Measurement of dynamic Stark polarizabilities by analyzing spectral lineshapes of forbidden transitions

# Measurement of dynamic Stark polarizabilities by analyzing spectral lineshapes of forbidden transitions

D. R. Dounas-Frazer    K. Tsigutkin    A. Family Department of Physics, University of California at Berkeley, Berkeley, CA 94720-7300    D. Budker Department of Physics, University of California at Berkeley, Berkeley, CA 94720-7300 Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720
July 17, 2019
###### Abstract

We present a measurement of the dynamic scalar and tensor polarizabilities of the excited state in atomic ytterbium. The polarizabilities were measured by analyzing the spectral lineshape of the 408-nm transition driven by a standing wave of resonant light in the presence of static electric and magnetic fields. Due to the interaction of atoms with the standing wave, the lineshape has a characteristic polarizability-dependent distortion. A theoretical model was used to simulate the lineshape and determine a combination of the polarizabilities of the ground and excited states by fitting the model to experimental data. This combination was measured with a 13% uncertainty, only 3% of which is due to uncertainty in the simulation and fitting procedure. The scalar and tensor polarizabilities of the state were measured for the first time by comparing two different combinations of polarizabilities. We show that this technique can be applied to similar atomic systems.

###### pacs:
32.90.+a, 32.70.Jz, 32.60.+i

## I Introduction

Static (dc) and dynamic (ac) electric dipole polarizabilities determine the response of neutral particles to applied electric fields. They are related to a host of atomic and molecular quantities, including the dielectric constant, refractive index, and Stark shift Autler and Townes (1955); Angel and Sandars (1968); Bonin and Kadar-Kallen (1994); Bonin and Kresin (1997), and are an important consideration for many current atomic, molecular, and optical physics experiments Mitroy et al. (2010). For example, polarizabilities play a vital role in the production of light traps for quantum information processing applications Ye et al. (2008). In the context of optical atomic clocks, Stark shifts constitute an important systematic effect that must be controlled Udem (2005); Takamoto et al. (2005); Barber et al. (2008); Chou et al. (2010). Similarly, Stark shifts also contribute to systematic effects in atomic parity violation (APV) measurements Wieman et al. (1987); Wood et al. (1999); Tsigutkin et al. (2009, 2010). Hence the determination of polarizabilities is a priority for high-precision atomic physics.

Present experiments typically rely on theoretical calculations of electric dipole polarizabilities Dzuba and Derevianko (2010); Safronova and Safronova (2010); Beck and Pan (2010). Several methods for measuring polarizabilities also exist. Early schemes involved the deflection of atoms in an inhomogeneous electric field Bonin and Kadar-Kallen (1994). More recent techniques include absolute frequency measurements Barber et al. (2008), atom interferometry Morinaga et al. (1993); Ekstrom et al. (1995); Deissler et al. (2008), and a technique that uses light force Kadar-Kallen and Bonin (1992, 1994). However, these methods typically provide information about the polarizability of an atom in its ground state Mitroy et al. (2010). Therefore, they are inappropriate for high-precision experiments where the polarizabilities of excited states are relevant.

As part of an ongoing investigation of parity violation in atomic ytterbium (Yb) Tsigutkin et al. (2009, 2010), a scheme for measuring a combination of polarizabilities of the ground and an excited state of Yb was developed Stalnaker et al. (2006). The scheme involves the simulation and measurement of the spectral lineshape of a forbidden electric dipole transition driven by a standing wave of light in the presence of a dc electric field. Due to the standing wave, the ac Stark shifts of the upper and lower states introduce a polarizability-dependent distortion in the lineshape, a phenomenon which was first observed and characterized during a search for APV in cesium Wieman et al. (1987). The difference of polarizabilities of the two states is treated as a variable parameter in the simulation and is measured by fitting the simulated lineshape to experimental data. We call this scheme the Lineshape Simulation Method (LSM). The LSM can be generalized to an arbitrary atomic species.

In this paper, we present the next generation of the LSM. The numerical procedures accommodate for a broad domain of values of input parameters, e.g., the intensity of the standing wave. In addition, the independent dimensionless parameters that determine the lineshape have been explicitly identified, thus facilitating error analysis. In general, the LSM is compatible with a variety of atomic species and field geometries. The LSM is sensitive to the difference of polarizabilities of the atom in its ground and excited states. Nevertheless, this method may yield unambiguous measurements of the vector and tensor components of the excited state polarizability, as we will show. In this sense, we present a versatile method for measuring the excited-state polarizabilities of atoms.

