## Abstract

Optical trapping of atoms employs high-intensity fields that necessarily alter atomic level structure. The calculation of light shifts by perturbation theory fails for scenarios that arise, for example, when the trapping light is near an excited-state transition or for polychromatic fields. We show here that non-perturbative methods based on Floquet’s theorem elegantly handle such scenarios. We compare our calculation to precision absorption spectroscopy on cold ^{87}Rb atoms in a bichromatic optical dipole trap at 1560 + 1529 nm. Proximity to excited-state resonances induces highly nonlinear level shifts, providing a strong test of theory. The good theory-experiment agreement suggests a new method for accurate measurements of excited-state electric-dipole matrix elements and a precision tool for engineering custom atomic level structures.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Light shifts, or ac Stark shifts, are ubiquitous in optical trapping of atoms [1]. They can be exploited to determine atomic properties for fundamental physics [2,3], in sensing applications such as optical magnetometry they can be detrimental [4,5] or beneficial [6], they can be used to characterize optical traps [7], and recently have been exploited for fine control and addressing of individual qubits in a trapped-ion quantum information processor [8,9]. Light shifts due to both blackbody radiation and probe light are a limiting factor in the accuracy of modern optical atomic clocks [10,11].

Light shifts are often calculated using second-order perturbation theory [9,12–18], however this is not adequate in situations with strong nonlinear light shifts and non-negligible mixing of different hyperfine energy levels. Here we describe a non-perturbative semiclassical theory for calculating light shifts based on Floquet’s theorem. Floquet’s theorem has a long history of being successfully applied to strongly periodically driven quantum systems [19], for example in the context of Rydberg atoms ionised by microwave fields [20,21].

The theory can accurately describe light shifts in a regime analogous to the magnetic Paschen-Back regime, i.e. a regime where the light shifts are large, nonlinear, and there is strong mixing of the hyperfine levels. It can describe light shifts due to multiple lasers of arbitrary polarization with wavelengths close to atomic resonances, with the limitation that the different wavelengths must be related by a rational fraction. At the same time, the mathematics is considerably simpler than in perturbative treatments [17,18] and handles strong level mixing in a natural way, thus extending the possibilities of light-shift engineering, e.g. for state preparation [22], and light-shift compensation techniques such as that demonstrated in [23,24].

We test our theory by performing absorption spectroscopy on the light-shifted |5P_{3/2}, *F*, *M*〉 magnetic sublevels in optically trapped ^{87}Rb. Our experiment can resolve light shifts of individual magnetic sublevels, and we find that our theory correctly predicts the positions of all levels after calibration of the in-situ light intensity and polarization. We use a simple model of atoms in a dipole trap to explain the observed spectrum. The spectrum is sensitive to both the trapping light intensity and polarization and can be used for calibration of both. There is nothing in the theory specific to this particular transition or atom, so we expect our theory to be useful for calculating light shifts in any system involving optically trapped atoms. Indeed it could greatly expand the range of experimental possibilties with conventional optically-trapped atoms, optically-trapped molecules [25], atoms in optical lattices [26], or atoms trapped close to optical fibers [27].

The theory presented here has a potential application to measuring excited-state electric-dipole matrix elements. Precise knowledge of dipole matrix elements is important for e.g. optical clocks, testing atomic structure calculations [28], and atomic parity non-conservation measurements [29,30]. While the idea of using light shifts to measure dipole matrix elements is not new [31], our experiment shows for the first time that Floquet’s theorem enables the possibility of quantitative comparison between theory and experiment in a regime of strong light shifts, where errors in electric-dipole transition matrix elements should manifest as a larger discrepancy between theory and experiment.

## 2. Floquet theory of light shifts

Floquet’s theorem states that the Schrödinger equation

with time-periodic Hamiltonian*H*(

*t*) =

*H*(

*t*+

*T*) has solutions of the form

*ψ*(

*t*) =

*φ(t*)

*e*, where

^{−iωt}*φ*(

*t*) =

*φ*(

*t*+

*T*) has the same periodicity as

*H*(

*t*). In the case of an atom in an oscillating external field, we have

*H*(

*t*) =

*H*

_{0}+

*V*(

*t*), where

*H*

_{0}is the free-atom Hamiltonian and

*V*(

*t*) =

*V*(

*t*+

*T*) is a periodic [19].

