Microwave to optical conversion with atoms on a superconducting chip
aa r X i v : . [ qu a n t - ph ] J u l Microwave to optical conversion with atoms on a superconducting chip
David Petrosyan,
Klaus Mølmer, J´ozsef Fort´agh, and Mark Saffman Institute of Electronic Structure and Laser, FORTH, GR-71110 Heraklion, Crete, Greece Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA (Dated: July 25, 2019)We describe a scheme to coherently convert a microwave photon of a superconducting co-planarwaveguide resonator to an optical photon emitted into a well-defined temporal and spatial mode.The conversion is realized by a cold atomic ensemble trapped close the surface of the superconductingatom chip, near the antinode of the microwave cavity. The microwave photon couples to a strongRydberg transition of the atoms that are also driven by a pair of laser fields with appropriatefrequencies and wavevectors for an efficient wave-mixing process. With only several thousand atomsin an ensemble of moderate density, the microwave photon can be completely converted into anoptical photon emitted with high probability into the phase matched direction and, e.g., fed intoa fiber waveguide. This scheme operates in a free-space configuration, without requiring strongcoupling of the atoms to a resonant optical cavity.
I. INTRODUCTION
Superconducting quantum circuits, which operate inthe microwave frequency range, are promising systemsfor quantum information processing [1, 2], as attested bythe immense recent interest of academia and industry.On the other hand, photons in the optical and telecom-munication frequency range are the best and fastest car-riers of quantum information over long distances [3, 4].Hence there is an urgent need for efficient, coherent andreversible conversion between microwave and optical sig-nals at the single quantum level [5]. Here we describesuch a scheme, which is compatible with both super-conducting quantum information processing and opticalquantum communication technologies.Previous work on the microwave to optical conversionincludes studies of optically active dopants in solids [6, 7],as well as electro-optical [8] and opto-mechanical [9] sys-tems. Cold atomic systems, however, have unique advan-tages over the other approaches. Atomic (spin) ensem-bles can couple to superconducting microwave resonatorsto realize quantum memory in the long-lived hyperfinemanifold of levels [10, 11]. Using stimulated Raman tech-niques [12, 13], such spin-wave excitations stored in thehyperfine transition can be reversibly converted into op-tical photons. Here we propose and analyze an efficientwave-mixing scheme for microwave to optical conversionon a integrated superconducting atom chip. In our setup,the microwave photon is confined in a coplanar waveg-uide resonator, while a cold atomic ensemble is trappednear the antinode of the microwave cavity mode at a dis-tance of several tens of microns from the surface of theatom chip. We employ a Rydberg transition between theatomic states that strongly couple to the microwave cav-ity field [14–17]. The coupling strength of the atoms tothe evanescent field of the cavity depends on the atomicposition, while the proximity of the atoms to the chipsurface leads to inhomogeneous Rydberg level shifts andthereby position-dependent detuning of the atomic reso- nance. This reduces the effective number of atoms par-ticipating in four-wave mixing in the presence of a pair oflaser fields with appropriate frequencies and wavevectors.Nevertheless, we show that high-efficiency conversion ofa microwave photon to an optical photon emitted into awell-defined spatial and temporal mode is still possiblein this setup. The coplanar waveguide resonator can alsocontain superconducting qubits, and hence our schemecan serve to interface them with optical photons.We note a related work [18] on microwave to opti-cal conversion using free-space six-wave mixing involv-ing Rydberg states. The achieved photon conversion ef-ficiency was, however, low, as only a small portion ofthe free-space microwave field could interact with the ac-tive atomic medium. Confining the microwave field in