Tripartite correlations over two octaves from cascaded harmonic generation
M. K. Olsen
quant-ph









mko]M.K. Olsen
mko@physics.uq.edu.au



School of Mathematics and Physics, University of Queensland, Brisbane, 
Queensland 4072, Australia.




We analyse the output quantum tripartite correlations from an intracavity nonlinear optical system which uses cascaded nonlinearities to produce both second and fourth harmonic outputs from an input field at the fundamental frequency. Using fully quantum equations of motion, we investigate two parameter regimes and show that  the system produces 
tripartite inseparability, entanglement and EPR steering, with the detection of these depending on the correlations being considered.





Cascaded systems, entanglement, steering.







§ INTRODUCTION



Fourth harmonic generation has not received a huge amount of attention in the scientific literature, possibly because materials with the nonlinearity needed for a five wave mixing process are difficult to find. Despite this inherent problem, and the difficulty of finding materials that are transparent over two octaves, Komatsu have successfully produced fourth harmonic from Li_2B_4O_7 crystal, with a conversion efficiency of 20% <cit.>. The advent of quasi-periodic superlattices meant that higher than second order processes were now available, with Zhu producing third harmonic by coupling second harmonic (SHG) and sum-frequency generation in 1997 <cit.>. Using CsLiB_6O_10, Kojima were able to produce fourth harmonic at a 10 kHz repetition rate by 2000 <cit.>. Broderick have produced fourth harmonic from a cascaded SHG process using a  HeXLN crystal <cit.> tuneable for both processes at the same temperature. Südmeyer produced fields at both second and fourth harmonics using an intracavity cascaded process with LBO and BBO crystals, with greater than 50% efficiency in 2007 <cit.>. More recently, Ji have generated light at 263 nm from a 1053 nm input, using KD^*P and NH_4H_2PO_4 crystals with non-critical phase matching <cit.>.

The theoretical examination of the quantum statistical properties of fourth harmonic generation began with Kheruntsyan , who analysed an intracavity cascaded frequency doubler process <cit.>. The authors adiabatically eliminated the highest frequency mode to calculate squeezing in the lower modes, also finding self-pulsing in the intensities. Yu and Wang <cit.> performed an analysis of the system without any elimination, starting with the full positive-P representation <cit.> equations of motion. Linearising around the steady-state solutions of the semi-classical equations, they performed a stability analysis and examined the entanglement properties using the method of symplectic eigenvalues <cit.>. More recently, Olsen has examined the quantum statistical properties of the system <cit.>, finding that quadrature squeezing and bipartite entanglement and asymmetric Einstein-Podolsky-Rosen (EPR) steering <cit.> are available for some of the possible bipartitions.  

In this work we extend previous analyses by examining various correlations often used in continuous variable systems to detect tripartite inseparability, entanglement, and steering. We begin with the two types of van Loock-Furusawa (vLF) inequalities <cit.> and their refinements by Teh and Reid <cit.> for mixed states. Following from that, we will use three mode EPR <cit.> inequalities developed by Olsen, Bradley and Reid (OBR) <cit.>, to investigate whether two members of a possible tripartition can combine to steer the third and whether our results can indicate genuine multipartite steering <cit.>. We investigate two different regimes with changes in the pumping rate, the loss rates, and the ratio of the two χ^(2) nonlinearities.




§ HAMILTONIAN AND EQUATIONS OF MOTION



The system consists of three optical fields interacting in nonlinear media, which could either be a periodically poled dielectric or two separate nonlinear crystals held in the same optical cavity. The equations of motion are the same for both. The fundamental field at ω_1, which will be externally pumped, is represented by â_1. The second harmonic, at ω_2=2ω_1, is represented by â_2, and the fourth harmonic, at ω_3=4ω_1, is represented by â_3. The nonlinearity κ_1 couples the fields at ω_1 and ω_2, while κ_2 couples those at ω_2 and ω_3. The unitary interaction Hamiltonian in a rotating frame is then written as 

    H_int = iħ/2[ κ_1(â_1^2â_2^†-â_1^† 2â_2)+κ_2(â_2^2â_3^†-â_2^† 2â_3) ].

The cavity pumping Hamiltonian is

    H_pump = iħ(ϵâ_1^†-ϵ^∗â_1),

where ϵ represents an external pumping field which is usually taken as coherent, although this is not necessary <cit.>. The damping of the cavity into a zero temperature Markovian reservoir is described by the Lindblad superoperator 

    Lρ = ∑_i=1^3γ_i(2â_iρâ_i^†-â_i^†â_iρ-ρâ_i^†â_i),

where ρ is the system density matrix and γ_i is the cavity loss rate at ω_i. We will treat all three fields as being at resonance with the optical cavity, which means that we do not need to examine quadrature correlations at all angles, but that the canonical X̂ and Ŷ quadratures are sufficient.

