Collective Enhancement and Suppression of Excitation Decay in Optical Lattices
aa r X i v : . [ qu a n t - ph ] O c t Collective Enhancement and Suppression of Excitation Decay in Optical Lattices
Hashem Zoubi, and Helmut Ritsch
Institut f¨ur Theoretische Physik, Universit¨at Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria (Dated: 23 October, 2009)We calculate radiative lifetimes of collective electronic excitations of atoms in an infinite one di-mensional lattice. The translational symmetry along the lattice restricts the photon wave vectorcomponent parallel to the lattice to the exciton wave number and thus the possible emission direc-tions. The resulting radiation damping rate and emission pattern of the exciton strongly deviatesfrom independent atom. For some wave numbers and polarizations the excitons superradiantly decayvery fast, while other excitons show zero radiation damping rate and form propagating meta-stableexcitations. Such states could be directly coupled via tailored evanescent fields from a nearby fiber.
PACS numbers: 37.10.Jk, 42.50.-p, 71.35.-y
Ultracold atoms in optical lattices are now one of themost flourishing directions of experimental and theoret-ical quantum physics [1]. Beside fundamental physicsissues on entanglement, measurement and decoherence,more and more applications and connections appear astest-systems for a wide range of puzzling effects in solidstate physics. The trapped atoms can be considered asartificial crystals, which differ from solid crystals in theirprecise and easy controllability of occupation number,lattice constant, lattice depth and symmetry. The latticeproperties can be controlled through the laser intensity,wave length and polarizations, and through laser coolingthe motion can be cooled and confined to the lowest statein each potential well corresponding to the first Blochband. The atomic dynamics is generally well described bya Bose-Hubbard model [2], which includes atom hoppingamong nearest neighbor sites and the on-site repulsiveatom-atom interactions. As central prediction one gets aquantum phase transition from a superfluid into a Mottinsulator phase [2, 3] with a fixed number of atoms persite for a deep lattice. Other condensed matter phenom-ena appearing in more complex situations can be studiedin generalized optical lattice setups as well [4].Optical properties of crystal solids are a well studiedsubject. One of the phenomena that strongly charac-terize the optical properties of solids is the formation ofcollective electronic excitations (excitons). In molecularcrystals a local electronic excitation delocalizes amongthe lattice molecules due to electrostatic interactions.The corresponding delocalized eigenstates are Frenkel ex-citons [5], which are quasi-particles that propagate in thelattice. In previous work we showed that Frenkel like-excitons also exist for ultracold atom lattices in the Mottinsulator phase [6, 7]. Enclosing the light field within acavity, such excitons and photons with the same wavevectors coherently couple to form cavity polaritons [6, 8].In the present work we study the radiative properties ofsuch excitons, which are formed due to resonant dipole-dipole interactions. Each exciton is represented by a wavethat propagates in the lattice with a given wave number[6] and coupled to the free space radiation field modesinto which it can decay. For the moment we restrictourselves to an infinite one dimensional lattice with one atom per site. This allows one to directly calculate the farfield radiation pattern and the resulting effective damp-ing rate of the exciton explicitly. As a result of the latticesymmetry an exciton with a fixed wave number can emita free photon with the same wave number componentalong the lattice axis. In particular we investigate, howthe damping rate, which significantly differs from the in-dependent atom result, changes with the wave numberand polarization direction.We consider a one-dimensional string of two-levelatoms with an electronic transition energy E a = ~ ω a with lattice constant a and one atom per site (see fig-ure (1)). Energy is transferred among atoms at differentsites by resonant dipole-dipole interactions [6], so thatthe electronic excitation Hamiltonian contains two terms,the on-site excitation and the energy transfer, it reads H ex = X n ~ ω a B † n B n + X nm ~ J nm ( θ ) B † n B m , (1)where B † n and B n are the electronic excitation creationand annihilation operators at site n , respectively, whichare assumed to be boson operators