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Analysis of dephasing mechanisms in a standing wave dipole trap

S. Kuhr,? W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede

Institut f¨ ur Angewandte Physik, Universit¨ at Bonn, Wegelerstr. 8, D-53115 Bonn, Germany (Dated: May 2, 2005) We study in detail the mechanisms causing dephasing of hyper?ne coherences of cesium atoms con?ned by a far o?-resonant standing wave optical dipole trap [S. Kuhr et al., Phys. Rev. Lett. 91, 213002 (2003)]. Using Ramsey spectroscopy and spin echo techniques, we measure the reversible and irreversible dephasing times of the ground state coherences. We present an analytical model to interpret the experimental data and identify the homogeneous and inhomogeneous dephasing mechanisms. Our scheme to prepare and detect the atomic hyper?ne state is applied at the level of a single atom as well as for ensembles of up to 50 atoms.

arXiv:quant-ph/0410037v2 2 May 2005

PACS numbers: 32.80.Lg, 32.80.Pj, 42.50.Vk

I.

INTRODUCTION

The coherent manipulation of isolated quantum systems has received increased attention in the recent years, especially due to its importance in the ?eld of quantum computing. A possible quantum computer relies on the coherent manipulation of quantum bits (qubits), in which information is also encoded in the quantum phases. The coherence time of the quantum state superpositions is therefore a crucial parameter to judge the usefulness of a system for storage and manipulation of quantum information. Moreover, long coherence times are of great importance for applications in precision spectroscopy such as atomic clocks. Information cannot be lost in a closed quantum system since its evolution is unitary and thus reversible. However, a quantum system can never be perfectly isolated from its environment. It is thus to some extent an open quantum system, characterized by the coupling to the environment [1]. This coupling causes decoherence, i. e. the evolution of a pure quantum state into a statistical mixture of states. Decoherence constitutes the boundary between quantum and classical physics [2], as demonstrated in experiments in Paris and Boulder [3, 4, 5]. There, decoherence was observed as the decay of macroscopic superposition states (Schr¨ odinger cats) to statistical mixtures. We can distinguish decoherence due to the progressive entanglement with the environment from dephasing effects caused by classical ?uctuations. This dephasing of quantum states of trapped particles has recently been studied both with ions [6] and neutral atoms in optical traps [7, 8]. In this work, we have analyzed measurements of the dephasing mechanisms acting on the hyper?ne ground states of cesium atoms in a standing wave dipole trap. More speci?cally, we use the two Zeeman sublevels | F = 4, mF = 0 and | F = 3, mF = 0 which are coupled by microwave radiation at ωhfs /2π = 9.2 GHz. We present our setup and the relevant experimental

tools in Sec. II, with special regard to the coherent manipulation of single neutral atoms. Our formalism and the notation of the dephasing/decoherence times are brie?y introduced in Sec. III. Finally, in Secs. IV and V we experimentally and theoretically analyze the inhomogeneous and homogeneous dephasing e?ects.

II.

EXPERIMENTAL TOOLS A. Setup

We trap and manipulate cesium atoms in a red detuned standing wave dipole trap. Our trap is formed of two counterpropagating Gaussian laser beams with waist 2w0 = 40 ?m and a power of max. 2 W per beam (see Fig. 1), derived from a single Nd:YAG laser (λ = 1064 nm). Typical trap depths are on the order of U0 = 1 mK. The laser beams have parallel linear polarization and thus produce a standing wave interference pattern. Both laser beams are sent through acoustooptic modulators (AOMs), to mutually detune them for the realization of a moving standing wave. This “optical conveyor-belt” was introduced in previous experiments [9, 10] and has been used for the demonstration of quantum state transportation [11]. For the experiments in this paper, however, we do not transport the trapped

? Electronic

address: kuhr@lkb.ens.fr

FIG. 1: Experimental setup.

2 atoms. To eliminate any heating e?ect arising due to the phase noise of the AOM drivers [10, 12], we used the non-de?ected beams (0th order of the AOMs) to form the dipole trap. The AOMs are only used to vary the dipole trap laser intensity by removing power from the trap laser beams. Cold atoms are loaded into the dipole trap from a high gradient magneto-optical trap (MOT). The high ?eld gradient of the MOT (?B/?z = 340 G/cm) is produced by water cooled magnetic coils, placed at a distance of 2 cm away from the trap. The magnetic ?eld can be switched to zero within 60 ms (limited by eddy currents in the conducting materials surrounding the vacuum chamber) and it can be switched back on within 30 ms. Our vacuum chamber consists of a glass cell, with the cesium reservoir being separated from the main chamber by a valve. Cesium atoms are loaded into the MOT at random from the background gas vapor. To speed up the loading process, we temporarily lower the magnetic ?eld gradient to ?B/?z = 25 G/cm during a time tlow . The low ?eld gradient results in a larger capture cross section which signi?cantly increases the loading rate. Then, the ?eld gradient is returned to its initial value, con?ning the trapped atoms at the center of the MOT. Varying tlow enables us to select a speci?c average atom number ranging from 1 to 50. The required time depends on the cesium partial pressure, which was kept at a level such that we load typically 50 atoms within tlow = 100 ms in these experiments. In order to transfer cold atoms from the MOT into the dipole trap, both traps are simultaneously operated for some tens of milliseconds before we switch o? the MOT. After an experiment in the dipole trap the atoms are transferred back into the MOT by the reverse procedure. All our measurements rely on counting the number of atoms in the MOT before and after any intermediate experiment in the dipole trap. For this purpose we collect their ?uorescence light by a home-built di?ractionlimited objective [15] and detect the photons with an avalanche photodiode (APD). Three diode lasers are employed in this experiment which are set up in Littrow con?guration and locked by polarization spectroscopy. The MOT cooling laser is stabilized to the F = 4 → F ′ = 3/F ′ = 5 crossover transition and shifted by an AOM to the red side of the cooling transition F = 4 → F ′ = 5. The MOT repumping laser is locked to the F = 3 → F ′ = 4 transition, it is π -polarized and is shined in along the dipole trap axis. To optically pump the atoms into the | F = 4, mF = 0 state, we use the unshifted MOT cooling laser, which is only detuned by +25 MHz from the required F = 4 → F ′ = 4 transition. This small detuning is partly compensated for by the light shift of the dipole trap. We shine in the laser along the dipole trap axis with π -polarization together with the MOT repumper. We found that 80% of the atoms are pumped into the | F = 4, mF = 0 state, presumably limited by polarization imperfections of the optical pumping lasers. For the state selective detection (see below) we use a “push-out” laser, resonant to the F = 4 → F ′ = 5 transition. It is σ + -polarized and shined in perpendicular to the trapping beams (z-axis in Fig. 1). To generate microwave pulses at the frequency of 9.2 GHz we use a synthesizer (Agilent 83751A), which is locked to an external rubidium frequency standard (Stanford Research Systems, PRS10). The ampli?ed signal (P = +36 dBm = 4.0 W) is radiated by a dipole antenna, placed at a distance 5 cm away from the MOT. Compensation of the earth’s magnetic ?eld and stray ?elds created by magnetized objects close to the vacuum cell is achieved with three orthogonal pairs of coils. For the compensation, we minimize the Zeeman splitting of the hyper?ne ground state mF manifold which is probed by microwave spectroscopy. Using this method we achieve residual ?elds of Bres < 0.4 ?T (4 mG). The coils of the z -axis also serve to produce a guiding ?eld, which de?nes the quantization axis.