*ψ*(

*t*) describes a dressed state of the Hamiltonian, with energy

*ћω*.

To find the dressed states, it suffices to consider $\mathcal{U}\left(T,0\right)$, the time-evolution operator for one period of the potential, for which $\mathcal{U}\left(t+T,t\right)\psi \left(t\right)=\psi \left(t\right){e}^{-i\omega T}$. The eigenstates of this operator are thus the dressed states *ψ _{i}*(

*t*), with eigenvalues exp[−

*ψ*]. This determines

_{i}T*ω*up to additive multiples of 2π/T. When 2π/T is large relative to fine- and hyperfine-structure splittings,

_{i}*ћω*can be unambiguously assigned by comparison against the bare energies.

_{i}To compute $\mathcal{U}\left(T,0\right)$, we use a numerical Euler method [32]. First we partition $\mathcal{U}\left(T,0\right)$ into *N* subintervals

*t*=

_{k}*kT*/

*N*, and the order of the product must be as in Eq. (2).

Now we calculate the two terms in the Hamiltonian. We work in the basis |*nJFM*〉, in which the free-atom Hamiltonian *H*_{0} is diagonal, with different *M* states degenerate

*K*≡

*F*(

*F*+ 1) −

*I*(

*I*+ 1) −

*J*− (

*J*+ 1) and the hyperfine constants

*A*and

_{nJ}*B*for

_{nJ}^{87}Rb are taken [28]. Fine-structure energies 〈

*nJ*|

*H*

_{0}|

*nJ*〉 are taken from the NIST atomic spectra database [33].

We describe the interaction between the atoms and the light in the electric-dipole approximation, so

where**E**(

*t*) is the electric field of a laser, and

**d**=

*e*

**r**is the electric-dipole operator.

To find the matrix elements of this interaction in our basis, it is convenient to work in Cartesian coordinates. Choosing *z* as the quantization axis, we first find *d _{z}*, the

*z*-component of

**d**, which describes ∆

*m*= 0 or

_{F}*π*transitions.

*nJ*|

*e*

**r**||

*n*′

*J*′〉 are known from the literature [34]. The

*d*matrix can be rotated to find the

_{z}*d*and

_{x}*d*matrices:

_{y}*F*= (

_{x}*F*

_{+}+

*F*

_{−})/2 and

*F*= −

_{y}*i*(

*F*

_{+}−

*F*

_{−})/2 are total angular momentum components, given in terms of the ladder operators F

_{±}with matrix elements [35]

The electric field is similarly described in Cartesian coordinates. As examples, if the incident optical field is monochromatic and polarized along $\widehat{\mathbf{z}}$, the electric field is

*ω*= 2

*πc*/

*λ*is the optical frequency,

*c*is the speed of light, and

*λ*is the wavelength. Circularly-polarized light has the field

**n**

*frequencies*

_{i}*ω*,

_{i}*i*∈ {1,2} can be written

*T*in Eq. (3) refers to one period of the

*total*electric field, so we can calculate the light shifts due to multiple wavelengths as long as they are related by rational fractions. E.g. if

*λ*

_{1}/

*λ*

_{2}=

*a*/

*b*, where

*a*and

*b*are positive integers, the period of the total electric field is the lowest common multiple of

*T*

_{1}and

*T*

_{2}, where

*T*= 2

_{i}*π*/

*ω*=

_{i}*λ*is the optical period.

_{i}/cIn this formulation *H*_{0} can be readily extended to include static magnetic and/or electric fields, and *V* can be adapted to include magnetic and higher electric multipole transitions, provided the matrix elements are known. We acknowledge that here we neglect any possible additional vacuum field, relaxation, continuum, or relativistic effects.

