Standing wave diffraction with a beam of slow atoms

journal of modern optics, 1997, vol. 44, no. 10, 1863± 1881

Stan d in g w av e d i€ rac tion w ith a be am of slow atom s È RR, K. DIECKMANN³ , M. ELBS § , S. KUNZE² , S. DU U. ERNST , A. HARDELL ¶ , S. WOLFi , and G. REMPE FakultaÈt fuÈr Physik, UniversitaÈt Konstanz, D-78457 Konstanz, Germany ( Received 6 June 1997 ) We report on atom optical experiments employing a magnetooptical trap as a source of a pulsed beam of slow atoms. After turning o€ the trap, experiments are performed on the cloud of atoms in free fall over a distance of 45 cm. We give a detailed description of the apparatus and discuss experimental results on the di€ raction of atoms from a standing light wave in the regimes of short, intermediate and long interaction times. Abstrac t.

PACS: 03.75Be, 32.80Pj, 42.50Vk.

1.

In trod u c tion T he coherent manipulation of atomic de Broglie waves has undergone rapid experimental progress during the last 15 years, and atom optics is now a well established research ® eld [1, 2]. A major area of activity in this ® eld is the investigation of interference and di€ raction phenomena of atomic de Broglie waves. Just as in photon optics, an atomic beam source with good spatial and temporal coherence properties is required. However, most atomic beam experiments so far have employed thermal sources, and considerable e€ orts had to be made in order to increase the coherence length. T ypically this was achieved by using techniques of spatial and spectral ® ltering analogous to light optics. T he use of atoms instead of photons however, opens up new possibilities, because, in contrast to photons, the wave properties of atoms can easily be manipulated by means of dissipative light forces [3]. In addition, the internal states of the atoms can easily be addressed. T his allows one to realize atomic beams with drastically improved coherence properties. T his paper describes our experimental approach in which we use a vapour cell magneto-optical trap (MOT ) [4] in order to prepare a sample of cold atoms. After trapping, the magnetic ® eld is turned o€ and the atoms are further cooled in an optical molasses. Next, the light beams are switched o€ and the atoms travel through the apparatus in free fall. Experiments are performed on the sample of atoms over a distance of 45 cm. T hese slow atoms can be used to study phenomena in the regime of long interaction times with externally applied ® elds. In our Present address: KPMG Unternehmensberatung , Berlin, Germany. Present address: FOM-Institute of Atomic and Molecular Physics (AMOLF), Amsterdam, T he Netherlands. § Present address: Institut fuÈr Quantenoptik, UniversitaÈt Hannover, Germany. ¶ Present address: Endres & Hauser, Maulburg, Germany. i Present address: FakultaÈt fuÈr Physik, UniversitaÈt Freiburg, Germany. ² ³

0950 ± 0340/97 $12´00

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1997 T aylor & Francis Ltd.

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experiments we investigate di€ raction of the atoms from a standing light wave. In addition, microwave transitions between long lived atomic ground states can be induced with the help of a microwave resonator surrounding the interaction region. Dropping atoms out of a MOT provides an excellent atomic beam source for a variety of experiments, for example see the work in [5]. T he paper is organized as follows: after a detailed description of the experimental setup in section 2, section 3 describes the `snapshot method’ we use to measure the temperature of the atomic cloud. Section 4 focuses on the coherence properties of the atomic beam, and section 5 describes the optical pumping mechanism used to prepare the atoms in a well-de® ned Zeeman sublevel. Section 6 presents results on the di€ raction of atoms from a near resonant standing light wave, where the interaction time between the atoms and the light ® eld is varied over a wide range. Finally, the last section gives a summary.

2.

Expe r im e n tal se tu p A scheme of the experimental setup is shown in ® gure 1. T he apparatus consists of two stainless steel vacuum chambers on top of each other, connected by

Figure 1. Schematic view of the experiment. A MOT serves as a source for cold rubidium atoms. After turning o€ the trap, the atoms fall through the interaction region and are ® nally detected by spatially resolved ¯ uoresence detection.

