Frequency Shifts due to Stark Effects on a Rb two-photon transition

TL;DR

This paper provides a detailed calculation and measurement of Stark shifts in a rubidium two-photon transition, crucial for developing stable compact optical atomic clocks, including strategies to mitigate these effects.

Contribution

It offers the first rigorous calculation of ac-Stark and blackbody radiation shifts for the Rb two-photon transition, with experimental validation and analysis of mitigation techniques.

Findings

Measured ac-Stark shift agrees with calculations

Identified temperature conditions for zero BBR shift or sensitivity

Quantified dc-Stark shift and clock sensitivity to alignment variations

Abstract

The $5S_{1/2}\rightarrow 5D_{5/2}$ two-photon transition in Rb is of interest for the development of a compact optical atomic clock. Here we present a rigorous calculation of the 778.1~nm ac-Stark shift ($2.30(4) \times10^{-13}$(mW/mm$^2$)$^{-1}$) that is in good agreement with our measured value of $2.5(2) \times10^{-13}$(mW/mm$^2$)$^{-1}$. We include a calculation of the temperature-dependent blackbody radiation shift, we predict that the clock could be operated either with zero net BBR shift ($T=495.9(27)$~K) or with zero first-order sensitivity ($T=368.1(14)$~K). Also described is the calculation of the dc-Stark shift of 5.5(1)$\times 10^{-15}$/(V/cm$^2$) as well as clock sensitivities to optical alignment variations in both a cat's eye and flat mirror retro-reflector. Finally, we characterize these Stark effects discussing mitigation techniques necessary to reduce final clock instabilities.

Tables

Table 1. Table 1: Values for the reduced electric-dipole matrix elements, ⟨ J | d | J ′ ⟩ quantum-operator-product 𝐽 𝑑 superscript 𝐽 ′ \langle J|d|J^{\prime}\rangle , taken from Safronova  et al. ( 2004 ) , are presented in a.u., the transition energies are taken from Ralchenko  et al. ( 2011 ) , and the Einstein A coefficients are calculated utilizing Equation 3 , except as noted. The chosen sign convention yields negative energies for the 5 ​ D 5 / 2 → 5 ​ P 3 / 2 → 5 subscript 𝐷 5 2 5 subscript 𝑃 3 2 5D_{5/2}\rightarrow 5P_{3/2} and 5 ​ D 5 / 2 → 6 ​ P 3 / 2 → 5 subscript 𝐷 5 2 6 subscript 𝑃 3 2 5D_{5/2}\rightarrow 6P_{3/2} transitions, indicating that 5 ​ D 5 / 2 5 subscript 𝐷 5 2 5D_{5/2} is the higher state.

Transition	$⟨J\left|d\right|J^{\prime }⟩$	Energy (cm-1)	$A_{ki}$ (MCyc/sec)	Transition	$⟨J\left|d\right|J^{\prime }⟩$	Energy (cm-1)	$A_{ki}$ (MCyc/sec)
$5S_{1/2}\to 5P_{1/2}$	4.233(6)	12578.950(2)	36.129(52) 111Lifetime measured in Refs. Volz and Schmoranzer (1996); Simsarian et al. (1998); Gutterres et al. (2002)	$5D_{5/2}\to 6F_{5/2}$	1.373(22) 5	4924.48(2)	0.076(2)
$5S_{1/2}\to 5P_{3/2}$	5.979(9)	12816.545(2)	38.12(13) 222Lifetime measured in Refs. Volz and Schmoranzer (1996); Simsarian et al. (1998); Boesten et al. (1997); Gutterres et al. (2002)	$5D_{5/2}\to 7F_{5/2}$	0.871(38) 5	5738.23(2)	0.048(4)
$5S_{1/2}\to 6P_{1/2}$	0.3235(9) 333Matrix elements measured in Herold et al. (2012)	23715.081(10)	1.498(8)	$5D_{5/2}\to 8F_{5/2}$	0.641(24) 5	6266.12(2)	0.034(3)
$5S_{1/2}\to 6P_{3/2}$	0.5239(8) 3	23792.591(10)	1.873(6)	$5D_{5/2}\to 9F_{5/2}$	0.513(26) 5	6629(3)	0.026(3)
$5S_{1/2}\to 7P_{1/2}$	0.101(5) 444Matrix element calculated in Ref. Safronova et al. (2004)	27835.02(1)	0.223(22)	$5D_{5/2}\to 10F_{5/2}$	0.440(22) 5	6889(3)	0.021(2)
$5S_{1/2}\to 7P_{3/2}$	0.202(10) 4	27870.11(1)	0.447(44)	$5D_{5/2}\to 11F_{5/2}$	0.450(23) 5	7080(3)	0.024(2)
$5S_{1/2}\to 8P_{1/2}$	0.059(3) 4	29834.94(1)	0.094(10)	$5D_{5/2}\to 12F_{5/2}$	0.472(71) 5	7225(3)	0.028(8)
$5S_{1/2}\to 8P_{3/2}$	0.111(6) 4	29853.79(1)	0.166(18)	$5D_{5/2}\to 13F_{5/2}$	0.478(72) 5	7338(3)	0.030(9)
$5D_{5/2}\to 5P_{3/2}$	1.999(70) 4	-12886.95(5)	2.89(20)	$5D_{5/2}\to 4F_{7/2}$	30.316(64) 5	1088.59(2)	0.300(1)
$5D_{5/2}\to 6P_{3/2}$	24.621(79) 4	-1910.907(52)	1.428(8)	$5D_{5/2}\to 5F_{7/2}$	11.24(32) 5	3574.27(2)	1.461(83)
$5D_{5/2}\to 7P_{3/2}$	13.82(33) 4	2166.61(2)	0.984(47)	$5D_{5/2}\to 6F_{7/2}$	6.140(97) 5	4924.46(2)	1.140(36)
$5D_{5/2}\to 8P_{3/2}$	3.292(11) 4	4150.29(2)	0.392(3)	$5D_{5/2}\to 7F_{7/2}$	3.90(17) 5	5738.22(2)	0.726(63)
$5D_{5/2}\to 9P_{3/2}$	1.691(85) 555Matrix elements derived using the method described in Reference Safronova et al. (2004).	5266.69(2)	0.212(21)	$5D_{5/2}\to 8F_{7/2}$	2.87(11) 5	6266.12(2)	0.512(39)
$5D_{5/2}\to 10P_{3/2}$	1.099(55) 5	5957.66(2)	0.129(13)	$5D_{5/2}\to 9F_{7/2}$	2.29(12) 5	6629(3)	0.388(41)
$5D_{5/2}\to 11P_{3/2}$	0.799(40) 5	6415.02(2)	0.085(8)	$5D_{5/2}\to 10F_{7/2}$	1.968(98) 5	6889(3)	0.321(32)
$5D_{5/2}\to 12P_{3/2}$	0.627(94) 5	6733.54(6)	0.061(18)	$5D_{5/2}\to 11F_{7/2}$	2.01(10) 5	7080(3)	0.365(4)
$5D_{5/2}\to 13P_{3/2}$	0.578(87) 5	6964.13(6)	0.057(17)	$5D_{5/2}\to 12F_{7/2}$	2.11(32) 5	7225(3)	0.425(129)
$5D_{5/2}\to 4F_{5/2}$	6.779(14) 5	1088.62(2)	0.020(1)	$5D_{5/2}\to 13F_{7/2}$	2.14(32) 5	7338(3)	0.429(128)
$5D_{5/2}\to 5F_{5/2}$	2.513(72) 5	3574.29(2)	0.097(6)