For our calculations we used only the energy levels shown in Fig. 1, comprising 136 distinct states. We performed several representative calculations with levels up to *n* = 10 and found these extra states contributed less than 1 MHz to the calculated light shifts which is below the resolution of our experiment, as explained below. We computed $\mathcal{U}$ numerically with Eq. (3) and cut off *N* at some finite value, but making sure it is sufficiently high such that the result has converged. For calculations with the 1560 nm beam only we used *N* = 200. With this number of time steps the calculated shifts were within 100 kHz of their value with *N* = 10^{4}. The shifts converge exponentially with a gradient of about 40 dB/decade, i.e. the shifts are 100 times more accurate with 10 times the number of time steps. All calculations were performed in MATLAB and our code is available online [36].

## 3. Experiment

To validate the above numerical technique we perform spectroscopy of the D_{2} hyperfine transitions in a cloud of cold ^{87}Rb atoms in the presence of strong nonlinear light shifts. A schematic of the experiment is shown in Fig. 2. To trap the atoms we use an optical dipole trap consisting of a single linearly polarized 10 W beam locked with < 100 kHz stability to 1560.492 nm (the second harmonic of which is locked to a transition of the ^{87}Rb D_{2} line at 780.246 nm), and focused to a spot size of ∼ 44 µm. A second beam near 1529 nm is mode-matched to the 1560 nm beam, with a controllable power from 0–100 mW. The two beams are combined on a polarizing beamsplitter, with the 1560 nm beam reflected and the 1529 nm beam transmitted, so the polarizations are linear vertical and horizontal, respectively. The 1560 nm beam is not perfectly linear before the beamsplitter, and the polarization is not perfectly cleaned on reflection from the cube, so there is some residual ellipticity. The 1529 nm beam can be scanned across the 5P_{3/2} →4D_{3/2(5 2}) excited-state resonances at 1529.26 (1529.36) nm, so we can induce strong light shifts in the 5P_{3/2} states with relatively low intensities. We measured two datasets, one with the 1529 laser at 1529.282 nm and another at 1529.269 nm. The 1529 nm laser was not frequency-stabilised, and the wavelength was measured with a calibrated wavemeter to drift by ±0.001 nm from the nominal wavelength over the duration of the measurements. A probe beam at 780 nm propagates at an angle of 60° relative to the trap axis, to reduce the chance of producing states that are “dark” to the probe light. The probe laser is stable to less than 100 kHz, and can be scanned up to 1 GHz to the red side of the D_{2} transition.

The experimental sequence is as follows: We trap approximately 3 × 10^{6} atoms in the *F* = 1 ground state in the 1560 nm optical dipole trap. Initially the trap depth is about 270 µK and the atoms have a temperature of about 40 µK. To ensure the atoms experience as homogeneous a light intensity as possible, we reduce the temperature and therefore the spatial extent of the cloud by performing an evaporation sequence followed by adiabatic increase of the trap depth back up to about 270 µK, obtaining 10^{5} atoms at 11 µK. We then pump the atoms into the *F* = 2 ground state and measure absorption of a probe laser as a function of the frequency of the probe beam and intensity of the 1529 nm beam. The probe has an intensity of around 5*I _{sat}*, where

*I*= 1.7 mWcm

_{sat}^{−2}, and the duration of the probe pulse is 20 µs. We estimate, using calculations from [37], that heating and movement during the probe pulse are negligible under these conditions.

## 4. Results and discussion

#### 4.1. Light shifts @ 1560 nm

We first consider in detail the absorption spectrum of atoms in the trap with no other incident light, i.e. light shifts induced just by the 1560.492 nm trap itself. Fig. 3 shows relative optical depth as function of probe beam frequency at zero 1529 nm beam intensity. We say “relative” as our image processing was calibrated for measuring the density of atoms in free space, correcting for saturation as described in [38]. The image processing technique described in [39] was found to help in detecting weak absorption signals. We adapt an equation from [7] as a model for our signal. Equation (12) describes theoretical optical depth as a function of probe detuning of a mixture of populations of non-interacting two-level atoms with differential light shift at thermal equilibrium in a harmonic potential. The model treats each possible transition as a separate population of two-level atoms, so the transition frequency of each population corresponds to a transition from the five near-degenerate 5S_{1/2}, *F* = 2 ground states to each of the seven light-shifted 5P_{3/2}, *F* = 3 excited states.

