QUANTUM-DEFORMED CANONICAL TRANSFORMATIONS, W∞ ALGEBRAS AND UNITARY TRANSFORMATIONS

Within the framework of probability distributions on projective Hilbert space a scheme for the calculation of multitime correlation functions is developed. The starting point is the Markovian stochastic wave function description of an open quantum system coupled to an environment consisting of an ensemble of harmonic oscillators in arbitrary pure or mixed states. It is shown that matrix elements of reduced Heisenberg picture operators and general time-ordered correlation functions can be expressed by time-symmetric expectation values of extended operators in a doubled Hilbert space. This representation allows the construction of a stochastic process in the doubled Hilbert space which enables the determination of arbitrary matrix elements and correlation functions. The numerical efficiency of the resulting stochastic simulation algorithm is investigated and compared with an alternative Monte Carlo wave function method proposed first by Dalibard et al. [Phys. Rev. Lett. {\bf 68}, 580 (1992)]. By means of a standard example the suggested algorithm is shown to be more efficient numerically and to converge faster. Finally, some specific examples from quantum optics are presented in order to illustrate the proposed method, such as the coupling of a system to a vacuum, a squeezed vacuum within a finite solid angle, and a thermal mixture of coherent states.

We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in Okuyama (2024) [7]. We show that the 1-particle Hilbert space introduced by Lin and Stanford (2023) [6] is related to the doubled Hilbert space H0⊗H0 of the 0-particle Hilbert space H0 by some linear isomorphism C. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of C−1 and C.

This thesis contributes to the study of quantum chaos and thermalization of closed quantum systems by applying tools from quantum information theory. We study scrambling – the delocalization of quantum information – in one-dimensional quantum many-body systems using the observable-independent tripartite information 𝐼3. We consider closed quantum systems that undergo unitary time evolution governed by a Hamiltonian, and compare them to the strong scrambling caused by random unitary operators (Haar scrambling). The tripartite information is a tool from quantum information theory that allows us to quantify correlations in space and time by considering the unitary operator evolving the system as a state |𝑈 (𝑡)⟩ in a doubled Hilbert space H ⊗ H. We found a ballistic spreading of information in translation invariant systems of spinless fermions that causes Haar scrambling in interacting models. For strongly disordered versions of the XXZ chain and the transverse-field Ising chain we found a complete breakdown of in- formation transport in non-interacting Anderson localized systems, as well as a logarithmically slow spreading of the information signal in systems that are said to be many-body localized (MBL).

The pseudomode model effectively captures the nonperturbative dynamics of open quantum systems with significantly reduced degrees of freedom. However, it is limited by the exponential growth of the Hilbert space dimension. To overcome the computational challenges, we propose a novel method that combines the multiple Davydov Ansatz with the Choi-Jamiolkowski isomorphism. Within this framework, the Lindblad equation is transformed into the non-Hermitian Schrödinger equation in a double Hilbert space, with its dynamics determined using the time-dependent variational principle. Three cases are calculated to demonstrate the effectiveness of the proposed method. We first discuss how the Davydov Ansatz works for the model with a single pseudomode. Extending the method to multiple pseudomodes, we show that the Ansatz effectively circumvents the exponential growth of the Hilbert space. Additionally, the method is also capable of addressing potential intersections that emerge in multibath scenarios. This approach offers potential applicability to various types of pseudomode models and other dissipative systems, providing a promising tool for the studies of open quantum dynamics.

Hamiltonians which are inaccessible in static systems can be engineered in periodically driven many-body systems, i.e., Floquet many-body systems. We propose to use interacting particles in a one-dimensional (1D) harmonic potential with periodic kicking to investigate two-dimensional topological and many-body physics. Depending on the driving parameters, the Floquet Hamiltonian of single kicked harmonic oscillator has various lattice structures in phase space. The noncommutative geometry of phase space gives rise to the topology of the system. We investigate the effective interactions of particles in phase space and find that the point-like contact interaction in quasi-1D real space becomes a long-rang Coulomb-like interaction in phase space, while the hardcore interaction in pure-1D real space becomes a confinement quark-like potential in phase space. We also find that the Floquet exchange interaction does not disappear even in the classical limit, and can be viewed as an effective long-range spin–spin interaction induced by collision. Our proposal may provide platforms to explore new physics and exotic phases by Floquet many-body engineering.