acavity would be a valuable route to enhance the conver-sion efficiency. A microwave to optical conversion schemeusing a single (Cs) atom that interacts with a supercon-ducting microwave resonator on the Rydberg transitionand with an optical cavity on an optical transition wasdiscussed in [19]. The advantage of the single atom ap-proach is that it requires moderate laser power for atomtrapping and leads to less light scattering and pertur-bation of the superconducting resonator. It relies, how-ever, on the technically demanding strong coupling ofthe single atom to both microwave and optical cavities.Reference [20] discusses the conversion of a microwavephoton to an optical telecommunication (E-band) pho-ton employing four-wave mixing in a small ensemble of(Yb) atoms in a copper microwave resonator and a high-finesse optical cavity. In contrast, our present approachuses a large ensemble of atoms with collectively enhancedcoupling to the microwave cavity and it leads to a co-herent, directional emission of the optical photon evenwithout an optical cavity. In a previous publication [21],we have employed a similar scheme to deterministicallyproduce single photons from a Rydberg excitation of asingle source atom coupled to the atomic ensemble viaresonant dipole-dipole interaction. ge xy z i s δ s k c η c − k ΩΩΩ p dd Ω k ˆ E ˆ E d pp ∆ FIG. 1. Schematics of the system: An ensemble of atomstrapped on a superconducting chip near a coplanar waveg-uide cavity converts the microwave photon of the cavity toan optical photon fed into a fiber waveguide, as shown in thelower part of the figure. The inset shows the atomic levelscheme. All the atoms are initially in the ground state | g i .A laser pulse couples | g i to the intermediate Rydberg state | i i with the Rabi frequency Ω p and detuning ∆ p ≃ ∆. Themicrowave cavity mode ˆ c is coupled non-resonantly to theRydberg transition | i i → | s i of the atoms with a position-dependent coupling strength η and detuning ∆ c ≃ − ∆. Withlarge one-photon detunings | ∆ p,c | ≫ | Ω p | , η , the two-photontransition | g i → | s i to the Rydberg state | s i is detuned by δ s = ∆ p + ∆ c . A strong laser field Ω d drives the transitionfrom | s i to the electronically excited state | e i that rapidlydecays with rate Γ e > Ω d to the ground state | g i and emitsa photon E predominantly into the phase-matched directiondetermined by the wave vector k = k p − k d . Our setup is primarily intended for optical communica-tion between microwave operated quantum sub-registers.As such, we consider the case of at most one microwavephoton encoding a qubit state at a time. The conversionof a microwave photon is accompanied by a Rydberg ex-citation of the atomic ensemble. But since at most onlya single atom is excited to the Rydberg state, the inter-atomic interactions and the resulting Rydberg excitationblockade [22, 23] do not play a role in our scheme, ir-respective of whether the atomic ensemble is larger ornot than any (irrelevant) blockade distance. This allowsus to restrict the analysis to the linear regime of conver-sion, greatly simplifying the corresponding calculationspresented below.
II. THE SYSTEM
Consider the system shown schematically in Fig. 1. Anintegrated superconducting atom chip incorporates a mi-crowave resonator, possibly containing superconductingqubits, and wires for magnetic trapping of the atoms.An ensemble of N ≫ | g i , a lower electronically excited state | e i and a pair of highly-excited Rydberg states | i i and | s i (see the inset of Fig. 1). A laser field of frequency ω p couples the ground state | g i to the Rydberg state | i i with time-dependent Rabi frequency Ω p and large de-tuning ∆ p ≡ ω p − ω ig ≫ | Ω p | . The atoms interactnon-resonantly with the microwave cavity mode ˆ c onthe strong dipole-allowed transition between the Rydbergstates | i i and | s i . The corresponding coupling strength(vacuum Rabi frequency) η = ( ℘ si / ¯ h ) ε c u ( r ) is propor-tional to the dipole moment ℘ si of the atomic transition,the field per photon ε c in the cavity, and the cavity modefunction u ( r ) at the atomic position r . The Rydbergtransition is