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arXiv:1911.04194 · Journal of the Optical Society of America B

From a posteriori to a priori solutions for a two-level system interacting with a single-photon wavepacket

Anita Magdalena Dąbrowska

Published Mar 5, 2020• 11 citations
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Abstract

We present the analytical formulas for the conditional and unconditional states of a two-level atom interacting with a single-photon wavepacket. We express thea prioristate of the system by means of the quantum trajectories related to the process of detection of photons in the output field. We give the formulas for the mean number of photons detected up to the given time, and we derive the expressions for the mean time of detection of the photons.

DOI
10.1364/josab.383561

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We present the analytical formulas for the conditional and unconditional states of a two-level atom interacting with a single-photon wavepacket. We express thea prioristate of the system by means of the quantum trajectories related to the process of detection of photons in the output field. We give the formulas for the mean number of photons detected up to the given time, and we derive the expressions for the mean time of detection of the photons.
In this paper, photon-counting statistics for an output field of a single-photon wave packet interacting with a quantum system was derived. The problem was formulated using a collision model with a bidirectional field approximated by two sequences of qubits. It was shown that the collision model formulated for the light with the temporal correlations allows deriving formulae for quantum trajectories in an effective and intuitive way. Starting from the discrete in-time evolution of the composed system, we determined the conditional and unconditional evolution of the quantum system, provided a physical interpretation of quantum trajectories, and determined the analytical formulae for the exclusive probability densities for the reflected and transmitted fields. Finally, our results were applied to a two-level atom, and we also determined analytical expressions for probability densities of arrival times of the successive counts.
The excitation of atomic and molecular systems by propagating light in a two-photon state within the Wigner-Weisskopf approximation has been described using stochastic tools. The problem of a stochastic evolution of the quantum system, depending on the results of the measurement of the output field, was formulated and solved making use of the model of repeated interactions and measurement. We defined the discrete in-time interaction between the quantum system and its environment being the electromagnetic field approximated by a chain or chains of harmonic oscillators. We determined analytical formulae for quantum trajectories associated with one-dimensional and two-dimensional counting processes, corresponding respectively to unidirectional or bidirectional input field prepared in the two-photon states. We derived the formulae for the exclusive probability densities of photon counts that allow us to completely characterize the photon statistics of the output field. Finally, we showed how to apply the quantum trajectories to obtain the formula for the probability of the two-photon absorption for a three-level atom in a ladder configuration. The paper also includes a discussion on the optimal two-photon state that maximizes the two-photon absorption probability.
We study the interaction of a V-type atom in a cavity with incident single- and two-photon wavepackets and derive an exact formula, valid in all parameter regimes, relating the spectrum of the outgoing wavepackets to the incident one. We present detailed results for several special input pulses and consider the potential performance of the system as a CPHASE gate for initial pulses in a product state. We find values of the cavity, atomic, and pulse parameters that yield a conditional phase shift of π, albeit with a relatively small overlap between the incoming and outgoing pulse forms.
Abstract The excitation of atomic and molecular systems by propagating light in a two-photon state within the Wigner–Weisskopf approximation has been described using stochastic tools. The problem of a stochastic evolution of the quantum system, depending on the results of the measurement of the output field, was formulated and solved making use of the model of repeated interactions and measurement. We defined the discrete in-time interaction between the quantum system and its environment being the electromagnetic field approximated by a chain or chains of harmonic oscillators. We determined analytical formulae for quantum trajectories associated with one-dimensional and two-dimensional counting processes, corresponding respectively to unidirectional or bidirectional input field prepared in the two-photon states. We derived the formulae for the exclusive probability densities of photon counts that allow us to completely characterize the photon statistics of the output field. Finally, we showed how to apply the quantum trajectories to obtain the formula for the probability of the two-photon absorption for a three-level atom in a ladder configuration. The paper also includes a discussion on the optimal two-photon state that maximizes the two-photon absorption probability.
We present a full multimode treatment of the interaction of the quantized radiation field with a single two-level atom in a one-dimensional waveguide configuration. Starting with an incident pulse consisting of an arbitrary (finite) number of photons in a general initial state, we derive the equations of motion and a closed-form expression for the shape of the pulse after its interaction with the atom. We then specialize our results to the two-photon case where a number of analytical results can be derived, for both unidirectional and bidirectional systems. We study the effects of different pulse shapes, the manifestations of the entangled, so-called bound state of the two photons, single- and two-photon detection probabilities, and provide simple approximate results for the strong-coupling (or adiabatic, or long-pulse) regime. We also discuss the requirements for a true unidirectional setup, and the application of such a setup to photon sorting proposed by Witthaut, Lukin, and S\o{}rensen [Europhys. Lett. 97, 50007 (2012)].
We study how the transient excitation probability of a two-level atom by a quantized field depends on the temporal profile of the incident pulse, in the presence of external losses, for both coherent and Fock states, and in two complementary limits: when the pulse contains only one photon (on average), and when the number of photons NNN is large. For the latter case we derive analytical expressions for the scaling of the excitation probability with NNN that can be easily evaluated for any pulse shape.