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Physical Review A

Pathway dynamics in the optimal quantum control of rubidium: Cooperation and competition

Fang Gao, Roberto Rey-de-Castro, Ashley Donovan, Jian Xu, Yaoxiong Wang, HERSCHEL A. RABITZ, Feng Shuang

Published Feb 13, 2014• 13 citations
Publisher

Abstract

The dynamics that take place in the optimal quantum control of atomic rubidium upon population transfer from state 5${S}_{1/2}$ to state 5${D}_{3/2}$ are investigated with Hamiltonian-encoding--observable-decoding (HE-OD). For modest laser powers two second-order pathways, 5${S}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}$5${P}_{3/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}$5${D}_{3/2}$ (pathway 1) and 5${S}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}$5${P}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}$5${D}_{3/2}$ (pathway 2), govern the population transfer process. Pathway 1 has larger transition dipoles than pathway 2. However, state 5${P}_{3/2}$ along pathway 1 may also be excited to an undesired state 5${D}_{5/2}$, which can result in population ``leakage.'' Thus, the two pathways may either cooperate or compete with each other in various dynamical regimes. An important feature in the case of cooperation is that the ratio between the amplitudes of pathways 1 and 2 oscillates over time with a frequency equal to the detuning between transitions 5${S}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}$5${P}_{3/2}$ and 5${P}_{3/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}$5${D}_{3/2}$. We also study the regime in which pathway 2 dominates the dynamics when the larger transition dipoles of pathway 1 can no longer compensate for its population leakage. The overall analysis illustrates the utility of HE-OD as a tool to reveal the quantum control mechanism.

DOI
10.1103/physreva.89.023416

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The dynamics that take place in the optimal quantum control of atomic rubidium upon population transfer from state 5S1/2{S}_{1/2}S1/2​ to state 5D3/2{D}_{3/2}D3/2​ are investigated with Hamiltonian-encoding--observable-decoding (HE-OD). For modest laser powers two second-order pathways, 5{S}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}5{P}_{3/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}5D3/2{D}_{3/2}D3/2​ (pathway 1) and 5{S}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}5{P}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}5D3/2{D}_{3/2}D3/2​ (pathway 2), govern the population transfer process. Pathway 1 has larger transition dipoles than pathway 2. However, state 5P3/2{P}_{3/2}P3/2​ along pathway 1 may also be excited to an undesired state 5D5/2{D}_{5/2}D5/2​, which can result in population ``leakage.'' Thus, the two pathways may either cooperate or compete with each other in various dynamical regimes. An important feature in the case of cooperation is that the ratio between the amplitudes of pathways 1 and 2 oscillates over time with a frequency equal to the detuning between transitions 5{S}_{1/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}5P3/2{P}_{3/2}P3/2​ and 5{P}_{3/2}$$\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}5D3/2{D}_{3/2}D3/2​. We also study the regime in which pathway 2 dominates the dynamics when the larger transition dipoles of pathway 1 can no longer compensate for its population leakage. The overall analysis illustrates the utility of HE-OD as a tool to reveal the quantum control mechanism.
There is an increasing interest in the extraction and control of the interfering quantum pathway amplitudes induced by control fields during laser-matter interactions. The Hamiltonian-encoding and observable-decoding (HE-OD) technique has been introduced for extracting the amplitudes of the pathways present in the dynamics and has recently been experimentally applied to the pathway manipulation of atomic rubidium. This paper theoretically explores various strategies for manipulating pathway amplitudes in the context of a laser field interacting with a multilevel system similar to atomic rubidium for both narrow-band and broadband ultrafast fields. In the perturbation regime, two second-order quantum pathways connecting the Rb states 5S1/25{S}_{1/2}5S1/2​ and 5D3/25{D}_{3/2}5D3/2​ dominate the dynamics, namely, $5{S}_{1/2}\ensuremath{\rightarrow}5{P}_{3/2}\ensuremath{\rightarrow}5{D}_{3/2}$ (pathway 1) and $5{S}_{1/2}\ensuremath{\rightarrow}5{P}_{1/2}\ensuremath{\rightarrow}5{D}_{3/2}$ (pathway 2). For narrow-band field control, the analysis is carried out in the time domain with the laser field including only four narrow-band envelope subpulses centered at the resonant frequencies. When the two pathways cooperate constructively, temporal oscillations appear in the ratio of the two pathway amplitudes, and we conclude in this case that the period corresponds to the detuning between transitions $5{S}_{1/2}\ensuremath{\rightarrow}5{P}_{3/2}$ and $5{P}_{3/2}\ensuremath{\rightarrow}5{D}_{3/2}$. For broadband field control, the dynamics are treated in the frequency domain with the laser field including both resonant and continuous nonresonant frequency components. Various control strategies based on manipulating the phase of selected spectral components are tested. Compared to the outcome from a transform limited pulse, a $\frac{\ensuremath{\pi}}{2}$ step scheme can increase the dynamic range of the ratio between the two pathway amplitudes by a factor of $\ensuremath{\sim}3$. A scheme that manipulates eight spectral blocks, in which the spectral boundaries depend on the resonant frequencies, can increase the ratio by several orders of magnitude. Numerical simulations show that further dividing the spectrum into hundreds of evenly spaced blocks does not significantly enhance the pathway ratio over the eight-block scheme. The quantum control of pathways investigated in this work provides valuable insights on how to incorporate known information about the structure of quantum systems for the effective reduction of quantum control complexity.
The expectation value of a quantum system observable can be written as a sum over interfering pathway amplitudes. In this Letter, we demonstrate for the fist time adaptive manipulation of quantum pathways using the Hamiltonian encoding-observable decoding (HE-OD) technique. The principles of HE-OD are illustrated for population transfer in atomic rubidium using shaped femtosecond laser pulses. The ability to manipulate multiple pathway amplitudes is of fundamental importance in all quantum control applications.
We report an experimental investigation of the excitation-transfer collision Cs(5D5/2{D}_{5/2}D5/2​)+Cs(6S)\ensuremath{\rightarrow}CS ((5D3/2{D}_{3/2}D3/2​)+Cs(6S). The upper 5D5/2{D}_{5/2}D5/2​ state was excited by a cw dye laser tuned to the one-photon, quadrupole-allowed 6S\ensuremath{\rightarrow}5D5/2{D}_{5/2}D5/2​ transition. Since the direct 5D\ensuremath{\rightarrow}6P fluorescence could not be detected with our apparatus, we monitored instead the cascade 6P\ensuremath{\rightarrow}6S fluorescence. The ratio of 6P1/2{P}_{1/2}P1/2​ to 6P3/2{P}_{3/2}P3/2​ fluorescence contains information on the collisional mixing that takes place in the 5D levels but also includes a significant contribution from mixing in the 6P levels. This latter contribution could effectively be subtracted out using the results of a second experiment in which a tunable cw diode laser was used to pump the 6P3/2{P}_{3/2}P3/2​ state, and the same fluorescence ratio monitored. The 5D mixing cross section we obtain, 70 A˚2{\r{}}^{2}˚2, is significantly larger than previous indirect determinations.