Subradiant Bell states in distant atomic arrays
P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch, and P. Zoller Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
We study collective ‘free-space’ radiation properties of two distant single-layer arrays of quantum emitters as two-level atoms. We show that this system can support a long-lived Bell superposition state of atomic excitations exhibiting strong subradiance, which corresponds to a non-local excitation of the two arrays. We describe the preparation of these states and their application in quantum information as resource of non-local entanglement, including deterministic quantum state transfer with high fidelity between the arrays representing quantum memories. We discuss experimental realizations using cold atoms in optical trap arrays with subwavelength spacing, and analyze the role of imperfections.
Introduction. — Recent advances in preparing regular arrays of atoms with optical traps [1–4] offer new oppor- tunities to engineer strong collective coupling between atoms and light, with applications in quantum informa- tion science. In particular, a single layer of atoms loaded into a regular 2D array with sub-wavelength spacing has been proposed as an atomic mirror with high reflectivity [5–9], as quantum memory with efficient storage and re- trieval [10], and to implement topological quantum optics [11, 12]; in addition, emission of single photons from bi- layer atomic arrays can be engineered to be highly direc- tional in free-space [13]. Moreover, single-layered atomic arrays have been shown to support subradiant collective excitations [14–16], which consist of excited superposi- tion states of atoms decaying much slower than a single isolated excited atom, due to interference in spontaneous emission [17–23]. FIG. 1. ‘Dark’ and ‘bright’ states in two distant atomic ar- Here we show that the composite quantum system rays. (a) Sketch of a single 2D atomic array, with light consisting of two distant single-layered arrays of atoms emitted perpendicular to atomic plane (corresponding to a [cf. Figs.1(a-c)] can support an atomic Bell superposition ‘bright’, i.e. radiating state). (b) Two-level scheme. (c) Setup state exhibiting strong subradiance. Remarkably, this with two distant atomic arrays: we plot the electric field non-radiating ‘dark’ state is a non-local entangled state, profile |ψ(r)| of photonic modes (blue) associated with the i.e. a superposition state of a collective excitation living ‘dark’ and ‘bright’ states as excitations in the two arrays (red). (d) Decay rates γn as imaginary part of eigenener- in the first or second array, where the two arrays can be gies of the non-hermitian effective Hamiltonian H [Eq. (1)], separated by a distance L much larger than the trans- in units of the single atom decay rate γe, ordered according to verse size L of each individual array. This phenomenon their quasi-momentum q (see text). The white (black) color relies on two⊥ ingredients. First, spontaneous emission denotes even (odd) parity. A pair of ‘dark’ and ‘bright’ states from a collective atomic excitation in a single layer can be are identified as the left-most dots. (e) Atomic wavefunction arXiv:1901.02665v2 [quant-ph] 7 Mar 2019 amplitudes in each array |(v ) | associated with the dark directional, with a proper phasing of the atomic dipoles, n j⊥ corresponding to light emission in both directions per- and bright state. In (c-e) δ⊥ = 0.75λ0, N⊥ = 10, L = 20λ0 (see text). pendicular to the atomic array, as in Fig.1(a) [5]. Sec- ond, radiation from two distant atomic arrays can – pro- vided the separation length L is commensurate with half the optical wavelength [upper panel in Fig.1(c)] – lead to atomic superposition states can be prepared naturally in destructive interference of light emitted to the left and to setups involving two – or more – atomic arrays, and pro- the right of the two arrays, corresponding to a subradiant vide a source of entanglement shared between the two state, i.e. this ‘dark’ state will show strongly suppressed atomic arrays, with applications for quantum network- radiative loss to the outside world. In contrast, the lower ing [24]. In particular, quantum information can be ex- panel in Fig.1(c) displays a ‘bright’ (i.e., radiating) state changed between the arrays representing ‘local’ quantum due to constructive interference. memories, in a coherent and deterministic process, with Below we will show that these non-local subradiant dark states acting as mediators. 2
Quantum optical model. — Our setup consists of two 2D arrays of N = N N atomic emitters with lat- tice spacing δ and⊥ size× L⊥ N δ , separated by a distance L along⊥ z. Each⊥ atom≡ ⊥ has⊥ a ground and an excited state, g j and e j , and is coupled to free- space modes of the| i radiation| i field via a dipole tran- sition with frequency ω0 = ck0 = 2πc/λ0. Here the multi-index j = (j , jz), where jz = 1, 2 labels the ar- ⊥ rays, while j = (jx, jy) label the atoms within each ⊥ array, with 1 jx, jy N . Atomic positions are ≤ ≤ ⊥ denoted by rj = (xj , yj , zj ). We start by studying the dynamics of a single excitation with wave function P + R P ψ(t) = j cj (t)σj 0 + dk λ ψλ(k, t) k, λ . | i + |Gi | i |Gi | i Here σj = e j g , = j g j , 0 is the photonic vac- uum state and| i hk|, λ|Githe⊗ state| i with| i a single photon with FIG. 2. Dark and bright state properties. (a) Ratio of dark and bright states decay rates for 1 (blue) and 2 (red) exci- wave vector k and| polarizationi λ. We extend our results tations, with δ⊥ = λ0/2 and L = mλ0/2 with integer m. below to states with multiple excitations. (b) Collective frequency shifts (dashed blue) and decay rates The atomic dynamics, due to successive photon emis- (red) of parity-symmetric (s) and anti-symmetric (a) single- sions and reabsorptions, is obtained by integrating out excitation states, with δ⊥ = 0.8λ0 and N⊥ = 12. (c) Dark the dynamics of the radiation modes ψλ(k, t) in a Born- and bright state decay rates, and (d) collective frequency shift Markov approximation. Assuming the field initially of the dark state, for L = 2λ0, δ⊥ = λ0/2, and N⊥ as in (a) for 1 excitation. in the vacuum state ψλ(k, 0) = 0, this yieldsc ˙j = P i 0 j j0 cj0 , where in a frame rotating with ω [25– − j H , 0 27], momentum q, which is obtained from the discrete Fourier transform of the corresponding eigenvectors (˜vn)q = 0 p Gˆ r r 0 p j,j i(γe/2) ∗ ( j j ) (1) P iδ⊥j q P 2 H ≡ − · − · (vn)j e ⊥· /√N as q = (˜vn)q q , with j⊥ ⊥ q | | | | is a non-hermitian effective Hamiltonian, whose hermi- discrete quasi-momentum q = (qx, qy) where qx,y = tian part describes coherent exchanges of atomic excita- π/δ + 2πnx,y/L (nx,y = 0, 1, ..., N 1). Two states −have a⊥ distinctly low⊥ quasi-momentum⊥q− k : the dark tions, while the non-hermitian part corresponds to dis- 0 sipation accounting for radiation of photons. Here γe is state, as well as a ‘bright’ state, which radiates photons the spontaneous decay rate of each atom, and the dyadic with a rate γb comparable to γe. We contrast our dark Green’s tensor Gˆ (r), representing the electric field at po- states with the q > k0 subradiant states in single layer sition r generated by a dipole located at the origin, is the setups, studied e.g. in Refs. [16, 20]. In Fig.1(e) we show 2 solution of ∇ ∇ Gˆ (r) k Gˆ (r) + (6πi/k )δ(r) = 0 the probability amplitude of the eigenvectors (vn)j⊥ for × × − 0 0 | | with Gˆ 0 1 accounting for independent single-atom the two states with lowest decay rates [28]. decay (see details≡ in [28]). The atomic transition polar- This pair of dark and bright states can be under- ization p is taken circular, with z as quantization axis. stood by considering first the situation where the arrays Dark and bright eigenstates. — The dynamics of are infinite (N ), and the eigenstates are plane ⊥ → ∞ iδ⊥j⊥ qn √ atomic excitations, including their radiative properties, waves (vn)j⊥ = e · / N with continuous quasi- can be understood by studying the spectrum of . De- momentum qn. We now make two assumptions: First, H noting its eigenvalues as = ∆ iγ /2 (with n = the lattice spacing satisfies δ < λ0. Under this condi- n n − n ⊥ 1,..., 2N), ∆ is interpreted as the self-energy of the tion, we obtain, provided qn 2π/δ k0 [28], n | | ≤ ⊥ − collective atomic excitation given by the corresponding k2+ q 2/2 γ = Γ[1 + p cos(k L)] z | n| , (2) eigenstate cn, while γn is its spontaneous emission rate. n n z k0qz In particular, an eigenstate is subradiant (or ‘dark’) if spontaneous emission occurs with a rate suppressed be- 2 p 2 2 with Γ = 3πγe/(k0δ ) , kz = k0 qn . Considering ⊥ − | | low the single-atom decay rate γe. In view of the mir- in particular the symmetric (pn = 1) and antisymmetric ror symmetry of the system, all eigenstates have a def- (p = 1) eigenstates with q = 0, we obtain a pair of n − n inite parity, i.e., they can be written as (cn)(j ,1) = states with decay rates γ = Γ[1 cos(k L)]. Similarly, ⊥ s/a ± 0 p (c ) (v )j /√2, with parity p = 1. their self-energies are ∆ = (Γ/2) sin(k L) + ∆ , as n n (j⊥,2) ≡ n ⊥ n ± s/a ± 0 d In Fig.1(d) we plot the decay rates [for the setup depicted in Fig.2(b), where ∆ d is a collective Lamb shift of Fig.1(c)], with parameters chosen as explained be- evaluated numerically. Our second assumption is that low. One of the eigenstates is remarkably subradiant, k0L = mπ with integer m, so that either γs or γa vanishes 3 with a decay rate of γ 10− γ . We also repre- due to interference in the emission of the two arrays, d ∼ e sent the mean absolute value of the transverse quasi- while the other reduces to γb = 2Γ. The corresponding 3
Bell states collective shift ∆d on the other hand is typically of the order of γe [c.f. Fig.2(d)]. It can be positive or negative 1 X h + m + i ψd/b = σ ( 1) σ (3) depending on δ , and vanishes around δ = 0.2λ0 and | i √2N (j⊥,1) ∓ − (j⊥,2) |Gi ⊥ ⊥ j⊥ δ = 0.8λ0 (see also Ref. [9]). ⊥Dark state preparation and quantum state transfer. — are thus respectively ‘dark’ and ‘bright’. In order to prepare the atoms in the dark state, we
For finite-sized arrays, the eigenstates (vn)j⊥ are con- consider the setup represented in Fig.3(a), where the fined, which has two consequences yielding a finite decay atomic level structure now includes a third state s . | i rate γd for the dark state. First, photon emission in We assume that the system is initially in a super- transverse directions is not perfectly cancelled. Second, position state of the first array S+ , with S+ = P 1 |Gi 1 photons emitted along z have a finite spread of trans- (vd)j s g . This could be realized for in- j⊥ ⊥ (j⊥,1) verse momentum, and thus diffract when propagating be- stance using| laser-dressedi h | Rydberg-Rydberg interactions tween the two arrays, thereby hindering the interference [13, 32], or single photon pulses [28]. Moreover, we as- of emission. This can be mitigated by curving the arrays sume a coherent field drives the s e transition in according to the phase profile of a Gaussian mode (r) the first array with Rabi frequency| i →Ω, | resonantlyi with E propagating along z [as shown in Fig.1(c)], in analogy the collective shift ∆d. The atoms are thus driven from + to the mirrors of an optical cavity. As represented in state S1 to a superposition of dark and bright states P |Gi + √ + + Fig.1(e), the spatial distribution of the dark state (as j (vd)j⊥ σ = (1/ 2)(σ + σ ) , where the ⊥ (j⊥,1) |Gi b d |Gi well as the bright state) is then (v )j (r )[29]. + P + m + d ⊥ (j⊥,1) operators σ (v ) (σ [ 1] σ )/√2 ∝ E d/b j d j⊥ (j ,1) (j ,2) Alternatively, one can add optical elements between the ≡ ⊥ ⊥ ∓ − ⊥ create a dark/bright atomic excitation. The decay rate arrays, such as lenses or fibers. of bright excitations can be orders of magnitude larger The spatial profile of the electric field, generated by than for dark excitations, such that their contribution to (virtual) photon exchanges between the atomic dipoles P the dynamics is vastly different. If γb Ω, the bright in the dark state, reads ψ(r) cj Gˆ (r rj ) p, and ∼ j − · mode can be adiabatically eliminated, and contributes forms a standing wave [see Fig.1(c)]. We emphasize that 2 an effective loss with rate Ω /γb, which can vanish in the – although the system resembles a cavity with each array spirit of a quantum Zeno effect. On the other hand, if acting as a mirror – we are interested here in the quantum Ω γd, the dynamics will yield oscillations between the state of the atoms. More precisely, the ratio of atomic to initial state and the non-local dark state. photonic excitations in the dark state is given by ΓL/(2c) This mechanism can be exploited for quantum state with speed of light c [28], which is assumed negligible transfer between the two arrays. Here, an initial qubit when integrating the field dynamics above, amounting superposition state in the first array ψi = cg + to neglecting retardation effects in the atomic dynamics. + 2 2 | i |Gi csS1 (with cg + cs = 1) is transferred deter- This is in analogy to atomic cavities built from strings of ministically|Gi to the| | second| | array. That is, we realize the atoms coupled to a 1D waveguide [30, 31]. + + process ψi ψf = cg + csS , where S = We now discuss how the geometric parameters P | i → | i |Gi 2 |Gi 2 j (vd)j⊥ s j g , with high fidelity 1 [33]. (N , L, δ ) affect the spectral properties of the system. ⊥ | i( ⊥,2)h | F ≈ ⊥ ⊥ By driving atoms in both arrays with Rabi frequency Ω, In Fig.2(a) we show the scaling of γd/γb as the relevant + the state S2 is coupled to the opposite superposition figure of merit, with the waist of (r) minimizing this P +|Gi + + (vd)j σ = (1/√2)(σ σ ) , and we can E 2 j⊥ ⊥ (j⊥,2) b d ratio. Low ratios can be achieved for L . L /λ0, a con- |Gi − |Gi ⊥ write an effective model, where the system is described by dition set by the diffraction limit, i.e. the spot size of four excitation modes: two ‘local’ modes, with creation the Gaussian mode must be smaller than the surface of operators S+ and S+, which represent quantum memo- the arrays. Remarkably, this condition allows to achieve 1 2 ries in ψi and ψf ; and two ‘non-local’ bright and dark strong subradiance even when the characteristic size of modes,| withi creation| i operators σ+ and σ+, connecting each array L is much smaller than their separation L, b d ⊥ the two memories. The dynamics can then be described i.e., the subradiant state is ‘non-local’. As an example, by a Lindblad master equation for the density matrix of for N = 20 and δ = 0.8λ0 (i.e., L = 16λ0), we obtain ⊥ 2 ⊥ ⊥ the atoms ρ, asρ ˙ = i[Heff, ρ] + γd [σd−]ρ + γb [σb−]ρ, γd/γb 10− for L 130λ0. − D D ∼ ∼ where [a]ρ aρa† (1/2)(a†aρ + ρa†a), and with an In Fig.2(b) we observe that the interference mecha- effectiveD Hamiltonian≡ − nism is quite sensitive to the separation between arrays, Ω + + as small deviations of L compared to λ0 will greatly in- Heff = σ (S− + S−) + σ (S− S−) + h.c. (4) √2 b 1 2 d 1 − 2 crease the decay rate γd [see Eq. (2)]. In Fig.2(c) we show the effect of the lattice spacing on the saturation The evolution of the system is shown in Fig.3(b), value of Fig.2(a) for small L. The ratio of dark to bright demonstrating transfer at time t = π/Ω[34]. We em- state decay rates is minimal for δ = λ0/2, for which the phasize that our protocol does not require tailoring the emission in transverse directions⊥ is best cancelled, and temporal shape of exchanged photons, in contrast to de- 4 scales with the atom number as γd/γb 1/N . The terministic quantum state transfer protocols with ‘flying’ ∼ ⊥ 4
