We propose and solve a simple but very general quantum model of a SU (2) spin interacting with a large external system with N states. The coupling is described by a random Hamiltonian in a new general Gaussian SU (2) × U ( N ) random matrix ensemble, that we introduce in this paper. We solve the model in the large N limit, for any value of the spin j and for any choice of the coupling matrix element distributions in the different possible angular momentum channels l (and provided that the internal dynamics of the spin is slow). Besides its mathematical interest as a non-trivial random matrix model, it allows us to study and illustrate in a simple framework various phenomena: the decoherence dynamics, the conditions of emergence of the classical phase space for the spin, and the properties of quantum diffusion in phase space. The large time evolution for the spin is shown to be non-Markovian in general, the Markov property emerging in some specific cases for the dynamics and the initial conditions.
We propose and solve a simple but very general quantum model of a SU (2) spin interacting with a large external system with N states. The coupling is described by a random Hamiltonian in a new general Gaussian SU (2) × U ( N ) random matrix ensemble, that we introduce in this paper. We solve the model in the large N limit, for any value of the spin j and for any choice of the coupling matrix element distributions in the different possible angular momentum channels l (and provided that the internal dynamics of the spin is slow). Besides its mathematical interest as a non-trivial random matrix model, it allows us to study and illustrate in a simple framework various phenomena: the decoherence dynamics, the conditions of emergence of the classical phase space for the spin, and the properties of quantum diffusion in phase space. The large time evolution for the spin is shown to be non-Markovian in general, the Markov property emerging in some specific cases for the dynamics and the initial conditions.
We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment.We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso-and even nanosize.We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition.We then workout an analog of Born-Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord.We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits.In particular, we find the vanishing of the entanglement at finite moments and its subsequent revivals (entanglement sudden death and entanglement sudden birth).Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.
We consider the decoherence of a quantum system S coupled to a quantum environment E. For states chosen uniformly at random from the unit hypersphere in the Hilbert space of the closed system S+E we derive a scaling relationship for the sum of the off-diagonal elements of the reduced density matrix of S as a function of the size DE of the Hilbert space of E. This sum decreases as 1/DE as long as ${D}_{E}\ensuremath{\gg}1$. We test this scaling prediction by performing large-scale simulations which solve the time-dependent Schr\"odinger equation for a ring of spin-1/2 particles, four of them belonging to S and the others to E, and for this ring with small world bonds added in E and/or between S and E. The spin-1/2 particles experience nearest-neighbor interactions that are identical for the interactions within S and random for the interactions within E and between S and E, or that are all identical. Provided that the time evolution drives the whole system from the initial state toward a scaling state, a state which has similar properties as states belonging to the class of quantum states for which we derived the scaling relationship, the scaling prediction holds. We examine various interaction parameters and initial states for our model system to find whether or not the time evolution reaches the class of states that have the scaling property. For the homogeneous ring we find that the evolution for select initial states does not reach these scaling states. This conclusion is not modified if we add some homogeneous random connections. For a ring we find that some randomness in the interaction parameters is required so that most initial configurations are driven toward the scaling state. Furthermore, if the amount of randomness is small the time required to reach the scaling states may be very large. For the case of all random interactions in E the ring is driven toward the scaling state. Adding small world bonds between S and E with random interaction strengths may decrease the time required to reach the scaling state or may prevent the scaling state from being reached. For the latter case we show that increasing the complexity of the environment by adding extra connections within the environment suffices to observe the predicted scaling behavior.
We study the decoherence of a qubit weakly coupled to frustrated spin baths. We focus on spin baths described by the classical Ising spin glass and the quantum random transverse Ising model which are known to have complex thermodynamic phase diagrams as a function of an external magnetic field and temperature. Using a combination of numerical and analytical methods, we show that for baths initially in thermal equilibrium, the resulting decoherence is highly sensitive to the nature of the coupling to the environment and is qualitatively different in different parts of the phase diagram. We find an unexpected strong non-Markovian decay of the coherence when the random transverse Ising model bath is prepared in an initial state characterized by a finite temperature paramagnet. This is contrary to the usual case of exponential decay (Markovian) expected for spin baths in finite temperature paramagnetic phases, thereby illustrating the importance of the underlying nontrivial dynamics of interacting quantum spinbaths.
This paper is a continuation of our previous paper [8], in which we have studied the dynamics of quantum correlations of two qubits embedded each into its own disordered multiconnected environment. We modeled the environment by random matrices of large size allowing for a possibility to describe meso- and even nanoenvironments. In this paper we also study the dynamics of quantum correlations of two qubits but embedded into a common environment which we also model by random matrices of large size. We obtain the large size limit of the reduced density matrix of two qubits. We then use an analog of the Bogolyubov-van Hove (also known as the Born-Markov) approximation of the theory of open systems and statistical mechanics. The approximation does not imply in general the Markovian evolution in our model but allows for sufficiently detailed analysis both analytical and numerical of the evolution of several widely used quantifiers of quantum correlation, mainly entanglement. We find a number of new patterns of qubits dynamics comparing with the case of independent environments studied in [8] and displaying the role of dynamical (indirect, via the environment) correlations in the enhancing and diversification of qubit evolution. Our results, (announced in [9]), can be viewed as a manifestation of the universality of certain properties of the decoherent qubit evolution which have been found previously in various exact and approximate versions of two-qubit models with macroscopic bosonic environment.
We study decoherence of one, two, and n non-interacting qubits. Decoherence, measured in terms of purity, is calculated in linear response approximation, making use of the spectator configuration. The environment and its interaction with the qubits are modelled by random matrices. For two qubits, numerical studies reveal a simple one to one correspondence between its decoherence and its internal entanglement decay. Using this relation we are able to give a formula for concurrence decay. For large environments the evolution induces a unital channel in the two qubits, providing a partial explanation for the relation above. Using a kicked Ising spin network, we study the exact evolution of two non-interacting qubits in the presence of a spin bath. We find that the entanglement (as measured by concurrence) of the two qubits has a close relation to the purity of the pair, and closely follows an analytic relation derived for Werner states. As a collateral result we find that an integrable environment causes quadratic decay of concurrence as well as of purity, while a chaotic environment causes linear decay. Both quantities display recurrences in some integrable environments. Good agreement with the results found using random matrix theory is obtained. Finally, we analyze decoherence of a quantum register in the absence of non-local operations. The problem is solved in terms of a sum rule which implies linear scaling in the number of qubits. Each term involves a single qubit and its entanglement with the remaining ones. Two conditions are essential: decoherence must be small and the coupling of different qubits must be uncorrelated in the interaction picture. We apply the result to the random matrix model, and illustrate its reach considering a GHZ state coupled to a spin bath.
Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).