SciPost Physics Submission Quantum Atomic Arrays: Fractional Filling and Trapping Pengfei Zhang Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA * [email protected] August 26, 2021 Abstract Quantum emitters, in particular, atomic arrays with subwavelength lattice constant, have been proposed to be an ideal platform for study the interplay between photons and electric dipoles. In this work, motivated by the recent experiment [1], we develop a microscopic quantum treatment using annihilation and creation operator of atoms in deep optical lat- tices. Using a diagrammatic approach on the Keldysh contour, we derive the cooperative scattering of the light and obtain the general formula for the S matrix. We apply our for- mulism to study two effects beyond previous treatment with spin operators, the effect of fractional filling and trapping mismatch. Both effects can lead to the imperfectness of atomic mirrors. For the fractional filling case, we find the cooperative linewidth is linear in filling fraction n. When there is a mismatch between the trapping potentials for atoms in the ground state and the excited state, multiple resonances can appear in the response function. Our results are consistent with existing experiments. Contents 1 Introduction1 2 Model 3 3 Diagrammatic Expansion4 4 Fractional Filling6 5 Trapping Mismatch8 6 Summary and Overlook9 References 10 A The Analytical Formula for π(!) 13 1 Introduction The ability to coherently storing photons and controlling their interaction with quantum matters is of vital importance for quantum science. Although single atoms and photons usually interact 1 SciPost Physics Submission p <latexit sha1_base64="quJIPSDL6wA4x5W37zoHg2ojV7A=">AAACz3icjVHLSsNAFD2Nr1pfVZdugkVwVdIq6EoKIrhswbZCW8pkOm1D8yKZKKVU3PoDbvWvxD/Qv/DOmIJaRCckOXPuPWfm3muHrhNLy3rNGAuLS8sr2dXc2vrG5lZ+e6cRB0nERZ0HbhBd2ywWruOLunSkK67DSDDPdkXTHp2rePNGRLET+FdyHIqOxwa+03c4k0S12x6TQ7s/ueha026+YBUtvcx5UEpBAemqBvkXtNFDAI4EHgR8SMIuGGJ6WijBQkhcBxPiIkKOjgtMkSNtQlmCMhixI/oOaNdKWZ/2yjPWak6nuPRGpDRxQJqA8iLC6jRTxxPtrNjfvCfaU91tTH879fKIlRgS+5dulvlfnapFoo9TXYNDNYWaUdXx1CXRXVE3N79UJckhJE7hHsUjwlwrZ302tSbWtaveMh1/05mKVXue5iZ4V7ekAZd+jnMeNMrF0lGxXDsuVM7SUWexh30c0jxPUMElqqiTd4hHPOHZqBm3xp1x/5lqZFLNLr4t4+ED9lOUDQ==</latexit> E0 <latexit sha1_base64="N9K+YtdcRzpl8XPEyGWUUULpTEk=">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</latexit> RE0 p <latexit sha1_base64="MHBpV/FXyuXZTcxFEyC+WyxC6cg=">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</latexit> T E0 Figure 1: Schematics of the model considered in this work: the atomic array in the optical lattices at fractional filling. less efficiently, ensembles of atoms can show a cooperative response of photons. As an example, superradiance can be realized when the radiations between atoms interfere constructively [2–8]. Recently, atomic arrays with subwavelength lattice structures are found to be an ideal platform where electric dipole-dipole interactions between atoms are mediated by photons [9–18]. The analysis shows the atomic arrays exhibit subradiance and are nearly perfect mirrors for a wide range of incident angles [16], as observed in recent experiments [1]. Later, there are many the- oretical studies on the fruitful physics in atomic arrays [19–26, 26–37]. For example, there are proposals for realizing non-trivial topology in atomic arrays [19–21], controlling atom-photon in- teraction using atomic arrays [22–26], and efforts in understanding their subradiant behaviors and ability of photon storage [26–32]. In most of these works, atoms are treated as point-like with no motional degree of freedom. The evolution of the system is described by using non-Hermitian Hamiltonian or Lindblad master − equation [16, 17], with spin degree of freedom σim = jrm; gihrm; eij. Here jrm; gi is the s-wave ground state for the atom at position rm. jrm; eii is the p-wave excited labeled by the dipole moment d = d ei of the corresponding transition g ! ei. However, in real experiments, the system consists of atoms moving in optical lattices. For deep optical lattices, although atoms are trapped near the potential minimum, the wave function for the motional degree of freedom may still play a role. Moreover, the consequence of fractional filling has been studied