Table 2. Table 2: Contribution to the atomic polarizability from the atomic core and continuum in atomic units. These numbers were included in the final calculation of ac and dc Stark shift.

Contribution	Static Scalar Polarizability666Polarizability numbers are taken from Safronova and Safronova (2011).
${\alpha }_{5S_{1/2}}\left(c\right)$	9.1
${\alpha }_{5S_{1/2}}\left(tail\right)$	1.24
${\alpha }_{5D_{5/2}}\left(c\right)$	9.0
${\alpha }_{5D_{5/2}}\left(>7P_{3/2}\right)$	88(0)
${\alpha }_{5D_{5/2}}\left(>6F_{5/2}\right)$	13(0)
${\alpha }_{5D_{5/2}}\left(>13F_{7/2}\right)$	100(0)
Contribution	Dynamic Scalar Polarizability
	at $\lambda =778.1$ nm
${\alpha }_{5S_{1/2}}\left(c\right)$	8.7
${\alpha }_{5S_{1/2}}\left(tail\right)$	0.5
${\alpha }_{5D_{5/2}}\left(c\right)$	9.0
${\alpha }_{5D_{5/2}}\left(>13P_{3/2}\right)$	-4(74)
${\alpha }_{5D_{5/2}}\left(>13F_{5/2}\right)$	-1.8(5)
${\alpha }_{5D_{5/2}}\left(>13F_{7/2}\right)$	-33(110)

Table 3. Table 3: The extended ray trace matrices used to calculate misalignment effects from the cat’s eye optic

Freespace propagation of distance $N$
Thin lens with focal length f at angle $\xi$
Concave mirror, radius $f\left(1+\Delta f\right)$, at angle $\xi +\beta$
Flat mirror, at angle $\xi$

Table 4. Table 4: Proposed wavelengths, associated shifts and required power (mW) per mW of 778 nm for a ac-Stark mitigation laser.

Wavelength	ac-Stark shift	Power multiplier
(nm)	(Hz/(mW/mm2))	$\times$P778
1556.2	-16.5	10.8
808	-30.9	5.7
785	-62.5	2.9
778	178.5	X

Equations

$$\delta \nu \left(r,z\right)=\frac{\Delta \alpha }{2c{ϵ}_{0}h}I\left(r,z\right),$$

$$\alpha \left(\omega ,J\right)=-\frac{2}{3\left(2J+1\right)}\sum _{J^{\prime }}\frac{{\omega }_{J^{\prime },J}|⟨J|d|J^{\prime }⟩{|}^2}{{\omega }_{J^{\prime },J}^2-{\omega }^2}.$$

$$A_{J,J^{\prime }}=\frac{2{\omega }^3(e_c{a}_0⟨J|d|J^{\prime }⟩{)}^2}{3{ϵ}_{0}h{c}^3(2J+1)},$$

$$E=\frac{A}{(n-d{)}^2},$$

$$\overline{\delta \nu }=\frac{\Delta \alpha }{2c{ϵ}_{0}h}\overline{I_{tot}},$$

$$\overline{I_{tot}}=\frac{\int I_{tot}{I}_1{I}_{2}dV}{\int I_1{I}_{2}dV},$$

$$\left(\begin{array}{ccc}A& B& \delta \\ C& D& \gamma \\ 0& 0& 1\end{array}\right),$$

$$I_{incident}=I_0{e}^{-2(x^2+y^2)/w_0^2},$$

$$M=\left(\begin{array}{ccc}1& 2N_{2}f& N_{2}f\xi \\ 0& 1& \xi \\ 0& 0& 1\end{array}\right).$$

