Particle emission from open quantum systems

Kevin A. Fischer, Rahul Trivedi, and Daniil Lukin

Phys. Rev. A 98, 023853 – Published 28 August 2018

Abstract

In this work, we discuss connections between different theoretical physics communities and their works, all related to systems that act as sources of particles such as photons, phonons, or electrons. Our interest is to understand how a low-dimensional quantum system driven by coherent fields, e.g., a two-level system, a Jaynes-Cummings system, or a photon pair source driven by a laser pulse, emits photons into a waveguide. Of particular relevance to solid-state sources is that we provide a way to include dissipation into the formalism for temporal-mode quantum optics. We will discuss the connections between temporal-mode quantum optics, scattering matrices, quantum stochastic calculus, continuous matrix product states and operators, and very traditional quantum optical concepts such as the Mandel photon counting formula and the Lindblad form of the quantum-optical master equation. We close with an example of how our formalism relates to single-photon sources with dephasing.

Received 13 March 2018
Revised 30 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.023853

Physics Subject Headings (PhySH)

Photoemission Quantum field theory Quantum optics Scattering amplitudes Scattering theory Single photon sources

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalParticles & Fields

Authors & Affiliations

Kevin A. Fischer^*, Rahul Trivedi, and Daniil Lukin

E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA

^*kevinf@stanford.edu

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Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
The general problem we solve in this paper is to compute the field scattered into a unidirectional (chiral) 1- $d$ field (waveguide) from an energy-nonconserving 0- $d$ Hamiltonian. This class of Hamiltonian is often used to represent coherent laser pulses scattering off quantum-optical systems such as a two-level system, a Jaynes-Cummings system, or an entangled photon pair source. First, we discuss just a single waveguide (a) and later extend to the same system coupled to a bath of modes that induce dissipation (b).
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Figure 2
General relationship between concepts in the theory of particle emission into a waveguide. Starting from the initial state of $| e^{'}, 0 〉$ , which represents the system beginning in a local eigenstate $| e^{'} 〉$ and the waveguide in vacuum, the arrows indicate increased embedding or calculation based on the previous object. Although this diagram is drawn for the case of just a single waveguide, the relationships can easily be extended to include the additional loss channels of the bath [as in Fig. 1] by changing ${tr}_{sys} \to {tr}_{sys} {tr}_{bath}$ and ${tr}_{wg} \to {tr}_{wg} {tr}_{bath}$ (the Kraus sum technique is also required). The dotted lines indicate that two concepts are connected via specific types of techniques discussed in the paper. Blue indicates the contributions of this work.
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Figure 3
Examples of local quantum systems as particle sources, with their $(L, H)$ tuples, corresponding non-Hermitian effective Hamiltonians, and relevant propagators for photon emission. (a) Two-level system, driven by a laser pulse. Because there is no dissipation, the scattered photonic state can be written and solved either as a pure state $| Ψ 〉$ or as a density matrix $χ$ . (b) Two-level system dispersively coupled to a bath, causing dephasing, and driven by a laser pulse. (c) Three-level cascade, driven by a laser pulse, where only emission from the transition $| i 〉 \to | g 〉$ is of interest. Due to the dissipation for (b) and (c), the photonic state can only be a density matrix $χ$ . All laser pulses are resonant with the $| g 〉 \leftrightarrow | e 〉$ transitions in (a)–(c).
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Figure 4
Photon emission from a two-level system with dephasing under excitation by a short laser pulse [shown schematically in Fig. 3]. The laser pulse has a temporal width $τ_{p} = 0.1 / γ_{0}$ that injects approximately one quanta of energy into the photon field, $P_{1} = 0.95$ . Trace purity $P$ of a single-photon component of emission (solid green curve) and Hong-Ou-Mandel (HOM) interference parameter $v$ (solid blue curve) are shown.
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