in the experiment. It is difficult to analyze the absence of an atom in the spin-operator language, and consequently, theoretical predictions for the fractional filling case are still absent. In this work, we overcome this difficulty by using a microscopic model for the coupled system consisting of atoms in deep optical lattices and photons. After making plausible assumptions, we derive the cooperative response of the system using a diagrammatic approach on the Keldysh contour. By summing up bubble diagrams with dressed Green’s function, we obtain neat results for the transmission coefficient and the reflection coefficient, with the contribution from motional wave function. Our result matches the previous analysis for unit filling when the potential of the excited state atoms are the same as that of the ground state. For fractional fillings, we find the cooperative linewidth is linear in n, consistent with the experimental observation. We then study the effect of the discrepancy of optical lattices for the ground state and excited state atoms, where the transition of the internal state can be accompanied by transitions in the motional degree of freedom. In particular, we find that multiple resonances can exist in the response function. 2 SciPost Physics Submission 2 Model We consider coupled systems with atoms and photons. Th Hamiltonian reads H = HEM + HA + Hint: (1) Here the first term is the Hamiltonian of the electromagnetic field Z µ H = dr 0 E(r)2 − 0 H(r)2 ; (2) EM 2 2 The second term describes the motion of atoms in optical lattices Z 2 ! Z 2 ! X y r y r HA = dr (r) !0 − + Ve(r) e (r) + dr (r) − + Vg(r) g(r); (3) ei 2 i g 2 i Vg=e(r) describes the optical lattice potential for ground/excited-state atoms. We have set ~ = 1 and m = 1. We assume each site is occupied by at most one atom, which corresponds to y 0 0 choosing fermionic commutation relation f a(r); a(r )g = δabδ(r − r ). The last term describes the interaction between atoms Z X + − Hint = − dr Pi (r) + Pi (r) ei · E(r); (4) i with + y − y Pi (r) = d ei (r) g(r) = Pi (r) : (5) The full Hamiltonian (1) describes general model with interaction between atoms and light to the order of electric dipole transition. For atomic arrays, the ground state particle is always tightly trapped near the local minimum of optical lattices, with a spread of wave function σ a, where a is the lattice constant [1]. Assuming the excited state is also trapped, we expand X X (r) ≈ ' (r − r ) a(r ); (r) ≈ '0 (r − r ) a (r ): g a n g n ei a n ei n (6) n n 0 Here 'a(r)='a(r) is the motional wave function for ground/excited-state atoms near the local min- 0 imum rn = 0 with the energy ea=ea, and we have rn = a(n1e1 + n2e2). The commutation relation a a;y b for η(rn) now becomes f η (rm); ξ(rn)g = δηξδabδmn. Using (6), the Hamiltonian HA and Hint can be simplified. We have 2 3 X 6X 0 a;y a a;y a 7 H = 6 e (r ) + e (r )7 ; (7) A 46 a ei ei n a g g n 57 na i and (4) becomes X + Hint = − pi (rn) ei · E(rn) + H.C. ; (8) ni with Z + X ∗ a;y b pi (rn) = d dr 'a(r)'b(r) ei (rn) g(rn): (9) ab Equation (2), (7) and (8) describe the dynamics of the atomic array. Initially, we prepare all atoms in the s-wave internal ground state jgi with motional ground state '0(r). The number of atoms in the excited states are suppressed due to the violation of energy conservation. We further add an external probe light, at fixed frequency !, which is near-resonant with δ ≡ !−!0 !; !0. ik·r The electric field reads E0(r) = E0e with cjkj = !. We take c = 1 from now on for conciseness. 3 SciPost Physics Submission This probe corresponds to the incident light in the scattering experiment. Its coupling to atoms reads X + −i!t δH = − pi (rn)e ei · E0(rn) + H.C. : (10) ni We assume the field strength E0 is weak and the response can be analyzed using the linear response theory. The total electric field including the incident light and the scattered light then reads Etot(!; r) = E0(r) + hE(!; r)i : (11) Far from the atomic array, when only a single diffraction order exists, we expect ikzz ikzjzj ikk·rk Etot(!; r) = 1e + S(!; kk)e · E0e ; (12) and S(!; kk) is the corresponding S matrix. 3 Diagrammatic Expansion The contribution to the scattered light hE(!; r)i can be efficiently organized