$$q=q_1+i{q}_2=\frac{A{w}_0^2\pi i/\lambda +B}{C{w}_0^2\pi i/\lambda +D},$$

$$q=w_0^2\pi i/\lambda +2N_{2}f,$$

$$I_{retro}=I_0{e}^{-2(x^2+y^2+2x(N_{2}f+z)(\xi -\theta )+(N_{2}f+z{)}^2(\xi -\theta {)}^2)/w_0^2},$$

$$\overline{I}=\frac{\overline{I_{tot}}}{\overline{I_{tot}}(\xi =\theta /2)}=\frac{\sqrt{3}}{2}\frac{\mathrm{erf}\left(2(y_1+y_2)/\sqrt{3}\right)-\mathrm{erf}\left(2y_2/\sqrt{3}\right)}{\mathrm{erf}\left(y_1+y_2\right)-\mathrm{erf}\left(y_2\right)},$$

$$\overline{I}\approx 1-\frac{\left(a^2+3a{N}_{2}f+3N_2^2{f}^2\right){\left(\xi -\theta \right)}^2}{9{\omega }_0^2},$$

$$\frac{d\overline{I}}{d\theta }\approx \frac{-2\left(a^2+3a{N}_{2}f+3N_2^2{f}^2\right)\theta }{9{\omega }_0^2}.$$

$$\frac{d\overline{I}}{d\theta }\approx \frac{-2f^2\left(N_1+N_2\right)\left(\xi +\left(N_1+N_2\right)\theta \right)}{3{\omega }_0^2}$$

$$\frac{d\overline{I}}{d\xi }\approx \frac{-2f^2\left(\xi +\left(N_1+N_2\right)\theta \right)}{3{\omega }_0^2}$$

$$⟨E\left(\omega \right){⟩}^2=\frac{\hslash }{{\pi }^2{ϵ}_0{c}^3}\frac{{\omega }^3}{e^{\hslash \omega /k_{B}T}-1}.$$

$$\delta \nu =\frac{1}{2h}{\int }_0^{\infty }\Delta \alpha (\omega )E{\left(\omega \right)}^{2}d\omega ,$$

$$\delta \nu =\frac{\Delta \alpha }{2h}{\int }_0^{\infty }E{\left(\omega \right)}^{2}d\omega \approx \frac{\Delta \alpha }{2h}{\left(8.3\frac{V}{cm}\right)}^2{\left(T/300K\right)}^4.$$

$$\alpha (\omega ,J)=-\frac{2}{3(2J+1)\hslash }\sum _{J^{\prime }}\frac{{\omega }_{J^{\prime },J}|⟨J|d|J^{\prime }⟩{|}^2({\omega }_{J^{\prime },J}^2-{\omega }^2)}{({\omega }_{J^{\prime },J}^2-{\omega }^2{)}^2+{\Gamma }_J^2{\omega }^2}.$$

$$\left(\begin{array}{c}{X}_f\\ Y_f\\ 1\end{array}\right)=\frac{1}{1+\Delta f}\left(\begin{array}{ccc}A& B& E\\ C& D& F\\ 0& 0& 1+\Delta f\end{array}\right)\left(\begin{array}{c}f\Delta X_i\\ \theta \\ 1\end{array}\right),$$

$$A=$$

$$B=$$

$$+(2(N_2-1)((N_2-1)\Delta d-1)+(2(N_2-1)\Delta d-1)N_2)\Delta f),$$

$$C=$$

$$D=$$

$$E=$$

$$F=$$

$$\overline{I}=\frac{\overline{I_{tot}}}{\overline{I_{tot}}(0)}=e^{\frac{-f^2(2x_i(1+\Delta f)+\xi +3\Delta f\xi +\beta +\Delta f\beta +(N_1+N_2)\theta -\Delta f\theta +(N_1+N_2)\Delta f\theta {)}^2}{3(1+\Delta f{)}^2{w}_0^2}}.$$

$$\overline{I}\approx$$

$$\Delta d^2\left(4-4N_2+\frac{88N_2^2}{9}+\frac{37a^2}{27f^2}-\frac{a}{f}+\frac{40a{N}_2}{9f}-2N_1+\frac{32N_2{N}_1}{3}+\frac{4a{N}_1}{3f}+4N_1^2\text{-}\frac{2{\pi }^2{\omega }_0^4}{f^2{\lambda }^2}\right).$$

Full text

††preprint: APS/123-QED

I Introduction

Narrow-line transitions can be realized through two-photon spectroscopy to explore a wide array of scientific phenomena. Two-photon transitions have been successfully leveraged for measuring fine and hyperfine structures Hänsch et al. (1974), Zeeman Bloembergen et al. (1974) and Stark Harvey et al. (1975) splittings, hot vapour collisional effects Biraben et al. (1975); Zameroski et al. (2014), and are important in precision Hydrogen spectroscopy Parthey et al. (2011). A common technique in two-photon spectroscopy is the degenerate two-photon method which uses photons derived from the same source, resulting in identical frequencies. The advantage of using degenerate photons in two beams of opposite $\vv{\bm{k}}$ vector is that all atoms, regardless of velocity, can contribute to a Doppler-free signal Cagnac et al. (1973); Biraben et al. (1974); Levenson and Bloembergen (1974); Bjorkholm and Liao (1974); Hänsch et al. (1974). As employed here Martin et al. (2018) the Rb $5S_{1/2}\rightarrow 5D_{5/2}$ transition can be leveraged to create a high stability optical clock with a simple vapour cell architecture, without the need for laser cooling, offering an alternative to saturated absorption systems such as molecular iodine Schkolnik, Vladimir et al. (2017), or pulsed optically pumped microwave systems Yun et al. (2017); Micalizio et al. (2012); Lin et al. (2017). Offering a simpler and more compact approach than more complicated (albeit higher stability) optical lattice clocks Brehm and Mullin (1989); Brusch et al. (2006); Dzuba and Derevianko (2010); Guo et al. (2010); Derevianko and Katori (2011); Ludlow et al. (2015); Doppler free degenerate two-photon spectroscopy provides an appealing architecture for which to build a compact optical atomic frequency standard Martin et al. (2018); Perrella et al. (2013a); Phelps et al. (2018) and has already shown promise for very small packaging Newman et al. (2019).