We demonstrate the LSM using the 408-nm transition in atomic Yb. Whereas the previous results Stalnaker et al. (2006) were obtained in the absence of a magnetic field, the present work uses a magnetic field to isolate Zeeman sublevels of the excited state. The ac Stark shifts of the sublevels are characterized by different combinations of scalar, vector, and tensor polarizabilities. This approach allows for the unambiguous determination of these polarizabilities. The ac scalar and tensor polarizabilities of the excited state in Yb are measured for the first time. Due to improvements in the experimental apparatus, the signal-to-noise ratio of the observed lineshape is an order of magnitude larger than for the previous implementation. In the present work, the statistical error introduced by the LSM is negligible compared to the systematic uncertainty of the experiment.

This paper is organized as follows. In Section II, we introduce our conventions for the polarizabilities and the Stark shift. The theoretical model, numerical procedure, and results of the simulation are discussed in Section III. In Section IV we apply the LSM to the Yb system and present the results. Finally, a summary of the results and an outlook for future experiments are given in Section V.

## Ii Atomic system

Throughout this work, we consider a Stark-induced transition between two atomic states of the same parity. The transition is induced by applying a uniform dc electric field . We assume that the transition is driven by a standing wave of light formed by two counter-propagating waves with the same polarization traveling in the directions. In this case, the electric field of the light is given by

 E(r,t)=E(r)cos(ωt)ϵ, (1)

where

 (2)

Here , , , and are the amplitude, wave-vector, angular frequency, and polarization of the electric field, respectively, is the radius of the standing wave, and is the perpendicular distance from the center of the standing wave. In addition to these parameters, we define the wave-number and the wavelength , where is the speed of light. We assume that since the overall sign of the field can be incorporated into the polarization . Equation (2) is appropriate for the case of a light field with a Gaussian profile. The discussion is limited to optical frequencies. In this regime, is uniform over atomic length scales. In order to study the influence of the magnetic structure of the transition, we also assume the presence of a uniform dc magnetic field . The quantization axis (-axis) is chosen so that for .

The dynamics of an atom in the presence of the external magnetic and electric fields described above is governed by the total Hamiltonian

 H=H0+H1(t), (3)

where and are the time-independent and time-dependent parts of . Here is the atomic Hamiltonian, is the Zeeman Hamiltonian, and are the dc and ac Stark Hamiltonians, and and are the magnetic and electric dipole moments of the atom, respectively. We assume that , , and are sufficiently weak that , , and can be treated as successive perturbations to .

Let and represent the degenerate eigenstates of the atomic Hamiltonian and their corresponding energies, respectively. Here is the total angular momentum quantum number, is the magnetic quantum number corresponding to the projection of the total angular momentum along the -axis, and is a set of other quantum numbers. Then, to lowest order in the perturbing fields and , the eigenstates of are

 |¯ℓ⟩=|ℓ⟩+∑ℓ′≠ℓ|ℓ′⟩⟨ℓ′|Hdc||ℓ⟩EA(ℓ)−EA(ℓ′), (4)

with corresponding energies

 E(¯ℓ)=EA(ℓ)+EZ(ℓ)+Edc(ℓ). (5)

Here and represent the Zeeman and dc Stark shifts, respectively. The Zeeman shift is given by , where is the Landé factor of the state and is the Bohr magneton. Throughout this work, we assume that is sufficiently strong to completely isolate the Zeeman sublevels of . The dc Stark shift is given by , where is the magnitude of the dc electric field, and is the dc polarizability of the atom in state . To derive Eq. (4), we neglected mixing of atomic eigenstates due to the magnetic field.

