Optimal collection of radiation emitted by a trapped atomic ensemble

Trapped atomic ensembles are convenient systems for quantum information storage in the long-lived sublevels of the electronic ground state and its conversion to propagating optical photons via stimulated Raman processes. Here we investigate a phase-matched emission of photons from a coherently prepared atomic ensemble. We consider an ensemble of cold atoms in an elongated harmonic trap with normal density distribution, and determine the parameters of paraxial optics to match the mode geometry of the emitted radiation and optimally collect it into an optical waveguide.


Introduction
An important yet difficult task in quantum information and communication is deterministic generation of optical photons in well-defined spatial and temporal modes. Photons can serve as flying qubits to encode quantum information and reliably transmit it over long-distances via quantum channels [1,2]. For optical photons, such quantum channels are free space or fiber waveguides. On the other hand, atomic ensembles have good coherence properties and strong dipole transitions for efficient coupling to optical photons [3][4][5]. Moreover, atoms can couple to microwave fields and thereby be interfaced with superconducting circuits, which are among the most advanced quantum processors [6]. The atomic ensembles can then play the role of the quantum memories and microwave to optical transducers [6][7][8] to interconnect distant superconducting quantum circuits via optical fibers.
Here we consider photon emission from an atomic ensemble coherently prepared in a collective storage state corresponding to a single symmetric spin-wave excitation. This preparation step may be the result of collective coupling of alkali atoms on the ground state hyperfine transition [9,10] or on the highly excited Rydberg transition [11,12] to a microwave resonator that in turn contains superconducting qubits. A stimulated Raman process triggered by a coupling laser pulse converts the collective spin excitation of the atoms to an optical photon emitted predominantly in the phase-matched direction. We examine the spatio-temporal profile of the emitted photon and determine the parameters of the paraxial optical elements to optimally collect this photon into a single-transverse-mode optical waveguide. The waveguide can then transmit the photon encoding the quantum information to a distant atomic ensemble, where a reverse process, equivalent to dynamical light storage in an electromagnetically induced transparency (EIT) medium [3], will coherently convert the optical excitation to a collective microwave spin excitation.
The paper is organized as follows. In Sect. 2 and in Appendices A and B, we present the mathematical formalism to describe the photon emission by coherently prepared atoms and photon collection by Gaussian optics. In Sect. 3 we present the results of analytic and numerical calculations for optimal photon collection, followed by conclusions in Sect. 4.

Interaction of atoms with a radiation field
Consider an ensemble of N three-level atoms with the long-lived ground-state sublevels |g and |s and an electronically excited level |e , as shown in Fig. 1(a). We assume that the atoms initially in the ground state |G ≡ |g 1 , g 2 , . . . , g N are transferred to the collective single-excitation storage state |S = 1 √ N N j=1 |g 1 , g 2 , . . . , s j , . . . , g N by a weak microwave or Raman process that acts symmetrically on all the atoms. A spatially uniform laser pulse couples near-resonantly the storage state |s to the excited state |e with Rabi frequency . An atom at position r in the excited state |e then emits a photon into the free-space radiation fieldÊ(r) = kâ k u k (r), with modes {u k (r)} forming a complete basis, and decays to the ground state |g . The Hamiltonian of the system is where the first term on the r.h.s. is the Hamiltonian for the field modes with energies ω k , the second term corresponds to the Bohr energies ω μ of the atomic levels |μ (μ = g, s, e), the third term describes the interaction of the atoms with the coupling laser with frequency ω c and wavevector k c ẑ, and the last term describes the coupling of the atoms to the free-space radiation field with the dipole moment ℘ eg on the transition |e → |g . We set the energy of the ground state to zero, ω g = 0, and assume that ω s ω e (ω c ω e ). The state vector of the system with the single atomic or photonic excitation can be expanded as where |s j ≡ |g 1 , g 2 , . . . , s j , . . . , g N , |e j ≡ |g 1 , g 2 , . . . , e j , . . . , g N , |G ≡ |g 1 , g 2 , . . . , g N , and |1 k ≡â † k |0 is the state of the radiation field with a single photon in mode k. The state vector is normalized as | = j (|c j | 2 + |b j | 2 ) + k |a k | 2 = 1, and it evolves according to the Schrödinger equation ∂ t | = -i H| , leading to a set of equations for the atomic amplitudes ( 2 a ) with c = ω cω e , and an equation for the field amplitudes cast in the integral form where g k (r j ) = ℘ eg ·u k (r j ) .

