# Experimental observation of magic-wavelength behavior in optical lattice-trapped $^{87}$Rb

## Abstract

We demonstrate the cancellation of the differential ac Stark shift of the microwave hyperfine clock transition in trapped Rb atoms. Recent progress in metrology exploits so-called “magic wavelengths,” whereby an atomic ensemble can be trapped with laser light whose wavelength is chosen so that both levels of an optical atomic transition experience identical ac Stark shifts. Similar magic-wavelength techniques are not possible for the microwave hyperfine transitions in the alkalis, due to their simple electronic structure. We show, however, that ac Stark shift cancellation is indeed achievable for certain values of wavelength, polarization, and magnetic field. The cancellation comes at the expense of a small magnetic-field sensitivity. The technique demonstrated here has implications for experiments involving the precise control of optically-trapped neutral atoms.

###### pacs:

Optical trapping is widely used for the manipulation of cold and ultracold neutral atoms. It is often desirable to optically trap neutral atoms without affecting the internal energy-level spacing of the atoms, particularly in cases (such as in atomic clocks or quantum computing) where the accurate measurement of an atomic transition frequency is paramount. Unfortunately, for any two internal states of an atom, optical trapping generally induces a differential shift: this shift presents uncertainty in atomic transition frequencies and also introduces inhomogeneity in the transition, associated with the spatial profile of the trapping potential. Approaches have been proposed and demonstrated Ye et al. (2008) to minimize these differential shifts in optical transitions, but it is desirable to extend these approaches to other atoms and transitions. Here we demonstrate a technique (similar to ones recently proposed Flambaum et al. (2008); Choi and Cho (2007)) to eliminate the trapping-light intensity dependence of the ground-state hyperfine transition in alkali atoms.

The differential shift arises from the different frequency-dependent ac polarizabilities of the ground () and excited () states of a transition. Trapping light with frequency and electric field shifts the transition frequency by

(1) |

where the light shifts depend on the light polarization through the vector nature of . Depending on the polarizability of the the two states, can sometimes be made to vanish (independent of the intensity ) for appropriate choice of .

To determine the ways that the differential shift can be cancelled, we consider the light-atom interaction. For dipole-allowed transitions in the low magnetic field limit, where the total angular momentum is a good quantum number, the light-induced Hamiltonian for states can be separated into three contributions characterized by the scalar, vector and tensor polarizabilities , and Happer and Mathur (1967); Deutsch and Jessen (2009):

(2) | |||||

where is the total atomic angular moment operator, and is a unit-normalized polarization vector. The vector contribution can be treated as arising from an effective magnetic field , and vanishes for linearly polarized light, where . The light shift of a given eigenstate is given by . The contributions to the total differential shift from the scalar, vector, and tensor polarizabilities are , , and , respectively.

If the two states and have different electronic configurations , the frequency dependence of the polarizabilities can be sufficiently different that a “magic” frequency may occur even when the light shift is dominated by just the scalar term, so that . Such a cancellation has been found for optical transitions, permitting high accuracy interrogation of narrow optical transitions for optically trapped atoms Brusch et al. (2006); Takamoto et al. (2005); Boyd et al. (2006).

For transitions within the same electronic configuration, such as microwave clock transitions, cancellation is less obvious. The slightly different polarizabilities have nearly identical frequency dependences, and the light shifts tend to track together as a function of , without crossing. (This is true in the far-detuned limit appropriate for optical dipole traps. In the limit where the detuning is small with respect to hyperfine transitions, cancellation is possible, but at the expense of a substantially increased scattering rate Kaplan et al. (2002); Chaudhury et al. (2006)). For example, Rosenbusch et al. Rosenbusch et al. (2009) showed that for linearly polarized light on alkali atoms, where the vector and tensor contributions to the differential shift are negligible, there is no magic wavelength such that the scalar polarizabilities cancel. (Although for atoms such as aluminum or gallium that have enough internal electronic structure to allow for a tensor contribution, magic wavelengths can be found where the scalar and tensor parts cancel each other Beloy et al. (2009).)