The atomic energy levels are also shifted by the ac Stark shift Autler and Townes (1955); Angel and Sandars (1968); Bonin and Kadar-Kallen (1994); Bonin and Kresin (1997), which is induced by the dynamic field . We assume that the frequency of the standing wave satisfies , where is the resonant frequency of the electric-dipole (E1) transition from the perturbed electronic ground state to a perturbed excited state . Thus it is appropriate to make a two-level approximation that involves neglecting dynamic interactions between states other than and . However, such an approximation can only account for ac Stark shifts that arise due to mixing of the states and with each other. To address this situation, we modify the energy of the perturbed ground state as follows:

 E(¯¯¯g)→E(¯¯¯g)+Eac(¯¯¯g), (6)

where

 Eac(¯¯¯g)=−12α¯gE(r)2, (7)

is the ac Stark shift of due to mixing of with states other than , and is the corresponding ac polarizability of the state . An analogous modification is made to the energy of the perturbed state .

We further assume that the unperturbed ground and excited states and have the same parity. In this case, the transition is induced by the dc electric field Bouchiat and Bouchiat (1975) and the mixing of the states and with each other is characterized by the induced dipole matrix element

 ⟨¯¯¯a|d⋅ϵ|¯¯¯g⟩≡din. (8)

We assume since any complex phase can be incorporated into the states and . Note that the value of depends on the dc field and the light polarization . In particular, as . Therefore, the dynamic field does not cause mixing of the states and in the absence of the electric field. The polarizability in Eq. (7) represents the ac polarizability of the unperturbed ground state , provided the effects of the dc field on the ac polarizability can be neglected. In this case, .

In general, ac polarizabilities depend on the polarization and frequency of the external light field. The polarizability of an arbitrary atomic state can be decomposed into three terms:

 αℓ =α0(γJ)+iα1(γJ)MJ(ϵ×ϵ∗)⋅^z +α2(γJ)3M2−J(J+1)J(2J−1)3|ϵ⋅^z|2−12. (9)

The quantities , , and are referred to as the scalar, vector, and tensor polarizabilities, respectively Bonin and Kadar-Kallen (1994). The scalar, vector, and tensor polarizabilities are independent of the magnetic quantum number and the polarization , and hence are independent of the choice of quantization axis and field geometry. However, they depend on the light frequency , as described in Appendix A.

Hereafter, we abandon the use of the overline to distinguish between perturbed and unperturbed atomic states. Despite the lack of an overline, quantum states should be interpreted as atomic states that have been perturbed by the static electric and magnetic fields and , unless otherwise noted.

We consider a system in which atoms in the excited state undergo spontaneous decay to the lower states and with corresponding rates and , where is the natural linewidth of the state . A schematic of the relevant energy level structure is shown in Fig. 1. As atoms decay from down to , they emit fluorescent light of frequency . The LSM involves both the simulation and measurement of the spectral lineshape of the transition. In this context, the “spectral lineshape” refers to the probability of emission of fluorescent light of frequency as a function of laser frequency . Although polarizabilities and depend on , we assume that they are effectively constant for . The LSM can be applied to any atomic system with the energy level structure shown in Fig. 1.

## Iii Spectral lineshape

The ac Stark shifts cause the resonant frequency of the transition to shift as atoms travel through the standing wave. As a result, the spectral lineshape depends heavily on the details of the light field. Because the light field amplitude is not spatially uniform [see Eq. (2) and the discussion thereafter], an atom with coordinate will experience a time-dependent electric field in its rest frame.

Assuming constant velocity, the atom’s position is where is the atomic velocity and is the position of the atom when . The origin is chosen to be at one of the nodes of the standing wave. A diagram of the geometry is shown in Fig. 2. The time-dependent field experienced by the atom is given by

 E[r(t)]=E0cos(kv∥t+kr0∥)e−(v2⊥t2+r20⊥)/r2b, (10)

where is given by Eq. (2). Here and are the components of the velocity that are perpendicular and parallel to . Similarly, and correspond to the perpendicular and parallel components of the position .

The total time dependence of the electric field in Eq. (1) is due to the fast oscillation of the light at frequency and the slow modulation of the amplitude with a frequency . The amplitude modulation is additionally characterized by a temporal Gaussian envelope with a characteristic width , which is the amount of time an atom spends within the radius of the standing wave. We consider non-relativistic atoms for which the conditions and are valid, and the optical oscillations are much faster than the modulation of the amplitude . In this case, the ac Stark shift is obtained by substituting Eq. (10) into Eq. (7).

