Inverse Design of Quantum Holograms in Three-Dimensional Nonlinear Photonic Crystals
Eyal Rozenberg, Aviv Karnieli, Ofir Yesharim, Sivan Trajtenberg-Mills, Daniel Freedman, Alex M. Bronstein, Ady Arie
IInverse Design of Quantum Holograms inThree-Dimensional Nonlinear Photonic Crystals
Eyal Rozenberg , Aviv Karnieli , Ofir Yesharim , Sivan Trajtenberg-Mills , DanielFreedman , Alex M. Bronstein and Ady Arie Department of Computer Science, Technion, Haifa, Israel Google Research, Haifa, Israel Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Israel School of Electrical Engineering, Fleischman Faculty of Engineering, Tel Aviv University, [email protected]
Abstract:
We introduce a systematic approach for designing 3D nonlinear photoniccrystals and pump beams for generating desired quantum correlations between structuredphoton-pairs. Our model is fully differentiable, allowing accurate and efficient learning anddiscovery of novel designs. © 2021 The Author(s)Quantum optics has proven to be an invaluable resource for the realization of many quantum technologies, suchas quantum communication, computing and cryptography. A prominent reason for this is the availability of sourcesgenerating nonclassical light, mainly based on nonlinear interactions; the most prevalent of which is spontaneousparametric down-conversion (SPDC) in second order nonlinear χ ( ) materials [1]. Many of these schemes focuson creating high-dimensional quantum states by employing the spatial degrees of freedom of free-space modes,such as Laguerre-Gauss (LG) beams carrying orbital angular momentum (OAM) [2, 3, 4] or Hermite-Gauss (HG)beams [5, 6]. Shaping of quantum correlations of entangled photons has received much attention, owing to possibleapplication in quantum communication. However, their manipulation via the spatial mode of the pump beam isquite limited [3, 4]. Recent breakthroughs in the fabrication of three-dimensional nonlinear photonic crystals [7]offer a promising new avenue for shaping and controlling arbitrary quantum correlations through the design ofthree-dimensional χ ( ) holograms (see Fig.1.a). Fig. 1:
Inverse design of quantum holograms model . a) A nonlinear photonic crystal hosts a three-dimensional hologram, and the pump fieldcan be constructed from different spatial modes. The resulting down-converted shaped photon pairs display second-order quantum correlations.The inverse problem is defined by learning the crystal hologram and pump modes that result in the desired quantum correlations. b) Anillustration of the architecture of the model. The pump and crystal-hologram, E p , χ ( ) , are parameterized by ϑ , ϕ which are learned parametersof the model. The stochastic nature of the SPDC interaction is set by the quantum vacuum , defined by a non-differentiable random node. Theforward model simulates the light-matter interaction and results in the 1st and 2nd quantum correlations, G ( ) , G ( ) . Their discrepancy againsta desired correlation is measured via an appropriate loss function. The model is fully differentiable and is optimized using backpropagation and first order optimization method Adam . The task of finding the right hologram (modulation pattern of the crystal nonlinearity) to provide the desiredquantum correlations falls into a broad family of inverse problems in optics. Given a forward model which mapsa hologram to the quantum correlations it produces, a loss function may be defined which measures the discrep-ancy between the quantum correlations produced by the forward model and the desired correlations. A commonapproach to solving the inverse problem then involves minimizing the loss function, see Figure 1. Although SPDCis a quantum mechanical process, simulation tools (i.e., forward models) nevertheless exist and have been shownto agree with experimental results [8, 9, 10]. Unfortunately, SPDC simulations are stochastic in nature, whichmakes it challenging to integrate them directly into optimization-based approaches to the inverse problem. Morespecifically, such approaches require the ability to compute the derivative of the parameters being optimized withrespect to the loss function, which is rendered difficult due to the stochasticity. Recent attempts to overcome thisissue [3] were not gradient-based, and were only used to learn the pump modes, disregarding the crystal structure.In this work, we discover novel designs of three-dimensional quantum holograms in nonlinear photonic crystalsfor generating desired quantum correlations between structured photon pairs, for example those of high-ordermaximally-entangled OAM states - sought after for applications in quantum information. We obtain these designsby solving the quantum optical inverse problem (Fig.1) using efficient inference and learning in the directed SPDCprobabilistic model, by employing the reparameterization approach [11]. Our implementation allows the gradientsof the loss function to be directly calculated with respect to the physical parameters of the model, and to thereby beused in the optimization process. In contrast to previous realizations, we do not need to differentiate with respectto non-deterministic parameters.The results for learning desired quantum correlations of photon pairs in the LG basis are visualized in Fig.2.As a target for learning, we provide the algorithm with a desired quantum state of the two-photon probability dis- a r X i v : . [ qu a n t - ph ] F e b ribution derived from the second-order quantum correlation P ( l i , l s ) = G ( ) ( l i , l s , l s , l i ) = (cid:104) ψ | a † l i a † l s a l s a l i | ψ (cid:105) ; where | ψ (cid:105) denotes the quantum state, a ( a † ) denote the photon annihilation (creation) operators, and l i , l s denote theOAM quantum number of the idler and signal photons. We perform simulations for a 1mm long LiNbO crystal,and with a 532nm, 1mW CW pump. The process is type-II quasi-phase-matched for an on-axis generation ofpolarization-distinguishable photon pairs at 1064nm. First, we let the algorithm learn a three-dimensional holo-gram inducing the quantum correlations associated with OAM qudit states, with d = d = | ψ (cid:105) = ∑ dl = e i ϕ l | l , l (cid:105) / √ d . In addition, we also learn correlations of high-order OAM qubits of the form | ψ (cid:105) = ( | l , − l (cid:105) + e i ϕ |− l , l (cid:105) ) / √
2, with l = Fig. 2: Results of learning quantum correlations in the Laguerre-Gauss basis. a-c
Learned two-photon quantum correlations of a) qutrit( d = b) ququint ( d =
5) and c) high-order qubit ( l = d) Learned three-dimensional crystal pattern for the qubit case (panel c ) for twocross-sections in the y coordinate. (Crystal patterns producing a-b not shown due to space constraints.) Similarly to the LG case, we now turn to the simultaneous learning of both the pump and crystal patterns,which will allow us to achieve desired correlations in the HG basis. We let the algorithm learn the quantum cor-relations of a maximally-entangled HG ququad ( d =
4) of the form | ψ (cid:105) = / ( | HG (cid:105) | HG (cid:105) + | HG (cid:105) | HG (cid:105) + | HG (cid:105) | HG (cid:105) + | HG (cid:105) | HG (cid:105) ) . The results of this learning of pump and crystal structure are depicted in Fig. 3. Fig. 3: Results of learning quantum correlations in the Hermite-Gauss basis. a) Learned correlations of a ququad ( n s , n i are the quantum modenumbers of the signal and idler). b) Learned pump magnitude (a.u.). c) Learned crystal profile at z = d) Learned crystal pattern along z . To conclude, we have shown the inverse design of three-dimensional nonlinear photonic crystals and pumpbeam profiles for shaping quantum correlations between spatial modes of SPDC photon pairs, for several high-dimensional quantum states used in quantum information. The run time was about 15 minutes on an NVIDIATITAN Xp 12Gb GPU. We intend to extend our work to allow for learning of the full density matrix of a targetquantum state, as well as the longitudinal variation of the nonlinear crystal controlling the joint spectral amplitude.
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