A study of atom localization in an optical lattice by analysis of the scattered light C. I. Westbrook, C. Jurczak1, G. Birkl2, B. Desruelle, W. D. Phillips3, A. Aspect Institut d’Optique, UA14 du CNRS, B.P.147 91403 Orsay CEDEX 2 We present an experimental study of a four beam optical lattice using the light 1 0 scatteredbytheatomsinthelattice. Weusebothintensitycorrelationsandobserva- 2 tions of the transient behavior of the scattering when the lattice is suddenly switched n a J on. We compare results for 3 different configurations of the optical lattice. We create 3 1 situations in which the Lamb-Dicke effect is negligible and show that, in contrast to ] what has been stated in some of the literature, the damping rate of the ’coherent’ h p atomic oscillations can be much smaller than the inelastic photon scattering rate. - m o PACS numbers: t a . s c si I. INTRODUCTION y h p In the past five years, optical lattices have attracted a great deal of experimental and [ 1 theoretical interest [1]. Beginning with studies of the quantized motion of atoms in the sub- v 6 micron sized potential wells [2, 3], many workers have succeeded in observing a large variety 7 7 of effects and are now beginning to investigate the transport properties, both classical and 2 . quantum in these systems [4–7]. 1 0 There are several methods of observation of optical lattices, and among them two groups, 2 1 : including ours, recently demonstrated that intensity correlation, a well established technique v Xi inotherfields[8], canbeusedtogaininformation[9,10]. Althoughthebulkofournewresults r has been related to the transport properties of atoms in these lattices (i.e. the motion from a well to well) [5], our experiment, like many others, also gives information about the motion of atoms inside the wells. In this paper we will detail our results on this motion and use our data to confirm recent ideas [11, 12] about the damping mechanisms which affect the motion of atoms inside the lattice potential wells. A novel feature in our experiment is that we can study the behavior of atoms in a tetra- hedral optical lattice for a large range of beam angles. This feature has already permitted 2 us to study transport by density waves in lattices as a function of lattice angle[13]. Here we will discuss a study of the intrawell dynamics, i.e. the oscillation of atoms inside a single well as a function of lattice angle. In addition to the intensity correlation measurements, we have also studied the same dynamics by observing the atomic fluorescence as a function of time after a homogeneous atomic sample is suddenly subjected to an optical lattice. Although not as powerful a method as Bragg scattering for studying the localization of atoms in individual wells, we demonstrate in the appendix that a fairly simple experiment is indeed sensitive to the degree of localization in a single well. II. THEORETICAL REMARKS In our experiments we used a 4-beam optical lattice configuration introduced in Ref. [14]. The configuration of laser beams and polarizations is shown in Fig. 1. We denote by θ the half angle between the lattice beams in the x − z and y − z planes. The x − z and y − z angles were always the same. The resulting electric field and optical potential are discussed in detail in Ref. [15]. Here we will simply state the results of this work which are important for our study. For an atom with an F =3 ground state and an F =4 excited g e state, Montecarlo wavefunction simulations[16][17] have shown that atoms in the lattice are rapidly pumped into the extreme magnetic sublevels m = ±3. Therefore we will neglect the F potentials not corresponding to these two states. The three dimensional light shift potential for atoms in the m =±3 state is given by: F U = −U0 (cid:18)29 (cid:16)cos2(k x)+cos2(k y)(cid:17)∓ 27 (2cos(k x)cos(k y)cos(2k z))(cid:19) (1) ± x y x y z 4 28 28 Where k = k = ksinθ, k = kcosθ , k = 2π/λ is the wavevector of the light and U is the x y z 0 light shift at the bottom of the potential well. The numerical factors come from the values of the Clebsch Gordan coefficients. In the large detuning limit this light shift is related to the saturation parameter s = Ω2R/(cid:16)∆2 + Γ2(cid:17) of a single laser beam by U = 4h¯∆s , with Ω 0 2 4 0 0 R being the Rabi frequency, ∆ the detuning and Γ the natural linewidth. From the curvature of the bottom of these wells one can calculate an oscillation frequency along each axis for an atom near the bottom of each well: (cid:115) 2U 0 Ω = ω sinθ (2) x,y R E R 3 (cid:115) 28 U 0 Ω = 2ω cosθ (3) z R 27E R Here E = ¯h2k2 denotes the recoil energy, and ω = E /h¯. R 2m R R These oscillation