Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator
aa r X i v : . [ qu a n t - ph ] S e p Analysis of a continuous-variable quadripartite cluster state froma single optical parametric oscillator
S. L. W. Midgley and M. K. Olsen
ARC Centre of Excellence for Quantum-Atom Optics, School ofMathematics and Physics, University of Queensland, QLD 4072, Australia.
A. S. Bradley
Jack Dodd Centre for Quantum Technology, Department of Physics,University of Otago, P. O. Box 56, Dunedin, New Zealand
O. Pfister
Department of Physics, University of Virginia, 382McCormick Road, Charlottesville, Virginia 22904-4714, USA (Dated: November 15, 2018)
Abstract
We examine the feasibility of generating continuous-variable multipartite entanglement in anintra-cavity quadruply concurrent downconversion scheme that has been proposed for the gen-eration of cluster states by Menicucci et al. [Physical Review Letters , 130501 (2008)]. Bycalculating optimized versions of the van Loock-Furusawa correlations we demonstrate genuinequadripartite entanglement and investigate the degree of entanglement present. Above the oscil-lation threshold the basic cluster state geometry under consideration suffers from phase diffusion.We alleviate this problem by incorporating a small injected signal into our analysis. Finally, weinvestigate squeezed joint operators. While the squeezed joint operators approach zero in the un-depleted regime, we find that this is not the case when we consider the full interaction Hamiltonianand the presence of a cavity. In fact, we find that the decay of these operators is minimal in acavity, and even depletion alone inhibits cluster state formation.
PACS numbers: 42.50.Dv,42.65.Yj,03.65.Ud,03.67.Mn,03.67.Lx . INTRODUCTION Cluster states are a class of graph states [1] which are of central importance as a re-source state for use in one-way, or measurement-based, quantum computing [2]. This typeof quantum computation, proposed in 2001 by Raussendorf and Briegel [3–5], differs signifi-cantly from the traditional circuit model of quantum computing in which unitary evolutionis achieved via a sequence of operations on single qubits [6, 7]. In order to realize one-wayquantum computing, a cluster state is generated and then a sequence of measurements areperformed on this highly entangled multipartite state [8].Most quantum computing proposals are based on qubits. Experiments have also beenperformed with qubits. In particular, Grover’s algorithm has been implemented using anoptical one-way quantum computer [9]. This approach relied on generating a four-qubitcluster state using a number of independent optical parametric oscillators (OPOs) and beamsplitters, and then performing measurements on this state.Lloyd and Braunstein [10] were the first to highlight the potential use of continuous-variables (CV) in quantum information. Multipartite CV entanglement has also been ex-tensively studied [11–14]. Since then, with the development of one-way quantum computing,the notion of using CV cluster states as a potential resource has arisen. Proposals specific toCV cluster state quantum computing and the generation of CV cluster states are numerous[8, 15–20]. They include schemes based on using a combination of single-mode squeezers andquantum non-demolition (QND) gates [17] or schemes that rely on single-mode squeezersand a network of beam splitters [18]. This field continues to attract interest and recently an-other proposal, based on realizing a CV cluster state using only a single QND gate, was putforward [21]. Experimental efforts to generate CV cluster states have also taken place, withthe first quadripartite cluster state generated in 2007 by Su et al. [22]. Similar experimentshave been performed by Yukawa et al. [23].Recent efforts [20, 24] have also focussed on the possibility of using a single OPO as ameans of generating CV multipartite entanglement, and in turn, a CV cluster state. In thisscheme, a single optical cavity is pumped by a number of field modes. The different modes ofthe resonator represent the multipartite entangled systems in the scheme. According to [20],the method generates a cluster state with the quadratures of the optical frequency comb ofthe OPO acting as a quantum computer register. In this article we extend this analysis and2onsider the feasibility of such a scheme.In [24] a single multimode OPA pumped by