Tripartite Continuous-variable Entanglement Research Articles

Effect of initial squeezing on tripartite continuous-variable entanglement of dynamical processes

In this paper, we investigate the effect of initial squeezing on tripartite continuous-variable (CV) Hamiltonians based on real physical systems. The Hamiltonians are based on the three-mode magnons excited by an external field, the one-dimensional interconnected tripartite spring-like system, and two spin ensembles coupled to a cavity resonator in the nondispersive regime in the CV limit. We find that for the three-mode magnons and the spring-like system, implementation of initial squeezing invariably enhances the two-way and three-way entanglement, while for the spin-resonator system, it depends on the alignment of the spin ensembles relative to the resonator field.

Continuous variable tripartite entanglement and steering using a third-order nonlinear optical interaction

Nonlinear optical devices are very useful for generating entanglement. One proposal for the generation of genuine tripartite entanglement [Phys. Rev. Lett. 120, 043601 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.043601] is to use a third-order nonlinear optical interaction. Here, we investigate a generalization of this, where we consider the quantum nature of all modes; moreover, we also investigate quantum steering, which is a stronger and asymmetric non-local correlation. We show that the final state presents not only tripartite continuous variable entanglement but also tripartite steering.

Tripartite continuous-variable entanglement of NOPA system**Projected supported by the National Natural Science Foundation of China (Grant No. 11504074) and the Science Fund from the State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Shanxi, China (Grant No. KF201601).

We investigate the fundamental limits to the achievable tripartite continuous-variable (CV) entanglement criterion of a generalized V1 criterion. Our numerical simulation results show that the non-degenerate eigenvalues do effect the performances of the estimated minimum variances. From below the threshold to above the threshold, with the increase of the pump parameter, the tripartite CV entanglement gradually disappears. The different off-diagonal elements seriously distort the weights for entanglement. We can obtain a good tripartite CV entanglement by appropriately controlling the values of off-diagonal elements εij.

The tripartite entanglement in a phase-mismatched frequency non-degenerated optical parametric amplifier

We present an analytical solution for Fokker–Planck equation of phase-mismatched frequency non-degenerated optical parametric amplifier (FNOPA). We investigate the continuous-variable (CV) tripartite entanglement characteristic of FNOPA working at phase-matched and phase-mismatched condition. From below threshold to above threshold, we numerically calculate the dependence of CV tripartite entanglement on relative loss parameter η and phase-mismatched factor Δω. The larger η and the smaller Δω can bring a good entanglement.

Versatility of continuous-variable asymmetric tripartite entanglement allows Alice and Clare to keep secrets from Bob

The fully symmetric Gaussian tripartite entangled pure states will not exhibit two-mode Einstein Podolsky-Rosen (EPR)-steering. This means that any two participants cannot share quantum secrets using the security of one-sided device independent quantum key distribution (1SDI-QKD) without involving the third. They are restricted at most to standard quantum key distribution (S-QKD), which is less secure. Here we show that least some asymmetric tripartite systems can exhibit bipartite EPR-steering, so that two of the participants can use 1SDI-QKD without involving the other. This is possible because the promiscuity relations of continuous-variable tripartite entanglement are different from those of discrete-variable systems. We analyse these properties for two different systems, showing that the asymmetric system has an extra degree of flexibility not found in the symmetric one.

