Partial Reversibility and Counterdiabatic Driving in Nearly Integrable Systems

Abstract

Adiabatic (or reversible) processes are the key concept unifying our understanding of thermodynamics and dynamical systems. Reversibility in the thermodynamic sense is understood as entropy-preserving processes, such as in the idealized Carnot engine, whereas in integrable dynamical systems it is understood as the conservation of the action variables. Between these two idealized limits, however, where the phase space can become mixed, things are much less clear. In this work, we first determine the extent to which reversible processes are even possible in this regime. We then explore how the dissipative losses resulting from rapidly driving these kinds of systems can be fought by approximate counterdiabatic driving. Finally, we argue that much of the phenomenology should be the same for quantum many-body systems with large degeneracy in the presence of integrability breaking perturbations.

Figures (6)

• Figure 1: The final energy variance for an initially microcanonical ensemble, after performing all three protocols at various rates $\tau$. In the upper panel, we drive from $\beta_i$ to $\beta_f$, using the ramp function in Eq. \ref{['eq:rampfunction']}. The dashed lines are analytical estimates in the slow-driving limit obtained from the effective SW Hamiltonian. They become asymptotically exact at small $\beta_f$. In the bottom panel, the same is plotted but for the cyclic protocol. The I-I protocol shows complete reversibility at long $\tau$. The I-N protocol leads to small but finite energy variance in the slow limit and hence is only partially reversible. For the N-N protocol the energy variance slowly decays with the protocol time, consistent with the asymptotic reversibility at $\tau\to\infty$.
• Figure 2: The energy fluctuations induced by the forward and cyclic versions of each protocol (dashed and solid lines respectively), using increasingly more sophisticated counterdiabatic driving protocols. Colours indicate different orders in the variational CD driving ansatz. The fluctuations from the I-I and I-N forward protocols show a plateau as a result of the thermalization within the degeneracy sector, which can be replicated by CD driving in the fast limit. In all other cases, local CD driving can provide significant suppression of the fluctuations before reaching a plateau.
• Figure 3: Numerically obtained energy fluctuations after $\beta$ in $H_\mathrm{NI}$ (Eq. \ref{['eq:HNI']}) is linearly ramped from 0 to 1, and then immediately reversed. For the lines cut off by the bottom of the page, they saturate within numerical precision of zero at a finite $\beta$, roughly indicating the $\beta_c$ before which things become reversible. For a sufficiently strong nonlinearity, faster driving leads to more fluctuations, but there is an intermediate regime where slower driving leads to more fluctuations. We attribute this to the presence of avoided crossings when the degenerate levels split by the Schrieffer Wolff Hamiltonian cross, shown in the right panel. Since the excited branch after the avoided crossing is the one which corresponds to the original degenerate sector, sufficiently slow driving actually follows the reversible path leading to greater fluctuations.
• Figure 4: The Schrieffer-Wolff analytical predictions (solid lines), compared with numerical results (dots) for the energy fluctuations from degeneracy-lifting perturbation in the integrable and nonintegrable models. The agreement is excellent at smaller $\beta_f$. The vertical dashed line indicates $\beta_f$ for I-I protocol, $\beta_f$ for I-N protocol is at $\log_{10}(\beta_f/\beta_0) = 0$. We use dimensionless units of $\beta$, $\beta_0 = m^2\omega^4/E_0$.
• Figure 5: We perform the cyclic I-I protocol with a variety of waiting times. If there is no waiting time (blue line), then we fail to capture the diabatic effects of fast driving. If the waiting time is not random, then we get oscillations corresponding to fact that the system has some intrinsic period. If the waiting time is randomized, there is a smooth exponential decay of the fluctuations in the slow driving limit, as expected.