two field modes is considered in the un-depleted pump approximation. A correspondence is shown between the CV multipartiteentangled output from this scheme and a CV square-cluster state. It is the square-clusterOPO scheme proposed in [20, 24] that we consider in our work. Qubit graph states analo-gous to the CV square-cluster state have been studied extensively [25]. However, apart from[26] no study of the entanglement properties of the square-cluster OPO scheme has beenconducted. Specifically, we investigate this concurrent system in order to verify the pres-ence of CV quadripartite entanglement and determine whether or not a CV cluster state isproduced. This builds on our previous analysis of a similar scheme also based on concurrentnonlinearities [27], in which we showed that quadripartite entanglement is present for thecase of an OPO pumped by four field modes. Furthermore, our work here extends tripartiteschemes proposed in Refs. [28–30] by utilizing quadruply concurrent nonlinearities.This paper is organized as follows. Section II provides an overview of the defining re-lation for CV cluster states, presents the Hamiltonian for the scheme under considerationand describes the connection between this scheme and the generation of a cluster state.In addition, Sec. II C describes the van Loock-Furusawa (VLF) criteria which are used asa means of quantifying quadripartite entanglement. Section III considers the interactionHamiltonian in the undepleted pump approximation and gives the VLF correlations underthis approximation. In Sec. IV we present the full equations of motion for the system andcalculate the VLF correlations without a cavity present, using the positive- P method. Wealso find the field intensities and compare these to the intensities in the undepleted approx-imation. Section V provides an overview of the linearized fluctuation analysis used in thiswork to calculate the measurable output fluctuation spectra from the cavity. These outputspectra are also found in Sec. V and used to demonstrate violation of the optimized vanLoock-Furusawa criteria and hence, demonstrate quadripartite entanglement. The spectraare obtained above and below the oscillation threshold and the steady-state solutions aboveand below the threshold are also found along with an expression for the critical pumping.Finally, in Sec. VI we consider whether or not the entangled output beams produced inthe proposed scheme do in fact constitute a cluster state, by using the defining relation forcluster states and the squeezed joint quadrature operators.3 I. GENERATION OF A SQUARE-CLUSTER STATE FROM AN OPO
A CV multi-mode entangled state can be classified as a cluster state if the defining relationpresented in Refs. [18, 26] is satisfied. To consider this definition, we first define quadraturefield operators for each mode as,ˆ X i = ˆ a i + ˆ a † i , ˆ Y i = − i (ˆ a i − ˆ a † i ) (1)such that [ ˆ X i , ˆ Y i ] = 2 i . We also let X and Y represent column vectors of the amplitude andphase quadratures, respectively, for each field mode. The definition of a CV cluster state isthen any Gaussian state whose quadratures satisfy Y − A X −→ , (2)where A is the adjacency matrix representing the graph of a given CV state and the arrowspecifies that the condition holds in the limit of infinite (or large) squeezing. When thiscondition is satisfied for a particular A -matrix, the CV state is a cluster state. The adjacencymatrix can be weighted and represents the couplings between different nodes on the graphrepresenting the cluster state. A. Physical Description and Hamiltonian
The system we model in this paper is comprised of an optical cavity containing a χ (2) non-linear crystal. The optical cavity is pumped by two field modes to produce four low-frequency entangled output modes at frequencies ω , ω , ω , ω . Mode 1 is pumped at aparticular frequency and polarization such that it produces modes 3 and 6, as well as modes4 and 5. Mode 2 is pumped such that it gives rise to modes 5 and 6. A schematic of thesetup is shown in Fig. 1.The Hamiltonian for this six-mode system is given by H = H int + H free + H pump + H bath , (3)where the interaction Hamiltonian is H int = i ~ [ χ ˆ a ˆ a † ˆ a † + χ ˆ a ˆ a † ˆ a † + χ ˆ a ˆ a † ˆ a † ] + h.c. , (4)4 IG. 1: (Color online) A χ (2) crystal inside a pumped Fabry-P´erot cavity. Pump lasers drivetwo intracavity