Article

# Engineering a Kerr-Based Deterministic Cubic Phase Gate via Gaussian Operations

Authors:
• NTT Research and Cornell University
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## Abstract

We propose a deterministic, measurement-free implementation of a cubic phase gate for continuous- variable quantum information processing. In our scheme, the applications of displacement and squeezing operations allow us to engineer the effective evolution of the quantum state propagating through an optical Kerr nonlinearity. Under appropriate conditions, we show that the input state evolves according to a cubic phase Hamiltonian, and we find that the cubic phase gate error decreases inverse quartically with the amount of quadrature squeezing, even in the presence of linear loss. We also show how our scheme can be adapted to deterministically generate a nonclassical approximate cubic phase state with high fidelity using a ratio of native nonlinearity to linear loss of only 1e−4, indicating that our approach may be experimentally viable in the near term even on all-optical platforms, e.g., using quantum solitons in pulsed nonlinear nanophotonics.

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... 22 The fact that some of these very same systems can additionally support ultra-short pulse operation via advanced dispersion engineering, 23 then, gives significant weight to the potential of coherently generating and manipulating non-Gaussian states of light in these platforms within the near future. 24 To fully leverage the technological capabilities of current and future quantum optical devices, however, we need to overcome major theoretical and modeling difficulties. In the strongly interacting regime of broadband optics, both the multimode and the "discrete" nature of photons have to be properly captured, rendering naïve quantum models prohibitively expensive numerically. ...
... Furthermore, established approximations for the physics of quantum Kerr solitons, such as time-dependent Hartree-Fock, 85 suggest it may be possible to view the solitonic supermode as experiencing single-mode Kerr dynamics in phase space, which would allow an initial coherent-state soliton to evolve into a non-classical state. 24,86,87 As shown in Fig. 2(E), our exact quantum simulations of this system using MPS indeed produces a highly non-classical state with Wigner function negativities, 88 a manifestation of coherent non-Gaussian dynamics. However, we find structural derivations from pure Kerr dynamics in the phase-space portrait, as well as loss of purity due to the entanglement of the soliton with higher-order supermodes. ...
... This refinement in our understanding of quantum Kerr soliton dynamics could prove useful in harnessing these ultrafast coherent dynamics as high-bandwidth, all-optical resources for quantum information and engineering. 24 ...
Preprint
The advent of dispersion-engineered and highly nonlinear nanophotonics is expected to open up an all-optical path towards the strong-interaction regime of quantum optics by combining high transverse field confinement with ultra-short-pulse operation. Obtaining a full understanding of photon dynamics in such broadband devices, however, poses major challenges in the modeling and simulation of multimode non-Gaussian quantum physics, highlighting the need for sophisticated reduced models that facilitate efficient numerical study while providing useful physical insight. In this manuscript, we review our recent efforts in modeling broadband optical systems at varying levels of abstraction and generality, ranging from multimode extensions of quantum input-output theory for sync-pumped oscillators to the development of numerical methods based on a field-theoretic description of nonlinear waveguides. We expect our work not only to guide ongoing theoretical and experimental efforts towards next-generation quantum devices but also to uncover essential physics of broadband quantum photonics.
... Whereas GKP codeword states have recently been produced experimentally [24,35], the generation of a cubic phase state has proven elusive thus far, despite the considerable theoretical [22,[36][37][38][39][40] as well as experimental [41] effort. In this work, we provide viable solutions for the generation of a cubic phase state, exploiting a family of non-Gaussian Wigner-negative states that have recently been generated experimentally. ...
... As an additional remark, it is interesting to compare our conversion protocol to the probabilistic synthesis protocols in Ref. [39], aiming at generating a cubic phase state starting by means of tunable optical circuits with optimized parameters, and the deterministic protocol in Ref. [40]. In these protocols, the non-Gaussian element is respectively provided by the measurement (photon-number resolving detector) and by the nonlinear medium (self-Kerr effect). ...
... Since a similar analysis has shown to yield deceivingly small gate errors when the state onto which the gate is applied is a coherent state or a displaced squeezed state, following Ref. [40] we use instead a hard instance of an arbitrary state, namely a GKP state in the encoded |+ state, which is expressed in the 10. Sketch of the simplified gate-teleportation gadget to implement a cubic gate on an input GKP state |+ L using an ancillary state |ψ c . ...
Article
Full-text available
Universal quantum computing with continuous variables requires non-Gaussian resources, in addition to a Gaussian set of operations. A known resource enabling universal quantum computation is the cubic phase state, a non-Gaussian state whose experimental implementation has so far remained elusive. In this paper, we introduce two Gaussian conversion protocols that allow for the conversion of a non-Gaussian state that has been achieved experimentally, namely the trisqueezed state [Chang et al., Phys. Rev. X 10, 011011 (2020)], to a cubic phase state. The first protocol is deterministic and it involves active (inline) squeezing, achieving large fidelities that saturate the bound for deterministic Gaussian protocols. The second protocol is probabilistic and it involves an auxiliary squeezed state, thus removing the necessity of inline squeezing but still maintaining significant success probabilities and fidelities even larger than for the deterministic case. The success of these protocols provides strong evidence for using trisqueezed states as resources for universal quantum computation.
... In Sec. V we examine another approximate method which relies on using squeezing and displacement operations to assemble a single mode cubic-phase gate starting from a Kerr gate [31]. Finally we provide a comparison of the methods as well as some discussion in Sec. ...
... As opposed to the cubic phase gate (5d) which is the only non-Gaussian operator in the universal set, providing the non-linearity needed to allow for decompositions of arbitrarily higher order. This gate can be realized either by using optical nonlinearities [31] or measurements [32][33][34][35][36][37]. It can also be realized in the microwave domain using the Josephson nonlinearity [38]. ...
... In this section we discuss some recent ideas on how to decompose more complicated gates in which one mixes in the generators of position and momentum of the same mode. The method was introduced by Yanagimoto et al. [31]. Both squeezing and displacement are operations which can be implemented with linear optics. ...
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We gather and examine in detail gate decomposition techniques for continuous-variable quantum computers and also introduce some new techniques which expand on these methods. Both exact and approximate decomposition methods are studied and gate counts are compared for some common operations. While each having distinct advantages, we find that exact decompositions have lower gate counts whereas approximate techniques can cover decompositions for all continuous-variable operations but require significant circuit depth for a modest precision.
... In this context, the use of ultra-short pulses is an attractive technological option [17][18][19][20]. High peak intensities significantly reduce the average pump powers needed to generate non-classical light [21,22], mitigating the need for high-Q, low-bandwidth resonators ubiquitous in continuous-wave devices. Furthermore, the multimode nature of pulses enables manipulation of the quantum state of light over a high-dimensional set of spectraltemporal channels, e.g., via temporal pulse-shaping [23][24][25]. ...
... As indicated even by linear analysis of pulsed squeezing, photons in a broadband pulse experience complicated spectral-temporal entanglement [35,36], for which we only have analytic treatments in the regime of weak nonlinearities [37]. Multimode effects are also well known to cause difficulties for the realization of pulsed quantum logic gates [38], and careful dispersion control and pulse shaping techniques have been developed in various contexts to realize (quasi-)singlemode gate operations [20,22,23]. But in the mesoscopic regime, any similar attempts to harness ultrafast quantum coherence face a significant modeling problem: The combination of non-Gaussian features with multimode entanglement produce complex nonlinear dynamics that often require exponentially large Hilbert-space dimension to describe in conventional models [39,40]. ...