Bipartite entanglement in continuous-variable tripartite systems

In the field of continuous-variable tripartite entanglement, the systems utilised can be either asymmetric or symmetric. It is therefore of interest to examine the differences in the entanglement properties of these two types of system, using two examples that are known to produce tripartite entanglement. We examine one asymmetric and one fully symmetric Gaussian continuous-variable system in terms of their tripartite and bipartite entanglement properties. We first treat pure states and are able to find analytic solutions using the undepleted pump approximation for the Hamiltonian models. Our symmetric system exhibits perfect tripartite correlations, but only in the unphysical limit of infinite squeezing. For more realistic squeezing parameters, the two systems exhibit both tripartite and bipartite entanglement. Secondly we treat the more experimentally reasonable situation where the interactions take place inside optical cavities and we are dealing with mixed states. In these cases, where the criteria for genuine tripartite entanglement are more stringent, we find that tripartite entanglement is still available, although over smaller bandwidths than three-mode inseparability. In general, the spectral results are consistent with the analytical solutions. We conclude that none of the outputs are completely analogous to either GHZ or W states, but there are parameter regions of the Hamiltonian dynamics where they produce T states as introduced by Adesso et al. [1,2]. In the intracavity cases, both bipartite entanglement and tripartite inseparability are always present, with genuine tripartite entanglement appearing as the pumping rate is increased. The qualitative differences in the output states for different interaction parameters indicate that continuous-variable tripartite quantum information systems offer a versatility not found in two-mode bipartite systems.

Generation of bright three-color tripartite continuous variable entanglement via the sum-frequency generation

We theoretically show that bright three-color tripartite continuous-variable entanglement can be generated directly by the process of triply resonant sum-frequency generation using a periodically poled supperlattice. It is demonstrated that two reflected pump beams and the output sum frequency beam are all entangled with each other according to the sufficient inseparability criterion for multipartite continuous variable (CV) entanglement. The dependence of the steady state region and the entanglement on system parameters are discussed. By utilizing the up-conversion process, the generated three entanglement fields have macroscopic intensities as soon as the pump fields are turned on.

Tripartite entanglement of microwave radiation via nonlinear parametric interactions enhanced by quantum interference in superconducting quantum circuits

We present an efficient scheme for generating the tripartite continuous variable entanglement of microwave radiation in a Δ-type three-level artificial atom placed in a transmission line resonator, simultaneously driven resonantly by two strong electromagnetic fields. We show that it is possible to obtain the full tripartite entanglement with the different frequencies in the presence of atom and cavity decays. In our scheme, interestingly, the nonlinear process of simultaneous parametric downconversion and upconversion interactions between microwave modes can be achieved via quantum interference induced by two classical fields and can be greatly enhanced under the condition of resonant driving of the two classical fields. Such a nonlinear interaction is responsible for the generation of the strong tripartite entanglement among three microwave modes. In practice, the system considered here could offer an alternative for implementing the scalable information processing in the solid-state system.

Genuine tripartite continuous-variable entanglement with spatial degrees of freedom of photons

We propose an experimentally feasible scheme to prepare tripartite continuous-variable entanglement with the spatial degrees of freedom of photons. The scheme relies on postselection upon the state obtained when one photon of an entangled pair produced by spontaneous parametric down-conversion interferes with a single photon on a 50:50 beam splitter. We show that the presence of genuine tripartite entanglement can be detected by the van Loock--Furusawa criterion.

Tripartite entanglement via coherently enhanced six-wave mixing

In this paper, we show that the tripartite continuous-variable entanglement can be obtained by using coherently enhanced six-wave mixing process in a triple Λ atomic system. Multiphoton resonances condition must be satisfied for the scheme. Due to atomic coherence, two of the cavity modes are pulled into a quantum beat and in the mixing interaction with the third cavity mode. This yields entanglement between the quantum-beat components and the third cavity mode. The pair of the beat modes is not entangled with each other, but they are both entangled with the third cavity mode. The generated tripartite entanglement is significantly dependent on the normalized detunings of the external coherent fields and the cooperativity parameters.