### iii.1 Absorption profile

In order to gain a qualitative understanding of the physics, we make the simplifying assumption

 E(t)≈E0cos(kv∥t). (11)

In this case, the Stark shifts of the ground and excited states lead to the following shift of the resonant energy of the transition:

 Eac(a)−Eac(g)=−12αagE20cos2(kv∥t), (12)

where

 αag≡αa−αg, (13)

is the difference of the polarizabilities of the ground and excited states. From an atom’s perspective, this is equivalent to a polarizability-dependent frequency modulation of the two counterpropagting light fields. Thus, the features of the lineshape can be understood by studying a related system: stationary atoms with fixed energy levels in the presence of two counter-propagating, frequency-modulated electric fields. In this subsection, we turn our attention to such a system.

The frequency-modulated electric fields have instantaneous frequencies and given by

 ω±(t)=ω±kv∥+ωmAmcos(ωmt), (14)

where

 ωm≡2kv∥andAm≡αagE208ℏkv∥, (15)

are the modulation frequency and modulation index, respectively. Equation (14) includes the term which accounts for the Doppler shifts of the frequencies of the two counter-propagating waves. To derive Eq. (14), we used a trigonometric identity to write and we neglected the time-independent term because it can be interpreted as an overall shift of the optical frequency: . The instantaneous frequency is characteristic of a light field with a time-dependent phase Silver (1992). Such a field is given by

 E±=(1/2)E0ei[(ω±kv∥)t+Amsin(ωmt)], (16)

where the factor of 1/2 is included so that the total field, which is the sum of two traveling waves, has an amplitude of . The effective field can be decomposed in the following way:

 E±=∞∑n=−∞(1/2)E0Jn(Am)ei(ω±kv∥+nωm)t, (17)

where are Bessel functions of the first kind. Thus the effective field consists of a principal field () which oscillates at a frequency , and infinitely many sidebands which oscillate at frequencies . The amplitude of the electric field of th sideband is .

The total electric field seen by the atom is the sum of the two counter-propagating light fields. To add the fields, the summation index in Eq. (17) is changed from to and the field is expressed as

 E−=∞∑n=−∞(1/2)E0Jn+1(Am)ei(ω+kv∥+nωm)t,

where because the modulation frequency is exactly twice the Doppler shift. Hence the sidebands of the two counter-propagating waves overlap and the total field is given by

 Etot=E++E−=∞∑n=−∞Eneiωnt, (18)

where

 En≡E02[Jn+1+Jn]andωn≡ω+kv∥+2nkv∥. (19)

In particular, the first-order sidebands from one field correspond with the carrier of the other Stalnaker et al. (2006), resulting in an absorption profile with a polarizability-dependent distortion. The “absorption profile” is a plot of the transition rate as a function of .

The rate of the transition is given by

 R=∞∑n=−∞2πℏ2|dinEn|2Γa/2π(ωn−ωag)2+(Γa/2)2. (20)

Equation (20) is valid in the weak excitation limit, that is, when the excitation rate is much smaller than all other relevant rates. To derive Eq. (20), we neglected the interference of different harmonic components, e.g., and . Such terms contribute small corrections to the transition rate which do not affect the qualitative behavior of the absorption profile. A plot of the absorption profile is given in Fig. 3. The single-atom absorption profile is clearly asymmetric about the atomic resonance ().

For an ensemble of atoms, the absorption profile is obtained by averaging the transition rate (20) over the velocity distribution. The average rate is

 ¯¯¯¯¯R≡∫∞−∞R(v∥)wL(v∥,vC)dv∥, (21)

where is the appropriate probability distribution for the parallel velocity , and is a characteristic speed in the direction. A plot of the average transition rate is shown in the insert of Fig. 3. For the insert, is taken to be a Lorentzian distribution with a full-width at half the maximum value (FWHM) of , where is the overall Doppler broadening of the line. The resulting absorption profile is similar to experimentally observed lineshapes in Yb (Section IV).

The absorption profile exhibits a polarizability-dependent feature: a dip that separates the profile into two distinct peaks. The sign of the polarizability determines whether the peak on the left is larger or smaller than the peak on the right. The following conditions need to be met in order for the distortion to be observed:

 Γa/4≪ωDandℏωD≲|αag|E20. (22)

The former condition ensures that the sidebands of the FM waves can be resolved. The latter ensures that the amplitude of the first-order sidebands is not negligible compared to the amplitude of the carrier, that is, for . If either or , then the asymmetric distortion will be suppressed, as can be seen in the insert of Fig. 3. In this case, the methods described here cannot be used to measure the polarizability . However, the LSM can still be used to measure in the absence of the distortion by comparing displacements of the central peak of the lineshape.