detuned from the cavity mode resonance by∆ c ≡ ω c − ω si , | ∆ c | ≫ η . A strong driving field of fre-quency ω d acts on the transition from the Rydberg state | s i to the lower excited state | e i with Rabi frequency Ω d and detuning ∆ d = ω d − ω se . The transition from theexcited state | e i to the ground state | g i is coupled withstrengths g k ,σ to the free-space quantized radiation fieldmodes ˆ a k ,σ characterized by the wave vectors k , polar-ization σ and frequencies ω k = ck .In the frame rotating with the frequencies of all thefields, ω p , ω c , ω d , and ω k , dropping for simplicity thepolarization index, the Hamiltonian for the system reads H/ ¯ h = − N X j =1 h ∆ ( j ) p ˆ σ ( j ) ii + δ ( j ) s ˆ σ ( j ) ss + δ e ˆ σ ( j ) ee + (cid:16) Ω p e i k p · r j ˆ σ ( j ) ig − η ( r j )ˆ c ˆ σ ( j ) si + Ω d e i k d · r j ˆ σ ( j ) se + X k g k ˆ a k e i k · r j e − i ( ω k − ω eg ) t ˆ σ ( j ) eg + H . c . (cid:17)i , (1)where index j enumerates the atoms at positions r j ,ˆ σ ( j ) µν ≡ | µ i j h ν | are the atomic projection ( µ = ν ) or tran-sition ( µ = ν ) operators, k p and k d are the wave vectorsof the corresponding laser fields, δ ( j ) s ≡ ∆ ( j ) p + ∆ ( j ) c = ω p + ω c − ω ( j ) sg is the two-photon detuning of level | s i ,and δ e ≡ δ ( j ) s − ∆ ( j ) d = ω p + ω c − ω d − ω eg is the three-photon detuning of | e i . The energies of the Rydberg lev-els | i i , | s i , and thereby the corresponding detunings ∆ ( j ) p,c and δ ( j ) s , depend on the atomic distance ( x − x j ) fromthe chip surface at x , which may contain atomic adsor-bates producing an inhomogeneous electric field [24, 25].We neglect the level shift of the lower state | e i , since itis typically less sensitive to the electric fields and has alarge width Γ e (see below).We assume that initially all the atoms are preparedin the ground state, | G i ≡ | g , g , . . . , g N i , the mi-crowave cavity contains a single photon, | c i , and allthe free-space optical modes are empty, | i . We can ex-pand the state vector of the combined system as | Ψ i = b | G i ⊗ | c i ⊗ | i + P Nj =1 d j e i k p · r j | i j i ⊗ | c i ⊗ | i + P Nj =1 c j e i k p · r j | s j i⊗ | c i⊗ | i + P Nj =1 b j e i ( k p − k d ) · r j | e j i⊗| c i ⊗ | i + | G i ⊗ | c i ⊗ P k a k | k i , where | µ j i ≡| g , g , . . . , µ j , . . . , g N i denote the single excitation states, µ = i, s, e , and | k i ≡ ˆ a † k | i denotes the state of the radi-ation field with one photon in mode k . The evolution ofthe state vector is governed by the Schr¨odinger equation ∂ t | Ψ i = − i ¯ h H | Ψ i with the Hamiltonian (1), which leadsto the system of coupled equations for the slowly-varyingin space atomic amplitudes, ∂ t b = i N X j =1 Ω ∗ p d j , (2a) ∂ t d j = i ∆ ( j ) p d j + i Ω p b − iη ∗ ( r j ) c j , (2b) ∂ t c j = iδ ( j ) s c j − iη ( r j ) d j + i Ω d b j , (2c) ∂ t b j = iδ e b j + i Ω ∗ d c j + i X k g k e i ( k − k p + k d ) · r j a k e − i ( ω k − ω eg ) t , (2d)while the equation for the optical photon amplitudeswritten in the integral form is a k ( t ) = ig ∗ k X j e i ( k p − k d − k ) · r j Z t dt ′ b j ( t ′ ) e i ( ω k − ω eg ) t ′ . (3)The initial conditions for Eqs. (2), (3) are b (0) = 1, b j (0) , c j (0) , d j (0) = 0 ∀ j , and a k (0) = 0 ∀ k .We substitute Eq. (3) into the equation for atomic am-plitudes b j , assuming they vary slowly in time, and ob-tain the usual spontaneous decay of the atomic state | e i with rate Γ e and the Lamb shift that can be incorporatedinto ω eg [26]. We neglect the field–mediated interactions(multiple scattering) between the atoms [27–29], assum-ing random atomic positions and sufficiently large meaninteratomic distance ¯ r ij > ∼ λ/ π . To avoid the atomicexcitation in the absence of a microwave photon in thecavity, we assume that the intermediate Rydberg level | i i is strongly detuned, ∆ ( j ) p ≃ − ∆ ( j ) c ≫ | Ω p | , η, | δ ( j ) s | for allatoms in the ensemble. In addition, we assume that thevariation of ∆ ( j ) p (∆ ( j ) c ) across the atomic cloud is smallcompared to its mean value ∆ ( − ∆), which presumessmall enough Rydberg levels shifts in the inhomogeneouselectric field. We