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arXiv:1209.6569 · Journal of Modern Optics

Raman transitions without adiabatic elimination: a simple and accurate treatment

Rui Han, Hui Khoon Ng, Berthold‐Georg Englert

Published Feb 1, 2013• 13 citations
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Abstract

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.

DOI
10.1080/09500340.2013.770573

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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.
In multi-level systems, the commonly used adiabatic elimination is a method for approximating the dynamics of the system by eliminating irrelevant, non-resonantly coupled levels. This procedure is, however, somewhat ambiguous and it is not clear how to improve on it systematically. We use an integro-differential equation for the probability amplitudes of the levels of interest, which is equivalent to the original Schrodinger equation for all probability amplitudes. In conjunction with a Markov approximation, the integro-differential equation is then used to generate a hierarchy of approximations, in which the zeroth order is the adiabatic-elimination approximation. It works well with a proper choice of interaction picture; the procedure suggests criteria for optimizing this choice. The first-order approximation in the hierarchy provides significant improvements over standard adiabatic elimination, without much increase in complexity, and is furthermore not so sensitive to the choice of interaction picture. We illustrate these points with several examples.
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.
Various approaches have been used in the literature for eliminating nonresonant levels in atomic systems and deriving effective Hamiltonians. Important among these are elimination techniques at the level of probability amplitudes, operator techniques to project the dynamics on to the subspace of resonant levels, Green’s function techniques, the James’ effective Hamiltonian approach, etc. None of the previous approaches is suitable for deriving effective Hamiltonians in intracavity situations. However, the James’ approach does work in the case of only two-photon transitions in a cavity. A generalization of the James’ approach works in the case of three-photon transitions in a cavity, but only under Raman-like resonant conditions. Another important approach for adiabatic elimination is based on an adaptation of the Markov approximation well-known in the theory of system–bath interactions. However, this approach has not been shown to work in intracavity situations. In this paper, we present a method of adiabatic elimination for atoms inside cavities in the presence of multiphoton transitions. We work in the Heisenberg picture, and our approach has the advantage that it allows one to derive effective Hamiltonians even when Raman-like resonance conditions do not hold.
This paper begins with the premise that stimulated Raman adiabatic passage (STIRAP) from one state to another by coupling to an intermediate state necessarily requires some population of that intermediate state. The usual view of the process depends on having no population in the intermediate state, but a simple argument from the time-dependent Schr\"odinger equation shows that such adiabatic transfer cannot occur. From another perspective, removing the time dependence of the Hamiltonian by making the rotating frame transformation cancels the optical frequency oscillations but does not account for the envelopes of the pulses commonly used for STIRAP. By assuming a small but finite intermediate state amplitude ${\ensuremath{\varepsilon}}_{1}$ in the wave function, the propagator for the evolution between initial and final states is derived. Applying the propagator to the initial state produces a single compact formula for the final intermediate-state population that is readily evaluated for an arbitrary choice of laser pulse parameters. Several plots for various experimental conditions show that under the conditions normally used for STIRAP experiments the effects are typically a few percent or less, but in the end this effect puts an ultimate limit on total efficiency.
The technique of stimulated Raman adiabatic passage (STIRAP) is a highly efficient tool for coherent population transfer in a chainwise-connected three-state quantum system. It is widely accepted that the two-photon resonance between the two end states of the chain is a crucial condition for STIRAP. This assumption is indeed correct when the two driving fields, pump and Stokes, produce nearly equal couplings for their respective transitions. We show analytically, supported by numeric examples, that when the pump and Stokes couplings differ significantly, the two-photon population transfer profile is distorted and its center is shifted away from the two-photon resonance; hence, the optimal operation of STIRAP demands a two-photon detuning. We derive simple analytic estimates for the center and the width of the two-photon profile. The results are of potential importance in situations when the pump and Stokes fields are of different physical natures (e.g., laser and cavity fields) or when two-photon resonance is unwanted (e.g., in the presence of residual light).