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a thin tube which contains graphite materials with a circular hole in the centre. T his hole is 5 cm below the MOT , has a diameter of 1. 5 mm, is 5 mm long and allows one to establish a pressure gradient between the two parts of the vacuum system. In the upper chamber, a rubidium background vapour of about 3 ´ 10- 8 mbar is produced by heating a small piece of rubidium inside the chamber. T his makes it possible to operate a vapour cell MOT , while in the lower chamber a pressure of about 2 ´ 10- 10 mbar reduces the collision probability between the freely falling atoms and thermal background atoms. T he rubidium partial pressure can be measured via ¯ uorescence in both vacuum chambers. In the lower chamber, the rubidium partial pressure is below the detection limit which we estimate to be less than 10- 15 mbar. T he whole apparatus is about 1 m high. It is evacuated with a turbo-molecular pump connected to the lower chamber and a non-evaporable getter material inside the upper chamber. T hree pairs of square coils, forming a cube of size 1 m, are used to compensate the earth’ s magnetic ® eld in the central part of the vacuum system. T his is required for molasses cooling. In addition, these coils are used to apply a magnetic bias ® eld during and after optically pumping the laser cooled atoms into a well-de® ned Zeeman sublevel.

2.1. Magneto-optical trap T he vapour-cell MOT employs six circularly polarized laser beams in s + ± s con® guration. Laser light, near resonant with the Fg = 3 ® Fe = 4 transition of the D2 -line (g = 5s2 S 1 /2 ® e = 5p2 P3 /2 ) of 85 Rb, is sent into the upper chamber through six commercial viewports. T o compensate non-resonant optical pumping into the state Fg = 2, light of an additional repumping laser resonant with the transition Fg = 2 ® Fe = 3 is superimposed with the trapping laser light. All six laser beams, which are in pairs perpendicular to each other, form a 54ë angle with respect to the vertical axis. T his con® guration displays high symmetry and allows free access from the top and the bottom. T he magnetic quadrupole ® eld, which is required for the MOT , is produced by two coils located outside the vacuum chamber. T his ® eld has a gradient of 1 mT cm- 1 at the centre. With a laser beam waist of x = 3. 8 mm (1 /e2 ± radius of intensity), typically 1 ´ 107 atoms can be captured with a time constant of 200 ms (1 /e± time). We typically load the trap for 500 ms for each cycle of the experiment. In order to align the laser beams and measure the number of trapped atoms, the cloud size and the temperature, the atoms are observed with a standard CCD camera which is mounted on top of the apparatus. A small part of the atoms’ s ¯ uorescence light is collected by a lens (10- 3 ´ 4p sr solid angle) and imaged onto the CCD chip using a commercial camera objective. After trapping, the magentic trap ® eld is switched o€ in less than 1 ms and the atoms are further cooled in optical molasses (see section 3) during a time interval of 10 ms. T he atoms are then released by switching o€ all laser light, and move through the vertically aligned setup in free fall, thus forming a pulsed beam of slow and cold atoms. Although all atoms are in state Fg = 3, they are distributed over all Zeeman sublevels mF = - 3,- 2, . . . , + 3. During the free fall towards the interaction zone additional laser beams are switched on for a few milliseconds in order to optically pump the atoms into a well-de® ned Zeeman sublevel (see section 5).

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2.2. Interaction region After optical pumping, the atoms enter the lower vacuum chamber through the graphite tube. T he atomic cloud is then collimated with a 200 m m wide horizontal slit in order to generate transverse coherence of the atomic de Broglie waves. T he atoms then pass the interaction zone, which is located 20 cm below the trap. T he vertical velocity of the atoms is v x = 2 m s- 1 , corresponding to a de Broglie wavelength of 2. 3 nm. Our experiments employ a near resonant standing light wave. T his is realized by retrore¯ ecting a Gaussian laser beam, propagating along the horizontal z-direction, from a ¯ at mirror which is mounted inside the vacuum chamber. T he laser beam has a vertical waist x x and the mirror is located in its focal plane. T he atoms pass the light ® eld at a distance of about 1 mm in front of the mirror. T he interaction time can be controlled by either using the transit time of the atoms through the waist or by pulsing the light ® eld. In the second case, a large vertical beam waist is used in order to illuminate all atoms with light of the same intensity. Due to the absorption and subsequent stimulated emission of photons during the non-resonant atom-® eld interaction, the transverse momentum pz of the atoms along the standing wave is changed by multiples of 2prec , where prec = h /¸ is the recoil momentum onto the atom when absorbing or emitting a photon with wavelength ¸ = 780 nm. T his process can also be described as di€ raction of the atomic de Broglie wave from a phase grating, which is induced by the position dependent ac-Stark shift during the atom-® eld interaction. T o monitor the corresponding momentum change, the transverse momentum distribution of the atoms, q p z ( pz ) , is observed as a Fraunhofer di€ raction pattern in the far ® eld. T his is performed with spatially resolved ¯ uorescence detection 25 cm below the interaction region. In the Fraunhofer approximation, the spatial distribution in the detection region, q z ( z ) , is related to the transverse momentum distribution after the interaction, q pz ( pz ) , by q z ( z ) = q pz ( pz =

