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 | Universidad de Zaragoza, University of Tsukuba |
| Citations | 3 |
| Analysis | Full AI Review Included |
Technical Documentation & Analysis: Floquet Engineering in Diamond NV Centers
Section titled âTechnical Documentation & Analysis: Floquet Engineering in Diamond NV CentersâReference: Castro, A. and Sato, S. A. (2023). Optimizing Floquet engineering for non-equilibrium steady states with gradient-based methods. SciPost Phys. 15, 029.
Executive Summary
Section titled âExecutive SummaryâThis research demonstrates a powerful methodology for controlling the properties of quantum materials in non-equilibrium steady states (NESS) using optimized periodic driving fields. The findings are highly relevant for the development of advanced diamond-based quantum technologies, particularly those utilizing the Nitrogen-Vacancy (NV) center.
- Core Achievement: Successful application of Quantum Optimal Control Theory (QOCT) combined with a Lindblad master equation to optimally tune NESS properties in an open quantum system.
- Model System: The Nitrogen-Vacancy (NV) center in diamond, driven by time-periodic magnetic fields (Floquet engineering).
- Optimization Goal: Maximization and minimization of the time-averaged spin component, <<Sz>>, by optimizing the Fourier coefficients of the driving fields.
- Exotic State Preparation: The technique achieved âexoticâ NESSs, demonstrating spin expectation values (e.g., <<Sz>> â 0.38) significantly exceeding the maximum allowed in thermal equilibrium (â 0.14).
- Experimental Relevance: By utilizing the Lindblad equation, the framework naturally incorporates decoherence and dissipation (rate Îł), ensuring the optimized fields are relevant for experimentally realizable quantum control.
- Material Exploration: This work establishes periodic perturbations as a novel degree of freedom for engineering material functionalities, extending material exploration beyond equilibrium phases.
Technical Specifications
Section titled âTechnical SpecificationsâThe following hard data points were extracted from the theoretical model and optimization results concerning the NV center system.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Model Fixed Unit | Nz = 1 | Dimensionless | NV center Hamiltonian parameter (sets energy scale) |
| Transverse Coupling | Nxy = 0.05 | Dimensionless | NV center Hamiltonian parameter |
| Static Magnetic Field (z-axis) | Bz = 0.3 | Dimensionless | NV center Hamiltonian parameter |
| Driving Field Scaling Factor | B0 = 0.1 | Dimensionless | Amplitude scaling for periodic perturbation V(u, t) |
| Inverse Temperature (Fixed) | ÎČ = 3 | 1/kBT | Used for NESS calculations |
| Dissipation Rate Constant | Îł = 0.2 | Dimensionless | Standard rate constant used in the Lindblad equation |
| Thermal Equilibrium Max <Sz> | â 0.14 | Dimensionless | Maximum Sz achievable by temperature variation (ÎČ > 0) |
| Optimized Max <<Sz>> | â 0.38 | Dimensionless | Time-averaged spin component achieved via QOCT (Îș=4.0) |
| Amplitude Constraint (Max Sz) | Îș = 4.0 | Dimensionless | Maximum bound on Fourier coefficients |
| Cutoff Frequency | ÏM = 2.0 Nz | Dimensionless | Maximum frequency component used in the driving field |
Key Methodologies
Section titled âKey MethodologiesâThe optimization of Non-Equilibrium Steady States (NESS) was achieved by integrating Quantum Optimal Control Theory (QOCT) with open quantum system dynamics.
- Open Quantum System Modeling: The system dynamics were governed by a Lindblad-type master equation (Eq. 1) incorporating a time-periodic Hamiltonian H(t) and Lindblad operators Vij to account for dissipation and decoherence effects (Markovian approximation).
- Periodic Driving Parametrization: The external periodic magnetic fields, gx(t) and gy(t), were defined using truncated Fourier expansions (Eq. 24). The control parameters $u$ were the coefficients of these expansions, allowing for âmulticolor periodic driving.â
- Optimization Target Definition: The objective function G(u) was defined as the time-average of the expectation value of an observable A (specifically, the Sz spin component) over one Floquet period T.
- NESS Calculation in Floquet-Liouville Space: The periodic NESS solution Ïu(t) was found by solving the linear homogeneous equation in the extended Floquet-Liouville space, which accounts for the time-dependence of the Lindbladian.
- Gradient Derivation: A novel procedure was developed to compute the gradient (âG/âuk) of the objective function with respect to the control parameters, circumventing difficulties associated with standard adjoint methods for periodic equations.
- Constrained Optimization: The Sequential Least-Squares Quadratic Programming (SLSQP) algorithm was used to perform the gradient-based optimization, subject to constraints on the amplitude of the driving field Fourier coefficients (|uj| †Îș), mimicking experimental limitations.
6CCVD Solutions & Capabilities
Section titled â6CCVD Solutions & CapabilitiesâThis research highlights the critical role of high-quality diamond substrates and precision engineering in achieving advanced quantum control. 6CCVD is uniquely positioned to supply the necessary materials and customization services required to replicate and extend this Floquet engineering research into practical devices.
| Category | 6CCVD Solution & Value Proposition |
|---|---|
| Applicable Materials | Optical Grade Single Crystal Diamond (SCD): The NV center is the core component. Achieving the low dissipation rates (Îł) required for optimal control necessitates ultra-high purity SCD with long coherence times (T2). 6CCVD supplies high-quality SCD plates, essential for minimizing background defects and maximizing quantum performance. |
| Custom Dimensions & Geometry | Precision Plates and Wafers: We offer custom SCD plates (up to 10x10mm) and large-area PCD wafers (up to 125mm). This capability supports the integration of complex microwave delivery structures necessary to generate the optimized periodic magnetic fields gx(t) and gy(t). |
| Surface Engineering | Ultra-Smooth Polishing (Ra < 1nm): Surface quality is paramount for minimizing decoherence. 6CCVD guarantees Ra < 1nm polishing on SCD, ensuring minimal surface scattering losses and optimal coupling for microwave control fields. |
| Metalization Services | Integrated Quantum Control Circuits: The periodic driving fields are typically delivered via patterned microwave transmission lines. 6CCVD provides in-house custom metalization services, including Ti, Pt, Au, W, Pd, and Cu, allowing researchers to define and pattern the precise geometries required for QOCT field delivery directly onto the diamond substrate. |
| Thickness Control | Flexible Substrate Thickness: We offer SCD thicknesses ranging from 0.1”m to 500”m. This flexibility is crucial for researchers optimizing the placement of NV centers relative to surface control structures. |
| Engineering Support | Material Selection for Low Dissipation: The success of the QOCT scheme depends heavily on the dissipation rate (Îł). 6CCVDâs in-house PhD team provides expert consultation to select diamond materials (e.g., isotopic purity, surface termination) that align with the low-decoherence assumptions of advanced optimal control models for similar NV Center Quantum Control projects. |
| Global Logistics | Reliable Global Shipping: We ensure prompt and secure global delivery (DDU default, DDP available) of high-value diamond materials, supporting international research collaborations. |
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.
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.