A Kinetic Model of Quantum Jumps

A new class of models describing the dissipative dynamics of an open quantum system S by means of random time evolutions of pure states in its Hilbert space is considered. The random evolutions are linear and defined by Poisson processes. At the random Poissonian times, the wavefunction experiences discontinuous changes (quantum jumps). These changes are implemented by some non-unitary linear operators satisfying a locality condition. If the Hilbert space of S is infinite dimensional, the models involve an infinite number of independent Poisson processes and the total frequency of jumps may be infinite. We show that the random evolutions in are then given by some almost-surely defined unbounded random evolution operators obtained by a limit procedure. The average evolution of the observables of S is given by a quantum dynamical semigroup, its generator having the Lindblad form.⁽¹⁾ The relevance of the models in the field of electronic transport in Anderson insulators is emphasised.     

Entangled Qubit States and Linear Entropy in the Probability Representation of Quantum Mechanics

The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of the probability distributions describing the qubit states. The generalization of the entanglement properties to the case of superposition of two-mode oscillator states is discussed using the probability representation of quantum states.     

Return Probability of the Open Quantum Random Walk with Time-Dependence

We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"     

Probability Representation of Quantum States

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born’s rule and recently suggested method of dequantizer–quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical–like equations for the probability distributions determining the quantum system states. Relations to phase–space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.     

Initial Decoherence of Open Quantum Systems

We present a new short-time approximation scheme for evaluation of decoherence. At low temperatures, the approximation is argued to apply at intermediate times as well. It then provides a tractable approach complementary to Markovian-type approximations, and appropriate for evaluation of deviations from pure states in quantum computing models.     

Probability Current and Trajectory Representation

A unified form for real and complex wave functions is proposed for the stationary case, and the quantum Hamilton-Jacobi equation is derived in the three-dimensional space. The difficulties which appear in Bohm's theory like the vanishing value of the conjugate momentum in the real wave function case are surmounted. In one dimension, a new form of the general solution of the quantum Hamilton-Jacobi equation leading straightforwardly to the general form of the Schrödinger wave function is proposed. For unbound states, it is shown that the invariance of the reduced action under a dilatation plus a rotation of the wave function in the complex space implies that microstates do not appear. For bound states, it is shown that some freedom subsists and gives rise to the manifestation of microstates not detected by the Schrödinger wave function.     

Hydrodynamics for Quasi-Free Quantum Systems

We consider quasi-free quantum systems and we derive the Euler equation using the so-called hydrodynamic limit. We use Wigner's well-known distribution function and discuss an extension to band distribution functions for particles in a periodic potential. We investigate the bosonic system of hard rods and calculate fluctuations of the density.     

光纤耦合对量子密钥分配系统光子探测的影响

借助量子密码术和卫星可以实现全球性的保密通信网络。但使用现有的单光子探测模块搭建星地量子密钥分配(QKD)系统,接收端就面临着空间光-多模光纤耦合的技术挑战。空间光-多模光纤耦合条件对星地量子密钥分配系统的跟瞄精度提出了严格要求。理论分析和定量计算表明,跟瞄精度ε与光束发散角θdiv的比值ε/dθiv≤0.5时,星地量子密钥分配系统的光子探测概率较高,系统可以正常工作;ε/dθiv≤0.1时,系统处于量子密钥产生速率为几kb/s的更理想状况。采用短波长更有利于满足空间光-多模光纤耦合条件,同时有利于系统获得更高的密钥产生速率。     

A Non-Standard Coupling Between Quantum Systems Originated From Their Kinetic Energy

In standard quantum mechanics, the coupling between quantum systems is described by a potential interaction term in the Hamiltonian. This type of coupling is well-rooted in nature and shapes the universe around us, from the interactions between single photons to the attractive force between atoms that forms molecules. Quantum mechanics does not forbid other kinds of interactions to take place. In this paper, a non-standard quantum coupling between quantum systems is proposed, originated from the kinetic energy rather than the potential interaction in the Hamiltonian. Unlike the potential-based coupling, the proposed coupling changes the fundamental structure of quantum mechanics in the form of modified uncertainty relations that are shaped by the coupling between the particles in the system. Two prototypical examples of non-standard systems that perform such kinetic-based coupling are presented. In the first example, it considers a particle confined in a heterostructure, such as a quantum dot, where the coupling is between the particle and dynamic walls that determine the size of the heterostructure. The second example involves a particle in a 3D heterostructure with coupling between its position axes. It then discusses several future implications of the proposed type of non-standard coupling.     