zm /t) ,

( 1)

where m is the mass of the atom, t is the time of ¯ ight between the interaction zone and the detection region and z is the transverse position in the detection region. Equation (1) expresses the simple fact, that an atom with horizontal momentum pz covers the horizontal distance z = pz t /m in free ¯ ight during time t. For the parameters of our experiment, neighbouring di€ raction orders with momentum di€ erence 2prec are spatially separated by 1. 1 mm. T he vertical acceleration of the atomic beam due to gravity does not a€ ect the validity of the Fraunhofer approximation, as long as one refers to the times involved in the problem and not to the vertical distances. 2.3. Fluorescence detection For detection, the atoms are irradiated by laser light resonant with the Fg = 3 ® Fe = 4 transition. A part of the ¯ uorescence light is collected by a large numerical aperture lens (10- 2 ´ 4p sr solid angle) and focused onto a photomultiplier operated in single photon counting mode with a quantum e ciency of about 10%. Stray light from the detection laser beam, scattered for example from the vacuum windows, is reduced by focusing the ¯ uorescence light through an additional small diameter aperture between the lens and the photomultiplier. With all light beams on, but no atoms present, the photomultiplier count rate is about 150 s- 1 . As mentioned above, this count rate is not caused by rubidium background

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vapour. T he laser beam used to excite the atoms has an asymmetric Gaussian pro® le with horizontal and vertical beam waists of x z = 140 m m and x x = 450 m m, respectively. T he horizontal waist determines the momentum resolution of the detection system, which is 0. 3 prec (full-width at half maximum (FWHM) of the laser intensity pro® le). We estimate that the number of ¯ uorescence photons emitted by one atom is limited to about 1300 due to the following e€ ect: the absorption of photons accelerates the atom thereby Doppler shifting the laser frequency out of the atomic resonance [6]. We found experimentally, that the count rate increases by about 30% when the frequency of the detection laser is slightly blue detuned. We have estimated that on average each atom passing the laser beam induces 1. 3 photomultiplier count events. T he photomultiplier signal is recorded with a counter which is gated on for 10 ms or less. T he detection beam is re¯ ected into the vacuum chamber by using a mirror mounted on a temperature stabilized galvo drive. Scanning this laser beam along the horizontal z-direction allows one to monitor the spatial distribution of the atoms. T o measure a momentum distribution, the count rate is recorded at typically 50 di€ erent positions of the detection beam. Each data point represents an average over typically 20 cycles where one cycle consists of trapping and cooling the atoms and the free fall through the apparatus. With 500 ms for loading the trap and 300 ms of free fall, each cycle lasts about 800 ms. As the galvo is moved only between subsequent cycles and not within one cycle, it takes about 15 min to record one momentum distribution. 2.4. Microwave resonator Rubidium atoms have two long-lived hyper® ne structure ground states, Fg = 2 and Fg = 3. In order to induce transitions between these states e ciently, the apparatus houses a microwave resonator, which allows one to perform Ramsey spectroscopy experiments [7]. Recently, this spectroscopy scheme was combined with di€ raction of the atoms from a standing light wave, thus making it possible to demonstrate a `virtual grating’ for massive particles, as described in [8]. Our rectangular microwave resonator with walls made of copper is 12 cm long, 8 cm wide, and 4 cm high. It is placed inside the vacuum system and encloses the interaction region with the standing light wave. T he mirror for the standing light wave is mounted on a ceramic plate inside the resonator. T he atoms as well as the laser beam enter and leave the resonator through holes drilled into the copper walls. T he T E102 -mode is tuned into resonance with the magnetic dipole transition Fg = 2 ® Fg = 3 at 3. 035 GHz. T he oscillating magnetic ® eld of this mode is linearly polarized in the horizontal y-direction. T his allows one to induce transitions with D mF = 0 with respect to this quantization axis. T he small magnetic bias ® eld (see section 5) Zeeman shifts the magnetic sublevels so that transitions can occur only for one particular value of mF . We have chosen to use the transition Fg = 2, mF = 0 ® Fg = 3, mF = 0 for two reasons. First, neither level shows any linear Zeeman e€ ect, so that the atomic resonance frequency is less sensitive to ¯ uctuating magnetic stray ® elds. Second, the state Fg = 3, mF = 0 can be populated by optical pumping (see section 5) . Our resonator has a loaded quality factor of Q L = 530. T hus an input power of - 13 dB m coupled into the resonator induces Rabi oscillations with a frequency of 3. 1 kHz (see ® gure 5 discussed in section 5) . T he duration of the interaction with the microwave is controlled by switching the input power on and o€ .

