Time-dependent adiabatic elimination in matter-wave optics

Abstract

We show how the dynamics of a specific subset of states can be separated from the dynamic of the total quantum state via a time-dependent projector-based formalism of adiabatic elimination. Within our formalism, we assume explicit time dependency in the coupling between both subsystems. Additionally, we do not assume that the elements of the Hamiltonian commute, as in matter-wave optics this not given in general. Here the center-of-mass degrees of freedom frequently need to be taken into account. Our formalism allows to perform the adiabatic elimination in such a setting.

Figures (4)

• Figure 1: Comparison of methods for the adiabatic elimination of two ancilla states in a five level system. (a), (b) and (c) shows the population of the three states in the relevant states $\ket{\text{g}}$, $\ket{\text{m}}$ and $\ket{\text{e}}$ respectively. In black the numerical solution of the time evolution, in blue the Hamiltonian derived in Ref. Paulisch2014, Eq. \ref{['eq:Hamiltonian_Paulisch']}, in green the Hamiltonian derived in Ref. Sanz2016, Eq. \ref{['eq:Hamiltonain_Sanz']} and in red our Hamiltonian, Eq. \ref{['eq:Hamiltonian_Sam']}. (d) shows the relative error to the numerical solution. The exact parameters and the Hamiltonian can be found in Appendix \ref{['sec:five_level']} together with the formula for the relative error.
• Figure 2: (a) Level scheme of a $\Lambda$-system with two lasers for Raman diffraction (b) Level scheme of a four-level system with two counter propagating lasers for Double Raman diffraction. (c) Level scheme of a two-level system with two lasers for Bragg diffraction. (d) Inverted $\Lambda$-system with four counter propagating lasers for Double Bragg diffraction. Note, that all systems are far detuned form the transition of the ground state manifold to the ancillas.
• Figure 3: Raman $\pi$-pulse for the $\text{D}_2$-Line of ${}^{87}\text{Rb}$. a) shows the pulse shape of the laser in form of a sine squared pulse. b) and c) show the momentum density plot of a box wave function of the ground state in b) and the momentum shifted excited state in c). Additionally we plotted the evolution of the resonant momentum in d). These momenta are marked with the white dashed line in the density plot. For the $\pi$-pulse where the total pulse area is given by $\pi$ the population is fully inverted as long as the momentum is resonant. In the density plot the effect of velocity selectivity can clearly be seen. The parts of the wave function that are Doppler detuned from the transition oscillate with another frequency and we do not get a full population inversion. The horizontal axis is the effective pulse area that was covered by the pulse so far and ends here with $A = \pi$. The vertical axis shows the Doppler frequency $\nu$ in units of the effective Rabi frequency.
• Figure 4: Raman $\pi/2$-pulse for the $\text{D}_2$-Line of ${}^{87}\text{Rb}$. a) shows the pulse shape of the laser in form of a sine squared pulse. b) and c) show the momentum density plot of a box wave function of the ground state in b) and the momentum shifted excited state in c). Additionally we plotted the evolution of the resonant momentum in d). These momenta are marked with the white dashed line in the density plot. For the $\pi/2$-pulse where the total pulse area is given by $\pi/2$ the population is brought into an equal superposition for the resonant momenta. In the density plot the effect of velocity selectivity can be observed. The horizontal axis is the effective pulse area that was covered by the pulse so far and ends here with $A = \pi/2$. The vertical axis shows the Doppler frequency $\nu$ in units of the effective Rabi frequency.