B.

State selective detection of a single neutral atom

Sensitive experimental methods had to be developed in order to prepare and to detect the atomic hyper?ne state at the level of a single atom. State selective detection is performed by a laser which is resonant with the F = 4 → F ′ = 5 transition and thus pushes the atom out of the dipole trap if and only if it is in F = 4. An atom in the F = 3 state, however, is not in?uenced by this laser. Thus, it can be transferred back into the MOT and be detected there. Although this method appears complicated at ?rst, it is universal, since it works with many atoms as well as with a single one. Other methods, such as detecting ?uorescence photons in the dipole trap by illuminating the atom with a laser resonant to the F = 4 → F ′ = 5 transition, failed in our case because the number of photons detected before the atom decays into the F = 3 state is not su?cient. In order to achieve a high e?ciency of the state selective detection process, it is essential to remove the atom out of the dipole trap before it is o?-resonantly excited to F ′ = 4 and spontaneously decays into the F = 3 state. For this purpose, we use a σ + -polarized push-out laser, such that the atom is optically pumped into the cycling transition | F = 4, mF = 4 → | F ′ = 5, mF = 5 . In our setup, the polarization is not perfectly circular, since for technical reasons we had to shine in the laser beam at an angle of 2? with respect to the magnetic ?eld axis. This entails an increased probability of exciting the F ′ = 4 level from where the atom can decay into the F = 3 ground state. To prevent this, we remove the atom from the trap su?ciently fast by shining in the push-out laser from the radial direction with high intensity (I/I0 ≈ 100, with w0 = 100 ?m, P = 30 ?W, where I0 = 1.1 mW/cm2 is the saturation intensity). In this regime its radiation pressure force is stronger than the dipole force in the ra-

3

FIG. 3: Atom counting. Initial and ?nal numbers of atoms are inferred from their ?uorescence in the MOT. Shown are the integrated APD counts binned in time intervals of 1 ms and accumulated over 10 repetitions with 20 atoms each.

FIG. 2: State selective detection of a single atom. The graphs show the ?uorescence signal of the atom during state preparation and detection, binned in time intervals of 5 ms. The bars above the graphs show the timing of the lasers. Graphs (a)(i) and (b)(i) show the signals of a single atom, prepared in F = 3 and F = 4, respectively. Graphs (a)(ii) and (b)(ii) show the added signal of about 150 events.

dial direction, such that we push out the atom within half the radial oscillation period (≈ 1 ms). In this case, the atom receives a momentum corresponding to the sum of all individual photon momenta. This procedure is more e?cient than heating an atom out of the trap, which occurs when the radiation pressure force of the push-out laser is weaker than the dipole force, and the atom performs a random walk in momentum space while absorbing and emitting photons. If we adiabatically lower the trap to typically 0.12 mK prior to the application of the push-out laser, we need on average only 35 photons to push the atom out of the trap. This number is small enough to prevent o?-resonant excitation to F ′ = 4 and spontaneous decay to F = 3.

A typical experimental sequence to test the state selective detection is shown in Fig. 2. First, the atom is transferred from the MOT into the optical dipole trap. Using the cooling and repumping laser of the MOT, we optically pump the atom either in the F = 3 (Fig. 2a) or the F = 4 hyper?ne state (Fig. 2b). The push-out laser then removes all atoms in F = 4 from the trap. Any remaining atom in F = 3 is transferred back into the MOT, where it is detected. Figures 2a(i) and 2b(i) show the signals of a single atom, prepared in F = 3 and F = 4, respectively. Our signal-to-noise ratio enables us to unambiguously detect the surviving atom in F = 3, demonstrating the stateselective detection at the single atom level. We performed 157 repetitions with a single atom prepared in F = 3 and found that in 153 of the cases the atom remains .2 trapped, yielding a detection probability of 97.5+1 ?2.0 %. Similarly, only 2 out of 167 atoms prepared in F = 4 .6 remain trapped, yielding 1.2+1 ?0.8 %. The asymmetric errors are the Clopper-Pearson 68% con?dence limits [16]. These survival probabilities can also be inferred by directly adding the signals of the individual repetitions and by comparing the initial and ?nal ?uorescence levels in the MOT, see Figs. 2a(ii) and 2b(ii). All following experiments are performed in the same way. We initially prepare the atoms in the | F = 4, mF = 0 state and measure the population transfer to | F = 3, mF = 0 induced by the microwave radiation. After the application of one or a sequence of microwave pulses, the atom is in general in a superposition of both hyper?ne states, | ψ = c3 | F = 3, mF = 0 + c4 | F = 4, mF = 0 , (1) with complex probability amplitudes c3 and c4 . Our detection scheme only allows us to measure the population of the hyper?ne state F = 3: P3 = |c3 |2 = w+1 , 2 (2)

where w is the third component of the Bloch vector, see below. The number P3 is determined from the number of

4 changed the position of the microwave antenna for practical reasons. The maximum population detected in F = 3 is C = (60.4 ± 0.7)%. This reduction from 100% is caused by two e?ects. First, when we use many (> 40) atoms at a time, up to 20% of the atoms are lost during the transfer from the MOT into the dipole trap due to inelastic collisions, as veri?ed in an independent measurement. The remaining losses arise due to the non-perfect optical pumping process.

