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Journal of Physics Communications

Exponentially complex nonseparable states in planar arrays of nonlinearly coupled one-dimensional elastic waveguides

Pierre Alix Deymier, Keith Runge, M. Arif Hasan

Published Aug 20, 2020• 6 citations
Publisher

Abstract

Externally driven parallel arrays of elastic waveguides inelastically coupled along their length are shown to support inelastic modes that span exponentially complex Hilbert spaces. The nonlinear coupling takes the form of power functions of the relative displacement between waveguides. First order perturbation theory is employed to obtained closed form solutions for the contribution of nonlinearity to the displacement field in the form of nonseparable superpositions of product states of plane waves. When the system is dissipative, the coefficients of the superposition of product states are complex and can be varied by controlling the characteristics of the way the array is driven externally. The entropy of 'entanglement' of these nonseparable superpositions of product states is calculated from the complex amplitudes through the density matrix. Navigation of the exponentially complex Hilbert space of product states enables manipulation of the degree of nonseparability of the superpositions.

DOI
10.1088/2399-6528/abb0f0

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Externally driven parallel arrays of elastic waveguides inelastically coupled along their length are shown to support inelastic modes that span exponentially complex Hilbert spaces. The nonlinear coupling takes the form of power functions of the relative displacement between waveguides. First order perturbation theory is employed to obtained closed form solutions for the contribution of nonlinearity to the displacement field in the form of nonseparable superpositions of product states of plane waves. When the system is dissipative, the coefficients of the superposition of product states are complex and can be varied by controlling the characteristics of the way the array is driven externally. The entropy of 'entanglement' of these nonseparable superpositions of product states is calculated from the complex amplitudes through the density matrix. Navigation of the exponentially complex Hilbert space of product states enables manipulation of the degree of nonseparability of the superpositions.
We show that the directional projection of longitudinal waves propagating in a parallel array of N elastically coupled waveguides can be described by a nonlinear Dirac-like equation in a 2 N dimensional exponential space.This space spans the tensor product Hilbert space of the two-dimensional subspaces of N uncoupled waveguides grounded elastically to a rigid substrate (called φ-bits).The superposition of directional states of a φ-bit is analogous to that of a quantum spin.We can construct tensor product states of the elastically coupled system that are nonseparable on the basis of tensor product states of N φ-bits.We propose a system of coupled waveguides in a ring configuration that supports these nonseparable states.
Abstract We introduce a formalism that enables the calculation of elastic wave functions supported by parallel arrays of coupled one-dimensional elastic waveguides. These wave functions are expressed as tensor products of a spinor part associated with directional degrees of freedom and an orbital angular momentum (OAM) part associated with the phase of the coupled waveguides. We demonstrate that one can construct wave functions as a superposition of these elastic waves, which cannot be written as a tensor product of a spinor part and an OAM part. These elastic wave functions are not separable in the tensor product Hilbert space of directional and OAM subspaces. We show that we can construct maximally nonseparable states that are similar to Bell states.
We experimentally navigate the Hilbert space of two logical phi-bits supported by an externally driven nonlinear array of coupled acoustic waveguides by parametrically changing the relative phase of the drivers. We observe sharp phase jumps of approximately 180° in the individual phi-bit states as a result of the phase tuning of the drivers. The occurrence of these sharp phase jumps varies from phi-bit to phi-bit. All phi-bit phases also possess a common background dependency on the drivers’ phase. Within the context of multiple time scale perturbation theory, we develop a simple model of the nonlinear array of externally driven coupled acoustic waveguides to shed light on the possible mechanisms for the experimentally observed behavior of the logical phi-bit phase. Finally, we illustrate the ability to experimentally initialize the state of single- and multiple- phi-bit systems by exploiting the drivers’ phase as a tuning parameter. We also show that the nonlinear correlation between phi-bits enables parallelism in the manipulation of two- and multi-phi-bit superpositions of states.
Atom arrays are a new type of quantum light-matter interface.Here we propose to employ one-dimensional ordered arrays as atomic waveguides.These arrays support optical guided modes that do not decay into free space.We show that these modes can be harnessed to mediate tunable, long-range interactions between additional "impurity qubits" coupled to the chain, without need for photonic structures.The efficient coupling between qubits and atomic waveguides enables the realization of tunable qubit-qubit interactions, which can be short or long range, dissipative or coherent, as well as chiral.Moreover, owing to the two-level nature of atoms, these waveguides are intrinsically quantum.In contrast to classical waveguides, where photons do not interact with each other, atomic waveguides display strong nonlinearities, which create a tunable dissipative channel for qubit-qubit interactions, and opens the door to the exploration of many-body physics between guided photons.This physics is universal as it only relies on photon interference and can also be observed with other types of quantum emitters, such as those in molecular or solid-state systems.
The arrays of nonlinear waveguides are a powerful integrated photonics platform for studying and manipulating quantum states of light. Also, they are a valuable resource for various quantum technologies. In this work, we employed a supermode approach to obtain an analytic solution to the evolution of degenerate biphoton states in arrays of nonlinear waveguides. The solution accounts for arrays of an arbitrary number of waveguides and coupling profiles. In addition, it provides an explicit analytic expression without the need of computationally-expensive steps. Actually, it only relies on the calculation of the eigenvalues and eigenvectors of the coupling matrix. In general, this needs to be performed numerically, but there are relevant instances in which analytic expressions are available. Thus, in certain cases, the procedure proposed here does not require the use of any numerical method. We analyze the general properties of the solution and show some application examples. Particularly, simple solutions that can be obtained by special symmetric injection profiles, the results for small arrays, and the properties of the propagation when only the center waveguide is pumped in a symmetric odd array. In addition, we present a proof-of-principle example on how to use the analytic solution to tackle inversion problems. That is, obtaining the initial conditions required to achieve a desired quantum state -- which is a valuable technological application. A relevant aspect of the method presented here is its scalability for large arrays due to the lack of resource-intensive steps -- both for the direct and inverse problems.
We study a driven-dissipative array of coupled nonlinear optical resonators by numerically solving the von Neumann equation for the density matrix. We demonstrate that quantum correlated states of many photons can also be generated in the limit where the nonlinearity is much smaller than the losses, contrary to common expectations. Quantum correlations in this case arise from the interference between different pathways that the system can follow in the Hilbert space to reach its steady state under the effect of coherent driving fields. We characterize, in particular, two systems: a linear chain of three coupled cavities and an array of eight coupled cavities. We demonstrate the existence of a parameter range where the system emits photons with continuous-variable bipartite and quadripartite entanglement, in the case of the first and the second system, respectively. This entanglement is shown to survive realistic rates of pure dephasing and opens up a new perspective for the realization of quantum simulators or entangled photon sources without the challenging requirement of strong optical nonlinearities.