pled to the excited state, while avoiding spontaneous decay from e to s . This could be realized for ex- | i | i ample using a Rydberg state s = nS1/2, m = 1/2 , with higher energy [10], or another| i ground| state s =i | i 5S1/2,F = 1, mF = 1 , coupled to e via a two-photon transition| [36]. Alternatively,i one can| i use for the opti- cal transition atoms with a J = 0 J = 1 transition, e.g. 88Sr; while this introduces three→ excited states with orthogonal dipole matrix elements, our results for dark and bright state decay rates remain qualitatively similar [28]. The atomic trap is characterized by a finite temper- FIG. 3. Quantum state transfer between ‘local’ quantum mem- ature and Lamb-Dicke parameter η [37]. The resulting ories. (a) Sketch and atomic Λ-level structure for coupling quantum memories. A weak homogeneous field Ω, resonant spread of the atomic wavefunction yields a renormaliza- 2 with the collective atomic shift ∆d, drives the |ei → |si tran- tion of the decay rates as γd/b γd/b[1 η (2nth + 1)] + 2 → − sition. (b) Temporal evolution of the atomic populations for γeη (2nth + 1) [28], where nth is the thermal occupation + the initial state S |Gi, with N⊥ = 12, L = 30λ0, δ⊥ = 0.8λ0. number of trap states, and we assumed η√2n + 1 1 1 th Red (green): number of atoms in state |si in the first (second) and γeη√2nth + 1 ων , with ων the atomic motional array. Black: total number of atoms in state |ei. (c) Infidelity 2 frequency. We thus need η (2nth + 1) . γd/γe. The for quantum state transfer as function of dark and bright state effect of missing atoms is similar [28]; for a defect proba- decay rates. Blue dots: parameters of Fig.2(a) for 1 excita- bility p, we find γ γ (1 p) + γ p + (p2), i.e. we tion. Red curve: Eq. (5). d/b → d/b − e O require p . γd/γe. Multiple excitations. — For states with multiple ex- photonic qubits [13, 35]. Fig.3(c) represents in red the citations, the dynamics can be studied again by ana- optimal achievable fidelity for given γd,b, which reads lyzing the spectral properties of the non-hermitian ef- fective Hamiltonian, which now takes the form Hdip = π√2γd/γb P + e− , (5) 0 j j0 σ σ−0 [28]. Since each atom cannot be ex- F ≈ j,j H , j j cited more than once, the doubly-excited state (σ+)2 showing the requirement γb γd. The blue dots rep- d |Gi resent simulations for atomic arrays with the parame- cannot be an exact eigenstate of Hdip. An analytical ters of Fig.2(a), with the optimal drive given by Ω = expression for the resulting decay rates can, however, p be obtained by treating the non-linearity as perturba- γdγb/8 [28]. As noted above, our treatment neglects ef- fects of retardation in atomic dynamics; Eq. (5) remains, tion, where each excitation effectively acts as a defect however, valid even for large delay times, although at the for the other, with the ‘defect’ probability p identified P 4 as the inverse participation ratio p = (vd)j (see cost of a slowdown of the dynamics [28]. j⊥ | ⊥ | Probing the dark state. — The existence of the dark Ref. [28]). In Fig.2(a) we show in red, for the eigenstate + 2 state can be detected in the reflection of an external laser closest to (σd ) , the ratio of the decay rate per excita- (2) |Gi (see details in [28]). We consider here a weak probing tion γd and γb, which is well captured by this analytical field with frequency ω0 + ∆d, propagating along z in the approximation (dashed red curves). Gaussian mode (r), and driving atoms prepared in the (2) 2 E For large N , we thus expect γd γe/N , since ground state . Assuming the transition frequency of ⊥ ∼ ⊥ |Gi (vd)j 1/N . Two regimes can then be explored. the atoms in each array is additionally detuned, by ∆ for ⊥ ∼ ⊥ First, for γ , γ(2) γ the system becomes effectively atoms in the first array and either ∆ or ∆ for the second d d b array, the dark and bright states are then− revealed in the almost linear, and in particular the protocol for quan- width of the resonance peak of the reflectivity R(∆). We tum state transfer above remains valid, with the replace- (2) 2 2 2 ment γ γ . This can be used to transfer states with obtain R = (γb γd) /(γb +4∆ ) for symmetric detuning, d d − 2 2 2 → and R = (γb γd) /(γb +4∆ /γd) for opposite detuning, more than one excitation, e.g. quantum error correct- which both have− a peak at ∆ = 0 [28]; the widths of these ing states such as cat or binomial states [38], allowing peaks are given by γb and √γdγb, respectively, allowing in principle to reach fidelities beyond Eq. (5). Second, ∼ (2) for a direct probing of the dark state lifetime. if γd γd , γb, excitations of radiating two-excitation Experimental considerations. — The level structure states can be adiabatically eliminated, exploiting again can be implemented in neutral atoms using for instance the quantum Zeno effect. This mechanism can be used stretched states of 87Rb for g = 5S ,F = 2, m = 2 to effectively block the transfer from the memories to the | i | 1/2 F i and e = 5P3/2,F = 3, mF = 3 , along with a strong dark state, and thereby can operate as a controlled-phase magnetic| i field| to eliminate otheri hyperfine states from gate [39]. Moreover, by the same principle, weakly driv- the dynamics. The level s needs to be coherently cou- ing the optical transition of atoms in one of the arrays | i 5
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Supplemental Material for: Subradiant Bell states in distant atomic arrays
P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch and P. Zoller Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
3 2 I. QUANTUM OPTICAL MODEL with γe = k0d /(3πε0) the single-atom decay rate, G(r) = p∗ Gˆ (r) p, and the dyadic Green’s tensor taking · · Here we provide details on our model and the defini- the explicit form tions in Eq. (1). We first consider the full system com- ik0r h ˆ 3e 2 prising the atoms in the first array (labeled with j = 1), G(r) = (k0r) + ik0r 1 z 2i(k r)3 − in the second array (j = 2), and including the electro- 0 z r r i magnetic field. Each atom has a ground state g and 2 + (k0r) 3ik0r + 3 ⊗2 , an excited state e . The dynamics is governed| byi the − − r | i which represents the field at position r emitted by a Hamiltonian Htot = Ha + Hf + Vaf , where Ha acts on the atoms, and reads (~ = 1) dipole at the origin, with ˆ 2 ˆ 6πi X + ∇ ∇ G(r) k0G(r) = δ(r), (8) Ha = ω0σj σj− + Hdrive, × × − − k0 j and we define Gˆ (0) = 1. The last term in Eq. (6) reads with j = (j , jz), j = (jx, jy) and 1 jx, jy N . 1 n o ⊥ ⊥ + ⊥ X + + ≤ ≤ (ρ) = Im( )j j0 σ−0 ρσ σ σ−0 , ρ . Here ω0 is the atomic transition frequency, σj e j g , L H , j j − 2 j j ≡ | i h | j,j0 and Hdrive is an additional term accounting for possi- ble additional laser drivings. The electromagnetic field In writing the master equation we moved to a rotat- R P Hamiltonian reads Hf = dk λ ωkbλ,† kbλ,k, where ing frame with the atomic transition frequency ω0, and ωk = c k with c the speed of light, bλ,k is the annihila- we made the following assumptions. (i) Rotating wave | | + + tion operator for photons with helicity λ = 1 satisfy- approximation: counter-rotating terms (such as σ σ 0 ), ± j j ing [b k, b† 0 0 ] = δ 0 δ(k k0). Finally, the interaction λ, λ ,k λ,λ − which do not preserve the number of atomic excitations, Hamiltonian reads are neglected. (ii) Markov approximation: retardation effects due to finite light velocity are also neglected. X + V = d (σ p∗ + σ−p) Eˆ (rj ) + h.c., af − j j · We can notice from Eq. (6) that the collective emission j properties of the arrays are determined by the spectrum of Hdip. Since Hdip conserves the total number of atomic where d is the atomic dipole, p the atomic transition P + excitations σ σ−, we can evaluate its eigenstates in polarization (we assume circular polarization), and the j j j each excitation subspace separately. In particular, for electric field operator expresses as single excitation eigenstates this amounts to diagonal- Z izing . Let us assume the atomic arrays are initially ˆ X ik r H E(r) = i dk k bλ,ke · eλ,k, prepared in one of these eigenstates ψ , with complex | | n λ eigenvalue = ∆ iγ /2, containing| Ni atomic exci- n n − n exc tations, i.e. ρ(0) = ψn ψn . Assuming here Hdrive = 0, with eλ,k the polarization unit vector, | i h | p 3 the dynamics of Eq. (6) will yield k = ck/(2[2π] ε0), and ε0 the vacuum permittivity. γnt Assuming the electromagnetic field is initially in the ρ(t) = e− ψn ψn + ρ0(t), | i h | vacuum state, the field dynamics can be integrated, to where ρ0(t) contains strictly less than Nexc atomic excita- obtain a Lindblad master equation for the reduced sys- tions, and γn is thus interpreted as the decay rate of the tem of the atoms, within a Born-Markov approximation. eigenstate ψn . More generally, starting from an initial We obtain [25, 27] | i mixture on the subspace with Nexc excitations, we can dρ h i write = i Hdrive + Re(Hdip), ρ 2 (ρ). (6) dt − − L ρ(t) = ρNexc (t) + ρ0(t),