modes with frequencies ω and ω (represented by circles and squares), which aredown-converted to four output modes with frequencies ω , ω , ω and ω . with the χ i representing the effective nonlinearities and ˆ a i and ˆ a † i denoting the bosonicannihilation and creation operators, respectively, for the intra-cavity modes at frequencies ω i . The pumping Hamiltonian which describes the cavity driving fields, in the appropriaterotating frame is H pump = i ~ X i =1 h ǫ i ˆ a † i − ǫ ∗ i ˆ a i i , (5)and the cavity damping Hamiltonian is given by H bath = ~ X i =1 h ˆΓ i ˆ a † i + ˆΓ † i ˆ a i i , (6)where ǫ i are the classical pumping laser amplitudes for modes i , and the ˆΓ i are the annihi-lation operators for bath quanta, representing losses through the cavity mirrors. B. The Undepleted Pump Approximation
Prior to studying the full Hamiltonian in the presence of an optical cavity, it is useful toconsider the properties of the Hamiltonian within the undepleted pump approximation. Thisapproximation assumes that all the high frequency pump modes remain highly populatedthroughout the interaction process, with no depletion taking place. Specifically, with the5avity absent we set ξ = χ h ˆ a (0) i and ξ = χ h ˆ a (0) i where ξ i are positive, real constants.Under this approximation, the interaction Hamiltonian can be written as follows, H int = i ~ ξ X j = m,n G mn (cid:2) ˆ a † m ˆ a † n − ˆ a m ˆ a n (cid:3) , (7)where we assume ξ = ξ i ( i = 1 , j represents all permutations of the low frequency modesand G mn are the components of the matrix, G = . (8)Inspecting the form of this G -matrix for the system, it can be seen that it correspondsto the graph in Fig. 1. The four nodes represent the four low frequency modes and thelines connecting the nodes represent the nonlinear coupling of the modes. Such a graph,representing the Hamiltonian of Eq. (7), has been shown to be equivalent to a four nodesquare-cluster state [24, 26]. C. Criteria for Quadripartite Entanglement
To determine whether or not a square-cluster state generated by the single OPO schemepresented in Sec. II is fully inseparable, it is possible to construct multipartite entanglementwitnesses. These are observables that allow one to distinguish multipartite entangled statesfrom separable states. In order to detect CV multipartite entanglement we use the set ofsufficient conditions proposed by van Loock and Furusawa (VLF) [31], which are a gener-alization of the conditions for CV bipartite entanglement [32, 33]. As shown in [27], theseconditions may be optimized for the verification of genuine quadripartite entanglement. Itshould be noted that other multi-partite entanglement witnesses also exist [34].Using the quadrature definitions in Eq. (1), the optimized inequalities which must be si-multaneously violated by the low frequency modes in order to demonstrate CV quadripartiteentanglement are given by, 6 ( ˆ X − ˆ X ) + V ( ˆ Y + g ˆ Y + g ˆ Y + ˆ Y ) ≥ , (9) V ( ˆ X − ˆ X ) + V ( g ˆ Y + ˆ Y + ˆ Y + g ˆ Y ) ≥ , (10) V ( ˆ X − ˆ X ) + V ( g ˆ Y + g ˆ Y + ˆ Y + ˆ Y ) ≥ , (11)where V ( ˆ A ) = h ˆ A i − h ˆ A i denotes the variance and the g i ( i = 3 , , ,
6) are arbitraryreal parameters that are used to optimize the violation of these inequalities. Eq. (9) andEq. (10) are minimized with respect to g , and g , , respectively. We then solve the resultingequations to obtain the optimized expressions, g = V ( V + V ) − V ( V + V ) V − V V , (12) g = V ( V + V ) − V ( V + V ) V − V V , (13) g = V ( V + V ) − V ( V + V ) V − V V , (14) g = V ( V + V ) − V ( V + V ) V − V V , (15)where V ij = h ˆ Y i ˆ Y j i + h ˆ Y j ˆ Y i i − h ˆ Y i ih ˆ Y j i (16)represents the covariances. For the case where i = j the covariance, denoted V i , reduces tothe usual variance, V ( ˆ Y i ). III. THE HEISENBERG EQUATIONS