Atoms
We substitute Eq. (3) into Eq. (2b) and use the basis of the plane waves u k (r) = ε k,σ ω k 2 0 V e ik·r within the quantization volume V . We then replace the summation over the modes by an integration, and use the Born-Markov approximation to eliminate the radiation field [13,14], while neglecting the field-mediated interatomic interactions assuming sufficiently large mean interatomic distance k |r j -r i | 1 [15][16][17]. We thus obtain the usual spontaneous decay rate = 1 4π 0 4k 3 e |℘ eg | 2 3 and the Lamb shift of level |e that can be incorporated into ω e . Equations (2a) and (2b) reduce to ( 4 a ) We next assume a resonant laser c = 0 with sufficiently weak Rabi frequency | | and set ∂ t b j = 0, obtaining b j i /2 e ik c ·r j c j . Substituting this into Eq. (4a) and performing the integration, we finally obtain with the initial condition b j (0) = 0 and c j (0) = e iφ j √ N ∀j ∈ {1, N}, where we included a spatial phase φ j of the single-excitation spin wave. In Sect. 3 we will analyze the influence of this phase on the photon collection into Gaussian optical modes.

Field
We may define the wavefunction of the emitted single-photon field via E(r, t) ≡ 0| G|Ê(r)| (t) [13,14]. Assuming an isotropic dipole moment, in Appendix A we show , and for an atomic ensemble with density ρ(r), the emitted field is

Paraxial optics
We assume that N 1 atoms in a harmonic trap have a cylindrically symmetric, normal density distribution with the width σ ⊥ and length σ z (standard deviations). Consistent with the assumption of closely spaced energy levels |g and |s (ω c ω e ), we have k c k e such that |k ck e | 1/σ z . The radiation emitted in the phase-matched direction k e k c ẑ is collected by paraxial optics which feeds it into a waveguide with a single transverse mode and a continuum of 1D modes k, see Fig. 1(b). The atomic ensemble is placed at the focus of the collecting optics with the beam waist w 0 . The corresponding field modes that couple to the waveguide are then the Gaussian modes where A = πw 2 0 /2 is the cross-section at the focus z = 0 and L is the quantization length, ζ k = kw 2 0 /2 is the Rayleigh length and q k (z) = z + iζ k is the complex beam parameter, w(z) = w 0 1 + z 2 /ζ 2 k is the transverse beam size at position z, R(z) = z + ζ 2 k /z is the radius of curvature of the phase front, and φ Gouy = arctan(z/ζ k e ) is the Gouy phase.
Our aim is to maximize the collection of radiation emitted by the atoms by the paraxial optical elements. If we, however, attempt to calculate the overlap between the singlephoton field of Eq. (6) and the resonant k = k e Gaussian mode of Eq. (8a)-(8b) as a volume integral | d 3 r v * k e (r)E(r)| 2 , it will diverge inside the atomic ensemble, where ρ(r ) = 0, due to the |r -r | -1 term.

Photon number in the forward Gaussian modes
An alternative, more tractable approach that we use here is to calculate directly the total number of photons n(t) emitted into the 1D continuum {k} of the forward Gaussian modes v k (r) all having the same waist w 0 at z = 0. We have where g k (r j ) = ℘ eg v k (r j ) is now the coupling strength of atom j at position r j to the k's Gaussian mode of Eq. (8a)-(8b).
In Appendix B we show that for a single atom placed at the origin, r j = 0, the amount of radiation collected by the Gaussian paraxial optics, is proportional to the ratio of the atomic absorption cross-section ς = 3λ 2 e /2π to the crosssection A = πw 2 0 /2 of the focused Gaussian beam at the atomic position. For many atoms N with the density distribution of Eq. (7), we have Substituting here Eq. (5) with c r (0) = e iφ(r) / √ N , where φ(r) is a spatial phase profile of the stored spin-wave, we have We can proceed along the Weisskopf-Wigner approximation [13,17] by replacing ω k and k by their resonant values ω e and k e and pulling them out of the frequency integral, which, upon extending the lower limit of integration, reduces to 2πδ(tt ). We then obtain where B(t) ≡ t 0 dt β 2 (t ) contains the time-dependence of the photon envelope, while the overlap integral is where we used that k e k c (k c ẑ). Note that in Eq. (13) the geometric factor is Equations (13)- (14) are the central equations of this paper. Our aim is to find the optimal value for the Gaussian beam waist w 0 that maximizes the probability of collecting the photon n(∞) ∝ GN emitted by an ensemble of N trapped atoms with the given width and length σ ⊥ , σ z (standard deviations) of the density distribution.

Optimal photon collection
We now present the results of our calculations for three different cases of the initial spatial phase distribution in the atomic cloud centered at the beam focus.

