Quantum light by atomic arrays in optical resonators
aa r X i v : . [ qu a n t - ph ] A ug Quantum light by atomic arrays in optical resonators
Hessam Habibian,
1, 2
Stefano Zippilli,
1, 2, 3 and Giovanna Morigi
1, 2 Grup d’ ´Optica, Departament de F´ısica, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain Theoretische Physik, Universit¨at des Saarlandes, D-66041 Saarbr¨ucken, Germany Department of Physics, Technische Universit¨at Kaiserslautern, D-67663 Kaiserslautern, Germany (Dated: November 5, 2018)Light scattering by a periodic atomic array is studied when the atoms couple with the mode of ahigh-finesse optical resonator and are driven by a laser. When the von-Laue condition is not satified,there is no coherent emission into the cavity mode, and the latter is pumped via inelastic scatteringprocesses. We consider this situation and identify conditions for which different non-linear opticalprocesses can occur. We show that these processes can be controlled by suitably tuning the strengthof laser and cavity coupling, the angle between laser and cavity axis, and the array periodicity. Wecharacterize the coherence properties of the light when the system can either operate as degenerateparametric amplifier or as a source of antibunched-light. Our study permits us to identify theindividual multi-photon components of the nonlinear optical response of the atomic array and thecorresponding parameter regimes, thereby in principle allowing one for controlling the nonlinearoptical response of the medium.
PACS numbers: 42.50.Nn, 42.50.Pq, 42.65.Yj
I. INTRODUCTION
Resonance fluorescence from a single atom exhibitsnon-classical features [1], which become evident in thecorrelation functions of the emitted light [2]. Non-classical properties emerge from the quantum nature ofthe scatterer, such as the discrete spectrum of the elec-tronic bound states of the scattering atom. They can beenhanced or suppressed by several scatterers forming aregular array [3–5]. In this case, at the solid angles whichsatisfy the von-Laue condition [6], the light in the far fieldis in a squeezed coherent state, while for a large numberof atoms it can exhibit vacuum squeezing at scatteringangles, for which the elastic component of the scatteredlight is suppressed [3].When the atoms of the array are strongly coupled withthe mode of a high-finesse resonator, emission into thecavity mode is in general expected to be enhanced. Theproperties of the light at the cavity output will depend onthe phase-matching conditions, determined by the anglebetween laser and cavity wave vector and by the peri-odicity of the atomic array. The coherence propertiesof the light at the cavity output may however be signif-icantly different from the ones predicted in free space.An interesting example is found when the geometry ofthe setup is such that the atoms coherently scatter lightinto the cavity mode. In this case the intracavity-fieldintensity becomes independent of the number of atoms N as N increases, while inelastic scattering is suppressedover the whole solid angle in leading order in 1 /N [7].These dynamics have been confirmed by experimentalobservations [8, 9], and clearly differ from the behaviourin free space [3].When the geometry of the setup is such that the von-Laue condition is not satisfied, photons can only be in-elastically scattered into the cavity mode. The smallersystem size for which coherent scattering is suppressed is found for two atoms inside the resonator. The prop-erties of the light at the cavity output for this specificcase have been studied in Refs. [10, 11]. To the best ofour knowledge, however, the scaling of the dynamics withthe number of atoms N is still largely unexplored in thisregime.In this article we characterize the coherence propertiesof the light at the cavity output when the light is scat-tered from a laser into the resonator by an array of atomsand the geometry of the system is such that coherentscattering is suppressed. For the phase-matching condi-tions, at which in free space the light is in a squeezed-vacuum state [3], we find that inside a resonator andat large N the system behaves as an optical paramet-ric oscillator, which in certain regimes can operate abovethreshold [12]. For a small number of atoms N , on thecontrary, the medium can act as a source of antibunchedlight. In this case it can either behave as single-photonor, for the saturation parameters here considered, two-photon “gateway” [13]. The latter behaviour is found fora specific phase-matching condition. We identify the pa-rameter regimes which allow one to control the specificnonlinear optical response of the medium.This article is organized as follows. In Sec. II the theo-retical model is introduced and the basic approximationsare discussed, which lead to the derivation of an effectiveHamiltonian for the atomic and cavity excitations. InSec. III we analyze the light at the cavity output underthe condition that there is no coherent scattering intothe cavity. The conclusions are drawn and outlooks arediscussed in Sec. IV. II. THEORETICAL MODEL
