Atom trapping on a chip: a tool for metrology 

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Atom trapping on a chip: a tool for metrology

This chapter aims to provide the reader with an introduction to the founding concepts of the Trapped Atom Clock on a Chip (TACC). We will begin with a brief discussion of the basic principles and advantages of atomic time keeping and will include an overview of compact atomic clocks with speci c focus on the bene ts of trapped atoms for application in metrology. We will then describe the idea of atom trapping on microstructures, also called atom chips. In trapped atom clocks interactions play a leading role, this will be referred to and expanded upon in the concluding part of this chapter. In particular we will focus on two e ects that play crucial roles in our experiment: the collisional shift of the clock frequency and the identical spin rotation e ect.

Basic concepts of time metrology

Atomic clocks

Figure 1.1: Locking principle of an oscillator on a atomic resonance. This scheme illustrates the basic principle of atomic clocks. The locked local oscillator provides the useful signal.
An atomic clock is essentially constituted of two elements: a local oscillator and an atomic reference. The general idea is to lock the local oscillator frequency fLO on an atomic transition of frequency fat. The response of the atom gives the di erence between the two frequencies which is used as an error signal ( gure 1.1 ). Ideally, the frequency of the local oscillator reproduces the atomic frequency exactly.
An atomic transition is the most stable frequency reference currently available. This is be-cause it does not drift in time due to the fact that atoms are stable objects within the limit of their radioactive disintegration time (47:5 109 yr for 87Rb). The atomic transition is selected to have a very narrow natural linewidth such that the width of the spectroscopy is limited by the interrogation time (Fourier-limited). The atomic response is the dependance of the state populations on the detuning fat fLO, and changes with the interrogation scheme (Ramsey or Rabi spectroscopy). To make the atomic response as steep as possible, and thus provide the most sensitive frequency measurement, long interrogations times are needed.
Figure 1.2: Example of the atomic response in the case of Rabi interrogation. Long interrogation times are needed to make the atomic response steep and provide high sensitivity to frequency changes.
The resolution we are able to achieve when measuring an atomic frequency is fundamen-tally limited by the atomic shot noise. However, in the real world, the atomic line position can uctuate under the in uence of interatomic interactions or external elds causing uctuations of the local oscillator frequency.
Clock accuracy and clock stability When the clock is locked, the local oscillator frequency can be written fLO(t) = fat [1 + + y(t)] , (1.1) where y(t) may uctuate, but its average is equal to 0. The accuracy of the clock is the error of the o set : this denotes how well the clock reproduces the atomic frequency of the atom isolated from the outside world. The ability to build accurate 133Cs clocks is one reason for its choice as the international time reference. Primary frequency standards need to be accurate clocks.
The uctuating part y(t) characterizes the stability of the clock. It must be as small as possible. It is fundamentally limited by the atomic shot noise, which arises from the measure-ment process.
A clock with unknown accuracy but with y(t) of small amplitude and averaging to zero can be used as a secondary frequency standard. Such a clock delivers a signal at the clock frequency fat(1 + ). The o set can be calibrated against a primary standard. In fact most applications of atomic clocks require frequency stability rather than accuracy since they can be calibrated periodically. The Trapped Atom lock on a Chip aims to be a highly stable secondary frequency standard.

Atom- eld interaction

The interaction between the local oscillator and the atom is treated in the near-resonant case. The atom can be reduced to the 2 clock levels and the general theory of a two-level atom interacting with an electromagnetic eld applies. We call the Rabi frequency of the atom/ eld coupling, and = fLO fat the detuning.
Figure 1.3: Model of the two-level atom interacting with an electromagnetic eld. is the Rabi fre-quency.

