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arXiv:1708.01338 · Physical Review A

Photon-pair generation in a lossy microring resonator. II. Entanglement in the output mixed Gaussian squeezed state

Paul M. Alsing, Edwin E. Hach

Published Sep 27, 2017• 6 citations
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Abstract

In this work we examine the entanglement of the output signal-idler squeezed vacuum state in the Heisenberg picture as a function of the coupling and internal propagation loss parameters of a microring resonator. Using the log negativity as a measure of entanglement for a mixed Gaussian state, we examine the competitive effects of the transfer matrix that encodes the classical phenomenological loss, as well as the matrix that that incorporates the coupling and internal propagation losses due to the quantum Langevin noise fields required to preserve the unitarity of the composite system (signal-idler) and environment (noise) structure.

DOI
10.1103/physreva.96.033848

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In this work we examine the entanglement of the output signal-idler squeezed vacuum state in the Heisenberg picture as a function of the coupling and internal propagation loss parameters of a microring resonator. Using the log negativity as a measure of entanglement for a mixed Gaussian state, we examine the competitive effects of the transfer matrix that encodes the classical phenomenological loss, as well as the matrix that that incorporates the coupling and internal propagation losses due to the quantum Langevin noise fields required to preserve the unitarity of the composite system (signal-idler) and environment (noise) structure.
We investigate entangled photon-pair generation in a lossy microring resonator using an input-output formalism based on the work of M. G. Raymer and C. J. McKinstrie [Phys. Rev. A 88, 043819 (2013)] and P. M. Alsing et al. [Phys. Rev. A 95, 053828 (2017)] that incorporates circulation factors that account for the multiple round-trips of the fields within the cavity. We consider the nonlinear processes of spontaneous parametric down-conversion and spontaneous four-wave mixing, and we compute the generated biphoton signal-idler state from a single-bus microring resonator, along with the generation, coincidence-to-accidental, and heralding efficiency rates. We compare these generalized results to those obtained by previous works employing the standard Langevin input-output formalism.
Microring resonators (MRR) are becoming a standard resource for highly efficient entangled photon pair generation. We examine the nonlinear processes in MRR using an input-output formalism that explicitly accounts for the contribution of the round trip circulation of the fields inside the MRR. This formalism reduces to the standard Langevin operator input-output formalism for cavities in the high cavity Q limit, near cavity resonances. Our approach is more general, applicable for arbitrary internal and bus-ring coupling losses, and away from resonances. We compare/contrast these two formalisms, and apply our approach to the output quantum state resulting from spontaneous parametric down conversion and spontaneous four wave mixing within the MRR. We compare our results to similar calculations employing the standard IO formalism, and note the role of the commutators of the noise operators on the undesired accidental singles rate.
The intensity buildup of light inside a lossy microring resonator can be used to enhance the generation of squeezed states via spontaneous parametric downconversion (SPDC). In this work, we model the generation of squeezed light in a microring resonator that is pumped with a Gaussian pulse via a side-coupled channel waveguide. We theoretically determine the optimum pump pulse duration and ring-to-channel coupling constant to minimize the quadrature noise (maximize the squeezing) in the ring for a fixed input pump energy. We derive approximate analytic expressions for the optimal coupling and pump pulse duration as a function of scattering loss in the ring. These results will enable researchers to easily determine the optimal design of microring resonator systems for the generation of quadrature-squeezed states.
Entangled multi-mode squeezed states of light have a wide variety of applications in quantum information systems, particularly in the generation of non-Gaussian states of light, which are central to continuous-variable quantum computing. Although theoretical approaches exist to model the nonlinear generation of one- and two-mode entangled states of light in ring resonator systems, these approaches are difficult to implement in modeling more complicated many-cavity systems. In this work, we present an efficient and accurate theoretical approach to modeling the generation and propagation of quantum states of light in lossy coupled-cavity systems containing a two- or three-mode nonlinear resonant structure. Our approach is general and computationally viable even in systems with hundreds of modes. We apply our method to the design and modeling of a multimode photonic crystal coupled-cavity system for the generation of entangled squeezed states of light on-chip. The system consists of a three-mode resonant structure coupled to three coupled-resonator optical waveguides (CROWs) in a square lattice silicon photonic crystal slab. The computational speed of the method allows us to efficiently optimize the system such that the signal and idler light in the two output CROWs remains entangled even after propagating tens of cavities down the CROWs.
We explore how III-V semiconductor microring resonators can efficiently generate photon pairs and squeezed vacuum states via spontaneous parametric down-conversion by utilizing their built-in quasi-phase-matching and modal dispersion. We present an analytic expression for the biphoton wave function of photon pairs generated by weak pump pulses and characterize the squeezed states that result under stronger pumping conditions. Our model includes loss and captures the statistics of the scattered photons. A detailed sample calculation shows that for low pump power conversion efficiencies of ${10}^{\ensuremath{-}5}$, corresponding to a rate of 39 MHz for a pump power of $1\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{W}$, are attainable for rudimentary structures such as a simple microring coupled to a waveguide, in both the continuous-wave and pulsed-excitation regimes. Our results suggest that high levels of squeezing and pump depletion are attainable, possibly leading to the deterministic generation of non-Gaussian states.
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.