We present a Schrödinger (Liouville) type of equation for a quantum open system. It has a correlated part, and various master equations may be its special cases. It also has significant applications for constructing decoherence-free subspace for quantum computation. It is related to the original Schrödinger (Liouville) equation for the total system through a non-unitary similarity transformation. It is unnecessary for its correlated part to be self-adjoint, so there is a complex spectrum for the corresponding Hamiltonian (Liouvillian), which enables the time evolution of states to be asymmetric. This shows just the correlation to produce evolution of world.
We present a Schrödinger (Liouville) type of equation for a quantum open system. It has a correlated part, and various master equations may be its special cases. It also has significant applications for constructing decoherence-free subspace for quantum computation. It is related to the original Schrödinger (Liouville) equation for the total system through a non-unitary similarity transformation. It is unnecessary for its correlated part to be self-adjoint, so there is a complex spectrum for the corresponding Hamiltonian (Liouvillian), which enables the time evolution of states to be asymmetric. This shows just the correlation to produce evolution of world.
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 evolution of mixed states of a closed quantum system is described by a group of evolution superoperators whose infinitesimal generator (the quantum Liouville superoperator, or Liouvillian) determines the mixed-state counterpart of the Schrödinger equation: the Liouville–von Neumann equation. When the state space of the system is infinite-dimensional, the Liouville superoperator is unbounded whenever the corresponding Hamiltonian is. In this paper, we provide a rigorous, pedagogically-oriented, and self-contained introduction to the quantum Liouville formalism in the presence of unbounded operators. We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage; starting from that, we develop some simpler characterizations of the domain of the Liouvillian and its square. We also provide, with explicit proofs, some domains of essential selfadjointness (cores) of the Liouvillian.
In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system (S∪E) is isolated from the rest of the universe. So we have two independent Hamiltonians H^S, H^E and an interaction V^ between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction V^ with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time t0=0, with a Gaussian kernel, for 0≤t<tc where tc∈]0,∞] is a critical time such that reversibility is lost for t>tc if tc<+∞. Some manifestations of decoherence, for t>0 small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.
Quantum computing, leveraging the principles of superposition and entanglement, speeds up the computational process in dynamical systems. Quantum Phase Estimation (QPE) which is a core of many quantum algorithms is an oracle to determine a significant invariant of systems, namely the eigenvalues of the state transition matrix. In addition to showing the scalability of QPE, this paper illustrates the time evolution of phase in the Schrödinger’s equation through Python Code executed in the Qiskit/Aer simulator. In addition, this paper proposes an efficient method for phase estimation of a non-self-adjoint Hamiltonian, which offers great potential in solving complex quantum systems.
MulAtoLEG (Multi-Atom Liouville Equation Generator) is an open-source Mathematica package for generating Liouville superoperators and Liouville equations, specialized for multilevel atomic systems comprising an arbitrary number of atoms. This scheme is based on an extension to multilevel atomic systems, originally developed by Lehmberg [R. H. Lehmberg, Phys. Rev. A 2, 883 (1970)] as an adjoint master equation for ensembles of two-level emitters and later reformulated by Genes [M. Reitz, C. Sommer and C. Genes, PRX Quantum 3, 010201 (2022)] as a master equation. The package facilitates the generation of equations for complex transition configurations in alkali atoms. Although primarily designed for atomic systems, it can also generate the master and adjoint master equations for general Hamiltonians and Lindbladians. In addition, it includes functionalities to construct the differential equations in the dressed-state basis, where, in many cases, the non-unitary evolution operator can be determined explicitly. To maximize computational efficiency, the package leverages Mathematica's vectorization and sparse linear algebra capabilities. Since MulAtoLEG produces exact equations without approximations, the feasible system size is naturally limited by the available computational resources.
For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is L2(R). Then, an alternative operator for generating the time evolution that is manifestly unitary is suggested, regardless of the choice of Hamiltonian. The new construction involves an additional positive operator that normalizes the wave function locally and allows us to preserve unitarity, even when dealing with infinite dimensional or non-normed spaces. Our considerations show that Schrödinger and Liouville equations are, in fact, two sides of the same coin and together they provide a unified description for unbounded quantum systems.