Ramsey and Rabi spectroscopy

Two interrogation schemes are commonly used [12].
Rabi spectroscopy involves interrogating the atoms with one pulse of constant amplitude and duration T . The atomic response, de ned as the probability of the atom to transfer from state j1i to state j2i is given by P2 = 2 + 4 2 2 sin2 2+422 2 (1.2)
Ramsey interrogation consists of applying two short pulses separated by a free evolution time TR. The pulses used have an area of =2. If the pulse durations are omitted the atomic response is given by P2 = 1 [1 + cos (2 TR)] (1.3)
In the Bloch sphere picture, the rst pulse of the Ramsey interrogation is equivalent to placing the atom in the equatorial plane. During the Ramsey time, TR, the atom evolves freely corresponding to a precession of the pseudospin along the equator at the frequency fat. The second pulse converts the accumulated phase into population di erence of j1i and j2i.
For equal interrogation times the Ramsey method provides an atomic response 1:6 times more sensitive to frequency changes than the Rabi scheme. Another major advantage of the Ramsey interrogation is that the atom is not subject to the interrogation eld during the phase evolution (to our level of approximation does not appear).

Compact frequency references

In this section we provide an overview of the various di erent types of compact atomic clocks and their applications in order to give the reader a broader perspective of our continued interest in researching and building atomic clocks.

Applications of compact atomic clocks

Global positioning system Now available in almost every car or smartphone, GPS consists of a set of satellites that continuously broadcast their position and time, exact to a billionth of a second. A GPS receiver takes this information and uses it to calculate the car’s or phone’s position on the planet. For this it compares its own time with the time sent by three satel-lites. This method requires that both the satellites and the receiver carry clocks of remarkable accuracy. However, by picking up a signal from a fourth satellite the receiver can compute its position using only a relatively simple quartz clock. To ensure time accuracy each satellite carries four atomic clocks, which are periodically re-calibrated when passing over the control stations [13].
Very Large Baseline Interferometry This is a technique that uses distant antennas poin-ting to the same radiofrequency stellar source (for example quasars) to increase angular reso-lution. The useful information is contained in the di erence of the signal arrival times on each of the two antennas. These arrival times need to be known accurately on both remote devices. The needs, in terms of clock stability, are so stringent that most stations use hydrogen masers for the synchronization [14].
Geophysics Atomic clocks may be applied and utilized in studies of the Earth’s rotation and the movements of tectonic plates for earthquake detection. [9].
Other elds such as space missions, meteorology and environment (monitoring of the atmo-sphere) might also bene t from the development of compact atomic frequency references [9]. There is no doubt that further applications of compact and stable atomic clocks will appear in the future.

Current status of compact atomic clocks

In this subsection we do not provide a complete overview of the eld of compact atomic clocks, rather, we focus on a few projects that target performances similar to ours in terms of size and frequency stability.
Pulsed, optically pumped clock (INRIM) This clock is composed of a vapor cell placed in a microwave cavity. It uses the Ramsey scheme with interrogation times of a few milliseconds due to the short coherence time of the atoms. First, an intense laser pulse pumps the atoms into one of the two states. The microwave transition is driven and a second laser pulse detects the atomic population. Recently, a short-term stability of 1:7 10 13=p was demonstrated, with an integrated instability of 5 10 15 and drifts below 10 14 per day [15].
Coherent population trapping (CPT) These clocks also interrogate the hyper ne tran-sition in an atomic vapor. They do not involve microwaves but two phase-coherent laser beams that are detuned by the clock frequency. Under these conditions the atoms can be pumped into a dark state where their resonance peaks sharply and may be used for locking the local oscilla-tor. The SYRTE CPT clock is operated in pulsed mode for a reduced sensitivity to laser power. Its latest status is a short term stability of 7 10 13=p integrating down to 4 10 14=p [16].
Trapped mercury ion clock This project is being developed at the Jet Propulsion Labo-ratory. Mercury ions are captured in a linear multipole trap, where microwave spectroscopy of the hyper ne transition is performed. The population is detected with a discharge lamp. In the last publication (2009) [17], a short-term stability of 2 10 13=p was reported , integrating down to 10 15 in one day for a 3 L physics package.
Cold atoms in an isotropic cavity (HORACE) This project is being developed at SYRTE. The basic idea is to use a spherical cavity to both cool and interrogate the atoms. Optical molasses is created inside the cavity and a Ramsey spectroscopy is performed on the free falling atoms. Atoms are recaptured at the end of each cycle and cycle times of 80 ms can be achieved. The current status of this project is a short term stability of 2:2 10 13=p , limited by the atomic shot noise, and frequency instability of 3 10 15 after 104 seconds of integration [18].
The TACC project also targets a stability of & 10 13=p . As explained later in this thesis, the discovery of the e ect of spin self-rephasing [11, 19] gives hope that this target may even be surpassed. In the next section we discuss the advantages and drawbacks of using trapped atoms for metrology.

