Optimizing Floquet engineering for non-equilibrium steady states with gradient-based methods
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2023-07-26 |
| Journal | SciPost Physics |
| Authors | Alberto Castro, Shunsuke Sato |
| Institutions | University of Tsukuba, Max Planck Institute for the Structure and Dynamics of Matter |
| Citations | 3 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis research details a novel application of Quantum Optimal Control Theory (QOCT) to design periodic driving fields that optimize the properties of Non-Equilibrium Steady States (NESS) in open quantum systems.
- Core Value Proposition: Provides a computational framework to engineer material properties in the non-equilibrium phase, specifically targeting steady states that are robust against decoherence and dissipation (environment interaction).
- Methodology: Extends QOCT to open systems governed by the time-periodic Lindblad master equation, utilizing multicolor periodic perturbations (Fourier expansions) as control parameters.
- Key Achievement (NV Center): Successfully optimized the time-averaged z-spin component (Sz) of a diamond Nitrogen-Vacancy (NV) center model under periodic magnetic driving.
- Performance Enhancement: Achieved a time-averaged Sz value (approx. 0.38) significantly greater than the theoretical maximum achievable in thermal equilibrium (approx. 0.14) at any temperature.
- Exotic State Preparation: Demonstrated the ability to prepare âexoticâ NESSs, including states with negative time-averaged Sz values, which are strictly forbidden in thermal equilibrium.
- Technical Implementation: The optimization relies on calculating the gradient of the objective function in Floquet-Liouville space, requiring the solution of large algebraic linear systems (dimension M = d2N, where d is Hilbert space dimension and N is the number of frequencies).
Technical Specifications
Section titled âTechnical SpecificationsâThe following parameters and results are derived from the application of the optimization scheme to the Nitrogen-Vacancy (NV) center in diamond model. Note that Nz is used as the base unit for energy and frequency throughout the model.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Optimized Max Time-Averaged Sz | ~0.38 | Dimensionless | Achieved NESS value (Îș = 4.0). |
| Thermal Equilibrium Max Sz | ~0.14 | Dimensionless | Maximum Sz achievable by varying temperature (ÎČ). |
| Fixed Inverse Temperature (ÎČ) | 3 | Nz-1 | Operating temperature for all optimization runs. |
| Default Dissipation Rate (Îł) | 0.2 | Nz | Rate constant used in the Lindblad model. |
| Field Amplitude Constraint (Îș) | 4.0 | Nz | Maximum allowed amplitude for Fourier coefficients (uj). |
| Fundamental Driving Frequency (Ï0) | 0.5 | Nz | Base frequency of the external magnetic field. |
| Cutoff Frequency (ÏM) | 2.0 | Nz | Maximum frequency component used (M=4). |
| Field-Free Hamiltonian Parameter (Nz) | 1 | Unitless | Sets the energy scale of the model. |
| Field-Free Hamiltonian Parameter (Nxy) | 0.05 | Nz | Transverse coupling term. |
| Static Magnetic Field (Bs) | 0.3 | Nz | Static component of the z-field. |
| Driving Field Scale (Bd) | 0.1 | Nz | Scale factor for the periodic perturbation. |
| Optimization Iterations | ~100 | #Iter | Typical number of gradient/function evaluations required for convergence. |
Key Methodologies
Section titled âKey MethodologiesâThe optimization of Non-Equilibrium Steady States (NESS) properties is achieved through a gradient-based Quantum Optimal Control Theory (QOCT) framework:
- Master Equation Formulation: The system dynamics are modeled using a time-periodic Lindblad-type master equation (Eq. 1), which includes both coherent driving (Hamiltonian H(t)) and incoherent dissipation (Lindblad operators Vij).
- Control Parametrization: The time-dependent external fields (e.g., magnetic fields gx(t) and gy(t)) are represented by multicolor periodic perturbations using truncated Fourier series (Eq. 24). The Fourier coefficients (u0, u1, âŠ) constitute the control parameters u.
- Objective Function Definition: The target metric G(u) is defined as the time-average of the expectation value of a desired observable A (e.g., Sz) over one period T of the NESS (Eq. 6).
- NESS Calculation (Floquet-Liouville Space): The NESS (periodic solution Ïu(t)) is found by transforming the Lindblad equation into a linear homogeneous algebraic system in Floquet-Liouville space (Eq. 15).
- Gradient Computation: The gradient components (âG/âuk) are calculated by determining the derivatives of the NESS with respect to the control parameters (âÏu/âuk).
- Solving for Derivatives: The derivative (âÏu/âuk) is found by solving a linear equation (Eq. 17) derived from the variation of the NESS equation. Since the operator L(u) has a non-empty kernel, a least-squares method is used, combined with the normalization condition (Tr[âÏu/âuk] = 0) to ensure a unique solution (Eq. 19).
- Optimization Algorithm: The Sequential Least-Squares Quadratic Programming (SLSQP) algorithm is employed to maximize or minimize G(u), subject to constraints on the control amplitudes ( |uj| less than or equal to Îș).
Commercial Applications
Section titled âCommercial ApplicationsâThis methodology provides a robust theoretical tool for engineering quantum systems where environmental coupling (decoherence) is significant, extending the utility of Floquet engineering beyond idealized isolated systems.
| Industry / Field | Application | Specific Functionality |
|---|---|---|
| Quantum Sensing | NV Center Optimization | High-fidelity preparation of spin states (Sz) for enhanced magnetic field sensitivity, achieving states inaccessible via thermal methods. |
| Quantum Computing | Robust State Preparation | Designing control pulses that prepare specific quantum states (NESSs) while actively mitigating the effects of dissipation and decoherence. |
| Materials Engineering | Exotic Phase Creation | Using optimized periodic driving to induce novel material properties or phases (e.g., topological states) that are forbidden in thermodynamic equilibrium. |
| Open Quantum Systems | Control and Manipulation | General framework for controlling the time-averaged properties of any periodically driven system coupled to a bath (e.g., cold atoms, cavity QED). |
| Computational Physics | Inverse Design Protocols | Providing a differentiable optimization protocol for NESSs, enabling efficient inverse design of external fields or internal system parameters. |
View Original Abstract
Non-equilibrium steady states are created when a periodically driven quantum system is also incoherently interacting with an environment - as it is the case in most realistic situations. The notion of Floquet engineering refers to the manipulation of the properties of systems under periodic perturbations. Although it more frequently refers to the coherent states of isolated systems (or to the transient phase for states that are weakly coupled to the environment), it may sometimes be of more interest to consider the final steady states that are reached after decoherence and dissipation take place. In this work, we demonstrate how those final states can be optimally tuned with respect to a given predefined metric, such as for example the maximization of the temporal average value of some observable, by using multicolor periodic perturbations. We show a computational framework that can be used for that purpose, and exemplify the concept using a simple model for the nitrogen-vacancy center in diamond: the goal in this case is to find the driving periodic magnetic field that maximizes a time-averaged spin component. We show that, for example, this technique permits to prepare states whose spin values are forbidden in thermal equilibrium at any temperature.