... Such an operating regime is of primary interest from a technological perspective because it is squarely on the near-term horizon of ongoing efforts to increase power efficiency in nonlinear nanophotonics [72][73][74][75]. In addition, a mesoscopic system where non-Gaussian quantum states exist alongside significant mean-field/Gaussian excitations can provide an excellent platform for quantum photonics as multi-photon interactions can effectively enhance material nonlinearities [22,[76][77][78]. ...
Preprint
We study the emergence of non-Gaussian quantum features in pulsed squeezed light generation with a mesoscopic number (i.e., dozens to hundreds) of pump photons. Due to the strong optical nonlinearities necessarily involved in this regime, squeezing occurs alongside significant pump depletion, compromising the predictions made by conventional semiclassical models for squeezing. Furthermore, nonlinear interactions among multiple frequency modes render the system dynamics exponentially intractable in na\"ive quantum models, requiring a more sophisticated modeling framework. To this end, we construct a nonlinear Gaussian approximation to the squeezing dynamics, defining a "Gaussian interaction frame" (GIF) in which non-Gaussian quantum dynamics can be isolated and concisely described using a few dominant (i.e., principal) supermodes. Numerical simulations of our model reveal non-Gaussian distortions of squeezing in the mesoscopic regime, largely associated with signal-pump entanglement. We argue that the state of the art in nonlinear nanophotonics is quickly approaching this regime, providing an all-optical platform for experimental studies of the semiclassical-to-quantum transition in a rich paradigm of coherent, multimode nonlinear dynamics. Mesoscopic pulsed squeezing thus provides an intriguing case study of the rapid rise in dynamic complexity associated with semiclassical-to-quantum crossover, which we view as a correlate of the emergence of new information-processing capacities in the quantum regime.
... Recent efforts in the field of nonlinear nanophotonics have resulted in the development of ultra-low-loss and highly efficient platforms for nonlinear optics [7,8], with experimental numbers coming remarkably close to bridging the long-standing gap between classical optics and the "strong interaction regime" of quantum optics with single-photon-level nonlinearity [9][10][11][12][13]. In particular, advances in dispersion engineering on these platforms enable ultra-short-pulse operation [14,15], where the available peak power further leverages the material nonlinearities by orders of magnitude, bringing the possibility of engineering highly nonclassical states of light into the foreseeable future [16]. In the presence of such strong nonlinearities, the quantum nature of individual photons plays a critical role in the physical behavior of these systems [17], i.e., classical mean-field theories and semiclassical approximations are no longer valid in predicting the results of experiments [18,19]. ...
... The main feature of the approximate TDHF solution is that it is closed within the subspace S of the soliton supermode f (sech) , so that, e.g., the reduced density matrix of |Ψ(t) in the subspace S has unit purity throughout the dynamics of (26). Nevertheless, due to the Kerr-type nonlinear phase shifts, (26) deviates from a coherent state as it evolves, leading to a variety of interesting phase-space dynamics [16,45,64,65]. On the other hand, it is difficult to quantify the accuracy or regime of validity of the TDHF due to its non-perturbative nature, and phasespace dynamics of quantum solitons beyond TDHF in the few-photon regime remain largely unexplored. ...
Preprint
Ultra-short pulses propagating in nonlinear nanophotonic waveguides can simultaneously leverage both temporal and spatial field confinement, promising a route towards single-photon nonlinearities in an all-photonic platform. In this multimode quantum regime, however, faithful numerical simulations of pulse dynamics naïvely require a representation of the state in an exponentially large Hilbert space. Here, we employ a time-domain, matrix product state (MPS) representation to enable efficient simulations by exploiting the entanglement structure of the system. In order to extract physical insight from these simulations, we develop an algorithm to unravel the MPS quantum state into constituent temporal supermodes, enabling, e.g., access to the phase-space portraits of arbitrary pulse waveforms. As a demonstration, we perform exact numerical simulations of a Kerr soliton in the quantum regime. We observe the development of non-classical Wigner-function negativity in the solitonic mode as well as quantum corrections to the semiclassical dynamics of the pulse. A similar analysis of χ (2) simultons reveals a unique entanglement structure between the fundamental and second harmonic. Our approach is also readily compatible with quantum trajectory theory, allowing full quantum treatment of propagation loss and decoherence. We expect this work to establish the MPS technique as part of a unified engineering framework for the emerging field of broadband quantum photonics.
... The argument above shows that any experiment that can implement Gaussian transformations and a cubic phase gate can in principle generate any arbitrary non-Gaussian state. Even though many protocols have been proposed to experimentally realise a cubic phase gate [79][80][81][82][83], any convincing implementations have yet to be demonstrated. One of the key problems is that experimental imperfections and finite squeezing are detrimental for the most commonly proposed methods [84]. ...
Preprint
Full-text available
Gaussian states have played on important role in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states. We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems. We then use this framework in various ways to introduce extra structure in the state space. We explain how non-Gaussian states can be characterised not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank. For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations. To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state. Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states. On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations. Finally, we will demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage. At the end of the Tutorial, we also provide an overview of several experimental realisations of non-Gaussian quantum states in quantum optics and beyond.
... Let us now move towards generating states with the nonlinear squeezing. At first we consider states with cubic nonlinearity [15,18,29,63]. Such states can be, in the idealized scenario, generated by applying a unitary cubic nonlinear operation given by Hamiltonianˆ=ˆ3 onto a Gaussian squeezed state. ...
Preprint
Quantum states with nonlinear squeezing are a necessary resource for deterministic implementation of high-order quadrature phase gates that are, in turn, sufficient for advanced quantum information processing. We demonstrate that this class of states can be deterministically prepared by employing a single Kerr gate accompanied by suitable Gaussian processing. The required Kerr coupling depends on the energy of the initial system and can be made arbitrarily small. We also employ numerical simulations to analyze the effects of imperfections and to show to which extent can they be neglected.
... Therefore it is of paramount importance to include some non-Gaussian elements to achieve quantum computational advantages [12,13]. Non-Gaussianity can be achieved through measurements such as photon-number-resolved detection (PNRD) [14,15], non-Gaussian gates such as the cubic phase gate [16,17], or by the inclusion of * rnehra@caltech.edu † me3nq@virginia.edu ...
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Continuous-variable quantum information processing through quantum optics offers a promising platform for building the next generation of scalable fault-tolerant information processors. To achieve quantum computational advantages and fault tolerance, non-Gaussian resources are essential. In this work, we propose and analyze a method to generate a variety of non-Gaussian states using coherent photon subtraction from a two-mode squeezed state followed by photon-number-resolving measurements. The proposed method offers a promising way to generate rotation-symmetric states conventionally used for quantum error correction with binomial codes and truncated Schr\"{o}dinger cat codes. We consider the deleterious effects of experimental imperfections such as detection inefficiencies and losses in the state engineering protocol. Our method can be readily implemented with current quantum photonic technologies.
... where we refer to the parameter c as the cubicity. Due to its fundamental role in quantum computation over continuous variables, various theoretical proposals have been put forward to generate such a state [10,17,35,[49][50][51][52][53][54][55][56][57][58], and recently a cubic-phase state was implemented experimentally in microvawe cavities [41]. To chose relevant parameters, we use the Wigner logarithmic negativity [31,32] as a guide, such that the negativity of our target cubic-phase state is comparable to the one of the other states investigated in this work. ...