The two-photon transition in Rb can be driven with two degenerate 778.1 nm photons, which can be generated either directly with a 778.1 nm laser diode or through second harmonic generation (SHG) of mature telecommunications C-band lasers at 1556.2 nm Martin et al. (2018). The relatively small virtual state detuning (see Figure 1) results in significant atomic excitation rates at modest optical intensities Nez et al. (1993); Touahri et al. (1997); Hilico et al. (1998), which allows for a high signal-to-noise ratio of the 420 nm fluorescence, as shown in Figure 1, due to the simple detection scheme with spectral filtering of the incident probe beam from the fluorescence signal. Fractional frequency instabilities of $<4\times 10^{-13}/\sqrt{\tau}$ up to 10,000 seconds have been measured, and operation with increased SNR has shown short term stability of $1\times 10^{-13}$ at one second Martin et al. (2018). This performance is comparable to other commercially available compact clocks, specifically the laser cooled microwave Rb compact clocks whose fractional frequency instability is $<8\times 10^{-13}/\sqrt{\tau}$ Ascarrunz et al. (2018) , and the current global position system (GPS) rubidium atomic frequency standard (RAFS) whose instability is $<2\times 10^{-12}/\sqrt{\tau}$ Phelan et al. (2005) over the same time-frame.

While laboratory based optical lattice systems provide the highest stability Brehm and Mullin (1989); Brusch et al. (2006); Dzuba and Derevianko (2010); Guo et al. (2010); Derevianko and Katori (2011); Ludlow et al. (2015) and progress towards portable lattice clocks is ongoing Koller et al. (2017), a more compact, transportable lattice clock has yet to be fully realized. However, the Optical Rubidium Atomic Frequency Standard (O-RAFS) has potential to meet current needs for compact frequency standards whose stability exceed active hydrogen maser capabilities Peters and Owings (1997), fractional frequency instabilities $<1\times 10^{-13}/\sqrt{\tau}$. Mitigation of the ac-Stark shift will be an integral component to achieving these ambitious instability goals. Unlike its lattice clock counterparts which leverage well known “magic” wavelengths Katori et al. (1999); Ye et al. (1999) to eliminate this light shift, no common light shift mitigation techniques are used for vapour cell clocks. Because the atoms are sensitive to the average intensity across the vapour, imperfect or unstable optical alignment can also lead to Stark shift fluctuations and clock instabilities. Moreover, since the atomic vapour is heated to increase signal to noise, the atomic vapour is immersed in blackbody radiation (BBR) Beloy et al. (2012); Safronova et al. (2013); Middelmann et al. (2012). Normally BBR is treated as a dc shift as the blackbody spectrum is far off resonance from the atomic transitions. However, the $5D_{5/2}$ state in Rb has resonant transitions that are well within the blackbody spectrum requiring calculation as an ac-Stark effect. Careful calculations and measurements of both the ac- and dc-Stark shifts are required to make predictions of clock performance to determine feasible clock instability goals, and decide whether more complicated Stark shift mitigation techniques are required.

The paper is organized as follows: Section II.1 details a calculation of the ac-Stark shift at 778.1 nm, Section II.2 calculates misalignment contributions to the ac-Stark shift, Section II.3 describes a measurement of the ac-Stark shift at 778.1 nm, Section III investigates a two-photon two color approach to excitation of the atom, and Section IV calculates the blackbody radiation shift for the two-photon transition.

II Stark Shift at 778.1 nm

II.1 Calculation

Although many environmental variables impact the clock instability of O-RAFS, the inherently large ac-Stark shift motivates the most difficult requirements. Careful calculation and direct measurement of the ac-Stark shift magnitude are pivotal to understanding the overall impact on clock performance. The ac-Stark shift can be written as Grynberg and Cagnac (1977),

[TABLE]

with $h$ Planck’s constant, $c$ the speed of light, $\epsilon_{0}$ permittivity of free space, $I\left(r,z\right)$ is the laser intensity, $z$ is the optical axis of the beam and $r$ completes the cylindrical coordinate system, and $\Delta\alpha$, the differential atomic polarizability between the two clock states will need to be calculated. Although the atomic vapour will absorb light from the beam the scattering rate on resonance for the two-photon transition is small, and the laser intensity along the propagation axis can be approximated as constant, $I\left(r,z\right)\approx I\left(r\right)$.

The rank-2 atomic polarizability tensor can be separated into three irreducible components: the scalar (trace), the vector (free symmetric) and the tensor (anti-symmetric) polarizabilites. The two-photon transition is pumped with linearly polarized light, yielding zero vector shift, and each hyperfine state is addressed uniformly, leaving the atom orientation independent, netting zero tensor shift. The remaining scalar term is written below in atomic units Deutsch and Jessen (2010); Steck (2015),

[TABLE]

$\langle J|d|J^{\prime}\rangle$ is the dipole matrix element whose resonant frequency is $\omega_{J^{\prime},J}$; $J$ and $J^{\prime}$ are the associated angular momentum quantum numbers.

