Superdiffusive quantum work and adiabatic quantum evolution in finite temperature chaotic Fermi systems

We study the full distribution of quantum work in generic, noninteracting, disordered fermionic nanosystems at finite temperature. We derive an analytical determinant formula for the characteristic function of work statistics for quantum quenches starting from a thermal initial state. For work small compared to the thermal energy of the Fermi gas, work distribution is Gaussian, and the variance of work is proportional to the average work, while in the low-temperature or large-work limit, a non-Gaussian distribution with superdiffusive work fluctuations is observed. Similarly, the time dependence of the probability of adiabaticity crosses over from an exponential to a stretched exponential behavior. For large enough average work, the work distribution becomes universal, and depends only on the temperature and the mean work. Apart from initial low-temperature transients, work statistics are well captured by a Markovian energy-space diffusion process of hardcore particles, starting from a thermal initial state. Our findings can be verified by measurements on nanoscale circuits or via single qubit interferometry.

Figure 1

Left: Energy levels of a disordered nanograin during a quantum quench. External time-dependent gate voltages and magnetic fields induce quantum mechanical level-to-level transitions and generate work. A thermal average is performed over initial states. Right: Ladder model. Particles move diffusively between uniformly spaced energy levels. Initial, finite $T$ configurations are generated with appropriate Boltzmann weights.

Figure 2

Mean occupations ${\langle f_k\rangle }_{\mathrm{RM}}$ of instantaneous eigenstates for three different pairs $(\langle \langle w\rangle \rangle ,\tilde{T})$, corresponding to large, intermediate, and small work compared to the thermal energy, respectively, and for different velocities $\tilde{v}$. Quantum mechanical results (symbols), computed from Eq. (17), are well captured by the ladder model (black crosses) and analytical approximation Eq. (38), further simplified to Eq. (40) for large, and to Eq. (42) for small work (black-dashed lines). (a) Occupation profiles in the slow quench limit with $\tilde{v}=0.5$ for the GOE ensemble for ($\tilde{T}=0.5$, $\langle \langle w\rangle \rangle =5$), ($\tilde{T}=8$, $\langle \langle w\rangle \rangle =10$), and ($\tilde{T}=3,\langle \langle w\rangle \rangle =10$). Inset: average work as a function of time for the same temperatures $\tilde{T}$. (b) Occupation profiles for the three random matrix ensembles for different velocities $(\tilde{v}=0.8,\tilde{T}=1,\langle \langle w\rangle \rangle =10)$ for GOE, $(\tilde{v}=3,\tilde{T}=3,\langle \langle w\rangle \rangle =15)$ for the GSE, and $(\tilde{v}=5,\tilde{T}=8,\langle \langle w\rangle \rangle =20)$ for GUE.

Figure 3

Regular part of the distribution of quantum work for GOE (up) and GSE (bottom) at various temperatures, for average work $\langle \langle w\rangle \rangle =5$, corresponding to velocities $\tilde{v}=0.5$ and 0.8, respectively. (a) Comparison of random matrix simulations (symbols) and the ladder model (dashed lines) for GOE. Most features are well captured by the classical ladder model. Fingerprints of level repulsion are present for small positive values of work at low temperatures. (b) GSE results. Level repulsion signatures are enhanced, but disappear fast with increasing temperature. A fast convergence towards Gaussian statistics is observed as ${\tilde{T}}^2\gtrsim \langle \langle w\rangle \rangle$.

Figure 4

Collapse of work statistics for large average work values and for various velocities within the three major matrix ensembles. The distributions collapse to a single curve, and agree very well with the predictions of the “ladder” model (green plus signs). Notice the non-Gaussian distribution, characteristic for large works, $\langle \langle w\rangle \rangle \gg {\tilde{T}}^2$.

Figure 5

Probability of adiabaticity for GUE at temperatures $\tilde{T}=0.5,1.5,3$, for velocity $\tilde{v}=0.8$ (slow-quench limit), and average work $\langle \langle w\rangle \rangle =5.$ At finite $T$, the initial exponential decay crosses over to a stretched exponential for $\langle \langle w\rangle \rangle >{\tilde{T}}^2$. Dashed lines denote analytical exponential and stretched exponential fits. Inset: Logarithmic plot for $\tilde{T}=0.5$ and $\tilde{T}=3$, showing the distinct behaviors at low and high temperatures.

Figure 6

Crossover from superdiffusive to diffusive work fluctuations for $\langle \langle w\rangle \rangle >1$ in the GOE ensemble. For injected work large compared to the thermal energy, fluctuations preserve their superdiffusive behavior as in the $T=0$ case, $\langle \langle \delta w^2\rangle \rangle \sim {\langle \langle w\rangle \rangle }^{3/2}$. In contrast, for small average work $1<\langle \langle w\rangle \rangle \ll {\tilde{T}}^2$, they exhibit a diffusive character, $\langle \langle \delta w^2\rangle \rangle \sim \langle \langle w\rangle \rangle$.

Figure 7

Numerical verification of the Crooks fluctuation relation (35) for the three ensembles with different quench times, velocities, and temperatures as indicated in the legend. The GSE result was computed using a single random matrix trajectory, while the GOE, GUE results were averaged over $\sim 500$ disorder realizations.

Figure 8

Real (a) and imaginary (b) parts of the characteristic function for the GOE ensemble, computed numerically from Eq. (23) for dimensionless quench time $\tilde{t}=34$ and velocity $\tilde{v}=0.5$, corresponding to the slow quench limit. The zero-temperature average work is ${\langle \langle w\rangle \rangle }_{T=0}=5$, and we evaluate the characteristic function at dimensionless temperatures $\tilde{T}=0,2,5.$ The slope of the imaginary part at $u=0$ is unaffected by the increase of the temperature, implying that, keeping the quench time fixed, the average work of the system is independent of the temperature within numerical precision. Upon increasing the temperature, a clear convergence is seen towards the characteristic function of a Gaussian work distribution with mean $\langle \langle w\rangle \rangle$ and variance $\delta w^2,$ shown in black-dashed line.