at low excitation den-sity with the commutation relation [ B n , B † m ] = δ nm . TheHamiltonian can be easily diagonalized, in exploiting thelattice symmetry, by using the transformation B n = 1 √ N X k e ikz n B k , (2)where N is the number of sites, and z n = an is the posi-tion of site n . We get the collective electronic excitationHamiltonian H ex = X k ~ ω ex ( k, θ ) B † k B k , (3)with the dispersion ω ex ( k, θ ) = ω a + P L J ( L, θ ) e ikL ,and L = ( n − m ) a . Restricting the energy trans-fer to only nearest neighbor sites, we explicitly obtain ω ex ( k, θ ) = ω a + 2 J ( θ ) cos( ka ), where we used J ( θ ) = J ( a, θ ), and for the dipole-dipole interaction we have ~ J ( θ ) = µ πǫ a (cid:0) − θ (cid:1) , here µ is the electric tran-sition dipole, and θ is the angle between the transitiondipole and the lattice axis, as seen in figure (1). Includ-ing corrections from more distant neighboring atoms willchange this spectrum and the particular form of the ex-citons, which cannot be calculated explicitly any more.Nevertheless, the results and conclusions of the followinglife time calculations are generally only weakly influencedby these corrections as the wave nature of the exciton ismainly determined by symmetry properties of the lattice.In using periodic boundary condition, the wave number k takes the values k = πNa p , with p = 0 , ± , ± , · · · , ± N/ J ( θ ).Due to the energy transfer, an quasi-stationary elec-tronic excitation is delocalized in the lattice and best rep-resented by a wave that propagates to the left or the rightdirection with wave number k , which is a good quantumnumber. Such collective electronic excitations are called“excitons”. While some excitons directly couple to in-coming plane waves, for other excitons with a fixed wavenumber k and a prescribed transition dipole direction θ ,such a straightforward excitation is not possible. As hasbeen implemented recently, one way to overcome this is-sue is the use of a thin pulled fiber along the lattice [9].The atomic lattice is generated parallel to the fiber atgiven distance, where the fiber photons and the corre-sponding lattice excitons are coupled. Hence, by sendingphotons with prescribed wave number and polarizationthrough the fiber we can excite an exciton in the latticewith the same wave number and polarization through theevanescent wave coupling [10]. a µ θ FIG. 1: A one-dimensional lattice of lattice constant a , andwith one atom per site. The transition dipole µ is plottedwhich makes an angle θ with the lattice axis. Here we calculate the electric field and intensity of theradiation emitted by a 1D lattice exciton into free space.The free-space radiation field Hamiltonian is given by H rad = X q λ E ph ( q ) b † q λ b q λ , (4)where b † q λ and b q λ are the creation and annihilation op-erators of a photon with wave vector q and polarization λ , respectively. The photon energy is E ph ( q ) = ~ ω r ( q ) = ~ cq . The electric field operator isˆ E rad ( r ) = i X q λ r ~ cq ǫ V n b q λ e q λ e i q · r − b † q λ e ∗ q λ e − i q · r o , (5)where e q λ is the photon polarization unit vector, and V is the normalization volume.The light-matter coupling is given by the electric dipoleinteraction H I = − ˆ µ · ˆ E rad , where the atomic transition dipole operator is defined by ˆ µ = ~µ P n (cid:0) B n + B † n (cid:1) . Weseparate the photon wave vector into two components,one parallel to the lattice axis and the other orthogonalto it, namely we use q = q ′ + q z z , with q = q ′ + q z .The atomic site positions are given by r n = z n z . Werepresent the electronic excitations by excitons, in usingthe inverse transformation of equation (2), and by usingthe lattice symmetry property N P n e i ( q z − k ) z n = δ q z k .The coupling between the excitons and the free radiationfield in the rotating wave approximation and with linearpolarizations, is given by H I = X q ′ kλ i ~ g q λ n b q ′ k,λ B † k − b † q ′ k,λ B k o , (6)where now q = ( q ′ , k ), with the coupling parameter ~ g q λ = − r ~ cqN ǫ V ( ~µ · e q λ ) . (7)It is seen that the coupling is only between excitons andphotons with the same z component wave vectors. Thetranslational symmetry along the lattice axis results ina conservation of momentum along the axis. Hence anexciton with wave number k couples only to photons withthe same k parallel to the lattice. Such a result stronglyaffect both the radiation emitted by an exciton with afixed k , and the damping rate of such exciton, as we showin the following. The effect of