Quantum work fluctuations in connection with the Jarzynski equality

A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change $\mathrm{\Delta }F$ of a system at inverse temperature $\beta$ from an ensemble average of nonequilibrium exponential work, i.e., $\langle e^{-\beta W}\rangle =e^{-\beta \mathrm{\Delta }F}$. The number of experimental work values needed to reach a given accuracy of $\mathrm{\Delta }F$ is determined by the variance of $e^{-\beta W}$, denoted $\mathrm{var}\left(e^{-\beta W}\right)$. We discover in this work that $\mathrm{var}\left(e^{-\beta W}\right)$ in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging $\mathrm{var}\left(e^{-\beta W}\right)$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of $\mathrm{var}\left(e^{-\beta W}\right)$ indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.

Figure 1

$\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ vs $\hslash \beta$, with $Q^{*}$ chosen to be 1, 1.01, 1.1, 1.5 (from bottom to top), with ${\omega }_0=0.35$ and ${\omega }_1=1$. The adiabatic case (bottom curve) has no divergence. Nonadiabatic effects induce a blowup in the low-temperature regime. Solid curves are obtained from the compact expression (5) and dots from numerics using a truncation of the quantum numbers in (3) with $n_{max}=m_{max}=30$.

Figure 2

Domain of divergence (white) in $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ for a sudden quench work protocol, with gray areas indicating finite values of $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$. In the high-temperature regime (a) the domain of divergence is ${\omega }_0\ge \sqrt{2}{\omega }_1$. This domain dramatically grows as temperature decreases from (b) to (d), assuming temperatures $\hslash \beta =5,10,15$.

Figure 3

Effect of an additional quartic potential on the work fluctuations of a QHO with ${\omega }_0=0.35$ and ${\omega }_1=1$, as described by $\mathrm{var}[exp(-\beta W_{\mathrm{dis}})]$ vs $\hslash \beta$. Panel (a) is for $H_a$ with $J=0$ (solid), 0.01 (dashed), and 0.05 (dotted); the work protocol is of constant compression velocity: $d\omega /dt=0.0464$ (red) and $d\omega /dt=0.5652$ (blue). In the absence of the quartic potential term, these sweeping velocities of compression give rise to $Q^{*}=1.011$ and $Q^{*}=1.5$, respectively. Panel (b) is for $H_b$, with $J=0$ (solid), 0.01 (dashed), with the same work protocols as in (a).

Figure 4

Behavior of $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ for sudden quench driving, $Q_{sq}^{*}=(\kappa +1/\kappa )/2$. The panels on the right show the domains for divergence/convergence (white/gray area) at temperatures (b) $\hslash \beta =0$, (c) $\hslash \beta =5$, (d) $\hslash \beta =10$, (e) $\hslash \beta =15$. In the high-temperature limit the domain of divergence is ${\omega }_0\ge \sqrt{2}{\omega }_1$. The area of convergence decreases with lower temperatures, eventually collapsing around the line ${\omega }_0={\omega }_1$. In the panel on the left are the profiles for the cross section, ${\omega }_0=0.35$, at temperatures $\hslash \beta =10,15$; the solid lines are from the compact expression Eq. (C5) and the dots are from numerics based on truncation in Eq. (C2) with $n_{max}=m_{max}=100$.

Figure 5

Behavior of $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ for the dynamic protocol (C12) with time of driving, $\tau =15$. We consider the temperature $\hslash \beta =10$ in (a) and (b) and the temperature $\hslash \beta =15$ in (c) and (d). The panels (b) and (d) show the corresponding domains of divergence/convergence (white/gray area). The panels (a) and (c) show the profile of $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ and $(Q^{*}-1)$ along ${\omega }_1$, for the cross section ${\omega }_0=0.35$ which is the dotted line in (b) and (d). For the panels (a) and (c) the solid lines depict $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ based on the compact expression Eq. (C5), while the red dots are from numerics in Eq. (C2) based on truncation of the quantum numbers with $n_{max}=m_{max}=50$. The location of the convergent/divergent behavior is associated with the valleys/peaks of the Husimi parameter, $Q^{*}$, as shown in dashed gray in (a) and (c).

Figure 6

Plotted in panel (a) is $\mathrm{var}\left(e^{-\beta W_{\mathrm{dis}}}\right)$ along the inverse time of driving $1/\tau$, under the work protocol (C12) characterized by a time-independent $\frac{d\omega }{dt}/{\omega }^2\left(t\right)$. We fix ${\omega }_0=0.35,{\omega }_1=1$, and $\hslash \beta =10$. The solid curves are based on the compact expression Eq. (C5), while the red dots are from numerics based on Eq. (C2), with truncation on the quantum numbers: $n_{max}=m_{max}=30$. The minima in panel (a) are associated with finite-time adiabatic dynamics [45]. Plotted in panel (b) is $(Q^{*}-1)$ along the inverse time of driving for the same protocol.

Figure 7

Divergence behavior of the variance of exponential quantum work, for a compression work protocol applied to particle in a box, with the initial width and final width of the box given by $L_0=2$ and $L_1=1$, with a constant velocity of the moving wall: $v=0.5$. Numerics with truncation of the double summation in Eq. (C3) at $max\left(n\right)=max\left(m\right)=32$ (red), 31 (orange), 30 (black). There is no compact expression for the transition probability $P_{m,n}^{0\to \tau }$, depicted in (F2), as the expression involves an infinite sum that can only be done numerically. In panel (b) we plot the logarithm of $\mathrm{var}[exp(-\beta W_{\mathrm{dis}})]$. It shows a linear trend with different tilts for each truncation. This indicates clearly an exponential growth in panel (a) with arguments determined by the truncation. We believe this is a signature of divergence baring resemblance of truncation of a divergent series of the form $\gamma ={\sum }_{n=1}^{\max \left(n\right)<\infty }{e}^{-\epsilon n^2}\approx e^{-\epsilon \max {\left(n\right)}^2}, \log \left(\gamma \right)\approx -\epsilon \max {\left(n\right)}^2$, for some constant $\epsilon <0$.