Using trapped atoms for metrology

The interest of using trapped atoms for metrology lies in the long interrogation times that can be achieved whilst keeping the system compact. However, special traps must be engineered in order to disturb the two clock states energies in the same way, as we will see in this section. Traps also enable one to cancel the atom’s recoil from the interrogation photon as in optical clocks.

Extended interrogation times

In atomic fountains the atoms are under free fall. The upper limit of the interrogation time is given by the size of the apparatus. By launching the atoms up against gravity one can gain a factor of 2, but the slow scaling of the free fall time t = 2h=g with the size of the apparatus h makes it hard to gain. We note, however, that recently an atomic fountain exceeding 10 m was proposed for testing general relativity [20, 21].
By trapping the atoms one can achieve arbitrarily long interrogation times. The new li-mitations to the interrogation time become the coherence time of the superposition (T2 in the language of the nuclear magnetic resonance), the lifetime of the atomic trapped cloud, the natural linewidth of the transition or the coherence time of the local oscillator.

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Cancelation of the e ect of the trap on the clock frequency

Magic traps for accurate clocks Atom trapping consists of giving the atomic state’s ener-gy a spatial dependance while metrology implies insensitivity to external elds. The apparent contradiction can be solved if we consider situations where the energy varies with the external eld for both clock states in the same way. In such a trap the energy di erence between the two clock states becomes insensitive to the trapping eld to rst order, and the frequency of the trapped atom is identical to the atomic frequency in free space (see gure 1.4). Such traps are called magic traps, and are the primary requirement in achieving clock accuracy.
A magic optical trap can be created by choosing a magic wavelength [22] at which both clock states have identical electric polarizabilities. For microwave clocks (typically Cs or Rb) there have been proposals to combine the polarization of the trapping light with a magnetic eld in order to eliminate the e ect of the optical trap on the clock frequency [23], however, this is at the expense of an increased magnetic eld sensitivity.
Magic traps for stable clocks The clock stability at the standard quantum limit is pro-portional to 1=C where C is the fringe contrast. When operating with thermal atomic clouds one faces the issue of atom dephasing. In this regime the atoms are all independent and the precession speed in the Bloch sphere is di erent for each of them: it depends on the clock frequency landscape experienced by an atom during its trajectory. As time passes atoms will dephase from each other which will reduce the contrast of the Ramsey fringes. Dephasing is greatly reduced in a magic trap as a result of the clock frequency being independent of position (or atom’s coordinates). Magic traps are tools for building stable clocks.
A second feature of magic traps is that they make the clock frequency insensitive to uc-tuations of the external eld, leading to a reduction in the technical noise associated with these uctuations. Pseudo-magic traps We de ne a pseudo-magic trap as a trap that possesses the following two properties: (1) no dephasing and (2) clock frequency insensitivity to changes of the trap amplitude, but does not reproduce the free-space frequency (see gure 1.4). Such a trap is the starting point for constructing a secondary frequency standard: (1) high clock quality factors are accessible as a result of long dephasing times and (2) the clock frequency is insensitive to trap magnitude uctuations, which removes a source of technical noise.
This gives the philosophy of the Trapped Atom Clock on a Chip (TACC). As mentioned, TACC relies on the existence of a pseudo-magic magnetic trap for 87Rb, the details of which are elaborated on further in this manuscript.

