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Controlled Dicke Subradiance from a Large Cloud of Two-Level Systems Tom Bienaimé,1 Nicola Piovella,2 and Robin Kaiser1 1Université de Nice Sophia Antipolis, CNRS, Institut Non-Linéaire de Nice, UMR 7335, F-06560 Valbonne, France 2Dipartimento di Fisica, Universita` Degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy (Dated: January 12, 2012) Dickesuperradiancehasbeenobservedinmanysystemsandisbasedonconstructiveinterferences between many scattered waves. The counterpart of this enhanced dynamics, subradiance, is a destructive interference effect leading to the partial trapping of light in the system. In contrast to therobustsuperradiance,subradiantstatesarefragileandspuriousdecoherencephenomenahitherto obstructed the observation of such metastable states. We show that a dilute cloud of cold atoms is an ideal system to look for subradiance in free space and study various mechanisms to control this 2 1 subradiance. 0 PACSnumbers: 03.65.Aa,03.65.Yz,32.80.Qk,42.25.Bs,42.50.Gy,42.50.Nn 2 n a Interferences between scattered waves by many parti- time long photon storage in a system of N atoms in free J clescangiverisetoalargevarietyofphenomena,includ- space. Subradiance for two ions has been observed in 1 ingcollectiveeffectssuchasBraggorMiescattering,well the past [19] and a reduced decay rate into one radia- 1 knowninthecontextofclassicaloptics[1]. Moreintrigu- tion mode has been achieved for N atoms [20]. How- ] ing situations can arise in the mesoscopic regime where ever, it has not yet been possible to control and sup- h interferences are at the origin of coherent backscattering press the decay into all vacuum modes for N atoms in p andAndersonlocalization[2]. Interestintrappingwaves freespaceextendingthusthelifetimeoftheexcitationto - m in disordered media has spurred the efforts in multiple many times the natural lifetime of a single atom. Start- scattering of light, as photons seem to be an ideal can- ing from the Ansatz used in previous work by several o t didate for non-interacting waves. Several mechanisms authors [11, 21–24], we show that the exponential kernel a allowing to increase the time an optical excitation can ofthedipole-dipolecouplingyieldsanimportantfraction . s stay in a system are known to exist and are based on ofatomstobecoupledintosubradiantmodes. Following c i different physical phenomena [3, 4]. Multiple scattering [25],westudydifferentinhomogeneousbroadeningmech- s of light or radiation trapping for instance allows for in- anisms, which allow to go beyond the fundamental Fano y h creased photon trapping in the absence of interferences coupling by controlled coupling between super- and sub- p [5]. Anderson localization is another possibility where radiant states using Doppler broadening and inhomoge- [ interferencesformlocalizedstateswithexponentiallyde- neous light shifts. In this letter, we consider three differ- 1 creasing coupling to the environment and a dense sam- ent experimental parameters to control subradiance: the v ple fulfilling the Ioffe-Regel criterion [6] is assumed to optical thickness of the cloud, the driving laser intensity, 4 be required. This letter addresses another fundamental and the temperature of the cloud. Moreover, we show 7 mechanism leading to ‘long-lived’ modes of excitation: how to distinguish subradiance from multiple scattering 2 the study of subradiance in dilute clouds of cold atoms of light by tuning the driving laser frequency. 