Simulations of atomic trajectories near a dielectric surface
SSimulations of atomic tra jectories near a dielectricsurface
N. P. Stern , D. J. Alton , and H. J. Kimble Norman Bridge Laboratory of Physics 12-33, California Institute of Technology,Pasadena, California 91125, USAE-mail: [email protected]
Abstract.
We present a semiclassical model of an atom moving in the evanescentfield of a microtoroidal resonator. Atoms falling through whispering-gallery modes canachieve strong, coherent coupling with the cavity at distances of approximately 100nanometers from the surface; in this regime, surface-induced Casmir-Polder level shiftsbecome significant for atomic motion and detection. Atomic transit events detectedin recent experiments are analyzed with our simulation, which is extended to consideratom trapping in the evanescent field of a microtoroid.PACS numbers: 37.30.+i , 34.35.+a, 42.50.Ct, 37.10.Vz a r X i v : . [ qu a n t - ph ] A ug imulations of atomic trajectories near a dielectric surface
1. Introduction
Strong, coherent interactions between atoms and light are an attractive resource forstoring, manipulating, and retrieving quantum information in a quantum network withatoms serving as nodes for quantum processing and storage and with photons acting asa long-distance carrier for communication of quantum information [1]. One realizationof a quantum node is an optical cavity, where light-matter interactions are enhancedby confining optical fields to small mode volumes. In the canonical implementation,a Fabry-Perot resonator with intracavity trapped atoms enables a panoply of cavityquantum electrodynamics (cQED) phenomena using single photons and single atoms,and thereby, validates many aspects of a cQED quantum node [2, 3].Despite these achievements, high-quality Fabry-Perot mirror cavities typicallyrequire significant care to construct and complex experimental instrumentation tostabilize. These practical issues have begun to be addressed by atom chips [4, 5], in whichatoms are manipulated in integrated on-chip microcavity structures offering a scalableinterface between light and matter [6, 7, 8]. Owing to their high quality factors, low modevolumes, and efficient coupling to tapered optical fibers [9], microtoroidal resonators area promising example of microcavities well-suited for on-chip cQED with single atomsand single photons [10]. Strong coupling [11, 12] and non-classical regulation of opticalfields [13, 14] have been demonstrated with atoms and the whispering-gallery modes ofa silica microtoroidal resonator.In our experiments with microtoroids, Cs atoms are released from an optical trapand fall near a silica toroid, undergoing coherent interactions with cavity modes as eachatom individually transits through the evanescent field of the resonator. In the mostrecent work of [12], atom transits are triggered in real-time to enable measurementof the Rabi-split spectrum of a strongly-coupled cQED system. Whereas a singleatom is sufficient to modify the cavity dynamics, falling atoms are coupled to thecavity for only a few microseconds. Atom dropping experiments necessarily involvea large ensemble of individual atomic trajectories and represent, consequently, a farmore complex measurement result.Interactions between a neutral atom and a dielectric surface modify the radiativeenvironment of the atom resulting in an enhanced decay rate [15] and Casimir-Polder(CP) forces [16, 17]. These perturbative radiative surface interactions are usuallyinsignificant in cQED experiments with Fabry-Perot resonators where atoms are far frommirror surfaces, but in microcavity cQED, atoms are localized in evanescent fields withscale lengths λ/ π ∼
150 nm near a dielectric surface. The experimental conditions formicrotoroidal cQED with falling atoms in [12] necessarily involve significant CP forcesand level shifts while simultaneously addressing strong coupling to optical cavity modes.Theoretical analysis of this experiment requires addressing both the strong atom-cavityinteractions and atom interactions with the dielectric surface of the microtoroid. Asreported in [12], spectral and temporal measurements offer signatures of both strongcoupling to the cavity mode and the significant influence of surface interactions on imulations of atomic trajectories near a dielectric surface
2. Atoms in a microtoroidal cavity
We approach the motion of atoms moving under the influence of surface interactionsand coherent cavity dynamics with a semiclassical method to efficiently simulate a largenumber of atom trajectories. For surface interactions, dispersion forces are calculated perturbatively using the linear response functions of SiO and a multi-level atom. Fornearly-resonant non-perturbative coherent interactions between atom and cavity, theatomic internal state and the cavity field are treated quantum mechanically within thetwo-level and rotating-wave approximations.Simulations of atomic motion follow the semiclassical method detailed in [18].Mechanical effects of light are incorporated classically as a force (cid:126)F ( (cid:126)r ) on a point particleatom at location (cid:126)r . Trajectories (cid:126)r ( t ) are calculated with a Langevin equation approachto incorporate momentum diffusion from fluctuations. At each simulation time step t i ,the atomic velocity is calculated as: v i +1 j = v ij + F ij ∆ t/m Cs + (cid:113) D ijj ∆ t/m W ij (1)where (cid:126)v i is the velocity at the i time step, m Cs is the atomic mass, and ∆ t is thesimulation time step t i +1 − t i . The (cid:126)W i are normally distributed with zero mean andstandard deviation of 1. Given the force (cid:126)F and diffusion tensor D ij as discussed insection 3, the atom trajectory (cid:126)r ( t ) and cavity transmission and reflection coefficients, T ( t ) and R ( t ) are calculated. A single atom strongly coupled to the cavity mode has alarge effect on cavity fields and optical forces, requiring simultaneous solutions of atomicmotion and cQED dynamics.Full quantization of atomic motion leads to an unwieldy Hilbert space not conduciveto efficient simulation. In contrast, semiclassical methods are well-suited for simulatingatomic motion in experiments with falling atoms near resonators. The ratio of the recoil imulations of atomic trajectories near a dielectric surface S / → P / transition is less than 10 − . Further,the recoil velocity of ∼ . t and energy shifts from surface interactions as if atom the atom werestationary. The remainder of this section discusses the quantum mechanical equationsof motion for the atom and cavity fields in the low-probe intensity limit, to be followedlater by contributions to the force (cid:126)F used in (1). An idealized microtoroid has axial symmetry, so we work in a standard cylindricalcoordinate system (cid:126)r → ( ρ, φ, z ). The toroid is modeled as a circle of diameter D m with dielectric constant (cid:15) revolved around the z -axis to make a torus of major diameter D M (Fig. 1(a)). The toroid is therefore defined by its minor diameter D m and itsprincipal diameter D p = D M + D m . The fabrication and characterization of high-qualitymicrotoroids are described in detail elsewhere [9].The axisymmetric cavity modes of interest are whispering-gallery modes (WGM),which lie near the edge of the resonator surface and circulate in either a clockwiseor counter-clockwise direction. These modes are characterized by an azimuthal modenumber m , whose magnitude gives the periodicity around the toroid and whosesign indicates the direction of propagation. The WGMs for ± m are degenerate infrequency but travel around the toroid in opposite directions. The mode electricfields for the WGM traveling waves are written as (cid:126)E ( (cid:126)r ) = E max (cid:126)f ( ρ, z ) e imφ , where (cid:126)f ( ρ, z ) = (cid:126)E ( ρ, z ) /E max is the mode function in the ρ − z cross-section normalized bythe maximum electric field E max . In general, backscattering couples these two modesso that a more useful eigenbasis for the system consists of the normal, standing wavemodes characterized by a phase and the periodicity | m | . This backscattering coupling h is assumed to be real, with the phase absorbed into the definition of the origin ofthe coordinate φ . In addition, the mode’s field decays at a rate κ i through radiation,scattering, and absorption. In our simulations, a cavity mode is fully described by itsspatial mode function (cid:126)f ( (cid:126)r ), its azimuthal mode number m , its loss rate κ i , and thecoupling h to the counter-propagating mode with mode number − m .We model the microtoroid modes using a commercial finite-element softwarepackage (COMSOL) to solve numerically for the vector mode functions (cid:126)f ( ρ, z ) for modesof a given m [19]. Mode volumes are calculated from, V m = (cid:82) dV (cid:15) ( (cid:126)r ) | (cid:126)E ( (cid:126)r ) | E = 2 π (cid:90) dA (cid:15) ( ρ, z ) ρf ( ρ, z ) (2)In this notation [10], the coupling of a circularly polarized optical field to an atomic imulations of atomic trajectories near a dielectric surface g ( (cid:126)r ) = (cid:104) (cid:126)d · (cid:126)E (cid:105) = f ( ρ, z ) e imφ (cid:115) πc γω (0)a 2 V m (3)where (cid:126)d is the dipole operator and ω (0)a = 2 πc/λ is the vacuum transition frequency ofthe two-level atom with free-space wavelength λ . WGMs are predominantly linearlypolarized, and so we average