Cancelation of the photon recoil

When an atom emits or absorbs a photon of wave vector k from a plane wave, it recoils with ~ the momentum ~k. This recoil can provoke a Doppler shift of the atomic transition frequency and introduce a bias on the frequency measurement. This recoil e ect can be inhibited [24] if the trap frequency !=(2 ) and the mass of the atoms m obey (Lamb-Dicke regime): is the Lamb-Dicke parameter.
Operating in such a regime is particularly crucial for clocks based on optical transitions, for which the recoil momentum is 105 times larger than for a microwave clock. This is one reason for choosing optical traps for such clocks, with typical trap frequencies of 100 kHz.
For a microwave clock, the Lamb-Dicke condition is less stringent and magnetic traps, which are typically less con ning than optical traps, can be used. In the case of 87Rb, a trap frequency of 10 Hz gives a Lamb-Dicke parameter of 3 10 4.

Neutral atom trapping on a chip

This section will include a brief account of the principles of magnetic trapping of neutral atoms with particular consideration of 87Rb for which a pseudo-magic magnetic trap exists. We will also give an overview of the basic concept of atom trapping on chips including examples of some trap con gurations.

Magnetic trapping

Neutral atoms interact with the magnetic eld via their magnetic dipole moment . In low magnetic elds (i.e. causing energy shifts much smaller than the hyper ne splitting) the atomic dipole moment is directly proportional to the total angular momentum F with the proportionality constant BgF (gF is the Lande factor). The interaction energy in a magnetic eld B takes the form U =B = Bgf F B = BgF mF jBj. (1.5)
Maxwell’s equations allows only the existence of local minima of the magnetic eld B in space. Thus, only atoms with a magnetic dipole moment antiparallel to the eld (low eld seekers) can be trapped, in minima of the magnetic eld.
To keep the atoms in the trap, it is important that their dipole moment adiabatically follows the local direction of the magnetic eld. The criteria is that the rate of change of the eld’s direction (in the reference frame of the moving atom) must be smaller than the Larmor frequency [10]: d ! = BjgF jB . (1.6) dt L ~
In regions of very small magnetic elds this condition is violated, resulting in atom losses (Majorana losses).

A pseudo-magic trap for 87Rb

Equation 1.5 is only approximate, and a rigorous derivation of the magnetic energy must include the hyper ne splitting. For states of J = 1=2, the hamiltonian can be diagonalized analytically and leads to the Breit-Rabi formula, which gives the eigenenergies as a function of the magnetic eld. At low elds, the eigenstates are very close to the jF; mF i states, and in the rest of the manuscript they are considered as equal.
The magnetic energy of the two trappable states j1i = jF = 1; mF = 1i and j2i = jF = 2; mF = 1i can be calculated. In particular there is a eld Bm around which the energy of these two states have identical dependence to the magnetic eld to rst order. Around this magic eld the corresponding energies can be expanded:
U1(r) = mB(r) + h 1(B(r) Bm)2 (1.7)
U2(r) = A2 A1 + mB(r) + h 2(B(r) Bm)2.

Table of contents :