FIG. 4: Rabi oscillations on the | F = 4, mF = 0 → | F = 3, mF = 0 clock transition recorded at a trap depth U0 = 1.0 mK. Each data point results from 100 shots with about 60 initial atoms. The line is a ?t according to Eq. (6). III. PHENOMENOLOGICAL DESCRIPTION OF DECOHERENCE AND DEPHASING

atoms before (Ninitial ) and after (N?nal ) any experimental procedure in the dipole trap. Ninitial and N?nal are inferred from the measured photon count rates, Cinitial , C?nal and Cbackgr (see Fig. 3): Ninitial = and N?nal = C?nal ? Cbackgr . C1atom (4) Cinitial ? Cbackgr C1atom (3)

The ?uorescence rate of a single atom, C1atom , is measured independently. From the atom numbers we obtain the fraction of atoms transferred to F = 3, P3 = N?nal Ninitial (5)

In our experiment we observe quantum states in an ensemble average, and decoherence manifests as a decay or dephasing of the induced magnetic dipole moments. It is useful to distinguish between homogeneous and inhomogeneous e?ects. Whereas homogeneous dephasing mechanisms a?ect each atom in the same way, inhomogeneous dephasing only appears when observing an ensemble of many atoms possessing slightly di?erent resonance frequencies. As we will see later, the most important di?erence between the two mechanisms is the fact that inhomogeneous dephasing can be reversed, in contrast to the irreversible homogeneous dephasing. The interaction between the oscillating magnetic ?eld component of the microwave radiation, B cos ωt, and the magnetic dipole moment, ?, of the atom is well approximated by the optical Bloch equations [17]: ˙ = ?? × u u (7)

The measured number of atoms, Ninital can be larger than the actual number of atoms in the dipole trap, since we lose atoms during the transfer from the MOT into the dipole trap (see below).

C. Rabi oscillations

We induce Rabi oscillations by a single resonant microwave pulse at the maximum RF power. For the graph of Fig. 4 we varied the pulse length from 0 ?s to 225 ?s in steps of 5 ?s. Each point in the graph results from 100 shots with about 60 ± 10 atoms each. The corresponding statistical error is below 1% and is thus not shown in the graph. The error of the data points in Fig. 4 is dominated by systematic drifts of the storage probability and e?ciencies of the state preparation and detection. Since w(t) = ? cos ?R t, we ?t the graph with C P3 (t) = (1 ? cos ?R t) , 2 (6)

with the torque vector ? ≡ (?R , 0, δ ) and the Bloch vector u ≡ (u, v, w). Here, ?R = ?B/? h is the Rabi frequency and δ = ω ? ω0 is the detuning of the microwave from the atomic transition frequency ω0 . In the following, the initial quantum state | F = 4, mF = 0 corresponds to the Bloch vector u = (0, 0, ?1), whereas | F = 3, mF = 0 corresponds to u = (0, 0, 1). We include the decay rates as damping terms into the Bloch equations and use a notation of the di?erent times for population and polarization decay similar to the one of nuclear magnetic resonance: ˙ = δ v ? u u T2 ˙ v ˙ w = ? δ u + ?R w ? = ? ?R v ? v T2 (8a) (8b) (8c)

w ? wst , T1

which yields a Rabi frequency ?R /2π = (14.60 ± 0.02) kHz. Note that this Rabi frequency is higher than the one used later in this report (10 kHz) because we

where . . . denotes the ensemble average. The total homogeneous transverse decay time T2 is given by the polarization decay time T2 ′ and the reversible dephasing ? time T2 1 1 1 = ′ + ?. T2 T2 T2 (9)

5

? Inhomogeneous dephasing (T2 ) occurs because the atoms may have di?erent resonance frequencies depending on their environment. Thus the Bloch vectors of the individual atoms precess with di?erent angular velocities and lose their phase relationship, they dephase. In our case, inhomogeneous dephasing arises due to the energy distribution of the atoms in the trap. This results in a corresponding distribution of light shifts because hot and cold atoms experience di?erent average trapping laser intensities. The longitudinal relaxation time, T1 , describes the population decay to a stationary value wst . In our case, T1 is governed by the scattering of photons from the dipole trap laser, which couples the two hyper?ne ground states via a two-photon Raman transition. This e?ect is suppressed due to a destructive interference e?ect yielding relaxation times of several seconds (see Sec. V B). We do not include losses of atoms from the trap in the decay constants, which occur on the same timescale.

IV.

INHOMOGENEOUS DEPHASING

We measure the transverse decay time T2 by performing Ramsey spectroscopy, which consists of the application of two coherent rectangular microwave pulses, separated by a time interval t [18]. The initial Bloch vector u0 = (0, 0, ?1) corresponds to an atom prepared in the | F = 4, mF = 0 -state. A π/2-pulse rotates the Bloch vector into the state (0, ?1, 0), where the atom is in a superposition of both hyper?ne states. The Bloch vector freely precesses in the uv -plane with an angular frequency δ . Note that δ has to be small compared to the Rabi frequency and the spectral pulse width, such that the pulse can be approximated as near resonant, and complete population transfer can occur. After a free precession during t, a second π/2-pulse is applied. The measurement of the quantum state ?nally projects the Bloch vector onto the w axis. We recorded Ramsey fringes for two di?erent dipole trap depths, U0 = 0.1 mK and 0.04 mK (see Fig. 5). Each point in the graph of Fig. 5 corresponds to 30 shots with about 50 trapped atoms per shot, yielding errors (not shown) of less than 1%. The quoted values for U0 are calculated from the measured power and the waist of the dipole trap laser beam and have an estimated uncertainty of up to 50%. We initially transfer the atoms from the MOT into a deeper trap (U0 > 1 mK) to achieve a high transfer e?ciency. When the MOT is switched o?, we adiabatically lower the trap depth using the AOMs. Our Ramsey fringes show a characteristic decay, which is not exponential. This decay is due to inhomogeneous dephasing, which occurs because after the ?rst π/2pulse, the atomic pseudo-spins precess with di?erent angular frequencies. In the following, we derive analytic expressions for the observed Ramsey signal and we show that the envelope of the graphs of Fig. 5 is simply the Fourier transform of the atomic energy distribution.

FIG. 5: Ramsey fringes recorded for two di?erent trap depths (a) U0 = 0.1 mK and (b) 0.04 mK. Their decay with time ? constants T2 = 4.4 ± 0.1 ms and 20.4 ± 1.1 ms, respectively, is governed by inhomogeneous dephasing caused by the energy distribution in the trap. Each data point results from 30 shots with about 50 initial atoms. The damped oscillation is a ?t with P3,Ramsey (t) and the envelopes are the functions ? B ± Aα(t, T2 ) (see Eqs. (26), (29)).

A.

