Neutral atoms experience forces from the interaction of the atomic dipole moment with the radiation field. In the vicinity of a material surface, the mode structure of the full electromagnetic field is modified due to the dielectric properties of nearby objects. The integrand depends on the dielectric function of SiO2 evaluated at the frequency ωa of the atomic transition.
Here, α(iξn) is the atomic polarizability and rk,⊥(iξn, k⊥) the reflection coefficients of the dielectric material, evaluated for the imaginary frequency iξn.
Simulating atoms detected in real-time near microtoroids
The atomic surface potential Usg for the ground state of cesium near a SiO2 surface is shown in Fig. The effect of the force approximation for the cylindrical diameter is less than our calculation precision in the region near the toroid surface (d) 300 nm). Central to our simulations is the generation of a set of N representative atomic trajectories for the experimental conditions of atoms falling past a microtoroid meeting the criteria for real-time detection.
Similar to the experimental procedure, we impose that the bare cavity output flux is less than 0.4 cts/µs at critical coupling and at resonance. If the atom crashes into the surface of the toroid, then the coupling is set to. 6.2.7), we calculate pt=0(g) and compare with the results of the semiclassical simulation, which includes dipole and surface forces (Fig. 6.5).
The refinements introduced in the linearized semiclassical model, such as velocity-dependent forces, curvature corrections to the surface forces, and distance-dependent decay rates, are at a level comparable to the width of the lines for the curves drawn in the figures. The cavity transmission T varies as a function of the atomic azimuth coordinate θ = mφ, as shown in Figure . The phase of the cavity output field depends on θ, suggesting the possibility for future experiments to measure the distribution of Figure 2.
For P1 the cavity is detuned to red, while the cavity in P2 is detuned blue. In addition to the qualitative differences in detected atomic trajectories summarized here, the effects of Udand Us are clearly visible in the experimental quantities Texp(t) and Reexp(t).
Calculating the polarizability and dielectric response functions
The total atomic polarizability is composed of contributions from valence electron transitions (αv) and high-energy electron transitions from the nuclear shells to the continuum (αc), such that α=αv+αc. The valence polarizabilityαv accounts for 96% of the total static polarizability [59] in Cs, with αc only significant at high frequencies. We assume that ac is the same for both the ground and excited states of Cs, while αv is of course sensitive to the different electronic transition manifolds for the 6S1/2 and 6P3/2 states.
For simplicity, all core electron transitions are lumped into a single high-frequency term of the form used in (6.14). This term contains two free parameters, fcore andωcore, which are found from the following two conditions. Using the calculation of αv(ω) for the ground state of Cs, we enforce that the static polarizability of the ground state α(ω→0) corresponds to the known theoretically calculated value [6].
We also ensure that the ground state LJ constant for a Cs atom near a metal surface corresponds to the known value [59, 117]C3 =−4πd~3. These conditions are sufficient to establish the two free parameters inαc(ω) for this single oscillator core model, although the high-frequency structure of the core polarizability is lost.
Analytic model of falling atom detection distributions
The transmission of T and thus the probability of detection depends on θ; in general, if the atoms fall uniformly around the toroid, the most detected trajectories will be at values of θ that maximize T(θ) for the cavity parameters of interest (θ=π/2 for ∆ca/2π= +40 MHz , for example as in Figure 6.6). The probability density function for the entire group of detected falling atoms spfall(g, θ) can be estimated as the product of the probability that any atom has a particular g and the probability of a trigger event occurring for an atom with a coupling g,. An atom transition is triggered when the total number of detected photons exceeds a threshold number, Cth, within the detection time window ∆tth.
This expression assumes that the atom is moving slowly so that T(g, θ) at the trigger event is the only T(g, θ) contributing to the detection probability. From (6.15), patom(g) can be written as the product of the probability p(g|gc) of an atom in a trajectory with a given gc having coupling g and the probability of a trajectory having that gc, pmax ( gc), built on top of allgc, . For atoms falling uniformly on the ρ−φ plane, pmax(gc)dgc is proportional to the area of a ring of radius ρ and thickness dρ, pmax(gc)dgc ∼2πρ dρ.
To find p(g|gc) we note that the probability is proportional to the time an atom in the trajectory is at a given g. From (6.15) for a constant velocity v, this trajectory is Gaussian and the relative probability must be proportional to dz. In practice, pfall(g) is quite similar to pfall(g, θ) evaluated for theθ that maximizes transmission.