Introduction 
1 Atom trapping on a chip: a tool for metrology 
1.1 Basic concepts of time metrology
1.1.1 Atomic clocks
1.1.2 Atom-eld interaction
1.1.3 Ramsey and Rabi spectroscopy
1.1.4 Compact frequency references
1.1.5 Using trapped atoms for metrology
1.2 Neutral atom trapping on a chip
1.2.1 Magnetic trapping
1.2.2 A pseudo-magic trap for 87Rb
1.2.3 Magnetic microtraps
1.3 Interactions between cold atoms
1.3.1 General framework: collisions at low energy
1.3.2 Collisional shift
1.3.3 Identical spin rotation eect (ISRE)
2 Experimental methods 
2.1 Overview of the experimental setup
2.1.1 The vacuum system and the chip
2.1.2 Magnetic shielding and optical hat
2.1.3 The interrogation photons
2.1.4 Low noise current sources
2.1.5 Optical bench
2.2 Typical cycle
2.3 Double state detection methods
2.3.1 Double detection: detection with Repump light
2.3.2 Detection with adiabatic passage
2.3.3 Comparison of the two methods
2.4 Loading very shallow traps
2.4.1 Motivations for producing very dilute clouds
2.4.2 Adiabaticity
2.4.3 Canceling the oscillation along x
3 Clock frequency stability 
3.1 Frequency stability analysis
3.1.1 Allan variance
3.1.2 Principle of the characterization of TACC
3.2 Analysis of the sources of noise on the clock frequency
3.2.1 Quantum projection noise
3.2.2 Detection noise
3.2.3 Atom number uctuations
3.2.4 Temperature uctuations
3.2.5 Magnetic eld uctuations
3.2.6 Atomic losses
3.2.7 Rabi frequency uctuations
3.2.8 Local oscillator frequency
3.2.9 Noise added by the post-correction
3.3 Experimental investigation
3.3.1 Measurement of the uncertainty on P2
3.3.2 The best post-correction parameter
3.3.3 Cloud oscillation
3.3.4 Detectivity uctuations
3.3.5 Variation with the bottom magnetic eld
3.3.6 Optimizing the Ramsey time
3.3.7 Optimizing the atom number
3.4 Best frequency stability up-to-date
3.5 Long term thermal eects
3.6 Conclusion
4 Bose-Einstein condensates for time metrology 
4.1 Theory of a dual component BEC
4.1.1 The Gross-Pitaevskii equation for a single component
4.1.2 Gross-Pitaevskii system for a dual component BEC
4.1.3 State-dependent spatial dynamics
4.1.4 Numerical modeling
4.2 Preparing Bose-Einstein condensates
4.2.1 Condensed fraction measurements
4.2.2 Critical temperature
4.2.3 BEC lifetimes
4.3 State-dependent spatial dynamics
4.3.1 Experimental observations
4.3.2 Data modelling
4.4 Coherence of a BEC superposition
4.4.1 In time domain
4.4.2 In frequency domain
4.5 Evidence for increased noise on the atomic response
4.5.1 Estimation of the technical noise contributions
4.5.2 Non-linear spin dynamics in a dual component BEC
4.6 Perspectives
5 Coherent sideband transition by a eld gradient 
5.1 Theory of the sideband excitation by an inhomogeneous eld
5.1.1 Field inhomogeneity
5.1.2 Calculation of the total coupling element
5.2 Spectra of trapped thermal atoms under inhomogeneous excitation
5.2.1 Typical data
5.2.2 Transfer eciency
5.2.3 Observation of the sideband cancelation
5.2.4 Sideband dressing by the carrier
5.3 Cloud dynamics induced by sideband excitations
5.3.1 Non sideband-resolved regime
5.3.2 Sideband-resolved regime
5.3.3 Interpretation
5.4 Conclusion
6 An atomic microwave powermeter 
6.1 Rabi spectra
6.1.1 Principle of the experiment
6.1.2 Results
6.2 Temporal Rabi oscillations
6.2.1 Principle of the experiment
6.2.2 Typical experimental data
6.2.3 Results
6.3 Clock frequency shift measurements
6.3.1 Principle of the experiment
6.3.2 Results
6.4 Discussion
7 Fast alkali pressure modulation 
7.1 Optimizing the preparation of cold atomic clouds
7.1.1 Reminders: MOT loading and trap decay
7.1.2 Constant background pressures case
7.1.3 Solutions with a double-chamber setup
7.1.4 Fast pressure modulation: a solution for single-cell setups
7.2 Experimental methods
7.2.1 Vacuum system
7.2.2 Optics and coils
7.2.3 Pressure measurements
7.3 A device for sub-second alkali pressure modulation
7.3.1 Presentation and design
7.3.2 MOT loading by a pulse
7.3.3 Sensitive measurement of the pressure decay
7.3.4 Rate equations for the adsorption/desorption dynamics
7.3.5 Long term evolution of the pressure
7.4 Other fast sources
7.4.1 Local heating with a laser
7.4.2 Laser heating of a commercial Dispenser
7.4.3 Laser heating of the dispenser active powder
7.4.4 Light-induced atom desorption
7.4.5 Reduced thermal mass dispenser
7.5 Conclusions and perspectives
Conclusion 
A AC Zeeman shifts of the clock frequency
B List of abbreviations and symbols
Bibliography

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