Di?erential light shift and decay of Ramsey fringes

The light shift of the ground state due to the Nd:YAG laser is simply the trapping potential U0 (?) = ?Γ I Γ h . 8 I0 ? (10)

The detuning of the Nd:YAG-laser from the D-line of an atom in F = 4 is 9.2 GHz less than for an atom in F = 3. As a consequence, the F = 4 level experiences a slightly stronger light shift, resulting in a shift of the F = 3 → F = 4 microwave transition towards smaller resonance frequencies. This di?erential light shift, δ0 , can be approximated as hδ0 = U0 (?e? ) ? U0 (?e? + ωhfs ), ? (11)

where ?e? = ?1.2 × 107 Γ is an e?ective detuning, taking into account the weighted contributions of the D1 and D2

6 lines [10]. ωhfs = 2.0 × 103 Γ is the ground state hyper?ne splitting. Since ωhfs ? ?e? , we ?nd that the di?erential light shift is proportional to the total light shift U0 , hδ0 = ηU0 , ? (12) Note that this distribution is only valid in the regime kB T ? U0 , since the virial theorem was applied for the case of a harmonic potential. To model the action of the Ramsey pulse sequence, we express the solutions of Eq. (7) as rotation matrices acting on the Bloch vector. The Ramsey sequence then reads uRamsey (t) = Θπ/2 · Φfree (δ, t) · Θπ/2 · u0 , with the matrices describing ? 1 Θπ/2 = ?0 0 (17)

with a scaling factor η = ωhfs /?e? = 1.45 × 10?4 . For atoms trapped in the bottom of a potential of U0 = 1 mK, the di?erential light shift is δ0 = ?2π × 3.0 kHz. In the semiclassical limit, i. e. neglecting the quantized motion of the atom in the dipole trap potential, the free precession phase accumulated by an atomic superposition state between the two π/2-pulses depends on the average di?erential light shift only. In the following, we calculate the expected Ramsey signal using this semiclassical approach and obtain simple analytical expressions. Furthermore, we veri?ed the validity of the presented model by performing a quantum mechanical density matrix calculation (not presented here) which agrees to within one percent with the semiclassical results. The small deviation can be attributed to the occurrence of small oscillator quantum numbers nosc ? 5 in the sti? direction of the trap. Note that, strictly speaking, our model of a timeaveraged di?erential light shift is only correct if the atom carries out an integer number of oscillation periods in the trap between the two π/2-pulses. However, we have checked that the variable phase accumulated during the remaining fraction of an oscillation period does not cause a measurable reduction of the Ramsey fringe contrast and can therefore be neglected. Since a hot atom experiences a lower laser intensity than a cold one, its averaged di?erential light shift is smaller. The energy distribution of the atoms in the dipole trap obeys a three-dimensional Boltzmann distribution with probability density [13, 14] p(E ) = E2 E . exp ? 3 2(kB T ) kB T (13)

The total precession angle, φ(δ, t), represents the accumulated phase during the free evolution of the Bloch vect tor, φ(t) = 0 δ (t′ )dt′ . The detuning δ (t) may in general vary spatially and in time, depending on the energy shifts of the atomic levels. If the Bloch vector is initially in the state u0 = (0, 0, ?1), we obtain from Eq. (17): wRamsey (t) = cos δt, (20)

and the free precession around the w-axis with angular frequency δ during a time interval t, ? ? cos φ(δ, t) sin φ(δ, t) 0 Φfree (δ, t) = ?? sin φ(δ, t) cos φ(δ, t) 0 ? . (19) 0 0 1

the action of a π/2-pulse, ? 0 0 0 1?, (18) ?1 0

where δ = ω ? ω0 is the detuning of the microwave radiation with frequency ω from the atomic resonance ω0 . In order to see the Ramsey fringes, we purposely shift ω with respect to the ground state hyper?ne splitting, ωhfs , by a small detuning, δsynth , set at the frequency synthesizer ω = ωhfs + δsynth . (21)

Here E = Ekin + U is the sum of kinetic and potential energy. In a harmonic trap the virial theorem states that the average potential energy is half the total energy, U = E/2. Thus, the average di?erential light shift for an atom with energy E is given by: δls (E ) = δ0 + ηE 2? h (14)

The atomic resonance frequency ω0 is modi?ed due to external perturbations ω0 = ωhfs + δls + δB , (22)

where δ0 < 0 is the maximum di?erential light shift. As a consequence, the energy distribution p(E ) yields, except for a factor and an o?set, an identical distribution α ? (δls ) of di?erential light shifts [14]: K3 (δls ? δ0 )2 exp [?K (δls ? δ0 )] α ? (δls ) = 2 with 2? h K= . ηkB T (16) (15)

where δls is the energy dependent di?erential light shift, δB is the quadratic Zeeman shift. Now, the inhomogeneously broadened Ramsey signal is obtained by averaging over all di?erential light shifts δls :

∞

wRamsey,inh (t) =

δ0

α ? (δls ) × cos [(δsynth ? δB ? δls )t] dδls .

(23)

Eq. (23) shows that the shape of the Ramsey fringes is the Fourier(-Cosine)-Transform of the atomic energy distribution. Note that in the above integral we have set the

7 upper integration limit to ∞, instead of the maximum physically reasonable value, δ0 /2, to obtain the analytic solution

? ? ) cos [δ ′ t + κ(t, T2 )], wRamsey,inh (t) = α(t, T2

Fig. 5(a)

Fig. 5(b)

(24)

with the sum of the detunings, δ ′ = δsynth ? δB ? δ0 (25)

? and a time- dependent amplitude α(t, T2 ) and phase shift ? κ(t, T2 ) [14]

U0 (est.) 0.1 mK 0.04 mK δsynth /2π 2250 Hz 1050 Hz A 28.7 ± 0.5 % 13.6 ± 0.1 % B 30.5 ± 0.1 % 13.8 ± 0.1 % δ ′ /2π 2133.7 ± 1.5 Hz 722.5 ± 0.5 Hz ? 0.35 ± 0.02 0.13 ± 0.03 ? T2 4.4 ± 0.1 ms 20.4 ± 0.6 ms TABLE I: Fit parameters extracted from the Ramsey fringes of Fig. 5 using Eq. (29).

? ) α(t, T2

= 1 + 0.95

t ? T2

2 ?3/2

(26)

and

? κ(t, T2 ) = ?3 arctan 0.97

t ? T2

.

