How to Hold an Atom Without Squeezing It: Magic Ideas
DEPARTMENT OF PHYSICS,
KOREA UNIVERSITY, SEOUL 02841, KOREA
While the cooling and trapping of atoms using laser light produces extremely cold samples trapped at a specific location, their utility for precision spectroscopy is limited due to perturbation of the cooling and trapping process itself. There has been much effort expended to solve this problem for transitions in both optical and radio frequencies. Our laboratory has proposed solutions using a magic wavelength for an optical transition and magic polarization for a ground hyperfine transition in RF. We also experimentally demonstrated elimination of unwanted frequency shift and broadening using these magic ideas.
In the 1980s there were a series of landmark experiments that allowed researchers to tame atomic motion using laser light together with various arrangements of magnetic fields. Collectively known as "the cooling and trapping of atoms" in the community, the techniques developed in that era opened ways to produce very cold and dense atoms. Using these techniques, a fountain-type atomic clock was invented, which improved the precision of time-keeping tremendously. The efforts to achieve ever colder and denser atomic sample culminated in the realization of a Bose-Einstein condensate (BEC) of alkali-metal atoms in 1995 . More recently, ultra-cold atoms are used either as qubits in quantum information processing  or as samples for quantum simulators .
Although the realization of the BEC quantum-degenerate gas of either bosonic or fermionic atoms has now taken center stage of this field, the initial motivation for cooling and trapping atoms was to achieve high precision in atomic spectroscopy. If you can hold atoms in one place long enough and keep them at a low temperature, it is obviously advantageous for precision spectroscopy. However, it turned out that perturbation from the external fields which were used for the cooling and trapping limited precision seriously: the results of spectroscopy using trapped atoms was not much better than those using a conventional oven beam.
There are three types of atom traps. A magneto-optic trap (MOT) relies on radiation pressure from near-resonant laser beams. The radiation pressure obtains position dependence owing to an applied magnetic field gradient. A magnetic trap relies on the Stern-Gerlach force from a magnetic field gradient acting on a paramagnetic atom. Finally, an optical trap relies on a dipole force, which is from spatially varying AC Stark shift owing to an intensity gradient of a trapping light field. When the frequency of the trapping light is red-detuned from the atomic resonance frequency, AC Stark shift is negative. Atoms are trapped where the intensity is maximum, for example, at a focal point of a Gaussian beam.
Fig. 1: An optical trap of a Gaussian beam.
While a MOT is a workhorse for most of the cooling and trapping experiments, the atoms trapped in a MOT are constantly bombarded with near-resonant laser beams and consequently they are not ideal samples for precision spectroscopy. In a fountain clock, atoms are cooled and trapped in a MOT. However, the MOT is then turned off and the atoms are tossed up for Ramsey interrogation of its resonance frequency. The environment of a magnetic trap is not friendly to spectroscopy either because both the strength and the direction of the applied magnetic field vary over space. A magnetic dipole moment of an atom adiabatically follows a local magnetic field, and hence points to arbitrary directions. An optical trap provides the most promising environment for precision spectroscopy. Especially when the detuning of a trapping laser beam is large, atoms rarely absorb photons from the trapping beam and they experience almost pure conservative potential. Detunings as large as a few THz or more are typically used for this purpose and such traps are known as a far-off resonance optical trap (FORT) .
Although a FORT provides the best chance for precision spectroscopy, it also disturbs trapped atoms. When we carry out spectroscopy using a transition between two levels |φg> and |φe> of an atom trapped in a FORT, AC Stark shifts of the two states due to the trapping field are not generally the same. This differential AC Stark shift produces both a shift in its resonance frequency and inhomogeneous broadening for an ensemble of atoms. The following figure shows the differential AC Stark shift of a two-level atom in a red-detuned Gaussian beam.
In this article we describe two schemes to eliminate the differential AC Stark shift. The transition between the two levels of interest can be in the optical frequency range, like the D transition of alkali-metal atoms, or in microwave or radio frequency (RF) range, as in a clock transition of a ground-state cesium atom. We will discuss both cases but will place more emphasis on the case of RF transitions.
If one considers a two-level system interacting with a perturbation, when one of the levels is pushed down, the other one should be pushed up and vice versa. It is impossible to remove the differential AC Stark shift shown in Fig. 2. However, we noticed that if you consider a third level, it may be possible to eliminate the differential shift. The following figure shows the relevant three levels of a cesium atom to eliminate the differential shift between the 6S1/2 state and the 6P3/2 state . The shift of the 6P1/2 state due to its coupling with the 6S1/2 state is positive, but that due to its coupling with the 5D3/2 and 5D5/2 states is negative. Because energy difference between the 6S1/2 - 6P3/2 transition is larger than that of the 6P3/2 - 5D transition, it is possible to make the total AC Stark shift of the 6P3/2 state the same as that of the 6S1/2 state by using the proper wavelength for a FORT. Hence the name "magic wavelength".
Fig. 3: The three-level structure of a cesium atom that allows a magic wavelength.
We carried out an experiment to demonstrate the idea of the magic wavelength using cesium atoms in an optical trap . After cesium atoms were trapped we shined a strong kicking beam to remove them from the trap and measured the number of atoms that remained. The following figure shows spectra of the kicking experiment for two wavelengths of the optical trap with the signal from a saturated absorption spectrometer as a reference. When the wavelength is 935 nm, which is the magic number for the cesium transition, the spectrum shows no frequency shift or broadening.