frequencies have been observed using many different methods. Our intensitycorrelationtechniqueamountstoobservingthebeatnotebetweenthelightradiated by different atoms. Since the atoms are oscillating at well defined frequencies, the power spectrumoftheemittedelectricfieldcontainsa’carrier’atthelaserfrequencyandsidebands separated by multiples of the oscillation frequency. Thus the ’self-beating’, or intensity correlationspectrumcontainsbeatnotesatthesefrequencies. Insteadofself-beatingonecan also use a local oscillator (heterodyning) to observe the sidebands [3, 18, 19]. Both of these methods can also be considered as spontaneous Raman spectroscopy of the vibrational levels in the potential wells. In pump-probe spectroscopy, on the other hand, [2, 20] one observes FIG. 1: Configuration of laser beams and polarizations used to create the optical lattice used in this experiment. The heavy arrows show the beam propagation directions and the fine arrows the polarizations. 4 stimulatedRamantransitionbetweentheenergylevelsinthewells. Four-wavemixingsignals also contain motional sidebands [21]. Recently various groups have observed oscillations in a transient way, by non-adiabatically changing the potential wells and observing the motion either through the redistribution of photons in the lattice beams [22, 23] or using Bragg scattering [17, 24, 25]. In the first discussions of the motion of atoms inside the potential wells of an optical lattice,attentionwasdrawntothefactthatthemotionalsidebands,whichwerethesignature of a well defined oscillation of the atoms in the wells, appeared to be narrower than the simplest expectation [2, 3]. Naively, one expects the linewidth of such a system to be approximately equal to the total photon scattering rate (at the bottom of a well this rate is given approximately by U0∆). References [2, 3], however, pointed out that in fact only the ¯hΓ inelasticscatteringrate(therateofscatteringeventsinvolvingachangeofthequantumstate of the atom) should determine that linewidth. Because the experiments were in the Lamb- Dicke regime, i.e. the amplitude of the oscillation was much smaller than the wavelength of the emitted radiation, inelastic scattering was strongly suppressed in these experiments [26, 27], and this accounted for the observed width. This argument is roughly correct if there is no significant excitation of the harmonic oscillator, i.e. most of the atoms are in the ground state, as was indeed approximately true in the experiments of Refs. [2, 3]. Later, however, Ref. [11] pointed out that when the amplitude of the atomic oscillation is large enough that one must include the effects of many levels in the well, there is a transfer of coherence mechanism which suppresses the linewidth even further. Essentially, this mechanism involves the fact that although coherences between adjacent levels in a well do decay at a rate determined by the inelastic scattering rate, these coherences are also ’fed’ by other, higher lying coherences, provided that the higher lying coherences oscillate at approximately the same frequency as the lower lying ones, i.e. that the well is nearly harmonic. Indeed, in the case of a perfectly harmonic well, Ref. [28] demonstrates that the coherence transfer mechanism works in such a way that the linewidth of the oscillator is determined only by the rate at which energy is extracted from the oscillator through its coupling to a thermal reservoir, and not by the decay rates of the individual coherences. This extraction rate is exactly the same as the damping, or cooling rate of the classical oscillator. The case of the perfectly harmonic oscillator was adapted to the case of laser cooling in 5 Ref. [12] to explain the damping rates observed in an experiment reported in Ref. [22]. The authors of Ref. [22] used two beams crossing each other at a small angle to produce a one dimensional optical lattice with a very large period (about 5 times the optical wavelength). Thus there was no Lamb-Dicke effect. The experiment, however, showed clear oscillations which damped out much more slowly than the inelastic scattering rate. Because the laser beam polarizations were parallel to each other the atoms behaved nearly like two level systems in the lattice and a Doppler cooling model was enough to show that this result was not surprising. In the case of Doppler cooling one can show that the mean occupation number of the harmonic oscillator (cid:104)n(cid:105) is equal to the inelastic scattering rate divided by the energy damping rate due to Doppler cooling. Thus (cid:104)n(cid:105) (cid:29) 1 implies a damping rate much smaller than the inelastic scattering rate, and