(27)

is not impaired by these imperfections. From the ?t parameters we obtain V = 0.97 ± 0.01 and V = 1.00+0 ?0.03 for the two cases. As a check of consistency, we calculate the di?erential light shift δ0 from the ?tted detuning δ ′ and the experimental values of δB and δsynth , δ0 = δsynth ? δB ? δ ′ . (31)

Despite this non-exponential decay, we have introduced the inhomogeneous or reversible dephasing time ? T2 as the 1/e-time of the amplitude α(t):

? T2 =

e2/3 ? 1 K = 0.97

2? h . ηkB T

(28)

? Thus, the reversible dephasing time T2 is inversely proportional to the temperature of the atoms. ? The phase shift κ(t, T2 ) arises due to the asymmetry of the probability distribution α ? (δls ). The hot atoms in the tail of the energy distribution dephase faster than the cold atoms, due to their larger spread. The fact that these hot atoms no longer contribute to the Ramsey signal results in a weighting of the mean δls towards larger negative values. To ?t our experimental data, we derive the following expression from Eq. (24),

P3,Ramsey (t)

? = B + α(t, T2 ) ? ′ ×A cos [δ t + κ(t, T2 ) + ?],

(29)

where the amplitude A and the o?set B account for the imperfections of state preparation and detection. The ? other ?t parameters are δ ′ , T2 , and a phase o?set ?. The corresponding ?ts are shown in Fig. 5 and the resulting ?t parameters are summarized in Table I. For the two graphs, the maximum population detected in F = 3, P3,max = A + B is only about 60% and 30%, respectively. The reduction to 60% in Fig. 5(a) is again due to imperfections in the optical pumping process and due to losses by inelastic collisions, as discussed in Sec. II C. The additional reduction in Fig. 5(b) occurs during the lowering of the trap to U0 = 0.04 mK, where another 50% of the atoms are lost. Note, however, that the fringe visibility V = A B (30)

The calculated quadratic Zeeman shift in the externally applied guiding ?eld of B = 97.9 ± 1.5 ?T is δB /2π = 412 ± 13 Hz, where the error is due to the uncertainty of the calibration. We obtain δ0 /2π = ?268 ± 13 Hz and δ0 /2π = ?78 ± 13 Hz. From the values of δ0 we can formally deduce the potential depth corresponding to U0 = 0.090 ± 0.004 mK and U0 = 0.026 ± 0.004 mK, which almost match the expected trap depths estimated from the dipole trap laser power assuming purely linear polarization. The discrepancy for the lowest trap depth could arise from the fact that the energy distribution is truncated at E = U0 , since we have lost the atoms with the highest energy during the lowering of the trap. This truncation will reduce the e?ective δ0 and thus yield a smaller trap depth. Finally, the phase o?set ? occurs because the Bloch vector precesses around the w-axis even during the application of the two π/2-pulses. In contrast, our ansatz of Eq. (17) takes into account only the free precession in between the two pulses. The additional precession angle amounts to ? = 2 tπ/2 δ ′ . Given tπ/2 = 16 ?s and the ?tted value of δ ′ we obtain ? = 0.42 for Fig. 5(a) and ? = 0.14 for Fig. 5(b), which is close to the ?tted values of Table I.

V. HOMOGENEOUS DEPHASING MECHANISMS A. Spin echoes

The inhomogeneous dephasing can be reversed using a spin echo technique, i. e. by applying an additional π -pulse between the two Ramsey π/2-pulses. Although originally invented in the ?eld of nuclear magnetic resonance [19], this technique was recently also employed in optical dipole traps [20].

8

FIG. 6: Spin echoes. Shown are spin echoes recorded for three di?erent trap depths, (a) U0 = 1.0 mK, (b) 0.1 mK and (c) 0.04 mK. We observe a decrease of the maximum spin echo amplitude with increasing time of the π -pulse, with longer decay times in lower trap depths. All spin echoes are ?tted using Eq. (37). In (a) and (b), the ?rst curve is a Ramsey signal, recorded with otherwise identical parameters.

We recorded echo signals in three di?erent trap depths, U0 = 1.0 mK, 0.1 mK and 0.04 mK for di?erent times of the π -pulse, τπ (see Fig. 6). We observe that the visibility of the echo signals decreases if we increase τπ . A slower decrease of the visibility is obtained in lower traps. For U0 = 0.04 mK, τπ = 200 ms, we even observed oscillations that reappear at t = 400 ms. In order to interpret these results, we ?rst model the action of the microwave pulses for the spin echo, similar to the discussion in Sec. IV. After the ?rst π/2-pulse at t = 0, all Bloch vectors start at u(0) = (0, ?1, 0). Due to inhomogeneous dephasing, the Bloch vectors rotate at slightly di?erent frequencies around the w-axis. A π -pulse at time τπ rotates the ensemble of Bloch vectors around the u axis by 180? and induces a complete rephasing at 2τπ in the state u(2τπ ) = (0, 1, 0). The corresponding matrix equation reads uecho (t) = Θπ/2 · Φfree (δ, t ? τπ ) · Θπ · ·Φfree (δ, τπ ) · Θπ/2 · u0 , where we de?ned ? ? 1 0 0 ? ? Θπ = ?0 ?1 0 ? . 0 0 ?1 (33) (32)

Here, τπ is the time between the ?rst π/2- and the π pulse, and t > τπ is the time of the second π/2 pulse. As a result of Eq. (32), we obtain wecho (t) = ? cos[δ (t ? 2τπ )]. (34)

We calculate the shape of the inhomogeneously broadened echo signal, wecho,inh(t), by integrating over all differential light shifts δls

∞

wecho,inh (t) = ?

δ0

α ? (δls ) × (35)

× cos [(δsynth ? δls ? δB )(t ? 2τπ )] dδls . The integration yields a result similar to Eq. (23),

wecho,inh(t) = ?α(t ? 2τπ ) × cos [δ ′ (t ? 2τπ ) + κ(t ? 2τπ )], (36) with amplitude α(t) and phase shift κ(t) as de?ned in Eqs. (26) and (27). Eq. (36) shows that the amplitude of the echo signal regains its maximum at time 2τπ . Finally, the population in F = 3 reads:

? P3,echo (t) = B ? α(t ? 2τπ , T2 ) ? ′ ×A cos [δ (t ? 2τπ ) + κ(t ? 2τπ , T2 ) + ψ ].

(37)

9

Fig. 6(a) Fig. 6(b) Fig. 6(c)

U0 (est.) 1.0 mK 0.1 mK 0.04 mK ? T2 0.86 ± 0.05 ms 2.9 ± 0.1 ms 18.9 ± 1.7 ms T2 ′ 10.2 ± 0.4 ms 33.9 ± 1.0 ms 146.2 ± 6.6 ms T1 (calc.) 8.6 s 86 s 220 s

? TABLE II: Summary of dephasing times. T2 and T2 ′ are 1 obtained from the echo signals of Fig. 6. T1 = Γ? Raman is calculated using Eq. (63).

? This equation is used to extract dephasing times T2 from all spin echoes of Fig. 6. The average values are listed in ? Table II, where T2 was obtained by averaging over the respective datasets. From the amplitude, A, and o?set, B , of each echo signal we calculate the visibility V = A/B , plotted in Fig. 7 as a function of τπ . The phase shift ψ accounts for slow systematic phase drifts during the spin echo sequence. So far, we considered the detuning as constant during the experimental sequence. We now include in our model a time-varying detuning, δ (t), in order to account for a stochastic variation of the precession angles of the Bloch vector, τπ 2τπ

φ1 =

0

δ (t) dt

and φ2 =

τπ

δ (t) dt,

(38)

before and after the π -pulse. The phase di?erence φ2 ? φ1 is expressed as a mean di?erence of the detuning, ?δ = φ2 ? φ1 . τπ (39)

FIG. 7: Decay of the spin echo visibility, extracted from the signals of Fig. 6. The ?ts (red lines) are the Gaussians of Eq. (45). The dashed and dotted lines are the best and worst case predictions inferred from the measured pointing instability of the trapping laser shown in Fig. 9.