over the dipole matrix elements to obtain an effectivetraveling wave coupling g tw which is approximately ∼ . D p , D m (cid:29) λ , the azimuthal component is small and we assume that theoptical field is linear outside of the toroid. Since the cavity losses are dominated byabsorption and defect scattering rather than the radiative lifetime set by the toroidgeometry [10], we let κ i and h be experimental parameters. Fig. 1 shows the lowest-order mode with m = 118 for a toroid with { D p , D m } = { , } µ m. The index m ischosen so that the cavity frequency ω c is near the 6 S / → P / transition of Cs with ω (0)a / π = 351 . m outside of the toroid is that of aGaussian wrapped around the toroid’s surface that decays exponentially with distancescale set by the free space wavevector 1 /λ = 2 π/λ , f ( ρ, z ) ∼ e − d/λ e − ( ψ/ψ ) (4)where d ( ρ, z ) = (cid:112) ( ρ − D M / + z − D m / ψ ( ρ, z ) = arctan zρ − D M / is the angle around the toroid cross-section ( ψ = 0 at z = 0),and ψ is a characteristic mode width (see Fig. 1a). Higher order angular modes arecharacterized by additional nodes along the coordinate ψ . We consider a quantum model of a two-level atom at position (cid:126)r ( t ) coupled to anaxisymmetric resonator shown schematically in Fig. 2. The terminology used herefollows the supplemental material of [11], [13], and [12], but the general formalismcan be found in additional sources (see [20], for example). As described in section 2.1,an axisymmetric resonator supports two degenerate counter-propagating whispering-gallery modes at resonance frequency ω c to which we associate the annihilation (creation)operators a and b ( a † and b † ). Each traveling-wave mode has an intrinsic loss rate, κ i ;the modes are coupled via scattering at rate h . External optical access to the cavity isprovided by a tapered fiber carrying input fields { a in , b in } at probe frequency ω p . Fiberfields couple to the cavity modes with an external coupling rate κ ex . The output fields of imulations of atomic trajectories near a dielectric surface (a) (b) ρ (µm) z ( µ m ) −2
10 μm D p D m ψ z ρ D M Figure 1. (a) A scanning electron microscope image of a microtoroid with definitionsof the relevant parameters discussed in the text. (b) The lowest order mode function f ( ρ, z ) of a toroid mode with { D p , D m } = { , } µ m and m = 118 and λ = 852 mn. the fiber taper in each direction are the coherent sum of the input field and the leakingcavity field, { a out , b out } = −{ a in , b in } + √ κ ex { a, b } [11, 13].We specialize to the situation of single-sided excitation, where (cid:104) b in (cid:105) = 0. Theinput field a in drives the a mode with strength ε p = i √ κ ex (cid:104) a in (cid:105) so that the incidentphoton flux is P in = (cid:104) a † in a in (cid:105) = | ε p | / κ ex . Experimentally accessible quantities arethe transmitted and reflected photon fluxes, P T = (cid:104) a † out a out (cid:105) and P R = (cid:104) b † out b out (cid:105) ,respectively. In experiments, data is typically presented as normalized transmissionand reflection coefficients defined as T = P T /P in and R = P R /P in . In the absence of anatom, the functions T and R for the bare cavity depend on the detuning ∆ cp = ω c − ω p and the cavity rates h , κ i , and κ ex . At critical coupling, κ ex = (cid:112) κ + h , the bare cavity T → cp = 0 [21].The cavity modes { a, b } both couple to a single two-level atom with transitionfrequency ω a at location (cid:126)r . In the context of cQED, the atomic system is described bya single transition with frequency ω a with the associated raising and lowering operators σ + and σ − and an excited state field decay rate γ . The atomic frequency ω a ( (cid:126)r ) maybe shifted from the free-space value ω (0)a by frequency δ a ( (cid:126)r ) due to interactions with thedielectric surface. The coupling of the traveling-wave modes { a, b } to the atomic dipoleis described by the single-photon coupling rate g tw ( (cid:126)r ) = g maxtw f ( ρ, z )e ± iθ , where f ( ρ, z )is the cavity mode function and θ = mφ . A discussion of f ( ρ, z ) for the modes of amicrotoroid appears in section 2.1. For an atom in motion, ω a ( (cid:126)r ), γ ( (cid:126)r ), and g tw ( (cid:126)r ) arespatially-dependent quantities that depend on the atomic position (cid:126)r ( t ).To study the atom-cavity dynamics, we write the standard Jaynes-Cummings-stylecQED Hamiltonian for coupled field modes [22, 11]: H/ (cid:126) = ω a ( (cid:126)r ) σ + σ − + ω c (cid:0) a † a + b † b (cid:1) + h (cid:0) a † b + b † a (cid:1) + (cid:0) ε ∗ p e iω p t a + ε p e − iω p t a † (cid:1) + (cid:0) g ∗ tw ( (cid:126)r ) a † σ − + g tw ( (cid:126)r ) σ + a (cid:1) + (cid:0) g tw ( (cid:126)r ) b † σ − + g ∗ tw ( (cid:126)r ) σ + b (cid:1) . (5) imulations of atomic trajectories near a dielectric surface a in b in ba b out a out κ ex h h g ω a ρ z φ γ κ i κ i (a) (b) −200 −100 0 100 200−200−1000100200 I m ( Λ / ħ ) ( � M H z ) ∆/2� (MHz) Λ Λ + Λ - Figure 2. (a) Schematic of the atom-toroid system. Coherent optical fields in thetapered fiber couple into whispering-gallery cavity modes of an axisymmetric resonator.These fields can couple to an atomic transition with rate g , scatter to the counter-propagating mode ( h ), escape to the environment ( κ i ), or couple in/out of the fiber( κ ex ). The atom is described as a two-level system with transition frequency ω a andspontaneous emission rate γ . (b) Imaginary part of the eigenvalues Λ i of the linearizedsystems as a function of detuning ∆ = ω c − ω (0)a for a Cs atom at φ = π/ g = 60 MHz critically coupled to a cavity with parameters { κ i , h } / π = { , } MHz(Eqs. (9 a )). Following the rotating-wave approximation, we write the Hamiltonian in a framerotating at ω p [11, 13, 20]: H/ (cid:126) = ∆ ap ( (cid:126)r ) σ + σ − + ∆ cp (cid:0) a † a + b † b (cid:1) + h (cid:0) a † b + b † a (cid:1) + ε ∗ p a + ε p a † + (cid:0) g ∗ tw ( (cid:126)r ) a † σ − + g tw ( (cid:126)r ) σ + a (cid:1) + (cid:0) g tw ( (cid:126)r ) b † σ − + g ∗ tw ( (cid:126)r ) σ + b (cid:1) , (6)where ∆ ap ( (cid:126)r ) = ω a ( (cid:126)r ) − ω p and ∆ cp = ω c − ω p . Dissipation from coupling to externalmodes is treated using the master equation for the density operator of the system ρ :˙ ρ = − i (cid:126) [ H, ρ ] + κ (cid:0) aρa † − a † aρ − ρa † a (cid:1) + κ (cid:0) bρb † − b † bρ − ρb † b (cid:1) + γ (cid:0) σ − ρσ + − σ + σ − ρ − ρσ + σ − (cid:1) (7)Here, κ = κ i + κ ex is the total field decay rate of each cavity mode, and 2 γ ( (cid:126)r ) is theatomic dipole spontaneous emission rate, which is orientation dependent near a dielectricsurface (section 4.1).The Hamiltonian (6) can be rewritten in a standing wave basis using normal modes A = ( a + b ) / √ B = ( a − b ) / √ H/ (cid:126) = ∆ ap ( (cid:126)r ) σ + σ − + (∆ cp + h ) A † A + (∆ cp − h ) B † B + (cid:0) ε ∗ p A + ε p A † (cid:1) / √ (cid:0) ε ∗ p B + ε p B † (cid:1) / √ g A ( (cid:126)r ) (cid:0) A † σ − + σ + A (cid:1) − ig B ( (cid:126)r ) (cid:0) B † σ − − σ + B (cid:1) , (8)where g A ( (cid:126)r ) = g max f ( ρ, z ) cos θ , g B ( (cid:126)r ) = g max f ( ρ, z ) sin θ , and g max = √ g maxtw . Inthe absence of atomic coupling ( g tw = 0), these normal modes are eigenstates of (6).With g tw (cid:54) = 0, the eigenstates of the Hamiltonian are dressed states of atom-cavity imulations of atomic trajectories near a dielectric surface h = 0 and g tw (cid:54) = 0, the atom defines a natural basis in which itcouples to only a single standing wave mode. For the modes { A, B } defined above,coupling may occur predominantly, or even exclusively, to one of the two normal modesdepending the azimuthal coordinate θ . For such θ , the system can be interpreted as anatom coupled to one normal mode in a traditional Jaynes-Cummings model with dressed-state splitting given by the single-photon Rabi frequency Ω (1) = 2 g ≡ g max f ( ρ, z ), alongwith a second complementary cavity mode uncoupled to the atom. Approximately for g tw (cid:29) h , this interpretation is consistent for any arbitrary atomic coordinate θ . For h (cid:54) = 0 and comparable to κ i with a fixed phase convention (such as Im( h ) = 0 usedhere), this decomposition is not possible for arbitrary atomic coordinate θ ; the atom ingeneral couples to both normal modes as a function of φ [11, 20].The master equation (7) can be numerically solved using a truncated numberstate basis for the cavity modes. Alternatively, the system is linearized by treatingthe atom operators σ ± as approximate bosonic harmonic oscillator operators with[ σ − , σ + ] ≈
1. For a sufficiently weak probe field, the atomic excited state populationis small enough that the oscillator has negligible population above the first excitedlevel and the harmonic approximation is quite good. As part of this linearization,we factor expectation values of normally-ordered operator products into productsof operator expectation values [18, 23]. Reducing operators to complex numberssuppresses coherence, but numerical calculations confirm that this approximation isaccurate when calculating cavity output fields and classical forces for the weak drivingpower levels considered here. In particular, experiments typically utilize a photon flux P T = (cid:104) a † out a out (cid:105) ∼
15 cts/ µ s corresponding to an average cavity photon population of (cid:104) a † a (cid:105) ∼ .