Trapping of atoms near dielectric surfaces
Optical tweezer trap
While an optical tweezer trap for a single atom has been realized at a sub-micron scale, using diffraction-limited focused FORT beam with numerical aperture of 0.7 formed by multiple lenses with a working distance of 1 cm [211] (later, [ 234) ]), there are a number of physical and practical limitations present in this approach. First, scalability and integrability may require miniaturization of the imaging system used to form the tightly focused beam. In the microtoroid lab of our group, we (notably Scott Kelber and Cindy Regal) explored an approach to this problem using a lens fiber (from Nanonics Imaging) that creates a tightly focused beam with a numerical aperture of 0.7 (beam radius of 300) nm) with a working distance of 5 µm.
Second, the diffraction limit generally imposes a minimum distance at which an atomic trap can be located relative to the toroid surface ≈λdip/2, which is larger than the decay length 1/e of the evanescent field of the microtoroid in the whispering gallery mode λ/2π ≈ 136 nm for λ = 852 nm. Third, the positioning of the trap location relative to the on-chip toroid presents a practical limitation. Here, the size of the optical beam and the working distance of the focusing lens represent geometrical limitations.
Finally, assuming that the tweezer trap will not be "destroyed" by the presence of the silicon dioxide toroid (which preliminary analysis has shown may be the case for the system under consideration), then there is more. Shown here is a tweezer optical trap using a focusing lens with a numerical aperture of 0.43 where atoms are loaded from the magneto-optical trap. There are possible solutions that can be implemented to allow toroid-to-trap distances comparable to the decay lengths of the toroidal evanescent field.
Orbiting trap
The difference in vertical scale lengths (ψ0 in (3.3)) for modes with different wavelengths leads to a trap that is not completely confined if both the red- and blue-detuned trap modes are of the lowest order (as in Figure 3.1. 1.1b ). Also shown are the red and blue evanescent potentials of the two trapping modes Ut, respectively. c) Simulated trajectories for trapping simulations with an eFORT Ut activated “on” by atomic detection att = 0 with ∆ca = 0. This makes the overall quality factor Q of a microtoroidal cavity sensitive to the amount of electric field present at the surface boundary of the torus, leading to the tendency of the fundamental mode to have a higher Q than the higher order modes.
This resonant Qsignature of the cavity is our guide for exciting the fundamental mode of the toroid1. Note that we do not consider the finite line widths of the cavity modes, we only show the lines representing the resonant frequencies of the cavity modes. 6.3.2.1 we discussed an orbital trap scheme that requires excitation of the fundamental mode (mode 1 or 1') for the blue-tuned FORT as well as the 3rd-order mode (mode 3 or 3') for the FORT tuned to red. , in order to form a fully closed trap potential.
Although this may seem like a simple task, it requires the ability to excite special high-order states of the microtoroid. The lower panel in part c) of the figure shows the frequency response to temperature, shape= 118, z-polarized fundamental mode, Dp = 12 µm, λ corresponding to the first fundamental mode labeled with the number 1 (blue line) in Fig. 6.9 c). This temperature locking method maintains the resonant frequency of the toroid within a few MHz of the cesium transition frequency.
Toroid-fiber trap
For example, in the landmark experiment of [248], and in our state-insensitive nanofiber trapping experiment discussed in Chapter 7. The state-insensitiveness of our trapping scheme is of particular importance for the toroid fiber trapping scheme described in this section. , because it ensures that the transition frequencies of the trapped atoms remain within the resonant linewidth of the cavity while the atoms move within the traps. Consequently, it is not very practical to have a dedicated nanofiber for trapping and another independent nanofiber for optical coupling in and out of the toroid. 2Note that this involves spectral filtering of the capture beams (∼10 mW) to be decoupled from the probe beams (∼100 fW) at the input-output ports of the fibers, which requires an extinction ratio of more than 11 orders of magnitude.
The color of the contour plots indicates the magnitude of the electric field |E|, as indicated by the color bar at the bottom. In Fig. 6.11 b), the origin of the texax is located on the axis of the nanofiber, so the left edge of the graph corresponds to x1 =a= 215 nm, the radius of the nanofiber, and x6 = 615 nm is the surface of the microtoroid, with a 400 nm surface gap- surface. The set of curves labeled (i) in Figure 6.11 b) corresponds to the case with one fiber (away from the toroid) as shown in (i) in Figure 6.11 a), while the set of curves labeled (ii) corresponds to the closer approach case close to the system consisting of two parallel waveguides.
The line appearing at x2 .. in Figure 6.11 b) is part of the total potential U, which includes the Casimir-Polder potentials of the dielectric surface of the fiber and the toroid. These different examples of trapping beam power illustrate the sensitivity of the trapping potential to changes in power at the ≈10% level. For example, using the experimental parameters from Chapter 5, we show the profile of the atom-photon coupling strength in Fig. 6.11 d), which shows the exponential decay profile of the evanescent field of a toroid.