... Chief among the applications driving the pursuit for high-Q=V cavities is quantum information. Within the past year alone, numerous proposals [93][94][95][96][97] have explored the feasibility of photonic microcavity-based quantum gates using strong photon-photon interactions mediated by bulk material nonlinearities. Driven by recent developments in high-Q=V microcavities [54,98] and thin film nonlinear optical materials, current experiments are approaching 1% [99] of this so-called "qubit limit of cavity nonlinear optics" [57] where single-photon nonlinearities outpace cavity losses. ...
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We present a joint theoretical and experimental analysis of thermorefractive noise in high-quality-factor (Q), small-mode-volume (V) optical microcavities. Analogous to well-studied stability limits imposed by Brownian motion in macroscopic Fabry-Perot resonators, we show that microcavity thermorefractive noise gives rise to a mode-volume-dependent maximum effective quality factor. State-of-the-art fabricated microcavities are found to be within one order of magnitude of this bound. By measuring the first thermodynamically limited frequency noise spectra of wavelength-scale high-Q/V silicon photonic crystal cavities, we confirm the assumptions of our theory, demonstrate a broadband sub-μK/Hz temperature sensitivity, and unveil a new technique for discerning subwavelength changes in microcavity mode volumes. To illustrate the immediate implications of these results, we show that thermorefractive noise limits the optimal performance of recently proposed room-temperature, all-optical qubits using cavity-enhanced bulk material nonlinearities. Looking forward, we propose and analyze coherent thermo-optic noise cancellation as one potential avenue toward violating these bounds, thereby enabling continued development in quantum optical measurement, precision sensing, and low-noise integrated photonics.
... Therefore it is of paramount importance to include some non-Gaussian elements to achieve quantum computational advantages [12,13]. Non-Gaussianity can be achieved through measurements such as photon-number-resolved detection (PNRD) [14,15], non-Gaussian gates such as the cubic phase gate [16,17], or by the inclusion of * rnehra@caltech.edu † me3nq@virginia.edu ...
Article
Full-text available
Continuous-variable quantum information processing through quantum optics offers a promising platform for building the next generation of scalable fault-tolerant information processors. To achieve quantum computational advantages and fault tolerance, non-Gaussian resources are essential. In this work, we propose and analyze a method to generate a variety of non-Gaussian states using coherent photon subtraction from a two-mode squeezed state followed by photon-number-resolving measurements. The proposed method offers a promising way to generate rotation-symmetric states conventionally used for quantum error correction with binomial codes and truncated Schrödinger cat codes. We consider the deleterious effects of experimental imperfections such as detection inefficiencies and losses in the state engineering protocol. Our method can be readily implemented with current quantum photonic technologies.
... The argument above shows that any experiment that can implement Gaussian transformations and a cubic phase gate can in principle generate any arbitrary non-Gaussian state. Even though many protocols have been proposed to experimentally realize a cubic phase gate [28,[135][136][137][138], any convincing implementations have yet to be demonstrated. One of the key problems is that experimental imperfections and finite squeezing are detrimental for the most commonly proposed methods [30]. ...
Article
Full-text available
Gaussian states have played an important role in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states. We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems. We then use this framework in various ways to explore the structure of the state space. We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank. For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations. To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state. Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states. On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations. Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage. At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.
... In particular, much of the effort has been dedicated to engineer the so-called "cubic phase gate" or, alternatively, to generate a "cubic phase state" [23]. Availability of the latter state allows for engineering a cubic phase gate by gate teleportation [15,16,[22][23][24][25][26] and thereby promotes the set of Gaussian operations to a universal set [8,27]. Having at one's disposal such a cubic gate would allow one, in particular, to engineer Gottesman-Kitaev-Preskill (GKP) states [23,28], which have been shown to yield fault tolerance [23,[29][30][31][32]. ...
Article
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... We show that states used in error-correction protocols, such as the Fock, binomial [6,18], and cat states [5,19], can be prepared with fidelities around 0.96. We also show that complex quantum states enabling universal continuous-variable quantum computing when supplied to Gaussian circuits such as the Gottesman-Kitaev-Preskill (GKP) state [7,[20][21][22] and the cubic phase state [7,[23][24][25] can be generated. In particular, the latter state has to our knowledge not been successfully experimentally implemented before [26,27]. ...
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... While unitary operations that use such a combination of quadrature and photon number operators have not been realized, we believe that our results will provide the impetus for the experimental realization of such unitary operations. Toward the realization of a universal gate set for CVQC, Yanagimoto et al. [34] propose a promising deterministic implementation of the non-Gaussian cubic phase gate exp iαx 3 , an approach worth exploring for implementing the two-qumode higher-order combined gates we have introduced. ...
Preprint
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... Harnessing the quantum nature of light potentially holds the key to overcoming classical limitations of conventional photonics in applications ranging from fundamental science, where coherent light sources have long been ubiquitous, to the more recent but rapidly developing field of quantum engineering and information processing [1][2][3][4]. In this context, quantum photonics stands out among other quantum-enhanced hardware platforms in its potential for long-distance connectivity, wide-bandwidth capacity, and room-temperature operability [5][6][7][8]. Many state-of-the-art proposals for photonic quantum information processing are particularly reliant on the physical process of parametric downconversion (PDC) [9][10][11], in which a medium with some optical nonlinearity (e.g., χ (2) ) facilitates the spontaneous generation of entangled photons. ...
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... For example, continuous-variable implementations of optical quantum computing [78,79] suffer from the same tradeoff between linearity and determinism. Applied to these systems, strong coupling in a χ (2) cavity can enable deterministic non-Gaussian gate operations and resource state preparations [56,80,81], circumventing the need for probabilistic implementations using measurement and feedback [82,83]. Combined with the ability to manipulate temporal mode structures with optical pulse gating [84,85], deterministic quantum operations on arbitrary photon temporal modes could be realized. ...
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... where we refer to the parameter c as the cubicity. Due to its fundamental role in quantum computation over continuous variables, various theoretical proposals have been put forward to generate such a state [10,17,35,[49][50][51][52][53][54][55][56][57][58], and recently a cubic-phase state was implemented experimentally in microvawe cavities [41]. To chose relevant parameters, we use the Wigner logarithmic negativity [31,32] as a guide, such that the negativity of our target cubic-phase state is comparable to the one of the other states investigated in this work. ...
Preprint
Full-text available
In the context of quantum technologies over continuous variables, Gaussian states and operations are typically regarded as freely available, as they are relatively easily accessible experimentally. In contrast, the generation of non-Gaussian states, as well as the implementation of non-Gaussian operations, pose significant challenges. This divide has motivated the introduction of resource theories of non-Gaussianity. As for any resource theory, it is of practical relevance to identify free conversion protocols between resources, namely Gaussian conversion protocols between non-Gaussian states. Via systematic numerical investigations, we address the approximate conversion between experimentally relevant single-mode non-Gaussian states via arbitrary deterministic one-to-one mode Gaussian maps. First, we show that cat and binomial states are approximately equivalent for finite energy, while this equivalence was previously known only in the infinite-energy limit. Then we consider the generation of cat states from photon-added and photon-subtracted squeezed states, improving over known schemes by introducing additional squeezing operations. The numerical tools that we develop also allow to devise conversions of trisqueezed into cubic-phase states beyond previously reported performances. Finally, we identify various other conversions which instead are not viable.