The physical system is composed by N identical atomswhich are regularly distributed along the z axis. They (a) (b) (b) FIG. 1: (Color online) (a) An array of atoms, with inter-particle distance d , is confined along the axis of a standing-wave optical cavity at frequency ω c and is transversally drivenby a laser, whose wave vector forms the angle Θ with thecavity axis. The atomic internal transition and the relevantfrequency scales are given in (b), with | i and | i groundand excited state of an optical transition with frequency ω and natural linewidth γ . The frequencies ω z = ω − ω p and δ c = ω c − ω p denote the detunings between the laser fre-quency ω p and the atomic and cavity frequency, respectively.The other parameters are the laser Rabi frequency Ω, theatom-cavity coupling strength g , and the decay rate κ of theoptical cavity. are located at the positions z j = jd where j = 1 , . . . , N and d is the interparticle distance [14]. An optical dipoletransition of the confined atoms interacts with the modeof a standing wave cavity, whose wave vector k is parallelto the atomic array, as illustrated in Fig. 1. Moreover,the atoms are transversally driven by a laser and scatterphotons into the cavity mode. We denote by ω p the fre-quency of the laser mode, whose polarization is assumedto be linear and parallel to the polarization of the cavitymode, and a and a † the annihilation and creation op-erators, respectively, of a cavity photon at frequency ω c (with [ a, a † ] = 1). Cavity and laser mode couple to theatomic dipolar transition at frequency ω with ground and excited states | i and | i . The Hamiltonian govern-ing the coherent dynamics of cavity mode and the atomsis given by H = ~ ω c a † a + ~ ω N X j =1 S zj + ~ g N X j =1 cos ( kz j + ϕ )( S † j a + a † S j )+ i ~ Ω N X j =1 ( S † j e − i ω p t e i( k p z j cos Θ − φ L ) − H . c . ) , (1)where Ω is the strength of the coupling between laserand atomic transition, and g the cavity vacuum Rabifrequency. The operators S j = | i j h | and S † j indicatethe lowering and raising operators for the atom at theposition z j , and S zj = ( | i j h | − | i j h | ) is the z com-ponent of the pseudo-spin operator. In Eq. (1) we haveintroduced the angle ϕ , which is the phase offset of thestanding wave at the atomic positions, the phase of thelaser φ L , and the angle Θ between the laser and the cav-ity wave vector.We remark that in the present study we neglect theatomic motion, and consider that the size of the atomicwavepacket is much smaller than the laser wavelengthand interparticle distance. We refer the reader to [10]for a quantitative study on the effects of the mechanicalmotion on the nonlinear optical processes in this kind ofsystem.In the rest of this section we will introduce and discussthe approximations, which allow us to solve the dynam-ics and determine the properties of the cavity field. Forsimplicity we also set k = k p : The difference between thelaser and cavity wave numbers can in fact be neglectedfor the purpose of this study. A. Weak excitation limit
We resort to the Holstein-Primakoff representation forthe spin operator [15] S † j = b † j (1 − b † j b j ) / , (2) S j = (1 − b † j b j ) / b j ,S zj = b † j b j − , where b j ( b † j ) is the bosonic operator annihilating (creat-ing) an excitation of the atom at z j , such that [ b j , b † j ′ ] = δ jj ′ . In the limit in which the atomic dipoles are drivenbelow saturation, we treat saturation effects in the lowestnon-vanishing order of a perturbative expansion, whosesmall parameter is the total excited-state population ofthe atoms, denoted by N tot . We denote the detuning ofthe laser from the atomic transition by ω z = ω − ω p . (3)and by γ the natural linewidth. In the low saturationlimit, | ω z + i γ/ | ≫ √ N Ω, then N tot ≪ N and wecan expand the operators on the right-hand side of theequations (2) in second order in the small parameter h b † j b j i ≪
1, obtaining S † j ≈ b † j − b † j b † j b j , (4) S j ≈ b j − b † j b j b j . (5)For N ≫ z N +1 = z . The atomic excitations are studiedin the Fourier transformed variable q , quasimomentumof the lattice, which is defined in the Brillouin zone (BZ) q ∈ ( − G / , G /
2] with G = 2 π/d the primitive recip-rocal lattice vector. Correspondingly, we introduce theoperators b q and b † q , defined as b q = 1 √ N N X j =1 b j e − i qjd , (6) b † q = 1 √ N N X j =1 b † j e i qjd , (7)which annihilate and create, respectively, an excitationof the spin wave at quasimomentum q and fulfilling thecommutation relation [ b q , b † q ′ ] = δ q,q ′ . After rewriting theHamiltonian in Eq. (1) in terms of spin-wave operators,we find H ≈ − N ~ ω z H pump + H (2) + H (4) , (8)where the first term on the Right-Hand Side (RHS) is aconstant and will be discarded from now on, while H pump = i ~ Ω √ N (cid:16) b † Q ′ e − i( ω p t + φ L ) − b Q ′ e i( ω p t + φ L ) (cid:17) (9)is the linear term describing the coupling with the laser.Term H (2) = ~ ω c a † a + ~ ω X q ∈ BZ b † q b q + ~ g √ N h ( b † Q e iϕ + b †− Q e − iϕ ) a + H . c . i (10)determines the system dynamics when the linear pump isset to zero and the dipoles are approximated by harmonicoscillators (analog of the classical model of the elasticallybound electron), while H (4) = − ~ g √ N X q ,q ∈ BZ ( b † q b † q b q + q − Q a e iϕ + b † q b † q b q + q + Q a e − iϕ + H . c . ) − i ~ Ω2 √ N X q ,q ∈ BZ ( b † q b † q b q + q − Q ′ e − i( ω p t + φ L ) − H . c . )(11) accounts for the lowest-order corrections due to satura-tion. In Eqs. (10) and (11) we have denoted by ± Q and Q ′ the quasimomenta of the spin waves which couple tothe cavity and laser mode, respectively, and which fulfillthe phase matching conditions Q = k + G , (12) Q ′ = k cos Θ + G ′ , (13)with reciprocal vectors G, G ′ such that Q, Q ′ ∈ BZ. Theatoms scatter coherently into the cavity mode when thevon-Laue condition is satisfied, namely one of the tworelations is fulfilled:2 k sin (Θ /
