Density Matrix Of System Research Articles

Kinetics in the Two-Level System with Strong Time-De-pendent Coupling of Its States to the Phonon Bath: Spin-Boson Description

Using the methods of nonequilibrium statistical mechanics, the master equation for the density matrix of an open dissipative quantum system is obtained under conditions, when the coupling between the electronic states of the system and the nuclear displacements in it is controlled by the alternating field. A time-dependent polaron transformation is proposed, which made it possible to solve kinetic equations using an expansion in a parameter characterizing transitions between “phonon-dressed” electronic states of the system. As an example, a mechanism is shown that can control the kinetics in a two-level system by applying a periodic force field to electron-phonon coupling.

Telling different unravelings apart via nonlinear quantum-trajectory averages

The Gorini-Kossakowski-Sudarshan-Lindblad master equation (ME) governs the density matrix of open quantum systems (OQSs). When an OQS is subjected to weak continuous measurement, its state evolves as a stochastic quantum trajectory, whose statistical average solves the ME. The ensemble of such trajectories is termed an unraveling of the ME. We propose a method to operationally distinguish unravelings produced by the same ME in different measurement scenarios, using nonlinear averages of observables over trajectories. We apply the method to the paradigmatic quantum nonlinear system of resonance fluorescence in a two-level atom. We compare the Poisson-type unraveling, induced by direct detection of photons scattered from the two-level emitter, and the Wiener-type unraveling, induced by phase-sensitive detection of the emitted field. We show that a quantum-trajectory-averaged variance is able to distinguish these measurement scenarios. We evaluate the performance of the method, which can be readily extended to more complex OQSs, under a range of realistic experimental conditions. Published by the American Physical Society 2024

Readout error mitigated quantum state tomography tested on superconducting qubits

Quantum technologies rely heavily on accurate control and reliable readout of quantum systems. Current experiments are limited by numerous sources of noise that can only be partially captured by simple analytical models and additional characterization of the noise sources is required. We test the ability of readout error mitigation to correct noise found in systems composed of quantum two-level objects (qubits). To probe the limit of such methods, we designed a beyond-classical readout error mitigation protocol based on quantum state tomography (QST), which estimates the density matrix of a quantum system, and quantum detector tomography (QDT), which characterizes the measurement procedure. By treating readout error mitigation in the context of state tomography the method becomes largely readout mode-, architecture-, noise source-, and quantum state-independent. We implement this method on a superconducting qubit and evaluate the increase in reconstruction fidelity for QST. We characterize the performance of the method by varying important noise sources, such as suboptimal readout signal amplification, insufficient resonator photon population, off-resonant qubit drive, and effectively shortened T1 and T2 coherence. As a result, we identified noise sources for which readout error mitigation worked well, and observed decreases in readout infidelity by a factor of up to 30.

Many-Body Entropies and Entanglement from Polynomially Many Local Measurements

Estimating global properties of many-body quantum systems such as entropy or bipartite entanglement is a notoriously difficult task, typically requiring a number of measurements or classical postprocessing resources growing exponentially in the system size. In this work, we address the problem of estimating global entropies and mixed-state entanglement via partial-transposed (PT) moments and show that efficient estimation strategies exist under the assumption that all the spatial correlation lengths are finite. Focusing on one-dimensional systems, we identify a set of approximate factorization conditions (AFCs) on the system density matrix, which allow us to reconstruct entropies and PT moments from information on local subsystems. This identification yields a simple and efficient strategy for entropy and entanglement estimation. Our method could be implemented in different ways, depending on how information on local subsystems is extracted. Focusing on randomized measurements providing a practical and common measurement scheme, we prove that our protocol requires only polynomially many measurements and postprocessing operations, assuming that the state to be measured satisfies the AFCs. We prove that the AFCs hold for finite-depth quantum-circuit states and translation-invariant matrix-product density operators and provide numerical evidence that they are satisfied in more general, physically interesting cases, including thermal states of local Hamiltonians. We argue that our method could be practically useful to detect bipartite mixed-state entanglement for large numbers of qubits available in today’s quantum platforms. Published by the American Physical Society 2024

A Study of Spin 1 Unruh–De Witt Detectors

A study of the interaction of spin 1 Unruh–De Witt detectors with a relativistic scalar quantum field is presented here. After tracing out the field modes, the resulting density matrix for a bipartite qutrit system is employed to investigate the violation of the Bell–CHSH inequality. Unlike the case of spin 1/2, for which the effects of the quantum field result in a decrease in the size of violation, in the case of spin 1, both a decrease or an increase in the size of the violation may occur. This effect is ascribed to the fact that Tsirelson’s bound is not saturated in the case of qutrits.