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Entangled microwave photons form a fundamental resource for quantum information processing and sensing with continuous variables. We use a low-loss Josephson metamaterial comprising superconducting non-linear asymmetric inductive elements to generate frequency (colour) entangled photons from vacuum fluctuations at a rate of 11 mega entangled bits per second with a potential rate above gigabit per second. The device is operated as a traveling wave parametric amplifier under Kerr-relieving biasing conditions. Furthermore, we realize the first successfully demonstration of single-mode squeezing in such devices - $2.4\pm0.7$ dB below the zero-point level at half of modulation frequency.
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We calculate the energy levels and corresponding eigenstates of an interacting scalar quantum field theory on a lattice using a continuous-variable version of the quantum imaginary-time evolution algorithm. Only a single qumode is needed for the simulation of the field at each point on the lattice. Our quantum algorithm avoids the use of non-Gaussian quantum gates and relies, instead, on detectors projecting onto eigenstates of the photon-number operator. Using Xanadu's Strawberry Fields photonic simulator, we obtain results on energy levels that are in very good agreement with results from exact calculations. We propose an experimental setup that can be realized with existing technology.
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We introduce a quasiprobability phase space distribution with two pairs of azimuthal-angular coordinates. This representation is well adapted to describe quantum systems with discrete symmetry. Quantum error correction of states encoded in continuous variables using translationally invariant states is studied as an example of application. We also propose an experimental scheme for measuring such distribution.
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Silicon is well known for its strong third-order optical nonlinearity, exhibiting efficient supercontinuum and four-wave mixing processes. A strong second-order effect that is naturally inhibited in silicon can also be observed, for example, by electrically breaking the inversion symmetry and quasi-phase matching the pump and the signal. To generate an efficient broadband second-harmonic signal, however, the most promising technique requires matching the group velocities of the pump and the signal. In this work, we utilize dispersion engineering of a silicon waveguide to achieve group velocity matching between the pump and the signal, along with an additional degree of freedom to broaden the second harmonic through the strong third-order nonlinearity. We demonstrate that the strong self-phase modulation and cross-phase modulation in silicon help broaden the second harmonic by 200 nm in the O-band. Furthermore, we show a waveguide design that can be used to generate a second-harmonic signal in the entire near-infrared region. Our work paves the way for various applications, such as efficient and broadband complementary-metal oxide semiconductor based on—chip frequency synthesizers, entangled photon pair generators, and optical parametric oscillators.
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Advanced quantum technologies require scalable and controllable quantum resources1,2. Gaussian states of multimode light, such as squeezed states and cluster states, are scalable quantum systems3,4,5, which can be generated on demand. However, non-Gaussian features are indispensable in many quantum protocols, especially to reach a quantum computational advantage⁶. Embodying non-Gaussianity in a multimode quantum state remains a challenge as non-Gaussian operations generally cannot maintain coherence among multiple modes. Here, we generate non-Gaussian quantum states of a multimode light field by removing a single photon in a mode-selective manner from a Gaussian state⁷. To highlight the potential for continuous-variable quantum technologies, we first demonstrated the capability to generate negativity of the Wigner function in a controlled mode. Subsequently, we explored the interplay between non-Gaussianity and quantum entanglement and verify a theoretical prediction⁸ about the propagation of non-Gaussianity along the nodes of photon-subtracted cluster states. Our results demonstrate large-scale non-Gaussianity with great flexibility along with an ensured compatibility with quantum information protocols. This range of features makes our approach ideal to explore the physics of non-Gaussian entanglement9,10 and to develop quantum protocols, which range across quantum computing11,12, entanglement distillation¹³ and quantum simulations¹⁴.
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The sensitivity of classical Raman spectroscopy methods, such as coherent anti-stokes Raman spectroscopy (CARS) or stimulated Raman spectroscopy (SRS), is ultimately limited by shot-noise from the stimulating fields. We present the complete theoretical analysis of a squeezing-enhanced version of Raman spectroscopy that overcomes the shot-noise limit of sensitivity with enhancement of the Raman signal and inherent background suppression, while remaining fully compatible with standard Raman spectroscopy methods. By incorporating the Raman sample between two phase-sensitive parametric amplifiers that squeeze the light along orthogonal quadrature axes, the typical intensity measurement of the Raman response is converted into a quantum-limited, super-sensitive estimation of phase. The resonant Raman response in the sample induces a phase shift to signal-idler frequency-pairs within the fingerprint spectrum of the molecule, resulting in amplification of the resonant Raman signal by the squeezing factor of the parametric amplifiers, whereas the non-resonant background is annihilated by destructive interference. Seeding the interferometer with classical coherent light stimulates the Raman signal further without increasing the background, effectively forming squeezing-enhanced versions of CARS and SRS, where the quantum enhancement is achieved on top of the classical stimulation.
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Gottesman–Kitaev–Preskill (GKP) states appear to be amongst the leading candidates for correcting errors when encoding qubits into oscillators. However the preparation of GKP states remains a significant theoretical and experimental challenge. Until now, no clear definitions for fault-tolerantly preparing GKP states have been provided. Without careful consideration, a small number of faults can lead to large uncorrectable shift errors. After proposing a metric to compare approximate GKP states, we provide rigorous definitions of fault-tolerance and introduce a fault-tolerant phase estimation protocol for preparing such states. The fault-tolerant protocol uses one flag qubit and accepts only a subset of states in order to prevent measurement readout errors from causing large shift errors. We then show how the protocol can be implemented using circuit QED. In doing so, we derive analytic expressions which describe the leading order effects of the nonlinear dispersive shift and Kerr nonlinearity. Using these expressions, we show that to mitigate the nonlinear dispersive shift and Kerr terms would require the protocol to be implemented on time scales four orders of magnitude longer than the time scales relevant to the protocol for physically motivated parameters. Despite these restrictions, we numerically show that a subset of the accepted states of the fault-tolerant phase estimation protocol maintain good error correcting capabilities even in the presence of noise.
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The stable operation of quantum computers will rely on error correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator1–3. In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates4–6. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach7–11. Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped ⁴⁰Ca⁺ ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit¹². We prepare and reconstruct logical states with an average squared fidelity of 87.3 ± 0.7 per cent. Also, we demonstrate a universal logical single-qubit gate set, which we analyse using process tomography. For Pauli gates we reach process fidelities of about 97 per cent, whereas for continuous rotations we use gate teleportation and achieve fidelities of approximately 89 per cent. This control method opens a route for exploring continuous variable error correction as well as hybrid quantum information schemes using both discrete and continuous variables¹³. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum14,15.
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Laboratory optical atomic clocks achieve remarkable accuracy (now counted to 18 digits or more), opening possibilities to explore fundamental physics and enable new measurements. However, their size and use of bulk components prevent them from being more widely adopted in applications that require precision timing. By leveraging silicon-chip photonics for integration and to reduce component size and complexity, we demonstrate a compact optical-clock architecture. Here a semiconductor laser is stabilized to an optical transition in a microfabricated rubidium vapor cell, and a pair of interlocked Kerr-microresonator frequency combs provide fully coherent optical division of the clock laser to generate an electronic 22 GHz clock signal with a fractional frequency instability of one part in 10^(13). These results demonstrate key concepts of how to use silicon-chip devices in future portable and ultraprecise optical clocks.
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The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat-binomial-GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multiqubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a multiqudit code.