2) = nG , (14)2 k cos (Θ /
2) = n ′ G , (15)with n, n ′ integer numbers. In free space, the von-Lauecondition corresponds to Eq. (14): for these angles onefinds squeezed-coherent states in the far field [3]. Whenthe scattered mode for which the von-Laue conditionis fulfilled corresponds to a cavity mode, superradiantscattering enhances this behaviour, until the number ofatoms N is sufficiently large such that the cooperativityexceeds unity. In this limit one observes saturation ofthe intracavity-field intensity, which reaches an asymp-totic value whose amplitude is independent of N as N is further increased. In the limit N ≫ /N [7].When the von-Laue condition is not satisfied, classi-cal mechanics predicts that there is no scattering intothe cavity mode. These modes of the electromagneticfields are solely populated by inelastic scattering pro-cesses. Moreover, in free space, when2 k sin (Θ /
2) = (2 n + 1) G / , (16)then the inelastically scattered light is in a vacuum-squeezed state [3]. Inside a standing-wave resonator, onthe other hand, the mode is in a vacuum-squeezed stateprovided that either Eq. (16) or an additional relation,2 k cos (Θ /
2) = (2 n + 1) G / , (17)is satisfied.In the following we will study the field at the cavityoutput as determined by the dynamics of Hamiltonian (8)when Q ′ = ± Q , namely, when the scattering processeswhich pump the cavity are solely inelastic. We remarkthat throughout this treatment we do not make specificassumptions about the ratio between the array periodic-ity d and the light wavelength λ (and therefore also con-sider the situation in which λ = 2 d . This situation hasbeen experimentally realized for instance in Refs. [8, 16–18]). B. Linear response: Polaritonic modes
We first solve the dynamics governed by Hamiltonian H (2) in Eq. (10). In the diagonal form the quadratic partcan be rewritten as H (2) = X j =1 ~ ω j γ † j γ j + X q = Q s ,q ∈ BZ ~ ω b † q b q , (18)where Q s labels the spin wave which couples with thecavity mode, such that b Q s = b Q if Q = 0 , G / , (19) b Q s = b Q e − iϕ + b − Q e iϕ √ . (20)The resulting polaritonic eigenmodes are γ = − a cos X + b Q s sin X , (21) γ = a sin X + b Q s cos X , (22)with respective eigenfrequencies ω , = 12 ( ω c + ω ∓ δω ) , (23) δω = q ( ω − ω c ) + 4˜ g N , (24)and tan X = ˜ g √ N / ( ω − ω ) , (25)which defines the mixing angle X . The parameter ˜ g isproportional to the coupling strength. In particular, ˜ g = g cos ϕ when Q = 0 , G / b Q s = b Q , otherwise ˜ g = g/ √ Q ′ . When Q ′ = ± Q , photons arepumped into the cavity via inelastic processes, which inour model are accounted for by the Hamiltonian term inEq. (11). On the other had, when the dynamics is con-sidered up to the quadratic term (hence, inelastic pro-cesses are neglected), only the mode Q ′ is pumped andthe Heisenberg equation of motion for b Q ′ reads˙ b Q ′ = − i ω z b Q ′ − γ b Q ′ + Ω √ N e − i φ L + √ γb q, in ( t ) , (26)that has been written in the reference frame rotating atthe laser frequency ω p . Here, γ is the spontaneous de-cay rate and b q, in ( t ) is the corresponding Langevin forceoperator, such that h b q, in ( t ) i = 0 and h b q, in ( t ) b † q, in ( t ′ ) i = δ ( t − t ′ ) [12]. The general solution reduces, in the limitin which | ω z | ≫ γ , to the form b Q ′ ≃ − i Ω √ Nω z e − i φ L (27)which is consistent with the expansion to lowest orderin Eq. (4) provided that Ω N ≪ ω z . In the referenceframe rotating at the laser frequency the explicit fre-quency dependence of the Hamiltonian terms is dropped,and ω → ω − ω p , ω → ω − ω p , ω → ω z , and ω c → ω c − ω p ≡ δ c . C. Effective Hamiltonian
Under the assumptions discussed so far, we derive fromHamiltonian (8) an effective Hamiltonian for the polari-ton γ . The effective Hamiltonian is obtained by adiabat-ically eliminating the coupling with the other polaritons,according to the procedure sketched in Appendix A, andreads H eff = ~ δω γ † γ + ~ (cid:0) αγ † e iφ L + α ∗ γ e − iφ L (cid:1) + ~ χγ † γ † γ γ + i ~ ( νγ † γ e iφ L − ν ∗ γ † γ e − iφ L ) , (28)where [19] δω = ω − ω p + 2Ω ω z ˜ S + ˜ g √ Nω z ˜ S ˜ C ! , (29) α = − Ω ω z (cid:0) ˜ S + ˜ g √ Nω z ˜ S ˜ C (cid:1) (cid:16) δ Q ′ ,G/ + C k = G/ α (cid:17) , (30) χ = ˜ g √ N ˜ S ˜ C h C k = G/ χ i , (31) ν = − Ω4 √ N ˜ S + 3˜ g √ Nω z ˜ S ˜ C ! C k = G/ ν , (32)with ˜ S = sin X and ˜ C = cos X . The terms C k = G/ j donot vanish when k = G/
2, and their explicit form is C k = G/ χ = (cid:18)