Entanglement and Bell inequalities with boosted tt¯

The Large Hadron Collider provides a unique opportunity to study quantum entanglement and violation of Bell inequalities at the highest energy available today. In this paper, we will investigate these quantum correlations with top quark pair production, which represents a system of two-qubits. The semileptonic top pair channel has a factor of 6 increase in statistics and easier reconstruction with respect to the dileptonic channel. Although measuring the spin polarization of the hadronic top quark is known to be challenging, our study indicates that it is feasible to reconstruct the spin density matrix of the two-qubit system using an optimal hadronic polarimeter. This is achieved with the aid of jet substructure techniques and NN-inspired reconstruction methods, which improve the mapping between subjets and quarks. We find that entanglement can already be observed at more than the 5σ level with existing data, and violation of Bell inequalities may be probed above the 4σ level at the HL-LHC with 3 ab−1 of data. Published by the American Physical Society 2024

Correlated noise enhancement of coherence and fidelity in coupled qubits

ABSTRACT It is generally assumed that environmental noise arising from thermal fluctuations are detrimental to preserving coherence and entanglement in a quantum system. In the simplest sense, dephasing and decoherence are tied to energy fluctuations driven by coupling between the system and the normal modes of the bath. Here, we explore the role of noise correlation in an open-loop model quantum communication system whereby the ‘sender’ and the ‘receiver’ are subject to local environments with various degrees of correlation or anti-correlation. We introduce correlation within the spectral density by solving multidimensional stochastic differential equations and introduce these into the Redfield equations of motion for the system density matrix. We find that correlation can enhance both the fidelity and purity of a maximally entangled (Bell) state. Moreover, by comparing the evolution of different initial Bell states, we show that one can effectively probe the correlation between two local environments. These observations may be useful in the design of high-fidelity quantum gates and communication protocols.

Optimal superpositions for particle detection via quantum phase

Exploiting quantum mechanics for sensing offers unprecedented possibilities. State-of-the-art proposals for novel quantum sensors often rely on the creation of large superpositions and generally detect a field. However, what is the optimal superposition size for detecting an incident particle from a specific direction? This question is nontrivial as, in general, this incident particle will scatter off with varied momenta, imparting varied recoils to the sensor, resulting in decoherence rather than a well-defined measurable phase. By considering scattering interactions of directional particulate environments with a system in a quantum superposition, we find that there is an optimal superposition size for measuring particles via a relative phase. As a consequence of the anisotropy of the environment, we observe a feature in the limiting behavior of the real and imaginary parts of the system's density matrix, linking the optimality of the superposition size to the wavelength of the scatterer. Published by the American Physical Society 2024

Geometric phase and Wehrl phase entropy for two superconducting qubits in a coherent field system under the effect of nonlinear medium

In this paper, we study the dynamical behavior of coherence, geometric phase, Wehrl entropy (WE) for a quantum model consisting of two superconducting qubits (SC–Qs) in the presence of multi-photon transition. We develop the Hamiltonian model and solve the Schrödinger equation that leads to evaluate the density matrix of whole system as well as subsystems. We introduce the effects of nonlinearity and subsystems interaction and display the temporal behavior of different quantumness measures in the existence and absence of nonlinear Kerr medium and Ising interaction.

Zombie cats on the quantum-classical frontier: Wigner-Moyal and semiclassical limit dynamics of quantum coherence in molecules.

In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

Explicit expressions for stationary states of the Lindblad equation for a finite state space

The Lindblad equation describes the time evolution of a density matrix of a quantum mechanical system. Stationary solutions are obtained by time-averaging the solution, which will in general depend on the initial state. We provide an analytical expression for the steady states of the Lindblad equation using the quantum jump unraveling, a version of an ergodic theorem, and the stationary probabilities of the corresponding discrete-time Markov chains. Our result is valid when the number of states appearing the in quantum trajectory is finite. The classical case of a Markov jump-process is recovered as a special case, and differences between the two are discussed.

Quantum state tomography, entanglement detection and Bell violation prospects in weak decays of massive particles

A rather general method for determining the spin density matrix of a multi-particle system from angular decay data is presented. The method is based on a Bloch parameterisation of the d-dimensional generalised Gell-Mann representation of ρ and exploits the associated Wigner- and Weyl-transforms on the sphere. Each parameter of a (possibly multipartite) spin density matrix can be measured from a simple average over an appropriate set of experimental angular decay distributions. The general procedures for both projective and non-projective decays are described, and the Wigner P and Q symbols calculated for the cases of spin-half, spin-one, and spin-3/2 systems. The methods are used to examine Monte Carlo simulations of pp collisions for bipartite systems: pp → W+W−, pp → ZZ, pp → ZW+, pp → {W}^{+}overline{t} , toverline{t} , and those from the Higgs boson decays H → WW* and H → ZZ*. Measurements are proposed for entanglement detection, exchange symmetry detection and Bell inequality violation in bipartite systems.

