Quantum-classical Correspondence Research Articles

Behavior of Correlation Functions in the Dynamics of the Multiparticle Quantum Arnol'd Cat.

The multi-particle Arnol'd cat is a generalization of the Hamiltonian system, both classical and quantum, whose period evolution operator is the renowned map that bears its name. It is obtained following the Joos-Zeh prescription for decoherence by adding a number of scattering particles in the configuration space of the cat. Quantization follows swiftly if the Hamiltonian approach, rather than the semiclassical approach, is adopted. The author has studied this system in a series of previous works, focusing on the problem of quantum-classical correspondence. In this paper, the dynamics of this system are tested by two related yet different indicators: the time autocorrelation function of the canonical position and the out-of-time correlator of position and momentum.

Nonlinear solution of classical three-wave interaction via finite-dimensional quantum model

The quantum three-wave interaction, the lowest-order nonlinear interaction in plasma physics, describes energy–momentum transfer between three resonant waves in the quantum regime. We describe how it may also act as a finite-degree-of-freedom approximation to the classical three-wave interaction in certain circumstances. By promoting the field variables to operators, we quantize the classical system, show that the quantum system has more free parameters than the classical system and explain how these parameters may be selected to optimize either initial or long-term correspondence. We then numerically compare the long-time quantum–classical correspondence far from the fixed point dynamics. We discuss the Poincaré recurrence of the system and the mitigation of quantum scrambling.

Quantum–classical correspondence in spin–boson equilibrium states at arbitrary coupling

The equilibrium properties of nanoscale systems can deviate significantly from standard thermodynamics due to their coupling to an environment. We investigate this here for the θ-angled spin–boson model, where we first derive a compact and general form of the classical equilibrium state including environmental corrections to all orders. Secondly, for the quantum spin–boson model we prove, by carefully taking a large spin limit, that Bohr’s quantum–classical correspondence persists at all coupling strengths. This shows, for the first time, the validity of the quantum–classical correspondence for an open system and gives insight into the regimes where the quantum system is well-approximated by a classical one. Finally, we provide the first classification of the coupling parameter regimes for the spin–boson model, from weak to ultrastrong, both for the quantum case and the classical setting. Our results shed light on the interplay of quantum and mean force corrections in equilibrium states of the spin–boson model, and will help draw the quantum to classical boundary in a range of fields, such as magnetism and exciton dynamics.

Experimental Observation of the Yang-Lee Quantum Criticality in Open Quantum Systems.

The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.

Non-KAM classical chaos topology for electrons in superlattice minibands determines the inter-well quantum transition rates

We investigate the quantum-classical correspondence for a particle tunnelling through a periodic superlattice structure with an applied bias voltage and an additional tilted harmonic oscillator potential. We show that the quantum mechanical tunnelling rate between neighbouring quantum wells of the superlattice is determined by the topology of the phase trajectories of the analogous classical system. This result also enables us to estimate, with high accuracy, the tunnelling rate between two spatially displaced simple harmonic oscillator states using a classical model, and thus gain new insight into this generic quantum phenomenon. This finding opens new directions for exploring and understanding the quantum-classical correspondence principle and quantum jumps between displaced harmonic oscillators, which are important in many branches of natural science.

Does canonical quantization lead to GKSL dynamics?

We introduce a generalized classical model of Brownian motion for describing thermal relaxation processes which is thermodynamically consistent. Applying the canonical quantization to this model, a quantum equation for the density operator is obtained. This equation has a thermal equilibrium state as its stationary solution, but the time evolution is not necessarily a Completely Positive and Trace-Preserving (CPTP) map. In the application to the harmonic oscillator potential, however, the requirement of the CPTP map is shown to be satisfied by choosing parameters appropriately and then our equation reproduces a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation satisfying the detailed balance condition. This result suggests a quantum-classical correspondence in thermal relaxation processes and will provide a new insight to the study of decoherence.

Higher rank quantum-classical correspondence

Higher rank quantum-classical correspondence

Quantum chaos in the Dicke model and its variants

Recently, the out-of-time-ordered correlator (OTOC) has gained much attention as an indicator of quantum chaos. In the semi-classical limit, its exponential growth rate resembles the classical Lyapunov exponent. The quantum–classical correspondence has been supported for the one-body chaotic systems as well as realistic systems with interactions, as in the Dicke model, a model of multi-two-level atoms and cavity field interactions. To this end, we calculate the OTOC for different variations of the Dicke model in an open quantum system setting. The connection between the superradiant phase transition of the Dicke model and the OTOC is studied. Further, we establish a relation between the OTOC and the second-order coherence function. This becomes important for the experimental studies of the OTOC and quantum chaos in the models of quantum optics.

