Formation of supermodes in atom-microcavity chains
aa r X i v : . [ qu a n t - ph ] A ug Formation of supermodes in atom-microcavity chains
Sandra Isabelle Schmid and J¨org Evers
Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany (Dated: November 14, 2018)A chain of atom-microcavity systems coupled by a common fiber is considered. We analyze theformation of supermodes and focus on the dependence of this effect on the chain geometry and thenumber of atom-cavity subsystems. We show that the significance of supermodes to the transmissionincreases with the number of atom-cavity subsystems. We identify spectral ranges in which the chaingeometry decides whether supermodes are formed, and ranges which are insensitive to the geometry.Furthermore, we show that the reflection signal allows to identify which cavities couple to atoms,which is a crucial information in experimental realizations of longer atom-cavity chain systems.
PACS numbers: 42.50.Pq, 42.60.Da,42.50.Ar
Whispering gallery microresonators have gainedtremendous interest over the past few years due to theiroutstanding properties [1]. Particularly promising pho-tonic systems arise if quantum systems such as atomsare coupled to the evanescent field of microresonators.Already the simplest case of a single particle coupled toa single microcavity has led to a number of fascinatingproposals and experiments. For example, strong cou-pling of the microcavity was demonstrated to atoms [2]and quantum dots [3], a single photon turnstile was re-alized in [4], and the complex interplay of counterpropa-gating resonator modes already due to a classical particlewas observed in [5]. Small chains of atom-cavity systemshave been analyzed as well. For example, in [6], twocoupled atom-cavity systems are considered for applica-tions in quantum networks [7]. But in this work, lightcan propagate only in one direction through the system,and backward couplings and thus supermodes betweenthe subsystems are neglected. This restriction was liftedin [8], in which two atom-cavity systems coupled to afiber are considered including the scattering between thecavities. However, only a single cavity mode instead of apair of modes was considered in each resonator. The non-coupling eigenmode of the resonator found, e.g., in [2],which can have strong influence on the system’s opti-cal properties, is not considered. Furthermore, only thetransmission was analyzed, and cases with more than twoatom-cavity systems, the reflection, and the formation ofsupermodes were not studied. Meanwhile, recent exper-iments indicate that a realization of chains of coupledatom-cavity systems is within reach. For example, nextto the single atom-cavity systems [2, 4], it was also shownthat atoms can be trapped by the evanescent field of atapered fiber [9, 10]. Furthermore, real-time detectionand feedback to monitor single atoms near a microres-onator was recently achieved, which is an important steptowards larger networks of atom-cavity systems [11].Motivated by this, here, we investigate a chain of N coupled cavity-atom subsystems connected via a waveg-uide and probed by a weak input field, see Fig. 1. Wemainly study the formation of supermodes, which can arise due to the scattering of light between the differentsubsystems. To identify these modes, we define a “super-ness” measure, which is given by the difference in trans-mission for the complete system relative to the transmis-sion to a corresponding system of independent cavitieswithout backward coupling. We find that the forma-tion of supermodes crucially depends on the length of thechain, and the relative distances between the subsystems.While the overall transmission decreases with increasingnumber of atom-cavity subsystems in the chain, the rela-tive contribution of the supermodes increases. As in ex-periments it can be difficult to achieve simultaneous cou-pling of all resonators to individual atoms, we also showthat the atom-cavity coupling configuration of the chaincan be determined from the reflected light, whereas thetransmitted light does not contain this information. Inthis sense, the reflection and transmission contain com-plementary information.Our system consists of a chain of N microresonator-atom systems coupled via a fiber, see Fig. 1. Due to theintercavity distance direct couplings between cavities areneglected. Each resonator n ∈ { , . . . , N } is modeled bya pair of counterpropagating modes described by the an-nihilation operators a n , b n , with scattering between themodes of rate h n . The coupling strength to the fiber is κ ex,n , and each cavity has an internal loss rate κ i,n suchthat the total loss rate is κ i = κ i,n + κ ex,n . The de-tuning between the resonator frequency ω cav,n and theinput field frequency ω L is δ n = ω cav,n − ω L . The dis-tance between the coupling points of two resonators n and