Recently there have been proposals Choi and Cho (2007); Flambaum et al. (2008) to use the vector component of the light shift to cancel the scalar differential shift using elliptically or circularly polarized light on alkali atoms. Since the vector light shift acts like an effective Zeeman field, this approach requires field-sensitive transitions, and depends on the ellipticity of the light. Additionally, the approach depends on the orientation of the atom determined by the total magnetic field , where is the external bias field and points along the local axis of circular polarization. This broadens the discussion to a “magic surface” in a space of wavelength, polarization and field direction: for most wavelengths there would exist either a magic angle between and (or fraction of circular polarization) where the differential light shift is cancelled.

This proposal is challenging, though, due to the fact that the differential vector shift on field sensitive transitions, , is typically orders of magnitude larger than the differential scalar shift . This scale discrepancy can be seen as follows: at lowest order (second order in the light field), the magnitude of the scalar shift is independent of . The first non-zero contribution, therefore, to occurs at third order (quadratic in electric field and linear in the hyperfine interaction) Rosenbusch et al. (2009). On the other hand, even though the magnitude of the vector shift is independent of at second order, acts like a Zeeman shift, depending on the projections , . It results from fine structure coupling Happer and Mathur (1967); Deutsch and Jessen (2009) and has significant non-zero contribution at second order. In heavy atoms like rubidium or cesium, the fine structure is orders of magnitude larger than the hyperfine structure, and . Cancellation requires an accurate rescaling of the size of , either by controlling a small amount of circular polarization at the level of Choi and Cho (2007), or tuning the angle between and such that its cosine is precisely set to a value smaller than 10Flambaum et al. (2008).

In this paper, we show that by using states at nonzero magnetic field we can achieve the light shift cancellation without sensitive control of the polarization or the angle . Rather than relying upon a careful projection between two vectors, we use the small field sensitivity experienced at nonzero magnetic field by atoms in the ground states. The residual vector shift is the response of the transition frequency to a perturbation of the bias field . We parametrize the field sensitivity of the transition (which arises from the local slope of the so-called quadratic Zeeman shift) using an effective magnetic moment , a parameter which is easier to precisely control at small values (via the external bias field B) than or . By applying this small vector light shift on individual Rb atoms trapped in sites of an optical lattice, we show that we can eliminate the dependence of the clock transition frequency on the intensity of the trapping light, achieving the goal of magic-wavelength behavior in an alkali-metal microwave transition. For details, see theoretical work elsewhere Derevianko (2009).

One can estimate the field sensitivity required for alkali atoms in purely circularly polarized light by evaluating the ratio of the scalar differential shift to the differential vector shift . Ignoring contributions from excited states other than the lowest levels, is determined by the slightly different laser detuning from the excited state due to the ground state hyperfine splitting :

(3) |

where is the laser detuning from the states. Given , therefore scales as . The vector light shift, however, depends on the fine structure splitting of the excited states Deutsch and Jessen (1998). For a hypothetical field-sensitive transition with sensitivity , a similar argument as for Eq. 3 shows that , so that . The required field sensitivity to allow for cancellation is therefore given approximately by , which is independent of detuning. For Rb, , giving a required field sensitivity of kHz/mT. For cesium, the required sensitivity is 23 kHz/mT. This points to a limitation of this approach: obtaining insensitivity to light intensity requires a sensitivity to magnetic fields.

Our experiment uses an ensemble of single atoms trapped at the sites of a 3D optical lattice, created by adiabatically loading a Bose-Einstein condensate (BEC) of 10 Rb atoms into a lattice generated by a Ti:sapphire laser operating at nm. A linearly polarized “vertical” lattice along divides the BEC into a stack of independent 2D condensates, and a separate, deformable lattice created by beams along and completes the 3D lattice Sebby-Strabley et al. (2006). This lattice can be transformed from a standard linearly polarized -periodic lattice into a double-well configuration, where both the lattice intensity and polarization can differ significantly between “left” and “right” sides of the double well. The details of this lattice are described extensively elsewhere Sebby-Strabley et al. (2006); Lee et al. (2007). The applied bias field lies in the plane.

We use the transition in Rb, which at zero field lies near 6.834 682 611 GHz, and increases with field approximately as 57.5 kHz/(mT). To measure the differential shift, we perform microwave Ramsey interferometry on the ensemble of atoms using -pulses and a variable Ramsey delay, with the pulses detuned from free-space resonance (as calculated by the Breit-Rabi formula Breit and Rabi (1931)) by 1 kHz. The observed Ramsey fringe shifts from free-space resonance as a function of light intensity, yielding a Hz shift at high intensity (lattice depth 30 ). This is the shift we seek to eliminate.

