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A First-Principles Thermodynamic Uncertainty Relation for Shortcuts to Adiabaticity

Guillermo Perna, Federico Centrone, Esteban Calzetta

Abstract

We study the fundamental limitations of implementing time-dependent Hamiltonian protocols when ''time'' is provided by a quantum clock rather than an external classical parameter. For a parametric harmonic oscillator controlled through a shortcut-to-adiabaticity (STA) schedule and coupled to a minimal clock degree of freedom, tracing out the clock yields an effective reduced dynamics that is a mixture of unitary Gaussian trajectories. Within a noise-dominated regime, we compute the energetic deviation from the target STA outcome and its fluctuations, together with the fidelity to the target evolution and the purity loss of the reduced state, for vacuum and coherent initial states. Combining these observables produces a thermodynamic-uncertainty-type tradeoff that links achievable precision to an irreducible loss of purity set by the clock precision and the protocol sensitivity.

A First-Principles Thermodynamic Uncertainty Relation for Shortcuts to Adiabaticity

Abstract

We study the fundamental limitations of implementing time-dependent Hamiltonian protocols when ''time'' is provided by a quantum clock rather than an external classical parameter. For a parametric harmonic oscillator controlled through a shortcut-to-adiabaticity (STA) schedule and coupled to a minimal clock degree of freedom, tracing out the clock yields an effective reduced dynamics that is a mixture of unitary Gaussian trajectories. Within a noise-dominated regime, we compute the energetic deviation from the target STA outcome and its fluctuations, together with the fidelity to the target evolution and the purity loss of the reduced state, for vacuum and coherent initial states. Combining these observables produces a thermodynamic-uncertainty-type tradeoff that links achievable precision to an irreducible loss of purity set by the clock precision and the protocol sensitivity.
Paper Structure (42 sections, 113 equations, 4 figures)

This paper contains 42 sections, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Autonomous clock model and reduced dynamics
  3. Target protocol and STA benchmark
  4. Autonomous implementation with a quantum clock
  5. Influence functional representation
  6. Noise-dominated single-parameter reduction
  7. Assumptions and regime of validity
  8. Unitary dynamics for single trajectory
  9. Symplectic evolution of the quadratures
  10. Symplectic map and Bogoliubov coefficients
  11. Observables, clock averages, and TUR
  12. Energy deviation and its clock-induced fluctuations
  13. Irreversibility and target overlap
  14. Closed clock averages in the small-noise regime
  15. TUR-type tradeoff.
  16. ...and 27 more sections

Figures (4)

  • Figure 1: Frequency protocols for the finite-time driving schedule. The blue curve shows the prescribed frequency squared $\omega^2\left(t\right)$ defined in Eq. \ref{['eq:fin_prot']}, which smoothly interpolates between the initial value $\omega_i$ and the final value $\omega_f$ during a total duration $\tau$. The orange curve shows the corresponding shortcut-to-adiabaticity (STA) frequency $\bar{\Omega}^2 \left( t \right)$ constructed from Eq. \ref{['eq:Omegabar']}, which guarantees that the oscillator follows the adiabatic mapping in finite time when time is treated as an external parameter. The green curve shows the STA deviation $\delta \Omega^2$ as defined in Eq. \ref{['eq:deltaOmega_main']}. Frequencies are shown in units where $\omega_i = 1$ and $\omega_f = 2$, and time is measured in units of the protocol duration $\tau$.
  • Figure 2: Figures of merit for the finite-time protocol as functions of the protocol duration $\tau$. Top panels: logarithm of the real and imaginary parts of the Bogoliubov coefficient $\beta_1$ (left) and logarithm of $\alpha_1$ (right), which characterize parametric excitations produced by the driving. Middle panels: fidelity with the target STA state (left) and purity of the reduced oscillator state (right), quantifying precision and irreversibility induced by discarding the clock degree of freedom. Bottom panels: logarithm of the thermodynamic-uncertainty-type ratio $\mathcal{R}_E$ defined in Eq. \ref{['eq:RE_def']} for different coherent-state amplitudes $\mu$ (left), and the phase-averaged ratio as a function of $\left| \mu \right|$ (right). These results illustrate the crossover between strong nonadiabatic effects at short durations and suppressed excitations in the slow-driving regime.
  • Figure 3: Frequency protocols for the infinite-time driving schedule defined in Eq. \ref{['eq:inf_prot']}. The blue curve shows a time window of the bare frequency squared $\omega^2\left(t\right)$, which asymptotically approaches constant values at early and late times. The orange curve shows the shortcut-to-adiabaticity frequency $\bar{\Omega}^2\left(t\right)$ obtained from Eq. \ref{['eq:Omegabar']},while the green curve shows the corresponding STA deviation $\delta\Omega^2$ as defined in Eq. \ref{['eq:deltaOmega_main']}. As in Fig. \ref{['Fig:finite_time']}, the STA protocol modifies the instantaneous frequency so that the system reproduces the adiabatic mapping in finite time. Frequencies are plotted in units where $\omega_i = 1$ and $\omega_f = 2$, with time measured in units of $\tau$.
  • Figure 4: Same quantities as in Fig. \ref{['Fig:finite_time_plots']} but for the infinite-time driving protocol defined in Eq. \ref{['eq:inf_prot']}. The top panels show the dependence on the characteristic timescale $\tau$ of the (first order expansion) Bogoliubov coefficients $\alpha_1$ and $\beta_1$, the middle panels the fidelity with the target STA state and the purity of the reduced oscillator state, and the bottom panels the thermodynamic-uncertainty-type ratio $\mathcal{R}_E$ for different values of the adiabatic parameter $\mu$ and its averaged counterpart in logarithmic scale. For short times the dynamics exhibits strong non-adiabatic excitations characterized by $\left|\beta_1\right| \sim \frac{1}{\tau}$, while increasing $\tau$ suppresses excitations and improve fidelity. The TUR ratio remains bounded from below, illustrating the trade-off between energetic fluctuations and irreversibility induced by tracing out the clock.