The Bloch vector at time 2τπ , when the inhomogeneous dephasing has been fully reversed, reads uecho (?δ, 2τπ ) = Θπ/2 · Φfree (δ + ?δ, τπ ) · Θπ · ·Φfree (δ, τπ ) · Θπ/2 · u0 , (40) which results in wecho (?δ, 2τπ ) = ? cos(?δ τπ ). (41)

For comparison with the experimental values, we ?t the spin echo visibility of Fig. 7 with a Gaussian, 1 2 2 V (2τπ ) = C0 exp ? τπ σexp 2 (45)

For a Gaussian distribution of ?uctuations with mean ?δ = 0 and variance σ (τπ )2 , pτπ (?δ ) = (?δ ) 1 √ exp ? , 2 σ (τπ )2 σ (τπ ) 2π

2

(42)

the average w-component of the Bloch vector is calculated,

∞

with a time-independent detuning ?uctuation σexp . We de?ne the homogeneous dephasing time T2 ′ as the 1/e decay time of the spin echo visibility: √ 2 ?1 ′ ′ . (46) V (2T2 ) = C0 e ? T2 = σexp

B. Origins of irreversible dephasing

wecho,hom (2τπ ) =

?∞

? cos (?δ τπ ) pτπ (?δ ) d?δ (43)

1 2 = exp ? 2 τπ σ (τπ )2 .

Thus, the spin-echo visibility, V , yields

2 2 V (2τπ ) = V0 exp ? 1 2 τπ σ (τπ ) .

(44)

Candidates for irreversible dephasing mechanisms include intensity ?uctuations (1) and pointing instability of the dipole trap laser (2), heating of the atoms (3), ?uctuating magnetic ?elds (4), ?uctuations of

10 the microwave power and pulse duration (5) and spin relaxation due to spontaneous Raman scattering from the dipole trap laser (6).

U0 σexp (meas.) (1) intensity ?uctuations pointing instability best case worst case heating (3) σh /2π (upper limit) photon scattering ′ σp (T2 )/2π magnetic ?eld ?uctuations σb (T2 ′ )/2π

1.0 mK

0.1 mK

0.04 mK

22.0 ± 0.9 Hz 6.6 ± 0.2 Hz 1.54 ± 0.07 Hz 5.9 Hz 0.67 Hz 0.17 Hz

(2)

10.6 21.6 5.3

Hz Hz Hz

2.4 6.7 1.6

Hz Hz Hz

1.3 3.7 2.0

Hz Hz Hz

FIG. 8: Allan deviation of the intensity ?uctuations according to Eq. (47).

(3a)

(3b)

4.5

Hz

1.5

Hz

1.4

Hz

(4)

1.7

Hz

0.35

Hz

0.17

Hz

TABLE III: Summary of dephasing mechanisms. Shown are ′ the ?uctuation amplitudes σ (T2 )/2π .

(1) Intensity ?uctuations of the trapping laser. The intensity ?uctuations are measured by shining the laser onto a photodiode and recording the resulting voltage as a function of time. From this signal we calculate σ (τπ )2 by means of the Allan variance, de?ned as [21]:

2 σA (τ ) =

1 m

m

k=1

(xτ,k+1 ? xτ,k )2 . 2

(47)

Here xτ,k denotes the average of the photodiode voltages over the k -th time interval τ , normalized to the mean voltage of the entire dataset. The resulting Allan deviation σA is a dimensionless number which expresses the relative ?uctuations. They directly translate into ?uctuations σ (τ ) of the detuning, √ σ (τ ) = 2δ0 σA (τ ). (48) √ The factor of 2 arises because σ (τ ) is the standard deviation of the di?erence of two detunings with standard deviation σA (τ ) each. The maximum di?erential light shift δ0 in Eq. (48) is calculated according to Eq. (31) using the measured values of δ ′ . As a result we ?nd relative intensity ?uctuations of σA (τ ) < 0.2% (see Fig. 8). The corresponding absolute ?uctuation amplitudes σ (T2 ′ )/2π (shown in Table III) are too weak to account for the observed decay of the spin echo visibility. (2) Pointing instability of the trapping laser. Any change of the relative position of the two interfering laser beams also changes the interference contrast, and hence the light shift δ0 . These position shifts can arise due to shifts of the laser beam itself, due to variations of the optical paths e. g. from acoustic vibrations of the mirrors

FIG. 9: Measuring the pointing instability. (a) The dipole trap beams having a frequency di?erence of ?ν = 10 MHz are overlapped on a fast photodiode. (b) Allan deviation of the amplitude of the resulting beat signal.

or from air ?ow. In order to measure the pointing instabilities we mutually detune the two dipole trap beams by ?ν = 10 MHz using the AOMs and overlap them on a fast photodiode (see Fig. 9(a)). The amplitude of the resulting beat signal directly measures the interference contrast of the two beams and is thus proportional to the depth of the potential wells of the standing wave dipole trap. We used a network analyzer (HP 3589A) operated in “zero span” mode to record the temporal variation of the beat signal amplitude within a ?lter bandwidth of 10 kHz. The resulting Allan deviation of the beat signal amplitudes is shown in Fig. 9(b). The lower curve shows the signal in the case of well overlapped beams, whereas for

11 the upper curve, we purposely misaligned the beams so that the beat signal amplitude is reduced by a factor of 2. In the latter case variations of the relative beam position cause a larger variation of the beat signal amplitude, since the beams overlap on the slopes of the Gaussian pro?le. These two curves measure the best and the worst cases of the ?uctuations. We found that the relative ?uctuations for long time scales of τ > 100 ms reach up to 3% in the worst case. They are thus one order of magnitude greater than the variations caused by intensity ?uctuations. The frequency ?uctuations σ (τ ) are again calculated using Eq. (48). This result is plotted together with the observed visibility in Fig. 7. Our data points lie in between these best and worst case predictions. (3a) Heating e?ects. Heating processes in the trap can also cause signi?cant irreversible decoherence, since they cause a variation of the atomic resonance frequency within the microwave pulse sequence. A constant heating ˙ increases the average energy of the atoms for the rate E second free precession interval [τπ , 2τπ ], compared to the ˙ π . The energy E of individual ?rst interval [0, τπ ], by Eτ atoms, however, can be changed by much more than this average energy gain. To estimate the e?ect we have to calculate the typical energy change of individual atoms during the free precession interval caused by the ?uctuating forces which are responsible for the heating. For this purpose we approximate the trap as harmonic and assume the following model of the heating process. Due to a random walk√ in momentum space, an initial atomic momentum p = 2mE evolves into a symmetric, Gaussian momentum distribution around p with a standard deviation of ˙ π ?prms given by the average energy gain Eτ

2 ˙ π = (?prms ) . Eτ 2m

over the initial energy E weighted by the ndimensional thermal energy distribution p(n) (E ) ∝ E n?1 exp (?E/kB T ):

∞

p(n) (?δ ) =

0

pE (?δ )p(n) (E )dE.