Within the undepleted pump approximation we can calculate the VLF criteria fromSec. II C in order to verify the presence of multipartite entanglement. The Heisenbergequations of motion for the field operators are given by,7 ˆ a dt = ξ ˆ a † , (17) d ˆ a dt = ξ ˆ a † , (18) d ˆ a dt = ξ ˆ a † + ξ ˆ a † , (19) d ˆ a dt = ξ ˆ a † + ξ ˆ a † , (20)and in turn these equations can be written in terms of the quadrature operators, d ˆ X dt = ξ ˆ X , (21) d ˆ Y dt = − ξ ˆ Y , (22) d ˆ X dt = ξ ˆ X , (23) d ˆ Y dt = − ξ ˆ Y , (24) d ˆ X dt = ξ ˆ X + ξ ˆ X , (25) d ˆ Y dt = − ξ ˆ Y − ξ ˆ Y , (26) d ˆ X dt = ξ ˆ X + ξ ˆ X , (27) d ˆ Y dt = − ξ ˆ Y − ξ ˆ Y . (28)We solve these equations to find analytic solutions for the quadrature operators as functionsof their initial values, and in turn the optimized VLF criteria can be calculated. Not a greatdeal is learnt from the exact form of these rather complicated analytic expressions. Hence,we plot the solutions for the optimized VLF criteria for the cases of equal and unequal valuesof ξ i .We first investigate solutions with both ξ i equal. That is, we assume that ξ = ξ , and plotthe optimized VLF criteria. The correlations I , I and I correspond to Eq. (9) - Eq. (11),respectively. Therefore, a value less than four violates the VLF inequalities. In Fig. 2 wesee that quadripartite entanglement is present since all three inequalities are simultaneouslyviolated. 8 ξ t V L F c o rr e l a ti on s I I I FIG. 2: (Color online) Optimized van Loock-Furusawa correlations, I and I (black solid line)and I (red dashed-dot line), found by solving the Heisenberg equations of motion in the un-depleted pump approximation. Having all three of the correlations drop below 4 is sufficient todemonstrate quadripartite entanglement. All quantities depicted here and in subsequent graphsare dimensionless. We also consider solutions with unequal ξ i , for the case where ξ = 0 . ξ . The VLFcorrelations are shown in Fig. 3. The violation of I and I in this case is less than in thesymmetric case shown in Fig. 2, however, we still observe a substantial violation of theseVLF inequalities. Furthermore, the violation of I is greater here than in the symmetriccase. ξ t V L F c o rr e l a ti on s I I I FIG. 3: (Color online) Optimized van Loock-Furusawa correlations, I and I (black solid line)and I (red dashed-dot line) with ξ = 0 . ξ . V. THE POSITIVE- P EQUATIONS
We now turn to an analysis of the setup introduced in Sec. II A by considering the fullinteraction Hamiltonian and introducing a pumped, resonant optical cavity to house thenonlinear media. The master equation for this system can be derived using a standardapproach [35] and is given by, ∂ ˆ ρ∂t = − i ~ h ˆ H pump + ˆ H int , ˆ ρ i + X i =1 γ i D i [ ˆ ρ ] (29)where γ i represent the cavity losses at each frequency and D i [ ˆ ρ ] = 2ˆ a i ˆ ρ ˆ a † i − ˆ a † i ˆ a i ˆ ρ − ˆ ρ ˆ a † i ˆ a i isthe Lindblad superoperator [35] under the zero-temperature Markov approximation. Fromthe master equation it is possible to derive a set of stochastic differential equations (SDEs)and then investigate the intra-cavity dynamics.We use the positive- P representation [36, 38] to perform a full quantum analysis. Inthis approach the master equation, Eq. (29), is mapped onto a Fokker-Planck equation(FPE) for the positive- P function to arrive at a set of c -number SDEs. To obtain the SDEsit is necessary that the diffusion matrix of the FPE is positive-definite. In the positive- P approach this is achieved by defining two independent stochastic fields α i and α + i and makinga correspondence between these operators and the mode operators ˆ a i and ˆ a † i , respectively.This approach allows us to perform stochastic calculations of normally-ordered operatormoments, for example, in the limit of a large number of trajectories ( α + j ) m α ni = h : (ˆ a † j ) m ˆ a ni : i .Therefore, despite being probabilistic, the positive- P method allows for a full quantumtreatment of the system when a sufficiently large number of trajectories is used. Followingthis approach, the resulting 12 ×
12 diffusion matrix is of the form, D = × × × d , (30)where r × c are null matrices and d is an 8 × = χ α
00 0 0 0 0 0 0 χ α +1 χ α χ α +1 χ α χ α
00 0 0 χ α +1 χ α +2 χ α χ α χ α +1 χ α +2 . (31)After factorizing D to find the noise terms, we find the set of Itˆo SDEs in the positive- P representation. The evolution equations for the high frequency fields are, dα dt = ǫ − χ α α − χ α α − γ α ,dα dt = ǫ − χ α α − γ α , (32)as well as the equations found by interchanging α i and α + i . For the low frequency fields wefind, dα dt = χ α α +6 − γ α + r χ α η ( t ) + iη ( t )) ,dα dt = χ α α +5 − γ α + r χ α η ( t ) + iη ( t )) ,dα dt = χ α α +4 + χ α α +6 − γ α + r χ α η ( t ) − iη ( t ))+ r χ α η ( t ) + iη ( t )) ,dα dt = χ α α +3 + χ α α +5 − γ α + r χ α η ( t ) − iη ( t ))+ r χ α η ( t ) − iη ( t )) , (33)11nd also the equations found by swapping α i with α + i and η i ( t ) with η i +2 ( t ). The η i ( t )are real, independent, Gaussian noise terms which have the correlations η i ( t ) = 0 and η i ( t ) η j ( t ′ ) = δ ij δ ( t − t ′ ). We assume throughout this article that all the intracavity modesare resonant with the cavity and as a result no detuning terms are included.An initial insight into the downconversion processes can be garnered by neglecting thepump and loss terms in Eqs. (32) and Eqs. (33) momentarily and simply looking at thedynamics with depletion present. The results of such a positive- P simulation are shown inFig. 