similar retarded interactionon the exciton spectrum at low dimensional molecularcrystals was studied by Agranovich et. al. [11].Now we calculate the electric field and the intensityof the light emitted from one dimensional optical latticeof cold atoms. We start from the radiation field opera-tor equation of motion, solving it formally and substitut-ing the source part back in the electric field operator ofequation (5), then we obtain for the positive part of theelectric field, for a fixed k and θ ,ˆ E + rad ( r , t ) = i X q ′ λ ω r ( q ) √ N ǫ V e i [ q ′ · ~ρ + kz − ω e ( k ) t ] × e q λ ( ~µ · e q λ ) Z t dt ′ ˜ B k ( t ′ ) e i [ ω e ( k ) − ω r ( q )]( t − t ′ ) , (8)where the exciton operator in a rotating frame is definedby B k ( t ) = ˜ B k ( t ) e − iω e ( k ) t , and the observation point isat r = ~ρ + z z . As the wave vector component parallelto the lattice is fixed, we have only to sum over wavevectors in the plane normal to the lattice direction. Thesum over q ′ casts into the integral X q ′ → S π Z d q ′ = S π Z + π − π dφ Z ∞ q ′ dq ′ , (9)where V = SL , and S is the normalization plane, withthe lattice length L = N a . We use also the summa-tion over the photon polarization P λ e q λ e q λ = 1 − qq q .For the different vectors we choose: the wave vector is q = ( q ′ cos φ, q ′ sin φ, k ), the transition dipole is ~µ =( µ sin θ, , µ cos θ ), and the observation point is chosento be at r = ( ρ, , z ), as seen in figure (2).After doing the angular integral we keep only the farfield terms proportional to 1 / √ ρ . As solvable examplewe consider long wave length (small wave number) exci-tons with ka ≪
1. Then we apply the Weisskopf-Wignerapproximation, which is in the spirit of the Markov ap-proximation [12]. The retarded electric field is given byˆ E + z ( r , t ) = (1 + i ) µω e ( k ) / cos θ πǫ ac / r πN ρ B k ( t − ρ/c ) e ikz , (10)and the intensity operator isˆ I = ˆ E − z ˆ E + z = (cid:18) µ cos θ ǫ a (cid:19) ω e ( k ) N πρc B † k ( t − ρ/c ) B k ( t − ρ/c ) . (11)The expectation values of the exciton operator is h B k ( t − ρ/c ) i = h B k (0) i e − iω e ( k )( t − ρ/c ) e − Γ k ( t − ρ/c ) / , (12)where Γ k is the damping rate of an exciton with wavenumber k . The expectation value of the intensity is h ˆ I i = (cid:18) µ cos θ ǫ a (cid:19) ω e ( k ) N πρc h B † k (0) B k (0) i e − Γ k ( t − ρ/c ) . (13)Our next task should be to calculate the exciton dampingrate, Γ k , into free space, and to compare the result withthe single atom damping rate in order to emphasize thecooperative effect among the atoms on the damping rate. q k θµ y x zq’ φ z (ρ,0, ) FIG. 2: The lattice is in the z direction, where the transitiondipole µ is in the ( x − z ) plane and makes and angle θ withthe lattice direction. The exciton has a wave number k alongthe lattice. The photon wave vector q has a component q ′ normal to the lattice in the ( x − y ) plane, and a z component k parallel to the lattice which is equal to the exciton wavenumber. The observation point is at ( ρ, , z ). The radiative exciton damping rate with a fixed k canbe evaluated using the Fermi Golden Rule, which readsΓ k = 2 π ~ X q ′ λ |h f | H I | i i| δ ( E ex ( k ) − E ph ( q )) . (14)The initial state is of a single exciton of wave number k in the lattice and an empty radiation field, that is | i i = | ex ( k ) , ph i , and the final state is of zero exci-tons and a single radiation field photon of wave vector q , that is | f i = | ex , ph ( q ) i . The matrix element isgiven by h f | H I | i i = i q ~ cqN ǫ V ( ~µ · e q λ ). As before theemitted photon has a wave vector component parallelto the lattice equals to the exciton wave number k . Thesummation is over the wave vectors q ′ normal to thelattice direction. The delta function cares for the con-servation of energy of the exciton and the emitted pho-ton. The summation over the photon polarizations isobtained by the relation P λ | ~µ · e q λ | = | ~µ | − | q · ~µ | q .The sum over q ′ is converted into the integral of equa-tion (9). As before we assume a transition dipole witha fixed direction of ~µ = ( µ sin θ, , µ cos θ ), and with q · ~µ = q ′ µ sin θ cos φ + kµ cos θ . After integration weobtain the resultΓ k = µ E ex ( k )4 ǫ a ~ c (cid:26) θ − ( ~ ck ) E ex ( k ) (cid:0) θ − sin θ (cid:1)(cid:27) . (15)Note that the free atom damping rate is Γ at = µ E a πǫ ~ c .As expected for long enough wave vectors k and cer-tain angles θ the damping rate can be much larger thanthe single atom one, as we get a superradiant enhance-ment analogous