Quantum Exciton-Polariton Networks through Inverse Four-Wave Mixing
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Quantum Exciton-Polariton Networks through Inverse Four-Wave Mixing
T. C. H. Liew and Y. G. Rubo
2, 3 Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, 21 Nanyang Link, Singapore 637371 Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea Instituto de Energ´ıas Renovables, Universidad Nacional Aut´onoma de M´exico, Temixco, Morelos 62580, Mexico (Dated: August 30, 2018)We demonstrate the potential of quantum operation using lattices of exciton-polaritons in pat-terned semiconductor microcavities. By introducing an inverse four-wave mixing scheme acting onlocalized modes, we show that it is possible to develop non-classical correlations between individualcondensates. This allows a concept of quantum exciton-polariton networks, characterized by theappearance of multimode entanglement even in the presence of realistic levels of dissipation.
Recently, there has been a significant attention devotedto the study of exciton-polaritons in lattices [1–7]. Assystems of nonlinear interacting bosons, they have oftenbeen suggested as potential candidates of quantum sim-ulators [8, 9] and indeed the minimization of the energyof a particular Hamiltonian on a graph was a problemconsidered recently [10]. While the majority of studiesof exciton-polaritons have been restricted to the clas-sical regime [11, 12], the quantum nature of polaritonshas received revived attention recently [13]. Therefore,it is natural to question whether exciton-polaritons canbe used to form lattices of entangled modes. Here wemust be aware that a lattice or graph of polaritons doesnot behave as a system of qubits. Instead each node ofa polariton network could be described by the quantumfield amplitude ˆ a n or the continuous amplitude and phasevariables associated with the operators:ˆ q n = ˆ a n + ˆ a † n √ , ˆ p n = ˆ a n − ˆ a † n i √ . (1)Since continuous variable modes can be entangled, net-works of continuous variable modes are highly relevantfor quantum applications. As an example, cluster statecomputation [14] based on continuous variables [15] is apotential route towards universal computation. It relieson producing a highly entangled state from an arbitrarylattice or graph of modes coupled by two-mode squeezingtype interactions, with a Hamiltonian of the form: H S = X nm w nm (cid:0) ˆ a n ˆ a m + ˆ a † n ˆ a † m (cid:1) , (2)where w nm describes the weights of different connectionsin the graph. Arranging such a Hamiltonian is already aproblem and it must be done making use of some inter-action process that is stronger than any detrimental pro-cesses in the system (dissipation, dephasing, etc.). Whileevidence of strongly interacting polaritons [18] was re-ported recently, it is not clear if any nonlinear interactionprocess in microcavities is sufficiently strong for the gen-eration of quantum resources. In the absence of strong interactions, exciton-polaritons tend to only demonstratenonlinear effects at high densities, when they are welldescribed by the classical physics corresponding to themean-field approximation. For this reason only a hand-ful of experimental reports of quantum exciton-polaritoneffects have appeared in the literature [16, 17]. Two-Mode Squeezing.—
Before considering how tobuild a polariton network, it is instructive to considerthe effect of the two-mode squeezing type Hamiltonian:ˆ H = − iα (cid:0) ˆ a ˆ a − ˆ a † ˆ a † (cid:1) . (3)Such a Hamiltonian generates entanglement, which canbe characterized by the violation of the inequality [19, 20]1 ≤ S = 12 [ V (ˆ q − ˆ q ) + V (ˆ p + ˆ p )] , (4)where the variances are defined by V ( ˆ O ) = h ˆ O i − h ˆ O i .The Heisenberg equations of motion give the evolutionof the quantum field operators ˆ a , ( t ) ≡ e i ˆ Ht ˆ a , e − i ˆ Ht (we set ~ = 1)ˆ a , ( t ) = cosh( αt/ a , + sinh( αt/ a † , . (5)To calculate the second order correlators, we consideroperators ˆ K = 1 + ˆ a † ˆ a + ˆ a † ˆ a , ˆ L = ˆ a ˆ a + ˆ a † ˆ a † , ˆ M = i (ˆ a ˆ a − ˆ a † ˆ a † ) and use the Lie algebra [ ˆ M , ˆ K ] = 2 i ˆ L ,[ ˆ M , ˆ L ] = 2 i ˆ K to obtainˆ K ( t ) = cosh( αt ) ˆ K + sinh( αt ) ˆ L, (6a)ˆ L ( t ) = cosh( αt ) ˆ L + sinh( αt ) ˆ K. (6b)Using these