1. At these photon numbers, cavity expectation values effectively factorizesuch that (cid:104) a † a (cid:105) ≈ (cid:104) a † (cid:105)(cid:104) a (cid:105) for the semiclassical treatment used here [24]. We use thisapproximation to write P T = (cid:104) a † out a out (cid:105) ≈ (cid:104) a † out (cid:105)(cid:104) a out (cid:105) , implying that we only needthe complex number (cid:104) a out (cid:105) = −(cid:104) a in (cid:105) + √ κ ex (cid:104) a (cid:105) and its conjugate to calculate thecavity transmission at these photon numbers. This approximation is not sufficientfor calculation of the g (2) ( τ ) correlation function where the nonlinearities must beincluded [12].The relevant equations of motion for the field amplitudes of the linearized masterequation are, (cid:104) ˙ a (cid:105) = − ( κ + i ∆ cp ) (cid:104) a (cid:105) − ih (cid:104) b (cid:105) − iε p − ig ∗ tw (cid:104) σ − (cid:105) , (9 a ) (cid:104) ˙ b (cid:105) = − ( κ + i ∆ cp ) (cid:104) b (cid:105) − ih (cid:104) a (cid:105) − ig tw (cid:104) σ − (cid:105) , (9 b ) (cid:104) ˙ σ − (cid:105) = − ( γ + i ∆ ap ) (cid:104) σ − (cid:105) − ig tw (cid:104) a (cid:105) − ig ∗ tw (cid:104) b (cid:105) . (9 c )Time and spectral dependence of this system of equations are governed by itseigenvalues Λ i . The imaginary part of the eigenvalues as a function of detuning∆ ≡ ∆ cp − ∆ ap = ω c − ω a are illustrated in Fig. 2b. For large ∆ (cid:29) | g tw | , the threeeigenvalues include one atom-like eigenvalue and two cavity-like eigenvalues separated bythe mode splitting h . For intermediate ∆, there is an anti-crossing of two dressed-stateeigenvalues Λ ± , while the third (cavity-like) Λ is uncoupled to the atom. imulations of atomic trajectories near a dielectric surface (cid:126)r ( t ). Analytic steady-state solutionsto (9 a ) for (cid:104) a (cid:105) ss and (cid:104) b (cid:105) ss are: (cid:104) a (cid:105) ss = iε p ( γ + i ∆ ap ) (cid:2) ( κ + i ∆ cp ) ( γ + i ∆ ap ) + | g tw | (cid:3) [ ih ( γ + i ∆ ap ) + ( g ∗ tw ) ] (cid:2) ih ( γ + i ∆ ap ) + g (cid:3) − [( κ + i ∆ cp )( γ + i ∆ ap ) + | g tw | ] (10 a ) (cid:104) b (cid:105) ss = − ih ( γ + i ∆ ap ) + g ( κ + i ∆ cp )( γ + i ∆ ap ) + | g tw | (cid:104) a (cid:105) ss (10 b ) (cid:104) σ − (cid:105) ss = − i g tw (cid:104) a (cid:105) ss + g ∗ tw (cid:104) b (cid:105) ss γ + i ∆ ap . (10 c )
3. Optical forces on an atom in a cavity
Neutral atoms experience forces from the interaction of the atomic dipole moment withthe radiation field. These optical dipole forces have a quantum mechanical interpretationas coherent photon scattering [25, 26]. For a light field near resonance with the atomicdipole transition, these optical forces can be quite strong, even at the single photon level;cavity-enhanced dipole forces [18, 27] have been exploited to trap [28] and localize [29]a single atom with the force generated by a single strongly-coupled photon. In thissection, we discuss how the optical forces, their first-order velocity dependence, andtheir fluctuations are included in our semiclassical simulation.
In a quantum mechanical treatment of light-matter interactions [26], the eigenstatesof the system are dressed states of atom and optical field. The quantum mechanicaloptical force on the atom at location (cid:126)r can be found from the commutator of the atommomentum (cid:126)p with the interaction Hamiltonian H int consisting of the last two termsfrom the Hamiltonian (6): (cid:126)F = d(cid:126)pdt = i (cid:126) [ H int , (cid:126)p ] = − (cid:126) ∇ g ∗ tw ( (cid:126)r ) (cid:0) a † σ − + σ + b (cid:1) − (cid:126) ∇ g tw ( (cid:126)r ) (cid:0) σ + a + b † σ − (cid:1) (11)The gradient from the position space representation of the momentum operator (cid:126)p onlyacts on g tw ( (cid:126)r ) and not on the field operators [30, 31]. The steady-state expectationvalues of (11) give the dipole force on the atom in the semiclassical approximation: (cid:104) (cid:126)F (cid:105) ss = − (cid:126) ∇ g ∗ tw ( (cid:126)r ) (cid:0) (cid:104) a † (cid:105) ss (cid:104) σ − (cid:105) ss + (cid:104) σ + (cid:105) ss (cid:104) b (cid:105) ss (cid:1) − (cid:126) ∇ g tw ( (cid:126)r ) (cid:0) (cid:104) σ + (cid:105) ss (cid:104) a (cid:105) ss + (cid:104) b † (cid:105) ss (cid:104) σ − (cid:105) ss (cid:1) (12)As described in section 2, the steady-state operator expressions are simplified byreducing expectation values of operator products to products of linearized steady-stateoperator expectation values. Ignoring fiber and spontaneous emission losses, an effectiveconservative dipole potential U d can be defined by integration of (12). imulations of atomic trajectories near a dielectric surface Non-zero velocity effects on the force (12) are found by including a first-order velocitycorrection in the steady state expectation values [23, 25, 31]. Consider a vector ofoperators (cid:126)O whose expectation values obey a linearized equation system such as (9 a ).Assuming a small velocity, we expand the operator expectation values (cid:104) (cid:126)O (cid:105) as: (cid:104) (cid:126)O (cid:105) = (cid:104) (cid:126)O (cid:105) + (cid:104) (cid:126)O (cid:105) + . . . , (13)where the subscripts denote the order of the velocity v in each term. If an atom ismoving through these fields, then the cavity parameters depend in general on atomicposition (cid:126)r . As (cid:126)r changes in time, the fields must evolve in response. Consequently, thetime derivative of the expectation value evolves not only from explicit time dependence,but from atomic motion as well. (cid:104) ˙ (cid:126)O (cid:105) = (cid:18) ∂∂t + (cid:126)v · (cid:126) ∇ (cid:19) (cid:104) (cid:126)O (cid:105) (14)Setting the explicit time derivatives in (14) to zero, the perturbative expansion of thetime derivative can be equated to the original linearized equation system. Collectingterms of each order in velocity gives an equation for the first-order term (cid:104) (cid:126)O (cid:105) in termsof the zero-velocity steady-state solution (cid:104) (cid:126)O (cid:105) . This procedure requires the spatialderivative of the zero-order steady-state solutions, where spatial dependence entersthrough the atomic transition frequency ω a ( (cid:126)r ), the spontaneous emission rate γ ( (cid:126)r ), andthe atom-cavity coupling g ( (cid:126)r ). Only terms linear in velocity are kept in the operatorproducts of the force (cid:126)F ( (cid:126)r ) in (12).In practice, first-order velocity corrections are small in our simulation. For example,Doppler shifts arising from spatial derivatives of the cavity modes are on the order of (cid:126)k · (cid:126)v , where (cid:126)k is the mode wavevector. For