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Journal of Physics B Atomic Molecular and Optical Physics

Adiabatic elimination of a nearly resonant quantum state

Boyan T. Torosov, Nikolay V. Vitanov

Published Jun 13, 2012• 14 citations
Publisher

Abstract

Quantum systems with multiple states are often reduced to simpler systems by adiabatic elimination of far-off-resonant states. This method provides accurate coupling and Stark shifts in the reduced system for very large values of the detunings only, far greater than that of the relevant couplings. We introduce two alternative techniques for exclusion of an off-resonant state based on the adiabatic and superadiabatic approximations, which provide far more accurate expressions for the coupling and the Stark shifts in the reduced system, and which are valid in much broader ranges of interaction parameters than the traditional adiabatic elimination method. The reason is that the conditions for the adiabatic and superadiabatic approximations only demand, for smooth pulse shapes, the detuning to be greater than the Fourier width of the pulse; hence, these approximations can be used even when the couplings exceed the detunings, i.e. when the excluded states can be described as 'nearly resonant'. The error of the adiabatic and superadiabatic approximations can be easily suppressed below the quantum computing benchmark of 10 −4 , an objective that is hard to achieve with the traditional adiabatic elimination method.

DOI
10.1088/0953-4075/45/13/135502

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Quantum systems with multiple states are often reduced to simpler systems by adiabatic elimination of far-off-resonant states. This method provides accurate coupling and Stark shifts in the reduced system for very large values of the detunings only, far greater than that of the relevant couplings. We introduce two alternative techniques for exclusion of an off-resonant state based on the adiabatic and superadiabatic approximations, which provide far more accurate expressions for the coupling and the Stark shifts in the reduced system, and which are valid in much broader ranges of interaction parameters than the traditional adiabatic elimination method. The reason is that the conditions for the adiabatic and superadiabatic approximations only demand, for smooth pulse shapes, the detuning to be greater than the Fourier width of the pulse; hence, these approximations can be used even when the couplings exceed the detunings, i.e. when the excluded states can be described as 'nearly resonant'. The error of the adiabatic and superadiabatic approximations can be easily suppressed below the quantum computing benchmark of 10 −4 , an objective that is hard to achieve with the traditional adiabatic elimination method.
Far-off-resonant pulsed laser fields produce negligible excitation between two atomic states but may induce considerable phase shifts. The acquired phases are usually calculated by using the adiabatic-elimination approximation. We analyze the accuracy of this approximation and derive the conditions for its applicability to the calculation of the phases. We account for various sources of imperfections, ranging from higher terms in the adiabatic-elimination expansion and irreversible population loss to couplings to additional states. We find that as far as the phase shifts are concerned, the adiabatic elimination is accurate only for a very large detuning. We show that the adiabatic approximation is a far more accurate method for evaluating the phase shifts, with a vast domain of validity; the accuracy is further enhanced by superadiabatic corrections, which reduce the error well below ${10}^{\ensuremath{-}4}$. Moreover, owing to the effect of adiabatic population return, the adiabatic and superadiabatic approximations allow one to calculate the phase shifts even for a moderately large detuning and even when the peak Rabi frequency is larger than the detuning; in these regimes the adiabatic elimination is completely inapplicable. We also derive several exact expressions for the phases using exactly soluble two-state and three-state analytical models.
Driven Raman processes – nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population – are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann–Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.
Driven Raman processes — nearly resonant two-photon transitions through an intermediate state that is non-resonantly coupled and does not acquire a sizeable population — are commonly treated with a simplified description in which the intermediate state is removed by adiabatic elimination. While the adiabatic-elimination approximation is reliable when the detuning of the intermediate state is quite large, it cannot be trusted in other situations, and it does not allow one to estimate the population in the eliminated state. We introduce an alternative method that keeps all states in the description, without increasing the complexity by much. An integro-differential equation of Lippmann-Schwinger type generates a hierarchy of approximations, but very accurate results are already obtained in the lowest order.
Abstract We consider adiabatic interaction of quantum systems with electromagnetic field in the presence of various dissipation processes. As it is known that in the presence of large intermediate detunings any n-level system can be reduced to an effective two-level system, we chose the two-level model as the basic model for a detailed analytical study. We demonstrate the possibility of reducing losses due to dephasing and non-adiabatic corrections by choosing an appropriate design of time-dependent interaction parameters. Simple analytical expressions are derived for both cold and hot atomic ensembles. The results obtained for the two-level system are applied to a three-level system by using the method of adiabatic elimination. Efficient population transfer is shown despite the relatively high dephasing rates.
Adiabatic elimination methods allow the reduction of the space dimension needed to describe systems whose dynamics exhibit separation of timescale. For open quantum systems, it consists in eliminating the fast part assuming it has almost instantaneously reached its steady state and obtaining an approximation of the evolution of the slow part. These methods can be applied to eliminate a linear subspace within the system Hilbert space or, alternatively, to eliminate a fast subsystem in a bipartite quantum system. In this work, we extend an adiabatic elimination method used for removing fast degrees of freedom within a system [Phys. Rev. A 101, 042102 (2020)] to eliminate a subsystem from an open bipartite quantum system. As an illustration, we apply our technique to a dispersively coupled two-qubit system and in the case of the open Rabi model.
Adiabatic process has found many important applications in modern physics, the distinct merit of which is that it does not need accurate control over the timing of the process. However, it is a slow process so that unwanted transitions will appear, which limits the application in quantum computation, due to the limited coherent times of quantum systems. Here, we propose a scheme to implement superadiabatic quantum state conversion in opto-electro-mechanical systems, where the process can be greatly speeded up while the precise timing control is still not necessary. In our scheme, only by modifying the coupling strength, we can achieve near perfect and fast quantum state conversion, where the adiabatic condition does not need to be met. In addition, the population of intermediate unwanted state can be further suppressed. Therefore, our protocol presents an important step towards practical state conversion between optical and microwave photons, and thus may find many important applications in hybrid quantum information processing.