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arXiv:1803.08327

Phonon as environmental disturbance in three level system

Dawit Hiluf Hailu, Yonatan Dubi

Published Mar 22, 2018• 2 citations
arXivHTMLPDF

Abstract

This work investigates the effect of phonon coupling on the transfer of population and creation of coherence using variant of stimulated Raman adiabatic passage (STIRAP) known as \emph{fractional} stimulated Raman adiabatic passage (FSTIRAP). The study is based on the Liouville equation, which is solved numerically in the adiabatic limit. Although the phonon is assumed to be coupled only to the intermediate state, it is coupled to the other two states by dipolar system-environment interaction, inducing phonon coupling to the other states which are not directly in contact with the phonon. At zero temperature the STIRAP pulse protocol's efficiency of the transfer decreases exponentially with the electron-phonon coupling, until the coupling strength is strong enough to make the process fully incoherent, in which case the population transfer is $\frac{1}{3}$ in each level. For the FSTIRAP protocol we find that the transferred population to target state decreases, leaving some population on the intermediate state. Consequently, there is an increase in the magnitude of the coherences $ρ_{01}, ρ_{12}$, albeit small. Furthermore population transfer for non-zero temperature and effect of coupling strength is investigated, it is observed that while both parameters negatively influence the efficiency of transfer the former decrease the transfer exponentially, thereby equilibrating the system fast, while the latter seen to decrease the transfer monotonically, and hence equilibrates slowly.

DOI
10.48550/arxiv.1803.08327

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This work investigates the effect of phonon coupling on the transfer of population and creation of coherence using variant of stimulated Raman adiabatic passage (STIRAP) known as \emph{fractional} stimulated Raman adiabatic passage (FSTIRAP). The study is based on the Liouville equation, which is solved numerically in the adiabatic limit. Although the phonon is assumed to be coupled only to the intermediate state, it is coupled to the other two states by dipolar system-environment interaction, inducing phonon coupling to the other states which are not directly in contact with the phonon. At zero temperature the STIRAP pulse protocol's efficiency of the transfer decreases exponentially with the electron-phonon coupling, until the coupling strength is strong enough to make the process fully incoherent, in which case the population transfer is 13\frac{1}{3}31​ in each level. For the FSTIRAP protocol we find that the transferred population to target state decreases, leaving some population on the intermediate state. Consequently, there is an increase in the magnitude of the coherences ρ01,ρ12ρ_{01}, ρ_{12}ρ01​,ρ12​, albeit small. Furthermore population transfer for non-zero temperature and effect of coupling strength is investigated, it is observed that while both parameters negatively influence the efficiency of transfer the former decrease the transfer exponentially, thereby equilibrating the system fast, while the latter seen to decrease the transfer monotonically, and hence equilibrates slowly.
This work explores the effect of phase relaxation on the population transfer efficiency in stimulated Raman adiabatic passage (STIRAP). The study is based on the Liouville equation, which is solved analytically in the adiabatic limit. The transfer efficiency of STIRAP is found to decrease exponentially with the dephasing rate; this effect is stronger for shorter pulse delays and weaker for larger delays, since the transition time is found to be inversely proportional to the pulse delay. Moreover, it is found that the transfer efficiency of STIRAP in the presence of dephasing does not depend on the peak Rabi frequencies at all, as long as they are sufficiently large to enforce adiabatic evolution; hence increasing the field intensity cannot reduce the dephasing losses. It is shown also that for any dephasing rate, the final populations of the initial state and the intermediate state are equal. For strong dephasing all three populations tend to 13\frac{1}{3}31​.
This work explores the effect of spontaneous emission on the population transfer efficiency in stimulated Raman adiabatic passage (STIRAP). The approach uses adiabatic elimination of weakly coupled density matrix elements in the Liouville equation, from which a very accurate analytic approximation is derived. The loss of population transfer efficiency is found to decrease exponentially with the factor ${\ensuremath{\Omega}}_{0}^{2}∕\ensuremath{\Gamma}$, where $\ensuremath{\Gamma}$ is the spontaneous emission rate and ${\ensuremath{\Omega}}_{0}$ is the peak Rabi frequency. The transfer efficiency increases with the pulse delay and reaches a steady value. For large pulse delay and large spontaneous emission rate STIRAP degenerates into optical pumping.
Stimulated Raman scattering involving adiabatic passage (STIRAP) is a proven technique for population transfer in three-level systems with strong oscillator strengths. We show that the STIRAP process should also yield high transfer efficiencies when the densities of states near the intermediate and final levels are high, provided certain criteria for the experimentally controllable parameters are met. We also show that high transfer efficiencies may even be possible when the pump and Stokes lasers can access levels outside of the three-level system. The criteria are derived with an approach that emphasizes the adiabatic nature of the transfer process in an eigenvector of the interaction Hamiltonian that resembles a trapped state. The results are compared to density matrix calculations.
Stimulated Raman adiabatic passage is a quantum protocol that can be used for robust state preparation in a three-level system. It has been commonly employed in quantum optics, but recently this technique has drawn attention also in circuit quantum electrodynamics. The protocol relies on two slowly varying drive pulses that couple the initial and the target state via an intermediate state, which remains unpopulated. Here we study the detrimental effect of the parasitic couplings of the drives into transitions other than those required by the protocol. The effect is most prominent in systems with almost harmonic energy level structure, such as the transmon. We show that under these conditions in the presence of decoherence there exists an optimal STIRAP amplitude for population transfer.
We study the process of stimulated Raman adiabatic passage (STIRAP) under the influence of a nontrivial solid-state environment, particularly the effect of two-level fluctuators (TLFs) as they are frequently present in solid-state devices. When the amplitudes of the driving pulses used in STIRAP are in resonance with the level spacing of the fluctuators, the quality of the protocol, i.e., the transferred population, decreases sharply. In general, the effect can not be reduced by speeding up the STIRAP process. We also discuss the effect of a structured noise environment on the process of coherent tunneling by adiabatic passage (CTAP). The effect of a weakly structured environment or TLFs with short coherence times on STIRAP and CTAP can be described by the Bloch-Redfield theory. For a strongly structured environment, a higher-dimensional approach must be used, where the TLFs are treated as part of the system.
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.