can then adiabatically eliminate theintermediate Rydberg level | i i , obtaining finally ∂ t b = i N X j =1 ˜ η j c j , (4a) ∂ t c j = ( i ˜ δ ( j ) s − Γ s / c j + i ˜ η j b + i Ω d b j , (4b) ∂ t b j = ( i ˜ δ e − Γ e / b j + i Ω ∗ d c j , (4c)where ˜ η j ≡ η ( r j )Ω p ∆ (cid:2) δ ( j ) s (cid:3) is the second-order couplingbetween | g j i ⊗ | c i and | s j i ⊗ | c i , while the second-order level shifts of | g j i and | s j i are incorporated into thedetunings ˜ δ ( j ) s ≡ δ ( j ) s + | Ω p | −| η ( r j ) | ∆ and ˜ δ e = δ e + | Ω p | ∆ .We have also included the typically slow decay Γ s of state | s i corresponding to the loss of Rydberg atoms [16, 30].Before presenting the results of numerical simulations,we can derive an approximate analytic solution of theabove equations and discuss its implications. We takea time-dependent pump field Ω p ( t ) (and thereby ˜ η j ( t ))and a constant driving field Ω d < Γ e /
2, which resultsin an effective broadening of the Rydberg state | s i by γ = | Ω d | Γ e / . Assuming γ ≫ Γ s / , ˜ δ e , we then obtain b j ( t ) = − γ Ω d ˜ η j ( t ) γ − i ˜ δ ( j ) s b ( t ) , (5a) b ( t ) = b (0) exp − Z t dt ′ N X j =1 | ˜ η j ( t ′ ) | γ − i ˜ δ ( j ) s . (5b)Substituting these into Eq. (3) and separating the tem-poral and spatial dependence, we obtain a k ( t ) = − i γ Ω d A k ( t ) × B k , (6)where A k ( t ) = Z t dt ′ Ω p ( t ′ ) e i ( ω k − ω eg ) t ′ e − β R t ′ dt ′′ | Ω p ( t ′′ ) | , (7a) B k = g ∗ k ∆ N X j =1 η ( r j ) γ − i ˜ δ ( j ) s e i ( k p − k d − k ) · r j , (7b)with β = P Nj =1 | η ( r j ) | γ − i ˜ δ ( j ) s .Equation (7a) shows that for a sufficiently smooth en-velope of the pump field Ω p ( t ), the optical photon is emit-ted within a narrow bandwidth β | Ω p | around frequency ω k = ω eg , which is a manifestation of the energy con-servation. The temporal profile of the photon field atthis frequency is ǫ ( t ) = ∂ t A k ( t ) = Ω p ( t ) e − β R t dt ′ | Ω p ( t ′ ) | ,where k = ω eg /c . The envelope of the emitted ra-diation can be tailored to the desired profile ǫ ( t ) byshaping the pump pulse according to Ω p ( t ) = ǫ ( t ) (cid:2) − β R t dt ′ | ǫ ( t ′ ) | (cid:3) − / [31], which can facilitate the photontransmission and its coherent re-absorption in a reverseprocess at a distant location [32–34]. Neglecting the pho-ton dispersion during the propagation from the sendingto the receiving node, and assuming the same or similarphysical setup at the receiving node containing an atomicensemble driven by a constant field Ω d , the complete con-version of the incoming optical photon to the cavity mi-crowave photon is achieved by using the receiving laserpulse of the shape Ω p ( t ) = − ǫ ( t ) (cid:2) β R t dt ′ | ǫ ( t ′ ) | (cid:3) − / [31].The spatial profile of the emitted radiation in Eq. (7b)is determined by the geometry of the atomic cloud,the excitation amplitudes of the atoms at differentpositions, and the phase matching. We assume anatomic cloud with normal density distribution ρ ( r ) = ρ e − x / σ x − y / σ y − z / σ z in an elongated harmonic trap, σ z > σ x,y . To maximize the resonant emission at fre-quency ω k = c | k p − k d | = ω eg into the phase matched di-rection k = k p − k d , we assume the (nearly) collinear ge-ometry k p , k d k e z . In an ideal case of all the atoms hav-ing the same excitation amplitude b j ∝ / √ N ∀ j , andhence B k ∝ R d rρ ( r ) e i ( k p − k d − k ) · r , the photon would beemitted predominantly into an (elliptic) Gaussian mode E ( r ) ∝ P | k | = k B k e i k · r with the waists w x, y = 2 σ x,y ,namely E ( x, y, z ) = (cid:18) πw x w y (cid:19) / e ik ( z + x / q ∗ x + y / q ∗ y ) , (8)where w x,y = w x, y h (cid:0) zζ x,y (cid:1) i / and q x,y = z − iζ x,y with ζ x,y = πw x, y λ . The corresponding angular spread(divergence) of the beam is ∆ θ x,y = λ πw x, y = k σ x,y ,which spans the solid angle ∆Ω = π ∆ θ x ∆ θ y . The proba-bility of the phase-matched, cooperative photon emissioninto this solid angle is P ∆Ω ∝ N ∆Ω, while the probabil-ity of spontaneous, uncorrelated photon emission into arandom direction is P π ∝ π . With P ∆Ω + P π = 1,we obtain P ∆Ω = N ∆Ω N ∆Ω+4 π which