Semiclassical Phase Research Articles

Improving metrology with quantum scrambling in a spin-1 Bose-Einstein condensate coupled to a cavity.

Spinor Bose-Einstein condensate is an ideal candidate for implementing the many-body entanglement, quantum measurement and quantum information processing owing to its inherent spin-mixing dynamics. Here we present a system of an 87Rb atomic spin-1 Bose-Einstein condensate coupled to an optical ring cavity, in which cavity-mediated nonlinear interactions give rise to saddle points in the semiclassical phase space, providing a general mechanism for exponential fast scrambling and metrological gain augment. We theoretically study metrological gain and fidelity out-of-time-ordered correlator based on time-reversal protocols and demonstrate that exponential rapid scrambling dynamics can enhance quantum metrology. In addition, we use the out-of-time-ordered correlator to probe dynamical phase transitions. This work is useful to understand the intrinsic relation between the concepts from different subfields of quantum science.

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Asymptotic approximations for the phase space Schrödinger equation

We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the anisotropic Gaussian approximation, related to the nearby orbit approximation, which is identified with the phase space representation of the Littlejohn semi-classical propagator. By means of the semi-classical propagator, we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the complex WKB theory developed by Maslov, on the basis of a canonical system in double phase space, analogous to the Berezin–Shubin–Marinov Hamilton–Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in .

Quantum Models à la Gabor for the Space-Time Metric.

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space into Hilbertian operators. The values are space-time variables, and the values are their conjugate frequency-wave vector variables. The procedure is first applied to the variables and produces essentially canonically conjugate self-adjoint operators. It is next applied to the metric field of general relativity and yields regularized semi-classical phase space portraits . The latter give rise to modified tensor energy density. Examples are given with the uniformly accelerated reference system and the Schwarzschild metric. Interesting probabilistic aspects are discussed.

General chaotic behaviors of heavy ion collisions at intermediate energy based on dynamical transport model

Motivated to explore the dynamical mechanism of multi-fragmentation phase transition and expand our knowledge on the deterministic chaos in nonlinear dynamic process, we study systematically the nonlinear dynamical behaviors of heavy ion collisions at intermediate energy. In order to highlight the general characteristics of the collision dynamics, in the present paper, we simulate a simple collision system, Ca+Ca, and obverse the performances of the typical nonlinear characteristic quantities such as the production of generalized entropy and the fractal structure of the semiclassical phase space during the reaction process. The multifragmentation entropy, information dimension and the dynamical fluctuations of fragment mass distribution in the final state of the reaction are also evaluated. Our results about the intermediate processes and the final products of the reaction are verified mutually so that confirm strongly the existence of nonlinear chaos in the heavy ion collisions at intermediate energy. Meanwhile, we find that the typical characteristics of the critical phase transition obtained in the analysis of the final production of the reaction are relatively more clear in the vicinity of the incident energies E=100 MeV.

Tractor atom interferometry

We propose a tractor atom interferometer (TAI) based on three-dimensional (3D) confinement and transport of split atomic wave-function components in potential wells that follow programmed paths. The paths are programmed to split and recombine atomic wave functions at well-defined space-time points, guaranteeing closure of the interferometer. Uninterrupted 3D confinement of the interfering wave-function components in the tractor wells eliminates coherence loss due to wave-packet dispersion. Using Crank-Nicolson simulation of the time-dependent Schr\"odinger equation, we compute the quantum evolution of scalar and spinor wave functions in several TAI sample scenarios. The interferometric phases extracted from the wave functions allow us to quantify gravimeter sensitivity for the TAI scenarios studied. We show that spinor TAI supports matter-wave beam splitters that are more robust against nonadiabatic effects than their scalar-TAI counterparts. We confirm the validity of semiclassical path-integral phases taken along the programmed paths of the TAI. Aspects for future experimental realizations of TAI are discussed.

