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arXiv:2311.03597

Using system-reservoir methods to derive effective field theories for broadband nonlinear quantum optics: a case study on cascaded quadratic nonlinearities

Chris Gustin, Ryotatsu Yanagimoto, Edwin Ng, Tatsuhiro Onodera, Hideo Mabuchi

Published Nov 6, 2023• 1 citation
arXivHTMLPDF

Abstract

In broadband quantum optical systems, nonlinear interactions among a large number of frequency components induce complex dynamics that may defy heuristic analysis. In this work we introduce a perturbative framework for factoring out reservoir degrees of freedom and establishing a concise effective model (effective field theory) for the remaining system. Our approach combines approximate diagonalization of judiciously partitioned subsystems with master equation techniques. We consider cascaded optical $χ^{(2)}$ (quadratic) nonlinearities as an example and show that the dynamics can be construed (to leading order) as self-phase modulations of dressed fundamental modes plus cross-phase modulations of dressed fundamental and second-harmonic modes. We then formally eliminate the second-harmonic degrees of freedom and identify emergent features of the fundamental wave dynamics, such as two-photon loss channels, and examine conditions for accuracy of the reduced model in dispersive and dissipative parameter regimes. Our results highlight the utility of system-reservoir methods for deriving accurate, intuitive reduced models for complex dynamics in broadband nonlinear quantum photonics.

