Michael Khasin

Spectrum of an Oscillator with Jumping Frequency and the Interference of Partial Susceptibilities

We study an underdamped oscillator with random frequency jumps. We describe the oscillator spectrum in terms of coupled susceptibilities for different-frequency states. Depending on the parameters, the spectrum has a fine structure or displays a single asymmetric peak. For nanomechanical resonators with a fluctuating number of attached molecules, it is found in a simple analytical form. The results bear on dephasing in various types of systems with jumping frequency.

Speeding up disease extinction with a limited amount of vaccine

We consider optimal vaccination protocol where the vaccine is in short supply. In this case, the endemic state remains dynamically stable; disease extinction happens at random and requires a large fluctuation, which can come from the intrinsic randomness of the population dynamics. We show that vaccination can exponentially increase the disease extinction rate. For a time-periodic vaccination with fixed average rate, the optimal vaccination protocol is model independent and presents a sequence of short pulses. The effect can be resonantly enhanced if the vaccination pulse period coincides with the characteristic period of the disease dynamics or its multiples. This resonant effect is illustrated using a simple epidemic model. The analysis is based on the theory of fluctuation-induced population extinction in periodically modulated systems that we develop. If the system is strongly modulated (for example, by seasonal variations) and vaccination has the same period, the vaccination pulses must be properly synchronized; a wrong vaccination phase can impede disease extinction.

Minimizing the population extinction risk by migration.

Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes. A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here, we answer this question for a connected network of model habitat patches with different carrying capacities.

Rise and fall of quantum and classical correlations in open-system dynamics

Interacting quantum systems evolving from an uncorrelated composite initial state generically develop quantum correlations--entanglement. As a consequence, a local description of interacting quantum systems is impossible as a rule. A unitarily evolving (isolated) quantum system generically develops extensive entanglement: the magnitude of the generated entanglement will increase without bounds with the effective Hilbert space dimension of the system. It is conceivable that coupling of the interacting subsystems to local dephasing environments will restrict the generation of entanglement to such extent that the evolving composite system may be considered as approximately disentangled. This conjecture is addressed in the context of some common models of a bipartite system with linear and nonlinear interactions and local coupling to dephasing environments. Analytical and numerical results obtained imply that the conjecture is generally false. Open dynamics of the quantum correlations is compared to the corresponding evolution of the classical correlations and a qualitative difference is found.

Noise and controllability: suppression of controllability in large quantum systems.

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise. Can one design control fields such that the effect of noise is negligible on the timescale of the transformation? Complete controllability in practice requires that the effect of noise can be suppressed for an arbitrary transformation. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases, determined by the dimension of the Hilbert space representation of the algebra. We show that for large quantum systems, generic noise in the controls dominates for a typical class of target transformation; i.e., complete controllability is destroyed by the noise.

Time-resolved extinction rates of stochastic populations

Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasistationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover-renewal and removal--is much slower than all other processes. In this case there is a time-scale separation in the system which enables one to introduce a short-time quasistationary extinction rate W1 and a long-time quasistationary extinction rate W2, and to develop a time-dependent theory of the transition between the two rates. It is shown that W1 and W2 coincide with the extinction rates when the population turnover is absent and present, but very slow, respectively. The exponentially large disparity between the two rates reflects fragility of the extinction rate in the population dynamics without turnover.

Algorithm for simulation of quantum many-body dynamics using dynamical coarse-graining

An algorithm for simulation of quantum many-body dynamics having su(2) spectrum-generating algebra is developed. The algorithm is based on the idea of dynamical coarse-graining. The original unitary dynamics of the target observables--the elements of the spectrum-generating algebra--is simulated by a surrogate open-system dynamics, which can be interpreted as weak measurement of the target observables, performed on the evolving system. The open-system state can be represented by a mixture of pure states, localized in the phase space. The localization reduces the scaling of the computational resources with the Hilbert-space dimension n by factor n{sup 3/2}(ln n){sup -1} compared to conventional sparse-matrix methods. The guidelines for the choice of parameters for the simulation are presented and the scaling of the computational resources with the Hilbert-space dimension of the system is estimated. The algorithm is applied to the simulation of the dynamics of systems of 2x10{sup 4} and 2x10{sup 6} cold atoms in a double-well trap, described by the two-site Bose-Hubbard model.

The globally stable solution of a stochastic nonlinear Schrödinger equation

Weak measurement of a subset of noncommuting observables of a quantum system can be modeled by the open-system evolution, governed by the master equation in the Lindblad form. The open-system density operator can be represented as a statistical mixture over non-unitarily evolving pure states, driven by the stochastic nonlinear Schrodinger equation (sNLSE). The globally stable solution of the sNLSE is obtained in the case where the measured subset of observables comprises the spectrum-generating algebra of the system. This solution is a generalized coherent state (GCS), associated with the algebra. The result is based on proving that the GCS minimizes the trace-norm of the covariance matrix, associated with the spectrum-generating algebra.

Removal of resonances by rotation in linearly degenerate two-dimensional oscillator systems

A system of two nonlinearly interacting, resonant harmonic oscillators is investigated, seeking transformation to approximate action-angle variables in the vicinity of the equilibrium via the canonical perturbation theory. A variety of polynomial perturbations dependent on parameters is considered. The freedom of choice of the zero-order approximation characteristic of a linearly degenerate (resonant) system is used to cancel lower-order resonant terms in the canonical perturbation series. It is found that the cancellation of the resonant terms is only possible for particular values of parameters of the interaction term. These special sets of parameters include all the cases with the Panleve property.

Temperature dependence of interaction-induced entanglement

Both direct and indirect weak nonresonant interactions are shown to produce entanglement between two initially disentangled systems prepared as a tensor product of thermal states, provided the initial temperature is sufficiently low. Entanglement is determined by the Peres-Horodecki criterion, which establishes that a composite state is entangled if its partial transpose is not positive. If the initial temperature of the thermal states is higher than an upper critical value T{sub uc} the minimal eigenvalue of the partially transposed density matrix of the composite state remains positive in the course of the evolution. If the initial temperature of the thermal states is lower than a lower critical value T{sub lc}{<=}T{sub uc} the minimal eigenvalue of the partially transposed density matrix of the composite state becomes negative, which means that entanglement develops. We calculate the lower bound T{sub lb} for T{sub lc} and show that the negativity of the composite state is negligibly small in the interval T{sub lb}<T<T{sub uc}. Therefore the lower-bound temperature T{sub lb} can be considered as the critical temperature for the generation of entanglement. It is conjectured that above this critical temperature a composite quantum system could be simulated using classical computers.