4 and compared to the undepleted pump approximation results for the low frequencymodes. Specifically, we plot the intensities of the fields where the horizontal axis is a scaledinteraction time, with ζ = χ | α , (0) | and χ = χ i ( i = 1 , P results as depletion becomes significant. ζ t | α i | | α | | α | | α | | α | FIG. 4: (Color online) Intensities of the high (red dashed-dot and blue solid lines) and low frequencymodes (dashed green and small dashed purple lines) calculated using the positive- P equationswithout a cavity present. The number of trajectories is 300,000. The parameters used are χ , =0 . α , (0) = 1 × and α , , , (0) = 0. The black solid lines show the low frequency modes inthe undepleted pump approximation. We also calculate the VLF correlations using the positive- P equations without a cavitypresent. In particular, we use 300,000 trajectories and all other parameters are the same asin Fig. 4. The results compare well with the undepleted pump results shown in Fig. 2 andare not visibly different for a large number of trajectories.12 . LINEARIZED FLUCTUATION ANALYSIS Returning to our analysis of the more experimentally relevant system introduced inSec. II A, we can consider the full quantum dynamics with depletion present and wherethe interaction occurs inside an optical cavity. This type of scheme is currently under ex-perimental study by one of us [24, 26] at the University of Virginia.We undertake a linearized fluctuation analysis [35] to obtain output spectral correlationsfor the cavity from the intracavity spectra. This is achieved by first linearizing the Eqs. (32)and (33) around the classical steady-state solutions. In the usual manner [35, 37], we thenfind a set of evolution equations for the fluctuations. To begin we neglect the noise termsin Eq. (33) so that α + i → α ∗ i . We then set α i = ¯ α i + δα i , where ¯ α i is a mean value and δα i represents the fluctuations. This gives a set of classical equations for the mean valuesand from these it is possible to obtain steady-state solutions. It also allows one to obtainlinearized fluctuation equations from which spectral correlations can be obtained.We find that an oscillation threshold is present in our symmetric system. Below thisthreshold we solve the set of classical equations for the mean steady-state values. We findthat the stationary solutions below the threshold value are¯ α i = ǫ i γ i for i ∈ { , } , ¯ α i = 0 for i ∈ { , , , } . (34)Returning to the linearized fluctuation analysis, to first order in the fluctuations theequations of motion for the fluctuations, δα = [ δα , δα +1 , δα , δα +2 , . . . , δα , δα +6 ] T , are givenby, d δα = − ¯ Aδα dt + ¯ B d W , (35)where ¯B is the noise matrix of Eq. (33) with the steady-state values inserted, d W is a vectorof Wiener increments [38] and ¯ A is the drift matrix with the steady-state values inserted asfollows, 13 A = A A − ( A ∗ ) T A , (36)where A = γ γ γ
00 0 0 γ , (37) A = χ ¯ α χ ¯ α χ ¯ α χ ¯ α χ ¯ α ∗ χ ¯ α ∗ χ ¯ α ∗ χ ¯ α ∗ χ ¯ α χ ¯ α
00 0 0 0 0 χ ¯ α ∗ χ ¯ α ∗ , (38)and A = γ − χ ¯ α γ − χ ¯ α ∗
00 0 γ − χ ¯ α γ − χ ¯ α ∗ − χ ¯ α γ − χ ¯ α − χ ¯ α ∗ γ − χ ¯ α ∗ − χ ¯ α − χ ¯ α γ − χ ¯ α ∗ − χ ¯ α ∗ γ . (39)For the linearized fluctuation analysis to be valid the fluctuations must remain smallcompared to the mean values and the eigenvalues of the drift matrix ¯ A must have nonegative real part. The eigenvalues are given by,14 , = γ ,λ = γ − h ǫ χ γ + r(cid:16) ǫ χ γ (cid:17) + (cid:16) ǫ χ γ (cid:17) i ,λ = γ + 12 h ǫ χ γ − r(cid:16) ǫ χ γ (cid:17) + (cid:16) ǫ χ γ (cid:17) i ,λ = γ + 12 h − ǫ χ γ + r(cid:16) ǫ χ γ (cid:17) + (cid:16) ǫ χ γ (cid:17) i ,λ = γ + 12 h ǫ χ γ + r(cid:16) ǫ χ γ (cid:17) + (cid:16) ǫ χ γ (cid:17) i . (40)There are six other eigenvalues but each of these is degenerate with