By omitting the Gaussian envelope in Eq. (11), we neglect effects of the atom’s finite transit time, such as broadening of the spectral line Demtröder (2003). Nonetheless, the transit time must satisfy the following restrictions:

 v⊥/rb≲dinE0/ℏandv⊥/rb≪ωD, (23)

where is the Rabi frequency of the transition. The former condition represents a system in which atoms have enough time to undergo excitation to the upper state , as will be discussed in Section III.2. The latter condition ensures that the Doppler broadening is sufficiently large that most atoms travel through many nodes and antinodes of the standing wave during their transit.

Although the absorption profile provides a satisfactory illustration of the physics, it cannot be used to measure the ac polarizabilities. The transition rate presented in Eq. (20) is valid only in the weak excitation limit and therefore cannot account for saturation effects. Moreover, Eq. (20) does not properly take into account interference of different probability amplitudes, nor does it include finite transit-time effects. A more complete picture is required to generate a theoretical lineshape that can be fitted to experimental data. Such a picture is achieved by the following model.

### iii.2 Fluorescence probability

Hereafter, we return our attention to the original system: moving atoms illuminated by light with a fixed frequency . The spectral lineshape of the transition is modeled by computing the probability of emission of fluorescent light of frequency as a function of laser frequency . The computation involves three steps. First, the time-dependent population of the state is computed by numerically solving the optical Bloch equations (OBE) for the case of atoms traveling through a non-uniform field (see Fig. 2). Second, the probability of fluorescence is determined by integrating the time-dependent decay rate with respect to time. Finally, the average fluorescence probability is computed by taking a weighted average of with respect to the atomic velocity and the offset .

Although the LSM can be used in conjunction with any atomic source, our model makes use of distributions that are appropriate for a collimated beam of thermal atoms traveling in a direction that is orthogonal to the standing wave. In this case, represents the component of the atom’s velocity along the atomic beam, and represents the angular spread of the beam. The corresponding velocity distribution is

 w(v)=wT(v⊥,vT)wL(v∥,vC), (24)

where

 wT(v⊥,vT)=2(v3⊥/v4T)e−(v⊥/vT)2, (25)

is the distribution of velocities appropriate for thermal atoms escaping from a hole, and is the velocity distribution appropriate for a collimated atomic beam. Here is the thermal speed of the atom, is the temperature of the oven, is the atom’s mass, is Boltzmann’s constant, and is the characteristic speed determined by the atomic-beam collimator. To model the effects of a vane collimator, we approximate the spread of parallel velocities by a Lorentzian distribution with a FWHM of , where is the overall Doppler broadening of the line.

In the following model, we use dimensionless parameters. Dimensionless parameters ease computation, and potentially facilitate the application of the model to several different atomic systems. Throughout, we make the approximation which is valid in the near-resonant regime ().

Time is measured in units . We define the dimensionless time and decay rates , , , and . We further define the dimensionless perpendicular and parallel velocities and , and the dimensionless perpendicular and parallel offsets and , respectively.

Consistent with the discussion in Section III.1, we introduce the following dimensionless parameters: the saturation parameter , characteristic modulation index , and Doppler parameter , defined by

 S0 ≡[dinE0/(ℏΓa)]2, (26) S1 ≡αagE20/(8ℏωD), (27)

and

 S2≡2ωD/Γa, (28)

respectively. We define an additional parameter by

 S3≡vT/(rbΓa). (29)

Note that is the time that an atom spends within the radius of the light field and hence represents the dimensionless transit time.

In terms of the dimensionless parameters, the conditions presented in expressions (22) reduce to and . When either of these conditions is violated, the characteristic dip in the lineshape is suppressed, as can be seen in Fig. 4. Likewise, conditions (23) reduce to and . Whereas the absorption profile discussed in Section III.1 was valid only in the weak excitation limit , the model of the fluorescence can accommodate large saturation parameters.