Three-photon energy–time entanglement

Entangled quantum particles have correlations stronger than those allowed by classical physics. These correlations are the focus of of the deepest issues in quantum mechanics [1-3] and are the basis of many quantum technologies. The entanglement of discrete particle properties has been studied extensively in the context of quantum computing [4], cryptography [5], and quantum repeaters [6] while entanglement between the continuous properties of particles may play a critical role in improving the sensitivity of gravitational wave detectors [7,8], atomic clocks [9], and other high precision instruments. The attributes of three or more entangled particles are fundamentally different from those of two entangled particles [10-14]. While the discrete variables of up to 14 ions [15] and the continuous variables between three intense optical beams [16, 17] have been entangled, it has remained an open challenge to entangle the continuous properties of more than two individual particles. Here we experimentally demonstrate genuine tripartite continuous-variable entanglement between three separated particles. In our setup the three particles are photons created directly from a single input photon; the creation process leads to quantum correlations between the colours, or energies, and emission times of the photons. The entanglement between our three photons is the three-party generalization of the Einstein-Podolsky-Rosen (EPR) [1] correlations for continuous variables, and allows for new fundamental tests of quantum mechanics to be carried out. Our scheme can be extended to carry out multi-particle Franson interferometry [18,19], and opens the possibility of using additional degrees of freedom in our photons to simultaneously engineer discrete and continuous-variable hyper-entangled states that could serve as a valuable resource in a wide variety of quantum information tasks.

Dynamics of nonclassical light in integrated nonlinear waveguide arrays and generation of robust continuous-variable entanglement

We study a class of nonlinear waveguide arrays in which the waveguides are endowed with ${\ensuremath{\chi}}^{(2)}$ nonlinearity and coupled through the evanescent overlap of the guided modes. We investigate the viability of such an array as a platform for generating both bipartite and tripartite continuous-variable entanglement. We explicitly address the effect of realistic losses on the entanglement produced, discuss the possible types of nonlinear materials that could be used, and suggest solutions for the possible phase matching issues in the waveguides. The simultaneous generation and manipulation of the light on a single waveguide chip circumvents the usual bandwidth problems associated with the use of external bulky optical elements and makes this avenue promising for further investigation.

Entanglement of tripartite microwave radiation via quantum coherence in single molecular magnets

We present an alternative scheme for the generation of the entangled microwave radiation with amplification in single triple-Λ five-level molecular magnets driven by two strong control fields and one relative weak pump field. By numerically simulating the dynamics of the system, we show that three fully entangled modes with different frequencies can be obtained in the presence of the relaxations. Our numerical results also show that the entanglement occurs among the three cavity fields with a large number of photons. Physically, the responsible mechanism is quantum coherence. Two strong control fields induce two cavity modes coupled with the strong-driven levels to a quantum beat, and then the relative weak pump field leads to the parametric interaction between such a beat with the third mode coupled with the weak-driven level. The present work provides an efficient approach to achieve tripartite continuous-variable entanglement in the molecular magnets system, which may be necessary for the progress of solid-state quantum information.

Steady-state entanglement and normal-mode splitting in an atom-assisted optomechanical system with intensity-dependent coupling

In this paper, we study theoretically the bipartite and tripartite continuous variable entanglement as well as the normal-mode splitting in a single-atom cavity optomechanical system with intensity-dependent coupling. The system under consideration is formed by a Fabry-Perot cavity with a thin vibrating end mirror and a two-level atom in the Gaussian standing-wave of the cavity mode. We first derive the general form of Hamiltonian describing the tripartite intensity-dependent atom-field-mirror coupling due to the presence of cavity mode structure. We then restrict our treatment to the first vibrational sideband of the mechanical resonator and derive a novel form of tripartite atom-field-mirror Hamiltonian. We show that when the optical cavity is intensely driven one can generate bipartite entanglement between any pair of the tripartite system, and that, due to entanglement sharing, the atom-mirror entanglement is efficiently generated at the expense of optical-mechanical and optical-atom entanglement. We also find that in such a system, when the Lamb-Dicke parameter is large enough one can simultaneously observe the normal mode splitting into three modes.

