General Quantum Systems Research Articles

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Superquantization rule for multistability in driven-dissipative quantum systems

Abstract We present a superquantization rule which indicates the possible robust stationary states of a generic driven-dissipative quantum system. Multistability in a driven cavity mode interacting with a qudit is revealed hence within a simple intuitive picture. The accuracy of the superquantization approach is confirmed by numerical simulations of the underlying quantum model. In the case when the qudit is composed of several two-level emitters coupled homogeneously to the cavity, we demonstrate the robustness of the superquantized steady states to single-emitter decay.

Inferring general links between energetics and information with unknown environment

Identifying general links between energetics and information in realistic open quantum systems is a long-standing problem in quantum thermodynamics and quantum information processing. However, the generality of existing efforts is often impeded by their specific assumptions about environments. Here we address the problem by developing a trajectory-level thermodynamic inference theory to establish general links using just the knowledge of the system, enabling a single framework applicable to a diverse range of environments. Underpinning the framework is a notion of excess energy introduced for inferring the net energy gain of the system after completing an information processing trajectory. We show that fluctuation behaviors of the excess energy encode general links between energetics and information with a conceptual advantage that they completely avoid assumptions about environments and system-environment coupling forms. Crucially, we obtain a single thermodynamic inequality that integrates upper bounds on heat dissipation and extracted work in terms of system's information content change, providing complementary constraints that greatly expand the context of existing well-adopted results based on the second law of thermodynamics. We also uncover lower bounds on the precision of the fluctuating system's energy and information content changes in terms of their Fano factors and a correlation function between them. By extending relations between energetics and information to higher-order fluctuations, we thus reveal a trade-off that a more precise inference of energy or information content changes requires a looser energetic-information link. We showcase the implications of these general links in a number of quantum information and thermodynamic tasks of application relevance. Our framework provides a toolkit for analyzing the interplay between energetics and information from the trajectory level in generic quantum systems, thereby adding an indispensable structure to the thermodynamics of information. Published by the American Physical Society 2024

RealTimeTransport: An open-source C++ library for quantum transport simulations in the strong coupling regime.

The description of quantum transport in the strong system-reservoir coupling regime poses a significant theoretical and computational challenge that demands specialized tools for accurate analysis. RealTimeTransport is a new open-source C++ library that enables the computation of both stationary and transient transport observables for generic quantum systems connected to metallic reservoirs. It computes the Nakajima-Zwanzig memory kernels for both dynamics and transport in real-time, going beyond traditional expansions in the bare system-reservoir couplings. Currently, several methods are available as follows: (i) A renormalized perturbation theory in leading and next-to-leading order, which avoids the low-temperature breakdown that limits the traditional theory. (ii) Starting from this well-behaved reference solution, a two- and three-loop, self-consistent renormalization-group transformation of the memory kernels is implemented. This allows refined quantitative predictions even in the presence of many body resonances, such as the Kondo enhancement of cotunneling. This paper provides an overview of the theory, the architecture of RealTimeTransport, and practical demonstrations of the currently implemented methods. In particular, we analyze the stationary transport through a serial double quantum dot and showcase for the T = 0 interacting Anderson model the complete time-development of single-electron tunneling (SET), cotunneling-assisted SET, and inelastic cotunneling resonances throughout the entire gate-bias stability diagram. We discuss the range of applicability of the implemented methods and benchmark them against other advanced approaches.

Recovering an accurate Lindblad equation from the Bloch-Redfield equation for general open quantum systems

Recovering an accurate Lindblad equation from the Bloch-Redfield equation for general open quantum systems

Practical Trainable Temporal Postprocessor for Multistate Quantum Measurement

We develop and demonstrate a trainable temporal postprocessor (TPP) harnessing a simple but versatile machine learning algorithm to provide optimal processing of quantum measurement data subject to arbitrary noise processes for the readout of an arbitrary number of quantum states. We demonstrate the TPP on the essential task of qubit state readout, which has historically relied on temporal processing via matched filters in spite of their applicability for only specific noise conditions. Our results show that the TPP can reliably outperform standard filtering approaches under complex readout conditions, such as high-power readout. Using simulations of quantum measurement noise sources, we show that this advantage relies on the TPP’s ability to learn optimal linear filters that account for general quantum noise correlations in data, such as those due to quantum jumps, or correlated noise added by a phase-preserving quantum amplifier. Furthermore, we derive an exact analytic form for the optimal TPP weights: this positions the TPP as a linearly scaling generalization of matched filtering, valid for an arbitrary number of states under the most general readout noise conditions, all while preserving a training complexity that is essentially negligible in comparison with that of training neural networks for processing temporal quantum measurement data. The TPP can be autonomously and reliably trained on measurement data and requires only linear operations, making it ideal for field-programmable gate array implementations in circuit QED for real-time processing of measurement data from general quantum systems. Published by the American Physical Society 2024

Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives.