typical azimuthal velocities of less than0.1 m/s, the Doppler shift is less than 1 MHz. The effect becomes more significant asatoms accelerate to high velocities near the surface, but atomic level shifts from surfaceinteractions are more significant in this regime than the Doppler shifts. Quantum fluctuations of optical forces are treated by adding a stochastic momentumdiffusion contribution to the atomic velocity in the Langevin equations of motion. Wecalculate the diffusion tensor components used in (1), D ii , using general expressions fordiffusion in an atom-cavity system generalized for the two-mode cavity of a toroid [32]:2 D ii = ( (cid:126) k ) γ (cid:12)(cid:12)(cid:10) σ − (cid:11) ss (cid:12)(cid:12) + (cid:12)(cid:12) (cid:126) ∇ i (cid:10) σ − (cid:11) ss (cid:12)(cid:12) γ + 2 κ (cid:0) | (cid:126) ∇ i (cid:104) a (cid:105) ss | + | (cid:126) ∇ i (cid:104) b (cid:105) ss | (cid:1) (15)for i = x, y, z , where γ is the atomic field spontaneous decay rate. The firstterm represents fluctuations from spontaneous emission, the second term describes afluctuating atomic dipole coupled to a cavity field, and the third represents a fluctuatingcavity field coupled to an atomic dipole. (15) is approximated using steady-state fieldscalculated from the linearized solutions to the master equation (10 a ). Although included imulations of atomic trajectories near a dielectric surface
4. Effects of surfaces on atoms near dielectrics
In the vicinity of a material surface, the mode structure of the full electromagneticfield is modified due to the dielectric properties of nearby objects. These off-resonantradiative interactions modify the dipole decay rate of atomic states and shift electronicenergy levels. This surface interaction varies spatially as the relative atom-surfaceconfiguration changes. The surface phenomena are dispersive and depend on the multi-level description of the atom’s electronic structure; they are calculated using traditionalperturbation theory with the full electromagnetic field without focusing on a few selectmodes enhanced by a cavity in cQED.
When a classical oscillating dipole is placed near a surface, its radiation pattern ismodified by the time-lagged reflected field from the dielectric surface. The spontaneousemission rate oscillates with distance d from the surface, which can be interpreted asinterference between the radiation field of the dipole and its reflection. The variation ofthe emission rate depends on whether the dipole vector is parallel or perpendicular tothe surface, as intuitively expected from the asymmetry of image dipole orientations ofdipoles aligned parallel and perpendicular to the surface normal. For either orientation,the spontaneous emission rate features a marked increase within a wavelength of thesurface due to surface evanescent modes that become available for decay for d (cid:46) λ .We calculate the surface-modified dipole decay rates γ ( (cid:107) )s ( d ) and γ ( ⊥ )s ( d ) for acesium atom near an SiO surface following the methods of Refs. [15, 33] (see Fig. 3).This calculation involves an integration of surface reflection coefficients over possiblewavevectors of radiated light. The integrand depends on the dielectric function of SiO evaluated at the frequency ω a of the atomic transition. The orientations refer to thealignment of a classical dipole relative to the surface plane. Radiative interactions with a surface are important components of motion for neutralatoms within a few hundred nm of a surface, with the potential for manipulating atomicmotion through attractive [16] or repulsive forces [34]. Depending on the theoreticalframework, these forces are naturally thought of as radiative self-interactions betweentwo polarizable objects, fluctuations of virtual electromagnetic excitations, or as amanifestation of vacuum energy of the electromagnetic field. These surface interactions,represented by a conservative potential U s , are sensitive to the frequency dispersion ofthe electromagnetic response properties of the atoms and surfaces. imulations of atomic trajectories near a dielectric surface d (μm) γ s / γ γ s( ⊥ ) γ s (||) Figure 3.
Variations of the dipole decay rate γ s ( d ) for a dipole oriented parallel ( (cid:107) )and perpendicular ( ⊥ ) to the surface normal as a function of distance d from a semi-infinite region of SiO . The decay rate is in units of the vacuum decay rate γ and thewavelength of the transition is λ = 852 nm. For an atom located a short distance d (cid:28) λ from a dielectric, the fluctuating dipoleof the atom interacts with its own surface image dipole in the well-known nonretardedvan der Waals interaction. Using only classical electrodynamics with a fluctuatingdipole, the surface interaction potential is found to take the Lennard-Jones (LJ) form U LJs = − C /d , where C is a constant that depends on the atomic polarizability anddielectric permittivity of the surface [35, 36, 37, 38]. At larger separations, virtualphotons exchanged between atoms and surfaces cannot travel the distance in time t ∼ /ω due to the finite speed of light. Consequently, the interaction potential isreduced, as first calculated in the 1948 paper by Casimir and Polder [39]. The retardedsurface potential takes the form U rets = − C /d for a constant C , where C dependson both c and (cid:126) as this is fundamentally both a relativistic and quantum phenomenon.The full theory of surface forces for real materials with dispersive dielectric functionscame with the work of Lifshitz [40, 41]. This framework reduces to both the abovesituations for the proper limits, and, importantly, it accounts for finite temperatures,predicting a U ths ∝ d − potential caused by thermal photons dominant at large distancesfor d (cid:29) (cid:126) c/k B T [42]. In our discussion, we refer to these generalized dispersion forcesas Casimir-Polder (CP) forces, whereas U LJs , U rets , and U ths refer to the appropriatedistance limits.In microcavity cQED, evanescent field distance scales are set by the scale length ofthe evanescent field, λ = λ / π = 136 nm (for the Cs D < d (cid:46)
300 nm) span both the LJ and retarded regimes, but are much shorter thanthe thermal regime ( d > µ m). In the transition region, the limiting power laws do notfully describe U s over the relevant range of d . In our modeling, we utilize a calculationof U s with the Lifshitz approach. The Lennard-Jones, retarded, and thermal limits arisenaturally from the Lifshitz formalism [42]. imulations of atomic trajectories near a dielectric surface
012 x 10 −24 ξ (rad/s) α ( i ξ ) ( c m - ) ξ (rad/s) ε ( i ξ ) (a) (b) Figure 4.