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We propose an experimentally-feasible method for enhancing the atom-field coupling as well as the ratio between this coupling and dissipation (i.e., cooperativity) of two atoms in an optical cavity. Our method also enables the generation of steady-state nearly-maximal quantum entanglement. Our approach exploits optical parametric amplification to exponentially enhance the atom-cavity interaction and, hence, the cooperativity of the system, with the squeezing-induced fluctuation noise being completely eliminated. Thus, an effective cooperativity much larger than $100$ can be achieved even for modest values of a squeezing parameter. We demonstrate that the entanglement infidelity (which quantifies the deviation of the generated state from a maximally-entangled state) is exponentially smaller than the lower bound on the infidelities obtained in other dissipative entanglement preparations without applying squeezing. Thus, this infidelity can be arbitrarily small. Our generic method for enhancing atom-cavity cooperativities can be implemented in a wide range of physical systems, and provides diverse applications for quantum information processing based on entanglement.
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We propose general methodology of deterministic single-mode quantum interaction nonlinearly modifying single quadrature variable of a continuous variable system. The methodology is based on linear coupling of the system to ancillary systems subsequently measured by quadrature detectors. The nonlinear interaction is obtained by using the data from the quadrature detection for dynamical manipulation of the coupling parameters. This measurement-induced methodology enables direct realization of arbitrary nonlinear quadrature interactions without the need to construct them from the lowest-order gates. Such nonlinear interactions are crucial for more practical and efficient manipulation of continuous quadrature variables as well as qubits encoded in continuous variable systems.
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We propose a scalable scheme for optical quantum computing using measurement-induced continuous-variable quantum gates in a loop-based architecture. Here, time-bin-encoded quantum information in a single spatial mode are deterministically processed in a nested loop by an electrically programmable gate sequence. This architecture can process any input state and an arbitrary number of modes with almost minimum resources, and offers a universal gate set for both qubits and continuous variables. Furthermore, quantum computing can be performed fault-tolerantly by a known scheme for encoding a qubit in an infinite dimensional Hilbert space of a single light mode.
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Integrated waveguides exhibiting efficient second-order nonlinearities are crucial to obtain compact and low power optical signal processing devices. Silicon nitride (SiN) has shown second harmonic generation (SHG) capabilities in resonant structures and single-pass devices leveraging intermodal phase matching, which is defined by waveguide design. Lithium niobate allows compensating for the phase mismatch using periodically poled waveguides, however the latter are not reconfigurable and remain difficult to integrate with SiN or silicon (Si) circuits. Here we show the all-optical enhancement of SHG in SiN waveguides by more than 30 dB. We demonstrate that a Watt-level laser causes a periodic modification of the waveguide second-order susceptibility. The resulting second order nonlinear grating has a periodicity allowing for quasi phase matching (QPM) between the pump and SH mode. Moreover, changing the pump wavelength or polarization updates the period, relaxing phase matching constraints imposed by the waveguide geometry. We show that the grating is long term inscribed in the waveguides, and we estimate a second order nonlinearity of the order of 0.3 pm/V, while a maximum conversion efficiency (CE) of 1.8x10-6 W-1 cm-2 is reached.
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The combination of nonlinear and integrated photonics enables Kerr frequency comb generation in stable chip-based micro- resonators. Such a comb system will revolutionize applications, including multi-wavelength lasers, metrology, and spectroscopy. Aluminum gallium arsenide (AlGaAs) exhibits very high material nonlinearity and low nonlinear loss. However, difficulties in device processing and low device effective nonlinearity made Kerr frequency comb generation elusive. Here, we demonstrate AlGaAs-on-insulator as a nonlinear platform at telecom wave- lengths with an ultra-high device nonlinearity. We show high-quality-factor (Q > 100,000) micro-resonators where optical parametric oscillations are achieved with milliwatt-level pump threshold powers, which paves the way for on-chip pumped comb generation.
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We present a concept of non-Gaussian measurement composed of a non-Gaussian ancillary state, linear optics and adaptive heterodyne measurement, and on the basis of this we also propose a simple scheme of implementing a quantum cubic gate on a traveling light beam. In analysis of the cubic gate in the Heisenberg representation, we find that nonlinearity of the gate is independent from nonclassicality; the nonlinearity is generated solely by a classical nonlinear adaptive control in a measurement-and-feedforward process while the nonclassicality is attached by the non-Gaussian ancilla that suppresses excess noise in the output. By exploiting the noise term as a figure of merit, we consider the optimum non-Gaussian ancilla that can be prepared within reach of current technologies and discuss performance of the gate. It is a crucial step towards experimental implementation of the quantum cubic gate.
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In order to achieve universal quantum computation using continuous variables, one needs to jump out of the set of Gaussian operations and have a non-Gaussian element, such as the cubic phase gate. However, such a gate is currently very difficult to implement in practice. Here we introduce an experimentally viable 'repeat-until-success' approach to generating the cubic phase gate, which is achieved using sequential photon subtractions and Gaussian operations. We find that our scheme offers benefits in terms of the expected time until success, although we require a primitive quantum memory.
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Unitary non-Gaussian nonlinearity is one of the key components required for quantum computation and other developing applications of quantum information processing. Sufficient operation of this kind is still not available, but it can be approximatively implemented with the help of a specifically engineered resource state constructed from individual photons. We present experimental realization and thorough analysis of such quantum resource states and confirm that the state does indeed possess properties of a state produced by unitary dynamics driven by cubic nonlinearity.
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Optical $\chi^{(2)}$ non-linearity can be used for parametric amplification and producing down-converted entangled photon pairs that have broad applications. It is known that weak non-linear media exhibit dispersion and produce a frequency response. It is therefore of interest to know how spectral effects of a strong $\chi^{(2)}$ crystal affect the performance. Here we model the spectral effects of the dispersion of a strong $\chi^{(2)}$ crystal and illustrate how this affects its ability to perform Bell measurements and influence the performance of a quantum gates that employ such a Bell measurement. We show that a Dyson series expansion of the unitary operator is necessary in general, leading to unwanted spectral entanglement. We identify a limiting situation employing periodic poling, in which a Taylor series expansion is a good approximation and this entanglement can be removed.
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We investigate non-Gaussian states of light as ancillary inputs for generating nonlinear transformations required for quantum computing with continuous variables. We consider a recent proposal for preparing a cubic phase state, find the exact form of the prepared state and perform a detailed comparison to the ideal cubic phase state. We thereby identify the main challenges to preparing an ideal cubic phase state and describe the gates implemented with the non-ideal prepared state. We also find the general form of operations that can be implemented with ancilla Fock states, together with Gaussian input states, linear optics and squeezing transformations, and homodyne detection with feed forward, and discuss the feasibility of continuous variable quantum computing using ancilla Fock states.
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Recently a scheme has been proposed for constructing quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. One of the difficult steps in this scheme is the preparation of the encoded states. We show how these states can be generated by coupling a continuous quantum variable to a single qubit. An ion trap quantum computer provides a natural setting for a continuous system coupled to a qubit. We discuss how encoded states may be generated in an ion trap.
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The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.
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We propose a deterministic implementation of weak cubic nonlinearity, which is a basic building block of a full scale CV quantum computation. Our proposal relies on preparation of a specific ancillary state and transferring its nonlinear properties onto the desired target by means of deterministic Gaussian operations and feed-forward. We show that, despite the imperfections arising from the deterministic nature of the operation, the weak quantum nonlinearity can be implemented and verified with the current level of technology.