12 + 12 δ Q, ± G / cos (4 ϕ ) (cid:19) (1 − δ k,G/ ) , C k = G/ α = 12 (cid:0) δ Q ′ ,Q + G/ e − iϕ + δ Q ′ , − Q + G/ e iϕ (cid:1) × (1 − δ k,G/ ) , C k = G/ ν = 1 √ (cid:16) δ Q ′ , Q e − iϕ + δ Q ′ , − Q e iϕ (cid:17) (1 − δ k,G/ ) . The coefficients have been evaluated under the require-ment Q ′ = ± Q .We now comment on the condition Ω N ≪ ω z , onwhich the validity of Eq. (27) is based. When this isnot fulfilled, such that |h b Q ′ i| ∼
1, Eq. (26) must con-tain further non-anharmonic terms from the expansionof Eq. (2) and which account for the saturation effects in b Q ′ . Since this spin mode is weakly coupled to the othermodes, which are initially empty, we expect that the po-laritons γ and γ will remain weakly populated and thestructure of their effective Hamiltonian will qualitativelynot change.We also note that in the resonant case, when ω z = 0,the form of Hamiltonian in Eq. (28) remains unchanged,while in the coefficients δω , α , ν , χ the following sub-stitution ω z → ( ω z + γ / /ω z must be performed . Afirst consequence is that α = 0, which implies that thereare no processes in this order for which polaritons arecreated (annihilated) in pairs. A further consequence isthat spontaneous decay plays a prominent role in the dy-namics. We refer the reader to Sec. II E for a discussionof the related dissipative effects. D. Discussion
Let us now discuss the individual terms on the RHS ofEq. (28). For this purpose it is useful to consider multi-level schemes, which allow one to illustrate the relevantnonlinear processes. The multilevel schemes are depictedin Fig. 2: state | ˜ n i is the polariton number state with ˜ n excitations. The blue arrows indicate transitions whichare coupled by the laser, for which the polariton stateis not changed; The red arrows denote transitions whichare coupled by the cavity field, for which the polaritonstate is modified by one excitation.Using this level scheme, one can explain the dynamicalStark shift δω of the polariton frequency in Eq. (29) asdue to higher-order scattering processes, in which laser-and cavity-induced transition creates and then annihi-lates, in inverse sequential order, a polariton.The second term on the RHS has coupling strengthgiven in Eq. (30), it generates squeezing of the polari-ton and does not vanish provided that Q ′ = G/ Q ′ = ± Q + G/
2. The latter condition is equivalent tothe free-space condition (16), while the first arises fromthe fact that the cavity mode couples with the symmet-ric superposition b Q s in Eq. (19). The correspondingphase-matched scattering event is a four-photon process,in which two laser photons are absorbed (emitted) andtwo polaritonic quanta are created (annihilated). For Q ′ = G/ Q ′ = ± Q the polaritons are createdin pairs with quasimomentum Q and − Q (the relation Q = G/ b − Q = b Q ). This specific termis also present when the geometry of the setup is suchthat von-Laue condition is fulfilled, and at this order isresponsible for the squeezing present in the light at thecavity output.The Kerr-nonlinearity (third term on the RHS) givesrise to an effective interaction between the polaritons andemerges from processes in which polaritons are absorbedand emitted in pairs. It is depicted in Fig. 2(b) for ageneric case. This term is directly proportional to thecavity coupling strength and inversely proportional to √ N . In this order, it is the term that gives rise to anti-bunching.The last term on the RHS, finally, is a nonlinear pumpof the polariton mode, whose strength depends on thenumber of polaritonic excitations. It is found when thephase-matching condition Q ′ = ± Q + G is satisfied,which is equivalent to the relation cos θ = (3 + nλ/d )when laser and optical resonator have the same wave-length, as in the case here considered. This relation canbe fulfilled for n = 0 and specific ratios λ/d . This termvanishes over the vacuum state, and it pumps a polari-ton at a time with strength proportional to the numberof polariton excitations. (a)(b)(c) FIG. 2: (Color online) Schematic diagram of transitions whichfulfill the phase-matching condition till fourth order. State | ˜ n i denote the number state of the polariton mode γ . The blue(light gray) arrows denote the laser-induced couplings and thered (dark gray) arrows denote the creation (annihilation) ofpolaritons due to the coupling with the cavity field. See textfor a detailed discussion. In general, photons into the cavity mode are pumpedprovided that either (i) Q ′ = ± Q or (ii) (for Q ′ = ± Q )one of the two conditions are satisfied: Q ′ = G/ Q ′ = ± Q + G/
2. We note that the strength of the Rabifrequency and of the cavity Rabi coupling may allowone to tune the relative weight of the various terms inHamiltonian (28). Their ratio scales differently with thenumber of atoms in different regimes, which we will dis-cuss below. Moreover, the interparticle distance of theatomic array constitutes an additional control parame-ter over the nonlinear optical response of the medium.Further phase-matching conditions are found when con-sidering higher-order terms in the expansion of the spinoperators in harmonic-oscillator operators from Eq. (2).Their role in the dynamics will be relevant, as long asthey compete with the dissipative rates, here constitutedby the cavity loss rate and spontaneous emission.