sidebands narrow compared to the inelastic scattering rate. This result was also stated in Ref. [29]. For lattices in which the internal structure of the atom plays a significant role in the cool- ing process (e.g. Sisyphus cooling), the Doppler model is clearly not adequate. But as long as the motion of the atoms at the bottom of the wells in an optical lattice is well described by a damped harmonic oscillator, this damping rate determines the fundamental limit to the width of the sidebands. (Note that Ref. [17] observed a damping rate which may be this fundamental rate. It is not yet clear, however, that the motion can be described simply as a harmonic oscillator.) In what follows we will show experiments in an optical lattice with Sisyphus cooling which confirms these ideas. We show that the inelastic scattering rate, i.e. the Lamb-Dicke parameter, has little to do with the width of the sidebands observed. III. INTENSITY CORRELATION SPECTROSCOPY As we have already mentioned, in correlation spectroscopy, one observes the beating between the light emitted by different atoms in the source. Here we will briefly review some of the ideas underlying intensity correlation spectroscopy; the reader is referred to Refs. [9, 30–32] for more detailed information. We use a spectrum analyser to record the power spectrum P (ω) of a detected photocurrent i(t). The Wiener-Khinchine theorem states that i P (ω) is proportional to the Fourier transform of the correlation function of the photocurrent i (cid:104)i(t)i(t+τ)(cid:105). If we consider the light as a classical field, the photocurrent is proportional to the incident light intensity I and our observation amounts to a measurement of the 6 FIG. 2: (a) Intensity correlation spectrum of atoms in an optical lattice, showing a peak at zero frequency corresponding to elastic scattering and a second peak (marked Ω ) at 100 kHz due to x spontaneous Raman scattering by atoms changing their level in the potential well. The angle of the lattice was θ = 30◦, the observation direction was along x, the detetected light polarization was along y, and the detuning was ∆ = −5Γ. (b) The autocorrelation function of the light as recorded by a digital correlator under similar conditions. normalized correlation function g (τ) of the scattered light intensity: 2 (cid:104)I(t)I(t+τ)(cid:105) g (τ) = 2 (cid:104)I(t)2(cid:105) where (cid:104)(cid:105) denotes a statistical average. Examples of the spectral and time domain data are shown in Fig. 2. The spectrum (a) and the correlation function (b) were taken using the same optical lattice, in one case using a spectrum analyser and in the other using a digital correlator. In 2(a) we have normalized the spectrum to that of a shot-noise limited light source of the same average intensity, using a procedure described in Ref. [9]. Thus a power density greater than 1.0 represents an ’excess noise power’ which is due to the atoms. In 2(b) we plot g (τ)−1. For large values of τ, this function tends to zero. 2 If one is observing a large number of independent scatterers one can show that the 7 intensity correlation function is related to the correlation function of the electric field E by g (τ) = 1+|g (τ)|2 (4) 2 1 (cid:104)E−(t)E+(t+τ)(cid:105) g (τ) = 1 (cid:104)E−(t)E+(t)(cid:105) where E± refers to the positive and negative frequency components of the electric field. The Fourier transform of g (τ) is proportional to the optical power spectrum, and this is 1 the quantity that is measured in heterodyne experiments. From Eq. 4 it is easy to see why the power spectrum of the photocurrent is proportional to the autoconvolution of the optical power spectrum. The fact that we are dealing with an auto-convolution, however, presents some difficulties in the quantitative interpretation of the spectra. For example, the zero frequency peak in our spectrum consists of the autoconvolution of the carrier plus the autoconvolution of the sidebands. Thus its lineshape is complex. In what follows we shall use a Montecarlo simulation to extract quantitative information from our spectra. IV. DESCRIPTION OF THE EXPERIMENT We begin by collecting atoms in a vapor cell Magneto-Optical Trap (MOT) with a Rb partialpressureoforder10−8hPa. ThelasersaretunedslightlytotheredoftheF =3→F =4 g e component of the D resonance line of 85Rb (λ = 780 nm, Γ/2π = 5.9 MHz). We can load 2 of order 3×107 atoms into the trap with a time constant of a few seconds. After loading the MOT, we switch off both the magnetic field and the trapping beams and turn on the 4 lattice beams. These beams come from a separate diode laser which is injected by the same master laser as the trap laser. The density