Investigation of the magnetic phase diagram of the 2D ferrimagnetic J 1–J 2 model

The zero-temperature phase diagram and spin dynamics of the 2D ferrimagnetic J 1–J 2 model with (S 1, S 2) = (1/2, 1) are investigated using the time-dependent cluster mean-field theory (t-CMFT). The t-CMFT enables the investigation of the quantum-mechanical as well as semi-classical phase diagram and spin dynamics by control of the entanglement. For the characterization of the ferrimagnetic system, the magnetization, the energy per atom, the cluster quantum states and the von Neumann entropy have been determined.

Slow Quantum Thermalization and Many-Body Revivals from Mixed Phase Space

Relaxation of few-body quantum systems can strongly depend on the initial state when the system's semiclassical phase space is mixed, i.e., regions of chaotic motion coexist with regular islands. In recent years, there has been much effort to understand the process of thermalization in strongly interacting quantum systems that often lack an obvious semiclassical limit. Time-dependent variational principle (TDVP) allows to systematically derive an effective classical (nonlinear) dynamical system by projecting unitary many-body dynamics onto a manifold of weakly-entangled variational states. We demonstrate that such dynamical systems generally possess mixed phase space. When TDVP errors are small, the mixed phase space leaves a footprint on the exact dynamics of the quantum model. For example, when the system is initialized in a state belonging to a stable periodic orbit or the surrounding regular region, it exhibits persistent many-body quantum revivals. As a proof of principle, we identify new types of "quantum many-body scars", i.e., initial states that lead to long-time oscillations in a model of interacting Rydberg atoms in one and two dimensions. Intriguingly, the initial states that give rise to most robust revivals are typically entangled states. On the other hand, even when TDVP errors are large, as in the thermalizing tilted-field Ising model, initializing the system in a regular region of phase space leads to slowdown of thermalization. Our work establishes TDVP as a method for identifying interacting quantum systems with anomalous dynamics in arbitrary dimensions. Moreover, the mixed-phase space classical variational equations allow to find slowly-thermalizing initial conditions in interacting models. Our results shed light on a link between classical and quantum chaos, pointing towards possible extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.

Semiclassical optimization of entrainment stability and phase coherence in weakly forced quantum limit-cycle oscillators.

Optimal entrainment of a quantum nonlinear oscillator to a periodically modulated weak harmonic drive is studied in the semiclassical regime. By using the semiclassical phase-reduction theory recently developed for quantum nonlinear oscillators [Y. Kato, N. Yamamoto, and H. Nakao, Phys. Rev. Res. 1, 033012 (2019)10.1103/PhysRevResearch.1.033012], two types of optimization problems, one for the stability and the other for the phase coherence of the entrained state, are considered. The optimal waveforms of the periodic amplitude modulation can be derived by applying the classical optimization methods to the semiclassical phase equation that approximately describes the quantum limit-cycle dynamics. Using a quantum van der Pol oscillator with squeezing and Kerr effects as an example, the performance of optimization is numerically analyzed. It is shown that the optimized waveform for the entrainment stability yields faster entrainment to the driving signal than the case with a simple sinusoidal waveform, while that for the phase coherence yields little improvement from the sinusoidal case. These results are explained from the properties of the phase sensitivity function.

Quantum Gravity on a Torus

Causal Dynamical Triangulations (CDT) is a non-perturbative lattice approach to quantum gravity where one assumes space-time foliation into spatial hyper-surfaces of fixed topology. Most of the CDT results were obtained for the spatial topology of the 3-sphere. It was shown that CDT has rich phase structure, including the semiclassical phase consistent with Einstein's general relativity. Some of the phase transitions were found to be second (or higher) order which makes a possibility of taking continuum limit viable. Here we present new results of changing the spatial topology to that of the 3-torus. We argue that the topology change does not change the phase structure nor the order of the phase transitions. Therefore CDT results seem to be universal independent of the topology chosen.

Quantum Transducer Using a Parametric Driven-Dissipative Phase Transition.

We study a dissipative Kerr resonator subject to both single- and two-photon detuned drives. Beyond a critical detuning threshold, the Kerr resonator exhibits a semiclassical first-order dissipative phase transition between two different steady states that are characterized by a π phase switch of the cavity field. This transition is shown to persist deep into the quantum limit of low photon numbers. Remarkably, the detuning frequency at which this transition occurs depends almost linearly on the amplitude of the single-photon drive. Based on this phase-switching feature, we devise a sensitive quantum transducer that translates the observed frequency of the parametric quantum phase transition to the detected single-photon amplitude signal. The effects of noise and temperature on the corresponding sensing protocol are addressed, and a realistic circuit-QED implementation is discussed.