Dispersive response functions for SiO and Cesium atoms. (a) Thedielectric function (cid:15) ( iξ ) for SiO evaluated for frequency ξ along the imaginary axis.(b) Total atomic polarizability α ( iξ ) for SiO evaluated for frequency ξ along theimaginary axis for the 6 S / ground state (red) and the 6 P / excited state calculatedas described in Appendix A. The potential U s enters our simulation in two ways. First, the transition frequency ω a of the two-level atomic system shifts away from the vacuum frequency by δ a =( U exs ( (cid:126)r ) − U gs ( (cid:126)r )) / (cid:126) , where U gs ( (cid:126)r ) and U exs ( (cid:126)r ) are the surface potentials for the ground andexcited states, respectively. Since the atom transitions between the ground and excitedstates during its passage through the mode, the average net force used in calculationsis found by weighting each contribution by the steady-state atomic state populations, F s = F gs (cid:0) − (cid:104) σ † (cid:105) ss (cid:104) σ (cid:105) ss (cid:1) + F exs (cid:104) σ † (cid:105) ss (cid:104) σ (cid:105) ss .We calculate U gs and U exs for a cesium atom near a glass SiO surface using theLifshitz approach. This calculation depends on the dispersion properties of the responsefunctions of materials, in this case the polarizability of the Cs ground state α ( ω ) andthe complex dielectric function (cid:15) ( ω ) of the silica surface. Modeling of these functions isdiscussed in Appendix A. In particular, these response functions must be evaluated onthe imaginary frequency axis ω = iξ , as shown in Figure 4.Following the method of [43], curvature of the toroid surface is implemented bytreating the toroid as a cylinder with radius of curvature R = D m /
2. The majoraxis curvature is neglected because for all relevant distances d (cid:28) D M /
2. The resultingformula can be interpreted as a sum over discrete Matsubara frequencies ξ n = 2 πnk B T / (cid:126) with an integration over transverse wave vectors, which we quote without derivation: [43] U surf ( d ) = − k B T (cid:114) RR + d ∞ (cid:88) n =0 (cid:48) α ( iξ n ) (cid:90) ∞ k ⊥ dk ⊥ e − q n d (cid:20) q n − R + d ) (cid:21)(cid:26) r (cid:107) ( iξ n , k ⊥ ) + ξ n q n c (cid:2) r ⊥ ( iξ n , k ⊥ ) − r (cid:107) ( iξ n , k ⊥ ) (cid:3)(cid:27) (16)Here, α ( iξ n ) is the atomic polarizability and r (cid:107) , ⊥ ( iξ n , k ⊥ ) are the reflection coefficients ofthe dielectric material evaluated for imaginary frequency iξ n . The primed summationimplies a factor of 1 / n = 0 term. The reflection coefficients for the two imulations of atomic trajectories near a dielectric surface r (cid:107) ( iξ n , k ⊥ ) = (cid:15) ( iξ n ) q n − k n (cid:15) ( iξ n ) q n + k n (17) r ⊥ ( iξ n , k ⊥ ) = k n − q n k n + q n (18)where q n = (cid:114) k ⊥ + ξ n c , k n = (cid:114) k ⊥ + (cid:15) ξ n c (19)and (cid:15) ( iξ n ) is the complex dielectric function evaluated for imaginary frequencies iξ n .Depending on the author, r (cid:107) ( r ⊥ ) is sometimes referred to as r TM ( r TE ). U gs is calculated by numerical evaluation of (16). U exs is also calculated using (16),but with an additional contribution accounting for real photon exchange from the excitedstate with the surface, which is proportional to Re (cid:104) (cid:15) ( ω a ) − (cid:15) ( ω a )+1 (cid:105) in the LJ limit [38, 44].The atom-surface potential U gs for the ground state of cesium near a SiO surface isshown in Fig. 5, including calculations for both a planar and a cylindrical surface.Without the cylindrical correction, the potential approaches the LJ, retarded, andthermal limits at appropriate distance scales. For the planar dielectric, our calculationyields C /h = 1178 Hz µ m and C /h = 158 Hz µ m for the LJ and retarded limits.Note that the transition region between LJ and retarded regimes occurs around d ∼ d > D m , theperturbative method accounting for the curvature is no longer accurate [43], but at thesedistances, the surface forces are insignificant to atomic motion due to their steep powerlaw fall-off. The excited state potential U exs has a similar form to U gs .
5. Simulating atoms detected in real-time near microtoroids
In order for the semiclassical model to be applied to our falling atom experiments, wemust simulate the atom detection processes. In particular, in [12], falling Cs atoms aredetected with real-time photon counting using a field programmable gate array (FPGA),with subsequent probe modulation triggered by atom detection.A microtoroidal cavity with frequency ω c is locked near the 6 S / , F = 4 → P / , F = 5 atomic transition of Cs at ω at desired detuning ∆ ca = ω c − ω (0)a . Fiber-cavity coupling is tuned to critical coupling where the bare cavity transmission vanishes, T (cid:46) T min (cid:39) .
01. For atom detection, a probe field at frequency ω p = ω c and flux P in ∼
15 cts/ µ s is launched in the fiber taper and the transmitted output power P T is monitored by a series of single photon detectors. Photoelectric events in a runningtime window of length ∆ t th are counted and compared to a threshold count C th . Asingle atom disturbs the critical coupling balance so that T /T min >
1, resulting in aburst of photons which correspond to a possible trigger event. Extensive details of theexperimental procedure are given in [12] and the associated Supplementary Information.Whereas only a single atom is required to produce a trigger, spectral and temporaldata are accumulated over many thousands of trigger events since each individual atom imulations of atomic trajectories near a dielectric surface d (μm) d U s ( d ) / h ( k H z μ m ) −3 −2 −1 −10 −10 −10 −1 −10 −2 U s LJ ~1/ d U s th ~1/ d U s g U s ex U s r e t ~ / d Figure 5.
Atom-surface potentials U gs (red) and U exs (blue) for a cesium atom atdistance d from a SiO surface. The solid lines are for a planar surface whereas thedashed lines are for a curved surface with radius of curvature R = D m / . µ m. Thelimiting regimes for U gs with a planar surface are shown as dotted lines, each calculatedfrom analytic expressions not using the Lifshitz formalism. The cylindrical surfacecorrection weakens the potential, which is noticeable in the retarded and thermalregimes. is only coupled to the cavity for a few microseconds. Simulation is a valuable techniqueto disentangle atomic dynamics from the aggregate data and offer insights into theatomic motion which underlies the experimental measurements. Central to our simulations is the generation of a set of N representative atomictrajectories for the experimental conditions of atoms falling past a microtoroid fulfillingthe criteria for real-time detection. Since experimental triggering is stochastic, thetrajectory set is generated randomly as well. For each desired collection of experimentalparameters P , a set of semiclassical atomic trajectories { (cid:126)r j ( t ) } P is generated thatsatisfies the detection trigger criteria. This ensemble is used to extract the cavity outputfunctions T ( t, ∆ ap ) and R ( t, ∆ ap ). For each individual trajectory, t = 0 is defined to bethe time when the trajectory is experimentally triggered by the FPGA. For each set P , N is chosen large enough for a sufficient ensemble average to be obtained for the finaloutput functions, which is typically at least 400 unique triggered trajectories.Within each simulation, the probe field is fixed to a given ω p . Cavity behavioris determined by the parameters ω c , h , κ i , and κ ex . h and κ i are determined frommeasurements of the bare cavity with no atoms present. Low-bandwidth fluctuationsin κ ex and ω c from mechanical vibration and temperature locking are modeled asnormally distributed random variations with standard deviations of 3 MHz and 1.5 imulations of atomic trajectories near a dielectric surface µ s at critical coupling and on resonance. This rate would be identicallyzero for ∆ cp = 0 and critical coupling in the absence of these fluctuations. If the noisethreshold is not met, then the particular trajectory is thrown out as it would have beenin experiments.The atomic cloud is characterized by its temperature, size, and its height abovethe microtoroid. Its shape is assumed to be Gaussian in each direction with parametersdetermined by florescence imaging. For each simulation loop, an initial atomic position (cid:126)r in is selected from the cloud and the initial velocity (cid:126)v in is selected from a Maxwell-Boltzmann distribution of temperature T . The trajectory is propagated forward in timeunder the influence of gravity until it crosses the toroid equatorial plane at z = 0.Only trajectories which pass within 1 µ m of the toroid surface at z = 0 are kept asa candidate for detection, as other atoms couple too weakly for triggering. Once anacceptable set of initial conditions is obtained, the trajectory (cid:126)r ( t ) is calculated over a 50 µ s time window around its crossing of z = 0, this time with the gravity, optical dipoleforces, and surface interactions included. As the atom moves through the cavity mode,the atom-cavity coupling g , level shifts, decay rates, and forces change with position (cid:126)r , causing deviations of the trajectory from the preliminary free-fall trajectory. If theatom crashes into the surface of the toroid, then the coupling is set to g = 0 onwardsand the trajectory effectively ends (except for random ‘noise’ photon counts arising fromthe non-zero background transmission).Using (cid:126)r ( t ) and the steady-state expressions for the fields (section 2), we find thetransmission T ( t ). The photon count record C i ( t j ) on each photodetector i for timestep t j is generated from a time-dependent Poisson process with mean count per bin of C i ( t j ) = T ( t j ) P in ∆ t , where ∆ t = t j +1 − t j = 1 ns and P in is the input flux. Since thetypical flux is P in ∼