approaches unity for N ∆Ω ≫ π or N ≫ k σ x σ y [35]. Hence, for the prod-uct N ∆Ω, and thereby P ∆Ω , to be large, we should takean elongated atomic cloud with large σ z (to have manyatoms N at a given atom density) and small σ x , σ y (tohave large solid angle ∆Ω).In our case, however, not all the atoms participateequally in the photon emission, since the atomic am-plitudes b j ∝ η ( r j ) γ − i ˜ δ ( j ) s depend strongly on the distance( x − x j ) from the chip surface via both the atom-cavitycoupling strength η ( r j ) ≃ η e − ( x − x j ) /D , and, more sen-sitively, the Rydberg state detuning ˜ δ ( j ) s ≃ αx j (see be-low). This detuning results in a phase gradient for theatomic amplitudes in the x direction, which will lead toa small inclination of k with respect to k p − k d in the x − z plane. More importantly, for strongly varying de-tuning, α > γ/σ x , only the atoms within a finite-widthlayer ∆ x < σ x are significantly excited to contributeto the photon emission. This reduces the cooperativ-ity via N → ξN with the effective participation frac-tion ξ ≃ ∆ xσ x <
1, but also leads to larger divergence∆ θ x ≃ k ∆ x in the x − z plane. III. RESULTS
We have verified these arguments via exact numericalsimulations of the dynamics of the system. We place N ground state | g j i atoms in an elongated volume at ran-dom positions r j normally distributed around the ori-gin, x, y, z = 0, with standard deviations σ z ≫ σ x,y .With the peak density ρ = 2 . µ m − and σ x,y = 4 µ m, σ z = 24 µ m, we have N = 15000 atoms in the trap inter-acting with the co-planar waveguide resonator at posi-tion x ≃ µ m (see Fig. 1). Taking the strip-line length L = 10 . D = 10 µ m, the effective cavity volume is V c ≃ πD L [10] yielding the field per photon ε c = p ¯ hω c /ǫ V c ≃ . ω c / π = c/L √ ǫ r ≃
12 GHz ( ǫ r ≃ . u ( r ) ≃ e − ( x − x ) /D . The cavity field variesvery little along the longitudinal z direction of the atomiccloud since the cloud dimension σ z is much smaller thanthe wavelength of the microwave radiation λ c = L . Wechoose the Rydberg states | i i = | P / , m J = 1 / i and | s i = | S / , m J = 1 / i of Rb with the quantum de-fects δ P = 2 .
651 and δ S = 3 .
131 [36], leading to thetransition frequency ω si / π ≃ . ℘ si ≃ a e . This results in the vacuum Rabifrequency η (0) / π ≃
190 kHz at the cloud center r = 0.We take a sufficiently large intermediate state detun-ing ∆ / π ≃
10 MHz, and time-dependent pump fieldΩ p ( t ) = Ω (cid:2) (cid:0) t − t √ σ t (cid:1)(cid:3) of duration t end ≃ µ swith t = t end / σ t = t end / / π ≃
200 kHz (see Fig. 2(a)). The wavelength ofthe pump field is λ p ≃
297 nm corresponding to a single-photon transition from | g i = | S / , F = 2 , m F = 2 i to | i i ; alternatively, a three-photon transition betweenstates | g i and | i i via the intermediate 5 P / and 6 S / states is possible. The single photon transition | g i → | i i has a small dipole moment ℘ gi = 2 . × − a e , andthe required peak intensity of the UV field to attain theRabi frequency Ω is I = ǫ c h Ω ℘ gi = 650 W/cm , whichcan be delivered by a laser pulse of 1 . w p ≃ µ m. The two-photon detuning ˜ δ s of the Rydberg level | s i is taken to be zero at the cloudcenter r = 0 and it varies with the atomic position alongthe x axis as ˜ δ s ( x ) = αx with α = 2 π × . µ m − dueto the residual or uncompensated surface charges on theatom chip [37, 38]. The strong laser field with wavelength λ d = 480 nm is driving the transition from the Rydbergstate | s i to | e i = | P / , F = 2 , m F = 2 i with a con-stant Rabi frequency Ω d / π = 1 MHz. The calculateddipole moment for this transition is ℘ gi = 3 . × − a e and the required intensity of the driving field is I d =55 W/cm . The decay rates of states | s i ( | i i ) and | e i are Γ s ( i ) / π = 1 . e / π = 6 . e − from its peak value at thecloud center, which means that about 3000 photons willhit the chip surface above the atomic cloud. In addition,the atoms in the cloud will scatter the UV photons in all4 π direction with the rate N Γ i (Ω p / ∆) , but this leadsto only 3 × − scattered photons per pulse. Assum-ing the surface reflectivity of 0 .
