We propose an alternative method to factorize an integer by using three harmonic oscillators. These oscillators are coupled together via specific Kerr nonlinear interactions. This method can be applied even if two harmonic oscillators are prepared in mixed states. As simple examples, we show how to factorize N=15 and 35 using this approach. The effect of dissipation of the harmonic oscillators on the performance of this method is studied. We also study the realization of nonlinear interactions between the coupled oscillators. However, the probability of finding the factors of a number is inversely proportional to its input size. The probability becomes low when this number is large. We discuss the limitations of this approach.
Introducing a Kerr medium in a cavity coupled to a harmonically moving mirror, we reproduce known and solvable interactions such as two-coupled harmonic oscillators and ion-laser like interactions for specific conditions. This is achieved by a unitary transformation allowing us to tune off the Kerr medium in order to simplify the Hamiltonian.
We show two examples in which the dynamical Casimir effect can be achieved by modulating the Kerr or higher order nonlinearities. In the first case the cavity field is coupled to an arbitrary number of qubits or an harmonic oscillator via the dipole interaction. In the second case, the modulation of the nonlinearities is accompanied by the off-resonance modulation of the cavity frequency. We present the analytic description of the phenomenon and supplement it with numeric simulations, demonstrating that photons can be created from vacuum and the resulting hyper-Poissonian photon statistics is …
We discuss a simple and experimentally realizable model for creation of enhanced Kerr nonlinearities accompanied by vanishing absorption. The model involves a V-type atom subjected to a strong drive laser, a weak probe laser and coupled to a single-mode cavity field. Working in the bad-cavity limit, we find that the simultaneous coupling of the cavity field to both atomic transitions creates a coherence between the transitions and thus can lead to quantum interference effects. We investigate the influences of the cavity field frequency, the cavity field-atom coupling constants and the atomic d…
We propose the use of a superconducting charge qubit capacitively coupled to two resonant nanomechanical resonators to generate Yurke-Stoler states, i.e. quantum superpositions of pairs of distinguishable coherent states 180$^\circ$ out of phase with each other. This is achieved by effectively implementing Kerr nonlinearities induced through application of a strong external driving field in one of the resonators. A simple study of the effect of dissipation on our scheme is also presented, and lower bounds of fidelity and purity of the generated state are calculated. Our procedure to implement …
The very small size of optical nonlinearities places wide ranging restrictions on the types of novel physics one can explore. For an ensemble of multi-level systems one can synthesize a large effective optical nonlinearity using quantum coherence effects but such non-linearities are technically extremely challenging to demonstrate at the single atom level. In this work we describe how a single artificial multi-level Cooper Pair Box molecule, interacting with a superconducting microwave coplanar waveguide resonator, when suitably driven, can generate extremely large optical nonlinearities at mi…
A fundamental and open question is whether cross-Kerr nonlinearities can be used to construct a controlled-phase (CPHASE) gate. Here we propose a gate constructed from a discrete set of atom-mediated cross-Kerr interaction sites with counter-propagating photons. We show that the average gate fidelity $F$ between a CPHASE and our proposed gate increases as the number of interaction sites increases and the spectral width of the photon decreases, e.g. with 12 sites we find $F > 99\%$.