10 MHz and the timescale of quantum correlations is ∼
10 ns, thephoton count process is assumed to be Poissonian on the relevant timescale of a fewhundred nanoseconds for atom detection. The count record C i ( t j ) is compared to thedesired threshold of C th in a time window ∆ t th [12]. If the trigger condition is met, theinitial conditions (cid:126)r in ,j , (cid:126)v in ,j , the random cavity parameters ω c and κ ex , and the randomnumber seed used to generate (cid:126)W i for diffusion processes are stored for later use. Thesemiclassical trajectory (cid:126)r j ( t ) can be fully reconstructed from these parameters. Thetime coordinate is shifted so that the trigger event occurs at t = 0. This process isrepeated to acquire N triggered trajectories.Cavity output functions such as the experimentally measurable transmission T exp ( t, P ) for each simulation parameter set P are calculated from the set of trajectories { (cid:126)r j ( t ) } : T exp ( t, P ) = 1 N N (cid:88) j T ( (cid:126)r j ( t ) , P ) (20)Reflection coefficients R exp ( t, P ) are calculated similarly. Spectra are calculated by imulations of atomic trajectories near a dielectric surface t < t < t for each probe frequency ω p .The times t and t are chosen to be the same as in our experiments, which is typically t = 250 ns and t = 750 ns. The set of triggered trajectories { (cid:126)r j ( t ) } is valid for agiven set of conditions P and detection criteria { C th , ∆ t th } until the trigger at t = 0. Inexperiments, the probe frequency ω p can be changed in power and detuning upon FPGAtrigger. Although the same set of trajectories is valid before t = 0 for each detuning,the trajectory set must be recalculated for t > T ( t ) in (20); the linearizedmodel is only used to calculate the trajectory (cid:126)r ( t ) and efficiently generate triggers.Whereas experiments give access only to ensemble averaged output functions,simulations contain the full trajectory paths. Provided that the simulation offers areasonable approximation of the true ensemble of trajectories, then these results providea window into the atomic dynamics underlying the cQED measurement of falling atomswhich are not readily clear from observations. The experimentally measurable cavity transmission T exp ( t ) is obtained in (20) as anaverage over the trajectory set { (cid:126)r j ( t ) } at each time t . Eq. (20) can formally be written asan integration over the probability distribution of coupling constants at time t , p t ( g, θ ),for the given experimental parameters P : T exp ( t, P ) = (cid:90) dg dθ T ( g, θ, P ) p t ( g, θ ) (21)The function T ( g, θ, P ) is shown in Fig. 6 for the parameters P relevant to experiments,specifically with ∆ ca / π = 0 ,
60 MHz. For this discussion, we assume all frequenciesare fixed and neglect surface shifts. In this perspective, T exp ( t ) is not directly related tothe trajectory set { (cid:126)r j ( t ) } but rather the probability distribution p t ( g, θ ) at time t . Thetime dependence of p t evolves based on the underlying trajectory ensemble. Δ ca / π = 0 MHza. b. T θ g tw � � � � T � � � � Δ ca / π = 60 MHz g tw θ Figure 6.
Plots of T ( g tw , θ, P ) for (a) and (b) calculated numerically from (7). Atomswith higher g tw generally have higher T and a larger probability for detection. Thevariation of T with θ is evident, with a different periodicity for the two cavity detunings. imulations of atomic trajectories near a dielectric surface p t ( g, θ ) in more detail sinceit is the formal output of the simulations. We consider only the distribution p t =0 ( g )over the coupling parameter g at the trigger time t = 0 by integrating out the angulardependence. Through a reasonably simple analytic model (detailed in Appendix B),we calculate p t =0 ( g ) and compare to the results of the semiclassical simulation, whichincludes dipole and surface forces (Fig. 7). For a cavity on resonance with the atomtransition, ∆ ca / π = 0, the analytic model agrees well with a simulation when dipole andsurface forces are not included. In this case, atom trajectories are nearly straight andvertical near the toroid, and the approximations of Appendix B are sufficient. Whenthe full forces are included in the semiclassical model, the additional forces shift thedistribution toward lower g . This effect is more significant for ∆ ca / π = 60 MHz. Thecorresponding experimental cQED spectra confirm that the semiclassical model withdipole and surface forces is necessary to reproduce spectral features in the real-timeexperiments (Fig. 7c). −100 0 100 20000. Δ pa /2 π (MHz) T (b) p t = ( g ) g /2 � (a) Δ ca = ca / = +60 MHz g /2 � (c) p t = ( g ) Δ ca / = +60 MHz Figure 7.
Distributions p t =0 ( g ) of coupling constants calculated for (a) ∆ ca / π = 0and (b) ∆ ca / π = +60 MHz. Distributions from the analytic model (red), semiclassicaltrajectory simulation with no dipole or surface forces (blue), and the simulation withall forces (black) are shown for comparison. (c) Experimental cQED spectra data forcavity detuning ∆ ca / π = 60 MHz (blue points) from [12] plotted with model spectracalculated from the distributions p t =0 ( g ) in panel (b). The red is the analytic modelof Appendix B and black is the semiclassical simulation. The cavity transmission T varies as a function of the atomic azimuthal coordinate θ = mφ , as evident in Fig. 6. This biases atomic detection towards specific locationsaround the toroid and leads to a non-uniform angular distribution p t =0 ( θ ) for atom imulations of atomic trajectories near a dielectric surface t = 0 for three simulation conditions relevant to the experimentsof [12]. Although averaged spectra do not explicitly measure the coordinate θ , thesesimulation makes clear that trajectories passing through certain regions around thetoroid are preferentially detected. The phase of the cavity output field depends on θ ,suggesting the possibility for future experiments to measure the distribution of Fig. 8. p t = π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π θ Δ ca /2π = -40 MHzΔ ca / = +40 MHzΔ ca = (a)(b) p t = Figure 8.
Probability distribution p t =0 ( θ ) of atomic azimuthal angle θ = mφ mod 2 π at transit detection time t = 0 presented as histograms of simulation runs. Shown arethe cases for cavity detunings (a) ∆ ca = 0 and (b) ∆ ca / π = ±
40 MHz. Normalizationis such that the sum across all θ is unity. We now turn to the simulated trajectories { (cid:126)r j ( t ) } . In contrast to experiments, insimulations we have the capability of turning certain forces selectively on and off. Inparticular, we can adjust the surface potential U s and the dipole forces, referred tosymbolically as U d (despite them not being strictly derivable from a potential). Toinvestigate the effects these optical phenomena have on atomic trajectories, we runsimulations for four cases: the full semiclassical model, the model without surface forces( U s = 0), the model without dipole forces ( U d = 0), and the model without any radiativeforces ( U d = U s = 0).Considering conditions relevant to [12], we plot simulations for two sets ofexperimental parameters P , in Fig. 9. For P , the cavity is detuned to the red,whereas the cavity is blue-detuned in P (∆ ca / π = −
40 MHz for P and +40 MHz imulations of atomic trajectories near a dielectric surface P ). In each set of conditions, the probe field is on resonance with the cavityfor high signal-to-noise atom detection (∆ cp = 0) and the average bare-cavity modepopulation of a is ≈ .