Coherently prepared nondegenerate Y-shaped four-level correlated emission laser: A source of tripartite entangled light

A detailed derivation of the master equation of the cavity radiation of a coherently prepared $Y$-shaped four-level correlated emission laser is presented. The outline of the procedures that can be employed in analytically solving the stochastic differential equations and the rate equations of various correlations are also provided. It is shown that coherently preparing the atoms in the upper two energy levels and the lower, initially, can lead to a genuine continuous variable tripartite entanglement. Moreover, preparing the atoms in the coherent superposition, other than the possible maximum or minimum, of the upper two energy levels, leaving the lower unpopulated, may lead to a similar observation. With the possibility of the atom at the intermediate energy level to take three different transition roots guided by the induced coherence, this system in general is found to encompass versatile options for practical utilization. In particular, coupling at least one of the dipole forbidden transitions by an external radiation is expected to enhance the degree of detectable entanglement.

Generation of three-mode light entanglement via double-channel resonant four-wave mixing

We present a scheme to generate tripartite continuous-variable entanglement at the steady state by using an ensemble of N a five-level double- Λ − V atoms embedded in a three-mode cavity. The scheme is based on the two-channel resonant four-wave mixing, which leads to the parametric interaction between the sum mode of two fields from the upper double- Λ system and the mode from the lower V system. It is for such reason that three cavity modes are correlated and entangled. The present scheme has the following two striking features. One is that the steady-state tripartite entanglement is achieved under the full-resonance condition. Another feature is the fact that all the fields are coupled to the different atomic transitions, respectively. In particular, the tripartite entanglement occurs among the fields with the large frequency differences, which may be necessary for the progress of quantum information.

Generation of multicolored tripartite entanglement by frequency doubling in a two-port resonator

Multicolored multipartite entanglement is of great importance in quantum communication and quantum information networks. In this paper, we calculate the quantum fluctuations of the fundamental frequency pump beam and second-harmonic beams in a two-port frequency doubling resonator, and investigate the tripartite continuous-variable entanglement generated by this device for the first time, to our knowledge. The quantum correlation among fundamental frequency pump beam and two harmonic beams is studied using a necessary and sufficient criterion for Gaussian entanglement states, the positivity under partial transposition. It is found that two-color tripartite entanglement exists in a large range of pump intensities and analysis frequencies.

Three-mode entanglement and amplification in correlated spontaneous emission lasers

We propose a scheme for establishing three-coloured correlated spontaneous emission lasers and generating tripartite continuous-variable entanglement. The interaction of three cavity fields is formed by beating two of them in a two-channel Raman or two-photon transition and by correlating the two-mode beat and the third cavity field in the successive transitions. Using the sufficient inseparability criterion proposed by van Loock and Furusawa, we show that the three cavity fields are entangled when the numbers of photons are large and the saturation is weak. The upper limits of the photon numbers for the entanglement are dependent on the frequencies and amplitudes of the external coherent fields.

Tripartite entanglement from the cavity with second-order harmonic generation

In this paper, tripartite entanglement among the two pump fields and the second-order harmonic field is established in the process of type-II second-order harmonic generation (SHG) with a triply resonant optical cavity below threshold. A sufficient inseparability criterion for continuous-variable tripartite entanglement proposed by van Loock and Furusawa is used to evaluate the degree of the quadrature--phase-amplitude correlations between the three modes. The dependence of the entanglement on the pump parameter and analysis frequency is also discussed. It is shown that the best entanglement appears at the appropriate pump power and analysis frequency. These three entangled states with different frequencies generated directly from a simple SHG process make the scheme useful for the application in quantum communication and network.

Continuous variable tripartite entanglement and Einstein-Podolsky-Rosen correlations from triple nonlinearities

We develop three-mode generalizations of the Einstein-Podolsky-Rosen paradox, allowing us to define inequalities which may be used to indicate continuous-variable tripartite entanglement. We use these inequalities and an appropriate version of the previously developed van Loock-Furusawa inequalities to theoretically compare the continuous-variable tripartite entanglement available from the use of three concurrent x (2) nonlinearities and from three independent squeezed states mixed on beamsplitters.