one of the eigenvalues inEq. (40). From these expressions it is clear that only λ , can have a negative real part. Forour chosen parameters and in the pump range 0 < ǫ , < λ that hasa real part which go from positive to negative. This is depicted in Fig. 5 where we observea plateau of stability for a range of pump values and the transition to an unstable region,where the negative real part of λ is plotted. In the latter region the linearized fluctuationanalysis is not valid.In the following we consider the completely symmetric case where the two pumpinginputs, ǫ , , are equal and given by ǫ , the two nonlinearities, χ , , are equal and denoted χ and finally we assume all the cavity losses are equal and given by γ = γ i for i = 1 , . . . , ǫ c , at which the oscillation threshold is reached. Furthermore, we confirm via a positive- P simulation that this is the threshold pumping value for which downconversion begins topopulate the low frequency modes. For our symmetric system, we find the critical pumpingamplitude to be ǫ c = γ χ h
21 + √ i , (41)where ǫ c = 61 .
10 20 30 40 50 0 10 20 30 40 50−0.6−0.4−0.20 ε ε R e [ λ − ] FIG. 5: (Color online) The region of stability (the plateau) and the transition to instability for arange of pump amplitudes, ǫ and ǫ , found by investigating the behavior of the negative real partof the eigenvalue λ , denoted Re[ λ − ]. S ( ω ) = ( ¯ A + iω ¯ B ¯ B T ( ¯ A T − iω − . (42)All the correlations required to study the measurable extracavity spectra are contained in thisintracavity spectral matrix. Equation (42) is related to the measurable output fluctuationspectra by standard input-output relations for optical cavities [39]. In particular, the spectralvariances and covariances have the general form,S outX i ( ω ) = 1 + 2 γ i S X i ( ω ) , S outX i ,X j ( ω ) = 2 √ γ i γ j S X i ,X j ( ω ) . (43)Similar expressions can be derived for the ˆ Y quadratures. For brevity, we use I outij (ie. anyof I out , I out , I out ) to represent the three output spectral correlations corresponding to theoptimized VLF correlations, I ij , of Eq. (9) to Eq. (11). That is, the same inequalities asgiven in Sec. II C in terms of variances also hold when expressed in terms of the outputspectra. It is these quantities that can be measured in experiments and that we calculate inthe remainder of this article. A. Output spectra below threshold
We now use these steady-state values to calculate the spectra. In Fig. 6 we plot theoutput spectral correlations, I outij , as a function of frequency below threshold for a particular16umping rate. The pump rate is chosen to be ǫ = 0 . ǫ c as this gives the best violation ofthe inequalities for our choice of parameters. The correlations I out and I out are equal and givethe maximum violation of the inequalities. The correlation I out is also shown. We observethat the spectra bifurcates such that no entanglement is present close to zero frequency. Forlarge frequencies I outij ( ω ) →
4. This is the uncorrelated limit for the optimized expressions.Collectively, however, these three output spectral correlations confirm that quadripartiteentanglement is present below threshold, since all I outij drop below four over a range offrequencies. −15 −10 −5 0 5 10 1511.522.533.544.55 ω (units of γ ) I ou t ij ( ω ) FIG. 6: (Color online) The spectral correlations below threshold, I outij ( ω ), for the intracavity quadru-ply concurrent scheme. The parameter values are χ = 0 . γ = 1 and ǫ = 0 . ǫ c . This value of ǫ gives best violation of the inequalities. The correlation I out is given by the red dashed-dot line,while I out and I out are equal and given by the black solid line. B. Output spectra above threshold
It is not as straightforward to obtain analytic expressions for the low and high frequencymodes above threshold. As was seen from the graph in Fig. 1, modes 3 and 4 are onlycoupled to 6 and 5, respectively, while 5 and 6 are also coupled with each other. Thisasymmetry results in an undefined overall phase so that, above the oscillation threshold, thelow frequency modes suffer phase diffusion, degrading the measurable correlations. We avoidthis by injecting a small signal [40, 41] in one of the low frequency modes, thus stabilizingthe overall phase. In particular, we find the steady-state solutions numerically using the17ositive- P equations with an injected signal, ǫ = 0 .