Let be the elements of the density matrix in the atom’s rest frame for states . We assume that the rotating wave approximation holds and dynamic interactions between states other than and can be neglected. In this case, the dimensionless optical Bloch equations (OBE) for the configuration shown in Fig. 1 are Loudon (2000)

 ˙ρaa =−i2Ω(ρag−ρga)−ρaa, (30a) ˙ρag =+i2Ω(ρgg−ρaa)−12(1−2iΔ)ρag, (30b) ˙ρbb =−Gbρbb+Gabρaa, (30c) ˙ρcc =−Gcρcc+Gacρaa, (30d)

where . The remaining density matrix elements and are determined from and . Here

 Ω≡1Γa[⟨a|d⋅ϵ|g⟩E(t)ℏ]=√S0f(ξ,u), (31)

is the Rabi frequency,

 Δ≡1Γa[ω−E(a)−E(g)ℏ]=δ+2S1S2f(ξ,u)2, (32)

is the detuning of the laser light from the resonance, , and the function is defined by

 f(ξ,u)≡cos[(S2/2)u∥τ+φ]e−[(S3u⊥τ)2+ξ2], (33)

where and . We further assume that all atoms initially occupy the ground state.

The probability that an atom will emit fluorescent light of frequency in a time interval is given by

 F=∫τfτiGbρbb(τ)dτ, (34)

where is the time-dependent rate of the decay. The choice of integration interval depends on both the transit time and the characteristic time of the fluorescent decay after the atoms leave the light field. In the case where the decay time is shorter than the transit time, it is appropriate to define the integration limits by . The factor of 3 ensures that the atom is “far” from the standing wave at the integration limits. To model a system with slower decays, the integration limit must be extended.

The fluorescence probability depends on the parameters of the atomic trajectory, that is, . We define the average probability of fluorescence by

 ¯¯¯¯¯F=∬F(ξ,u)w(ξ,u)dξdu, (35)

where is the probability distribution associated with the atom’s initial position and velocity. In our model, we assume , where the distribution is a uniform distribution over the intervals and . The finite integration limits are justified by the following properties of the system: First, the amplitude of the standing wave drops to less than 0.01% of its maximum value when . Therefore, atoms will only pass through the light if . Second, constitutes a phase shift of the electric field which is unique only for . Consistent with Eqs. (24) and (25), the velocity distribution satisfies , where is the velocity distribution for atoms escaping from a hole with unit thermal speed, and is a Lorentzian distribution with FWHM of 1.

The fluorescence probability is a function of the detuning from the atomic resonance, that is,

 ¯¯¯¯¯F=¯¯¯¯¯F(δ;S,G), (36)

where and are parameters. We refer to a plot of as a function of as the “simulated lineshape” of the transition. Three such plots are shown in Fig. 4. In Fig. 4, curves (a) and (b) demonstrate the suppression of the polarizability-dependent distortion when either the Doppler parameter or the modulation index is too small. The distortion is most pronounced in curve (c), for which and . The simulated lineshapes in Fig. 4 are qualitatively similar to the absorption profiles in the insert of Fig. 3, as expected.

### iii.3 Numerical procedure

The numerical procedures described here are valid for a variety of atomic species. However, the simulations were performed with parameter values appropriate for the Yb system described in Section IV.

We used a stiffly stable Rosenback method Press et al. (2007) to numerically solve a system of equations related to Eqs. (30) and (34), with . This system of equations is described in Appendix B. The Rosenback method involves two tolerances–denoted atol and rtol in Ref. Press et al. (2007)–which were both set to . The multi-dimensional integral in Eq. (35) was computed using an adaptive Monte Carlo routine Press et al. (2007). In our implementation, the integration routine involves evaluations of the integrand. For various values of , the average estimated error was less than 1% of the value of the integral.

For computational purposes, we restricted the integration to the following finite domain: , , , and . The subdomains for and were discussed after Eq. (35). The finite integration subdomains for and are justified as follows: Atoms with a large parallel speed experience a Doppler shift that is much larger than the characteristic Doppler broadening of the spectral line. Such atoms only contribute to the wings of the lineshape, where is large and the probability of fluorescence is very small. Moreover, for a Lorentzian velocity distribution with unit FWHM, for about 95% of atoms.