The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star‐product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many‐body potential scattering problem simplifies to a statistical‐mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ħ. We present semiclassical expansion of quantum characteristics for many‐body scattering problem and provide tools for calculation of average values of time‐dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non‐locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum‐mechanical extensions of the Liouville theorem on conservation of the phase‐space volume and the Poincaré theorem on conservation of 2p‐forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.

Phase space and path integral methods have been applied to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one-way Helmholtz equation. These methods are particularly appropriate for extended, strongly inhomogenenous, multidimensional channeling environments. Operator symbols play a pivotal role in the analytical constructions and numerical algorithms. The construction of the operator symbol requires the exact (or approximate) solution of the (Helmholtz) Weyl composition equation in the Weyl pseudodifferential operator calculus. The exact symbol for the profile K2(q) = K02 + A tanhρq + B sech2 ρq is constructed and briefly analyzed. This family of profiles is particularly appropriate for ocean seismo-acoustic modeling, as it includes (1) both symmetric and asymmetric wells with trapped modes and (2) sharp gradient features. Two limiting cases are the delta and discontinuity profiles. The results are used to illustrate several points pertinent to wide-angle propagation modeling and the refractive index profile reconstruction problem. [Work supported by NSF, AFOSR, and ONR.]

Hilbert space may be regarded as a convenient standard approximation by interpolation and extrapolation to a unitary space, in general decomposable into a sum of semisimple spaces. In the limit we expect a particle theory expanded in powers of h according to the number of interacting particles. In the classical limit h tending to zero, phase space, with equations of motion reducible to Hamiltonian form, replaces Hilbert space. The modification of states by observing them is taken care of by considering probability distributions of petty ensembles. If the equations for any single observer can be made autonomous by replacing empirical time by universal time with an arrow we have a causal system. We then obtain the relation between probability and negentropy required for the second law of thermodynamics. An approximate Newtonian theory provides proximate particles with internal and external variables and admits the Poincaré group. For the internal variables we have approximately the Breit interaction. For the external variables we have equivalence for observers of the homogeneous Lorentz group of relativity. We introduce grand ensembles of ultimate particles, and nebulae as proximate particles in the large. We assume the Einstein principle of equivalence for the ten parameter set of observers suggested by relativity and suppose the second law of thermodynamics holds for each observer. The Einstein law of gravitation follows in classical theory to the order of the reciprocal of the large constant, in general with positive natural curvature as well as that corresponding to mass. Replacing particle interactions by fields we include them in classical theory.

This paper considers states on the Weyl algebra of the canonical commutation relations over the phase space R2n. We show that a state is regular if and only if its classical limit is a countably additive Borel probability measure on R2n. It follows that one can “reduce” the state space of the Weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical.

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature β become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2πħ)3 are reduced to points in the phase space, the partition functionTr{exp(−βĤ) for N identical bosons (fermions) approaches (2πħ)−3N(N!) ∫ ... ∫ d3r1 d3p1 ... d3rN d3pNexp(−βH). The two factors, (2πħ)−3N and (N!)−1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles.

In the classical mechanical limit in which quantum cells of volume (2πħ)2 are reduced to points in phase space, the Bose and Fermi distributions are shown to approach the Boltzmann distribution. This limit is a continuous limit and therefore does not alter the indistinguishability of bosons (fermions) which is defined in terms of a discontinuous permutation symmetry of the density operator and Hamiltonian for the system. This means that the classical particles which correspond to the identical boson (fermions) remain indistinguishable. In the same limit, the partition function for N identical bosons approaches (2πħ)−3N (N!)−1 f. f d3r1 d3p1…d3rN d3pN exp (–βH)  Zc. The two factors, (2πħ)−3N and (N!)−1 which are often added in an ad hoc manner in many books on statistical mechanics, are thus derived from first principles. In the classical statistical limit in which the number density n and the reciprocal temperature β become small, the quantum statistical distributions also approach the Boltzmann distribution. This is shown by examining the chemical potential μ as a function of n and β. The classical approximation is valid if the thermal de Broglie wavelength is much shorter than the interparticle distance irrespective of any interparticle interaction.

Quantization of linear maps on a torus-fresnel diffraction by a periodic grating

Phase-space representations play an increasingly important role in several branches of physics. Here, we review the author's studies of the Bohr-Heisenberg correspondence principle within the Weyl-Wigner phase-space representation. The analysis leads to refined correspondence rules that can be successfully used far away from the classical limit considered by Bohr and Heisenberg.

A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.