With the assistance of weak cross-Kerr nonlinearities, we introduce an optical scheme to fuse two small-size polarization entangled W states into a large-scale W state without qubit loss, i.e.,$\mathrm{W}_{n+m}$ state can be generated from an $n$-qubit W state and a $m$-qubit W state. To complete the fusion task, two polarization entanglement processes and one spatial entanglement process are applied. The fulfillments of the above processes are contributed by a cross-Kerr nonlinear interaction between the signal photons and a coherent state via Kerr media. We analyze the resource cost and the …
Bistable mechanical vibration is observed in a cavity magnomechanical system, which consists of a microwave cavity mode, a magnon mode, and a mechanical vibration mode of a ferrimagnetic yttrium-iron-garnet (YIG) sphere. The bistability manifests itself in both the mechanical frequency and linewidth under a strong microwave drive field, which simultaneously activates three different kinds of nonlinearities, namely, magnetostriction, magnon self-Kerr, and magnon-phonon cross-Kerr nonlinearities. The magnon-phonon cross-Kerr nonlinearity is first predicted and measured in magnomechanics. The sys…
We investigate a hybrid electro-optomechanical system that allows us to obtain controllable strong Kerr nonlinearities in the weak-coupling regime. We show that when the controllable electromechanical subsystem is close to its quantum critical point, strong photon-photon interactions can be generated by adjusting the intensity (or frequency) of the microwave driving field. Nonlinear optical phenomena, such as the appearance of the photon blockade and the generation of nonclassical states (e.g., Schrödinger cat states), are predicted in the weak-coupling regime, which is feasible for most curre…
We predict that the effective nonlinear optical susceptibility can be tailored using the Purcell effect. While this is a general physical principle that applies to a wide variety of nonlinearities, we specifically investigate the Kerr nonlinearity. We show theoretically that using the Purcell effect for frequencies close to an atomic resonance can substantially influence the resultant Kerr nonlinearity for light of all (even highly detuned) frequencies. For example, in realistic physical systems, enhancement of the Kerr coefficient by one to two orders of magnitude could be achieved.
We develop the method of adiabatic tracking for photo- and magneto-association of Bose-Einstein atomic condensates with models that include Kerr type nonlinearities. We show that the inclusion of these terms can produce qualitatively important modifications in the adiabatic dynamics, like the appearance of bifurcations, in which the trajectory that is being tracked loses its stability. As a consequence the adiabatic theorem does not apply and the adiabatic transfer can be strongly degraded. This degradation can be compensated by using fields that are strong enough compared with the values of t…
The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the number of photons in the field. At present, many proposals use nonlinearities to generate photon-photon interactions. For propagating fields these interactions result in undesirable features such as spectral correlation between the photons. Here, we engineer a discrete network composed of cross-Kerr interaction regions to simulate a self-Kerr medium. The medium has effective long-range interactions implemented in a physically local way. We compute the one- and two-photon S matrice…
We consider the Jaynes-Cummings model describing the interaction of a Cooper pair box (CPB) and a nanoresonator (NR) in the presence of a Kerr medium and losses The evolution of the entropy of both subsystems and the CPB population inversion were calculated numerically. It is found that these properties increase when the NR frequency is time-dependent, even in the presence of losses; the effect is very sensitive to detuning and disappears in the resonant regime. The roles played by the losses affecting the CPB and the NR are also compared.
In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, an optomechanical system with an asymmetric anharmonic mechanical resonator is studied. The cross-Kerr interaction and the Kerr-like self-interaction of photons and vibration quanta arise in the Hamiltonian. These interactions are induced by both cubic and quartic nonlinearities of oscillations of the mechanical resonator and the cavity-resonator interaction that is linear in mechanical displacements. We demonstrate a bistable behavior of the number of vibration quanta and find that this behavior i…
We discuss general properties of the equilibrium state of parametric down-conversion in superconducting quantum circuits with detunings and Kerr anharmonicities, in the strongly nonlinear regime. By comparing moments of the steady state and those of a Schrödinger cat, we show that true Schrödinger cats cannot survive in the steady state if there is any single-photon loss. A delta-function 'cat-like' steady-state distribution can be formed, but this only exists in the limit of an extremely large nonlinearity. The steady state is a mixed state, which is more complex than a mixture or linear comb…
We considered the multiphoton resonance in the periodically driven quantum oscillator with Kerr nonlinearity in the presence of weak high-order nonlinearities. Multiphoton resonance leads to the emergence of peaks and dips in the dependence of the stationary occupations of the stable states on detuning. We demonstrated that due to high-order nonlinearities, these peaks and dips acquire additional fine structure and split into several closely spaced ones. Quasiclassically, multiphoton resonance is treated as tunneling between the regions of the oscillator phase portrait, and the fine structure …
Entanglement represents an important resource for quantum information processing, but its generation itself requires physical resources that are limited. We propose a scheme for generating a wide class of entangled qudit-type states of optical field modes at sites separated by noisy medium when only weak optical nonlinearities are available at both sites. The protocol is also based on exploiting a weak probe field, transmitted between the sites and used for generation of quantum correlations between two spatially separated field modes. The idea of probabilistic entanglement enhancement by meas…
It is well-known that the nonlinear coupling between a mechanical oscillator and a superconducting oscillator or optical cavity can be used to generate a Kerr-nonlinearity for the cavity mode. We show that the strength of this Kerr-nonlinearity, as well as the effect of the photon-pressure force can be enormously increased by modulating the strength of the nonlinear coupling. We present an electromechanical circuit in which this enhancement can be readily realized.