(53)

Finally we obtain the rms detuning ?uctuations σheat from the resulting distribution of ?δ as σheat (τπ )

(n) (n) 2 ∞

=

?∞

?δ 2 p(n) (?δ )d?δ.

(54)

Evaluation of σheat for the experimentally relevant time ′ scale τπ = T2 /2 yields σheat =

(n)

η ? h

n ˙ ′ ET2 kB T . 2

(55)

Heating e?ects in our trap have been investigated in detail in Ref. [12]. An upper limit for the heating rate ˙ = 2 · 10?2 mK/s is obtained from the trap lifetime of E ˙ of 50 s (for U0 = 1.0 mK). When we linearly scale E to our trap depths of U0 = 1.0 mK, U0 = 0.1 mK, and U0 = 0.04 mK, and assume temperatures of T = 0.1 mK, T = 0.06 mK, and T = 0.02 mK, we obtain ?uctuation (3) amplitudes for the 3D-case (n = 3) of σheat = 5.3 Hz, (3) (3) σheat = 1.6 Hz and σheat = 2.0 Hz, respectively. We stress however, that these values for σheat are upper ˙ for the limits since we did not measure heating rates E ˙ trap depths we used. The actual values of E and the resulting values for σheat could be orders of magnitude smaller than the upper limits inferred from the life time because the heating rate strongly depends on the oscillation frequencies and the details of the laser noise spectrum. (3b) Photon recoil. Our model of the heating process also gives an estimate of the dephasing due to photon recoil. If we had one photon scattered per time interval τπ giving two recoils, we would obtain a heating rate ? 2 k2 1 ˙ = h E . m τπ Inserting into Eq. (55) (n = 3) yields (56)

(49)

˙ π we can linearly approximate the Assuming E ? Eτ energy-momentum relationship at E . In this approximation the momentum distribution is therefore equivalent to a Gaussian distribution of the energies with ˙ π. ?Erms = 2 E Eτ (50)

According to Eq. (14) the corresponding standard deviation σheat,E of the detunings ?δ is η σheat,E (τπ ) = h ? ˙ π, E Eτ (51)

1ph σheat = ηk

3kB T . m

(57)

Scattering of nph photons would yield σph (nph ) = √ 1ph nph σheat . (58)

depending on the initial energy E. We now integrate the distribution of the detunings pE (?δ ) = ?δ 2 1 1 exp ? = √ 2π σheat,E (τπ ) 2 σheat,E (τπ )

2

Given a scattering rate Γs , the number of scattered photons obeys a Poissonian distribution. Since for our parameters, the probability of scattering more than one photon is negligible, we obtain σph (τπ ) = ηk 3kB T Γs τπ Γs τπ . exp ? m 2 (59)

(52)

12 We use the temperatures of the previous paragraph and the photon scattering rates (see below) of Γs = 10.6 s?1 , ′ Γs = 1.06 s?1 and Γs = 0.41 s?1 . With τπ = T2 /2 ′ ′ we obtain σph (T2 ) = 4.5 Hz, σph (T2 ) = 1.5 Hz and ′ σph (T2 ) = 1.4 Hz, respectively. (4) Fluctuating magnetic ?elds. Using a ?uxgate magnetometer we measured a peak-to-peak value of the magnetic ?eld ?uctuations of ?B = 0.13 ?T, dominated by components at ν = 50 Hz. The resulting frequency shift on the | F = 4, mF = 0 → | F = 3, mF = 0 transition is: ?ω = 2 ?ω0→0 B0 ?B, (60) Raman processes are strongly reduced due to a destructive interference of the transition amplitudes. Thus, the spin relaxation rate is much larger than the spontaneous scattering rate. This e?ect was ?rst observed on optically trapped Rubidium atoms in the group of D. Heinzen [22] and was also veri?ed in experiments in our group [23]. The corresponding transition rate is calculated by means of the Kramers-Heisenberg formula [24], which is a result from second order perturbation theory. We obtain for the rate of spontaneous transitions, Γs , from the ground state | F, m to the ground state | F ′′ , m′′ : Γs =

3 3 c2 ω L I a(1/2) a(3/2) + 4 4? hd ?1/2 ?3/2 2

,

(62)

where B0 = 97.9 ?T is the o?set ?eld and ?ω0→0 /2π = 43 mHz/(?T)2 is the quadratic Zeeman shift. For our case, we obtain ?ω = 1.1 Hz. The e?ect of the magnetic ?uctuations depends on the time interval between the microwave pulses. If this time is large compared to 1/ν , all ?uctuations cancel except for those of the last oscillation period. As a consequence, the e?ect on the detuning ?uctuations σ also decreases. We calculate this e?ect by computing the Allan deviation σA (τ ) of a 50 Hz sine√ signal. The detuning ?uctuations then read σb (τ ) = 2?ω σA (τ ). The resulting σb (T2 ′ ), shown in Table III, is too small to account for the decay of the spin echo amplitude. (5) Fluctuation of microwave power and pulse durations. The application of two π/2-pulses and one π pulse results in wecho (2τπ ) = ?1. Any ?uctuations of the amplitude (??R /?R ) or pulse duration (?τ /τ ) result in variations of the amplitude of the spin echo signal, i. e. wecho (2τπ ) = ? cos ?φ according to: ?φ 2π

2

where ?J′ = ωL ? ωJ′ is the detuning of the dipole trap laser from the 6PJ′ -state and d = 4, 4 | ?+1 | 5, 5 , with the dipole operator ?+1 for ?m = +1 transitions. The transition amplitudes a(J’) are obtained by summing over all possible intermediate states | F ′ , m′ of the relevant 6PJ′ manifold [22]. For Rayleigh scattering processes, which do not change the hyper?ne state (F, M = F ′′ , M ′′ ), the amplitudes add up, a(3/2) = 2a(1/2) . However, for state changing Raman processes (F, M = F ′′ , M ′′ ), the two transition amplitudes are equal but have opposite sign, a(3/2) = ?a(1/2) . Then the two terms in Eq. (62) almost cancel in the case of far detuning, ?1/2 ≈ ?3/2 . As a result the spontaneous Raman scattering rate scales as 1/?4 whereas the Rayleigh scattering rate scales as 1/?2 . The suppression factor can be expressed using the ?ne structure splitting ?fs = ?3/2 ? ?1/2 as ΓRaman = β Γs with β = ?fs 3?1/2

2

.