*i*indicates the

*i*state, ${t}_{i}=\frac{{k}_{B}T}{\hslash \mathrm{\Gamma}}\left(\frac{{\alpha}_{e,i}}{{\alpha}_{g}}-1\right)$, ${v}_{i}=\frac{U}{\hslash \mathrm{\Gamma}}\left(\frac{{\alpha}_{e,i}}{{\alpha}_{g}}-1\right)$, and $\delta =\frac{\omega -{\omega}_{0}}{\mathrm{\Gamma}}$ are normalised temperature, trap depth, and probe detuning, respectively.

^{th}*k*is Boltzmann’s constant,

_{B}*T*is the cloud temperature (11 µK, from time-of-flight measurements), Γ is the natural linewidth,

*α*

_{e}_{,}

*is the polarizability of the*

_{i}*i*excited state,

^{th}*α*the ground state (the tensor light shift of the different 5S

_{g}_{1/2},

*F*= 2 ground states is on the order of kHz),

*ω*is the probe laser frequency,

*ω*

_{0}is the free-space transition frequency, and

*m*is the mass of the

^{87}Rb atom. The differential polarizability of a transition is equal to the differential light shift, i.e.

*α*/

_{e}*α*= ∆

_{g}*f*/∆

_{e}*f*, where ∆

_{g}*f*

_{e}_{(}

_{g}_{)}is the light shift of the excited (ground) state,

*U*is the trap depth, and

*C*is a fitting parameter depending on the number of atoms measured, and the absorption cross-section of the

_{i}*i*level for the probe beam. We note that the definition of

^{th}*ν*is slightly different from ref. [7], where the “−1” term is missing.

_{i}To fit Eq. (12) to the data shown in Fig. 3 we model the electric field of the 1560 nm laser as

*α*/

_{e}*α*. We include the quadrature phase

_{g}*ϕ*to account for a slight ellipticity of the 1560 nm light after reflection at a polarizing beamsplitter as discussed in section 3. If

*ϕ*= 0 this simply describes a linearly polarized electric field oscillating in the $\widehat{\mathbf{x}}+\widehat{\mathbf{y}}$ plane. The coefficients

*C*, electric field ${\mathcal{E}}_{1}$, and quadrature phase

_{i}*ϕ*were free parameters in the fit [40]. The light intensity is related to the electric field by where

*ϵ*

_{0}is the permittivity of free space. From the fit we extracted

*I*

_{1560}= 3.01 ± 0.01 × 10

^{9}Wm

^{−2}, which agrees well with power meter measurements, and

*ϕ*= 0.11 ± 0.01 [41]. By using colder atoms and/or a deeper trap, these quantities could be known more accurately. The trap depth

*U*is equal to the light shift of the ground state at peak light intensity at the center of the trap. We obtained

*U*=

*h*· 5.87 ± 0.05 MHz (=

*k*· 282 ± 2 µK. We can compare the

_{B}*U*obtained from the fit to

*U*calculated from the measured trap oscillation frequency

_{calc}*f*= 1.22 kHz and the beam waist measured with a beam profiler w = 44 µm as

_{osc}*U*= (2

_{calc}*π*w

*f*)

_{osc}^{2}

*m*/4 =

*h*· 6.2 MHz. The difference between the two can be explained with an error in the measurement of the beam waist of 2 µm.

The arrows in Fig. 3 show calculated light shifts of atomic transitions at the bottom of the trap, i.e. ∆*f _{e}*

_{,}

*− ∆*

_{i}*f*at peak light intensity. The data peaks are slightly offset from the theoretical peaks due to the finite temperature of the atoms: atomic density peaks above the bottom of the trap.

_{g}#### 4.2. Light shifts @ 1560 nm + 1529 nm

Figures 4(a) and 4(b) show absorption of the probe beam as a function of probe frequency and 1529 nm beam intensity. The black lines are calculated transition frequencies relative to the free-space 5S_{1/2}, *F* = 2 → 5P_{3/2}, *F* = 3 transition. We used the data shown in Fig. 3 as a calibration of the experimental parameters to then perform the calculation of energy level shifts as a function of 1529 nm beam intensity, so the only fitting parameter here is the calibration of the 1529 intensity. The left-most column in Fig. 4(a) shows the same data as that shown in Fig. 3. Figure 4(b) was measured on a different day and the experimental parameters had drifted slightly.