In this paper, the interaction between a $Λ$-type three-level atom and two-mode cavity field is discussed. The detuning parameters and cross-Kerr nonlinearity are taken into account and it is assumed that atom-field coupling and Kerr medium to be $f$-deformed. Even though the system seems to be complicated, the analytical form of the state vector of the entire system for considered model is exactly obtained. The time evolution of nonclassical properties such as quantum entanglement and position-momentum entropic uncertainty relation (entropy squeezing) of the field are investigated. In each ca…
High-Q microwave cavity modes coupled to transmon ancillas provide a hardware-efficient platform for quantum computing. Due to their coupling, the cavity modes inherit finite nonlinearity from the transmons. In this work, we theoretically and experimentally investigate how an off-resonant drive on the transmon ancilla modifies the nonlinearities of cavity modes in qualitatively different ways, depending on the interrelation among cavity-transmon detuning, drive-transmon detuning and transmon anharmonicity. For a cavity-transmon detuning that is smaller than or comparable to the drive-transmon …
Weak cross-Kerr nonlinearities between single photons and coherent states are the basis for many applications in quantum information processing. These nonlinearities have so far mainly been discussed in terms of highly idealized single-mode models. We develop a general theory of the interaction between continuous-mode photonic pulses and apply it to the case of a single photon interacting with a coherent state. We quantitatively study the validity of the usual single-mode approximation using the concepts of fidelity and conditional phase. We show that high fidelities, non-zero conditional phas…
Continuous-variable systems realized in high-coherence microwave cavities are a promising platform for quantum information processing. While strong dynamic nonlinear interactions are desired to implement fast and high-fidelity quantum operations, static cavity nonlinearities typically limit the performance of bosonic quantum error-correcting codes. Here we study theoretical models of nonlinear oscillators describing superconducting quantum circuits with asymmetric Josephson-junctions loops. Treating the nonlinearity as a perturbation, we derive effective Hamiltonians using the Schrieffer-Wolff…
We study the properties of an array of QED-cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinear couplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally, correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage, and the external driving, and show the emergence of a number of different quantum phases. A photon crystal associated…
Fast and robust cat state preparation utilizing higher order nonlinearities in Rydberg ensembles
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Freie Universität Berlin（柏林自由大学）；CESQ/ISIS (UMR 7006), CNRS and University of Strasbourg（CNRS与斯特拉斯堡大学 CESQ/ISIS (UMR 7006)）；Nordita, KTH Royal Institute of Technology and Stockholm University（诺尔迪塔，皇家理工学院与斯德哥尔摩大学）；Department of Physics, Stockholm University（斯德哥尔摩大学物理系）；Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院凝聚态理论研究所）
We propose a fruitful scheme for exploring multiphoton entangled states based on linear optics and weak nonlinearities. Compared with the previous schemes the present method is more feasible because there are only small phase shifts instead of a series of related functions of photon numbers in the process of interaction with Kerr nonlinearities. In the absence of decoherence we analyze the error probabilities induced by homodyne measurement and show that the maximal error probability can be made small enough even when the number of photons is large. This implies that the present scheme is quit…
We propose a new method to produce self- and cross-Kerr photonic nonlinearities, using light-induced Stark shifts due to the interaction of a cavity mode with atoms. The proposed experimental set-up is considerably simpler than in previous approaches, while the strength of the nonlinearity obtained with a single atom is the same as in the setting based on electromagnetically induced transparency. Furthermore our scheme can be applied to engineer effective photonic nonlinear interactions whose strength increases with the number of atoms coupled to the cavity mode, leading to photon-photon inter…