On the other hand, atoms with a perpendicular speed that satisfies are moving so fast that the transit time is much smaller than the inverse Rabi frequency. Such atoms do not spend enough time in the light field for the transition to be realized. Since most atoms travel at or near the thermal speed , the condition represents a system in which most atoms have enough time to interact with the light. We assume that atoms with speed do not contribute significantly to the lineshape. Note that for only about 0.1% of atoms. Finally, we ignore counterflow of atoms in the atomic beam by requiring .

For fixed and , we computed the average fluorescence for 100 discrete values of , and various discrete values of , , and . Here . The results were interpolated using cubic splines to approximate the continuous function . Three curves which are typical of those produced by this procedure are presented in Fig. 4.

The LSM involves fitting the simulated curve to the observed lineshape to determine best-fit values of , , and . From the best-fit values, the following three quantities can be calculated: the polarizability difference , the circulating power of the standing wave111The standing wave is formed by two counter-propagating waves of light. The circulating power of the standing wave is defined as the average power of a single traveling wave. The electric field of the wave propagating in the direction is given by . The corresponding time-averaged intensity in SI units is , where the time average is taken over a single period of oscillation. Therefore, the circulating power of the standing wave is given by ., and the Doppler broadening of the transition which are given by

 αag =[4d2inℏΓa]×S1S2/S0, (37) P =[116πr2bcϵ0ℏ2Γ2ad2in]×S0, (38)

and

 ωD=(Γa/2)×S2, (39)

respectively. Here is the permittivity of free space. The present implementation of the LSM uses Mathematica’s nonlinear regression routine to determine the best-fit values of , , and . Alternatively, Eq. (38) can be solved for in terms of and . Thus the LSM can also be used to measure the induced dipole moment when the power is known, as was done in the previous application of the LSM Stalnaker et al. (2006).

The treatment of systematic errors and statistical uncertainties is straightforward. In Eq. (37), for instance, the uncertainty in the dimensionful quantity is due solely to systematic effects, whereas the uncertainty in the term arises from statistical uncertainties in both the observed signal and the fitting algorithm. The total uncertainty of the quantity is obtained by adding these independent uncertainties in quadrature. If the signal-to-noise ratio (SNR) of the observed lineshape is sufficiently high, then the error of the measurement of will be dominated by the uncertainties of the known quantities and .

## Iv Application to Ytterbium

The electronic structure of Yb is shown in Fig. 5. The low-lying energy eigenstates of Yb match the structure shown in Fig. 1 under the following mapping: , , , and . Therefore, the LSM can be used to measure the difference in ac polarizabilities of the upper state and the ground state at 408 nm by analyzing the lineshape of the 408-nm transition. In this case, the lineshape is measured by observing the 556-nm fluorescence of the decay. The intermediate state is metastable and hence .

There is an additional decay of to the metastable state , which is not shown in Fig. 1. The state can represent multiple metastable states, including both and . In this interpretation, is the rate of decay of to all metastable states.

The highly forbidden transition is induced by the Stark mixing technique. This technique involves the application of a static, uniform electric field which mixes the upper state with opposite-parity states, predominantly the state. The effective dipole moment associated with the Stark-induced transition is given by Stalnaker et al. (2006)

 din=βag|(Edc×ϵ)(1)−Ma|, (40)

where is the vector transition polarizability of the transition, is the elementary charge, is the Bohr radius, and is a spherical tensor of rank one222Let be the rank-one spherical tensor associated with the Cartesian vector . Then the components of are given by and .. Here is the magnetic quantum number of the state.

Because the angular momentum of the ground state is , only the scalar term in Eq. (9) contributes to the polarizability of . That is, and so is independent of the geometry of the applied fields. Hence the dependence of on the field geometry is due entirely to the vector and tensor polarizabilities and of the excited state . In this case, the LSM is sensitive to the difference of the scalar polarizabilities of the ground and excited states. However, the vector and tensor polarizabilities of the excited state can be measured unambiguously by varying the polarization of the standing wave.

A recent calculation Derevianko and Dzuba () of the polarizability of the ground state at 408 nm yielded333Equation (7) implies that has units of energy per squared electric field. However, in this work is normalized by and presented in units of frequency per squared electric field. For a more thorough discussion of unit conventions, see Ref. Mitroy et al. (2010).