For calculating light shifts with both the 1560 nm and 1529 nm beams present we model the electric field as

*ϕ*= 0. The wavelength of the 1560 nm trapping beam was

*λ*

_{1}= 1560.492 nm, so for one measurement we set the wavelength of the 1529 beam to be ${\lambda}_{2}=\frac{49}{50}{\lambda}_{1}\approx 1529.282\text{nm}$. For another measurement we set

*λ*

_{2}= 1529.269 nm, and modelled the 1560 nm beam wavelength as ${\lambda}_{1}=\frac{51}{50}{\lambda}_{2}\approx 1559.854\text{nm}$, as the ratio of real experimental wavelengths does not form a simple rational fraction. We mostly compensate for this mismatch between the real and assumed wavelength of the 1560 nm light by reducing the intensity of light in the calculation by 2.0%. There is still an estimated error of up to 150 kHz in the calculated light shifts, however this is below the uncertainty introduced by the unstable 1529 nm laser.

The arrows in Figs. 4(a) and 4(b) show lines representing the light shifts for a representative level given a change in the 1529 nm laser wavelength of ±0.001 nm, showing the data and theory agree to within experimental error given the uncertainty of the laser wavelength.

## 5. Measuring electric-dipole matrix elements

The theory presented here could be useful in measurement of electric-dipole matrix elements. Techniques using Stark shifts to measure matrix elements already exist, such as comparing shifts at two different wavelengths [31], or using tune-out wavelengths [42]. The theory presented here however allows orders-of-magnitude improvements in the size of the shifts that can be predicted, with a potentially equal gain in the accuracy of measurements of matrix elements.

Additionally, with the possibility of accurate calculations with multiple wavelengths, excited-state transitions can be coupled to a transition from the ground state with multiple near-resonant lasers, making the shifts of the imaging transition depend on an arbitrarily chosen electric-dipole matrix element.

## 6. Conclusion

We have presented a theory for the calculation of strong atomic light shifts due to multiple wavelengths, where the shifts can be nonlinear and larger than the hyperfine splitting. We validated our theory by predicting and measuring light shifts of the D_{2} transition in ^{87}Rb caused by incident light nearly resonant with the 5P_{3/2} → 4D transitions. As far as we know, this is the first such calculation and measurement of light shifts outside of the linear perturbative regime. Moreover, our results are not specific to any particular transition or atom, and should be widely applicable in any system involving optical trapping.

## Funding

MINECO/FEDER, MINECO projects MAQRO (Ref. FIS2015-68039-P); XPLICA (FIS2014-62181-EXP); Severo Ochoa (SEV-2015-0522); Catalan (2014-SGR-1295); European Union Project QUIC (641122); European Research Council project AQUMET (280169); ERIDIAN (713682); Fundació Privada CELLEX.

## Acknowledgments

The authors would like to thank J. Douglas and R. Jiménez-Martínez for useful feedback on the manuscript. The authors declare that there are no conflicts of interest related to this article.

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**40. **Quantitative prediction of C_{i} is feasible but would require use of the optical Bloch equations to solve for atomic dynamics in the presence of the 1560 nm beam, the single probe beam at 780 nm, and repump light also at 780 nm which is emitted from six directions toward the centre of the trap.

**41. **We estimate the uncertainties on the fitted values as the square root of the diagonal elements in the p × p matrix (J^{T} J)^{−1}R^{T} R/(N_{pts} − p). Where J is the Jacobian of the model fit function, R is a vector of the fit residuals, N_{pts} is the number of measurements, and p the number of fitting parameters. In this case p = 9: the light intensity I, quadrature phase ϕ, and the seven peak amplitudes C_{i}.

**42. **A. Fallon and C. Sackett, “Obtaining atomic matrix elements from vector tune-out wavelengths using atom interferometry,” Atoms **4**(2) 12 (2016). [CrossRef]