5, in the relevant evolution equations.The pump is far more intense than the injected signal, with ǫ chosen to be approximatelyone per cent of the pump strength.It is also necessary to make use of an injected signal when calculating the output spec-tra above threshold. This enables us to calculate the spectra for a range of pump valuesabove threshold. In Fig. 7 we plot the output spectral correlations above threshold for apumping rate of ǫ = 1 . ǫ c . We see that all three of the VLF inequalities are violated for arange of frequencies and therefore quadripartite entanglement is present above the thresholdcondition. As in the below threshold case, we also see the spectra bifurcating close to zerofrequency and hence not demonstrating entanglement in this region. −15 −10 −5 0 5 10 1511.522.533.544.55 ω (units of γ ) I ou t ij ( ω ) FIG. 7: (Color online) The spectral correlations above threshold for the intracavity quadruplyconcurrent scheme. The parameter values are χ = 0 . γ = 1 and ǫ = 1 . ǫ c . The otherparameters are the same as in the below threshold case. The correlation I out is repesented by thered dashed-dot line. The correlations I out and I out are equal and represented by the black solidline. We also calculate the maximum quadripartite entanglement for a range of pump fieldamplitudes below and above threshold. This result is shown in Fig. 8, where we plot theminimum value of the output spectra at any frequency, as a function of ǫ/ǫ c . From this plotit is clear that quadripartite entanglement persists below threshold and well above threshold.In Fig. 9 and Fig. 10 we plot the VLF correlations as a function of frequency and pumpingrate. In Fig. 9, we crop the peak of the spectra for visualization purposes. The value of thiscorrelation would otherwise increase to a maximum value of I out , ≈
60. From these surface18lots we can also see the onset of the bifurcation. ε / ε c m i n [I ou t ij ( ω )] FIG. 8: (Color online) Maximum quadripartite entanglement as a function of the ratio of thecavity pumping to the pumping threshold. All other cavity parameters are the same as in Fig. 6and Fig. 7. Again, the correlations I out and I out are equal for these parameters and hence cannotbe differentiated on the graph. Note that for ǫ/ǫ c = 1 the validity of the results is limited as thelinearized analysis is no longer valid.FIG. 9: (Color online) Plot of the spectral correlations I out and I out as a function of the pumpingrate ǫ/ǫ c and frequency ω (in units of γ ). All other cavity parameters are the same as in Fig. 8.Note that we crop the peak of the spectra for visualization purposes. IG. 10: (Color online) Plot of the spectral correlation I out as a function of the pumping rate ǫ/ǫ c and frequency ω (in units of γ ). All other cavity parameters are the same as in Fig. 8. VI. CONFIRMING THE CLUSTER STATE DEFINING RELATION
Now that we have demonstrated genuine quadripartite entanglement below and abovethreshold, we turn our attention to whether or not the state generated by the scheme is infact a cluster state. To do this we employ the defining relation for cluster states, given byEq. (2). In the undepleted pump regime, there exists a connection between this conditionand the squeezed joint operators for the system [23, 24, 26, 42]. Under certain mode rotationsand in the limit of infinite (or large) squeezing the operators for the system give rise to thecluster state equation of Eq. (2) for a square-cluster state. We stress that this equivalence hasonly been shown to hold within the undepleted pump approximation. We briefly overviewthe equivalence here. The squeezed joint quadrature operators are simply the eigenvectors ofthe system found by solving the Heisenberg equations of motion for the system representedby the Hamiltonian in Eq. (7) with the adjacency matrix given by Eq. (8). For our schemethe joint quadrature operators are, O = ( − c ˆ X + c ˆ X − ˆ X + ˆ X ) e − c r ,O = ( − c ˆ X − c ˆ X + ˆ X + ˆ X ) e − c r ,O = ( c ˆ Y + c ˆ Y + ˆ Y + ˆ Y ) e − c r ,O = ( c ˆ Y − c ˆ Y − ˆ Y + ˆ Y ) e − c r , (44)20here c = ( √ − / c = ( √ / r = ξt is the squeezing parameter. Thecommon eigenstate of these joint operators is a quadripartite entangled state that tendstowards a cluster state when r → ∞ . That is, any squeezing operator combinations that areproportional to the squeezing factor e − r will automatically satisfy the cluster state conditionof Eq. (2) in the limit of infinite (or large) squeezing. We investigate this by example andcompare these operators to the cluster state defining relation, Y − A X −→ . We chooseone possible solution for the adjacency matrix A , given by A = 12 √