Phase space perspective on a model for isomerization in an optical cavity.

Explanation for the modification of rates and mechanism of reactions carried out in optical cavities still eludes us. Several studies indicate that the cavity-mediated changes in the nature of vibrational energy flow within a molecule may play a significant role. Here, we study a model polaritonic system, proposed and analyzed earlier by Fischer et al., J. Chem. Phys. 156, 154305 (2022), comprising a one-dimensional isomerization mode coupled to a single photon mode in a lossless cavity. We show that the isomerization probability in the presence of virtual photons, for specific cavity-system coupling strengths and cavity frequencies, can exhibit suppression or enhancement for different choices of the initial reactant vibropolariton wavepacket. We observe a qualitative agreement between the classical and quantum average isomerization probabilities in the virtual photon case. A significant part of the effects due to coupling to the cavity can be rationalized in terms of a "chaos-order-chaos" transition of the classical phase space and the phase space localization nature of the polariton states that dominantly participate in the quantum isomerization dynamics. On the other hand, for initial states with zero photons (i.e., a "dark cavity"), the isomerization probability is suppressed when the cavity frequency is tuned near to the fundamental frequency of the reactive mode. The classical-quantum correspondence in the zero photon case is unsatisfactory. In this simple model, we find that the suppression or enhancement of isomerization arises due to the interplay between cavity-system energy flow dynamics and quantum tunneling.

Fast adaptive synchronization of discrete quantum chaotic maps

By using linear entropy, a discrete quantum logistic chaotic map is introduced and the dynamical evolution of the quantum radiation field interacting with the matter is studied by changing the independent parameters. Its typical effect is the chaotic property of the quantum optics and a quantum classical correspondence is obtained, and the chaos and the regular structure of phase space can be better revealed by using the coherence dynamics of phase space. In addition, an approach for fast adaptive synchronization of the discrete quantum maps is proposed. Results validate the effectiveness of the proposed scheme.

Correspondence between excited energy eigenstates and local minima of the energy landscape in quantum spin systems

The quantum-classical correspondence between local minima on the classical energy landscape and excited eigenstates in the energy spectrum is studied within the context of many-body quantum spin systems. In mean-field approximations of a quantum problem, one usually focuses on attaining the global minimum of the resulting energy function, while other minimum solutions are usually ignored. For frustrated systems, a strict distinction between global and local minimum is often not tenable since first-order-type transitions can interchange the roles played by two different minima. This begs the question of whether there is any physical interpretation for the local minima encountered in mean-field approximations of quantum systems. We look at the problem from the perspective of quantum spin systems. Two models are studied, a frustrated model with quenched disorder and a pure system without frustration. Accurate classical energies of the minima are compared with the full spectrum of energy levels, allowing us to search for signs of correspondence between them. It is found that the local minima can generally be interpreted as excited energy eigenstates. Instances of spurious minima are also reported.

Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in dd spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the (2+1)(2+1)-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner’s lattice models M_{(d,n)}M(d,n) that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary.Our procedure can be generalized to the simulation of Kitaev’s Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model M_{(d,n)}M(d,n) is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.

Cyclotron radiation from shaped electron wavefunctions

Recent years have shown increasing interest in understanding the role of the wavefunction of a quantum source on the characteristics of its emitted radiation. In this work, we demonstrate that shaping the wavefunction of the source can drastically change the instantaneous emission. We exemplify this concept by examining an electron in cyclotron motion, calculating the angular power distribution of emission by an electron in a Schrodinger cat state. The emitted cyclotron radiation reveals a breakdown of classical–quantum correspondence. The short-time dynamics of the radiation process shows deviations in the power and electron trajectory that disappear at long times, where the predictions of classical electrodynamics are recovered.