n + 1 to the fiber is L n . The atom at resonator n ismodeled as two-level systems with resonance frequency ω at,n , decay rate γ n , atomic raising (lowering) operators S ± n and detuning ∆ n = ω at,n − ω L to the incident light.The position-dependent coupling constants of the atomto modes a n and b n are g n,a and g n,b , respectively. Dueto the scattering h n it is convenient to introduce normalmodes A n = ( a n + b n ) / √ B n = ( a n − b n ) / √ PSfrag replacements g g g N − g N h h hN − hNκi, κi, κi,N − κi,Nκex, κex, κex,N − κex,N L N − L N -1 N aout, bout, a in, b out, aout, aout, bout, aout,N − bout,Nain,N − a out,N b in,N FIG. 1. Chain of N cavity-atom subsystems connected bi-directionally by a waveguide. the Hamiltonian reads H N = N X n =1 − ~ ∆ n S − n S + n + ( δ n + | h n | ) A + n A n + ~ ( δ n − | h n | ) B + n B n + i ~ p κ ex,n (cid:0) A in,n A † n + B in,n B † n + H. c. (cid:1) + ~ (cid:0) g A n A † n S − n − ig B n B † n S − n + H. c. (cid:1) . (1)where we define S zn = [ S + n , S − n ], and input fields to res-onator n are denoted {A in,n , B in,n } . For our calcula-tions we use a semiclassical treatment and replace oper-ators with their respective expectation values [3]. Thecoupling of the different cavities arises since the inputflux of cavity n depends on the outputs of the neigh-boring cavities via a in,n = a out,n − exp( ikL n − ) and b in,n = b out,n +1 exp( ikL n +1 ). Then together with theinput-output relations [12] a out,n = − a in,n + p κ ex,n a n ,all input and output fluxes can be determined.In general, solving the coupled system for N cavi-ties is a demanding task due to the large dimension ofthe Hilbert space. One approach to solve a system oftwo coupled atom-cavity systems was presented in [8],based on a real space wavefunction approach for a sin-gle photon wave packet [13]. However, it is well knownthat resonators without atoms can be described by trans-fer matrices relating inputs and outputs of a single res-onator [14]. We found that a related approach usingtransfer matrices M n is also possible for resonators cou-pled to an atom in certain parameter regimes. Thus,each atom-cavity system can be solved separately, avoid-ing the large Hilbert space of the combined N -cavitysystem. We verified using exact calculations of smallermultiple cavity systems that this approach is possibleas long as the atoms are far from saturation. Then,the n th cavity-atom system can be described via thefrequency-dependent transmission t n and reflection r n determined by a out,sc = t n (∆ n ) · a in,sc + r n (∆ n ) · b in,sc and b out,sc = r n (∆ n ) · a in,sc + t n (∆ n ) · b in,sc , out ofwhich the transfer matrix M n can be formed. In thisformalism, the optical path between the cavities is char-acterized by a diagonal matrix M φ n with eigenvaluesexp[ iφ n ] and exp[ − iφ n ], where the phase angles φ n are defined by the intercavity distances L n as φ n = 2 πL n /λ where λ is the wavelength of the incident light. Thetotal system is then governed by the matrix M total = M N · M φ N − · M N − · · · M · M φ · M , which connectsthe input fluxes a in, and b in,N with the output fluxes a out,N and b out, .We now turn to a discussion of our observables. As-suming driving of the system from the left side only( b N,in = 0), the transmission T and reflection R are givenby T = |h a out,N i| / | a in, | and R = |h b out, i| / | a in, | .Next, we are interested in studying the formation of su-permodes. These are modes which receive nontrivial con-tribution from multiple cavities, thus having propertieswhich go beyond the combination of the properties of theindividual subsystems. Hence, we define the “superness” of a mode, ∆ T = T − T ind,N , which is the differencebetween the transmission obtained for the full N -cavitysystem with backward coupling to the transmission for achain of N independent cavities T ind,N = T · T · · · T N without backward coupling.For our numerical results we assume that the atomsare located such that they couple to the modes B n andset g A n = 0. Note that the normal mode A still con-tributes to the optical properties as it couples to the in-cident probe beam. Furthermore, identical spontaneousemission rates for the atoms γ n = γ are assumed.We start with N = 2. Figure 2 depicts our results for∆ T for two subsystems, h n = 50 γ and different distances L n . We observe that for specific detunings the transmis-sion behavior crucially depends on the intercavity dis-tance whereas for other detunings the systems behavesalmost as a chain of independent cavity-atom systems.In particular, at ∆ n ≈ γ and ∆ n ≈ − γ , the trans-mission of the full system is equal to the combination ofthe individual systems for all distances. This can be ex-plained by noting that for these detunings, either | t n | (for∆ n ≈ γ ) or | r n | (for ∆ n ≈ − γ ) is small. This