to the Dicke model. However, we seethat for some k and θ the damping rate can be muchsmaller than the single atom one or even completely van-ish. Such metastable states (dark states), which appearfor lattices with two atoms per site [8], thus also existin infinite chains of atoms as treated. Quite generally at θ = 0 o we can calculate a critical wave vector k c , givenby E ex ( k c ) = ~ ck c , above which the damping rate is zeroand the excitations can no longer decay radiatively. Forgeneral polarization such a critical wave vector exists ifthe following equation has a solution( ~ ck c ) E ex ( k c ) = 1 + cos θ θ − sin θ . (16) D a m p i ng R a t e [ H z ] ka (a) D a m p i ng R a t e [ H z ] ka (b) FIG. 3: The damping rate vs. ka , for (a) θ = 0, and (b) θ = 90. The dashed-line is for a single atom damping rate. Let us now exhibit this behavior in several plots of thedamping rate. We use the parameters: the transitionenergy is E a = 1 eV , the lattice constant is a = 1000 A ,and the transition dipole is µ = 1 eA . In figure (3.a)we plot the damping rate as a function of ka for θ = 0 o ,the plot includes also the damping rate of a single atom(dashed line). Note that for small wave vectors, ka ∼ k c the damping rate becomes zeroand no damping is obtained beyond k c . In figure (3.b)the plot is for θ = 90 o , here the damping rates are largerthan the single atom one for small wave vectors, ka ∼
0, but now the damping rate increases with increasingthe wave vector. Namely, the large wave vector statesbecame more superradiant states. Exist an angle, around θ = 54 . o , where the damping rate changes its behavior. D a m p i ng R a t e [ H z ] θ FIG. 4: The damping rate vs. θ (full-line), for ka = 0 . D a m p i ng R a t e [ H z ] θ (a) 0 45 90 135 1800246810x 10 D a m p i ng R a t e [ H z ] θ (b) FIG. 5: The damping rate vs. θ , for (a) ka = 0 .
5, and (b) ka = 1. The dashed-line is for a single atom damping rate. Next for different values of ka we plot the dampingrate as a function of θ . In figure (4) we use ka = 0 . θ = 90 o the damping rate is lower than that of θ = 0 o . In figure (5.a) we use ka = 0 . θ ∼ o and increases to a maximum at θ = 90 o , that is muchlarger than the single atom case and which is stronglysuperradiant state. For larger ka , in figure (5.b) we use ka = 1, now for angles up to 45 o the damping rate iszero, and they are nonzero for θ ∼ o − o with a highmaximum at θ = 90 o . For ka = π at the boundary ofthe Brillouin zone, the non-zero region is for θ ∼ . o − . o , and with higher rate at θ = 90 o than the previouscase of figure (5.b).We showed that the damping rate and the radia-tion emitted of an exciton in one dimensional lattice isstrongly different from independent atoms. The excitonsformed by the energy transfer among the lattice atomsdue to dipole-dipole interactions can be characterized bytheir wave numbers along the lattice. This restricts thespontaneously emitted photons to the same wave vectorcomponent parallel to the lattice. The radiative damp-ing rate of such excitons into free space strongly dependson both, the wave number and the polarization direc-tion of the exciton. In close analogy to superfluorescentscattering, some wave numbers and polarizations exhibita much larger damping rate than an independent atom(superradiant excitons), while for other wave numbersand polarizations the damping rate is much smaller andeven can vanish completely, so that the excitons now aremetastable. The present results can be adopted to anyperiodic distribution of optically active materials, e.g. alattice of quantum dots, one dimensional molecular crys-tals, periodic semiconductor nanostructures, or a chain oftrapped ions, etc. Interestingly such excitons might stillbe able to couple to and decay into a nearby materialstructure if it supports the corresponding wave vectors.As a single atom can only bear one excitation, several ofthese excitons will interact and might open possibilitiesof nonlinear light interactions at very low intensities.The work was supported by the Austrian ScienceFunds (FWF), via the project (P21101 and S40130). Wethank A. Rauschenbeutel for communicating his recentexperimental results [9]. [1] I Bloch, et. al., Rev. Mod. Phys. , 885 (2008).[2] D Jaksch, et. al., Phys. Rev. Lett. , 3108 (1998).[3] M Greiner, et. al., Nature , 39 (2002).[4] M Lewenstein, et. al.,
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Phys. Rev. A , 13817 (2007).[7] M Antezza, et. al., Phys. Rev. Lett. , 123903 (2009).[8] H Zoubi, and H Ritsch,