DOI
10.48550/arxiv.2311.03597

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In broadband quantum optical systems, nonlinear interactions among a large number of frequency components induce complex dynamics that may defy heuristic analysis. In this work we introduce a perturbative framework for factoring out reservoir degrees of freedom and establishing a concise effective model (effective field theory) for the remaining system. Our approach combines approximate diagonalization of judiciously partitioned subsystems with master equation techniques. We consider cascaded optical χ(2)χ^{(2)}χ(2) (quadratic) nonlinearities as an example and show that the dynamics can be construed (to leading order) as self-phase modulations of dressed fundamental modes plus cross-phase modulations of dressed fundamental and second-harmonic modes. We then formally eliminate the second-harmonic degrees of freedom and identify emergent features of the fundamental wave dynamics, such as two-photon loss channels, and examine conditions for accuracy of the reduced model in dispersive and dissipative parameter regimes. Our results highlight the utility of system-reservoir methods for deriving accurate, intuitive reduced models for complex dynamics in broadband nonlinear quantum photonics.
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We introduce two approaches to modeling the quantum dynamics of optical χ(3)χ^{(3)}χ(3) solitons. Taking an open-system viewpoint, we project the underlying quantum field into system (soliton) and residual reservoir components. The reservoir is treated as either (i) a discrete ``Lanczos supermode'' (LSM) expansion which localizes dynamics to a few-supermode basis, or (ii) a non-local environment which can be traced out by deriving a Markovian master equation (ME). Using these methods, we analyze and identify the quantum structure of both the soliton's stability and its hierarchy of perturbations. Through numerical simulations, we confirm both methods effectively capture quantum-induced soliton phase shifts in a concise few-mode (single-mode for ME) basis, and the LSM approach also captures photon loss which arises from non-Markovian dispersive couplings. As neither method is limited to the linearized regime, our approaches provide powerful computational tools to analyze complex non-Gaussian quantum dynamics of solitons where other commonly-used methods fail, providing insight into such non-perturbative regimes. We also investigate radiation that occurs in the presence of higher-order dispersion with ultrashort pulses, deriving a ME that predicts photon loss consistent with classical theory, but find that both classical and ME theory dramatically underestimate the actual amount of dissipation, which we explain in terms of dispersive coupling-induced soliton broadening.
The theoretical description of broadband, multimode quantum pulses undergoing a second-order χ(2)χ^{(2)}χ(2)-nonlinear interaction can be quite intricate, due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples input states: A Fock state in a single broadband mode and a two-mode squeezed vacuum, both in the THz-frequency regime, up-converted to a single output broadband mode of optical frequencies. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast time scales.
The quantum features of ultrashort-pulse optical parametric oscillators (OPOs) are investigated theoretically in the nonlinear regime near and above threshold. Viewing the pulsed OPO as a multimode open quantum system, we rigorously derive a general input-output model that features nonlinear coupling among many cavity (i.e., system) signal modes and a broadband single-pass (i.e., reservoir) pump field. Under appropriate assumptions, our model produces a Lindblad master equation with multimode nonlinear Lindblad operators describing two-photon dissipation and a multimode four-wave-mixing Hamiltonian describing a broadband, dispersive optical cascade, which we show is required to preserve causality. To simplify the multimode complexity of the model, we employ a supermode decomposition to perform numerical simulations in the regime where the pulsed supermodes experience strong single-photon nonlinearity. We find that the quantum nonlinear dynamics induces pump depletion as well as corrections to the below-threshold squeezing spectrum predicted by linearized models. We also observe the formation of non-Gaussian states with Wigner-function negativity and show that the multimode interactions with the pump, both dissipative and dispersive, can act as effective decoherence channels. Finally, we briefly discuss some experimental considerations for potentially observing such quantum nonlinear phenomena with ultrashort-pulse OPOs on nonlinear nanophotonic platforms.
The dispersive regime of nnn-photon qubit-oscillator interactions is analyzed using Schrieffer-Wolff perturbation theory. Effective Hamiltonians are derived up to the second order in the perturbation parameters. These effective descriptions reveal higher-order qubit-oscillator cross-Kerr and oscillator self-Kerr terms. The cross-Kerr term combines a qubit Pauli operator with an nnn-degree polynomial in the oscillator photon number operator, while the self-Kerr term is an $(n\ensuremath{-}1)$-degree polynomial in the oscillator photon number operator. In addition to the higher-order Kerr terms, a qubit-conditional 2n2n2n-photon squeezing term appears in the effective non-rotating-wave-approximation Hamiltonian. Furthermore, perturbation theory is applied to the case of multiple qubits coupled to a shared oscillator. A photon-number-dependent qubit-qubit interaction emerges in this case, which can be leveraged to tune the effective multiqubit system parameters using the oscillator state. Results for the converse setup of multiple oscillators and a single qubit are also derived. In this case, a qubit-conditional oscillator-oscillator nonlinear interaction is found. The spectral instabilities plaguing multiphoton qubit-oscillator models are carefully treated by introducing stabilizing higher-order terms in the Hamiltonian. The stabilizing terms preserve low-photon subspaces, avoid negative infinite energies, and facilitate reliable numerical calculations used to validate analytical predictions. The effective descriptions developed here offer a simple and intuitive physical picture of dispersive multiphoton qubit-oscillator interactions that can aid in the design of implementations harnessing various nonlinear effects.
The simplest open, nonlinear optical system consists of a single bosonic mode with nonlinear self-interactions coupled to driving fields and reservoirs. In this chapter, we will examine a number of systems of this type using the methods that have been developed in the preceding chapters. Systems of this type can exhibit a variety of interesting physical effects. For example, in the case of a driven nonlinear oscillator, there can be more than one steady state (bistability), and quantum statistical effects can manifest themselves by affecting the stability of these states. Other quantum statistical effects have also been predicted in a variety of related systems – such as photon anti-bunching, squeezing and changes to spectra. In general, the size of these effects scales inversely with the size of the system. This ‘system size’ refers to a threshold photon number, a number of atoms, or some similar quantity. The nature of the steady states themselves can become less than straightforward. Nonequilibrium dissipative systems, of which the systems considered in this chapter are examples, have parameters that describe the energy input to the system, and their steady states depend on these parameters. When driven far from equilibrium, such systems can exhibit bifurcations in their steady states. In more complex cases, this eventually leads to periodic oscillations and chaos. Devices like this can be realized with multiple types of nonlinearity, ranging from nonlinear dielectrics through to cavity QED (with near-resonant atoms), cavity optomechanics (with nanomechanical oscillators) and circuit QED (with superconducting Josephson junctions). We will start with cases where all the interactions are with the reservoirs, which may be linear or nonlinear, while the cavity itself is harmonic in its response.