E. Cavity input-output formalism
We consider the full system dynamics, including theatomic spontaneous emission and the cavity quantumnoise due to the coupling to the external modes of theelectromagnetic field via the finite transmittivity of thecavity mirrors. Denoting by κ and γ the linewidth of thecavity mode and of the atomic transition, respectively,the Heisenberg-Langevin equations for the operator a and b q read ˙ a = 1i ~ [ a, H ] − κa + √ κa in ( t ) , (33)˙ b q = 1i ~ [ b q , H ] − γ b q + √ γb q, in ( t ) , (34)where a in , b q, in are the Langevin operators, which aredecorrelated one from the other and fulfill the relations h a in ( t ) i = 0 and h a in ( t ) a † in ( t ′ ) i = δ ( t − t ′ ). Here, theaverage h·i is taken over the density matrix at time t = 0of the system composed by the atomic spins and by theelectromagnetic field. The output field a out at the cavitymirror is given by the relation a out + a in = √ κa ( t ) . (35)Let us now consider the scattering processes occurringin the system. They can be classified into three types:(i) a laser photon can be scattered into the modes of theexternal electromagnetic field (emf) by the atoms, with-out the resonator being pumped in an intermediate time;(ii) a laser photon can be scattered into the cavity modeby the atom and then dissipated by cavity decay; (iii) alaser photon can be scattered into the cavity mode by theatom, then been reabsorbed and emitted into the modesof the external emf. Processes of kind (i) include elas-tic scattering. They can be the fastest processes, but donot affect the properties of the light at the cavity out-put. Processes of kind (ii) are the ones which outcouplethe intracavity field, but need to be sufficiently slow in order to allow for the build-up of the intracavity field.Processes of kind (iii) are detrimental for the nonlinearoptical dynamics we intend to observe, as they introduceadditional dissipation (see for instance [20, 21] for an ex-tensive discussion and [10] for a system like the one hereconsidered but composed by two atoms).Processes (iii), i.e., reabsorption of cavity photons fol-lowed by spontaneous emission, can be neglected assum-ing that the laser and cavity mode are far-off resonancefrom the atomic transition. In this limit, the cavity ispumped by coherent Raman scattering processes and aneffective Heisenberg-Langevin equation for the polariton γ can be derived assuming that its effective linewidth κ = κ cos X + ( γ/
2) sin X fulfilling the inequality κ ≪ δω (that corresponds to the condition for whichthe vacuum Rabi splitting is visible in the spectrum oftransmission [17, 18, 22–24]). We find˙ γ = 1i ~ [ γ , H eff ] − κ γ + √ κ ˜ Ca in ( t ) + √ γ ˜ Sb q, in ( t ) , (36)which determines the dissipative dynamics of the polari-ton. The field at the cavity output is determined usingthe solution of the Heisenberg Langevin equation (36)with Eqs. (21)-(22) in Eq. (35). In some calculations,when appropriate we solved the corresponding masterequation for the density matrix of the polaritonic modes γ and γ .Some remarks are in order at this point. Nonlinear-optical effects in an atomic ensemble, which is reso-nantly pumped by laser fields, have been studied for in-stance [25], where the nonlinearity is at the single atomlevel and is generated by appropriately driving a four-level atomic transitions [26].It is important to note, moreover, that Eq. (36) is validas long as the loss mechanisms occur on a rate whichis of the same order, if not smaller, than the inelasticprocesses. This leads to the requirement that the atom-cavity system be in the strong-coupling regime. III. RESULTS
We now study the properties of the light at the cav-ity output as a function of various parameters, assumingthat Q ′ = G/ Q ′ = ± Q and Q ′ = ± Q + G hold. Under these conditions the effectiveHamiltonian in Eq. (28) contains solely the squeezing andthe Kerr-nonlinearity terms, while ν = 0. Moreover, weassume the condition κ ≃ κ > γ .The possible regimes which may be encountered canbe classified according to whether the ratio ε = | α/χ | is larger or smaller than unity. In the first case themedium response is essentially the one of a parametricamplifier. In the second case the Kerr non-linearity dom-inates, and polaritons can only be pumped in pairs pro-vided that the emission of two polaritons is a resonantprocess.Let us now focus on the regime in which the systemacts as a parametric amplifier, namely, ε ≫
