigma }_i=0.5$ (green), and ${\sigma }_f/{\sigma }_i=0.75$ (blue). (a) presents the high temperature limit (37) and (b) the zero temperature limit (40).

Figure 4

The scaled averaged work distribution ${\sigma }_i{\langle p\left(w\right)\rangle }_{i,f}$ given by Eq. (27) is displayed as a function of $w/{\sigma }_i$ for different inverse temperatures $\beta =0.1/{\sigma }_f$ (red), $\beta =1/{\sigma }_f$ (green), and $\beta =10/{\sigma }_f$ (blue) and different ratios of the GUE widths: ${\sigma }_f/{\sigma }_i=0.25$ in (a), ${\sigma }_f/{\sigma }_i=0.5$ in (b), ${\sigma }_f/{\sigma }_i=0.75$ in (c). The black crosses in (a) are estimates of the work pdf Eq.(1) at the initial inverse temperature $\beta {\sigma }_i=0.1$ from a sample of ${10}^8$ pairs of energy eigenvalues drawn from two GUEs with ${\mu }_f={\mu }_i=0$ and ${\sigma }_f/{\sigma }_i=0.25$. The agreement of the simulation with the numerical integration of Eq. (27) with Eqs. (33) and (35) is excellent.

Figure 5

Average scaled work $\langle w\rangle /{\sigma }_i$ in (a) and variance of the work ${\sigma }_w^2/{\sigma }_i^2$ in (b) as functions of the scaled inverse temperature $\beta {\sigma }_i$ for different ratios of the GUE widths: ${\sigma }_f/{\sigma }_i=0.25$ (red), ${\sigma }_f/{\sigma }_i=0.5$ (green), and ${\sigma }_f/{\sigma }_i=0.75$ (blue). Dashed black lines indicate the asymptotic zero temperature limit.