The non-Markovian dynamics of a three-level quantum system coupled to a bosonic environment is a difficult problem due to the lack of an exact dynamic equation such as a master equation. We present for the first time an exact quantum trajectory approach to a dissipative three-level model. We have established a convolutionless stochastic Schrödinger equation called time-local quantum state diffusion (QSD) equation without any approximations, in particular, without Markov approximation. Our exact time-local QSD equation opens a new avenue for exploring quantum dynamics for a higher dimensional quantum system coupled to a non-Markovian environment.PACS numbers: 42.50.Lc Every small quantum system, such as a two-level atom (qubit), a three-level atom (qutrit) or a cavity mode (simple harmonic oscillator), should be regarded as an open quantum system due to the inevitable interaction with its environment. The effect from the environment may bring about some grave problems, such as quantum decoherence and disentanglement, to the dynamics of the open system [1]. Much has been studied in the Markov limit where the dynamics of open quantum system is typically described by a standard Lindblad master equation which is equivalent to a quantum state diffusion (QSD) equation for a pure state [2][3][4][5][6][7] (setting = 1):where the notation ∆ t (A) ≡ A − A t for any operator A and A t ≡ ψ t |A|ψ t denotes the quantum average. L is the system operator coupled to the environment, often called Lindblad operator. The dynamics of the open system is driven by the classical stochastic process z t . The reduced density matrix for the system of interest can be recovered by averaging quantum trajectories generated by the QSD equation (1):Here M [·] denotes the ensemble average over the classical noise. Besides many appealing features exhibited by pure state quantum trajectories, Eq.(1) provides a very efficient numerical tool in solving quantum dynamics of Markov open systems. Non-Markovian environments have become increasingly important in recent times due to their relevance in explaining new experimental advances in high-Q microwave cavities, photonic crystals and atom laser in BEC [8][9][10][11][12][13][14]. It is also evident from the recent progress in quantum information processing, that environmental memory may be utilized to generate or modulate entanglement evolution of an open quantum system [15,16]. Clearly, an approach that is capable of dealing with nonMarkovian quantum system is highly desirable. A nonMarkovian QSD equation for a general non-Markovian open system by Diósi, Gisin and Strunz has provided a powerful tool in dealing with the exact dynamics of quantum open systems irrespective of the coupling strength, the correlation time and the spectral density of the bosonic environments [17][18][19][20][21][22][23][24]. Despite extensive research, the explicit non-Markovian QSD equations only exist for a single two-level system, the quantum Brownian motion model and optical cavities due to the intricate functional d...
Dynamical decoupling operations have been shown to reduce errors in quantum information processing. Leakage from an encoded subspace to the rest of the system space is a particularly serious problem for which leakage elimination operators (LEO) were introduced. These are a particular type of decoupling which are designed to eliminate such leakage errors. Here, we provide an analysis of non-ideal pulses, rather than the well-understood ideal pulses or bang-bang controls. We show that under realistic conditions for experiments these controls will provide protection from errors. Furthermore, we find that the effect of LEOs depends exclusively on the integral of the pulse sequence in the time domain with proper ratio of pulse duration time and its period. When these two key parameters are chosen within certain bounds, leakage errors of the open system (exemplified by a three-level system for the nitrogen-vacancy centers under external magnetic field) would be dramatically decreased. The results are illustrated by the fidelity dynamics of LEO sequences, ranging from regular rectangular pulses, random pulses and even disordered (noisy) pulses.Introduction.-Leakage from a subspace encoding a qubit into the larger Hilbert space of a system's Hilbert space is particularly damaging since it removes any benefit the encoding may provide. Leakage elimination operators (LEOs) were originally proposed to counteract the influence of leakage operators (denoted L) in a two-level system which encodes one logical qubit in a multi-level Hilbert space [1][2][3][4]. The leakage elimination was achieved by employing unbounded fast and strong "bang-bang" (BB) pulses [5] which apply to first order corrections of the evolution. In general, the total Hamiltonian for system and bath can be written aswhere the operator of type P represents the operations only acting within the qubit subspace, i.e., the subspace of interest. The operator of type Q has no effect on the qubit subspace because it acts only within the remaining subspace of the whole Hilbert space perpendicular to the subspace of P , and L represents the diffusion between the P -and Q-subspaces
The adiabatic theorem addresses the dynamics of a target instantaneous eigenstate of a time-dependent Hamiltonian. We use a Feshbach P-Q partitioning technique to derive a closed one-component integro-differential equation. The resultant equation properly traces the footprint of the target eigenstate. The physical significance of the derived dynamical equation is illustrated by both general analysis and concrete examples. We find an interesting phenomenon showing that a dephasing white noise can enhance and even induce adiabaticity. This phenomenon, distinguishing itself from any artificial control process, may occur in natural physical processes. We also show that particular white noises can shorten the total duration of dynamic processing, such as in adiabatic quantum computing.
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