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arXiv:2312.04394 · Physical Review A

Parametric amplification of a quantum pulse

Offek Tziperman, Victor Rueskov Christiansen, Ido E. Kaminer, Klaus Mølmer

Published Nov 20, 2024• 3 citations
arXivHTMLPDF

Abstract

Creating and manipulating quantum states of light requires nonlinear interactions. We present here a multimode theory for the transformation of an arbitrary quantum pulse by Hamiltonians that are quadratic in field creation and annihilation operators. We show, in particular, that any input quantum pulse will feed only one or two distinct output modes. Our result readily provides both the output modes and the quantum states after transformation of arbitrary input wave packets. While being central for applications in quantum information processing with traveling bosonic modes, e.g., in quantum networks, our theoretical method and its results are not anticipated by the conventional Bogoliubov diagonalization of the problem in many input and output eigenmodes.

DOI
10.1103/physreva.110.053712

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Creating and manipulating quantum states of light requires nonlinear interactions. We present here a multimode theory for the transformation of an arbitrary quantum pulse by Hamiltonians that are quadratic in field creation and annihilation operators. We show, in particular, that any input quantum pulse will feed only one or two distinct output modes. Our result readily provides both the output modes and the quantum states after transformation of arbitrary input wave packets. While being central for applications in quantum information processing with traveling bosonic modes, e.g., in quantum networks, our theoretical method and its results are not anticipated by the conventional Bogoliubov diagonalization of the problem in many input and output eigenmodes.
A single quantum pulse undergoing parametric amplification feeds into at most two pulses in the output. In this work, we present an efficient, analytical method for finding the quantum state of these output modes. Our method applies the amplification to the vacuum rather than to the input state, and subsequently applies a transformed version of the operator that creates the input state from vacuum. Given the input and output pulse mode functions, the method is analytical, and therefore computationally very efficient, and it can be readily generalized to multiple non-vacuum input modes. We exemplify the method by computing the output quantum states resulting from the input of a coherent, a Schrödinger cat, and a single photon input quantum state. We further employ the method to obtain the quantum state in one of the two output modes heralded upon detection of vacuum in the other, least populated, mode.
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.
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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.