The resulting non-hermitian dipole-dipole interaction where ρNexc (t) is a density matrix with Nexc excitations P + Hamiltonian reads H = 0 j j0 σ σ−0 , where satisfying dip j,j H , j j dρNexc † j j0 = i(γ /2)G(rj rj0 ), (7) = iHdipρNexc + iρNexc Hdip. H , − e − dt − 7
p 2 2 II. SPECTRAL ANALYSIS OF H where r = (r , z), and qz = k0 q . This allows us to rewrite the⊥ Green’s tensor as − | |
Here we discuss the spectrum of (i.e., the spectrum 3 Z eiq r⊥ eiqz z H Gˆ (r) = dq k21 Q Q · | | , (9) of H in the single-excitation subspace), in the cases 3 0 dip 4πk0 − ⊗ qz of infinite and finite arrays. We then explain how the system can be probed with a laser to measure the bright where Q = (q, qzsgn(z)). and dark state decay rates. Due to translational invariance and parity symmetry, the eigenstates of are plane waves H pn−1 iδ⊥j⊥ qn iπ (jz 1) (cn)j = e · e 2 − /√2N,
A. Infinite planar arrays with qn in the first Brillouin zone and pn = 1 the eigen- state parity. Next we use the relation ±
2 We first derive analytical expressions for the spectrum X iδ⊥j q 2π X (2) e ⊥· = δ (q g), (10) of in the case of infinite planar arrays (N ). We δ − H ⊥ → ∞ j g use the identity [27] ⊥ ⊥ where the sum on the right-hand side runs over vectors
ik0r Z iq r⊥ iqz z g of the reciprocal lattice, i.e., gx,y = (2π/δ )mx,y with e i e · e | | ⊥ = dq , integer mx,y. We thus obtain from Eq. (7) r 2π qz
X j j0 (c )j0 H , n j0 0 Z iδ⊥j (q q) iqz zj zj0 3iγe X e ⊥· n− e | − | pn−1 0 iδ⊥j⊥ q 2 2 iπ 2 (jz 1) = 3 dqe · k0 q p e − −8πk0 0 0 − | · | qz√2N j⊥,jz
2 2 (11) 2 i k q g zj z 0 √ 0 n j pn−1 0 3(2π) iγe X X 2 2 e −| − | | − | iπ (j jz ) = (cn)j k (q g) p e 2 z − 8πk (k δ )2 0 n p 2 2 − 0 0 0 g − | − · | k0 qn g ⊥ jz − | − | 2 2 k (q g) p 2 2 X 0 n i√k qn g L = (cn)j i(Γ/2) p− | − · | 1 + pne 0 −| − | , − k k2 q g 2 g 0 0 − | n − |
2 where Γ = 3πγe/(k0δ ) . We distinguish between two contributions, i.e. write From Eq. (11), a⊥ finite number of diffraction orders P P P g = g + g, where the second sum accounts for (i.e., vectors g) contribute to the decay rate γn, as for these vectors. Similarly, we write ∆n = ∆n + ∆n with qn g k0 the eigenvalue becomes purely real. We | − | ≥ 2 2 thus obtain X k0 (qn g) p ∆n = pn(Γ/2) − | − · | p 2 2 2 2 g k0 k0 qn g X k0 (qn g) p − | − | γn = Γ − | − · | p 2 2 q g k0 k0 qn g 2 2 − | − | (12) sin k0 qn g L , q − | − | 2 2 1 + pn cos k0 qn g L − | − | while ∆n is obtained numerically from the eigenvalues of . This last term is independent of L as the exponential P H where g is restricted to vectors g of the reciprocal lat- term in Eq. (11) vanishes and is thus identical to the tice satisfying q g < k0. In particular, for q = 0, self-energy of a single 2D array [9]. | n − | n a single order (mx = my = 0) contributes, provided δ < λ0, in which case we obtain Eq. (2). Moreover, ⊥ B. Finite-sized (curved) arrays this becomes valid for all qn if δ < λ0/2. ⊥ The self-energies ∆n can be similarly evaluated, how- ever with a bit of caution. Indeed the real part in Eq. (11) We now consider the case of finite atomic arrays. As diverges as all g with q g k now contribute. discussed in the main text, in order to mitigate the | n − | ≥ 0 8 spreading of wavepackets for photons propagating be- 2. Analytical expressions tween the arrays, we assume the atoms in each array are located along the phase profile of a Gaussian mode We now derive expressions for the eigenvalues of . (r). In the following we provide details on this Gaussian H E Due to the finite array size, plane waves are not longer mode, derive analytical expressions for the spectrum of eigenstates of , however parity remains a symmetry of , and provide in the end a numerical study of the eigen- the system. WeH thus write the eigenstates as (c ) = H n (j⊥,1) state distribution. √ pn(cn)(j⊥,2) (vn)j⊥ / 2 with pn = 1. The matrix can then be decomposed≡ into 2 matrices± of size N, namelyH 1. Definition of the Hermite-Gaussian modes and array ( 0)j ,j0 ( )(j ,1),(j0 ,1) = ( )(j ,2),(j0 ,2) (17) curvature H ⊥ ⊥ ≡ H ⊥ ⊥ H ⊥ ⊥ accounting for the dipole-dipole interaction within each Here we summarize the properties and notations of the array, and Hermite-Gauss modes, which are solutions of the parax- ( ) 0 ( ) 0 = ( ) 0 , (18) 2 1 j⊥,j⊥ (j⊥,1),(j⊥,2) (j⊥,2),(j⊥,1) ial equation for light [∂z (i/2k0) ]TEMj,k(r) = 0 for H ≡ H H − ∇⊥2 2 2 modes propagating along z, with ∂x + ∂y . These accounting for the effective interaction between different modes represent a natural basis for∇⊥ the≡ field generated arrays, with v being an eigenstate of + p , with n H0 nH1 by the atomic arrays. Assuming the focal point is here the same eigenvalue n. located at r = 0, these modes are defined by the waist The Green’s tensor in Eq. (7) can be formally decom- w0 as [27] posed as
q 2 3π TEM (r) = 2 H √2x/w(z) H √2y/w(z) j,k πw(z) i j G(r) = 2 Gpar(r) + G0(r), (19) k0 2 2 2 2 2 (x +y )/w(z) i(k0(x +y )/[2R(z)] ψj,k(z)) e− e − , where Gpar(r) is the Green’s function for paraxial modes, (13) reading with j, k = (0, 1, ...), Hj is the Hermite polynomial of 2 order j, k0 ik0[ z + r⊥ /(2 z )] Gpar(r) = e | | | | | | , (20) p 2 2πi z w(z) = w0 1 + (z/zR) (14) | | with the mode width, Z 0 ik0z ik0z R(z) = z 1 + (z /z)2 (15) dr0 Gpar(r r0)TEMj,k(r0)e = TEMj,k(r)e R ⊥ − the radius of curvature, (21) for z0 > z. 1 ψj,k(z) = (j + k + 1)tan− (z/zR) We define the Fourier transform 2 1 the Gouy phase and zR = πw /λ0the Rayleigh length, X iδ⊥j q 0 (˜v ) = (v ) e ⊥· , n q √ n j⊥ and are normalized as N j Z ⊥ dr TEMj,k(r)(TEMj0,k0 (r))∗ = δj,j0 δk,k0 . and the mean absolute quasi-momentum ⊥ X 2 In particular, the phase profile of the Gaussian mode, q = (˜v )q q , | n | | | which determines the curvature of the atomic arrays, is q ik0z taken as (r) TEM0,0(r)e . Specifically, for a given E ∼ where qx, qy = π/δ + 2πnx,y/L , with nx,y = separation distance L between arrays and mode waist w , ⊥ ⊥ 0 0, 1, ..., N 1. We− consider in particular eigenstates with the longitudinal position z of atom j satisfies ⊥ j low quasi-momentum− q. As we saw in Sec.IIA, provided 2 2 k0zj + k0(xj + yj )/[2R(zj )] ψ0,0(zj ) = k0L/2, (16) δ < λ0 only a single diffraction order contributes to the − ± spontaneous⊥ emission, meaning that photons are emit- where the + and signs correspond respectively to − ted mostly in the direction normal to the arrays, and atoms in the second array (jz = 2) and in the first array as such can be treated within a paraxial approximation. (j = 1), such that the phase of (rj ) only depends on z E This motivates us to look for eigenvectors distributed as jz. From Eq. (15), the curvature radius for the arrays is ik0z(j ,1) r ⊥ (v(j,k))j⊥ TEMj,k( (j⊥,1))e . From Eqs. (6), maximal when L 2zR, yielding a displacement along (19) and (21∼ ), we then have z of L2 /(2L) at≈ the corners of the arrays, which can ∼ ⊥ X take values of the order of the wavelength λ0. As we ( ) 0 (v ) 0 0 j⊥,j⊥ (j,k) j⊥ show in the next section, the condition of Eq. (16) allows 0 H j⊥ us to construct non-local eigenstates of with Gaussian H ∆(j,k) iΓ/2 iγ0 /2 (v(j,k))j0 , distribution. ≈ − − (j,k) ⊥ 9 a 4 b 100 is approximately constant for atoms within the same ar- ray. We then have 0