05 photons. The toroid cavity parameters are those of [12], { g max , h, κ in , κ ex } / π = { , , , } MHz. Comparing the full model, we see thattrajectories for P primarily crash into the surface, whereas those from P both crashand are repelled from the toroid. This asymmetry is due to the repulsive or attractivedipole force for different probe detunings relative to the atomic transition. The largesteffect of turning surface forces off is seen in the blue-detuned trajectories, which have alower crash rate when U s = 0. With U d = 0, both P and P trajectories look nominallythe same; the detuning ∆ ca only affects cQED spectra, with a minor imperceptible effectarising from CP potentials initially shifting the atomic transition either closer to (red)or further from (blue) the cavity field.In addition to the qualitative differences in detected atom trajectories summarizedhere, the effects of U d and U s are evident in the experimental quantities T exp ( t ) and R exp ( t ). Since here we focus specifically on trajectory calculations, the reader is referredto [12] for detailed comparisons of spectral and temporal simulations to experimentaldata. imulations of atomic trajectories near a dielectric surface U s = U s = U d = ρ ( μ m ) z (μm) −
012 12 ρ ( μ m ) ρ ( μ m ) − z (μm) F u ll m od e l U d = ρ ( μ m ) U s = U s = U d = U d = Δ ca / � = + M H z Δ ca / � = - M H z F i g u r e . S i m u l a t e d t r a j ec t o r i e s f o r m o d e l p a r a m e t e r s P , ( ∆ c a / π = M H z ) p l o tt e d f o r f o u r m o d e l s o f r a d i a t i v e f o r ce s : t h e f u ll s e m i c l a ss i c a l m o d e l, U s = , U d = , a nd U s = U d = . F o r t h e f u ll m o d e l, a t h r ee - d i m e n s i o n a l r e p r e s e n t a t i o n i ss h o w n , w h il e t r a j ec t o r i e s a r e p r o j ec t e d o n t o t h e t w o - d i m e n s i o n a l ρ − z p l a n e f o r a ll c o nd i t i o n s . M ag e n t a t r a j ec t o r i e s r e p r e s e n t u n - t r i gge r e d a t o m s , b l u e p a t h s a r e d e t ec t e d a t o m s f o r t < nd r e dp a t h s r e p r e s e n t a t o m t r a j ec t o r i e s a f t e r t h e t r i gg e r f o r t > . imulations of atomic trajectories near a dielectric surface
6. Trapping atoms in the evanescent field of a microtoroid
Our trajectory simulation can be extended to study trapping of atoms in a two-colorevanescent far off-resonant trap (eFORT) near a microtoroidal resonator. An evanescentfield trap takes advantage of the wavelength dependence of scale lengths for the opticaldipole force of two optical fields with frequencies far-detuned from the atomic transitionto limit scattering [45, 46, 47]. The relative powers of the two fields are set so that nearthe surface, the blue-detuned, repulsive field is stronger than a red-detuned attractivefield. As each field falls off with a decay constant of roughly λ = 2 π/λ , at some distance,the red, attractive field will dominate and the atom will be attracted to the surfaceforming a potential minimum. Recently, evanescent fields have been harnessed to trapatoms in a two-color eFORT around a tapered optical fiber [48], where the fiber enablesefficient optical access to deliver both high intensity trapping fields and weaker probefields to the trapped atoms in a single structure. The tapered fiber can be positionedas desired, bringing the trapped atoms near a device for atomic coupling.The tapered nanofiber eFORT is a remarkable achievement toward integratingatom traps with solid-state resonators, but the nanofiber scheme does not allow directintegration with a cavity for achieving strong, coherent coupling between light andtrapped atoms. Another disadvantage is that trap depth is limited by the large totalpower required to achieve trapping with evanescent fields. The high quality factorsand monolithic structure of WGM resonators allow evanescent field traps free fromthese problems while maintaining efficient optical access from tapered fiber coupling.Two-color evanescent field traps in WGM resonators have been analyzed in detail forspheres [49] and microdisks [50]. In this section, we extend our simulations of atomsin the evanescent field of a microtoroid to an eFORT that can capture single falling Csatoms triggered upon an atom detection event.Unlike nanofibers, a microtoroid cannot be placed directly in a magneto-opticaltrap for a source of cold atoms. As shown in [12], we have the experimental capabilityto detect a single atom falling by a microtoroid and trigger optical fields while that atomremains coupled to the cavity mode. The semiclassical simulations described here areideal for investigating the capture of falling atoms in a trap triggered upon experimentalatom detection.We add an additional eFORT potential U t to our semiclassical trajectory model inaddition to the dipole forces and surface potential U s . For our simulation, U t is formedfrom a red (blue)-detuned mode near 898 nm (848 nm) with powers ∼ µ W to give atrap depth of ∼ . d ∼
150 nm from the surface (Fig. 10a). The red (blue) fieldsinteract primarily with the 6 S / → P / (6 S / → P / ) transition. The trap depthis limited by the total power in vacuum that can propagate in the tapered fibers of [12].Power handling can be improved with specific attention to taper cleanliness, so withexperimental care the trap depth can be increased reasonably from the discussion here,although we simulate under the conditions given to illustrate that this trap is alreadyexperimentally accessible. imulations of atomic trajectories near a dielectric surface −5 d (µm) U ( d ) / k B ( m K ) U tblue U tred U t ρ (µm) z ( µ m ) −2
02 9 11 13 Δ ca /
2π = 0 MHz a.b. c.d.
Figure 10. (a) The trapping potential U t along the z = 0 axis with the CP potentialincluded. Also shown are the red and blue evanescent potentials of the two trappingmodes, U t , respectively. (b) The mode function used in U t for the 898 nm modewith m = 106. (c) Simulated trajectories for trapping simulations with a eFORT U t triggered “on” by atom detection at t = 0 with ∆ ca = 0. Falling atoms with the FORTbeams “off” ( t <
0) are colored blue, whereas trajectories after the trap is triggeredare red. Trajectories are colored pink for t > µ s to illustrate the timescale. Roughly25% of the triggered trajectories become trapped. (d) Same as (c) showing only thetrapped trajectories and a clearer view of atom orbits in the evanescent trap. The difference in vertical scale lengths ( ψ in (4)) for modes of different wavelengthleads to a trap that is not fully confined if both the red- and blue-detuned trap modesare of the lowest order (as in Fig. 2.1b). As | ψ | increases, the repulsive blue-detunedlight weakens faster than the red-detuned field, and atoms can crash into the toroidsurface. This problem is alleviated by exciting a higher-order mode for the 898 nmlight, as shown in Fig. 10b. The modal pattern confines atoms near z = 0 and preventstrap leakage along ψ . This problem is not present in the microdisk eFORT of [50]because the optical mode extent is determined by structural confinement and not theoptical scale length. Use of a higher-order mode was also used to form an atom-galleryin a microsphere [49].During the detection phase of the simulation, U t = 0. At t = 0 conditioned on anatom detection trigger, U t is turned on. The kinetic energy of an atom with typical fall imulations of atomic trajectories near a dielectric surface v ∼ . g/ π > t = 10 µ s, approximately25% of triggered atom trajectories are captured when the trapping potential is turnedon. Simulated trapping times exceed 50 µ s, limited not by heating from trapping lightbut by the radiation pressure from the unbalanced traveling whispering-gallery modesof nearly-resonant optical probe field. This probe field can be turned off so that theatoms remain trapped beyond the simulation time.In contrast to the standing-wave structure of a typical eFORT or Fabry-Perot cavitytrap [51], microtoroidal resonators offer the tantalizing possibility of radially confiningan atom in a circular orbit around the toroid [49, 52]. The U t = 0 outlined heredoes not confine the atoms azimuthally, forming circular atom-gallery orbits around themicrotoroid [52] (Fig. 10c,d). In the same manner as [48], a localized trap can be achievedby exciting a red-detuned standing wave for three-dimensional trap confinement.This trapping simulation outlines how real-time atom detection can be utilized totrap a falling atom in a microtoroidal eFORT. In practice, microtoroidal traps presentsome serious practical challenges. Notably, because the trap quality is sensitive to theparticular whispering-gallery mode, the excited optical mode must be experimentallycontrolled. The success of an eFORT for Cs atoms around a tapered nanofiber [48]strongly suggests that similar trap performance might be achieved for an eFORTaround a high- Q WGM cavity, localizing atoms in a region of strong coupling to amicroresonator.
7. Conclusion
We have presented simulations of atomic motion near a dielectric surface in the regime ofstrong coupling to a cavity with weak atomic excitation. As required by experimentaldistance scales, this simulation includes surface interactions, which manifest throughtransition level shifts and center-of-mass Casimir-Polder forces. Analysis of thesimulated trajectories gives insight into the atomic motion underlying experimentalmeasurements of ensemble-averaged spectral and temporal measurements for singleatoms detected in real-time. We have adapted our simulations to investigate thecapturing of atoms in an evanescent field far off-resonant optical trap in a microtoroid.Our simulations suggest that falling atoms can be captured into an eFORT arounda microtoroid, offering an experimental route towards trapping a single atom in atom-chip trap in a regime with both strong cQED interactions and significant Casimir-Polderforces simultaneously. In this system, the sensitivity afforded by coherent cQED can beused not only for atom-chip devices, but also as a tool for precision measurements ofoptical phenomena near surfaces. imulations of atomic trajectories near a dielectric surface Acknowledgments
We acknowledge support from NSF, DoD NSSEFF program, Northrop GrummanAerospace Systems, and previous funding from ARO and IARPA. N.P.S. acknowledgessupport of the Caltech Tolman Postdoctoral Fellowship.