relations and taking the vacuum state as aninitial condition one arrives at S = (cid:10) ˆ K ( t ) − ˆ L ( t ) (cid:11) = e − αt < , (7)indicating evolution of the system towards the entangledco-eigenstate of the EPR pair of operators ˆ q − ˆ q andˆ p + ˆ p . This result, S = e − αt , remains unchangedfor initial coherent states of the fields. We note that theEPR pair of operators [21] needed to demostrate the non-classical correlations depends on the Hamiltonian. Otherpossible pairs can be obtained with the gauge transfor-mation of operators in (4), ˆ a , → ˆ a , e iφ , , with subse-quent optimization over the phases φ , . Inverse Four-Wave Mixing. —Let us now consider asingle cavity with a four-wave mixing (parametric) typeresonance, described with the Hamiltonian H = α (cid:16) ˆ a † ˆ a † ˆ a L ˆ a U + ˆ a † L ˆ a † U ˆ a ˆ a (cid:17) , (8)where α describes the strength of the four-wave mixingprocess. Physical realizations of the above Hamiltoniancould be made in exciton-polariton micropillars [23] ora Kerr nonlinear photonic crystal cavity [24]. Finding aparametric resonance would however require careful tun-ing [25], which suggests that systems compatible withpost-growth tuning would be the most realistic choices.For example, dipolariton based setups allow electricalcontrol of mode energies [26]. Regardless of the mech-anism of introducing Hamiltonian (8), it is typically thecase that α will be weak compared to the system lossesΓ, that is, typical optical systems are only weakly non-linear ( α ≪ Γ).The Hamiltonian (8) is usually considered for gener-ating the fields ˆ a L and ˆ a U from initial excitation of thefield ˆ a , however, we can also consider the inverse processillustrated in Fig. 1a. Namely, if the modes ˆ a L and ˆ a U a ` a L a U LaserLaser a ` a ` Ja L b L c L FIG. 1: (color online) a) Scheme of inverse four-wave mix-ing. b) Coupling of spatially separated modes, each drivenby the inverse four-wave mixing scheme of (a). c) Potentialgeneralization into lattices and arbitrary graphs. are driven by coherent laser fields then particles scatterin pairs from ˆ a L and ˆ a U to the mode ˆ a . It is true thatunder such conditions the modes ˆ a L and ˆ a U should be-have only classically, such that their physics can not gobeyond what is expected from making the mean-field ap-proximation on these modes, but doing so leaves still areduced quantum Hamiltonian acting on the mode ˆ a : H = α (cid:0) ˆ a † ˆ a † + ˆ a ˆ a (cid:1) , (9)where α = α h a U ih a L i . While this is just the Hamil-tonian of two particle creation, by considering its intro-duction via the aforementioned inverse four-wave mixing process we have a way to make this a strong effect: since h a L i and h a U i can be increased by the resonant drivingintensity, one can reach the regime α ≫ Γ.Considering exciton-polariton systems the regime α ≫ Γ has essentially been realized previously under differentconditions, where the blueshift due to polariton-polaritoninteractions may exceed the linewidth and cause bista-bility [27]. It is worth mentioning that four-wave mixingexperiments also revealed an interesting polarization de-pendence [28, 29], which allow the signal mode ˆ a to havea different linear polarization to that of the others (ˆ a U and ˆ a L ), useful for better resolution and limiting otherscattering processes. Coupled cavities. —If we now consider a pair of cou-pled cavities, which could be made with the techniques ofRef. [23], the model Hamiltonian becomes (see Fig. 1b): H = α (cid:16) ˆ a + ˆ a + ˆ a † + ˆ a † (cid:17) − J (cid:16) ˆ a † ˆ a + ˆ a † ˆ a (cid:17) , (10)where J is the coupling constant between the cavitiesand we can set α > a and ˆ a .It is convenient to define new operators, representinga symmetric-antisymmetric basisˆ a = ˆ a + + ˆ a − √ , ˆ a = ˆ a + − ˆ a − √ , (11)decoupling the Hamiltonian into two parts:ˆ H = 12 X σ = ± (cid:2) α (ˆ a σ + ˆ a † σ ) − σJ ˆ a † σ ˆ a σ (cid:3) . (12)We can then consider the Bogoliubov transformˆ a σ = cosh( x/ b σ + σ sinh( x/ b † σ , (13)which in the case | J | > α and tanh( x ) = α/J reducesthe Hamiltonian into the simple form H = ω (cid:0) ˆ b † + ˆ b + − ˆ b †− ˆ b − (cid:1) , (14)where ω = √ J − α .If we take the vacuum state as the initial condition thenonly the