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Single photons provide excellent quantum information carriers, but current schemes for preparing, processing and measuring them are inefficient. For example, down-conversion provides heralded, but randomly timed single photons, while linear-optics gates are inherently probabilistic. Here, we introduce a deterministic scheme for photonic quantum information. Our single, versatile process---coherent photon conversion---provides a full suite of photonic quantum processing tools, from creating high-quality heralded single- and multiphoton states free of higher-order imperfections to implementing deterministic multiqubit entanglement gates and high-efficiency detection. It fulfils all requirements for a scalable photonic quantum computing architecture. Using photonic crystal fibres, we experimentally demonstrate a four-colour nonlinear process usable for coherent photon conversion and show that current technology provides a feasible path towards deterministic operation. Our scheme, based on interacting bosonic fields, is not restricted to optical systems, but could also be implemented in optomechanical, electromechanical and superconducting systems which exhibit extremely strong intrinsic nonlinearities.
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Full control over the spatio-temporal structure of quantum states of light is an important goal in quantum optics, to generate for instance single-mode quantum pulses or to encode information on multiple modes, enhancing channel capacities. Quantum light pulses feature an inherent, rich spectral broadband-mode structure. In recent years, exploring the use of integrated optics as well as source-engineering has led to a deep understanding of the pulse-mode structure of guided quantum states of light. In addition, several groups have started to investigate the manipulation of quantum states by means of single-photon frequency conversion. In this paper we explore new routes towards complete control of the inherent pulse-modes of ultrafast pulsed quantum states by employing specifically designed nonlinear waveguides with adapted dispersion properties. Starting from our recently proposed quantum pulse gate (QPG) we further generalize the concept of spatio-spectral engineering for arbitrary $\chitwo$-based quantum processes. We analyse the sum-frequency generation based QPG and introduce the difference-frequency generation based quantum pulse shaper (QPS). Together, these versatile and robust integrated optics devices allow for arbitrary manipulations of the pulse-mode structure of ultrafast pulsed quantum states. The QPG can be utilized to select an arbitrary pulse mode from a multimode input state, whereas the QPS enables the generation of specific pulse modes from an input wavepacket with Gaussian-shaped spectrum.
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Most quantum computation schemes propose encoding qubits in two-level systems. Others exploit the use of an infinite-dimensional system. In "Encoding a qubit in an oscillator" [Phys. Rev. A 64, 012310 (2001)], Gottesman, Kitaev, and Preskill (GKP) combined these approaches when they proposed a fault-tolerant quantum computation scheme in which a qubit is encoded in the continuous position and momentum degrees of freedom of an oscillator. One advantage of this scheme is that it can be performed by use of relatively simple linear optical devices, squeezing, and homodyne detection. However, we lack a practical method to prepare the initial GKP states. Here we propose the generation of an approximate GKP state by using superpositions of optical coherent states (sometimes called "Schrödinger cat states"), squeezing, linear optical devices, and homodyne detection.
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Silicon waveguides are promising chi(3)-based photon pair sources. Demonstrations so far have been based on picosecond pulsed lasers. Here, we present the first investigation of photon pair generation in silicon waveguides in a continuous regime. The source is characterized by coincidence measurements. We uncover the presence of unexpected noise which had not been noticed in earlier experiments. Subsequently, we present advances towards integration of the photon pair source with other components on the chip. This is demonstrated by photon pair generation in a Sagnac loop interferometer and inside a micro-ring cavity. Comparison with the straight waveguide shows that these are promising avenues for improving the source. In particular photon pair generation in the micro-ring cavity yields a source with a spectral width of approximately 150 pm resulting in a spectral brightness increased by more than 2 orders of magnitude.
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Quantum computers, which harness the superposition and entanglement of physical states, could outperform their classical counterparts in solving problems with technological impact-such as factoring large numbers and searching databases. A quantum processor executes algorithms by applying a programmable sequence of gates to an initialized register of qubits, which coherently evolves into a final state containing the result of the computation. Building a quantum processor is challenging because of the need to meet simultaneously requirements that are in conflict: state preparation, long coherence times, universal gate operations and qubit readout. Processors based on a few qubits have been demonstrated using nuclear magnetic resonance, cold ion trap and optical systems, but a solid-state realization has remained an outstanding challenge. Here we demonstrate a two-qubit superconducting processor and the implementation of the Grover search and Deutsch-Jozsa quantum algorithms. We use a two-qubit interaction, tunable in strength by two orders of magnitude on nanosecond timescales, which is mediated by a cavity bus in a circuit quantum electrodynamics architecture. This interaction allows the generation of highly entangled states with concurrence up to 94 per cent. Although this processor constitutes an important step in quantum computing with integrated circuits, continuing efforts to increase qubit coherence times, gate performance and register size will be required to fulfil the promise of a scalable technology.
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Continuous-variable cluster states offer a potentially promising method of implementing a quantum computer. This paper extends and further refines theoretical foundations and protocols for experimental implementation. We give a cluster-state implementation of the cubic phase gate through photon detection, which, together with homodyne detection, facilitates universal quantum computation. In addition, we characterize the offline squeezed resources required to generate an arbitrary graph state through passive linear optics. Most significantly, we prove that there are universal states for which the offline squeezing per mode does not increase with the size of the cluster. Simple representations of continuous-variable graph states are introduced to analyze graph state transformations under measurement and the existence of universal continuous-variable resource states. Comment: 17 pages, 5 figures
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We show that relatively simple integrated photonic circuits have the potential to realize a high fidelity deterministic controlled-phase gate between photonic qubits using bulk optical nonlinearities. The gate is enabled by converting travelling continuous-mode photons into stationary cavity modes using strong classical control fields that dynamically change the effective cavity-waveguide coupling rate. This architecture succeeds because it reduces the wave packet distortions that otherwise accompany the action of optical nonlinearities [J. Shapiro, Phys. Rev. A 73, 062305 (2006); J. Gea-Banacloche, Phys. Rev. A 81, 043823 (2010)]. We show that high-fidelity gates can be achieved with self-phase modulation in χ(3) materials as well as second-harmonic generation in χ(2) materials. The gate fidelity asymptotically approaches unity with increasing storage time for an incident photon wave packet with fixed duration. We also show that dynamically coupled cavities enable a trade-off between errors due to loss and wave packet distortion. Our proposed architecture represents a new approach to practical implementation of quantum gates that is room-temperature compatible and only relies on components that have been individually demonstrated.
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We study theoretically the interaction between two photons in a nonlinear cavity. The photons are absorbed into the cavity by an effective tuning of its input-output coupling via external control of a coupling to a second, strongly output-coupled cavity mode. Such “dynamically coupled” cavities, which can be implemented using bulk χ(2) and χ(3) nonlinearities, enable incoming photon wave packets to be absorbed into the cavity with high fidelity when the duration of the control is similar to that of the wave packets. Further, this configuration can be used to avoid limitations in the photon-photon interaction time set by the delay-bandwidth product of passive cavities and enables the elimination of wave-packet distortions caused by dispersive cavity transmission and reflection. We consider three kinds of nonlinearities, two arising from χ(2) and χ(3) materials and one due to an interaction with a two-level emitter. To analyze the input and output of few-photon wave packets, we use a Schrödinger-picture formalism in which traveling-wave fields are discretized into infinitesimal time bins. We suggest that dynamically coupled cavities provide a very useful tool for improving the performance of quantum devices relying on cavity-enhanced light-matter interactions such as single-photon sources and atomlike quantum memories with photon interfaces. As an example, we present simulation results showing that high-fidelity two-qubit entangling gates may be constructed using any of the considered nonlinear interactions.
