The equivalence between classical epidemic model and nondissipative and dissipative quantum tight-binding model is derived. Classical epidemic model can reproduce the quantum entanglement emerging in the case of electrostatically coupled qubits described by von-Neumann entropy both in non-dissipative and dissipative case. The obtained results shows that quantum mechanical phenomena might be almost entirely simulated by classical statistical model. It includes the quantum like entanglement and superposition of states. Therefore coupled epidemic models expressed by classical systems in terms of classical physics can be the base for possible incorporation of quantum technologies and in particular for quantum like computation and quantum like communication. The classical density matrix is derived and described by the equation of motion in terms of anticommutator. Existence of Rabi like oscillations is pointed in classical epidemic model. Furthermore the existence of Aharonov-Bohm effect in quantum systems can also be reproduced by the classical epidemic model. Every quantum system made from quantum dots and described by simplistic tight-binding model by use of position-based qubits can be effectively described by classical model with very specific structure of S matrix that has twice bigger size as it is the case of quantum matrix Hamiltonian. Obtained results partly question fundamental and unique character of quantum mechanics and are placing ontology of quantum mechanics much in the framework of classical statistical physics what can bring motivation for emergence of other fundamental theories bringing suggestion that quantum mechanical is only effective and phenomenological but not fundamental picture of reality.
The quantum dynamics of arbitrary
N-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary
N-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension
is general for any
N. The time evolution of an
N-level system [generated by a complex element of the SU(
N) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in
n Euclidean dimensions [the rotation group SO(
n)] is then mapped directly to
n oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent
N-level systems scales as
N2 for the density matrix formalism but increases only linearly with
N for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
The dissipation and decoherence (for example, the effects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum system with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an Langevin-Schroedinger type stochastic differential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) model L-Sh SDE is a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for different operators are calculated. In particular, the expression for uncertain relations depending on parameter of interaction with environment is obtained. The Weyl transformation for stochastic operators is specified. Ground state Winger function is developed in detail.
Starting from a many-body classical system governed by a trace-form entropy we derive, in the stochastic quantization picture, a family of non linear and non-Hermitian Schroedinger equations describing, in the mean filed approximation, a quantum system of interacting particles. The time evolution of the main physical observables is analyzed by means of the Ehrenfest equations showing that, in general, this family of equations takes into account dissipative and damped effects due to the interaction of the system with the background. We explore the presence of steady states by means of solitons, describing conservative solutions. The results are specialized to the case of a system governed by the Boltzmann-Gibbs entropy.