999 (the field propagationdirection is parallel to the surface), we have only a fewabsorbed photons per pulse, which is negligible comparedto the cooling rate of the cryogenic environment. Similarestimates for the driving field show that only about 700photons will hit the surface of the atom chip, and lessthan one will be absorbed during the conversion cycle,while the scattering from the cloud is negligible since atmost only one atom can be excited to state | s i at a time. −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60Position z ( µ m) P z Time t ( µ s) | a k ( t ) | Ω p / π ( M H z ) −10 −5 0 5 10 y P y −10 −5 0 5 10 x P x t p e ( t ) (a)(b)(c) FIG. 2. (a) Rabi frequency of the pump field. (b) Probabil-ity | a k ( t ) | (unnormalized) of photon emission into the reso-nant ω k = ω eg , phase-matched k = k p − k d mode, obtainedfrom the exact numerical (black solid line) and analytical (reddashed line) solutions. The inset shows the total population p e ( t ) of the atoms in state | e i . (c) Spatial distribution of time-integrated photon emission probability (in any direction) P ,as obtained from a single realization of the atomic ensemble.Main panel shows P z along the z axis (integrated over the x and y direction), while the insets show P x and P y along x and y (black solid lines). For comparison, we show the Gaussians P z = √ πσ z e − z / σ z and P y = √ πσ y e − y / σ y for the z and y directions, and P x = N | η ( x ) | γ + δ s ( x ) e − x / σ x along x , with N the normalization (red dashed lines). In Fig. 2 we show the results of our numerical simula-tions of the dynamics of the system and compare themwith the analytical solutions. In the inset of Fig. 2(b)we show the time dependence of the total population p e ( t ) = P Nj =1 | b j ( t ) | of the atoms in the excited state | e i . As atoms decay from state | e i to the ground state | g i , they emit a photon with rate Γ e | b j ( t ) | . The spatialdistribution of time-integrated photon emission probabil-ity (in any direction) P ( r j ) = Γ e R t end | b j ( t ) | dt is shownin Fig. 2(c). This probability follows the Gaussian den-sity profile of the atoms along the y and z directions,but in the x direction it is modified by an approximateLorentzian factor | η ( x ) | γ + δ s ( x ) (if we neglect the x depen-dence of η ( x )) due to the position-dependent detuning δ s ( x ). Only part of the radiation is coherently emit-ted into the phase-matched direction k = k p − k d , withprobability | a k ( t ) | of photon emission into the resonant ω k = ω eg mode shown in Fig. 2(b). Note that for non-resonant modes ω k = ω eg with the rapidly oscillatingphase factor e i ( ω k − ω eg ) t ′ in Eq. (3) or (7a), the photonamplitude a k ( t ) ∝ P j b j ( t ) tends to zero at large times t end (as do b j ( t )’s), even for the phase-matched direction -0.1 0 0.1-0.1 0 0.1-0.1 0 0.1-0.1 0 0.1 0 0.5 1 φ θ /π x a k θ / π y (b) x θ = x a k θ / π y (a) θ =0 y θ /π xy θ z k FIG. 3. Angular probability distribution | a k | (unnormalized)of the photon emitted along the z direction, as a function of θ x = θ cos( φ ) and θ y = θ sin( φ ) with θ the polar and φ theazimuthal angles, as shown in middle top inset. Panel (a)corresponds to the case of the collinear geometry k p , k d k ˆ z ,with the radiation emitted at a small angle θ x = k ˆ z ≃ . π . Panel (b) shows the case with a small inclination k d ˆ z = 0 . π and k p k ˆ z , leading to k k ˆ z ( θ x = 0). Thered dashed lines in the insets of each density plot show theGaussian B ( θ x , θ x ; θ y , θ y ) of Eq. (9) with θ y = 0 (upperinsets) and θ x = 0 (right insets), while θ y = 0 and θ x as percases (a) and (b). k ≃ k p − k d .In Fig. 3 we show the angular probability distri-bution of the emitted photon. The beam divergence∆ θ x = k ∆ x ≃ . π in the x − z plane is almosttwice