of atoms in the lattice is approximately 2×109 cm−3. We allow the atoms to equilibrate in the lattice for 5 ms before beginning the data acquisition period. All the lattice beams enter the vacuum chamber by one of two large windows on the sides. Our geometry permits angles θ as large as 40◦. This angle could be changed fairly easily and in our experiments we compared values of 20◦, 30◦ and 40◦. These angles were measured with a precision of approximately 1◦. We measured the temperature of the atoms in our lattice by releasing the atoms and monitoring their time of flight to a probe beam placed 18 mm below the lattice (the y-axis is vertical). For a lattice angles of 30◦ we recorded the mean energy as a function of the depth of the potential. We determined the potential depth by measuring the oscillation frequency 8 of the atoms and using Eq. 2 and including a 10% anharmonicity correction (see below). The results are shown in Fig. 3. As expected, the temperature depends linearly on the light shift. A linear fit to the data gives: k T B = 0.17U +60E (5) 0 R 2 This result corresponds closely to the results of Ref. [18, 33], which studied a 3D lattice at θ = 45◦ for Cs; both the slope and the intercept seem to be somewhat higher in our case. This discrepancy my be due to differences in the angle of the lattice. Although Ref. [33] reported that the temperatures were isotropic to within 20%, our 2D simulations of the motion of atoms with a J = 1/2 → J = 3/2 structure [30, 34] show, for lattice angles of g e 20◦ or 30◦, that the temperature is anisotropic with the x direction a factor of 1.5 hotter than the z direction. The fact that our measurements of the temperature along x are above those reported in Ref. [18, 33] seems to bear this out [35]. Inthecorrelationexperiments, thelightscatteredbytheatomswasdetectedbyavalanche photodiodes with integrated amplifiers and discriminators. These signals were sent either to an FFT spectrum analyser, or a digital correlator. The detection solid angle was approxi- mately 10−6 sr. Typical count rates were of order 105 to 106 s−1. The detectors were capable FIG. 3: Temperature of atoms in a lattice with θ = 30◦ measured by time of flight as a function of the depth of the potential wells. 9 FIG. 4: Intensity correlation spectrum of a lattice with θ = 40◦. (a) Observation along the z-axis, (b) Observation direction in the x−z plane at 10 degrees with respect to the z-axis. The detected polarization was linear along y, and ∆ = −5Γ of a maximum count rate of 106 s−1 without significant saturation. The detector dark count rate was approximately 250 s−1 and the background count rate, mostly due to scattering of the laser beams by the vapor was of order 104 s−1. When acquiring correlation data, we typically loaded the MOT for 200 ms and used a data acquisition period of 100 ms during which atom losses were negligible. The measurement cycle was repeated and averaged 102 to 105 times. V. RESULTS A. Measurement of the oscillation frequencies Figure4showstwointensitycorrelationspectratakenfromanopticallatticewithθ = 40◦. In (a) the detection direction was the z axis of Fig.1. In (b) we used a detection angle of 10◦ from the z direction. We see that when one observes the light scattered along one of the 10 FIG. 5: Measurements of the observed oscillation frequency as a function of the square root of the potential well depth. We show the results for two lattice angles. The dependence is linear as expected. The straight lines are fits to the data (see text). axes of symmetry of the lattice one sees only an oscillation in the direction of that axis (see, however, Ref. [13] for an exception). When another observation direction is chosen one sees both of the eigenfrequencies of the lattice. Note that the positions of the resonances do not change when the observation angle is varied. When the lattice angle is changed the two eigenfrequencies change according to Eq. 2, smaller lattice angles leading to more widely spaced frequencies. Fig. 5 shows our measurements of the dependence of oscillation frequency predicted by Eq.2. When we fit the (cid:113) datatoastraightlineΩ = Cω 2E /U sinθ, wefindC = 0.8andC = 0.9forθ = 40◦ and x R R 0 θ = 20◦ respectively. Our precision is limited by our 10% uncertainty in the laser intensity. Thus our measurements are marginally consistent with Eq. 2 in which C = 1. However, we expect that the anharmonicity of the potential should reduce the expected value of C by of order 20%[18]. We have also made a systematic study of the relative positions of the two resonances when they are measured simultaneously by observing the spectrum at a small angle relative