Uniform spatial phase
Assume that the initial atomic spin-wave has a uniform spatial phase φ(r) = 0 ∀r.
Consider first the case of the atom cloud with the spatial dimensions much smaller than the Rayleigh length, σ ⊥ , σ z ζ k e . The mode function appearing in Eq. (14) can then be approximated as exp -ik e x 2 + y 2 2q k e (z) exp - where we have used that near the focus the beam waist is w 0 , the beam radius of curvature R tends to infinity, while the Gouy phase can be linearized as φ Gouy z/ζ k e . Equation (14) reduces to a Gaussian integral and we obtain where the dimensionless quantities are defined asσ z,⊥ ≡ k e σ z,⊥ andw 0 ≡ k e w 0 . The optimal beam waist that maximizes the geometric factor G = 6|ξ | 2 /w 2 0 is In the limit of σ z → 0, the photon collection efficiency is maximized for w max 0 = √ 2σ ⊥ , with |ξ | 2 → 1/4 and thus G 3/4σ 2 ⊥ . This is of course an intuitive result meaning that the waist of the Gaussian beam should be equal to the width of the atomic ensemble. As will be seen below, this result also applies to the atomic ensembles with sufficiently large width, which is, however, not optimal for photon collection by the paraxial Gaussian optics.
When the atomic cloud is larger than the Rayleigh length, the integral of Eq. (14) can be evaluated as In Fig. 2(a1), we show the optimal values ofw max 0 that maximize G for various values of σ z ,σ ⊥ . The corresponding maxima for G are shown in Fig. 2(b1). We notice that for a wide range of values ofσ z ,σ ⊥ the optimal waist of the Gaussian beam is againw max 0 √ 2σ ⊥ and the corresponding maximum for the geometric factor is |ξ | 2 1/4 and thus G = 6|ξ | 2 /(w max 0 ) 2 3/4σ 2 ⊥ . Only for sufficiently narrow and not too long ensembles,σ ⊥ √ 2σ ⊥ we also have |ξ | 2 1/4 and thus G 3/4σ 2 ⊥ 10 (σ ⊥ ∼ λ e ) andσ z 200 (σ z 30λ e ), we obtain a significantly better photon collection efficiency n(∞) ∝ GN 5 (assuming N = 10 3 atoms) into the appropriately focused (w 0 = w max 0 ) Gaussian beams.

Spatial phase compensating the Gouy phase
Recall that the Gouy phase φ Gouy (z) = arctan(z/ζ k e ) of the Gaussian beam varies with z. We may therefore further improve the photon collection efficiency if we imprint onto the stored atomic spin-wave a spatial phase that would compensate the Gouy phase, This can be achieved, e.g., by using spatially varying electric or magnetic fields or ac Stark shifts induced by off-resonant lasers [18]. Now the integral of Eq. (14) can be evaluated as Alternatively, we can write We numerically maximize |ξ | 2 /(w 0 ) 2 for different values ofσ z ,σ ⊥ , and show the resultinḡ w max 0 in Fig. 2(a2) and the corresponding maxima for G in Fig. 2(b2). Note that while the overall behavior of the geometric factor is similar to the previous case of uniform spatial phase, we nevertheless obtain good photon collection efficiency n(∞) ∝ GN 5 (N = 10 3 ) even for longer atomic cloudsσ z 300 (σ z 50λ e ) since we compensate for the Gouy phase that changes the sign across the focal point z = 0. In other words, when a photon emitted from a highly elongated atomic ensemble, σ z ζ k (σ z w 2 0 /2), with a uniform spatial phase φ(r) = 0 is collected into a focused Gaussian mode, the photon amplitudes originating from the regions of z < 0 and z > 0 interfere partially destructively, while for the spatial phase φ(r) = -φ Gouy (z) this interference is constructive.

Full spatial phase compensation of a Gaussian beam
We finally examine the case of the spatial phase of the atomic spin wave having the same spatial dependence as that of a focused Gaussian beam, which would correspond to a photon from a Gaussian mode stored in an atomic ensemble, e.g., via stopping a dark-state polariton in the EIT regime [3]. For the integral of Eq. (14) we then obtain (24) For a small atomic cloud, w(z) w 0 , we have the analytic result |ξ | 2 = w 4 0 /(w 2 0 + 2σ 2 ⊥ ) 2 , which is the same as in Eq. (17) in the limit of σ z → 0. For larger cloud sizes, we again maximize numerically |ξ | 2 /w 2 0 for different values ofσ z ,σ ⊥ , obtainingw max 0 and the corresponding maxima for G which are shown, respectively, in Fig. 2(a3) and (b3). Even though w max 0 √ 2σ ⊥ for even smaller values ofσ ⊥ as compared to case 2 above, the maximum for the photon collection efficiency n(∞) ∝ GN has nearly the same dependence onσ z , σ ⊥ as that with only the Gouy phase compensation. The physical reason for this result is that the curvature of the phase front of a Gaussian mode plays only a minor role in narrow atomic ensembles yielding the largest photon collection efficiency. where ς = 3λ 2 e /2π is the resonant absorption cross section of the atom. This is a very simple and intuitive result: For an initially excited atom, |b(t )| 2 = et , we have that n(t → ∞) = ς/(πw 2 0 ), i.e., the amount of radiation collected by the Gaussian paraxial optics is proportional to the ratio of the atomic absorption cross-section to the cross-section of the focused Gaussian beam at the atomic position.

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