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Terahertz (THz)-bandwidth continuous-wave (CW) squeezed light is essential for integrating quantum processors with time-domain multiplexing (TDM) by using optical delay line interferometers. Here, we utilize a single-pass optical parametric amplifier (OPA) based on a single-spatial-mode periodically poled ZnO:LiNbO3 waveguide, which is directly bonded onto a LiTaO3 substrate. The single-pass OPA allows THz bandwidth, and the absence of higher-order spatial modes in the single-spatial-mode structure helps avoid degradation of squeezing. In addition, the directly bonded ZnO-doped waveguide has durability for high-power pump and shows small photorefractive damage. Using this waveguide, we observe CW 6.3-dB squeezing at 20-MHz sideband by balanced homodyne detection. This is the first realization of CW squeezing with a single-pass OPA at a level exceeding 4.5 dB, which is required for the generation of a two-dimensional cluster state. Furthermore, the squeezed light shows 2.5-THz spectral bandwidth. The squeezed light will lead to the development of a high-speed on-chip quantum processor using TDM with a centimeter-order optical delay line.
Article
Quasi-phase-matched interactions in waveguides with quadratic nonlinearities enable highly efficient nonlinear frequency conversion. In this paper, we demonstrate the first generation of devices that combine the dispersion engineering available in nanophotonic waveguides with quasi-phase-matched nonlinear interactions available in periodically poled lithium niobate (PPLN). This combination enables quasi-static interactions of femtosecond pulses, reducing the pulse energy requirements by several orders of magnitude compared to conventional devices, from picojoules to femtojoules. We experimentally demonstrate two effects associated with second harmonic generation (SHG). First, we observe efficient quasi-phase-matched SHG with ${\lt} {100}\;{\rm fJ}$<100fJ of pulse energy. Second, in the limit of strong phase-mismatch, we observe spectral broadening of both harmonics with as little as 2 pJ of pulse energy. These results lay a foundation for a new class of nonlinear devices, in which coengineering of dispersion with quasi-phase-matching enables efficient nonlinear optics at the femtojoule level.
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The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.
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Entanglement is the key resource for measurement-based quantum computing. It is stored in quantum states known as cluster states, which are prepared offline and enable quantum computing by means of purely local measurements. Universal quantum computing requires cluster states that are both large and possess (at least) a two-dimensional topology. Continuous-variable cluster states—based on bosonic modes rather than qubits—have previously been generated on a scale exceeding one million modes, but only in one dimension. Here, we report generation of a large-scale two-dimensional continuous-variable cluster state. Its structure consists of a 5- by 1240-site square lattice that was tailored to our highly scalable time-multiplexed experimental platform. It is compatible with Bosonic error-correcting codes that, with higher squeezing, enable fault-tolerant quantum computation.
Article
We propose a theoretical scheme to deterministically generate Fock states in a Kerr cavity through adiabatic variation of the driving field strength and the cavity detuning. We show that the required time to generate an n-photon Fock state scales as the square root of n. Requirements for the Kerr coefficient relative to the system decoherence rate are provided as a function of desired state fidelity, indicating that the scheme is potentially realizable with the present state of the art in microwave superconducting circuits.
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Photonic quantum computing is one of the leading approaches to universal quantum computation. However, large-scale implementation of photonic quantum computing has been hindered by its intrinsic difficulties, such as probabilistic entangling gates for photonic qubits and lack of scalable ways to build photonic circuits. Here, we discuss how to overcome these limitations by taking advantage of two key ideas which have recently emerged. One is a hybrid qubit-continuous variable approach for realizing a deterministic universal gate set for photonic qubits. The other is the time-domain multiplexing technique to perform arbitrarily large-scale quantum computing without changing the configuration of photonic circuits. These ideas together will enable scalable implementation of universal photonic quantum computers in which hardware-efficient error correcting codes can be incorporated. Furthermore, all-optical implementation of such systems can increase the operational bandwidth beyond terahertz in principle, ultimately enabling large-scale fault-tolerant universal quantum computers with ultrahigh operation frequency.
Article
We demonstrate that nondegenerate four-wave mixing in a Si3N4 microring resonator can result in a nonlinear coupling rate between two optical fields exceeding their energy dissipation rate in the resonator, corresponding to strong nonlinear coupling. We demonstrate that this leads to a Rabi-like splitting, for which we provide a theoretical description in agreement with our experimental results. This yields new insight into the dynamics of nonlinear optical interactions in microresonators and access to novel phenomena.
Article
Periodically poled lithium niobate (PPLN) waveguides are a powerful platform for efficient wavelength conversion. Conventional PPLN converters, however, typically require long device lengths and high pump powers due to the limited nonlinear interaction strength. Here we use a nanostructured PPLN waveguide to demonstrate an ultrahigh normalized efficiency of 2600%/W−cm2 for second-harmonic generation of 1.5 μm radiation, more than 20 times higher than that in state-of-the-art diffused waveguides. This is achieved by a combination of sub-wavelength optical confinement and high-fidelity periodic poling at a first-order poling period of 4 μm. Our highly integrated PPLN waveguides are promising for future chip-scale integration of classical and quantum photonic systems.
Article
On-chip optical waveguides with low propagation losses and precisely engineered group velocity dispersion are important to nonlinear photonic devices such as soliton microcombs, and likewise can be employed for on-chip gyroscopes, delay lines, or Brillouin lasers. Yet, despite intensive research efforts, nonlinear integrated photonic platforms still feature propagation losses orders of magnitude higher than in standard optical fiber. The tight confinement and high index contrast of integrated waveguides make them highly susceptible to fabrication-induced surface roughness, causing dominant scattering losses. Therefore, microresonators with ultra-high-Q-factors are, to date, attainable only in polished bulk crystalline or chemically etched silica-based devices, which pose, however, challenges for full photonic integration. Here, we demonstrate the fabrication of silicon nitride (Si3N4) waveguides with unprecedentedly smooth sidewalls and tight confinement with record-low propagation losses. This is achieved by combining the photonic Damascene process with a novel reflow process, which reduces etching roughness, while sufficiently preserving dimensional accuracy. This leads to previously unattainable mean scattering Q-factors of 12 × 10⁶ for tightly confining waveguides with anomalous dispersion. Via systematic process step variation and two independent characterization techniques, we differentiate the scattering and absorption loss contributions and reveal metal-impurity-related absorption to be an important loss origin. Although such impurities are known to limit optical fibers, this is the first time, to the best of our knowledge, they are identified—and play a tangible role—in absorption of integrated microresonators. Taken together, our work provides new insights into the origins of propagation losses in Si3N4 waveguides and provides the technological basis for integrated nonlinear photonics in the ultra-high-Q regime.
Article
We present and analyze a method where parametric (two-photon) driving of a cavity is used to exponentially enhance the light-matter coupling in a generic cavity QED setup, with time-dependent control. Our method allows one to enhance weak-coupling systems, such that they enter the strong coupling regime (where the coupling exceeds dissipative rates) and even the ultrastrong coupling regime (where the coupling is comparable to the cavity frequency). As an example, we show how the scheme allows one to use a weak-coupling system to adiabatically prepare the highly entangled ground state of the ultrastrong coupling system. The resulting state could be used for remote entanglement applications.
Article
Intense light beams propagating in a lossless, dispersionless, single-mode optical fiber are subject to the Kerr effect, i.e., to the intensity-dependent refractive index of the fiber’s fused-silica core. Classically, Kerr-effectinduced self-phase modulation (SPM) can be used for spectral broadening of a picosecond pulse for grating-pair pulse compression down to femtosecond duration. Quantum mechanically, Kerr-effect-induced four-wave mixing(FWM) has been used to produce squeezed-state light. We present a quantum propagation theory for a lossless, dispersionless fiber with the Kerr nonlinearity. The theory includes classical SPM and quantum FWM within their regions of validity. It introduces a material time constant for the Kerr interaction, limiting the quantum phase shifts caused by the broadband zero-point fluctuations that accompany any input field, to develop a coarse-grained time multitemporal mode field analysis. Explicit expressions are obtained for the first and the second output-field moments when the fiber’s input field is in an arbitrary Gaussian state. These results are used to obtain homodyne-detection noise spectra, which are employed, in turn, toseek experimentally accessible manifestations of the Kerr time constant.