50 0 √ √ √ , (45)and which corresponds to a weighted square graph CV cluster state. From Eq. (2) theresulting cluster state equations are, Y − X − √ X → ,Y − √ X − X → ,Y − X − √ X → ,Y − √ X − X → , (46)where the arrow again represents the limit r → ∞ . In the undepleted regime and in thelimit of infinite (or large) squeezing these equations are equivalent to Eqs. (44) under certainmode rotations. Specifically, if we rotate modes 5 and 6 by π/ X → Y and Y → − X the equivalence can be seen [24]. Hence, by simply verifying that the squeezedjoint quadrature operators approach zero in the limit r → ∞ , we can determine whether ornot the proposed scheme gives rise to a cluster state. As an aside, we actually calculate thevariances of the squeezed operators of Eqs. (44), and ensure that these approach zero, asthese are the quantities we have access to in our simulations.We first confirm that the squeezed operators approach zero as expected [24] in the unde-pleted case. In Fig. 11 we calculate the squeezed operators from the Heisenberg equations(solid lines). We see that O = O and O = O and both sets of operators approach zero as21 = ξt approaches infinity. Therefore, the cluster state equation is confirmed in the unde-pleted regime. In Fig. 11 we also plot the squeezed operators calculated from the positive- P equations in the absence of an optical cavity (dashed lines). We note that the positive- P results go to zero but start to increase from zero at longer times. Moreover, as O , ap-proaches zero, O , increases and does not approach zero. For the cluster state relation tobe satisfied, all squeezing operators must approach zero simultaneously. As seen here this isnot the case, except at around ξt = 2 . ξ t S qu eeze d J o i n t O p e r a t o r s FIG. 11: (Color online) Squeezed joint operators, O i , in the undepleted pump approximation(black solid lines) and positive- P results without a cavity (colored dashed lines). The number oftrajectories for the stochastic simulation is 300,000 with χ = 0 .
01 and α , (0) = 1 × . In Fig. 12 we plot the squeezing operators for the full interaction Hamiltonian in thepresence of a cavity, based on positive- P simulations. This result is more relevant to thescheme that would be realized in the proposed experiment. Unlike the undepleted case, thesqueezing operators calculated in this case do not approach zero. Instead, the squeezed oper-ators plateau at non-zero values in the steady-state. We find that the decay of the squeezedjoint operators is in fact quite minimal (of order 20%). This indicates that generating acluster state from the output of a single OPO may present a challenge.22 t (units of γ −1 ) S qu eeze d J o i n t O p e r a t o r s FIG. 12: (Color online) Squeezed joint operators, O i , calculated using the positive- P equations inthe presence of a cavity. The operators are calculated for the below threshold case using 50,000trajectories, with ǫ = 0 . ǫ c , χ = 0 .
01 and γ = 1. VII. CONCLUSIONS
We have examined an experimentally feasible quadruply concurrent intracavity schemeas proposed in Refs. [20, 24]. We investigated this system as a potential source of CVquadripartite entanglement and as a candidate for the generation of a CV square-clusterstate. We verified the presence of quadripartite entanglement via optimized versions of thewell-known VLF correlations. The proposed scheme provided a source of bright entangledoutput beams above the critical pumping threshold when an injected signal was incorporatedinto the analysis. Below threshold we also detected quadripartite entanglement with themaximum entanglement predicted near threshold.We have also calculated the squeezed joint operators to determine if the state produced bythe proposed scheme is a CV cluster state. Within the undepleted pump approximation, weconfirmed that the squeezed joint operators approached zero in the limit of large squeezing.However, in our analysis of the more experimentally realistic case where depletion is presentand a cavity is used to house the nonlinear media, we did not observe that the squeezingoperators approached zero as required by the cluster state defining relation. Furthermore,including depletion alone was sufficient to inhibit cluster state formation and once the cavitywas also included the decay of the squeezing operators was found to be minimal. This leadsus to conclude that the utility of this system as a source of cluster states depends on the23egree to which less than perfect squeezing is acceptable. Overall, solution of the undepletedHeisenberg equations of motion can only be used as a general guide to the performance ofsuch a system once it is placed inside an optical cavity. Without the cavity, we also findthat an energy-conserving positive- P simulation of the basic downconversion process showsinhibited cluster state formation due to pump depletion. VIII. ACKNOWLEDGMENTS
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