leadsto a vanishing backcoupling between the cavities suchthat the system is similar to the uncoupled case. For∆ n ≈ − γ the population of the upper level of boththe atoms is comparably high and the presence of theatom supresses the reflection which results in | r n | ≈ n ≈ γ almost all incoming light is reflected at the -0.1 0 0.1 0.2 0.3 0.4-200 -150 -100 -50 0 50 100 150 200 PSfrag replacements(a)(b)(a)(b) a out,N ∆ T ∆ T ∆ n /γTT sc L N . λ . λ . λ . λ . λ PW1PW2PW3 t N − , r N − t N , r N | t || r | φ t /πφ r /π ∆ /γt and r FIG. 2. (Color Online) Formation of supermodes indicatedby large modification ∆ T of transmission compared to a chainof independent atom-cavity systems. Parameters are N = 2, h n = 50 γ , g B n = 70 γ , L tot = 200 . λ and intercavity distances L = 100 . λ , L = 100 . λ , L = 100 . λ , and L = 100 . λ . first subsystem and thus the cavity modes and the excitedstate of the atom next to cavity two are hardly populated.The low transmittivity and reflectivity are achieved sincethe respective detunings correspond to the eigenvalues ofthe Hamiltonian for a single subsystem. In contrast, forother detunings, pronounced resonances in the “super-ness” ∆ T can be observed at certain distances. Theseresonances exceed ∆ T ≈ . | t n | ≈ | r n | and thus the energy exchange in both directions is en-hanced. Interestingly, for the detunings with high ∆ T ,the modes B n coupled to the atoms are hardly populatedin both the cavities, whereas the non-coupling modes A n are highly populated, demonstrating the significance ofthe non-coupling normal mode.We now turn to larger arrays of atom-cavity systems.The dependence of the supermode identified in Fig. 2on the chain length from N = 2 up to N = 20 sub-systems is shown in Fig. 3. We found that in absoluteterms, the superness ∆ T reduces with increasing N un-til it almost vanishes for N = 20. This reduction canbe traced back to the overall reduction in transmission T for increasing N , since each resonator leads to a cer-tain amount of loss. In contrast, the ”relative superness“∆ T /T increases with N , as shown in the right panel ofFig. 3. The reason is that the reduction in transmissionbecomes stronger with increasing N for independent res-onators, such that eventually all relevant residual trans-mission must origin from an enhancement via construc-tive interference through the formation of a supermode.Next, we discuss the reflection properties. Figure 4shows results for R for N = 2 and L = 100 . λ . Butunlike in Figs. 2 and 3, we now additionally consider thecases in which only the first cavity couples to an atom( g B = 0), only the second cavity couples to an atom -0.2-0.1 0 0.1 0.2 0.3 0.4-150 -100 -50 0 50 100 150 -20 0 20 40 60 PSfrag replacements(a)(b)(a)(b) a out,N ∆ T ∆ T ∆ n /γ ∆ n /γTT sc L N t N − , r N − t N , r N ∆ TT φ t /πφ r /π ∆ /γt and r FIG. 3. (Color online) Dependence of the supermode contri-bution ∆ T on the chain length N . The left panel shows theabsolute “superness“ ∆ T , the right panel the relative value∆ T /T . The parameters are as in Fig. 2, with L n = 100 . λ . ( g B = 0), or both cavities without atom. We observethat the reflection crucially depends on the presence andposition of the coupling atoms. For example, if ∆ n = 0and only one atom is located close to cavity 2, no lightis transmitted at the first cavity, since | t (∆ n = 0) | = 0.Thus no light enters cavity 2 and therefore the dynamicsof the coupled chain is totally governed by subsystem 1.However, if the atom is located close to cavity 2, lightcan be transmitted for ∆ n = 0 at cavity 1, but not atcavity 2. In this case again all light is reflected, but bothsubsystems influence the reflection. Thus the reflectiondepends on the coupling position of a single atom. Fromthe example in Fig. 4 we find that if the atom couplesto cavity 1, zero reflection can be observed at ∆ n ≈ γ .However, if the atom couples to cavity 2, there is onlya slight local minimum at this detuning. Furthermore,the cases with no nearby atom or an atom at both thecavities can be well distinguished by measuring the re-flection intensity, or by comparing the reflection at twofrequencies. In an experiment, this behavior could be ex-ploited to determine how many atoms couple to whichresonators, which is particularly useful for experimentsin which atom-cavity coupling is based on a falling cloudof cold atoms [2, 4, 11], or if the atoms cannot be trappedreliably over a long time compared to a measurement.The results for ∆ T and R can be explained in termsof interference between different pathways the lightcan take through the system. The contribution to a out,N for the simplest possible pathway P W N cavities reads a N,P W = a in, Q Nn =1 t n exp[ iφ n ]. However, the fully coupled sys-tem also allows for extended pathways, e.g., throughcavities 1 . . . N , back to N −