Following the usual procedures <cit.>, we proceed via the von Neumann and Fokker-Planck equations to derive equations of motion in the positive-P representation <cit.>,

    dα_1/dt   =   ϵ-γ_1α_1+κ_1α_1^+α_2+√(κ_1α_2) η_1, 
    
    dα_1^+/dt   =   ϵ^∗-γ_1α_1^++κ_1α_1α_2^++√(κ_1α_2^+) η_2, 
    
    dα_2/dt   =    -γ_2α_2+κ_2α_2^+α_3-κ_1/2α_1^2+√(κ_2α_3) η_3, 
    
    dα_2^+/dt   =    -γ_2α_2^++κ_2α_2α_3^+-κ_1/2α_1^+ 2+√(κ_2α_3^+) η_4, 
    
    dα_3/dt   =    -γ_3α_3-κ_2/2α_2^2, 
    
    dα_3^+/dt   =    -γ_3α_3^+-κ_2/2α_2^+ 2,

noting that these have the same form in either Itô or Stratonovich calculus <cit.>.
The complex variable pairs (α_i,α_j^+) correspond to the operator pairs (â_i,â_j^†) in the sense that stochastic averages of products converge to normally-ordered operator expectation values, e.g. α_i^+ mα_j^n→⟨â_i^† mâ_j^n⟩. The η_j are Gaussian noise terms with the properties η_i=0 and η_j(t)η_k(t')=δ_jkδ(t-t').

  


§ QUANTUM CORRELATIONS

      

Before defining the inequalities we will use, we define the amplitude quadratures of the three interacting fields as

    X̂_i   =   â_i+â_i^†, 
    
    Ŷ_i   =    -i(â_i-â_i^†),

with the Heisenberg uncertainty principal demanding that the product of the variances, V(X̂_i)V(Ŷ_i)≥ 1.

For three mode inseparability and entanglement, we use the van-Loock Furusawa inequalities <cit.>, which have proven useful for other cascaded optical systems <cit.>. The first of these is 

    V_ij = V(X̂_i-X̂_j)+V(Ŷ_i+Ŷ_j+g_kŶ_k) ≥ 4,

for which the violation of any two demonstrates tripartite inseparability. The g_j, which are arbitrary and real, can be optimised <cit.>, using the variances and covariances, as

    g_i = -V(Ŷ_i,Ŷ_j)+V(Ŷ_i,Ŷ_k)/V(Ŷ_i).

Teh and Reid <cit.> have shown that, for mixed states, tripartite entanglement is demonstrated if the sum of the three correlations is less than 8, with genuine tripartite EPR (Einstein-Podolsky-Rosen)-steering <cit.> requiring a sum of less than 4. 

The second set set of vLF inequalities,

    V_ijk = V(X̂_i-X̂_j+X̂_k/√(2))+V(Ŷ_i+Ŷ_j+Ŷ_k/√(2)) ≥ 4,

requires the violation of only one to prove tripartite inseparability. Teh and Reid <cit.> also showed that for mixed states any one of these less than 2 demonstrates genuine tripartite entanglement, while one of them less than 1 demonstrates genuine tripartite EPR steering. Because our nonlinear system is held in a cavity which is open to the environment, we are working with mixed states here. 

For multipartite EPR-steering, Wang showed that the steering of a given quantum mode is allowed when not less than half of the total number of modes take part in the steering group <cit.>. In a tripartite system, this means that measurements on two of the modes are needed to steer the third. In order to quantify this, we will use the correlation functions developed by Olsen, Bradley, and Reid <cit.>. With tripartite inferred variances as

    V_inf^(t)(X̂_i)    =    V(X̂_i)-[V(X̂_i,X̂_j±X̂_k)]^2/V(X̂_j±X̂_k), 
    
    V_inf^(t)(Ŷ_i)    =    V(Ŷ_i)-[V(Ŷ_i,Ŷ_j±Ŷ_k)]^2/V(Ŷ_j±Ŷ_k),

we define

    OBR_ijk = V_inf^(t)(X̂_i)V_inf^(t)(Ŷ_i),

so that a value of less than one means that mode i can be steered by the combined forces of modes j and k. According to the work of He and Reid <cit.>, genuine tripartite steering is demonstrated whenever

    OBR_ijk+OBR_jki+OBR_kij < 1.

In this work we will use only the plus signs in Eq. <ref>, which will be denoted on the figure axes as OBR_ijk^+. We found that this gave greater violations of the inequalities in some cases, although the results were not qualitatively different.




§ STEADY-STATE SPECTRAL CORRELATIONS



We find that the semi-classical and quantum solutions for the intensities are identical until a certain pump power, after which the system enters a self-pulsing regime <cit.>. Below this pump power, the steady-state solutions for the field amplitudes found from the integration of the full positive-P equations and their semiclassical equivalents  are identical. The semiclassical equations are found by removing the noise terms from Eq. <ref>, and have been solved numerically here. 

The measured observables of an intracavity process are usually the output spectral correlations, which are accessible using homodyne measurement techniques <cit.>. These are readily calculated in the steady-state by treating the system as an Ornstein-Uhlenbeck process <cit.>. In order to do this, we begin by expanding the positive-P variables into their steady-state expectation values plus delta-correlated Gaussian fluctuation terms, e.g.

    α_ss→⟨â⟩_ss+δα.