Quantum-classical correspondence of strongly chaotic many-body spin models

Quantum-classical correspondence of strongly chaotic many-body spin models

A quantum-classical correspondence in the dynamics around higher order saddle points: a Bohmian perspective

A quantum-classical correspondence in the dynamics around higher order saddle points: a Bohmian perspective

Correlated collective excitation and quantum entanglement between two Rydberg superatoms in steady state

<sec>Owing to the unique physical characteristics of Rydberg atoms, which play an important role in quantum information and quantum computation, the theoretical and applied research of Rydberg atoms have become the hot spots of scientific research in recent years. With the large polarizability of Rydberg atoms, even a small electric field could cause a considerable electric dipole moment, resulting in a strong dipole-dipole interaction between Rydberg atoms. The multiple excitations of the Rydberg states are strongly inhibited because of the strong dipole interaction between atoms within a mesoscopic interaction (blockade) region. We call this phenomenon the dipole blockade effect. The dipole blockade effect makes it possible to build single-photon quantum devices, implement quantum gates, generate quantum entanglement, and simulate many-body quantum problems, etc.</sec><sec>A Rydberg atomic ensemble in the same blockade region can be regarded as a superatom. In the same way, if these atoms trapped in two optical dipole traps, each sub-ensemble can be considered as a sub-superatom which is closely related to the superatom. According to the fact that two Rydberg sub-superatoms can be strongly correlated due to sharing no more than one excited Rydberg atom, we study correlated collective excitation and quantum entanglement between two Rydberg sub-superatoms in a steady state. With the superatom model, the problem of exponentially increasing system size with the number of atoms can be circumvented to a certain extent in studying many-body physics. By solving the two-body Lindblad’s master equation accurately, we obtain the analytical expressions for the collective excitation probabilities of the two sub-superatoms, and the concurrence measuring the bipartite entanglement between them. Our results show that they are all sensitive to the number of atoms in each Rydberg superatom: the bigger (including more atoms) the Rydberg superatom, the higher the collective Rydberg excitation probability is. And that the maximally entangled state can only be obtained with two equal-sized Rydberg superatoms. When this condition is fulfilled, the mesoscopic entanglement can be generated by adding the number of atoms in each Rydberg superatom. This may provide an attractive platform for studying the quantum-classical correspondence and have potential promising applications in quantum information processing.</sec>

Asymptotic representations of Hamiltonian diffeomorphisms and quantization

We show that for a special class of geometric quantizations with “small” quantum errors, the quantum classical correspondence gives rise to an asymptotic projective unitary representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.

Quantum extraordinary-log universality of boundary critical behavior

The recent discovery of extraordinary-log universality has generated intense interest in classical and quantum boundary critical phenomena. Despite tremendous efforts, the existence of quantum extraordinary-log universality remains extremely controversial. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into the superfluid phase when the hopping strength is sufficiently enhanced on edges. At the bulk critical point, the open edges exhibit the special, ordinary, and extraordinary critical phases. In the extraordinary phase, logarithms are involved in the finite-size scaling of two-point correlation and superfluid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to modern quantum emulators for interacting bosons in lattices, the edge critical phases might be realized in experiments.

Classical-quantum correspondence of special and extraordinary-log criticality: Villain's bridge

There has been much recent progress on exotic surface critical behavior, yet the classical-quantum correspondence of special and extraordinary-log criticality remains largely unclear. Employing worm Monte Carlo simulations, we explore the surface criticality at an emergent superfluid-Mott insulator critical point in the Villain representation, which is believed to connect classical and quantum O(2) critical systems. We observe a special transition with the thermal and magnetic renormalization exponents $y_t=0.58(1)$ and $y_h=1.690(1)$ respectively, which are close to recent estimates from models with discrete spin variables. The existence of extraordinary-log universality is evidenced by the critical exponent $\hat{q}=0.58(2)$ from two-point correlation and the renormalization-group parameter $\alpha=0.28(1)$ from superfluid stiffness, which obey the scaling relation of extraordinary-log critical theory and recover the logarithmic finite-size scaling of critical superfluid stiffness in open-edge quantum Bose-Hubbard model. Our results bridge recent observations of surface critical behavior in the classical statistical mechanical models [Parisen Toldin, Phys. Rev. Lett. 126, 135701 (2021); Hu $et$ $al.$, $ibid.$ 127, 120603 (2021); Parisen Toldin $et$ $al.$, $ibid.$ 128, 215701 (2022)] and the open-edge quantum Bose-Hubbard model [Sun $et$ $al.$, Phys. Rev. B 106, 224502 (2022)].

Embedding the Yang-Lee quantum criticality in open quantum systems

The Yang-Lee edge singularity is a quintessential nonunitary critical phenomenon characterized by anomalous scaling. However, an imaginary magnetic field involved in this phenomenon makes its physical implementation nontrivial. By embedding the original classical system which exhibits the Yang-Lee edge singularity in a quantum system with an ancilla qubit and by invoking the quantum-classical correspondence, we demonstrate a physical realization of the nonunitary quantum criticality in an open quantum system, where the nonunitary criticality is identified with the singularity at an exceptional point caused by postselection of quantum measurement.