Appendix A. Calculating the Polarizability and Dielectric ResponseFunctions
Evaluation of Casimir-Polder interactions of atoms with the surface of the dielectricresonator requires evaluation of the atomic polarizability and of the dielectric functionas functions of a complex frequency. Here we outline our analytic model of the complexdielectric function for SiO and the atomic polarizability of Cesium atoms in the groundand excited states.The complex dielectric function (cid:15) ( ω ) = (cid:15) + i(cid:15) is modeled using a Lorentz oscillatormodel of the real and imaginary parts of the response function to analytically introducefrequency dependence and enforce causality, (cid:15) ( ω ) = (cid:15) ∞ + (cid:88) j f j ( ω j − ω ) + ω γ j (cid:0) ( ω j − ω ) + iωγ j (cid:1) (A.1)Here, ω j is the resonance frequency, γ j is the damping coefficient, and f j is the oscillatorstrength for each oscillator in the model. (cid:15) ∞ = (cid:15) ( ω → ∞ ) = 1. (cid:15) can be expressed interms of the complex index of refraction ˜ n = n + iκ as (cid:15) = ˜ n = n − κ + 2 inκ , where n is the refractive index and κ is the extinction coefficient. Experimental data for ˜ n forSiO is available over a wide frequency range [53], which is used to fit the parametersof (A.1) for a seven-oscillator model ( j = 1 − α s ( ω ) for Cesium in a state s iscalculated as a sum over transitions of the form, α s ( ω ) = (cid:88) n e f ns m e ω ns − ω , (A.2)where e is the electron charge, m e is the electron mass, ω ns is the transition frequency,and f ns is the signed oscillator strength for the transition of state n to the state s ( f ns > n is above s in energy). A more complete expression for the responsefunction α ( ω ) should include damping coefficients given by the transition linewidths.Since our calculations involve integrals over infinite frequency on the imaginary axisand atomic linewidths are generally narrow with respect to transition frequencies, weassume that the off-resonant form given by (A.2) without damping is sufficient. We alsonote that this expression does not account for the differences between magnetic sublevelsand hyperfine splitting, which again represent small corrections when these expressionsare integrated over the imaginary frequency axis. The general form of (A.2) applies tothe polarizabilities for both the 6 S / ground state and the 6 P / excited state, with anadditional tensor polarizability for the 6 P / state. imulations of atomic trajectories near a dielectric surface α v ) and high-energy electron transitions from the core shells to thecontinuum ( α c ), such that α = α v + α c . The valence polarizability α v constitutes 96%of the total static polarizability [54] in Cs, with α c only significant at high frequencies.We take α c to be the same for both the ground and excited states of Cs, whereas α v is obviously sensitive to the different electronic transition manifolds for 6 S / and 6 P / states. Valence electron oscillator strengths and transition frequencies are tabulatedin many sources [55, 56]. Our estimate of α v ( ω ) for the ground state includes all6 S / → N P / and 6 S / → N P / transitions, with N = 6 −
11. For the excitedstate, α v ( ω ) is calculated using 6 P / → (6 − S / , 6 P / → (5 − D / , and6 P / → (5 − D / transitions. Tensor polarizability contributions sum to zerowhen averaged over all angular momentum sublevels [57]. In agreement with [54], ourcalculation of α v comprises about 95% of the total static polarizability.For simplicity, all core electron transitions are lumped into a single high-frequencyterm of the form used in (A.2). This term contains two free parameters, f core and ω core ,which are found from the following two conditions. Using the calculation of α v ( ω ) forthe Cs ground state, we enforce that the ground state static polarizability α ( ω → α (0) = 5 . × − cm . Wealso ensure that the ground state LJ constant for a Cs atom near a metallic surfaceagrees with the known value [54, 59] C = − (cid:126) πd (cid:82) ∞ α ( iξ ) dξ = 4 . · h kHz µ m . Theseconditions are sufficient to fix the two free parameters in α c ( ω ) for this single oscillatorcore model, although the high-frequency structure of the core polarizability is lost. Forthe excited state calculation, we use the same α c ( ω ). Appendix B. Analytic model of falling atom detection distributions
Here we develop an analytic model of the distribution p fall ( g, θ ) of coupling parameters g and azimuthal coordinate θ = mφ . Atoms are assumed to fall at constant verticalvelocity with no forces, in contrast to the more complete semiclassical trajectories usedin this manuscript to generate p t ( g ). An abbreviated description of this model appearsin the Supplementary Information of [12].The linearized steady-state cavity transmission T (∆ ap , g ( (cid:126)r )) is a known function of∆ ap and (cid:126)r . We only consider the lowest order mode where the cavity mode functionis approximately Gaussian in z and exponential in distance from the surface d . Theapproximate temporal behavior of the coupling constant g for a single trajectory is, g ( ρ, z ( t )) = g c ( ρ ) e − ( z ( t ) /z ) . (B.1)where g c ( ρ ) is the maximum value of the g at the closest approach of its trajectory( z = 0), z is a characteristic width assumed to be independent of ρ , and z ( t ) = − vt . g c ( ρ ) decays exponentially from the maximum g max at the toroid surface, g c ( ρ ) ∼ g max e − ( ρ − D p ) /λ . The transmission T and hence the detection probability depend on θ ; in general, if atoms fall uniformly around the toroid, the most numerous trajectories imulations of atomic trajectories near a dielectric surface θ which maximize T ( θ ) for the cavity parameters ofinterest ( θ = π/ ca / π = +40 MHz, for example, as in Fig. 8).The probability density function for the full ensemble of detected falling atoms p fall ( g, θ ) can be estimated as the product of the probability of any atom having aparticular g and the probability of a trigger event occurring for an atom with coupling g , p fall ( g, θ ) ∼ p atom ( g ) p trigger ( g, θ ) . (B.2)An atom transit is triggered when the total detected photon counts exceeds a thresholdnumber, C th , within a detection time window ∆ t th . For a probe beam of input flux P in , the mean counts in this window are C = T ( g, θ ) P in ∆ t th . This expression assumesthat the atom is moving slowly so that the T ( g, θ ) at trigger event is the only T ( g, θ )that contributes to the detection probability. The detection probability p trigger ( g, θ ) isestimated from a Poisson distribution of mean count C .From (B.1), p atom ( g ) can be written as a product of the probability p ( g | g c ) of anatom in a trajectory with a given g c to have coupling g and the probability of a trajectoryto have that g c , p max ( g c ), integrated over all g c , p atom ( g ) = (cid:90) g max g p ( g | g c ) p max ( g c ) dg c (B.3)The integral has limits from g to g max since g c cannot be smaller than g .For atoms falling uniformly over the ρ − φ plane, p max ( g c ) dg c is proportionalto the area of a ring of radius ρ and thickness dρ , p max ( g c ) dg c ∼ πρ dρ . Using g c ( ρ ) ∼ e − ( ρ − D p / /λ , dg c g c ∼ − dρλ . Hence, p max( g c ) ∼ /g c for ( ρ − D p / (cid:28) D p / p ( g | g c ) we note that that the probability is proportional to the time an atom inthe trajectory is at a particular g . From (B.1) for a constant velocity v , this trajectoryis Gaussian and the relative probability must be proportional to dz . Finding thedifferential as a function of g gives p ( g | g c ) ∝ dz ∼ g √ ln( g c /g ) .Putting the results together in (B.3) gives p atom ( g ) ∼ (cid:90) g max g gg c dg c (cid:112) ln( g c /g ) ∼ (cid:114) ln (cid:16) g max g (cid:17) g (B.4)This result diverges as g goes to zero since there are infinite transits with small g c andinfinite time for atoms with small g for any transit regardless of g c for t → ±∞ . Thisdivergence is not problematic in calculating (B.2) since p trigger( g, θ ) cuts off for low g faster than the logarithmic divergence in p atom ( g ).The spectrum for given experimental parameters as a function of probe detuning∆ ap = ω p − ω (0)a can be written as: T (∆ ap ) = (cid:90) g max T (∆ ap , g, θ ) p fall ( g, θ ) dg dθ (B.5)where the normalization of p fall ( g, θ ) is chosen such that (cid:90) g max p fall ( g, θ ) dg dθ = 1 (B.6) imulations of atomic trajectories near a dielectric surface g , p fall ( g ) independent of θ , is found by integrating over θ .In practice, p fall ( g ) is quite similar to p fall ( g, θ ) evaluated for the θ which maximizesthe transmission. Fig. 7 compares this simple model for p fall ( g ) with the equivalentdistribution from the semiclassical trajectory simulation, p t =0 ( g ). References [1] Kimble H J 2008
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