Quantum smooth boundary forces from constrained geometries

We implement the so-called Weyl–Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy boundary, a position-dependent mass, and an extra potential on the quantum level. The consistency of our quantization is discussed by analyzing the semi-classical phase space portrait of the derived quantum dynamics, which is obtained as a regularization of its original classical counterpart.

Semiclassical phase reduction theory for quantum synchronization

We develop a general theoretical framework of semiclassical phase reduction for analyzing synchronization of quantum limit-cycle oscillators. The dynamics of quantum dissipative systems exhibiting limit-cycle oscillations are reduced to a simple, one-dimensional classical stochastic differential equation approximately describing the phase dynamics of the system under the semiclassical approximation. The density matrix and power spectrum of the original quantum system can be approximately reconstructed from the reduced phase equation. The developed framework enables us to analyze synchronization dynamics of quantum limit-cycle oscillators using the standard methods for classical limit-cycle oscillators in a quantitative way. As an example, we analyze synchronization of a quantum van der Pol oscillator under harmonic driving and squeezing, including the case that the squeezing is strong and the oscillation is asymmetric. The developed framework provides insights into the relation between quantum and classical synchronization and will facilitate systematic analysis and control of quantum nonlinear oscillators.

A generalized phase space approach for solving quantum spin dynamics

Numerical techniques to efficiently model out-of-equilibrium dynamics in interacting quantum many-body systems are key for advancing our capability to harness and understand complex quantum matter. Here we propose a new numerical approach which we refer to as generalized discrete truncated Wigner approximation (GDTWA). It is based on a discrete semi-classical phase space sampling and allows to investigate quantum dynamics in lattice spin systems with arbitrary S ≥ 1/2. We show that the GDTWA can accurately simulate dynamics of large ensembles in arbitrary dimensions. We apply it for S > 1/2 spin-models with dipolar long-range interactions, a scenario arising in recent experiments with magnetic atoms. We show that the method can capture beyond mean-field effects, not only at short times, but it also can correctly reproduce long time quantum-thermalization dynamics. We benchmark the method with exact diagonalization in small systems, with perturbation theory for short times, and with analytical predictions made for models which feature quantum-thermalization at long times. We apply our method to study dynamics in large S > 1/2 spin-models and compute experimentally accessible observables such as Zeeman level populations, contrast of spin coherence, spin squeezing, and entanglement quantified by single-spin Renyi entropies. We reveal that large S systems can feature larger entanglement than corresponding S = 1/2 systems. Our analyses demonstrate that the GDTWA can be a powerful tool for modeling complex spin dynamics in regimes where other state-of-the art numerical methods fail.

Power spectrum of primordial perturbations for an emergent universe in quantum reduced loop gravity

We study the dynamics and predictions of a new emergent-universe model recently derived within Quantum Reduced Loop Gravity and based on the so-called statistical regularization scheme. These effective geometries show a dynamical transition from a stationary spacetime, with nearly constant scale factor at very early times, to a late-time semiclassical phase well approximated by a classical Friedmann-Robertson-Walker spacetime. We show that this is always the case when the matter content is a minimally coupled scalar field subject to a quadratic potential, including the massless case. Besides, a finite period of (nearly) exponential expansion in the semiclassical region can take place. Hence, we incorporate cosmological scalar and tensor perturbations, with a well-defined dynamics, and compute their power spectra at the end of inflation. We show that they are nearly scale invariant up to some scale where scale invariance is broken. Besides, they show qualitative differences with respect to the bouncing scenario of Loop Quantum Cosmology at scales where the scale invariance is broken. Nevertheless, the tensor-to-scalar ratio remains approximately constant even for modes well affected by the background evolution.