λ −2 ik z ik L iφ 2 10 r 0 (j⊥,2) 0 j,k TEMj,k( (j⊥,2))e = e e (v(j,k))j⊥ , w/ Overlaps where 0 10−4 0 1 2 −2 −1 0 1 10 10 10 10 10 10 10 1 2 φj,k = 2(j + k)tan− [L/(2zR)], (23) L/λ0 Lλ0/L c ⊥ such that v is eigenstate of with eigenvalue (j,k) H0 ± H1 b ik0L iφj,k / = ∆ iΓ/2 1 e e iγ0 /2. (24) d (j,k) j,k (j,k)
L /L2 3. Numerical study
FIG. 4. (a) Optimal mode width w(z = L/2), with Here we provide details on the eigenstates v and p n L/(πλ0) in dashed red. (b) Overlaps On of eigenstates eigenvalues n of for finite arrays. The decay rates γd with Gaussian distribution, for dark and bright states (up- H and γb are obtained by diagonalizing and identifying per points) and all other states (lower points). δ⊥ = λ0/2, the dark and bright states as the eigenstatesH with low- N⊥ = 4, 8, 12, 16, 20 (light to dark blue). (c) Ratio of dark est quasi-momentum q. We minimize the ratio γd/γb by and bright state decay rates with N⊥ = 8, 12, 16 (light to dark blue) and δ⊥ = 0.5λ0, for curved (dots) and flat (crosses) ar- varying w0, which sets the longitudinal atomic according 2 1.25 rays. Dashed red: ∼ (Lλ0/L⊥) . to Eq. (16). In Fig.4(a) we show the optimal mode width w [from Eq. (14)] for the parameters of Fig.2(a) for a single ex- 2 2 p where Γ = 3πγe/(k0δ ) , and we approximated the sum citation. At large L & L /λ0, we have w = Lλ0/π, P ⊥ R 2 ⊥ as an integral j0 dr0 /δ . The term γ(0j,k) is a which is the minimal width achievable for fixed L within ⊥ ≈ ⊥ ⊥ phenomenological decay by photon emission into non- the diffraction limit, where zR = L/2. In this regime paraxial modes, which we add as a perturbation account- imperfections (i.e., finite γd) are mainly due to the array ing for G0(r) in Eq. (19). In analogy to Sec.IIA, the size being too small to fit a Gaussian mode connecting 2 self-energy ∆(j,k) on the other hand diverges due to the the arrays. At small L L /λ0, the width saturates ⊥ divergence in Eq. (20) when z 0, and must be evalu- to around w L /4. This is a trade-off between having ∼ ⊥ ated numerically. v is thus→ approximately eigenstate the Gaussian mode (r) fit the arrays, and increasing (j,k) E of 0. Similarly, we get the number of participating atoms in order to minimize H the emission to non-paraxial modes, both effects lead- X ik z ( ) 0 (v ) 0 i(Γ/2)TEM (r )e 0 (j⊥,2) , 1 j⊥,j⊥ (j,k) j⊥ j,k (j⊥,2) ing to a finite rate γd. In Fig.4(b) we represent the 0 H ≈ − j⊥ overlap of the eigenvectors vn with the Gaussian mode (r). For each eigenvectors, this overlap is computed as where the effect of G0(r) is here neglected in a paraxial E 2 P P 2 approximation for the photons exchanged between differ- On = j (r(j ,1))(vn)j∗ / j (r(j ,1)) . The ⊥ E ⊥ ⊥ ⊥ |E ⊥ | ent arrays. overlaps for the dark and bright modes are represented Using Eq. (16), we have as the upper points, and are close to 1. Conversely, the sum of the overlaps of all other eigenstates is represented ik0z(j ,2) ik0L TEM (r )e ⊥ = e (v )j , 0,0 (j⊥,2) (0,0) ⊥ as the lower points, and takes very small values, vanish- 2 ing for L L /λ0. such that v(0,0) is eigenstate of 0 1 with eigenvalue . H ± H The requirement⊥ for curving the atomic arrays is stud- ik0L ied in Fig.4(c), where we compare the ratio of γd/γb (0,0) = ∆0,0 iΓ/2 1 e iγ(00 ,0)/2. (22) − ± − between flat arrays (for which zj = L/2) and curved This is the expression of the eigenvalues for the sym- arrays (satisfying Eq. (16)). One sees± that the curvature metric and anti-symmetric eigenstates of the main text, can improve this ratio by several orders of magnitude for 2 where we identify γd γ(00 ,0) and ∆d ∆0,0. For L . L /λ0. ≡ ≡ ⊥ k0L = mπ with integer m one of these states is thus In Fig.5 we show the decay rates of all eigenstates γn ‘dark’ (with minimal decay γd), while the other state as well as their average quasi-momenta q for N = 10. ⊥ is ‘bright’ (as it decays with rate γb = 2Γ + γd). For On the left, we represent the same situation as in Fig.1. (j, k) = (0, 0) on the other hand, we get similar expres- We note first that the dark state, labeled 1, as well as 6 sions by considering that the Gouy phase ψj,k in Eq. (13) the bright state above, have the distribution of a TEM0,0 10
L λ δ λ L = 20λ0, δ⊥ = 0.75λ0 L = 2λ0, δ⊥ = 0.75λ0 = 10 0, ⊥ = 0.5 0
2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2 0 1 ...... 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 4
0 1 1
...... 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 3 1
2 arg(v ) 2 v 2 2 arg(v ) v
2 arg(v ) 2 2 2 arg(v ) v
FIG. 5. Spectrum of H. First row: decay rates γn and average quasi-momentumq ¯ of the eigenstates of H, with N⊥ = 10. An even (odd) parity is denoted with a white (resp. black) circle. Second and third rows: distribution of probability amplitude of eigenstates |vn| in each 2D array, and of phase arg(vn). mode. The second most subradiant state, labeled 2, cor- C. Dark and bright state probing responds to a TEM1,1 mode. We note that its parity is opposite to that of v1, which is due to the fact that here L Finally, here we show how to probe the dark and bright is large enough that zR = L/2, and φ1,1 = π in Eq. (23). state lifetimes in the reflectivity of a laser. The situation On the other hand, eigenvectors distributed according to is represented in Fig.6(a). We consider a weak laser with TEM1,0 and TEM0,1 modes cannot be subradiant as they polarization p propagating along z in the Gaussian mode are out of phase, with φ1,0 = φ0,1 = π/2. (r) at frequency ω + ∆ , and driving the system with E 0 d the atoms in their ground state = j g j . Assuming the field is weak enough, each photon|Gi ⊗ will| i be scattered by the system independently, and the reflectivity can be In the second column, we consider the situation with evaluated for single-photon pulses. We can thus write L = 2λ , which corresponds to the opposite extreme 0 the state of the system for a single excitation as regime where z L, and φ 0, such that now the R j,k ≈ subradiant states have the same parity, and eigenvectors Z X + X with TEM and TEM distributions can also be sub- ψ(t) = cj (t)σ 0 + dk ψλ(k, t) k, λ , 1,0 0,1 | i j |Gi | i |Gi | i radiant. The decay rate significantly increases with q, j λ which can be understood as a gradual breaking of the paraxial approximation, and can be seen from Eq. (12) with here cj (0) = 0. We assume moreover that atoms in as the interference between arrays becomes imperfect. each array jz additionally detuned by ∆jz . We consider
two situations, with either ∆jz = ∆ (i.e., with the same detuning for atoms in both arrays), or ∆z = 2∆(3/2 jz) (with opposite detuning between the two arrays). − Finally, in the third column we show that subradiance As in Sec.I, the field dynamics can be integrated, yield- can also appear in eigenvectors with large q when δ < ing for the atoms ⊥ λ0/√2. There, subradiance is due to the fact that these X guided modes have their momentum larger than k0, and c˙j = i j j0 + (∆ ∆ )δj j0 cj0 (t) − H , jz − d , as such reside outside of the light cone, and are studied j0 e.g. in Refs. [16, 20]. These modes are localized in each s (25) array, and are thus degenerate in energy, in contrast to 3πcγe in 2 ψ (rj , t), the non-local modes with low q. − 2k0 11