1. In thiscase one finds that the number of photons at the cavityoutput at time t is h a † out a out i t ≃ κ ˜ C h γ † γ i t , with h γ † γ i t = 12 α κ − α + e − κ t sinh ( αt ) (37)+ e − κ t (cid:18) − κ κ cosh(2 αt ) + α sinh(2 αt ) κ − α (cid:19) . Depending on whether α > κ or α < κ , one finds thatthe dynamics of the intracavity polariton corresponds toa parametric oscillator above or below threshold, respec-tively. In the following we focus on the case below thresh-old and evaluate the spectrum of squeezing. We firstobserve that the quadrature x ( θ ) = γ e − iθ + γ † e iθ hasminimum variance for θ = π/ h ∆ x ( π ) i st = κ κ + | α | , (38)where the subscript st refers to the expectation valuetaken over the steady-state density matrix. The squeez-ing spectrum of the maximally squeezed quadrature is S out ( ω ) =1 + Z + ∞−∞ h : x ( π ) out ( t + τ ) , x ( π ) out ( t ) : i st e − i ωτ dτ (39)=1 − κ ˜ C | α | ( κ + | α | ) + ω , (40)where h : : i st indicates the expectation value for thenormally-ordered operators over the steady state, with x ( θ ) out = a out e − iθ + a † out e iθ , and h A, B i st = h AB i st − h A i st h B i st .We now discuss the parameter regime in which thesedynamics can be encountered. The relation ε ≫ ≫ g . When | ω z | ≫ Ω √ N , in thislimit | α | ≃ Ω g N/ω z , and squeezing can be observedonly for very small values of κ . Far less demanding pa-rameter regimes can be accessed when relaxing the con-dition on the laser Rabi frequency, and assuming thatΩ √ N ∼ | ω z | . In this case squeezing in the light at thecavity output can be found provided that Ω ≫ κ when g √ N ∼ | ω z | [27].Figures 3(a) and (b) display the spectrum of squeez-ing when the system operates as a parametric amplifierbelow threshold. Here, one observes that squeezing in-creases with N . Comparison between Fig. 3(a) and 3(b) shows that squeezing increases also as the single-atom cooperativity increases (provided the correspond-ing phase-matching conditions are satisfied and the laserRabi frequency Ω ≫ g ). These results agree and extend −20 −15 −10 −5 0 5 10 15 2000.20.40.60.81 ω/κ S o u t ( ω ) (a) (a) −20 −15 −10 −5 0 5 10 15 2000.20.40.60.81 ω/κ S o u t ( ω ) (b) (b) FIG. 3: (Color online) Squeezing spectrum for the maximumsqueezed quadrature when k = G/ Q ′ = G/ Q ′ = Q for ϕ, φ L = 0. The parameters are Ω = 200 κ , ω z = 10 κ and N = 10 , ,
100 atoms (from top to bottom) for (a) g = 4 κ and (b) g = 10 κ . The detuning δ c is chosen such that δω is zero. The curves are evaluated from Eq. (39) by numeri-cally calculating the density matrix of the polariton field fora dissipative dynamics, whose coherent term is governed bythe effective Hamiltonian in Eq. (28). The value g = 4 κ isconsistent with the experimental data of Ref. [29]. the findings in Ref. [10], which were obtained for an arrayconsisting of 2 atoms.Let us now focus on the regime when ε ≪
1. Here,the polaritons may be only emitted in pairs into the res-onator. In order to characterize the occurrence of thesedynamics we evaluate the second-order correlation func-tion at zero time delay in the cavity output defined by[12] g (2) (0) = h a † a i st h a † out a out i st . (41)Function g (2) (0) quantifies the probability to measuretwo photon at the cavity output at the same time. There-fore, subpossonian (superpossonian) statistics are hereconnected to the value of g (2) (0) smaller (larger) thanone, while for a coherent state g (2) (0) = 1.Subpossonian photon statistics at the cavity outputcan be found as a result of the dynamics of Eq. (28).Here, for phase-matching conditions leading to ν = 0 and α = 0, polaritons can only be created in pairs. When theKerr-nonlinearity is sufficiently large, however, the con-dition can be reached in which only two polaritons can beemitted into the cavity, while emission of a larger numberis suppressed because of the blockade due to the Kerr-term. This is reminiscent of the two-photon gateway re-alized in Ref. [13], where injection of two photons insidea cavity, pumped by a laser, was realized by exploitingthe anharmonic properties of the spectrum of a cavitymode strongly coupled to an atom. In the case analysedin this paper, the anharmonicity arises from collectivescattering by the atomic array, when this is transversallydriven by a laser. Moreover, we note that the observa-tion of these dynamics requires Ω √ N ≪ | ω z | , g √ N and | α | > κ , which reduces to the condition Ω /ω z > κ when g √ N ∼ ω z .Figure 4(a) displays g (2) (0) as a function of the pumpfrequency ω p for the phase matching conditions giving ν = 0 and α = 0. Function g (2) (0) is evaluated by numer-ically integrating the master equation with cavity decay,where the coherent dynamics is governed by an effectiveHamiltonian which accounts for the effect of both polari-ton modes and is reported in Eq. (A1) in the Appendix.Antibunching is here observed over an interval of valuesof ω p , about which the cavity mode occupation has amaximum (blue curve in Fig. 4(b)). The maximum cor-responds to the value of ω p for which the emission of twopolaritons γ is resonant. Note that the spin-wave exci-tation, red curve in Fig. 4(b), is still sufficiently small tojustify the perturbative expansion at the basis of our the-oretical model. Figure 4(c) displays the amplitudes | χ | ,determining the strength of the Kerr-nonlinearity, and | α | , scaling the squeezing dynamics, in units of | δω | andas a function of ω p . One observes that for the chosen pa-rameters | χ | > | α | . Maximum antibunching is here foundwhen the cavity mean photon number is maximum.It is important to notice that emission of polaritons inpairs is possible when the collective dipole of the atomicarray is driven. For fixed values of Ω and g , we expectthat this effect is washed away as N is increased: this be-haviour is expected from the scaling of the ratio ε with N .Taking k = G/ X ≪