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3.

Te m pe r atu re of th e atom ic c lou d T his section describes in detailÐ the method [9] we use to measure temperature T of the atomic cloud after molasses cooling. T he temperature is an important parameter as it determines the longitudinal coherence as well as the ¯ ux of the useful atomic beam (see section 4). It can be determined by observing the free expansion of the cloud after turning o€ the optical molasses. A similar technique was used in the ® rst experiments on Bose± Einstein condensation [10]. In our case, the molasses laser beams are switched o€ and the atoms propagate freely. T his leads to a spherically symmetric expansion of the cloud due to the atom’s thermal velocity distribution. After an adjustable time delay t, the laser light is switched on again. At the same time the CCD-camera is triggered to take a snapshot picture of the expanded cloud. T he width of the cloud is found by ® tting a Gaussian to a one-dimensional slice of the picture, which re¯ ects the density distribution of the atomic cloud along the z-direction, q z ( z , t) . We assume that q z ( z , 0) is a Gaussian with standard deviation s z ( 0) and that the velocity distribution q v z ( v z ) is given by a Boltzmann distribution ( a Gaussian 1 with standard deviation s v z = ( kB T /m) /2 ) which is not correlated with q z ( z , 0) . At time t after release, q z ( z , t) is given by the convolution integral q z( z

, t) =

ò

q z(z

-

v z t , t = 0) q

vz ( v z )

dv z .

( 2)

T his yields a Gaussian with standard deviation s

z ( t)

=

( s

z ( 0)

2

+

kB T 2 t m

)

1 /2

.

( 3)

Figure 2 shows typical experimental data, recorded with this snapshot method. 1 T he plot displays the FWHM of the cloud D z ( t) = ( 8 ln 2) /2 s z ( t) as a function of the free propagation time t. During molasses cooling, the laser intensity was I < Isat and the laser frequency was detuned by ¢ = - 5C from the atomic resonance, where Isat = 1. 65 mW cm- 2 is the saturation intensity and C = 2p ´ 6 MHz is the natural line width. After 10 ms molasses cooling with these parameters, we typically achieve temperatures on the order of 10 m K. T he cloud size after molasses cooling is typically D z ( 0) = 0. 7 mm (FWHM). T he snapshot pictures are taken with the laser detuned by ¢ = - 2C . T his small detuning gives rise to heating, which increases the cloud size considerably after about 10 ms, much longer than the exposure time of 2 ms. Next, the trap is reloaded under identical conditions and the experiment is repeated in order to measure the cloud size for another free propagation time. Every data point in ® gure 2 is averaged over 6 measurements, with error bars re¯ ecting the statistical uncertainties. T he solid curve shows the theoretical prediction from equation (3) , where s z ( 0) and the temperature T are ® t parameters. T he vanishing slope for short expansion times re¯ ects the fact, that immediately after release most of the atoms in the cloud do not move radially away from the centre, but propagate in random directions, thus staying within the cloud’ s initial volume. As a consequence, the cloud expands slowly at the beginning, and the linear expansion due to the thermal motion of the atoms can only be observed after about 10 ms. It is apparent from ® gure 2 that the experimental data are in reasonably good agreement