which can be evaluated numerically, and provides the green and red curves in Fig.6(b). We stress that the expression of Eq. (28) is valid only within the paraxial approximation. This breaks down for configurations with w0 . λ0, which can occur for small arrays with L . ⊥ 4λ0. As we saw from Fig.4(b), only the dark and bright FIG. 6. Dark and bright state probing. (a) Level scheme and sketch for probing the dark and bright states. A weak states have significant overlap with the Gaussian distri- bution (rj ), such that we can restrict the vector space probe field E, resonant with the collective atomic shift ∆d and E propagating in the Gaussian mode E(r), drives the atomic of the matrix +∆z ∆d1 to these two states, and invert arrays, while the transition frequency of the atoms in the H − P P R r 2 it on this subspace. Approximating j jz d /δ first and second arrays is shifted by ∆, either symmetrically ≈ 2 ⊥ 2 ⊥ and using Eq. (16), we obtain R(∆) = (γb γd) /(γb + or with opposite sign. (b) The reflection probability displays 4∆2) for the case of symmetric detuning between− the two a peak with the width given by γb (for symmetric shift, in 2 2 2 √ arrays, and R(∆) = (γb γd) /(γb + 4∆ /γd) for the green) or γdγb (for opposite shift, in red), with N⊥ = 12, − L = 30λ0, δ⊥ = 0.8λ0. Dashed black: analytical expressions. case of opposite detuning, with R = 1 if ∆ = 0. These expressions are represented in the dashed black curves of Fig.6(b), which show excellent agreement with the numerical results. where we moved to a frame rotating with ω0 + ∆d, and defined the input field Z III. PHOTONIC LINK BETWEEN QUANTUM in 1 ik r i(ωk ω0 ∆d)t ψ (r, t) = p dke · e− − − MEMORIES (2π)3 X p∗ e kψ (k, 0). Here we write an effective model for the atomic dy- · λ, λ λ namics, retaining four modes as expressed in Eq. (4). The resulting field on the other hand reads, neglecting We then derive the expression for the fidelity of quan- retardation effects, tum state transfer in Eq. (5), and explain how to write and read from the local quantum memory states. We r 2 finally discuss how our results extend to non-markovian in γek0 X ψ(r, t) = ψ (r, t) i cj (t)G(r rj ) (26) − 6πc − regimes, where retardation effects due to the finite speed j of photons exchanged between arrays is no longer negli- For long pulses, i.e., varying over timescales much larger gible. than the atomic response time 1/γd, we can setc ˙j 0 in Eq. (25), and get from Eq. (26) ≈ A. Effective four mode model in γe X ψ(r, t) = ψ (r, t) G(r rj ) − 2 − j,j0 We now consider each atom has a Λ level structure, 1 in as represented in Fig.3(a). We assume at most a single ( + ∆ ∆ 1)− 0 ψ (rj0 , t), H z − d j,j atom is in state e or s at a time, and a laser drives (27) the e s transition| i | resonantlyi with the cooperative where (∆ ) 0 = ∆ δ 0 . | i → | i z j,j jz j,j shift ∆ , with homogeneous Rabi frequency Ω. In a ro- The reflectivity R is then obtained by taking d tating frame, the system is thus described by the master ψin(r, t) = (r) as the overlap between ψ(r) and a target equation in Eq. (6), with mode ψtar(rE) X X 2 + + Z Hdrive = ∆d σj σj− + Ω (sj σj− + h.c.), tar − R = dr (ψ (r))∗ψ(r) , j j ⊥ + where the target mode is the Gaussian mode propagat- with sj = s j g . We start from an initial state of the tar | i h | ing to the left, i.e., ψ (r) = ( (r))∗. Within a parax- form ial approximation for the GaussianE mode we replace the ψ = c + c S+ , (29) Green’s function by its paraxial counterpart in Eq. (19), | ii g |Gi s 1 |Gi and apply Eq. (21) to obtain + P + where S = (vd)j s creates an excitation of 1 j⊥ ⊥ (j⊥,1) 2 r 2 2 s in the first array, with (vd)j⊥ ( (j⊥,1)) the proba- 9π γe X 1 | i ∼ E R = (rj )( + ∆z ∆d1)− 0 (rj0 ) (28) bility amplitude corresponding to the dark (and bright) 4k4 E H − j,j E + P + 0 j j0 state. Similarly we define S = (vd)j s , which , 2 j⊥ ⊥ (j⊥,2) 12 creates an excitation of s in the second array. The dy- The general solution of Eq. (31) for c2 (t) can be written namics of Eq. (6) will then| i excite only the eigenstates of in the form with Gaussian distribution, i.e., the dark and bright H X iωb,it X iωd,it states, as shown in Fig.4(b). We thus obtain an effective c2 (t) = Cb,ie− + Cd,ie− , (32) i= 1 i= 1 model with only four modes: the two local modes, with ± ± + + creation operators S1 and S2 , as well as the dark and where, in the regime γ Ω γ , bright modes, created by the operators d b 1 + X + m + ωd, 1 = ( iγd g) , σ = (vd)j σ ( 1) σ /√2. ± 4 − ± b/d ⊥ (j⊥,1) ± − (j⊥,2) (33) j q ⊥ 1 2 2 ωb, 1 = iγb 16Ω γ , ± 4 − ± − b The atomic dynamics thus follows p with g 16Ω2 γ2 the frequency of the oscillations. dρ ≡ − d = i[Heff, ρ] + γb [σb−]ρ + γd [σd−]ρ, (30) The corresponding amplitudes can be readily obtained dt − D D using the initial conditions, and read with Ω i arctan[γd/g] Cd, 1 = e± ± Ω + + g Heff = σ S− + S− + σ S− S− + h.c. (34) √2 b 1 2 d 1 − 2 4Ω2 Cb, 1 = − . ± 2 2 p 2 2 16Ω γb γb γb 16Ω and [a]ρ = aρa† (1/2)(a†aρ + ρa†a). We note that − ± − D − Fig.3 provides a numerical verification of this four modes Using Eqs. (32), (33) and (34), the first maximum of c2 (t) model for the simulation of quantum state transfer, which is approximately at half the period of Rabi oscillations, we describe below. i.e. t (π arctan [γ /g]) /g. The fidelity , given max ≈ − d F by c2 (tmax), depends on the drive Ω. Expanding c2(tmax) up to the first order in Ω/γb and γd/Ω, we get B. Quantum state transfer Ωπ πγd 2 2 c2 (tmax) = 1 + + + (Ω/γb) + (γd/Ω) . We now provide an analytical derivation of the fidelity − γb 8Ω O O for quantum state transfer. Assuming the system is ini- p The optimal drive then reads Ω = γ γ /8 and the tially prepared in the pure state ψ of Eq. (29), the opt b d i corresponding optimal fidelity of the state transfer is atomic density matrix can be expressed| i as 2 p c (t ) = 1 π 2γ /γ + (γ /γ ), ρ(t) = ψ(t) ψ(t) + P (t) , Fopt ≡ | 2 max | − d b O d b | i h | g |Gi hG| which is Eq. (5). where
+ + + + ψ(t) = cg + c1(t)s + c2(t)s + cb(t)σ + cd(t)σ , C. Write and read of quantum memory using | i 1 2 b d |Gi single photon pulses with c1(0) = cs, c2(0) = cb(0) = cd(0) = Pg(0) = 0. We wish to transfer the quantum state ψ to the second | ii We now discuss how one can write and read from the array, i.e., have the system evolve to quantum memories in the arrays. In particular, assum- ing the atoms are in state ψ as in Eq. (29) while the ψ = c + c S+ . | ii | f i g |Gi s 2 |Gi photonic field is in the vacuum state 0 , we show that the atoms can be brought to their ground| i state while We define the fidelity of the state transfer as |Gi 2 emitting a photonic qubit cg 0 +cs 1 , where 1 denotes maxt c2 (t) for cs = 1. Eq. (30) then yields | i | i | i F ≡ | | a state with a single photon leaving the system in a well Ω defined spatio-temporal mode, propagating in a given di- c˙1 (t) = i (cb (t) + cd (t)) rection. The time-reversed process allows one to absorb − √2 a photonic qubit, thereby preparing the atoms in state Ω c˙2 (t) = i (cb (t) cd (t)) ψi . − √2 − | i (31) We make the following two additional assumptions. γb Ω First, the phase acquired by a photon propagating be- c˙b (t) = cb (t) i (c1 (t) + c2 (t)) − 2 − √2 tween the arrays k0L can be modified, e.g. by slightly γd Ω changing the distance L over a range of λ0/2. Second, c˙d (t) = cd (t) i (c1 (t) c2 (t)) . ∼ − 2 − √2 − the laser drive Ω can be turned off for the atoms in the 13 second array. For convenience, we consider the system The flux of photons emitted in the Gaussian mode (r) + E prepared in state S1 , with the laser driving only the (i.e., propagating to the right) is obtained as the overlap first array. The dynamics,|Gi and in particular the spatio- Z 2 temporal shape of the emitted photon, can be obtained P (t) = c dr ( (r))∗ ψ(r, t) following the steps in Sec.IIC. We write here the state → ⊥ E as 2 3πγ Ω2 X + + 2 e X 1 = c˜(t) ( (rj ))∗ ( ∆ 1)− 0 δ 0 (v )j0 ψ(t) = (cj (t)σj +c ˜j (t)sj ) 0 2 d j,j jz ,1 d ⊥ | i |Gi | i | | 2k0 0 E H − j j,j Z 2 X γd + dk ψλ(k, t) k, λ , =2 c˜(t) 2Ω2Γ , |Gi | i | | e2ik0LΓ2 (Γ + γ )2 λ − d withc ˜j (0) = (vd)j⊥ δjz ,1 and cj (0) = ψλ(k, 0) = 0. Inte- where in the second line we replace the Green’s func- grating the field dynamics, we get tion by its paraxial counterpart as in Eq. (19) and used X Eq. (21), and in the third line restricted ( ∆d1) to 0 0 