1, for instance,one finds ε ∼ √ N, indicating that the strength of theKerr nonlinearity decreases relative to the coupling α as N grows. This is also consistent with the results reportedin Fig. 3. In this context, the expected dynamics is rem-iniscent of the transition from antibunching to bunchingobserved as a function of the number of atoms in atomicensembles coupled with cavity QED setups [28].For the results here presented we have assumed thespontaneous emission rate to be smaller than κ . In gen-eral, the predicted nonlinear effects can be observed incavities with a large single-atom cooperativity and in the
130 132 134 136 138 14000.511.522.53 ( ω − ω p ) /κ g ( ) ( ) (a) (a)
130 132 134 136 138 1400.10.20.30.40.50.60.70.8 h a † a i , h b † Q b Q i ( ω − ω p ) /κ (b) (b)
130 132 134 136 138 14000.511.522.5 ( ω − ω p ) /κ | χ / δ ω | , | α / δ ω | (c) (c) FIG. 4: (Color online) (a) Second order correlation functionat zero time delay g (2) (0) versus ω p (in units of κ ) when thecavity is solely pumped by inelastic processes (here, k = G/ Q ′ = Q, Q and Q ′ = G ′ / ϕ, φ L = 0). The correlationfunction is evaluated numerically solving the master equationfor the polaritons in presence of cavity decay, with the co-herent dynamics given by Hamiltonian (A1) (solid line) andby Hamiltonian (28) (dashed line). (b) Corresponding av-erage number of intracavity photons h a † a i (blue upper line)and spin wave occupation h b † Q b Q i (red lower line). (c) Ratios | χ/δω | (blue upper line) and | α/δω | (red lower line) versus ω p . The parameters are g = 80 κ , Ω = 30 κ , ω z − δ c = 70 κ ,and N = 2 atoms. At the minimum of g (2) (0), ω z ≃ κ and δ c ≃ κ . so-called good cavity regime [22]. The required param-eter regimes for observing squeezing have been realizedin recent experiments [29]. The parameters required inorder to observe a two-photon gateway are rather de-manding for the regime in which the atoms are drivenwell below saturation. Nevertheless, a reliable quantita-tive prediction with an arbitrary number of atoms wouldrequire a numerical treatment going beyond the Holstein-Primakoff expansion here employed. IV. CONCLUSION
An array of two-level atoms coupling with the modeof a high-finesse resonator and driven transversally by alaser can operate as controllable nonlinear medium. Thedifferent orders of the nonlinear responses correspond todifferent nonlinear processes exciting collective modes ofthe array. Depending on the phase-matching conditionand on the strength of the driving laser field a nonlinearprocess can prevail over others, determining the dom-inant nonlinear response. These dynamics emerge fromthe backaction of the cavity mode on the scatterers prop-erties, and are enhanced for large single-atom coopera-tivities. We have focussed on the situation in which thescattering into the resonator is inelastic, and found thatat lowest order in the saturation parameter the light atthe cavity output can be either squeezed or antibunched.In the latter case, it can either operate as single-photon ortwo-photon gateway, depending on the phase-matchingconditions. Our analysis permits one to identify the pa-rameter regimes, in which a nonlinear-optical behaviourcan prevail over others, thereby controlling the mediumresponse. An interesting outlook is whether the consid-ered effects can be used in order to develop turnstile de-vices, like the one realised in [30], for quantum networks.In view of recent experiments coupling ultracold atomswith optical resonators [8, 9, 17, 18, 24, 29, 31–33], thesefindings show that the coherence properties at the cav-ity output can be used for monitoring the spatial atomicdistribution inside the resonator. Another related ques-tion is how the properties of the emitted light dependon whether the atomic distribution is bi- or multiperi-odic [34]. In this case, depending on the characteristicreciprocal wave vectors one expects a different nonlin-ear response at different pump frequency and possiblyalso wave mixing. When the interparticle distance isuniformly distributed, then coherent scattering will besuppressed. Nevertheless, the atoms will pump inelasti-cally photons into the cavity mode. While in free spacethe resonance fluorescence is expected to be the incoher-ent sum of the resonance fluorescence from each atom,inside a resonator one must consider the backaction dueto the strong coupling with the common cavity mode.Finally, in this article we neglect the atomic kinetic en-ergy, assuming that the spatial fluctuations of the atomiccenter of mass at the potential minima are much smallerthan the typical length scales determining the coupling with radiation [10]. It is important to consider, that whenthe mechanical effects of the scattered light on the atomsis taken into account, conditions could be found whereselforganized atomic patterns are observed [31, 32, 35–39], which are sustained by nonclassical light. The anal-ysis here presented sets the basis for studies towards thisdirection.