(63)

=

??R ?R

2

+

?τ τ

2

(61)

With ?τ /τ < 10?3 (measured) and ??R /?R < 10?2 (speci?cations of the synthesizer) we obtain ?φ/2π < 10?2 , which is too small to be observed. Moreover, this e?ect neither depends on the dipole trap depth nor on the time delay between the microwave pulses. The timing of the microwave pulses would be a?ected by a clock inaccuracy of the D/A-board of the computer control system which triggers the microwave pulses. Its speci?ed accuracy ?τ /τ = 10?4 , results in a phase ?uctuation δ ′ τπ ?τ /τ < 0.01 for all parameters δ ′ and τπ used in our experiment. Thus, the ?uctuations of microwave power, pulse duration, and timing do not account for the observed reduction of the spin echo visibility. (6) Spin relaxation due to light scattering. The population decay time, T1 , is governed by the scattering of photons from the dipole trap laser, which couples the two hyper?ne ground states via a two-photon Raman transition. In our case, the hyper?ne changing spontaneous

For the case of cesium, we obtain a suppression factor of β = 0.011. The Rayleigh scattering rate for an atom trapped in a potential of U0 = 1.0 mK is Γs = 11 s?1 . Then, the corresponding spontaneous Raman scattering rate is ΓRaman = 0.12 s?1 and the population decay time 1 T 1 = Γ? Raman = 8.6 s. Since in most of our experiments, the trap depth is signi?cantly smaller, T1 will be even larger. As a consequence, we neglect the population decay due to spontaneous scattering. Note that the experiments of Refs. [22, 23] were only sensitive to changes of the hyper?ne F -state, since the atoms were in a mixture of mF -sublevels. However, the theoretical treatment above predicts similarly long relaxation times for any particular mF -sublevel, which is consistent with our observations.

C.

Conclusions

We have developed an analytical model which treats the various decay mechanisms of the hyper?ne coherence

13 of trapped cesium atoms independently. This is justi?ed by the very di?erent time scales of the decay mechanisms ? (T2 ? T2 ′ ? T1 ). Our model reproduces the observed shapes of Ramsey and spin echo signals, whose envelopes are the Fourier transform of the energy distribution of the atoms in the trap. The irreversible decoherence rates manifest themselves in the decay of the spin echo visibility and are caused by ?uctuations of the atomic resonance frequency in between the microwave pulses. In the above analysis we have investigated various dephasing mechanisms and characterized them by the corresponding amplitude of the detuning ?uctuations which are summarized in Table III. We ?nd that a major mechanism of irreversible dephasing is the pointing instability of the dipole trap laser beams resulting in ?uctuations of the trap depth and thus the di?erential light shift. Signi?cant decoherence is also caused in the shallow dipole trap by heating due to photon scattering. Heating due to technical origin, such as ?uctuations of the depth and the position of the trap, cannot be excluded as an additional source of decoherence. Compared to our experiment, signi?cantly longer co? herence times (T2 = 4 s) were observed by N. Davidson and S. Chu in blue detuned traps in which the atoms are trapped at the minimum of electric ?elds [7]. In ? Ref. [7], T2 = 15 ms obtained with sodium atoms in a Nd:YAG dipole trap (U0 = 0.4 mK) was reported, which is comparable to our observation. In other experiments, the inhomogeneous broadening has been reduced by the addition of a weak light ?eld, spatially overlapped with the trapping laser ?eld and whose frequency is tuned in between the two hyper?ne levels [25]. Of course, cooling the atoms to the lowest vibrational level by using e. g. Raman sideband cooling techniques [26, 27], would also reduce inhomogeneous broadening. The magnetic ?eld ?uctuations could possibly be largely suppressed by triggering the experiment to the 50 Hz of the power line.

Acknowledgments

We thank Nir Davidson for valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft and the EC.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

W. H. Zurek, Phys. Rev. D 26, 1862 (1982). W. H. Zurek, Physics Today 44, 36 (October 1991). M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996). S. Haroche, Physics Today 51, 36 (July 1998). C. Monroe, D. M. Meekhof, B. E. King and D. J. Wineland, Science 272, 1131 (1996). F. Schmidt-Kaler et al., J. Phys. B: At. Mol. Opt. Phys. 36, 623 (2003). N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich and S. Chu, Phys. Rev. Lett. 74, 1311 (1995). R. Ozeri, L. Khaykovich and N. Davidson, Phys. Rev. A 59, R1750 (1999). S. Kuhr et al., Science 293, 278 (2001). D. Schrader et al., Appl. Phys. B 73, 819 (2001). S. Kuhr et al., Phys. Rev. Lett. 91, 213002 (2003). W. Alt et al., Phys. Rev. A 67, 033403 (2003). H. Metcalf, P. van der Straten, Laser Cooling and Trapping, Springer, 1st ed. (1999). In Refs. [11] and [12] we erroneously used √ the energy distribution p(E ) = 2(kB T )?3/2 π ?1/2 E exp (?E/kB T ), valid for particles in free space. We now use the correct distribution of Eq. (13). This implies that Eqs. (1) and (3) of Ref. [11] have to be replaced by Eqs. (15) and (26)

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

of this paper. Please see also the errata to Refs. [11] and [12] (to be published). W. Alt, Optik 113, 142 (2002). R. Barlow, Statistics , Wiley, New York (1989). L. Allen, J. H. Eberly, Optical resonance and two-level atoms , Wiley, New York (1975). N. Ramsey, Molecular Beams , Oxford University Press, London (1956). E. Hahn, Phys. Rev. 80, 580 (1950). M. F. Andersen, A. Kaplan, N. Davidson, Phys. Rev. Lett 90, 023001 (2003). D. W. Allan, Proc. IEEE 54, 221 (1966). R. A. Cline, J. D. Miller, M. R. Matthews, D. J. Heinzen, Opt. Lett. 19, 207 (1994). D. Frese et al., Phys. Rev. Lett. 85, 3777 (2000). R. Loudon, The Quantum Theory of Light , Clarendon, Oxford (1983). A. Kaplan, M. F. Andersen, N. Davidson, Phys. Rev. A 66, 045401 (2002). S. E. Hamann et al., Phys. Rev. Lett. 80, 4149 (1998). H. Perrin, A. Kuhn, I. Bouchoule and C. Salomon, Europhys. Lett. 42, 395 (1998).

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