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Figure 2. Determination of the temperature of the atomic cloud after molasses cooling. T he data points show the measured cloud size (FWHM) as a function of the free propagation time after turning o€ the optical molasses. T he solid line is the theoretical prediction from equation (3), where the temperature T and the initial width of the cloud s z ( 0) are ® t parameters. T he best-® t values are T = 10 m K and 1 /2 D z ( 0) = ( 8 ln 2) s z ( 0) = 0. 6 mm (FWHM).

with the theoretical prediction. T hus the technique allows one to perform reliable temperature measurements with a statistical error of about 0. 5 m K. Besides this snapshot method another technique is widely used to measure that temperature, namely the time-of-¯ ight technique [11], where at a ® xed distance below the MOT the distribution of the arrival times of the atoms is monitored. This technique di€ ers from the snapshot method which allows one to perform a quasicontinuous observation of the cloud’s expansion by scanning an adjustable time delay. In addition, the time-of-¯ ight technique allows one only to measure the temperature along the vertical direction, while the snapshot method can be used in any direction. To compare the two schemes expeirmentally, we have measured the temperature using the time-of-¯ ight technique in our detection zone 45 cm below the MOT. Figure 3 shows a time-of-¯ ight distribution where the count rate in a time window of 1 ms is plotted versus the time of ¯ ight after turning o€ the molasses.

Figure 3. T ime-of-¯ ight distribution of the atoms measured in the detection region. T he data show the count rate averaged over a gate time of 1 ms for di€ erent times t after release from the molasses. T he mean value of the distribution t = 297 ms corresponds to the distance of about 45 cm fallen. The temporal width s t = 2. 8 ms (standard deviation ) corresponds to a temperature of T = 8. 2 m K.

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T he mean arrival time t = 297 ms corresponds to the distance fallen, while the temporal width s t = 2. 8 ms (standard deviation) determines the size of the atomic cloud in the detection region, s x( t) = v xs t = 8. 4 mm, where v x = 3 m s- 1 is the vertical velocity 45 cm below the MOT . Inserting s x( t) into equation (3) yields a temperature of T = 8. 2 m K, where s x( 0) has been neglected. T his result is in reasonable agreement with the one obtained with the snapshot technique. In particular, the temperature along the vertical and the horizontal direction is the same, as can be expected for a purely thermal atomic motion.

4. Coh e re n c e prope rtie s of th e atom ic be am 4.1. L ongitudinal coherence We now turn to a discussion of the coherence properties of the pulsed beam of falling atoms. T he longitudinal coherence length [1]

l coh x =

ms

( 4)

vx

is determined by the standard deviation of the longitudinal velocity distribution s v x . In the following, we distinguish between two di€ erent cases which are both realized in the experiments described below. In case one, all atoms interact with pulsed laser ® elds at the same time, while in case two, a continuous-wave (cw) light ® eld at a ® xed distance below the MOT is used, so that the interaction starts at di€ erent times depending on the initial vertical velocity of the atom. First we discuss case one, where the interaction with the laser occurs at a given time. In this case, all atoms are accelerated during the same time interval. It follows that the velocity change due to the gravitational acceleration is the same for all atoms. T herefore, the velocity distribution remains unchanged with a constant value for the width 1 /2 ( 5) s v x = ( kB T /m) . For the parameters of our experiment, T = 10 m K, it follows that l coh x = 23 nm is independent of time. With a velocity of v x = 2 m s- 1 in the interaction region, we have v x /D v x = 27 (FWHM ). A completely di€ erent situation arises in case two, where the measurement is performed with a cw laser beam at a ® xed distance below the MOT . It follows that atoms with di€ erent initial velocity along the vertical direction reach the interaction zone with the laser at di€ erent times. T he velocity change due to the gravitational acceleration therefore depends on the atom’ s initial velocity. For example, atoms initially moving upward need a longer time to reach the laser than atoms moving downward at the beginning, therefore experiencing a larger velocity change. As a consequence, the velocity distribution is drastically compressed with respect to its initial value. T he change in s v x can be calculated by taking into account that the width of the energy distribution s E remains unchanged. For atoms with a mean velocity v x, the energy width is given by s

E

=

¶ E s ¶ vx

vx

= mv xs

vx .