Final calculation of Equation 2 uses the finite basis of B-splines as outlined in Ref. Johnson et al. (1988). Tables 1 and 2 summarize the states and extra considerations included in the polarizability calculation, displaying the transition energy difference, the dipole matrix elements, and the Einstein A coefficients calculated from (unless stated otherwise) Hilborn (1982); Corney (1977); Ditchburn (1976),

[TABLE]

where $e_{c}$ is the electron charge and $a_{0}$ is the Bohr radius. A large number of the parameters listed in Table 1 originate from Safronova et al. Safronova et al. (2004), however, the $5S_{1/2}\rightarrow 5P_{1/2}$ and $5S_{1/2}\rightarrow 5P_{3/2}$ dipole matrix elements are calculated utilizing measured lifetimes Volz and Schmoranzer (1996); Simsarian et al. (1998); Boesten et al. (1997); Gutterres et al. (2002). The final matrix elements were calculated utilizing the method described in Reference Safronova et al. (2004); these elements only account for a small fraction of the overall differential atomic polarizability. Uncertanities in the matrix elements were estimated according to References Safronova and Johnson (2008); Jiang et al. (2009). A majority of the energy levels and uncertainties were obtained from Ralchenko et al. (2011). However, the energy levels for the $\left(9-13\right)F$ states were calculated utilizing quantum defect theory(QDT) with a correction accounting for slight discrepancies between observed and predicted energy levels. QDT generalizes that energy deviations from the Rydberg atom can be written,

[TABLE]

where, E is the energy, n is the principal quantum number and d is the quantum defect. Equation 4 can be used to calculate the energies and thus the transition frequencies subtracting the result from the Rb ion limit in Sansonetti and Nave (2010). Necessary elements for the calculation are the Rydberg constant substituted for A from Mohr et al. (2012) and the defects for Rb which are: $S=3.13$, $P=2.64$, $D=1.35$, and $F=0.016$ Gallagher and Peram (2011). The uncertainty of the energy levels is extrapolated from lower states.

The differential polarizability was calculated for a range of incident wavelengths shown in Figure 1. Calculation of the ac-Stark shift at 385.284 THz yields a fractional frequency shift of $2.30(4)\times 10^{-13}$ /(mW/mm2).

Equation 1 shows an ac-Stark shift dependence on local intensity of the laser electric field. The atomic vapour effectively samples the laser intensity distribution, and the fluorescence spectrum is shifted on average by

[TABLE]

$\overline{I_{tot}}$ is the spatial weighted average intensity,

[TABLE]

with the total intensity as the sum of the two beam profiles $I_{tot}=I_{1}+I_{2}$. The weighting function $I_{1}I_{2}$ is utilized as a measure of the intensity in the two-photon region, the beam overlap in the vapour cell, the main contributor to the ac-Stark shift. Equation 5 details that variations in laser intensity will cause a clock shift, and these can arise from laser power fluctuations as well as slight optical alignment variations.

II.2 Alignment Shift

Typical optic mounts allow slight tip/tilt adjustments which are required for control of the beam. In the O-RAFS system, variations in alignment affect the total intensity of the two-photon excitation causing an ac-Stark instability. This section examines slight angular variations of the laser light emitted from the fiber launcher, as well as slight variation of the retro-reflector, and calculates the variation of $\bar{I_{tot}}$. Figure 2 presents what a small angular misalignment would look like for both a flat mirror (a) and a cat’s eye retro-reflector (b). These two separate retro-reflectors have been successfully employed in the experimental apparatus. The optimal retro-reflector can be determined through calculation of Equation 6.

Normally, the intensity of a Gaussian beam after interaction with a series of optical components can be calculated using the ABCD or M matrices and the complex beam parameter $q$ Siegman (1986); Steck (2015). However, the M matrices require that the optical elements are placed normal to the optical axis, which is not true for a general case. Instead, we use the extended ray trace matrices Siegman (1986) (see Table 3),

[TABLE]

where $\delta$ is a displacement from the optical axis, $\gamma$ is a rotation of the optic from normal incidence, and the ABCD elements are unchanged.

In the flat mirror case, the Gaussian beam originating from the fiber launcher can be written as

[TABLE]

with $w_{0}$ the $1/e^{2}$ intensity radius. The extended M matrix for the beam as it re-enters the vapour cell after reflection off the mirror is

[TABLE]

The complex beam paramter $q$ Siegman (1986); Steck (2015),

[TABLE]

can be used to to determine the retro reflected beam profile as it re-enters the cell. Substituting the parameters from Equation 9 yields,

[TABLE]

where $\lambda$ is the wavelength. Calculating the retro-reflected beam from these q-parameters and using the approximation that the free space propagation lengths are less than the Rayleigh length and thus $(q_{1}/q_{2})<<1$, yields,

[TABLE]

for the retro-reflected beam. Equation 6 is calculated and the normalized weighted average, $\overline{I}$, is given by,

[TABLE]

where $y_{1}=a\left(\xi-\theta\right)/\omega_{0}$ and $y_{2}=Mf\left(\xi-\theta\right)/\omega_{0}$. This result is plotted in Figure 3 for the experimental set-up described in Martin et al. (2018). The M matrix formulation leverages the small angle approximation where $\xi,\theta<<1$. If the higher order terms are ignored, the weighted average becomes

[TABLE]

which has been expanded to second order in $\theta$, $\xi$ and their cross terms.