In order to cancel the differential shift, we transform the lattice into a configuration placing an effective magnetic field (originating from an increased local elliptical polarization) on every other lattice site, where is aligned parallel to . After Ramsey interferometry, we record only the populations of atoms on the effective-field sites, discarding atoms on the unused sites through a “dumping” process Sebby-Strabley
et al. (2007); Lee et al. (2007). In Fig. 1 we show the measured differential shift of the transition as a function of the lattice intensity, both in the linearly polarized lattice and in the elliptically polarized lattice.
The latter clearly shows a strongly diminished dependence of on lattice-beam intensity.
The remaining Hz shift from bare vacuum resonance is due to the differential light shift caused by the confinement in the linearly polarized vertical lattice. We measured this shift using Ramsey interferometry and varying the intensity of the vertical lattice, yielding an offset value of 48(4) Hz^{1}

This work has implications for experiments requiring the accurate control of trapped neutral atoms. In particular, the use of optically trapped neutral atoms for quantum information applications can be adversely affected by fluctuations and inhomogeneity in the trapping and control lasers Lundblad et al. (2009). The strong reduction of the differential shift presented in this work offers a possible solution to these issues. More generally, this work shows that differential insensitivity to light can be obtained at the cost of slightly increased sensitivity to magnetic fields; it may be useful in situations where the light field fluctuations and inhomogeneity are more difficult to control than the global magnetic field.

###### Acknowledgements.

We thank Andrei Derevianko, Ivan Deutsch, Saijun Wu, and Stephen Maxwell for helpful discussion. This work was partially supported by IARPA and ONR. NL acknowledges support from the NRC.### Footnotes

- Unless otherwise stated, all uncertainties herein reflect the uncorrelated combination of 1 statistical and systematic uncertainties.

### References

- J. Ye, H. J. Kimble, and H. Katori, Science 320, 1734 (2008).
- V. V. Flambaum, V. A. Dzuba, and A. Derevianko, Phys. Rev. Lett. 101, 220801 (2008).
- J. M. Choi and D. Cho, Journal of Physics: Conference Series 80, 012037 (2007).
- W. Happer and B. Mathur, Physical Review 163, 12 (1967).
- I. H. Deutsch and P. S. Jessen, Opt. Comm. (2009).
- A. Brusch, R. L. Targat, X. Baillard, M. Fouché, and P. Lemonde, Phys. Rev. Lett. 96, 103003 (2006).
- M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, Nature 435, 321 (2005), 10.1038/nature03541.
- M. Boyd, T. Zelevinsky, A. Ludlow, S. Foreman, S. Blatt, T. Ido, and J. Ye, Science 314, 1430 (2006).
- A. Kaplan, M. F. Andersen, and N. Davidson, Phys Rev A 66, 045401 (2002).
- S. Chaudhury, G. Smith, K. Schulz, and P. Jessen, Phys. Rev. Lett. 96, 043001 (2006).
- P. Rosenbusch, S. Ghezali, V. A. Dzuba, V. V. Flambaum, K. Beloy, and A. Derevianko, Phys. Rev. A 79, 013404 (2009).
- K. Beloy, A. Derevianko, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. Lett. 102, 120801 (2009).
- A. Derevianko, arXiv:0912.3233 (2009).
- I. Deutsch and P. Jessen, Phys Rev A 57, 1972 (1998).
- J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto, Phys. Rev. A 73, 033605 (2006).
- P. J. Lee, M. Anderlini, B. L. Brown, J. Sebby-Strabley, W. D. Phillips, and J. V. Porto, Phys. Rev. Lett. 99, 020402 (2007).
- G. Breit and I. Rabi, Physical Review 38, 2082 (1931).
- J. Sebby-Strabley, B. L. Brown, M. Anderlini, P. J. Lee, W. D. Phillips, J. V. Porto, and P. R. Johnson, Phys. Rev. Lett. 98, 200405 (2007).
- N. Lundblad, J. M. Obrecht, I. B. Spielman, and J. V. Porto, Nat. Phys. 5, 575 (2009).