2 . i.e. the trapping of light by destructive interferences in 1 the single scattering regime. This mechanism is based 0 on the pioneering work by Dicke who studied enhanced 2 1 decay rates in small and large samples [7]. Typically : Dicke states are considered for an assembly of N two- v levelsystems, realizede.g. byatoms[8]orquantumdots Xi [9]. Different regimes can be studied, using either an We consider a Gaussian cloud, with root mean square size σ, of N two-level atoms (positions r , transition r initially fully inverted system with N photons stored by i a wavelength λ = 2π/k, excited state lifetime 1/Γ), ex- the N atoms or an initial state in which the whole sys- cited by an incident laser (Rabi frequency Ω , detun- tem shares a single excitation [10, 11]. These systems 0 ing ∆ , wavevector k ). We define the optical thickness have attracted an increasing attention in the context of b(∆ )0= b /(cid:2)1+4(∆0/Γ)2(cid:3) with b = 3N/(kσ)2 its res- quantum information science [12–14], where the accessi- 0 0 0 0 onant value. Restricting the atomic Hilbert space to the ble Hilbert space can be restricted to single excitations subspacespannedbythegroundstateoftheatoms|G(cid:105)≡ using e.g. the Rydberg blockade [15–18]. |g···g(cid:105) and the single excited states |i(cid:105) ≡ |g···e ···g(cid:105) i In this letter, we will show how it is possible to under- and tracing over the photon degrees of freedom, one ob- stand and control the coupling of light into metastable tain an effective Hamiltonian describing the time evolu- (cid:80) subradiant states, illustrating that large dilute clouds of tion of the atomic wavefunction |ψ(cid:105) = α|G(cid:105)+ β |i(cid:105). i i cold atoms are an ideal system to observe for the first The effective Hamiltonian using standard approxima- 2 tions [24, 26] can then be written as: on 100 Heff = (cid:126)Ω20 (cid:88)(cid:2)ei∆0t−ik0·riS−i +e−i∆0t+ik0·riS+i (cid:3) opulati Superradiance p i d −i(cid:126)2Γ(cid:88)S+i S−i − (cid:126)2Γ(cid:88)(cid:88)VijS+i S−j, (1) xcite10-1 e i i j(cid:54)=i d Subradiance e z where the first term describes the coupling to the laser ali m field, the second accounts for the finite lifetime of the r o excited states, the third one describes the dipole-dipole N10-2 0 1 2 3 4 5 6 7 8 interactions, with Vij = expk|irki|−rir−j|rj|, and S±i , Szi are Time (Γ−1) the usual pseudo-spin operators for the kets |g (cid:105) and i |e (cid:105). The effective Hamiltonian (1) describes dipole- FIG. 1: (color online). Time evolution of the normalized ex- i dipolecouplingsinthescalarlightapproximation,where cited state population (black solid curve) after switching off the laser for N = 2000 atoms, kσ = 10, ∆ = 10Γ (the near field and polarization effects are neglected since we 0 laserwasonbeforeduring50Γ−1 toletthesystemreachthe are considering dilute clouds, N(λ/σ)3 (cid:28) 1. At low steady state). At first, the population decreases faster than intensity (where the single excitation approximation is the single atom decay (black dashed line) and then slower. valid), α(cid:39)1 and the previous model describes a system Werespectivelyidentifythesephenomenaassuper-andsub- of N classical dipoles driven by an incident electric field radiance. The inset shows the emission diagrams of the su- as expected by linear optics [27]. We use these classi- perradiant ‘timed Dicke’ state |TD(cid:105) (blue) and subradiant cal equations to study single photon subradiance. The modes (red) averaged over 8 realizations (rescaled to allow convenient comparison). driven steady state solution |ψ(cid:105) (cid:39) |G(cid:105) + (cid:15)|TD(cid:105) bears the phenomenon of single photon superradiance with a ‘timed Dicke’ state |TD(cid:105) = √1 (cid:80) eik0ri|i(cid:105) and an am- √ N i and subradiant states, it should be possible to efficiently plitude (cid:15)(cid:39) NΩ /(2∆ +i(1+b /12)Γ) [24]. This was 0 0 0 store populations into ‘long-lived’ subradiant modes. e.g. exploited to explain the measured cooperative radiation pressure force in dilute clouds [23]. In Fig. 1, we show the normalized excited state popu- InordertostudyenhancedstoragetimebasedonDicke lation∝(cid:80)i|βi|2 ascomputedfromnumericalsolutionof subradiance, we will investigate the coupling between theeffectiveHamiltonianEq. (1)inthelinearregime. As ‘short-lived’ superradiant states, such as |TD(cid:105), and the precise initial conditions play a crucial role in the subse- otherstates|ϕ(cid:105)ofthesingleexcitationHilbertsubspace. quent fast and slow decay [27], we start initially with all Let us first consider the minimal coupling, based on