The final desired quantity is the sensitivity of the average weighted total intensity to angular misalignment in each optic, or the derivative of the intensity w.r.t. the angular variable. Taking the derivative of Equation 14 w.r.t. either $\theta$ or $\xi$ yield the same result,

[TABLE]

The cat’s eye calculation is more challenging. A cat’s eye retro-reflector consists of a convex lens with a focal length f, and a concave mirror with a radius of curvature f, placed at the focus of the lens. An ideal cat’s eye provides an anti-parallel retro-reflected beam Snyder (1975), whereas misalignment in the fiber launcher will yield non-parallel beams in the case of a flat mirror reflector. Although the incident beam can be recycled and the q-parameter formulation is the same, the M matrix becomes more complex. Various misalignment mechanisms in the cat’s eye optic calculation, shown in Figure 2, include small deviations of the location of the mirror (now located at $f(1+\Delta d)$) the radius of curvature of the mirror ($f(1+\Delta f)$), and also allowing the lens and mirror to have a misalignment angle $\beta$ between them. The following assumptions are enforced: $\xi<<1$, $\theta<<1$, $\Delta d<<1$, $\Delta f<<1$, $\Delta X_{i}<<1$, $\beta<<1$, and the initial beam is collimated. Three special cases were examined, and details on each special case can be found in the appendix:

Analytic derivation with $\Delta d=0$.

• 2.

One specific numeric case with $\Delta d\neq 0$.

• 3.

Analytic derivation with $\Delta d\neq 0$, $\Delta X_{i}=0$, $\beta=0$ and $\Delta f=0$

Figure 3 shows the variation in average intensity as a function of $\Delta X_{i}$, $\beta$, $\xi$, and $\theta$ with all other misalignment variables set explicitly to zero. The fiber launcher angular sensitivity dominates in this simple case. As in the flat mirror case the sensitivity of the average intensity w.r.t. angular misalignments of the fiber launcher and retro-reflector is the desired quantity. In the cat’s eye case these solutions differ,

[TABLE]

[TABLE]

where $\Delta f=0$, $\beta=0$ , and $\Delta X_{i}=0$. It is apparent that Equation 16 is larger than Equation 17 by a factor of $N_{1}+N_{2}$. By design, the cat’s eye suppresses angular motion of the retro-reflector optic over the flat mirror, but at the expense of sensitivity to angular misalignment of the fiber launcher.

The choice of retro-reflector will depend on environmental conditions. For a fully dynamic environment, the flat mirror case might be the best choice, however, if the collimator can be rigidly mounted, the cat’s eye retro-reflector is ideal for reducing alignment variations. Moreover, reduction of alignment sensitivity can be achieved by reducing the laser intensity or increasing the beam waist.

II.3 Measurement

Measuring the ac-Stark shift is important to confirm the accuracy of the theoretical result. An experiment was designed specifically to measure the effects of the ac-Stark shift on the output clock frequency. Two lasers were utilized in the Stark shift measurement (Figure 4): a Ti:Sapph laser tuned slightly away from the two-photon resonance to 385287.8 GHz, which is far enough from the resonant two-photon excitation frequency to introduce no measurable vapour excitation, and the clock laser tuned to be on resonance with the two-photon transition, described in Martin et al. (2018). After amplification and subsequent second harmonic generation of 778.1 nm, the clock laser and the Ti:Sapph laser separately pass through a half wave-plate followed by a quarter wave-plate. The remaining light in each beam is subsequently sent through a variable optical attenuator (VOA), which is used for laser power stabilization. Each beam then passes through another half wave-plate and polarizer, properly aligning the polarization to fiber couple each beam into two separate arms of a polarization maintaining (PM) $2\times 2$ 50:50 fiber splitter. A portion of the Ti:Sapph light is sampled before fiber coupling. This signal is used to feedback to the Ti:Sapph VOA and stabilize the optical power coupled into the beam splitter. One arm of the splitter is sent to a detector used for independent power measurements. The second arm of the splitter is sent to the vapour cell assembly through a $2\times 2$ 99:1 PM fiber splitter. Up to 30 mW of Ti:Sapph light and 30 mW of clock laser light were delivered to the vapour cell assembly.

The vapour cell assembly is enclosed in 5 mm thick single layer mu-metal magnetic shield to reduce Zeeman shifts and broadening. The atomic vapour is regulated to a constant temperature, ideally 100 *∘*C. The final vapour cell operational temperature is achieved though use of a dual stage temperature apparatus. The first temperature state is regulated to $60$ *∘*C, Temp stage 1 in Figure 4, providing a stable reference for the final stage regulated to 100 *∘*C, Temp stage 2 in Figure 4. The vapor cell, which is a rectangular prism with dimensions 5 mm $\times$ 5 mm $\times$ 25 mm, containing $>99\%$ isotopically enriched 87Rb, is placed such that it has a $1~{}$K thermal gradient along its length, forcing the cell’s cold spot on the pinched-off fill tube of the borosilicate glass cell. The vapor cell is oriented at Brewster’s angle with respect to the incident laser beam to reduce stray reflections. The vapour cell assembly is further described in Martin et al. (2018).

The two witness photodiodes, a Thorlabs SM05PD1A, labeled witness 1 in the diagram, and a Thorlabs PDA36A, labeled by the total power detector on in the diagram were independently calibrated to both a Thorlabs PM160 and an Ophir PD300-TP hand held silicon power meter at the ”X” marked in Figure 4. Differences in the calibrations are included in the combined statistical and systematic errorbars of Figure 5.