Acknowledgments
The authors are grateful to Sonia Fernandez-Vidal,Stefan Rist, Sergey Hritsevitch, and Belen Paredes forstimulating discussions and helpful comments. This workwas supported by the European Commission (EMALIMRTN-CT-2006-035369; Integrating project AQUTE;STREP PICC), by the European Science Foundation(EUROQUAM: CMMC), and by the Spanish Ministe-rio de Ciencia y Innovaci´on (QOIT, Consolider-Ingenio2010; QNLP, FIS2007-66944; Ramon-y-Cajal; Juan-de-la-Cierva). G. M. acknowledges the German ResearchFoundation (DFG) for support (MO1845/1-1). H. H. ac-knowledges support from the Spanish Ministerio de Cien-cia y Innovaci´on (FPI grant).0
Appendix A: Derivation of the effective Hamiltonian ¿From the general form of the Hamiltonian in Eq. (8)for the case in which there is no coherent scattering( Q ′ = ± Q ), in the weak excitation limit one can obtainthe effective dynamics for the polariton described by γ .We focus on the regime in which Ω √ N ≪ | ω z | . As weare interested in the dynamics of the mode b Q s and ofthe cavity mode a , the relevant terms determining theirdynamics are given in lowest order by H eff = H pump + ~ ω Q ′ b † Q ′ b Q ′ + ~ X σ =1 , ω σ γ † σ γ σ + H ′ (A1)with H ′ = − ~ g √ N n b † Q ′ b † Q ′ (cid:0) b − Q e i ϕ + b Q e − i ϕ (cid:1) δ Q ′ ,G/ +2 b † Q ′ (cid:16) b † Q e i ϕ + b †− Q e − i ϕ (cid:17) b Q ′ + (cid:16) b † Q b † Q b Q e i ϕ + b †− Q b †− Q b − Q e − i ϕ (cid:17) +(1 − δ k,G/ ) h b † Q b †− Q b − Q + δ Q, ± G / b †− Q b †− Q b Q +2 δ Q,Q ′ b †− Q b † Q ′ b Q + δ − Q,Q ′ b †− Q b †− Q b Q ′ + δ Q ′ ,Q + G/ b † Q ′ b † Q ′ b Q i e i ϕ +(1 − δ k,G/ ) h b † Q b †− Q b Q + δ Q, ± G / b † Q b † Q b − Q +2 δ − Q,Q ′ b † Q b † Q ′ b − Q + δ Q,Q ′ b † Q b † Q b Q ′ + + δ Q ′ , − Q + G/ b † Q ′ b † Q ′ b − Q i e − i ϕ o a − i ~ Ω2 √ N e − i φ L n b † Q ′ b † Q b Q + 2(1 − δ k,G/ ) b †− Q b † Q ′ b − Q + δ Q ′ ,G/ b † Q b †− Q b Q ′ (cid:2) − δ k,G/ ) (cid:3) +(1 − δ k,G/ ) h δ Q,Q ′ b † Q b † Q b − Q + δ − Q,Q ′ b †− Q b †− Q b Q + δ Q ′ ,Q + G/ b † Q b † Q b Q ′ + δ Q ′ , − Q + G/ b †− Q b †− Q b Q ′ io +H . C . By substituting b Q ′ with its mean value in Eq. (27), whichcorresponds to neglect the backaction on the mode Q ′ dueto the nonlinear coupling, one obtains closed equationsof motion for the modes b Q s and a (where we therebydiscard the effect of the nonlinear coupling with the othermodes, which are initially empty and which gives riseto higher order corrections). In this limit the effectiveHamiltonian (28) for the polariton γ is derived providedthat the detuning of the laser from the polariton γ ismuch larger than the strength of the nonlinear couplingwith polariton γ . [1] D. F. Walls and P. Zoller, Phys. Rev. Lett. , 709(1981); C. Cohen-Tannoudij, J. Dupont-Roc, and GilbertGrynberg, Atom-Photon Interactions , J. Wiley and SonsEds. (New York, 1992).[2] L. Mandel and E. Wolf, Optical Coherence and QuantumOptics (Cambridge University Press, Cambridge, U.K.,1995).[3] W. Vogel, D.-G. Welsch, Phys. Rev. Lett. , 1802(1985).[4] L. Mandel, Phys. Rev. A , 929 (1983).[5] C. Skornia, J. von Zanthier, G. S. Agarwal, E. Werner,and H. Walther, Phys. Rev. A , 063801 (2001); G. S.Agarwal, J. von Zanthier, C. Skornia, and H. Walther, Phys. Rev. A 65 , 053826 (2002).[6] N.W. Ashcroft, N.D. Mermin,
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