ra- atomsinthegroundstate|G(cid:105)andkeepthecoherentlaser diative dipole-dipole coupling. As the effective Hamilto- drive for 50Γ−1 before switching it off, realizing thus ex- nian Eq. (1) is non-Hermitian, its eigenstates are non perimentallyaccessibleconditions. Thefastinitialdecay orthogonal ((cid:104)ϕsuper|ϕsub(cid:105) =(cid:54) 0) and subradiant states ofthesuperradiantstateΓsuper =(1+b0/12)Γisclearly thus have common features with auto-ionizing states or seen. Moreover, after this initial fast decay, subradiance Fano resonances [28]. This Fano-type coupling between manifests itself in a slowly decaying excited population ‘long-lived’subradiantstatesand‘short-lived’superradi- with a rate well below the single atom decay rate. At ant states leads to additional decay channels in addition first,thesubradiantdecayisnotpurelyexponentialsince to direct decay of the subradiant states to the ground several modes decay simultaneously. For longer times, it state. This situation is reminiscent of the Hanle effect then ends up with a pure exponential decay (referred as [29] where a competition of direct decay and transverse subradiant decay in the following) when only one ‘long- coupling can lead to surprisingly narrow resonances. For lived’ mode dominates. The emission diagram of the su- instance, thesteadystatesolutionofopticalBlochequa- perradiant ‘timed Dicke’ state |TD(cid:105) is clearly forward tionsforasystemconsistingofthegroundstate|G(cid:105),one directed (see Fig. 1), a phenomenon reminiscent of Mie superradiant|ϕ (cid:105)andasinglesubradiantstate|ϕ (cid:105) scattering. On the other hand, subradiant modes show super sub reveals that in the absence of Fano coupling the steady isotropicdiagrams. Theydonotpossessthesymmetryof state solution of the subradiant state is zero. Neglecting thelaserexcitationsincetheyarenotdirectlycoupledto the direct decay of the subradiant state to the ground it, a feature which can be exploited in the experimental state on the other hand, one finds that for resonant ex- detection of subradiance. citation and small coupling between the excited states, In Fig. 2, the subradiant decay rates (due to Fano afteralongtransienttimeallatomsarepumpedintothe coupling) are plotted as a function of the inverse on- subradiant state |ϕ (cid:105). The optical Bloch equations of resonanceopticalthicknessofthesystem. Thedifference sub suchasimplifiedthree-levelmodelshowthat,byincreas- to multiple scattering of light which can also yield long ing in a controlled manner the coupling between super- photon trapping times, can be understood by looking at 3 0.8 3 Γ)0.6 Γ) e ( e (2 at at y r0.4 0.12 y r a a ec ec1 D D 0.2 0.11 4 2 0 2 4 Detuning (Γ) 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 / Optical thickness Doppler broadening (kσv/Γ) FIG.2: (coloronline). Subradiantdecayrateasafunctionof FIG. 3: (color online). Super- (blue curve) and subradiant 1/b (we kept N = 400 constant, ∆ = 10Γ and varied kσ). (red curve) decay rates as a function of Doppler broadening 0 0 At large optical thickness it scales as ∝ Γ/b (black dashed forN =200atoms,kσ=5andalaserdetuningof∆ =10Γ. 0 0 line) and saturates to Γ for dilute clouds. The insert shows Initially the laser is driving the atoms for 50Γ−1 and the thesubradiantdecayrateasafunctionofdetuningforb =3 subradiant decay rate is evaluated 50Γ−1 after the laser is 0 (N = 400, kσ = 20). The shaded area corresponds to the switched off. multiple scattering region b(∆ )>1. 