second-order correlators of a -fields contribute tothe inequality (4). It is easy to show that h ˆ b † σ ˆ b σ ( t ) i = sinh ( x/ , (15a) h ˆ b σ ( t ) i = − σ x ) e − iσωt , (15b)which results in h ˆ a † σ ˆ a σ ( t ) i = sinh ( x ) sin ( ωt ) , (16a) h ˆ a σ ( t ) i = sinh( x ) sin( ωt ) [ σ cosh( x ) sin( ωt ) + i cos( ωt )] . (16b)In the symmetric-antisymmetric basis we have S = 1 + h ˆ a † + ˆ a + + ˆ a †− ˆ a − − ℜ (cid:0) ˆ a − ˆ a − (cid:1) i , (17)where ℜ denotes the real part and the first order corre-lators vanish in our case. Substituting the explicit formof the correlators, we obtain: S = 1 + 2 sinh ( x ) sin ( ωt ) − sinh(2 x ) sin ( ωt )= J + α cos(2 ωt ) J + α . (18)While this expression can never reach the value of zero,for the case J > α , one can reach the value ( J − α ) / ( J + α )for the specific time when the cosine function evaluatesto −
1. Since J − α can be tuned to be small, one canthen in principle reach arbitrarily small values of S . Togive a visualization, Fig. 2a shows the variation in S atsome fixed time as a function of J . Figure 2b then showsthe minimal value of S obtainable for increasing valuesof α . ΑΤ=
ΑΤ= ΑΤ=
50 5 10 15 200.00.20.40.60.81.0 J Τ S ΑΤ S a L b L FIG. 2: (color online) a) Dependence of S on J . Here S is evaluated at t = τ and it can be seen that there is anoptimal value of J ≈ α . The minimum value of S canbecome smaller for increasing values of α . b) Dependence ofthe optimum value of S (with optimally chosen J ) on α .The dashed curve shows the function S = e − ατ obtainedfrom the ideal squeezing Hamiltonian (3) for comparison. The formation of entanglement in the above schememight be seen as a round about way to create entangle-ment from four-wave mixing, which could be obtainedalready from Hamiltonian (8). Indeed the usual methodof exciting the central mode ˆ a and looking at correla-tions between ˆ a L and ˆ a U has been considered before, indifferent contexts [30–34]. It should be stressed howeverthat the conventional method requires α to be signifi-cant compared to the dissipation rate and also α shouldbe stronger than other scattering processes (e.g., scatter-ing with acoustic phonons) that may resonantly couplethe modes to be entangled. In the scheme that we con-sider here α can become the dominant interaction in thesystem as it is enhanced by the density of modes a L and a U . Furthermore, local interactions, such as scatteringwith phonons and sample disorder are not able to couplespatially separated modes ˆ a and ˆ a . Dissipation. —We have shown so far that the systemof coupled cavities driven by parametric resonance cangenerate entangled states, which become asymptoticallyclose to the level of entanglement expected from a two-mode squeezing type operation, as measured by the vio-lation of inequality (4). As we have noted in the previoussection α can be controlled by the intensity of externallasers. In principle, J can also be controlled by exter-nal fields, for example, by using external electric [35] oroptical fields [36] to modify the potential between latticepoints.While we expect the regime α ≫ Γ to be experimen-tally accessible, given the parametric driving scheme, itis still instructive to consider the influence of dissipationin the system. This is readily introduced by modificationof the Heisenberg equations: d h ˆ O i dt = i h (cid:2) ˆ H , ˆ O (cid:3) i + Γ2 X n h a † n ˆ O ˆ a n − ˆ a † n ˆ a n ˆ O − ˆ O ˆ a † n ˆ a n i . (19)This introduces additional dissipation terms in our equa-tions of motion, which are solved in the SupplementaryMaterial [37]. The resulting effect of dissipation is illus-trated in Figs. 3(a,b). As one would expect, too much G= GΤ=
GΤ=
10 5 10 15 200.00.20.40.60.81.0 J Τ S ΑΤ=
ΑΤ= ΑΤ= ΑΤ= GΤ S a L b L FIG. 3: (color online) a) Dependence of S on J . As inFig. 2a S is evaluated at t = τ , but here we consider a fixedvalue of ατ = 5 and consider different values of dissipation Γ.b) Dependence of the optimum value of S (with optimallychosen J ) on Γ. dissipation results in a loss of entanglement. However,given the parametric pumping scheme it is in principlepossible to work in the limit where Γ ≪ α . At some shorttime such that τ ≪ / Γ one then obtains a high degreeof entanglement despite the presence of dissipation.