Representation Systems and Quantum Structures

Two important classes of quantum structures, namely orthomodular posets and orthomodular lattices, can be characterized in a classical context, using notions like partial information and points of view. Using the formalism of representation systems, we show that these quantum structures can be obtained by expressing conditions on the existence of particular points of view, of particular ways to observe a system. PACS: 02.70.Wz, 03.67.Lx.     

Moment Systems Derived from Relativistic Kinetic Equations

In this paper, we are interested in the derivation of macroscopic equations from kinetic ones using a moment method in a relativistic framework. More precisely, we establish the general form of moments that are compatible with the Lorentz invariance and derive a hierarchy of relativistic moment systems from a Boltzmann kinetic equation. The proof is based on the representation theory of Lie algebras. We then extend this derivation to the classical case and general families of moments that obey the Galilean invariance are also constructed. It is remarkable that the set of formal classical limits of the so-obtained relativistic moment systems is not identical to the set of classical moments quoted in Ref. 21 and one could use a new physically relevant criterion to derive suitable moment systems in the classical case. Finally, the ultra-relativistic limit is considered.     

New Challenges for Classical and Quantum Probability

The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.     

Extending Quantum Probability from Real Axis to Complex Plane

Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle’s random motion in the complex plane. The probability distribution ρc(t,x,y) of the particle’s position over the complex plane z=x+iy is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution ρc(t,x,y) is verified by the solution of the complex Fokker–Planck equation. It is shown that quantum probability |Ψ|2 and classical probability can be integrated under the framework of complex probability ρc(t,x,y), such that they can both be derived from ρc(t,x,y) by different statistical ways of collecting spatial points.     

Decay Process of Quantum Open System at Finite Temperatures

Starting from the formal solution to the Heisenberg equation, we revisit an universal model for a quantum open system with a harmonic oscillator linearly coupled to a boson bath. The analysis of the decay process for a Fock state and a coherent state demonstrate that this method is very useful in dealing with the problems in decay process of the open system. For finite temperatures, the calculations of the reduced density matrix and the mean excitation number for the open system show that an initial coherent state will evolve into a temperature-dependant coherent state after tracing over the bath variables. Also in short-time limit, a temperature-dependant effective Hamiltonian for the open system characterizes the decay process of the open system.     

Decay Process of Quantum Open System at Finite Temperatures

Starting from the formal solution to the Heisenberg equation, we revisit an universal model for a quantum open system with a harmonic oscillator linearly coupled to a boson bath. The analysis of the decay process for a Fock state and a coherent state demonstrate that this method is very useful in dealing with the problems in decay process of the open system. For finite temperatures, the calculations of the reduced density matrix and the mean excitation number for the open system show that an initial coherent state will evolve into a temperature-dependant coherent state after tracing over the bath variables. Also in short-time limit, a temperature-dependant effective Hamiltonian for the open system characterizes the decay process of the open system.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

Operator Correlations and Quantum Regression Theorem in Non-Markovian Lindblad Rate Equations

Non-Markovian Lindblad rate equations arise from alternative microscopic interactions such as quantum systems coupled to composite reservoirs, where extra degrees of freedom mediate the interaction between the system and a Markovian reservoir, as well as from systems coupled to complex structured reservoirs whose action can be well approximated by a direct sum of Markovian sub-reservoirs (Budini in Phys. Rev. A 74:053815 [2006]). The purpose of this paper is two fold. First, for both kinds of interactions we find general expressions for the system operator correlations written in terms of the Lindblad rate propagator. Secondly, we find the conditions under which a quantum regression hypothesis is valid. We show that a non-Markovian quantum regression theorem can only be granted in a stationary regime, being a necessary condition the fulfillment of a detailed balance condition. This result is independent of the underlying microscopic interaction, providing a criterion for the validity of the regression hypothesis in non-Markovian Lindblad-like master equations. As an example, we study the correlations of a two-level system coupled to different kind of reservoirs.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.