Holographic interferences in photoelectron spectra: different approaches

We perform a theoretical study of the holographic structures in photoelectron spectra for ionization of hydrogen atoms induced by short laser pulses. To elucidate the nature of the holographic structures present in the momentum distributions of photoelectrons, we use several quantum approximations, such as the strong field and Coulomb–Volkov approximations up to second order, as well as semiclassical Monte Carlo simulations. In a single-cycle pulse, we eliminate the intracycle interference from the spectra isolating the holographic structure formed in the photoionization process. We probe the different approaches and analyze the role of electron–core interaction numerically by solving the time dependent Schrodinger equation. We show that the two-step semiclassical model of Shvetsov-Shilovski et al. [Phys. Rev. A 94, 013415 (2016)] fully considers the effect of the Coulomb potential on the electron dynamics and semiclassical phase reproducing the holographic structure in full quantum calculations. Contrarily, perturbative quantum (strong field and Coulomb–Volkov) and semiclassical (quantum trajectory Monte Carlo) methods account only partially for some of the characteristics of the holographic interference pattern.

Electronic optics in graphene in the semiclassical approximation

We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameterh≪1. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to aneffective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.

Efficient continuous-wave noise spectroscopy beyond weak coupling

The optimization of quantum control for physical qubits relies on accurate noise characterization. Probing the spectral density $S(\omega)$ of semi-classical phase noise using a spin interacting with a continuous-wave (CW) resonant excitation field has recently gained attention. CW noise spectroscopy protocols have been based on the generalized Bloch equations (GBE) or the filter function formalism, assuming weak coupling to a Markovian bath. However, this standard protocol can substantially underestimate $S(\omega)$ at low frequencies when the CW pulse amplitude becomes comparable to $S(\omega)$. Here, we derive the coherence decay function more generally by extending it to higher orders in the noise strength and discarding the Markov approximation. Numerical simulations show that this provides a more accurate description of the spin dynamics compared to a simple exponential decay, especially on short timescales. Exploiting these results, we devise a protocol that uses an experiment at a single CW pulse amplitude to extend the spectral range over which $S(\omega)$ can be reliably determined to $\omega=0$.

On quantum approach to modeling of plasmon photovoltaic effect

Surface plasmons in metallic nanostructures including metallically nanomodified solar cells are conventionally studied and modeled by application of the Mie approach to plasmons or by the finite element solution of differential Maxwell equations with imposed boundary and material constraints (e.g., upon commercial COMSOL software system). Both approaches are essentially classical ones and neglect quantum particularities related to plasmon excitations in metallic components. We demonstrate that these quantum plasmon effects are of crucial importance especially in theoretical simulations of plasmon-aided photovoltaic phenomena. Quantum corrections considerably improve both the Mie and COMSOL approaches in this case. We present the semiclassical random phase approximation description of plasmons in metallic nanoparticles and apply the quantum Fermi golden rule scheme to assess the sunlight energy transfer to the semiconductor solar cell mediated by surface plasmons in metallic nanoparticles deposited on the top of the battery. In addition, short-ranged electron–electron interaction in metals is discussed in the framework of the semiclassical hydrodynamic model. The significance of the related quantum corrections is illustrated by the quantumly improved COMSOL simulations.

Exploring many-body localization and thermalization using semiclassical methods

The Discrete Truncated Wigner Approximation (DTWA) is a semi-classical phase space method useful for the exploration of Many-body quantum dynamics. In this work we investigate Many-Body Localization (MBL) and thermalization using DTWA and compare its performance to exact numerical solutions. By taking as a benchmark case a 1D random field Heisenberg spin chain with short range interactions, and by comparing to numerically exact techniques, we show that DTWA is able to reproduce dynamical signatures that characterize both the thermal and the MBL phases. It exhibits the best quantitative agreement at short times deep in each of the phases and larger mismatches close to the phase transition. The DTWA captures the logarithmic growth of entanglement in the MBL phase, even though a pure classical mean-field analysis would lead to no dynamics at all. Our results suggest the DTWA can become a useful method to investigate MBL and thermalization in experimentally relevant settings intractable with exact numerical techniques, such as systems with long range interactions and/or systems in higher dimensions.