Engineered Dissipation for Thermal-State Tracking Across Quantum Criticality

Technical Summary: Engineered Dissipation for Thermal-State Tracking Across Quantum Criticality

Problem Statement
The preparation of equilibrium states in quantum many-body systems is a central challenge in quantum technologies, ranging from quantum annealing to adiabatic computation. Standard protocols rely on adiabatic driving, where a control parameter is varied slowly to keep the system in the instantaneous eigenstate of the Hamiltonian. However, near a quantum critical point (QCP), the energy gap closes, causing the required adiabatic timescale to diverge. Consequently, finite-time driving inevitably generates nonadiabatic excitations and defects, a phenomenon described by the Kibble–Zurek (KZ) mechanism.

While environmental coupling (dissipation) is often viewed as a source of decoherence, it can also be a resource for state engineering. Existing approaches have largely focused on how a fixed, natural environment modifies critical dynamics or on coherent control strategies (e.g., shortcuts to adiabaticity) to suppress excitations. A significant gap remains in systematically engineering Markovian dissipative transition rates to actively steer a system along a prescribed finite-temperature thermal trajectory during finite-time driving across a QCP. Specifically, current methods do not fully address the dual requirement of controlling nonadiabatic excitations while simultaneously generating the population redistribution necessary to track a changing thermal target.

Methodology
The authors develop a Markovian dissipative framework designed to track an instantaneous Gibbs state, $\rho^\star(t)$, during a finite-time ramp of the system Hamiltonian $H_S(t)$.

1. Theoretical Framework:

   • The target state is defined as the instantaneous Gibbs state $\rho^\star(t) = e^{-\beta H_S(t)} / Z(t)$.
   • The time evolution of this target requires both a unitary rotation of the eigenbasis (handled by counterdiabatic driving) and a redistribution of occupation probabilities (populations).
   • Since unitary evolution cannot change eigenvalues, the authors construct a non-unitary dissipative term, $D[\rho]$, within the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation.
   • The core construction involves engineering transition currents between instantaneous energy eigenstates. The net population current is constrained to match the time derivative of the target populations, $\dot{\lambda}_n(t)$.
   • The authors derive a family of non-negative transition rates, $W_{ij}(t)$, that satisfy this constraint. This family includes an arbitrary common current term, $Q(t) \geq 0$, which allows for additional control over coherence damping without altering the net population transfer. The minimal realization ($Q=0$) activates only the necessary directional current (excitation or relaxation) dictated by the target trajectory.

2. Application to the Transverse-Field Ising Model (TFIM):

   • The framework is applied to the one-dimensional TFIM driven linearly across its quantum critical points ($g/J = \pm 1$).
   • Using the Jordan-Wigner and Fourier transformations, the model is decomposed into independent Bogoliubov-de Gennes (BdG) momentum sectors, each acting as an effective two-level system.
   • The engineered dissipator is applied mode-by-mode. The transition rates are calculated based on the instantaneous quasiparticle energy gap and the prescribed thermal populations for each mode $k$.

3. Comparative Analysis:

   • The performance of the engineered dynamics is benchmarked against two other scenarios: (1) purely unitary evolution (coherent driving) and (2) coupling to a natural thermal reservoir (Ohmic bath with fixed temperature).
   • Metrics include the excited-state population dynamics, the thermal excitation deviation (distance from the target Gibbs state), the suppression of dynamically generated coherence, and the scaling of residual errors with driving rate.

Key Results
Numerical simulations for the TFIM reveal the following:

• Suppression of Thermal Excitation Deviation: The engineered protocol significantly reduces the deviation of the actual state from the prescribed thermal trajectory compared to both unitary and natural-bath dynamics. While unitary and natural-bath evolutions exhibit persistent deviations after crossing the critical point, the engineered dynamics actively drive the populations back toward the target.
• Scaling Behavior:
  • Unitary Dynamics: Follow the conventional Kibble–Zurek scaling in the intermediate-rate regime, where the residual thermal excitation deviation scales as $\delta^{nth}_{ex} \sim (v/v_{ch})^{1/2}$. At very slow rates, finite-size effects modify this scaling.
  • Engineered Dynamics: The residual deviation is strongly suppressed and exhibits a substantially steeper scaling, approaching $\delta^{nth}_{ex} \sim (v/v_{ch})^2$ over a broad range of driving rates. This indicates a quadratic slow-driving behavior rather than the square-root KZ behavior.
  • Natural Bath: Exhibits a crossover behavior dependent on the competition between the ramp timescale and the fixed bath relaxation timescale, failing to adapt to the changing thermal target.
• Coherence Control: The engineered dynamics naturally suppress off-diagonal coherences generated during the drive. Furthermore, the free parameter $Q(t)$ allows for tunable enhancement of coherence damping (by increasing bidirectional transition rates) without disturbing the prescribed population tracking.
• Robustness: The protocol maintains its effectiveness when the system is driven across both critical points ($g/J = 1$ and $g/J = -1$), preventing the accumulation of errors that typically plague natural-bath dynamics in multi-critical traversals.

Significance and Claims
The paper claims that engineered dissipation offers a distinct route for controlling thermal-state preparation in driven quantum critical systems. The primary significance lies in the ability to actively reshape the finite-time approach to a thermal trajectory, rather than passively relying on a fixed environment or purely coherent control.

Key claims include:

1. Active Steering: Unlike natural baths, the engineered transition currents are constructed directly from the prescribed thermal trajectory, allowing the system to actively track the target state.
2. Improved Scaling: The protocol changes the finite-rate dependence of residual errors from the standard Kibble–Zurek behavior to a steeper quadratic scaling, suggesting higher efficiency for finite-time state preparation.
3. Dual Control: The framework provides simultaneous control over population redistribution (tracking the thermal target) and coherence damping (via the $Q(t)$ parameter).
4. Experimental Feasibility: The authors note that recent advances in reservoir engineering (e.g., in superconducting circuits and trapped ions) provide the necessary ingredients, such as programmable, energy-selective incoherent pumping and tunable dissipation rates, to potentially realize these protocols. However, they acknowledge that mapping the momentum-sector resolved rates to local spin controls in many-body systems presents experimental challenges regarding non-locality and bandwidth constraints.

The work concludes that while the current construction is formulated for independent quadratic sectors, it establishes a foundational framework for dissipative state preparation that could be extended to interacting many-body systems and optimized under specific experimental resource constraints.