0 alsobecomparedtoscalinglawsobtainedfromquantum the decay rates for different excitation frequencies. As chaoticscatteringtheory[31]: forlargeb (corresponding 0 shown in the inset of Fig. 2, close to resonance we ob- toalargenumberofatomsN comparedtothenumberof servereduceddecayrates,whichweassociatetothelarge outgoing modes M ∼ (kσ)2) the minimum width of the optical thickness for resonant photons [5]. For larger de- resonance is expected to scale as Γ ∼M/N ∼1/b . sub 0 tunings however, the decay rate becomes independent of We now turn to the possibility of controlled coupling the excitation frequency, consistent with the subradiant between super- and subradiant states, opened by cold nature of states weakly excited off-resonance. atoms. In atomic physics many different inhomogeneous The fundamental Fano coupling between sub- and su- broadening mechanisms are known. We will focus on perradiant states in the Dicke basis can be understood two such mechanisms, well adapted to become a control from the diagonal terms in the bare bases |i(cid:105). Indeed, knob to steer excitations into the subradiant state. Let the local field at i-th atom is the sum of the external us thus consider the impact of residual motion of the field E and the field scattered by all the other dipoles j atoms. TheeffectiveHamiltonian(1)canbeextendedto 0 at the location of the atom i: include Doppler shifts and time dependent positions of (cid:88) the atoms. The coupled equations for the dipole ampli- Etot(ri)=E0(ri)+ Ej(ri)(cid:39)E0(ri)(1+εieiϕi). (2) tudes β read in the linear regime i j(cid:54)=i (cid:20) (cid:21) We describe the scattered field by a small lo(cid:112)cal speckle β˙i = −Γ2 +i(∆0−k0·vi) βi− iΩ20 + i2Γ(cid:88)Vijβj, fieldwitharandomamplitudescalingasεi ∝ b(∆0)(cid:28) j(cid:54)=i 1 and a random phase ϕ . The amplitudes β depend on (4) i i thelocalfieldanddifferfromthedriventimedDickeam- pcolirtruedspeso.nTdihnegdtiopoalne iinihsotmhuosgednrievoeunswbirtohadaernainndgommefichelad- SwohlevriengVtihje(ts)e e=quatkeii|okrn|ir−si−rfjro+j+r((vviini−−cvvrjje))att||sein−gik0t·e[rmi−prejr+a(tvuir−ev,j)wt]e. nism [25], where we now have inhomogeneity in ampli- notice that the fast superradiant and the slow subradi- tude and phase. An estimation of the expected lifetimes antdecayratesdrawclosertothesingleatomdecayrate of the ‘long-lived’ subradiant modes is also an important Γ(seeFig. 3). Thisresultshowsthatthefragilesubradi- issue. We have checked numerically they scale as antmodesarequicklydestroyedevenbymoderateatomic motion. Cold atoms seem a well adapted system allow- 1 Γsub ∼ b , (3) ing to tune the atomic motion from being negligible to 0 becoming dominant. We also checked that the dominant for b >1 (see Fig. 2). This scaling is close to what can term in the reduced super and subradiance stems from 0 be obtained assuming that in the limit of large detun- thepositiondependentdipole-dipolecouplingV (t)term ij ing ∆ , where multiple scattering inside the sample can rather than from the random detuning term in Eq. (4). 0 be neglected, the escape rate of the excitation from the Thisdependenceontheatomicmotionexplainswhysub- sample is well approximated by the inverse of the ‘long- radianceofN atomshasnotbeenobservedinhotatomic lived’ mode lifetime, which scale as 1/b [30]. This can vapors, despite the efforts in this field in the 1970s [8]. 0 4 Exploitingthepossibilityofcontrolledcouplingviain- 3 )3 homogeneous broadening even further, we studied the Γ) (a) 30− (b) role of larger laser intensities on super- and subradiance. e ( 2 1x 2 As the effective Hamiltonian approach is only valid to y rat ac. ( filirnsetaorrodpertiicnstrheegiRmaeb.iWfieeldtchoeurepfloinreg,uistecdanaomnalystterreaetquthae- Deca 1 ub. fr1 tion approach, where the evolution of the atomic den- 0 S0 sity operator ρ in the electric-dipole, rotating-wave, and 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 Born-Markov approximations is given in the interaction Rabi frequency (Γ) Rabi frequency (Γ) picture by [32, 33] FIG.4: (coloronline). (a)Super-(bluecurve)andsubradiant 1 (cid:16) (cid:17) (cid:88)(cid:88) (red curve) decay rates as a function of laser intensity for ρ˙ = i(cid:126) Heffρ−ρHe†ff + γijS−jρS+i (5) N = 200 atoms, kσ = 5 (b0 = 24) and a laser detuning of i j ∆0 =10Γ. (b) Subradiant fraction for the same parameters. The laser is switched on during 50Γ−1 and the subradiant where in the scalar light limit γ = Γsink|ri−rj|. Pro- decayrateandsubradiantfraction(i.e. theremainingexcited ij k|ri−rj| statepopulation)arecomputed50Γ−1 afterswitchingoffthe jecting Eq. (5) on the different Fock states, we obtain laser. a set of coupled