After the clock laser is stabilized to be on resonance, the Ti:Sapph laser power is varied using the VOA in its optical train. The frequency shifts are measured and averaged over 100 seconds and are reported in Figure 5. The data was fit with a orthogonal distance regression (ODR) algorithm which weights the error bars in both the x and y coordinates, yielding a fractional frequency fit of -2.5(2)$\times 10^{-13}$ (mW/mm2)-1 with a reduced $\chi^{2}$ of 1.57. A gray shaded region shows associated error with the fit. The theoretical value is plotted on the same curve and shows good agreement with the experimentally measured values.

III Two Color

Reduction of the ac-Stark shift would be a powerful tool to reach the final long-term instability goal of $1\times 10^{-15}$ at one day. Alternative to the outlined approach in Martin et al. (2018), two lasers of different frequencies could be utilized to excite the atom in a two-color approach described in Perrella et al. (2013a); Gerginov and Beloy (2018); Perrella et al. (2013b).

However, this approach leads to residual Doppler broadening, because in the atomic frame the $\vv{\bm{k}}$ vectors of the excitation photons no longer match. The increased spectral width requires a higher excitation rate to achieve similar clock performance motivating the choice to operate this two-color scheme near resonance (see Figure 6). Short term stability has been shown to remain unchanged in this design. However, clock stability on timescales exceeding a few hours has not yet been measured. Instabilities on longer timescales can often be driven by technical noise, i.e. lock-point errors, reference voltage drifts, detector responsivity drift, etc. Broadening the transition could make control of these noise sources more difficult.

An alternative two-color approach introduces a second laser (off-resonance from the virtual and intermediate states) to the degenerate two photon Doppler free experiment. The wavelength of this mitigation laser would be chosen such that the differential polarizability sign is opposite that of a 778.1 nm photon. Normally, the clock laser intensity is measured and stabilized to a voltage reference. This scheme is sensitive to stabilities of the detector and voltage reference as well as drift in each device. The two-color approach would allow for the laser intensities to be stabilized with respect to each other reducing requirements on detector sensitivities and drift as well as eliminating a need for a precise voltage reference. Unfortunately, frequency drifts in the mitigation laser would cause variations in the ideal ratio, imposing requirements on the frequency stability. Volumetric Bragg grating stabilized lasers developed for Raman spectroscopy experiments May et al. (2008); Knorr et al. (2010); Vorobiev et al. (2008); Paillet et al. (2010) offer potential options for mitigation lasers. Table 4 displays three possible mitigating wavelengths, the clock laser and calculated shifts. For a system whose purpose is to minimize the required operational power while maintaining high short term clock stability the mitigation laser at 785 nm is the preferred choice.

Use of a mitigation laser could ease the challenge of power stabilizing the probe laser. O-RAFS has already demonstrated that clock laser power can be stabilized to 0.1% Martin et al. (2018). Current requirements on laser stability however, require absolute laser stabilization to 0.01%. If the cancellation can be maintained at a 0.1% level stabilization of the ac-Stark shift can be maintained at 0.01% allowing for final clock stability of $1\times 10^{-15}$ to be achieved.

IV Blackbody Radiation Shift

Plank’s Law describes the electromagnetic radiation emanating from an object at temperature $T$. The time averaged intensity of the radiated field can expressed, Brehm and Mullin (1989),

[TABLE]

An atom in a bath of electromagnetic radiation will experience an ac-Stark shift. Equation 18 describes an electromagnetic field whose intensity is dependent on the source temperature. Thus, fluctuating temperatures of a source that is radiatively coupled to the atoms can cause a clock shift via temperature driven ac-Stark interactions. The shift arising from blackbody radiation (BBR) can be calculated by,

[TABLE]

where $\Delta\alpha(\omega)=\alpha_{e}-\alpha_{g}$. Oftentimes, the resonance frequencies between atomic states connected to the ground and excited states involved in the clock transition are far from the blackbody spectrum. In this case the blackbody spectrum can be treated as a static polarizability field and Equation 19 can be simplified to be,

[TABLE]

Systems that require higher precision include a small dynamic contribution, $\eta$, to account for frequency dependence Beloy et al. (2012); Safronova et al. (2013); Middelmann et al. (2012). However, for the Rb two-photon transition the wide-band BBR spectrum has significant overlap with the $5D_{5/2}\rightarrow 4F_{7/2,5/2}$ transitions at operational temperatures making it necessary to fully integrate Equation 19.

This integral was calculated numerically two separate ways. First, the polarizability was calculated using Equation 2 and the integral was performed using the Cauchy’s principle value. The second method Steck (2015) introduced the decay rate , $\Gamma_{j}$, of each transition found in Marek and Munster (1980); Safronova and Safronova (2011),

[TABLE]

When the polarizability is written this way the function no longer diverges in the resonant cases. Equation 19 was then calculated with a deterministic adaptive integration technique. BBR shifts were examined under nominal operational conditions, specifically looking at the shifts produced in the temperature range of 300 K to 500 K. The fractional difference for each of these calculations for both the excited state and ground state were less then $1\times 10^{-5}$.