 α0(1S0)=0.218(11)Hz⋅(V/cm)−2. (41)

Calculations of the polarizability at 408 nm are complicated by the potential existence of odd-parity eigenstates with energy close to twice the energy of a 408-nm photon. Such states could lead to a resonantly enhanced polarizability of the state. The energy spectrum in this region (which is below the ionization limit) is very dense due to the excitation of orbitals. The knowledge of the energy spectrum is far from complete in this region. This provides one of the motivations for determining the polarizabilities experimentally.

The first implementation of the LSM Stalnaker et al. (2006) was used to measure the quantity

 αIag=−0.312(34)Hz⋅(V/cm)−2, (42)

were . Here the superscript “I” is introduced to distinguish this measurement from the results of the present work. To determine the tensor contribution unambiguously requires a second measurement of a different combination of scalar and tensor polarizabilities. This is accomplished in the present work by the application of a dc electric field that is parallel to the standing wave (Fig. 6), whereas the previous measurement was performed with a dc field that was perpendicular to the standing wave. In addition, the current experiment includes a strong magnetic field not present in the previous case. The magnetic field makes possible the measurement of the vector polarizability, as discussed in Section V.

### iv.1 Experimental apparatus and field geometry

The details of the experimental apparatus were reported elsewhere Tsigutkin et al. (2010), and only a brief description is provided here. A schematic of the setup is shown in Fig. 6. A beam of Yb atoms is produced by a stainless-steel oven loaded with Yb metal, operating at 500 C. The oven is outfitted with a multislit nozzle, and there is an external vane collimator reducing the spread of the atomic beam in the -direction. Downstream from the collimator, atoms enter a region with three external fields: a uniform, static magnetic field ; a uniform, static electric field ; and a non-uniform, dynamic electric field .

The magnetic field is generated by a pair of axially magnetized neodymium (Nd) magnets. These magnets produce a field with sufficient strength (more than 50 G) to completely isolate the Zeeman sublevels of the upper state . The electric field is generated by two wire-frame electrodes separated by 2 cm. The ac electric field is due to standing-wave light at the transition wavelength of 408.346 nm in vacuum, which is produced by doubling the frequency of the output of a Ti:sapphire laser (Coherent 899). About 7 mW of 408-nm light is coupled into a power buildup cavity (PBC) with finesse of approximately 15,000. The PBC is an asymmetric cavity with a flat input mirror and a curved back mirror with a 50-cm radius of curvature. The separation between the mirrors is 22 cm.

Fluorescent light with a wavelength of 556 nm is collected with a light guide and detected with a photomultiplier tube (PMT). With the exception of the PMT, the entire apparatus is contained within a vacuum chamber with a residual gas pressure of Torr.

As can be seen in Fig. 6, the fields and point in the direction. Likewise, the standing wave is oriented along the -axis. The light field lies in the -plane. For this geometry, the transition to the upper state is suppressed.

The polarization of the light field is of the form , where . To further characterize the polarization, we introduce three parameters: polarization angle , degree of ellipticity , and handedness , which are given by Auzinsh et al. (2010)

 tan2θ =2Re(ϵxϵ∗y)/(|ϵx|2−|ϵy|2), (43) sin2ϕ =2|Im(ϵxϵ∗y)|=|i(ϵ×ϵ∗)⋅^z|, (44)

and

 h=−sgn[i(ϵ×ϵ∗)⋅^z]. (45)

Linearly, circularly, and elliptically polarized light are described by , , and , respectively. The sense of rotation is determined by : left and right-handed polarizations correspond to and , respectively. Substituting and into Eqs. (40) and (9) yields

 din=βagEdc√(1/2)[1+hMasin(2ϕ)], (46)

and

 αa=α0(3D1)−hMasin(2ϕ)α1(3D1)−12α2(3D1), (47)

for . Here we have used Eqs. (44) and (45) to eliminate and in favor of the degree of ellipticity and the handedness . For this geometry, both and are independent of the polarization angle .

According to the geometry in Fig. 6, the component of the atom’s velocity that is perpendicular to the standing wave is . The output of the oven is about 6 mm in the -direction, and is located more than 20 cm away from the standing wave. In order for an atom to pass through the standing wave, its velocity components must satisfy . Therefore, the approximation is valid and the use of the thermal distribution given in Eq. (25) is justified. However, for a high-precision measurement, the effect of this approximation needs to be investigated.