equations for the density matrix ele- ments ρ ≡ (cid:104)G|ρ|G(cid:105), ρ ≡ (cid:104)i|ρ|G(cid:105) and ρ ≡ (cid:104)i|ρ|j(cid:105). G|G i|G i|j This approach is still restricted to the same Hilbert sub- field, is quickly reduced by a large amount. The inho- space with at most one excitation and is thus limited to mogeneous coupling is thus significantly decreased clos- moderate laser intensities or to situations where multi- ing the ‘door’ between the subradiant and superradiant ple excitations are suppressed as for instance in the case states. One can see this effect in Fig. 4 (a), where a de- of Rydberg blockade. However, it does not require the creaseinthesuperradiantdecayrateisobservedbecause ground state population to remain unaffected and can thereisstillsomeinhomogeneousbroadeningsourcejust take into account the light shifts of the states. We have afterswitchingoffthelaser. However,forlongertimethe checkedanalyticallyandnumericallythatatfirstorderin localfieldismuchlessintenseandalmostnovariationof Rabifrequencyofthelasercoupling,themasterequation thesubradiantdecayrateisseen-lightinducedinhomo- is equivalent to the equation used in the effective Hamil- geneous broadening source is no longer present. In that tonianapproachEq. (1)inthelinearregimeα(cid:39)1(with case the subradiant decay rate is just the same as the correspondence ρ ↔ 1, ρ ↔ β ). Here we exploit G|G i|G i one given by Fano coupling. In the same way, for low in- the small fluctuations induced by the random local field tensities, the inhomogeneous broadening induced by the driving the individual atoms i. As the local field has a laser is dominated by the Fano coupling. The subradi- randomspecklestructure(seeEq. (2)),therandomlight ant decay and subradiant fraction then remain almost shifts and phases can be understood as an inhomoge- unaffected, as illustrated in Fig. 4 for Ω <0.1Γ. neousbroadeningmechanism, dependingontheinterfer- 0 encetermbetweentheincidentfieldE (r )andthescat- In conclusion, we have shown that inhomogeneous 0 i tered field εieiϕiE0(ri). The advantage of the light shift broadeningschemesallowtounderstandandcontrolstor- coupling and dephasing is the flexibility it offers as the age of an optical excitation into ‘long-lived’ subradiant laser intensity can be easily and quickly controlled. Fig. modes. Wehaveproposedtousethecloudopticalthick- 4 shows the super- and subradiant decay rates as a func- ness,thedrivinglaserintensity,orthecloudtemperature tionoftheRabifrequencyΩ andthesubradiantfraction as possible experimental control parameters for subra- 0 (i.e. theremainingexcitedstatepopulation)50Γ−1 after diance, but further parameters, such as a far detuned switching off the laser (the laser was on during 50Γ−1). specklefieldormagneticfieldwouldyieldsimilarresults. WhentheRabifrequencywasvariedwecheckedthatthe This opens the door for the first observation of Dicke (cid:80) excited state population remains small ( ρ < 0.15) subradiance of photons in a cold cloud of N atoms in i i|i to ensure consistency with the model. We observed in free space. Inhomogeneous broadening schemes will also Fig. 4 (b) up to a three time increase of the subradi- be of interest to studies of Anderson localization of light antfractionwhentheintensityisraisedcomparedtothe in resonant two-level systems [10]. Mapping the inho- Ω → 0 value determined by Fano coupling. Note that mogeneous coupling schemes described in this letter to 0 the subradiant fraction would be the same for any inten- a lambda scheme as used in quantum information sci- sity with the effective Hamiltonian approach (due to lin- ence [34] will help to understand limitations of storage earity). Fig. 4illustrateshowchanginglaserintensityal- of qubits in atomic vapors in modes less exposed to fast lowscontrollingthecouplingstrengthbetweensuper-and decay in a decoherence free subspace [35]. Controlled subradiant modes and subsequently the subradiant pop- transfer to other modes can also be achieved with non- ulation. 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