The field of cavity optomechanics has achieved groundbreaking photonic control and detection of mechanical oscillators, based on their coupling to linear electromagnetic modes. Lately, however, there is an uprising interest in exploring cavity nonlinearities as a powerful new resource in radiation-pressure interacting systems. Here, we present a flux-mediated optomechanical device combining a nonlinear Josephson-based superconducting quantum interference cavity with a mechanical nanobeam. We demonstrate how the intrinsic Kerr nonlinearity of the microwave circuit can be used for a counter-intu…
We show how to generate an entangled pair of yttrium iron garnet (YIG) samples in a cavity-magnon system without using any nonlinearities which are typically very weak. This is against the conventional wisdom which necessarily requires strong Kerr like nonlinearity. Our key idea, which leads to entanglement, is to drive the cavity by a weak squeezed vacuum field generated by a flux-driven Josephson parametric amplifier (JPA). The two YIG samples interact via the cavity. For modest values of the squeezing of the pump, we obtain significant entanglement. This is the principal feature of our sche…
We present a method for implementing a weak optical Kerr interaction (single-mode Kerr Hamiltonian) in a measurement-based fashion using the common set of universal elementary interactions for continuous-variable quantum computation. Our scheme is a conceptually distinct alternative to the use of naturally occurring, weak Kerr nonlinearities or specially designed nonlinear media. Instead, we propose to exploit suitable offline prepared quartic ancilla states together with beam splitters, squeezers, and homodyne detectors. For perfect ancilla states and ideal operations, our decompositions for …
In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the $L^2$-subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit.
We study theoretically nonlinear effects arising from the presence of a Kerr-down conversion nonlinear crystal inside an optomechanical cavity. For this system we investigate the influences of the two nonlinearities, i.e., the Kerr nonlinearity and the parametric gain, on the dynamics of the oscillating mirror, the intensity and the squeezing spectra of the transmitted field, and the steady-state mirror-field entanglement. We show that in comparison with a bare optomechanical cavity, the combination of the cavity energy shift due to the Kerr nonlinearity and increase in the intracavity photon …
Cavity optomechanics has proven to be a field of research rich with possibilities for studying motional cooling, squeezing, quantum entanglement and metrology in solid state systems. While to date most studies have focused on the modulation of the cavity frequency by the moving element, the emergence of new materials will soon allow to explore the influences of nonlinear optical effects. We therefore study in this work the effects due to a nonlinear position-modulated self-Kerr interaction and find that this leads to an effective coupling that scales with the square of the photon number, meani…
For solid-state spin systems, the collective spin motion in a single crystal embodies multiple magnetostatic modes. Recently, it was found that the cross-Kerr interaction between the higher-order magnetostatic mode and the Kittel mode introduces a new operable degree of freedom. In this work, we propose a scheme to entangle two magnon modes via the cross-Kerr nonlinearity when the bias field is inhomogeneous and the system is driven. Quantum entanglement persists at the steady state, as demonstrated by numerical results using experimentally feasible parameters. Furthermore, we also demonstrate…
We consider the model of two coupled oscillators with Kerr nonlinearities in the rotating-wave approximation. We demonstrate that for a certain set of parameters of the model, the multi-photon transitions occur between many pairs of the oscillator states simultaneously. Also, the position of the multi-photon resonances does not depend on the coupling strength between two oscillators. We prove rigorously that this is a consequence of a certain symmetry of the perturbation theory series for the model. In addition, we analyse the model in the quasi-classical limit by considering the dynamics of t…
