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Journal of the Optical Society of America B

General model of counterpropagating continuous-variable entangled states in lossy waveguides

Hossein Seifoory, Marc M. Dignam

Published May 31, 2022• 1 citation
Publisher

Abstract

We present a general theoretical approach to model an integrated source of counterpropagating continuous-variable entangled states in lossy waveguides pumped by a classical pulsed source incident from above the waveguide. We use a backward Heisenberg approach to model the generation of the entangled state and then solve the adjoint master equation to model the propagation of the state in the lossy waveguide. We employ a numerical method to implement the Schmidt decomposition method for the biphoton wave function rather than pursuing analytical methods. This approach allows us to model a wide variety of waveguide systems and pump configurations. We apply our model to the nonlinear generation and propagation of continuous-variable entangled states in a coupled resonator optical waveguide under a variety of different pump conditions and derive the optimal pumping conditions for our system.

DOI
10.1364/josab.458074

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We present a general theoretical approach to model an integrated source of counterpropagating continuous-variable entangled states in lossy waveguides pumped by a classical pulsed source incident from above the waveguide. We use a backward Heisenberg approach to model the generation of the entangled state and then solve the adjoint master equation to model the propagation of the state in the lossy waveguide. We employ a numerical method to implement the Schmidt decomposition method for the biphoton wave function rather than pursuing analytical methods. This approach allows us to model a wide variety of waveguide systems and pump configurations. We apply our model to the nonlinear generation and propagation of continuous-variable entangled states in a coupled resonator optical waveguide under a variety of different pump conditions and derive the optimal pumping conditions for our system.
We present a general approach to model an integrated source of counterpropagating continuous-variable entangled states based on a coupled-resonator optical waveguide that is pumped by a classical pulsed source incident from above the waveguide. This paper is an extension of our previous work~(Ref. \cite{PhysRevA.100.033839}), where we analytically investigated the generation and propagation of continues-variable entangled states in this coupled-cavity system in the presence of intrinsic loss. However, in this work, we employ a numerical method to implement the Schmidt decomposition method rather than pursuing analytical methods. We show that not only this gives us a much higher degree of freedom in choosing the pumping parameters which were not possible to investigate analytically, but also it enables us to go beyond some of the approximations we had made to derive analytical expressions before.
We present an integrated source of counterpropagating entangled states based on a coupled resonator optical waveguide that is pumped by a classical pulsed source incident from above the waveguide. We investigate theoretically the generation and propagation of continuous-variable entangled states in this coupled-cavity system in the presence of intrinsic loss. Using a tight-binding approximation, we derive analytic time-dependent expressions for the number of photons in each cavity, as well as for the correlation variance between the photons in different pairs of cavities, to evaluate the degree of quantum entanglement. We also derive simple approximate expressions for these quantities that can be used to guide the design of such systems, and discuss how pumping configurations and physical properties of the system affect the photon statistics and the degree of quantum correlation.
Continuous-variable (CV) entanglement is a valuable resource in the field of quantum information. One source of CV entanglement is the correlations between the position and momentum of photons in a two-mode squeezed state of light. In this paper, we theoretically study the generation of squeezed states, via spontaneous parametric down conversion, inside a two-mode lossy cavity that is pumped with a classical optical pulse. The dynamics of the density operator in the cavity is modeled using the Lindblad master equation, and we show that the solution to this model is the density operator for a two-mode squeezed thermal state, with a time-dependent squeezing amplitude and average thermal photon number for each mode. We derive an expression for the minimum correlation variance inside the cavity that depends crucially on the difference in the losses between the two modes. We apply our analytic solution to the important example of a microring resonator that is pumped with a Gaussian pulse. The expressions that we derive will help researchers optimize CV entanglement in lossy cavities.
Encoding continuous-variable quantum information in the optical domain has recently enabled the generation of large entangled states, yet robust implementation remains a challenge. Here, we present a straightforward protocol for generating multipartite entanglement based on spontaneous parametric down-conversion in a circular array of quadratic nonlinear waveguides. We provide a rigorous theoretical framework, including comprehensive derivations of the propagation equations and the identification of regimes where analytical solutions are possible. Crucially, our approach identifies the pump and detection configurations required to sustain and measure multipartite full inseparability across arbitrary propagation distances and for any number of waveguides N=4nN=4 nN=4n. This regime, elusive to standard numerical methods, represents a key requirement for scalable quantum protocols. Our scheme is inherently robust as it relies on phase-matched propagation eigenmodes, making it resilient against variations in sample length, coupling, and nonlinearity.
Two-mode squeezed states have potential applications in quantum teleportation, quantum computation, and quantum information. Spontaneous parametric down conversion (SPDC) is one of the processes that can be used to generate two-mode squeezed optical states, both in bulk and integrated systems. In this work, we theoretically model SPDC in a vertically-pumped, lossy, coupled-resonator optical waveguide (CROW), shown in Fig. 1(a), to generate counterpropagating continuous variable (CV) entangled states. For a CROW in which the cavities are identical, using the nearest neighbour TB approximation, the CROW mode dispersion is given by ω̃Fk≈ ω̃F[1+ β̃1cos(kD)] ≡ ωFk-iγFk,, where ω̃Fis the individual-cavity complex mode frequency, β̃1is the complex coupling parameter, D is the periodicity of the CROW, and k is the Bloch vector. We consider the case of a pump that is Gaussian in time and space, with spatial and temporal full width at half maxima of ΔrFMHM= 2 √{l n2}/σ+and ΔtFMHM= 2 √{l n2}/τ/(σ_D), respectively, where σ+and σ_ are the k-space width parameters (see below), and τ = Re {1 / (ω̃Fβ̃1)} is the minimum time for the light to propagate one period.