Steady state coherence and entanglement in a dissipative two-qubit system and its application for decision making

Quantum decision theory was recently established based on the mathematics of Hilbert spaces to formalize the key concepts of quantum mechanics such as uncertainty and superposition that particularly manifest in cognitive processes. We aim to address how a two-qubit system interacting with a global squeezed vacuum field reservoir can be exploited as a black box model for quantum decision-making. To do so, we encode the input of the black box into the density matrix of the two-qubit system. The global reservoir drives the system towards its steady state via the decoherence process. The outcome of the black box can be obtained by performing measurement on the stationary state of the two-qubit system. We reveal that the squeezed reservoir can be used to control the coherence and entanglement of the black box’s output. Interestingly, it is shown that the steady state entanglement and coherence between two qubits, which have the same value, can be obtained by measuring the output state of the black box. Besides, we examine the potential application of the considered model for entanglement decision-making.

Correlated quantum dynamics of graphene clusters

Phase-space representations are a family of methods for dynamics of both bosonic and fermionic systems, that work by mapping the system's density matrix to a quasiprobability density and the Liouville-von Neumann equation of the Hamiltonian to a corresponding density differential equation for the probability. We investigate here the accuracy and the computational efficiency of one approximate phase-space representation, called the fermionic truncated Wigner approximation (fTWA), applied to the Fermi-Hubbard model. On a many-body 2D system, with hopping strength and Coulomb $U$ tuned to represent the electronic structure of graphene, the method is found to be able to capture the time evolution of first-order (site occupation) and second-order (correlation functions) moments significantly better than the mean-field, Hartree-Fock method. The fTWA was also compared to results from the exact diagonalization method for smaller systems, and in general the agreement was found to be good. The fully parallel computational requirement of fTWA scales in the same order as the Hartree-Fock method, and the largest system considered here contained 198 lattice sites.

Equivalence of quantum and classical third order response for weakly anharmonic coupled oscillators.

Two-dimensional (2D) infrared (IR) spectra are commonly interpreted using a quantum diagrammatic expansion that describes the changes to the density matrix of quantum systems in response to light-matter interactions. Although classical response functions (based on Newtonian dynamics) have shown promise in computational 2D IR modeling studies, a simple diagrammatic description has so far been lacking. Recently, we introduced a diagrammatic representation for the 2D IR response functions of a single, weakly anharmonic oscillator and showed that the classical and quantum 2D IR response functions for this system are identical. Here, we extend this result to systems with an arbitrary number of bilinearly coupled, weakly anharmonic oscillators. As in the single-oscillator case, quantum and classical response functions are found to be identical in the weakly anharmonic limit or, in experimental terms, when the anharmonicity is small relative to the optical linewidth. The final form of the weakly anharmonic response function is surprisingly simple and offers potential computational advantages for application to large, multi-oscillator systems.

Penentuan Density Matrix Sistem Kuantum Satu Partikel Dimensi Satu dengan Metode Discretized Path Integral (DPI)

A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.

Understanding Traffic Jams Using Lindblad Superoperators

We propose a model to simulate different traffic-flow conditions in terms of quantum graphs hosting an (N+1)-level dot at each site. Our model allows us to keep track of the type and of the destination of each vehicle. The traffic flow inside the system is encoded in a proper set of Lindbladian local dissipators that describe the time evolution of the system density matrix. Taking advantage of the invariance of the Lindblad master equation under inhomogeneous transformations we derive the quantum Hamiltonian for the bulk dynamics in a proper experimental setup.

Steady-state relation of a two-level system strongly coupled to a many-body quantum chaotic environment

We study the long-time average of the reduced density matrix (RDM) of a two-level system as the central system, which is locally coupled to a many-body quantum chaotic system as the environment, under an overall Schrödinger evolution. A phenomenological relation among elements of the RDM is proposed for a dissipative interaction in the strong coupling regime and is tested numerically with the environment as a defect Ising chain, as well as a mixed-field Ising chain.

Quantum Manifestation of the Classical Bifurcation in the Driven Dissipative Bose-Hubbard Dimer.

We analyze the classical and quantum dynamics of the driven dissipative Bose-Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose-Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed.

Semiclassical dynamics of electrons in a space-time crystal: Magnetization, polarization, and current response

A space-time crystal is defined as a quantum-mechanical system with both spatial and temporal periodicity. Such a system can be described by the Floquet-Bloch (FB) theory. We first formulate a semiclassical theory by constructing a wave packet through the superposition of the FB wave functions, and we derive the equations of motion of FB electrons subjected to slowly varying external fields (not to be confused with the fast-changing Floquet drive), revealing behaviors similar to ordinary Bloch electrons but with quantities modified in the Floquet context. Specifically, we study the local magnetic moment due to the self-rotation of the wave packet, a contribution to total magnetization from the Berry curvature in $\mathbf{k}$-space, and the polarization of a fully occupied FB band. Based on the semiclassical theory, we can also show the fingerprint of the energy flow in such an energy-nonconserved system. We then discuss the density matrix of a FB system attached to a thermal bath, which allows us to investigate quantities involving many electrons in the noninteracting limit. As an application, we calculate the intrinsic current response in an oblique space-time metal showing the nonequilibrium nature of the FB system. The current response can also be related to the acoustoelectric effect. Overall, we develop a systematic approach for studying space-time crystals, and we provide a powerful tool to explore the electronic properties of this exotic system with coupled space and time.