0 H − c˙j = i j,j ∆dδj,j cj (t) iΩδjz ,1c˜j (t), the subspace of symmetric and anti-symmetric Gaussian − H − − j0 P R 2 states, replaced j dr /δ and used the property ⊥ ≈ ⊥ ⊥ c˜˙j = iΩδj ,1cj (t), of Eq. (16). We note that P vanishes for γd 0. Sim- − z → ≈ (35) ilarly, the flux of photons emitted in the Gaussian mode where we moved to a frame rotating with ω + ∆ . As- ∗(r) (i.e., propagating to the left), reads 0 d E suming k0L = mπ with integer m, neither the symmetric 6 Z 2 or anti-symmetric Gaussian states are dark, and their P (t) =c dr (r)ψ(r, t) decay rate to paraxial modes is given by Γ(1 cos[k L]) ← ⊥E ± 0 2 [see Eq. (22)]. Thus, provided Ω Γ(1 cos[k0L]) 2ik0L | ± | 2 2 e Γ (Γ + γd) (ideally by setting cos[k0L] = 0), the population of state =2 c˜(t) Ω Γ 2ik L 2 − 2 . | | e 0 Γ (Γ + γd) e j can be adiabatically eliminated, i.e., we setc ˙j 0 − | i ≈ in Eq. (35), yielding 2 For γd 0 this reduces to P (t) =γ ˜ c˜(t) , showing ≈ ← | | X 1 that the photon is emitted in the left-propagating Gaus- cj (t) = Ω ( ∆ 1)− 0 δ 0 c˜j0 (t), − H − d j,j jz ,1 sian mode. This allows to perform a transfer from the j0 (36) quantum memory state ψi to a propagating photonic 2 X 1 ˙ 0 0 | i c˜j =iΩ δjz ,1 ( ∆d1)− 0 δj ,1c˜j (t). qubit, as well as the time-reversed process. H − j,j z j0
Next we note that for the initial condition above, from D. Beyond the Markov approximation
Eq. (36) we havec ˜j (t) =c ˜(t)(vd)j⊥ δjz ,1. Restricting ∆ 1 to the space spanned by the symmetric and anti-H− d Finally, we discuss the effects of time-delays in the symmetric Gaussian states, we get atomic dynamics, arising from the finite propagation time 2 of photons between arrays, which were neglected in the ˙ Ω (Γ + γd) c˜ = 2 2ik L 2 2 c˜(t), previous sections. These effects become relevant only e 0 Γ (Γ + γd) − when the arrays are separated by L & c/Γ 10m for i.e., the memory will spontaneous emit a photon with Γ in the MHz range, which can be realized by∼ mediating rate photons exchanged between arrays with optical lenses or 2 fibers. We first show how this affects the decay rates of Ω (Γ + γd) γ˜ = 4Re . dark and bright states, and then study its effect on the − e2ik0LΓ2 (Γ + γ )2 − d state transfer fidelity. We remark thatγ ˜ is independent of k0L if γd 0, and reduces toγ ˜ = 2Ω2/Γ. The spatio-temporal shape≈ of the outgoing photon is obtained from Eq. (26), as 1. Effect on dark and bright states r γ k2 Let us write the state of the system ψ(r, t) =ic˜(t)Ω e 0 6πc Z X 1 X + X G(r rj )( ∆d1)− 0 δj0 ,1(vd)j0 . ψ(t) = cj (t)σj 0 + dk ψλ(k, t) k, λ , − H − j,j z ⊥ | i |Gi | i |Gi | i j,j0 j λ We note that the temporal distribution can be tailored and integrate the field dynamics without neglecting retar- by varying Ω in time. dation in photon propagation between different arrays, 14 yielding [13], with the definitions of Eqs. (17) and (18) where z1 = L/2 and z2 = L/2 denote the position of the first and− second arrays along z. In particular, in the Xh c˙ (t) = i ( ) 0 c 0 (t) (j⊥,1) 0 j⊥,j⊥ (j⊥,1) dark state the system can reach a quasi-equilibrium if − 0 H j⊥ the exponent in the decay of the dark state amplitude in κL i Eq. (37) is much smaller than 1/τ, and we get (with + e− ( 1)j ,j0 c(j0 ,2)(t τ) , 2 2 H ⊥ ⊥ ⊥ − zR L for simplicity) E(z, t) Γ sin(k0z) cd(t) . Xh While this quantity shows that the≈ arrays continuously| | c˙(j ,2)(t) = i ( 0)j ,j0 c(j0 ,2)(t) ⊥ − H ⊥ ⊥ ⊥ exchange photons at a rate Γ even when Γτ 0, the j0 ⊥ total number of photons between∼ the arrays at any→ time, κL i + e− ( 1)j ,j0 c(j0 ,1)(t τ) , given by H ⊥ ⊥ ⊥ − Z z2 with τ = L/c. Here we added an attenuation coefficient κ 1 Γτ 2 Nph = E(z, t)dz = cd(t) , accounting for additional decay channels induced by pos- c z1 2 | | sible optical elements mediating the exchanged photons, but neglected any effect of dispersion. In particular, for vanishes in that limit. the dark and bright state amplitudes we get
Γ + γd Γ κL 2. Effect on state transfer fidelity c˙ (t) = c (t) e− c (t τ). b/d − 2 b/d ∓ 2 b/d − Defining the Laplace transform variablesc ˜(s) = The effect of retardation on the state transfer fidelity [c(t)](s), we arrive at can be studied in a similar way. Following the same pro- L cedure, with the notations of Sec.IIIB, we have sτ κL 1 Γ(1 e− − ) + γ c˜ (s) = s + − d − c (0), b/d 2 b/d Ω c˙1 (t) = i (cb (t) + cd (t)) − √2 which cannot be analytically inverted directly. An an- Ω alytical approximation can however be obtained by ex- c˙2 (t) = i (cb (t) cd (t)) sτ 2 − √ − panding e− = 1 sτ + (sτ) to lowest order in Γτ, 2 − O yielding Γ + γd Γ κL c˙ (t) = c (t) e− c (t τ) b − 2 b − 2 b − −κL (1+e )Γ+γd Ω exp 2 Γτe−κL t − − i (c1 (t) + c2 (t)) cb(t) =2 κL cb(0) − √2 2 Γτe− − −κL (37) (1 e )Γ+γd Γ + γd Γ κL exp − −κL t c˙d (t) = cd(t) + e− cd(t τ) − 2+Γτe − 2 2 − cd(t) =2 κL cd(0). 2 + Γτe− Ω i (c1 (t) c2 (t)) , From Eq. (37) increasing the retardation Γτ decreases − √2 − the decay of the dark state due to atomic losses with with initial conditions c (0) = 1, c (0) = c (0) = c (0) = rate γ . This is a consequence of the dark state being 1 2 d b d 0. The solution for the Laplace transform of c then reads now a superposition of field and atomic excitations, with 2 only the atomic part decaying if κL = 0. If we include 2 sτ κL 2Ω Γe− − a finite attenuation κL, the photonic component also in- c˜2(s) = , (2Ω2 + s(2s + Γ + γ ))2 s2Γ2e 2sτ 2κL duces losses. d − − − The photonic field is expressed as which is Laplace-inverted numerically.
r 2 The effect of retardation is represented for κ = 0 in γek0 X 3 3 5 1 ψ(r, t) = i cj (t τ r rj /L)G(r rj ), Fig.7 with Γ τ [10− , 10 ] and γd/Γ [10− , 10− ]. − 6πc − | − | − ∈ ∈ j In all these figures we see that the effect is to rescale the parameters of the system. In Figs.7(a,b) we see providing for the photonic flux E(z, t) = in particular that the fidelity for quantum state transfer c R dr ψ(r, t) 2, within the paraxial approximation ⊥| | is almost constant for all values of the retardation Γτ. for the Green’s tensor, This can be understood as while increasing Γτ increases Γ Z the state transfer time tmax, the decay rate of the dark E(z, t) dr state in Eq. (37) decreases, resulting in a constant overall ≈ 4 ⊥ loss probability. In Fig.7(c,d) we see that the required (r)[c (t τ z z /L) + c (t τ z z /L)] E b − | − 1| d − | − 1| optimal Ω decreases, such that the transfer time tmax 2 increases linearly with the delay at large Γτ. For finite + ∗(r)[c (t τ z z /L) c (t τ z z /L)] , E b − | − 2| − d − | − 2| attenuation κL this reduces the transfer fidelity. 15
(a) (b) grees of freedom, reads X Hdip =ωv aj†,αaj,α j,α
iγe X + σ σ−0 G rj rj0 + rˆj rˆj0 − 2 j j − − j,j0
X iγe X + =ω a† aj σ σ−0 G rj rj0 v j,α ,α − 2 j j − (c) (d) j,α j,j0
iγe X + σ σ−0 rˆj rˆj0 ∇ G rj rj0 − 2 j j − · − j,j0
iγe X + 2 σ σ−0 rˆj rˆj0 ∇ G rj rj0 − 4 j j − · − j,j0 + (η√2n + 1)3, O th (38) where we performed a Taylor expansion for the Green’s −4 FIG. 7. (a) Infidelity for state transfer, with 2γd = 10 Γ, tensor, with η = k0/√2mωv the Lamb-Dicke parameter, Γτ/2 ∈ [10−3, 103] (light to dark blue) and κ = 0. (b) Infi- aj,α the annihilation operator of the motional excitation delity for state transfer, with optimal Ω, (Γτ/2) ∈ [10−3, 103] of atom j along axis α (x, y, z), rˆj the quantized coor- (light to dark blue) and κ = 0. Dashed black: 1 − ∈ p dinates of atom j relative to its trap center position rj , exp(−π 2γd/γb). (c) Optimal value of Ω, with 2γd/Γ ∈ [10−5, 10−1] (light to dark blue) and κ = 0. Dashed black: and where ∇ acts on G(r). We assumed the unperturbed p density matrix of the system factorizes as ρ(0) = ρ ρ 1/ 1 + 3Γτ/4. (d) Transfer time tmax, with optimal Ω, at th −5 −1 ⊗ 2γd/Γ ∈ [10 , 10 ] (light to dark blue) and κ = 0. Dashed where ρth stands for a thermal distribution of phononic black: 1 + Γτ/2. modes with mean number nth, while ρat accounts for the internal atomic degrees of freedom.