1, and then to the out-put port via N . This pathway contributes a N,P W = a N,P W r N r N − exp(2 iφ N − ). In case of high ∆ T as forthe supermodes presented in Fig. 2, contributions arising PSfrag replacements(a)(b)(c) (d)(c) (b)(a) a out,N R ∆ TR ∆ n /γt , r L N L n L L N − PW1PW2PW3 t N − , r N − t N , r N | t || r | φ t /πφ r /π ∆ /γt and r FIG. 4. (Color online) Reflection for asymmetric atom-cavitycoupling constellations with N = 2. In (a) no atom couplesto the cavities, in (b) one atom couples to the first cavity, in(c) to the second, and in (d) both cavities have nearby atoms.Coupling atoms have g B i = 70 γ , and L = 100 . λ . from different pathways interfere constructively. For thepathways PW1 and PW2 the condition for constructiveinterference is φ r N + φ r N − + 2 φ N − = m · π with m ∈ Z .We emphasize that this analysis implies that the orderof the subsystems in the chain has no influence on T andthus also ∆ T as long as the input flux b in,N = 0. Incontrast, for the reflection R , changing the order of sub-systems within the chain in general leads to completelydifferent results, even though the sensitivity depends onthe intercavity lengths L n . The reason is that trans-mitted light necessarily must pass all cavities, whereasreflection can already occur at the first cavity such thatthe light does not reach the other cavities. Therefore, thedetermination of the positions at which atoms couple tothe chain as in Fig. 4 is only possible via the reflection.In this sense, the reflection provides information aboutthe system properties complementary to those obtainedfrom the transmission.We conclude with an analysis of the N = 3 case, whichis the simplest realization with two potentially differ-ent inter-cavity distances. If at least one cavity in thechain has no nearby atom, the transmission of the cou-pled system is suppressed in a large range of probe fielddetunings around ∆ n = 0 by the cavity without atom.Then also ∆ T has low values. Since we are interestedin supermodes, we focus on the case with atoms. Wekeep the distance between subsystem 1 and 3 fixed as L tot = L + L = 200 . λ and move the central cavity2, i.e., the distances L and L = L tot − L are varied.In Fig. 5 we show results for the “superness” ∆ T forthe parameters h n = 50 γ and g B n = 70 γ for differentpositions of cavity 2. For some detunings only by mov-ing the central cavity-atom subsystem, the transmissionchanges qualitatively, from a strong supermode charac-ter to properties governed by the individual cavities only. -0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-300 -200 -100 0 100 200 300 PSfrag replacements(a)(b)(a)(b) a out,N ∆ T ∆ T ∆ n /γTT sc L N . λ . λ . λ . λ PW1PW2PW3 t N − , r N − t N , r N | t || r | φ t /πφ r /π ∆ /γt and r FIG. 5. (Color online) Formation of supermodes as in Fig. 2,but for N = 3 subsystems. Parameters are h n = 50 γ , L tot =200 . λ and L = 100 λ , 100 . λ , 100 . λ , and 100 . λ . For example, the peak around ∆ = 25 γ has ∆ T ≈ . L = 100 . λ , but values up to 0.4 for L = 100 . λ .Interestingly, there are also structures which have smallsupermode character only slightly influenced by L , e.g.,around ∆ = − γ . These results again can be under-stood from a pathway analysis as before.In summary, we have analyzed the formation of super-modes in chains of atom-microcavity systems, focusingon the effect of the relative positioning and the chainlength on the “superness” ∆ T . Furthermore, we haveshown that in contrast to the transmission, the reflectionenables the identification of cavities coupling to an atom,which is crucial for an experimental realization of longeratom-cavity chains. [1] K. Vahala, Nature , 839 (2003).[2] T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S.Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kim-ble, Nature , 671 (2006).[3] K. Srinivasan and O. Painter, Nature , 862 (2007).[4] B. Dayan, A. S. Parkins, T. Aoki, H. J. Kimble, E. P.Ostby, and K. J. Vahala, Science , 1062 (2008).[5] A. Mazzei, S. G¨otzinger, L. de S. Menezes, G. Zumofen,O. Benson, and V. Sandoghdar, Phys. Rev. Lett. ,173603 (2007).[6] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys.Rev. Lett. , 3221 (1997).[7] H. J. Kimble, Nature , 1023 (2008).[8] X. Zang and C. Jiang, Journal of Physics B: Atomic,Molecular and Optical Physics , 065505 (2010).[9] E. Vetsch, D. Reitz, G. Sagu´e, R. Schmidt, S. T.Dawkins, and A. Rauschenbeutel, Phys. Rev. Lett. ,203603 (2010).[10] F. Le Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A , 063403 (2004).[11] D. J. Alton, N. P. Stern, T. Aoki, H. Lee, E. Ostby, K. J.Vahala, and H. J. Kimble, Nature Phys. , 159 (2011).[12] C. W. Gardiner and M. J. Collett, Phys. Rev. A , 3761 (1985).[13] J.-T. Shen and S. Fan, Phys. Rev. A , 023837 (2009). [14] Y.-F. Xiao, B. Min, X. Jiang, C.-H. Dong, and L. Yang,IEEE J. of Quantum Electronics44