The resultant BBR shift of the $5S_{1/2}$ ground state, $-$2.68(2) Hz, whose polarizabilities are far off resonance for the examined temperature range, yields a result that was consistent with the Farley et al. Farley and Wing (1981) calculations of $-2.789$ Hz at 300 K. The shift also is consistent with a $T^{4}$ temperature dependence and a static polarizability approximation. More interesting were the results from integration over the excited state polarizabilities. At $300$ K our value of $-$158.4(12) Hz differed from Farley et al. calculation of $-181.4$ Hz, likely due to differences in polarizability values. Regardless of this difference, the $5D_{5/2}$ state has resonant polarizablities in the temperature range of interest. Not only was the calculated ac-Stark shift no longer monotonic, the differential polarizability changed sign (see Figure 7). This result is not consistent with either a $T^{4}$ behavior or with the static polarizability approximation. The calculation yields two interesting “magic” temperatures. Around 495.9(27) K, $\delta\nu=0$ and around 368.1(14) K, the BBR shift is insensitive to changes in environmental temperature. Clock operation hoping to achieve greater stability could operate around 368 K to suppress environmental temperature dependence. However, to benefit from this reduced temperature sensitivity, the more significant temperature shift arising from Rb collisional effects would also require mitigation Martin et al. (2018); Zameroski et al. (2014).

V dc-Stark

The presence of unknown electric charges on dielectric surfaces in the vicinity of the atomic sample causes a DC Stark shift and could, at least in principle, cause a clock instability if the charge was to slowly migrate. In fact, the buildup of stray charge inside of a vacuum chamber on a mirror with a piezoelectric transducer was shown to be detrimental to the overall performance in an optical lattice clock Lodewyck et al. (2012). We calculate the DC polarizability for the Rb 2-photon transition from Equations 1 and 2, setting $\omega=0$. The polarizability of the $5D_{5/2}$ state dominates due to the presence of low-lying resonances, and we find the induced Stark shift to be 4.27(4) Hz/(V/cm)2 by summing over the matrix elements in Table 1 and inclusion of the core and continuum polarizabilites found in Table 2. When probed with a 778 nm laser, the 2-photon frequency standard displays a fractional sensitivity of 5.55(5) $\times$ 10*-15* /(V/cm)2.

Thus, in principle, a small electric field on the order of 1 V/cm with a slow time wander could cause a significant long-term stability for this frequency standard. However, since the glass vapor cell is embedded in a block of copper for thermal control, we expect this effect to be minimal. The inclusion of a UV LED to remove the charge Lodewyck et al. (2012) is one simple way to rule out this possibility in future investigations of long-term instabilities.

VI Conclusions

We presented a calculated ac-Stark shift of $2.30(4)\times 10^{-13}$(mW/mm2)-1 in good agreement with the measured value of $2.5(2)\times 10^{-13}$(mW/mm2)-1. Careful examination of alignment shows that the cat’s eye retro-reflector helps reduce sensitivity of the average intensity to variations in the retro-reflecting optic over the flat mirror reflector. However, the cat’s eye retro-reflector increases the sensitivity of the average intensity to angular misalignments of the fiber launcher, and introduces another very sensitive misalignment variable, the distance between the lens and mirror in the cat’s eye optic. While effort can be placed to reduce the dynamic response of the displacement $\Delta d$, care must be taken to ensure that the average intensity signal is maximized during initial alignment. For a dynamic system, however, the flat mirror retro-reflector might be the best choice to reduce complexity and sensitivity to motion. Ultimately, any effort to further reduce the overall ac-Stark shift will also reduce Stark shift related alignment sensitivities.

Two separate Stark shift mitigation techniques were discussed. Practical limitations of available power in a portable system make the two-color, two-photon technique less appealing. However, introduction of a Stark shift canceling laser could help reduce overall ac-Stark effects and allow for system fractional frequency instabilities as low $1\times 10^{-15}$.

The calculated BBR shift differs from extrapolated values. The calculation results in two interesting temperatures: the magic tune out temperature, where the BBR polarizability is zero, and the temperature where the sensitivity to temperature variations is zero, effectively removing the BBR driven clock instability.

acknowledgments

We thank Jordan Armstrong and Space Dynamics Laboratory for assistance in constructing the experimental apparatus. Research preformed by M.S.S. was performed under the sponsorship of the US Department of Commerce, National Institute of Standards and Technology. *

Appendix A Cat’s Eye Alignment Calculation

Calculation of the intensity weighted average for the cat’s eye case requires the Gaussian beam profile of the retro-reflected beam. After retro-reflecting off of the cat’s eye the beam re-enters the vapour cell with the following M matrix parameters:

[TABLE]

where,

[TABLE]

Here we utilize the same approximations as the flat mirror case, namely that all distances are much shorter than the incident beam Rayleigh length. Even leveraging this approximation yields a complicated Gaussian function. In order to simplify the analytical solution a few special cases were examined.

Setting $\Delta d=0$ yields

[TABLE]

This expression is shown in Figure 3 and is utilized to calculate Equations 16 and 17.

• 2.

A numerical integral was performed using a quadrature method. The numerical result is displayed in Figure 3 (b).

• 3.

The final case studied simplified the geometry to reduce the number of free parameters. $\Delta f$ in known to have small impacts on the retro-reflected beam profile Snyder (1975). Changes in $\Delta X_{i}$ and $\theta$ have similar effects on the beam propagation, a similar relationship exits between $\beta$ and $\Delta d$. With this in mind another analytical case was examined where $\Delta X_{i}=0$, $\Delta f=0$ and $\beta=0$. It was also necessary to expand the integrand in a Taylor series and ignore the higher order terms before final integration, yielding an average intensity of

[TABLE]

Taking the derivative of the above Equation w.r.t. either $\theta$ or $\xi$ yields the same result as the $\Delta d=0$ case. Figure 3 (b) shows average intensity given above plotted with the geometry presented in Reference Martin et al. (2018) along with the numeric results for the same geometry.