( 6)

T he energy width of the initial distribution consists of two components: the spread in kinetic energy kB T /2 and the spread in potential energy mgs x( 0) due to the ® nite

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initial size of the cloud. In our experiment the spread in kinetic energy is typically a factor of 6 smaller than the spread in potential energy. Hence the spread in kinetic energy can be neglected [12] and from equation (6) one calculates s

vx

=

gs x( 0) . vx

( 7)

1 With ( 8 ln 2) /2 s x( 0) = 0. 7 mm and v x = 2 m s- 1 , one ® nds l coh x = 510 nm and v x /D v x = 600 (FWHM).

4.2. Transverse coherence T he transverse coherence properties are determined by the collimation slit and the size of the source, which in our case is the cloud size after molasses cooling. We ® rst consider the case of a point-like source and return to the discussion of the size of the source later. Our arguments are based on purely geometrical considerations, because di€ raction of the atomic wave from the slit can be neglected [13]. With this approximation, the width D pz of the transverse momentum distribution of atoms passing the slit is given by: D pz = mD z /tslit ,

( 8)

where tslit denotes the ¯ ight time from the source to the slit and D z is the slit width. As was discussed above, di€ raction of atoms from a structure with spatial periodicity ¸ /2 creates a di€ raction pattern with neighbouring orders separated by 2prec . Obviously these orders can be resolved, if D pz

< 2prec .

( 9)

In order to ful® ll this condition, D pz is narrowed with a collimation slit just above the interaction zone, 18 cm below the trap, corresponding to tslit = 190 ms. It follows from equation (8) , that the slit must be smaller than 2. 3 mm. For reasons discussed below, we have chosen a 200 m m wide slit, which gives D pz = 0. 2prec for those atoms which have passed the slit, which is clearly small enough. T he spatial extension of the collimated atomic beam in the detection region is D z det = D pz ( tdet + tslit ) /m, where tdet is the ¯ ight time from the slit to the detection region. A change of the transverse momentum of an atom by an even multiple of prec , can be observed as a change of the position of the atom in the detection region equal to even multiples of z rec = prec tdet /m = 0. 55 mm. T his yields D z det

z rec

=

D pz

prec

´

tdet + tslit . tdet

( 10)

If the detection were performed in the far-® eld ( tdet @ tslit ) , this ratio would simply be D pz /prec . However in our experiment we have ( tdet + tslit ) /tdet = 2. 7, so that the ratio D z det /z rec increases signi® cantly from 0. 2 to 0. 54. We have compensated for this increase by using a smaller slit, as was mentioned above. We now turn to the case of a spatially extended source. Giving a detailed description of the physical processes, that determine the exact properties of our source, would be far beyond the scope of this paper. T herefore we idealize the situation by regarding the source to be made up of a large number of point sources which are mutually incoherent. T his is a common approximation in light optics. T he di€ raction patterns produced by each of these point sources are nearly

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identical but slightly shifted with respect to each other. T he total intensity pattern of the extended source is obtained by summing over all the intensity patterns originating from each of the individual point sources. T his incoherent overlap smears out the observed di€ raction pattern, and the condition, that neighbouring di€ raction orders shall be resolved, limits the size of the source. We ® rst consider the case, where the collimation slit is much smaller than the size of the source. T herefore the slit can be considered to be point like, so that the width D pz of the momentum distribution of atoms passing the slit is again given by equation (8), where now D z is the size of the source. T he di€ raction orders will be resolved, if the condition equation (9) is ful® lled [14], so that the source must be smaller than 2. 3 mm. In our experiment the size of the source D z = 0. 7 mm (FWHM) is small enough, so that a second collimation slit just below the source is not required. From equation (8) we expect a momentum which D pz = 0. 6prec (FWHM) [15]. Even though the size of the slit and the size of the source are both limited by the same condition, namely equation (8) combined with equation (9), there is a di€ erence in the physical signi® cance of these two limitations. If the source is point like but the slit is too large, the overlapping standing wave di€ raction orders interfere, while in the opposite case, where the slit is point like and the source is too large, the incoherent overlap of di€ raction orders just smears out the pattern. We now consider the realistic case, where the source and the slit both have a ® nite size. T he spatial distribution of the collimated atomic beam in the detection region is the convolution integral of the two distributions for a point-like source and for a point-like slit. Hence the total width is D z total