Ergodicity of quantum dynamics is often defined through statistical properties of energy eigenstates, as exemplified by Berry's conjecture in single-particle quantum chaos and the eigenstate thermalization hypothesis in many-body settings. In this work, we investigate whether quantum systems can exhibit a stronger form of ergodicity, wherein any time-evolved state uniformly visits the entire Hilbert space over time. We call such a phenomenon complete Hilbert-space ergodicity (CHSE), which is more akin to the intuitive notion of ergodicity as an inherently dynamical concept. CHSE cannot hold for time-independent or even time-periodic Hamiltonian dynamics, owing to the existence of (quasi)energy eigenstates which precludes exploration of the full Hilbert space. However, we find that there exists a family of aperiodic, yet deterministic drives with minimal symbolic complexity-generated by the Fibonacci word and its generalizations-for which CHSE can be proven to occur. Our results provide a basis for understanding thermalization in general time-dependent quantum systems.

Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space.

The properties of mixed eigenstates in a generic quantum system with a classical counterpart that has mixed-type phase space, although important to understand several fundamental questions that arise in both theoretical and experimental studies, are still not clear. Here, following a recent work [Č. Lozej, D. Lukman, and M. Robnik, Phys. Rev. E 106, 054203 (2022)2470-004510.1103/PhysRevE.106.054203], we perform an analysis of the features of mixed eigenstates in a time-dependent Hamiltonian system, the celebrated kicked top model. As a paradigmatic model for studying quantum chaos, the kicked top model is known to exhibit both classical and quantum chaos. The types of eigenstates are identified by means of the phase-space overlap index, which is defined as the overlap of the Husimi function with regular and chaotic regions in classical phase space. We show that the mixed eigenstates appear due to various tunneling precesses between different phase-space structures, while the regular and chaotic eigenstates are, respectively, associated with invariant tori and chaotic components in phase space. We examine how the probability distribution of the phase-space overlap index evolves with increasing system size for different kicking strengths. In particular, we find that the relative fraction of mixed states exhibits a power-law decay as the system size increases, indicating that only purely regular and chaotic eigenstates are left in the strict semiclassical limit. We thus provide further verification of the principle of uniform semiclassical condensation of Husimi functions and confirm the correctness of the Berry-Robnik picture.

Tuning quantum pathway interference in two-color laser photoemission using DC bias

Coherent control of quantum systems depends on the manipulation of quantum interference through external fields. In this work, we investigate the effects of DC bias field on coherent control of quantum pathways in two-color laser photoemission using exact analytical solutions of the one-dimensional time dependent Schrödinger equation. Increasing DC bias lowers and narrows the surface potential barrier, shifting the dominant emission to lower order multiphoton photoemission, photo-assisted tunneling and then direct tunneling. Those lower order photon absorption processes result in fewer possible pathways, and therefore modulation of photoemission current can be suppressed as DC field increases. It is shown that a maximum modulation depth of 99.4% can be achieved for a gold emitter at local DC bias F 0 = 0.5 V nm−1, fundamental (800 nm) laser field F 1 = 2.6 V nm−1 and second harmonic laser field F 2 = 0.25 V nm−1 . For a given set of input parameters, the total photoemission consists of different k-photon processes, each of which has their own different multiple possible pathways and interference effects. However, the quantum pathways and their interference for the dominant k-photon process and for the total photoemission probability show the same trends. This study demonstrates strong flexibility in tuning two-color lasers induced photoemission using a DC bias and provides insights into coherent control schemes of general quantum systems.

