3,286 research outputs found
Quantum trajectories for time-dependent adiabatic master equations
We develop a quantum trajectories technique for the unraveling of the quantum
adiabatic master equation in Lindblad form. By evolving a complex state vector
of dimension instead of a complex density matrix of dimension ,
simulations of larger system sizes become feasible. The cost of running many
trajectories, which is required to recover the master equation evolution, can
be minimized by running the trajectories in parallel, making this method
suitable for high performance computing clusters. In general, the trajectories
method can provide up to a factor advantage over directly solving the
master equation. In special cases where only the expectation values of certain
observables are desired, an advantage of up to a factor is possible. We
test the method by demonstrating agreement with direct solution of the quantum
adiabatic master equation for -qubit quantum annealing examples. We also
apply the quantum trajectories method to a -qubit example originally
introduced to demonstrate the role of tunneling in quantum annealing, which is
significantly more time consuming to solve directly using the master equation.
The quantum trajectories method provides insight into individual quantum jump
trajectories and their statistics, thus shedding light on open system quantum
adiabatic evolution beyond the master equation.Comment: 17 pages, 7 figure
Adaptive Backstepping Controller Design for Stochastic Jump Systems
In this technical note, we improve the results in a paper by Shi et al., in which problems of stochastic stability and sliding mode control for a class of linear continuous-time systems with stochastic jumps were considered. However, the system considered is switching stochastically between different subsystems, the dynamics of the jump system can not stay on each sliding surface of subsystems forever, therefore, it is difficult to determine whether the closed-loop system is stochastically stable. In this technical note, the backstepping techniques are adopted to overcome the problem in a paper by Shi et al.. The resulting closed-loop system is bounded in probability. It has been shown that the adaptive control problem for the Markovian jump systems is solvable if a set of coupled linear matrix inequalities (LMIs) have solutions. A numerical example is given to show the potential of the proposed techniques
Delayed feedback control in quantum transport
Feedback control in quantum transport has been predicted to give rise to
several interesting effects, amongst them quantum state stabilisation and the
realisation of a mesoscopic Maxwell's daemon. These results were derived under
the assumption that control operations on the system be affected
instantaneously after the measurement of electronic jumps through it. In this
contribution I describe how to include a delay between detection and control
operation in the master equation theory of feedback-controlled quantum
transport. I investigate the consequences of delay for the state-stabilisation
and Maxwell's-daemon schemes. Furthermore, I describe how delay can be used as
a tool to probe coherent oscillations of electrons within a transport system
and how this formalism can be used to model finite detector bandwidth.Comment: 13 pages, 5 figure
Dual and backward SDE representation for optimal control of non-Markovian SDEs
We study optimal stochastic control problem for non-Markovian stochastic
differential equations (SDEs) where the drift, diffusion coefficients, and gain
functionals are path-dependent, and importantly we do not make any ellipticity
assumption on the SDE. We develop a controls randomization approach, and prove
that the value function can be reformulated under a family of dominated
measures on an enlarged filtered probability space. This value function is then
characterized by a backward SDE with nonpositive jumps under a single
probability measure, which can be viewed as a path-dependent version of the
Hamilton-Jacobi-Bellman equation, and an extension to expectation
Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model
We study a family of mean field games with a state variable evolving as a
multivariate jump diffusion process. The jump component is driven by a Poisson
process with a time-dependent intensity function. All coefficients, i.e. drift,
volatility and jump size, are controlled. Under fairly general conditions, we
establish existence of a solution in a relaxed version of the mean field game
and give conditions under which the optimal strategies are in fact Markovian,
hence extending to a jump-diffusion setting previous results established in
[30]. The proofs rely upon the notions of relaxed controls and martingale
problems. Finally, to complement the abstract existence results, we study a
simple illiquid inter-bank market model, where the banks can change their
reserves only at the jump times of some exogenous Poisson processes with a
common constant intensity, and provide some numerical results.Comment: 37 pages, 6 figure
Quantum trajectories and open many-body quantum systems
The study of open quantum systems has become increasingly important in the
past years, as the ability to control quantum coherence on a single particle
level has been developed in a wide variety of physical systems. In quantum
optics, the study of open systems goes well beyond understanding the breakdown
of quantum coherence. There, the coupling to the environment is sufficiently
well understood that it can be manipulated to drive the system into desired
quantum states, or to project the system onto known states via feedback in
quantum measurements. Many mathematical frameworks have been developed to
describe such systems, which for atomic, molecular, and optical (AMO) systems
generally provide a very accurate description of the open quantum system on a
microscopic level. In recent years, AMO systems including cold atomic and
molecular gases and trapped ions have been applied heavily to the study of
many-body physics, and it has become important to extend previous understanding
of open system dynamics in single- and few-body systems to this many-body
context. A key formalism that has already proven very useful in this context is
the quantum trajectories technique. This was developed as a numerical tool for
studying dynamics in open quantum systems, and falls within a broader framework
of continuous measurement theory as a way to understand the dynamics of large
classes of open quantum systems. We review the progress that has been made in
studying open many-body systems in the AMO context, focussing on the
application of ideas from quantum optics, and on the implementation and
applications of quantum trajectories methods. Control over dissipative
processes promises many further tools to prepare interesting and important
states in strongly interacting systems, including the realisation of parameter
regimes in quantum simulators that are inaccessible via current techniques.Comment: 66 pages, 29 figures, review article submitted to Advances in Physics
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