Fig. 4: Results of loss spectroscopy using kicking beams.
Basically the same idea was independently proposed by Katori  for a strontium atom. Strontium atoms in an optical lattice at its magic wavelength play an important role in the optical frequency standard. Our idea of a magic wavelength for cesium D2 transitions was successfully applied to a cavity quantum electrodynamics experiment , where the cavity was tuned to both the magic wavelength and the D2 transition.
Alkali-metal atoms, for which the cooling and trapping techniques were first developed because of the cycling transitions at convenient wavelengths, have hyperfine structures due to the nuclear spins. The ground hyperfine splitting is the largest, typically in the GHz range, and the most useful. Cesium ground hyperfine splitting of 9.2 GHz serves as a standard for time and frequency. In quantum information processing (QIP), the two spin states, denoted as F and F+1 in Fig. 5, serve as a qubit.
When an alkali-metal atom is in an optical trap, however, the two ground hyperfine states F and F+1 experience different AC Stark shifts because they see different detunings owing to the hyperfine splitting (see Fig. 5). In general, the hyperfine splitting is much smaller than the detuning and the differential shift is not as large as in the case of an optical transition. For cesium atoms in a typical FORT, maximum differential shift is a few kHz. However, its applications such as an atomic clock or precision measurement of symmetry violation require much greater precision. In QIP applications, inhomogeneous broadening is translated into dephasing of a superposition state, and the differential shift limits utility of the spin qubit in an optical trap.
Fig. 5: Relevant level structure for ground hyperfine spectroscopy of alkali-metal atoms in an optical trap.
There have been many efforts to circumvent this problem. The first one was to use a blue-detuned optical bottle, where atoms were kept away from the light field . Due to diffraction, the isolation of trapped atoms from the light field was never good enough and trap volume was small. There were also approaches to use a two-color trap  or multiple spin echo pulses  to compensate for the differential shift, but they were difficult to implement and only reduced the shift without eliminating it. Our method of magic polarization goes to the heart of the problem to eliminate the differential AC Stark shift.
When an alkali-metal atom in its ground state |nS, F, mF> with mF being the angular momentum and the z component is in a laser field with an amplitude ɛ, its AC Stark shift is
where αF and βF are the scalar and vector polarizabilities, respectively, η is degree of circularity of the trap beam, and gF is the Lande g factor. In a typical optical trap, α is much larger than β. Different detunings for the F and F+1 states lead to different α and β, and the problematic differential AC Stark shift. Our proposal is to use the vector term to cancel the differential shift by tuning η . The vector term behaves like a Zeeman shift with an effective magnetic field strength proportional to both η and the trap beam intensity. For a transition from the |nS, F, mF> to the |nS, F+1, mF> state, magic condition is satisfied when
Hence, the name magic polarization. There was an independent proposal based on the same idea to adjust the angle between a quantization axis defined by a magnetic field and the propagation direction of the trap beam while the polarization is fixed to a circular one . In practice, polarization is much easier to tune than the angle.
The idea of magic polarization is applicable to all alkali-metal atoms. However, for heavy atoms like rubidium or cesium vector polarizability, which is proportional to the spin-orbit coupling strength, it is so large that any deviation of η from the magic value would introduce a big differential shift from the vector term itself. For our demonstration of magic polarization we chose lithium, whose vector polarizability is only 1% of that of rubidium . We use the transition from the |2S, F=1, mF=1> to the |2S, F=2, mF=2> state. The magic η was 0.413 and it is consistent with theoretic value. As shown in Fig. 6, we achieve the full-width at half maximum (FWHM) of only 0.6 Hz when the RF pulse duration is 2s. Uncertainty limited FWHM is 0.4 Hz. The extra broadening is from either inexact tuning of polarization or magnetic field noise.
Fig. 6: Line shape of the hyperfine transition at η = 0.413.
We also measured the coherence time of a linear superposition state using Ramsey spectroscopy. It is an important parameter for an application of the hyperfine states as a qubit for QIP. The result is shown in Fig. 7, where the coherence time is 0.82 s.
Fig. 7: Ramsey signal to measure the coherence time.
We note that the scheme relies on the Zeeman-like shift and it is applicable only to a Zeeman-sensitive transition. It cannot be used for the clock transition between two mF=0 states. It is, however, not a serious limit of the scheme. It is not likely that a clock based on optically trapped atoms, even with an interrogation time of a few seconds, would become competitive with a fountain clock. On the other hand, in the field of QIP there are more interesting applications of the magic polarization. The dephasing of a qubit, or a linear superposition state of two hyperfine states, due to inhomogeneous broadening has been a limiting factor in using optically trapped atoms as qubits.
Recently we successfully loaded and manipulated a single lithium atom at a specific site of a 1D optical lattice without disturbing an atom in the adjacent site. Using a single atom at a specific site with a long coherence time, we plan to construct a quantum gate.
Acknowledgements: This work was supported by the National Research Foundation of Korea (Grant No. 2009 -0080091).
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Donghyun Cho is a professor of physics at Korea University. After receiving a PhD from Yale University on the subject of time-reversal symmetry violation in a diatomic molecule, he worked at JILA, University of Colorado, on parity violation using cesium atoms. He joined the Department of Physics, Korea University, in 1994. His research interest is high precision spectroscopy and quantum information processing using optically trapped atoms.