Reviving product states in the disordered Heisenberg chain

When a generic quantum system is prepared in a simple initial condition, it typically equilibrates toward a state that can be described by a thermal ensemble. A known exception is localized systems that are non-ergodic and do not thermalize; however, local observables are still believed to become stationary. Here we demonstrate that this general picture is incomplete by constructing product states that feature periodic high-fidelity revivals of the full wavefunction and local observables that oscillate indefinitely. The system neither equilibrates nor thermalizes. This is analogous to the phenomenon of weak ergodicity breaking due to many-body scars and challenges aspects of the current phenomenology of many-body localization, such as the logarithmic growth of the entanglement entropy. To support our claim, we combine analytic arguments with large-scale tensor network numerics for the disordered Heisenberg chain. Our results hold for arbitrarily long times in chains of 160 sites up to machine precision.

Bounds on the smallest sets of quantum states with special quantum nonlocality

An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \cite{Halder19}. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in Cd1⊗Cd1⊗⋯⊗CdN for any di≥2 and 1≤i≤N?) raised in a recent paper \cite{Shi22N}. Compared with all previous relevant proofs, our proof here is quite concise.

Stationarity of quantum statistical ensembles at first-order phase transition points

We study the dynamics of quantum statistical ensembles at first-order phase transition points of finite macroscopic systems. First, we show that at the first-order phase transition point of the system, with an order parameter that does not commute with the Hamiltonian, any quantum state with an order parameter that has a nonzero value always evolves towards a macroscopically distinct state after a sufficiently long time. From this result, we argue that the stationarity required for statistical ensembles should be interpreted as stationarity on a sufficiently long but finite timescale. Finally, we prove that the density matrix of the squeezed ensemble, a class of generalized statistical ensembles proposed as the only concrete method of constructing phase coexistence states applicable to general quantum systems, is locally stationary on timescales diverging in the thermodynamic limit. Our results support the validity of the squeezed ensemble from a dynamical point of view and open the door to nonequilibrium statistical physics at the first-order phase transition point.

Information loss, mixing and emergent type III1 factors

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity.We call operators that cluster in time “mixing” and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III1 factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III1 factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III1 von Neumann factor.In a theory of Generalized Free Fields (GFF), we show that if the two-point function clusters in time, all operators are mixing, and the algebra is a type III1 factor. For example, in \U0001d4a9 = 4 SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III1 factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C∗-algebra and von Neumann subalgebras of GFF associated with time bands and, more generally, open sets of the bulk spacetime using the HKLL reconstruction map.

Locally distinguishing nonlocal sets with entanglement resource

Recently, much attention has been paid to the construction of nonlocal sets. The existence of nonlocal sets gives rise to the research on the entanglement-assisted discrimination. We first present a method to locally distinguish a nonlocal set of orthogonal product states in (ℂd)⊗n by utilizing only a 2⊗2 maximally entangled state. Then we generalize the method to general multipartite quantum system ⊗i=1nℂdi based on the same entangled resource. Fortunately, we find that a 2⊗2 maximally entangled state can be employed as a universal resource for perfectly distinguishing the nonlocal sets constructed by Zhen et al (2022). In particular, the protocols we designed could not only efficiently distinguish these states, but also show that the strength of these nonlocal sets is minimal from the perspective of auxiliary resources.

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to various coherently controlled closed and open quantum systems. In this work, we adopt the GRAPE method for optimizing objective functionals in open quantum systems driven by both coherent and incoherent controls. In our case, a tailored or engineered environment acts on the controlled system as control via time-dependent decoherence rates or, equivalently, via spectral density of the environment. To develop the GRAPE approach for this problem, we compute gradient of various objectives for general N-level open quantum systems for the piecewise constant class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a matrix determining the system’s dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano’s method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated. Robustness of the optimal controls is also studied.

Statistical ensembles for phase coexistence states specified by noncommutative additive observables

A phase coexistence state cannot be specified uniquely by any intensive parameters, such as the temperature and the magnetic field, because they take the same values over all coexisting phases. It can be specified uniquely only by an appropriate set of additive observables. Hence, to analyze phase coexistence states the statistical ensembles that are specified by additive observables have been employed, such as the microcanonical and restricted ensembles. However, such ensembles are ill-defined or ill-behaved when some of the additive observables do not commute with each other. Here, we solve this fundamental problem by extending a generalized ensemble in such a way that it is applicable to phase coexistence states that are specified by noncommutative additive observables. We prove that this ensemble correctly gives the density matrix corresponding to phase coexistence states of general quantum systems as well as the thermodynamic functions. Furthermore, these ensembles are convenient for practical calculations because of their good analytic properties and useful formulas by which temperature and other intensive parameters are directly obtained from the expectation values of the additive observables. As a demonstration, we apply our formulation to a two-dimensional system whose phase coexistence states are specified by an additive observable (order parameter) that does not commute with the Hamiltonian.