Given that we can calculate the ⟨â_i⟩_ss, we may then write the equations of motion for the fluctuation terms. The resulting equations are written for the vector of fluctuation terms as

    d/dtδα⃗ = -Aδα⃗+BdW⃗,

where A is the drift matrix containing the steady-state solutions, B is found from the factorisation of the diffusion matrix of the original Fokker-Planck equation, D=BB^T, with the steady-state values substituted in, and dW⃗ is a vector of Wiener increments. As long as the matrix A has no eigenvalues with negative real parts and the steady-state solutions are stationary, this method may be used to calculate the intracavity spectra via

    S(ω) = (A+iω)^-1D(A^-iω)^-1,

from which the output spectra are calculated using the standard input-output relations <cit.>.

In this case

    A =
    [       γ_1   -κ_1α_2 -κ_1α_1^∗         0         0         0; -κ_1α_2^∗       γ_1         0   -κ_1α_1         0         0;    κ_1α_1         0       γ_2   -κ_2α_3 -κ_2α_2^∗         0;         0  κ_1α_1^∗ -κ_2α_3^∗       γ_2         0   -κ_2α_2;         0         0    κ_2α_2         0       γ_3         0;         0         0         0  κ_2α_2^∗         0       γ_3 ],

and D is a 6× 6 matrix with [κ_1α_2,κ_1α_2^∗,κ_2α_3,κ_2α_3^∗,0,0] on the diagonal. In the above, the α_j should be read as their steady-state values.
Because we have set γ_1=1, the frequency ω is in units of γ_1. S(ω) then gives us products such as δα_iδα_j and  δα_i^∗δα_j^∗, from which we obtain the output variances and covariances for modes i and j as

    S^out(X_i,X_j) = δ_ij+√(γ_iγ_j)(S_ij+S_ji).




§ RESULTS





This system has a very rich parameter regime, with κ_1, κ_2, ϵ, γ_2 and γ_3 all capable of changing independently within any physical constraints. We have performed extensive numerical experiments and present the results for two representative regimes. The first was found to maximise violations of bipartite correlations and give asymmetric steering <cit.>, while the second is interesting because of the different predictions of the various correlations. 

The first parameter set we present is that used previously for bipartite correlations <cit.>, with κ_1=5× 10^-3, κ_2=4κ_1, ϵ=105, γ_1=1 and γ_2=γ_3=γ_1/2, which was shown to give bipartite steering, both symmetric and asymmetric, in all bipartitions. The intention of this parameter set, with lower loss rates at the higher frequencies, is to give the two higher frequency fields more time to interact within the cavity. We find, as shown in Fig. <ref>, that all three possible pairs can steer the remaining mode, but according to the criteria of He and Reid, since the minimum of the sum of the three is 1.44, genuine tripartite steering is not present. We note that the vLF inequalities were not violated and that since steering is a strict subclass of entanglement, these have missed tripartite entanglement that is present. The better sensitivity of EPR type measures for detecting entanglement has previously been found with bipartite systems <cit.>, where the Reid EPR correlations <cit.> have detected entanglement missed by the Duan-Simon positive partial transpose measure <cit.>.



In the second parameter regime, κ_1=10^-2, κ_2=0.5κ_1, ϵ=105, γ_1=1 and γ_2=2γ_1 and γ_3=γ_1/4. We see that only one of the V_ij correlations shown in Fig. <ref>, V_12, drops below 4. This result on its own could be taken to indicate that the system is not tripartite inseparable, but this is not the case. Two of the V_ijk shown in Fig. <ref> drop below a value of 4. Since only one of the possible three violating the inequality is sufficient to prove inseparability, this result is more than sufficient. It is not, however, sufficient to demonstrate genuine tripartite entanglement, since neither of the two V_ijk which violate the vLF inequality exhibit values of less than 2.




For this parameter set we find that the vLF correlations are more efficient at finding separability than the OBR_ijk, which are shown in Fig. <ref>. We see that only OBR_123 drops below one, and then by an insignificant amount. This means that, while the participants receiving modes 2 and 3 can combine to steer mode 1 in a marginal fashion which would quite possibly be destroyed by experimental noise, the other two pairings cannot perform steering in any fashion at all by way of Gaussian measurements. Whether steering via non-Gaussian measurements is possible is outside the scope of this article.






§ CONCLUSIONS



In conclusion, we have analysed a system of cascaded intracavity harmonic generation in terms of tripartite correlations for the detection of inseparability, entanglement, and EPR steering. We have examined two different parameter regimes and found non-classical quantum correlations across two octaves of frequency difference in both of these. In the first regime, EPR like correlations were found to be the best indicator of inseparability and entanglement. In the second, one set of the vLF correlations indicated tripartite inseparability which was missed by the other set, while the EPR correlations were inconclusive, finding only marginal steering in one of the three partitions. Our system, which could have possible applications in multiplexing, is a good physical example of the difficulty of finding versatile measures for tripartite entanglement in mixed systems, with different correlation measures being efficient in different regimes.   



§ ACKNOWLEDGMENTS


I would like to thank Margaret Reid for the invitation to contribute to this special issue. 

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