Article
The demand for nonlinear effects within a silicon platform to support photonic circuits requiring phase-only modulation, frequency doubling, and/or difference frequency generation, is becoming increasingly clear. However, the symmetry of the silicon crystal inhibits second order optical nonlinear susceptibility, $\chi^{(2)}$. Here, we show that the crystalline symmetry is broken when a DC field is present, inducing a $\chi^{(2)}$ in a silicon waveguide that is proportional to the large $\chi^{(3)}$ of silicon. First, Mach-Zehnder interferometers using the DC Kerr effect optical phase shifters in silicon ridge waveguides with p-i-n junctions are demonstrated with a $V_{\pi}L$ of $2.4Vcm$ in telecom bands $({\lambda}_{\omega}=1.58{\mu}m)$ without requiring to dope the silicon core. Second, the pump and second harmonic modes in silicon ridge waveguides are quasi-phase matched when the magnitude, spatial distribution of the DC field and $\chi^{(2)}$ are controlled with p-i-n junctions. Using these waveguides, second harmonic generation at multiple pump wavelengths are observed with a maximum efficiency of $P_{2{\omega}}/P_{\omega}^2$=12%/W at ${\lambda}_{\omega}=2.29{\mu}m$ in a 1mm long waveguide. This corresponds to a field-induced $\chi^{(2)}=41pm/V$, comparable to non-centrosymmetric media (LiNbO3, GaAs, GaN). The field-induced nonlinear silicon photonics will lead to a new class of CMOS compatible integrated devices spanning from near to mid infrared spectrum.
Article
A linearized quantum theory of soliton squeezing and detection is presented. The linearization reduces the quantum problem to a classical one. The classical formulation provides physical insight. It is shown that a quantized soliton exhibits uncertainties in photon number and phase, position (time), and momentum (frequency). Detectors for the measurement of all four operators are discussed. The squeezing of the soliton in the fiber is analyzed. An optimal homodyne detector for detection of the squeezing is presented that suppresses the noise associated with the continuum and the uncertainties in position and momentum.
Article
The extension of the conventional finite-difference time-domain solution of the full vector Maxwell equations to modeling femtosecond optical-pulse propagation in a nonlinear Kerr medium that exhibits a finite response time is presented. Numerical results are given for nonlinear self-focusing in two space dimensions and time; the technique can be generalized to three space dimensions with adequate computer resources. Comparisons with previously reported and anticipated results are made. Several novel phenomena that are not observed with scalar models of self-focusing and that can be attributed only to the complete solution of the vector Maxwell equations are discussed.
Article
We discuss the quantum theory of self-phase modulation as applied to optical fibers. We use a formalism that does not rely on a cavity and thereby resolves some anomalous length dependences present in earlier studies. We show that the exact expectation values and variances can be evaluated without the need for linearizing about a classical pump wave. The standard quantum equations for self-phase modulation are generalized in order to remove singularities in some expectation values whose origin is the instantaneous response time approximation.
Article
A symmetric nonlinear Mach—Zehnder interferometer containing a Kerr medium performs the operation of squeezing. The operation is performed by the interference of cotraveling waves, and the pump wave is removed. This mechanism is broadband. We analyze the operation of the interferometer and determine the degree of shot-noise reduction achieved in a balanced detector. A modification of the interferometer into a fiber ring reflector is described that accomplishes the squeezing and the pump separation.
Article
We present a canonical quantum theory of radiation in nonlinear media taking into account the effects of linear dispersion and absorption in a consistent way. We start from a microscopic model and apply recently developed concepts of the quantization of radiation in dispersive and absorbing linear media and extend the theory in order to include nonlinear optical processes. We derive propagation equations in space and time for the quantized radiation field, with special emphasis on the propagation of quantum-light pulses in Kerr media. In particular, the method enables us to derive systematically the noise sources that arise naturally in the nonlinear terms.
Article
By embedding an atom capable of electromagnetically induced transparency inside an appropriate photonic-crystal microcavity it may become possible to realize an optical nonlinearity that can impart a π-rad-peak phase shift in response to a single-photon excitation. Such a device, if it operated at high fidelity, would then complete a universal gate set for all-optical quantum computation. It is shown here that the causal, noninstantaneous behavior of any χ(3) nonlinearity is enough to preclude such a high-fidelity operation. In particular, when a single-photon-sensitive χ(3) nonlinearity has a response time that is much shorter than the duration of the quantum computer’s single-photon pulses, essentially no overall phase shift is imparted to these pulses by cross-phase modulation. Conversely, when this nonlinearity has a response time that is much longer than this pulse duration a single-photon pulse can induce a π-rad overall phase shift through cross-phase modulation, but the phase noise injected by the causal, noninstantaneous response function precludes this from being a high-fidelity operation.
Article
We develop a quantum theory of propagation in dispersive nonlinear media from the foundations of a correctly quantized field theory. Quantum fluctuations are handled by coherent-state expansions of localized field states. A stochastic nonlinear Schrödinger equation in the field variables is obtained for media with an intensity-dependent refractive index. This predicts squeezing for a continuous-wave input, over a wide bandwidth with anomalous dispersion, and over a gradually reducing bandwidth with normal dispersion. The equation is easily modified to include thermal-noise sources as well. For solitons, fluctuations are reduced over the soliton bandwidth. This leads to quantum solitons that have quadrature fluctuations less than the level of vacuum fluctuations. The complementary quadrature has a correspondingly increased fluctuation level.
Article
We develop a quantum theory of propagation in dispersive nonlinear media. Quantum fluctuations are handled via the coherent-state positive-P representation. A stochastic nonlinear Schr\"odinger equation in the field variables is obtained which predicts wide band squeezing in the region of anomalous dispersion. For soliton inputs, fluctuations are reduced over the soliton bandwidth. This leads to quantum solitons which have quadrature fluctuations less than the level of vacuum fluctuations.
Article
1. Classical nonlinear optics 2. Field quantization 3. Quantized fields in dielectric media 4. Microscopic description of media 5. Coherence and quantum dynamics in simple system 6. Decoherence and reservoirs 7. Phase-space representations 8. Single-mode devices 9. Degenerate parametric oscillator 10. Quantum fields in waveguides 11. Quantum propagation in nonlinear fibers 12. Quantum information.
Article
In the preceding paper [paper I of a two-part study; Lai and Haus, Phys. Rev. A 40, 844 (1989)] we have used the time-dependent Hartree approximation to solve the quantum nonlinear Schrödinger equation. In the present paper, the eigenstates of the Hamiltonian are constructed exactly by Bethe's ansatz method and are superimposed to construct exact soliton states. Both fundamental and higher-order soliton states are constructed and their mean fields are calculated. The quantum effects of soliton propagation and soliton collisions are studied in the framework of this construction. It is shown that a soliton experiences dispersion as well as phase spreading. The magnitude of this dispersion is estimated and is shown to be very small when the average photon number of the soliton is much larger than unity. The phase and position shifts due to a collision and the uncertainty of these shifts are also calculated.