Quantum neuromorphic approach to efficient sensing of gravity-induced entanglement

The detection of entanglement provides a definitive proof of quantumness. Its ascertainment might be challenging for hot or macroscopic objects, where entanglement is typically weak, but nevertheless present. Here we propose a platform for measuring entanglement by connecting the objects of interest to an uncontrolled quantum network, whose emission (readout) is processed to recognize the state of the former, and hence also the amount of entanglement. First, we demonstrate the platform and its features with generic quantum systems. As the network effectively learns to recognize quantum states, it is possible to sense the amount of entanglement after training with only nonentangled states. Furthermore, by taking into account measurement errors, we demonstrate entanglement sensing with precision that scales beyond the standard quantum limit and outperforms measurements performed directly on the objects. Finally, we utilize our platform for sensing gravity-induced entanglement between two masses and predict an improvement of two orders of magnitude in the precision of entanglement estimation compared to existing techniques.

Topological synchronization of quantum van der Pol oscillators

To observe synchronization in a large network of classical or quantum systems demands both excellent control of the interactions between the nodes and very accurate preparation of the initial conditions due to the involved nonlinearities and dissipation. This limits the applicability of this phenomenon for future devices. Here, we demonstrate a route towards significantly enhancing the robustness of synchronized behavior in open nonlinear systems that utilizes the power of topology. In a lattice of quantum van der Pol oscillators with topologically motivated couplings, boundary synchronization emerges in the classical mean field as well as the quantum model. In addition to its robustness against disorder and initial state perturbations, the observed dynamics is independent of the underlying topological insulator model provided the existence of zero-energy modes. Our work extends the notion of topology to the general nonlinear dynamics and open quantum system realm with applications to networks where specific nodes need special protection like power grids or quantum networks.

Rapid stabilization of stochastic quantum systems in a unified framework

Rapid stabilization of general stochastic quantum systems is investigated based on the rapid stability of stochastic differential equations. We introduce a Lyapunov–LaSalle-like theorem for a class of nonlinear stochastic systems first, based on which a unified framework of rapidly stabilizing stochastic quantum systems is proposed. According to the proposed unified framework, we design the switching state feedback controls to achieve the rapid stabilization of single-qubit systems, two-qubit systems, and N-qubit systems. From the unified framework, the state space is divided into two state subspaces, and the target state is located in one state subspace, while the other system equilibria are located in the other state subspace. Under the designed state feedback controls, the system state can only transit through the boundary between the two state subspaces no more than two times, and the target state is globally asymptotically stable in probability. In particular, the system state can converge exponentially in (all or part of) the state subspace where the target state is located. Moreover, the effectiveness and rapidity of the designed state feedback controls are shown in numerical simulations by stabilizing GHZ states for a three-qubit system.

Spin-phonon coupling and magnetic relaxation in single-molecule magnets.

Electron-phonon coupling is important in many physical phenomena, e.g. photosynthesis, catalysis and quantum information processing, but its impacts are difficult to grasp on the microscopic level. One area attracting wide interest is that of single-molecule magnets, which is motivated by searching for the ultimate limit in the miniaturisation of binary data storage media. The utility of a molecule to store magnetic information is quantified by the timescale of its magnetic reversal processes, also known as magnetic relaxation, which is limited by spin-phonon coupling. Several recent accomplishments of synthetic organometallic chemistry have led to the observation of molecular magnetic memory effects at temperatures above that of liquid nitrogen. These discoveries have highlighted how far chemical design strategies for maximising magnetic anisotropy have come, but have also highlighted the need to characterise the complex interplay between phonons and molecular spin states. The crucial step is to make a link between magnetic relaxation and chemical motifs, and so be able to produce design criteria to extend molecular magnetic memory. The basic physics associated with spin-phonon coupling and magnetic relaxation was outlined in the early 20th century using perturbation theory, and has more recently been recast in the form of a general open quantum systems formalism and tackled with different levels of approximations. It is the purpose of this Tutorial Review to introduce the topics of phonons, molecular spin-phonon coupling, and magnetic relaxation, and to outline the relevant theories in connection with both the traditional perturbative texts and the more modern open quantum systems methods.