Raman-type laser excitation of a trapped atom allows one to realize the quantum mechanical counterpart of phenomena of nonlinear optics, such as Kerr-type nonlinearities, parametric amplification, and multi-mode mixing. Additionally, huge nonlinearities emerge from the interference of the atomic wave function with the laser waves. They lead to a partitioning of the phase space accompanied by a significantly different action of the time evolution in neighboring phase-space zones. For example, a nonlinearly modified coherent "displacement" of the motional quantum state may induce strong amplitud…
A cross-Kerr interaction produces a phase shift on two modes of light proportional to the number of photons in both modes, and is sometimes called cross-phase modulation. Cross-Kerr nonlinearities have many applications in classical and quantum nonlinear optics, including the possibility of a deterministic and all-optical controlled-phase gate. We calculate the one- and two-photon S-matrix for fields propagating in a medium where the cross-Kerr interaction is spatially distributed at discrete interaction sites comprised of atoms. For the interactions considered, we analyze the cases where the …
We report a first-principles study of the driven dissipative dynamics for Kerr oscillators in the mesoscopic regime. This regime is characterized by large Kerr nonlinearity, realized here using the nonlinear kinetic inductance of a large array of Josephson junctions. The experimentally measured nonlinear resonance lineshapes of the junction array modes show significant deviations from steady-state numerical predictions, and necessitate time-dependent numerical simulations indicative of strong measurement-induced dephasing in the system arising from the large cross-Kerr effect between array mod…
Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein condensates under certain conditions, solves the unstructured search problem more efficiently than does the Schrödinger equation, because it includes a cubic nonlinearity, proportional to $|ψ|^2ψ$. This is not the only nonlinearity of the form $f(|ψ|^2)ψ$ that arises in effective equations for the evolution of real quantum physical systems, however: The cubic-quintic nonlinear Schrödinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrödinger eq…
We propose a bootstrapping approach to generation of maximally path-entangled states of photons, so called ``NOON states''. Strong atom-light interaction of cavity QED can be employed to generate NOON states with about 100 photons; which can then be used to boost the existing experimental Kerr nonlinearities based on quantum coherence effects to facilitate NOON generation with arbitrarily large number of photons all within the current experimental state of the art technology. We also offer an alternative scheme that uses an atom-cavity dispersive interaction to obtain sufficiently high Kerr-no…
Photon-pressure coupling between two superconducting circuits is a promising platform for investigating radiation-pressure coupling in novel parameter regimes and for the development of radio-frequency (RF) quantum photonics and quantum-limited RF sensing. So far, the intrinsic Josephson nonlinearity of photon-pressure coupled circuits has not been considered a potential resource for enhanced devices or novel experimental schemes. Here, we implement photon-pressure coupling between a RF circuit and a microwave cavity containing a superconducting quantum interference device (SQUID) which can be…
In this paper, we follow our presented model in J. Opt. Soc. Am. B {\bf 30}, 1109--1117 (2013), in which the interaction between a $Λ$-type three-level atom and a quantized two-mode radiation field in a cavity in the presence of nonlinearities is studied. After giving a brief review on the procedure of obtaining the state vector of the atom-field system, some further interesting and important physical features (which are of particular interest in the quantum optics field of research) of the whole system state, i.e., the number-phase entropic uncertainty relation (based on the two-mode Pegg-Bar…
We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrodinger (NLS) equations with generalized parity-time (PT)-symmetric Scarff II potentials. Particularly, a PT-symmetric k-wavenumber Scarff II potential and a multi-well Scarff II potential are considered, respectively. For the k-wavenumber Scarff II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT-symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multi-well Scarff II potenti…