larger than that ∆ θ y = k σ y ≃ . π in the y − z plane, consistent with the narrower spatial distribution∆ x ≃ . µ m < σ x = 4 µ m of the atomic excitation (oremission) probability P ( r ), as discussed above. In thecollinear geometry, k p , k d k ˆ z , the radiation is emitted ata small angle θ x = k ˆ z ≃ . π due to the detuninginduced phase gradient of the atomic amplitudes b j along x . With a small angle k d k p = 0 . π between the driveand the pump fields, the latter still propagating along z ,we can compensate this phase gradient, resulting in thephoton emission along z ( θ x = 0). We may approxi-mate the angular profile of the emitted radiation with aGaussian function B k ∝ B ( θ x , θ x ; θ y , θ y ) = e − ( θ x − θ x ) / ∆ θ x e − ( θ y − θ y ) / ∆ θ y . (9)We then see from Fig. 3 that in the y − z plane the angularprofile corresponds to a Gaussian mode with θ y = 0,but in the x − z plane the angular profile deviates fromthe Gaussian, the more so for the case of the correctedemission angle θ x = 0. To fully collect this radiation,we thus need to engineer an elliptic lens with appropriatenon-circular curvature along the x direction.The total probability of radiation emitted into thefree-space spatial mode E ( r ) subtending the solid angle∆Ω = π ∆ θ x ∆ θ y is P ∆Ω ≃ .
74. This probability can beincreased by optimizing the geometry of the sample, e.g.,making it narrower and longer, as discussed above. Alter-natively, we can enhance the collection efficiency of thecoherently emitted radiation by surrounding the atomsby a moderate finesse, one-sided optical cavity. Assum-ing a resonant cavity with frequency ω k , mode function u o ( r ) and length L o , the overlap v = √ L o R d r E ( r ) u ∗ o ( r )determines the fraction of the radiation emitted by theatomic ensemble into the cavity mode, while the cavityfinesse F determines the number of round trips of the ra-diation, n ≃ F/ π , and thereby the number of times it in-teracts with the atoms, before it escapes the cavity. Theprobability of coherent emission of radiation by N atomsinto the cavity output mode is then P out ≃ | v | nN | v | nN +4 π . IV. CONCLUSIONS
We have proposed a scheme for coherent microwaveto optical conversion of a photon of a superconductingresonator using an ensemble of atoms trapped on a su-perconducting atom chip. The converted optical photonwith tailored temporal and spatial profiles can be fedinto a waveguide and sent to a distant location, wherethe reverse process in a compatible physical setup cancoherently convert it back into a microwave photon and,e.g., map it onto a superconducting qubit.In our scheme, the atoms collectively interact with the microwave cavity via a strong, dipole-allowed Rydbergtransition. We have considered the conversion of at mostone microwave photon to an optical photon, for whichthe interatomic Rydberg-Rydberg interactions are ab-sent. In the case of multiple photons, however, the long-range interatomic interactions will induce strong non-linearities accompanied by the suppression of multipleRydberg excitations within the blockade volume associ-ated with each photon [39, 40]. This can potentially hin-der the microwave photon conversion and optical photoncollection due to distortion of the temporal and spatialprofile of the emitted radiation.
ACKNOWLEDGMENTS
We acknowledge support by the US ARL-CDQI pro-gram through cooperative agreement W911NF-15-2-0061, and by the DFG SPP 1929 GiRyd and DFG ProjectNo. 394243350. D.P. is partially supported by theHELLAS-CH (MIS Grant No. 5002735), and is gratefulto the Aarhus Institute of Advanced Studies for hospi-tality, and to the Alexander von Humboldt Foundationfor additional support in the framework of the ResearchGroup Linkage Programme. [1] J. Clarke and F.K. Wilhelm,
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