Using a group theoretical approach we derive an equation of motion for a mixed quantum-classical system. The quantum-classical bracket entering the equation preserves the Lie algebra structure of quantum and classical mechanics: The bracket is antisymmetric and satisfies the Jacobi identity, and, therefore, leads to a natural description of interaction between quantum and classical degrees of freedom. We apply the formalism to coupled quantum and classical oscillators and show how various approximations, such as the mean-field and the multiconfiguration mean-field approaches, can be obtained from the quantum-classical equation of motion.
We study the ergodicity and mixing of quantum kicked rotor (QKR) with two distinct approaches. In one approach, we use the definitions of quantum ergodicity and mixing recently proposed in [Phys. Rev. E 94, 022150 (2016)], which involve only eigen-energies (Floquet quasi-energies for QKR). In the other approach, we study ergodicity and mixing with quantum Poincarè section, which is plotted with a method that maps a wave function unitarily onto quantum phase space composed of Planck cells. Classical Poincarè section can be recovered with the effective Planck constant gradually diminishing. We d…
The investigation of the ultimate limits imposed by quantum mechanics on amplification represents an important topic both on a fundamental level and from the perspective of potential applications. We propose here a novel setup for an optomechanical amplifier, constituted by a mechanical resonator dispersively coupled to an optomechanical cavity asymmetrically driven around both mechanical sidebands. We show that, on general grounds, the present amplifier operates in a novel regime-- which we here call phase-mixing amplification. At the same time, for a suitable choice of parameters, the amplif…
We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks.
We propose an approach to generate strong quantum entanglement by the controllable four wave mixing mechanism in a single cavity, weak coupling optomechanical system. The optomechanical system is driven by a strong two tone pump field and a weak signal field, simultaneously. The two tone pump field consists of a lower and an upper sideband, which couple with the optical cavity and mechanical resonator, and generate the beam splitter and two mode squeezing interactions under the rotating wave approximation. This interaction mechanism modifies the effective susceptibility of the optomechanical c…
The thermodynamical entropy of a system which consists of different kinds of ideal gases is known to be defined successfully in the case when the differences are described by classical or quantum theory. Since these theories are special examples in the framework of generalized probabilistic theories (GPTs), it is natural to generalize the notion of thermodynamical entropy to systems where the internal degrees of particles are described by other possible theories. In this paper, we consider thermodynamical entropy of mixing in a specific series of theories of GPTs called the regular polygon the…
The Classical-to-Quantum Transition with Broadband Four-Wave Mixing
量子光学的一个关键问题是低功率下的非经典双光子关联如何演化为高功率下的经典相干。直接观察从量子到经典行为的交叉是可取的，但由于缺乏足够的实验技术，覆盖超宽的动态范围的光子通量从单光子制度的经典水平是困难的。我们研究了在\emph{large dynamic range of $\sim80dB$ in photon flux}上通过宽带四波混频（FWM）产生的光的光谱内的双光子相关性，该双光子相关性跨越经典到量子的跃迁，使用区分经典和量子行为的双光子干涉效应。我们探索量子经典性质的光通过观察内部损耗的干涉对比度的依赖性，并证明量子崩溃和复苏的干涉时，在光纤中的四波混频增益成为虚构的。
The Hellmann-Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive nonrelativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical st…
Transformation Relation of Two Descriptive Systems between Quantum Mechanics and Classical Statistical Mechanics and Dynamic Foundation of Classical Statistical Physics
Measurement theory in classical mechanics is investigated in the formulation of classical mechanics by Koopman and von Neumann (KvN), using Hilbert space. It is shown that the classical and the quantum measurements give different "relative interpretations" of the measurement state and the recording state of the measurement device. The uncertainty relation in classical mechanics is also derived.
By means of the examples of classical and Bohmian quantum mechanics, we illustrate the well-known ideas of Boltzmann as to how one gets from laws defined for the universe as a whole to dynamical relations describing the evolution of subsystems. We explain how probabilities enter into this process, what quantum and classical probabilities have in common and where exactly their difference lies.
In this paper we show that, via an extension of time, some metric structures naturally appear in both classical and quantum mechanics when both are formulated via path integrals. We calculate the various Ricci scalar and curvatures associated to these metrics and prove that they can be choosen to be zero in classical mechanics while this is not possible in quantum mechanics.
Integrated, monolithic nonlinear cavities are of high interest in both classical and quantum optics experiments for their high efficiency and stability. However, a general, analytic theory of classical three wave mixing in such systems that encompasses multiple monolithic designs, including both linear and nonlinear regions, as well as any three-wave mixing process has not yet been fully developed. In this paper, we present the analytic theory for a general, classical three wave mixing process in a cavity with arbitrary finesse and non-zero propagation losses, encompassing second harmonic, sum…
Quantum walks on graphs have been shown in certain cases to mix quadratically faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. Here, we construct a lifted Markov chain on a graph with $n^2 D(G)$ vertices that mixes exactly to the average mixing distribution of a quantum walk on the graph $G$ with $n$ vertices, where $D(G)$ is the diameter of $G$. Moreover, the mixing time of this chain is $D(G)$ timesteps, and we prove …
Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving a noncommutative operator algebraic variant of the Koopman-von Neumann approach. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechan…
The aim of this paper is to analyze the reconstructability of quantum mechanics from classical conditional probabilities representing measurement outcomes conditioned on measurement choices. We will investigate how the quantum mechanical representation of classical conditional probabilities is situated within the broader frame of noncommutative representations. To this goal, we adopt some parts of the quantum formalism and ask whether empirical data can constrain the rest of the representation to conform to quantum mechanics. We will show that as the set of empirical data grows conventional el…