Multimode Entanglement. —In comparison to conven-tional methods of entanglement generation with respectto four-wave mixing, a further advantage of our schemethat entangles polariton modes separated in real space isthat it is in principle scalable; by coupling more cavitiesin space, arbitrary networks could be considered such asthe one illustrated in Fig. 1c.As an example let us consider a system of four identicalcavity, which are subjected to the Hamiltonian: H = X n =1 α (cid:0) ˆ a † n ˆ a † n + ˆ a n ˆ a n (cid:1) − J A ( t ) (cid:16) ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a (cid:17) − J B ( t ) (cid:16) ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a (cid:17) . (20)This Hamiltonian is a generalization of Hamiltonian (10),where we assume that it is possible to control the lin-ear coupling in time. For simplicity, we will consider( J A ( t ) = J , J B ( t ) = 0) for the time 0 < t < τ and( J A ( t ) = 0, J B ( t ) = J ) for time τ < t < τ . It is possibleto write Heisenberg equations of motion and their timedependent solution can be obtained analytically [37]. Al-ternatively, in the absence of dissipation, it is more effi-cient to solve for the operator evolution in the Heisenbergpicture [37].While violation of inequality (4) is a sufficient condi-tion for entanglement, the definition given of S is notideal for all states. In particular, varying the phases ofmodes ˆ a and ˆ a changes the value of S and thus todemonstrate the entanglement we should minimize S over all choices of local phases. As we mentioned above,this is equivalent to finding the best EPR pair of opera-tors. The procedure is detailed in [37], where we define˜ S as the value of S minimized over phase rotations.The result is shown in Fig. 4a, where, in addition to char-acterizing the entanglement between modes ˆ a and ˆ a ,we find also entanglement between other pairs of modes,using similar definitions for S and S (other combina-tions of modes display identical entanglement character-istics due to symmetry). S Ž S Ž S Ž ΑΤ S Ž ΑΤ= GΤ I a L b L FIG. 4: (color online) a) Dependence of the optimum values of˜ S , ˜ S , and ˜ S (with optimally chosen J ) on α . Parameters:Γ = 0, t A = t B = τ . b) Dependence of the optimum value of I (with optimally chosen J ) on Γ, for different values of α . In addition to entanglement between pairs of modes,multimode entanglement, simultaneously between allfour modes of the system can be evidenced by the vi-olation of the inequality [38]:12 ( V (ˆ q − ˆ q ) + V (ˆ p + ˆ p + g ˆ p + g ˆ p )) = I ≥ , (21) where g is an arbitrary real parameter that should bechosen so as to optimize the violation of the inequality.In the general case of four modes, one should also breaktwo other inqualitites to evidence an entangled state, ob-tained by permuting the modes [38]. However, giventhe symmetry of our four mode example in a ring theseinequalities are equivalent and the violation of inequal-ity 21 is a sufficient condition. Following correct choiceof the parameter g and optimization over the phases ofthe modes [37], we indeed find that the quantity I candrop below one and even reach zero, as shown in Fig. 4for different values of α and Γ. Conclusion. —The evolution of polariton networksfrom the classical to quantum regime implies finding amechanism of generating quantum correlations that canovercome the dissipation of the system. Nonlinearity, inthe form of polariton-polariton interactions is tradition-ally weak, however, here we have shown theoretically thatan inverse four-wave mixing geometry allows enhance-ment to an effective strongly nonlinear regime. Localnonlinearity and standard Josephson coupling betweenspatially separated modes is then sufficient to generatequantum entanglement both between pairs of modes andmultiple modes. We hope this will stimulate further dis-cussions on polariton simulators, which have begun re-cently [10].This research was supported by the MOE (Singapore)grant 2015-T2-1-055, by IBS-R024-D1, and by CONA-CYT (Mexico) grant 251808. 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