B. Elimination of phonon modes IV. EFFECT OF FINITE LAMB-DICKE PARAMETER AND TEMPERATURE We now perform an adiabatic elimination of the mo- tional degrees of freedom in Eq. (38). We write Hdip = H0 + V as a sum of a free Hamiltonian H0 and an in- teraction term V . Assuming η√2n + 1 1, we can Here we discuss the effects of phononic degrees of free- th truncate the expansion in Eq. (38) to second order. We dom for atoms trapped in optical lattices with finite thus have Lamb-Dicke parameter η, and thermal phonon distribu- tion with mean number nth. We derive a correction to X iγe X + † 0 r r 0 , and in particular show that the spread of the atomic H0 =ωv aj,αaj,α σj σj−G j j − 2 0 − wavefunctionH leads to an additional individual decay of j,α j,j 2 iγ each atom of γeη (2nth + 1). e X + V = σ σ−0 rˆj rˆj0 ∇ G rj rj0 − 2 j j − · − j,j0
iγe X + 2 σ σ−0 rˆj rˆj0 ∇ G rj rj0 . − 4 j j − · − j,j0
Moving to an interaction picture with respect to H0, we get an effective Hamiltonian H˜ to second order pertur- bation in V , assuming further that γη√2n + 1 ω . th v A. Model We then obtain Z t H˜ = Tr [V (t) ρ ] i dsTr [V (t) V (s) ρ ] , ph th − ph th Assuming that each atom is trapped with a harmonic −∞ (39) potential with frequency ωv, the non-hermitian Hamil- where we denote Trph for the trace over phononic degrees tonian Hdip, including now the coupling to motional de- of freedom. 16
The first term in Eq. (39) is evaluated as 100
Trph [V (t) ρth] 10−1 iγe X + 2 = Trph σj σj−0 rˆj rˆj0 ∇ G rj rj0 ρth − 4 0 − · − b j,j γ −2 / 10 d iγe 2 X + γ = η (2n + 1) σ σ−0 (∇ ∇)G rj rj0 . −2k2 th j j · − 0 j=j0 10−3 6 Using the vector field identity ∇ (∇ V ) = × × ∇ (∇ V ) (∇ ∇)V and Eq. (8), we further get 10−4 · − · 10−3 10−2 10−1 Trph [V (t) ρth] pd 2 iγeη X + = (2n + 1) σ σ−0 G rj rj0 FIG. 8. Effect of probability p of having defects on each site, 2 th j j − j=j0 with N⊥ = 4, 8, 12, 16, 20 (light to dark blue), δ⊥ = 0.5, L = 6 30λ0. Solid: numerics. Dashed: expression from Eq. (43). iγe 2 X + ˆ η (2n + 1) σ σ−0 p∗ ∇ ∇ G rj rj0 p. − 2k2 th j j · · − · 0 j=j0 6 The last term in this equation stands for the variation This shows an additional individual decay for each atom, of the longitudinal part of the Green’s tensor, and in- and an additional rescaling of the interatomic interaction. troduces a renormalization of the near-field interaction. In order have a given ratio for γd/γb, we must thus satisfy 3 2 Dropping this near-field term, which decays as rj rj0 , the condition η (2nth + 1) . γd/γb. we finally have | − |
iγe 2 X + Tr [V (t) ρ ] = η (2n + 1) σ σ− ph th − 2 th j j V. EFFECT OF MISSING ATOMS j
iγe 2 X + + η (2n + 1) σ σ−0 G rj rj0 Here we discuss how the presence of defects in the 2 th j j − j,j0 atomic arrays affect the system. The effect of holes can (40) be accounted for by writing for the dipole-dipole interac- h i R t tion Hamiltonian of Eq. (7) Using the identity dsTrph αˆj (t)α ˆj0 0 (s) ρth = −∞ δα,α0 δj,j0 / (iωv), withα ˆj the component of rˆj along axis = ideal holes, (42) α, we get for the second term in Eq. (39) H H − H Z t ideal holes where is the matrix without defects, and j j0 = i dsTrph [V (t) V (s) ρth] , ideal H holes H − 0 if atom j or j0 is missing and 0 = 0 otherwise. −∞ j,j j,j 2 2 H H γe η X + We first consider the situation where a single atom, 00 00 0 = 2 ∇G rj rj ∇G rj rj σj σj−0 . −4ωvk − · − say atom i, is missing from the arrays. The dipole-dipole 0 j,j0,j00= j,j0 6 { } interaction Hamiltonian can be expressed from Eq. (42) (41) as This term can be evaluated in the limit of infinite arrays
(N ), by splitting the sum as γe X + γe + ⊥ 0 → ∞ Hdip = i σj σj−0 G(rj rj ) i σi σi− X X − 2 0 − − 2 = (1 δj j00 δj0 j00 + δj j00 δj0 j00 ). j,j − , − , , , j,j0,j00= j,j0 j,j0,j00 γe X + γe X + 6 { } + i σ σ−0 G(ri rj0 ) + i σ σ−G(rj ri). 2 i j − 2 j i − Using the representation of Eq. (9) for the Green’s tensor j0 j and the property of Eq. (10), we then find that the dark √ P + + P + and bright states (1/ 2N) j (σ(j ,1) σ(j ,2)) are For the state ψn j=i(cn)j σj , we then obtain ⊥ ⊥ ± ⊥ |Gi | i ∝ 6 |Gi eigenstates of the Hamiltonian in Eq. (41) with purely real eigenvalues, provided δ < λ and k L = mπ. γn 2 0 0 ψn Hdip ψn = ∆n i 1 (cn)i Therefore, for these states the⊥ term in Eq. (41) con- h | | i − 2 − | | γe 2 4 tributes only a frequency renormalization. i (cn)i + ( (cn)i ). Thus, only the term of Eq. (40) contributes to the ra- − 2 | | O | | diation, such that the decay rates are corrected as Let us now consider a situation where each atom i has γ γ η2 (2n + 1) + γ 1 η2 (2n + 1) a probability 0 p 1 of being missing, and denote the d/b → e th d/b − th ≤ ≤ 17
(a) (b) associated random variable as si. We then have
γe X + γe X + 0 r r 0
H = i σ σ−G( ) i s σ σ− e dip j j j j i i i e − 2 0 − − 2 / j,j i / b d
VI. MULTIPLE EXCITATIONS where σj−,i = g j ei , which now mixes states with dif- ferent polarizations.| i h | Diagonalizing this Hamiltonian, we obtain a degenerate pair of dark and bright states, po- For states with more than one atom in e or s the fact larized in the x y plane, with decay rates represented that each atom cannot support more than| i a single| i exci- in Fig.9. Notably,− these decay rates remain close to the tation generates an atomic non-linearity, which induces values obtained for two-level atoms. an additional decay rate. The state ψ(2) (σ+)2 | d i ∝ d |Gi for instance is not an eigenstate of Hdip, however we can treat this non-linearity in first order perturbation theory. We get (2) (2) γd X 4 ψ Hdip ψ = ∆d i 1 (vd)j h d | | d i − 2 − | | j⊥ γd γe X 4 + ∆ i i (v )j . d − 2 − 2 | d ⊥ | j⊥ (44) The decay rate per excitation, as represented in Fig.2(a), is then obtained as γ(2) = Im ψ(2) H ψ(2) . − h d | dip | d i From Eq. (43), we can interpret the result of Eq. (44) as one of the two excitations decays with rate γd, and acts as a defect for the other excitation with probability p = P 4 (vd)j identified as twice the inverse participation j⊥ ⊥ ratio| of the dark| state.
VII. IMPLEMENTATION WITH FOUR-LEVEL ATOMS
In all the calculations above and in the main text we treated the atoms as two-level systems with a circular