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arXiv:1806.01958 · Physical review. B./Physical review. B

Few-photon scattering and emission from low-dimensional quantum systems

Rahul Trivedi, Kevin A. Fischer, Shanshan Xu, Shanhui Fan, Jelena Vuckovic A

Published Oct 29, 2018• 50 citations
arXivHTMLPDF

Abstract

We show how to use the input-output formalism to compute the propagator for a low-dimensional quantum system coupled to an optical field under the rotating wave and Markovian approximation. The propagator for this coupled system is expressed in terms of the Green's functions of the low-dimensional system, and it is shown that these Green's functions can be computed entirely from the evolution of the low-dimensional system under an effective non-Hermitian Hamiltonian. Our formalism generalizes the previous works that have focused on time-independent Hamiltonians to systems with time-dependent Hamiltonians (e.g., coherently driven systems), making it a suitable computational tool for the analysis of a number of experimentally interesting quantum systems. We illustrate our formalism by applying it to analyze photon emission and scattering from the driven and undriven two-level system and the three-level lambda system.

DOI
10.1103/physrevb.98.144112

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We show how to use the input-output formalism to compute the propagator for a low-dimensional quantum system coupled to an optical field under the rotating wave and Markovian approximation. The propagator for this coupled system is expressed in terms of the Green's functions of the low-dimensional system, and it is shown that these Green's functions can be computed entirely from the evolution of the low-dimensional system under an effective non-Hermitian Hamiltonian. Our formalism generalizes the previous works that have focused on time-independent Hamiltonians to systems with time-dependent Hamiltonians (e.g., coherently driven systems), making it a suitable computational tool for the analysis of a number of experimentally interesting quantum systems. We illustrate our formalism by applying it to analyze photon emission and scattering from the driven and undriven two-level system and the three-level lambda system.
We provide a systematic treatment of NNN-photon transport in a waveguide coupled to a local system using the input-output formalism. The main result of the paper is a general connection between the NNN-photon S matrix and the Green's functions of the local system. We also show that the computation can be significantly simplified by exploiting the connectedness structure of both the S matrix and the Green's function, and by computing the Green's function using an effective Hamiltonian that involves only the degrees of freedom of the local system. We illustrate our formalism by computing NNN-photon transport through a cavity containing a medium with Kerr nonlinearity, with NNN up to 3.
We develop a new computational tool and framework for characterizing the scattering of photons by energy-nonconserving Hamiltonians into unidirectional (chiral) waveguides, for example, with coherent pulsed excitation. The temporal waveguide modes are a natural basis for characterizing scattering in quantum optics, and afford a powerful technique based on a coarse discretization of time. This overcomes limitations imposed by singularities in the waveguide-system coupling. Moreover, the integrated discretized equations can be faithfully converted to a continuous-time result by taking the appropriate limit. This approach provides a complete solution to the scattered photon field in the waveguide, and can also be used to track system-waveguide entanglement during evolution. We further develop a direct connection between quantum measurement theory and evolution of the scattered field, demonstrating the correspondence between quantum trajectories and the scattered photon state. Our method is most applicable when the number of photons scattered is known to be small, i.e. for a single-photon or photon-pair source. We illustrate two examples: analytical solutions for short laser pulses scattering off a two-level system and numerically exact solutions for short laser pulses scattering off a spontaneous parametric downconversion (SPDC) or spontaneous four-wave mixing (SFWM) source. Finally, we note that our technique can easily be extended to systems with multiple ground states and generalized scattering problems with both finite photon number input and coherent state drive, potentially enhancing the understanding of, e.g., light-matter entanglement and photon phase gates.
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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.
A set of coupled non-Markovian equations of motion is derived for the Green's function of a driven two-level atom. These equations can be expressed in terms of the density-matrix elements only in the Markovian limit, in which case the set reduces to optical Bloch equations. The non-Markovian effect arises from correlations between successive photon emissions, an effect which is neglected in the existing equations of motion. The resulting correction is of the order of ${\ensuremath{\gamma}}_{0}$/${\ensuremath{\Omega}}_{0}$ in the strong-field region (${\ensuremath{\gamma}}_{0}$${\ensuremath{\Omega}}_{0}$) and it vanishes in the weak-field region, where ${\ensuremath{\gamma}}_{0}$ is the Einstein A coefficient and ${\ensuremath{\Omega}}_{0}$ is the Rabi frequency.
We develop a quantum input-output theory for open photonic systems based on macroscopic quantum electrodynamics using the first-order electromagnetic Green's function. Quantum fields enter and leave the photonic system through waveguide ports, while the electromagnetic Green's function propagates the fields through the arbitrary interior of the photonic system, which may be dispersive and absorbing. The resulting relation contains a port-to-port scattering matrix and a Langevin-noise contribution from material absorption that together preserve the bosonic output commutation relations. For embedded emitters, the low-saturation and single-excitation scattering response is determined by Green's functions connecting the ports and emitters while retaining the non-Markovianity of the photonic environment. Using finite-difference time-domain simulation, we calculate the emitter-modified transmission for a nanophotonic cavity and inverse-designed coupler, providing a direct route from computed electromagnetic response to a quantum input-output model.