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We report on directly measuring the atom number in a Bose-Einstein condensate by the method of optical pumping. Only the branching ratio of the spontaneous decay in the system and the absorption energy of a probe laser beam are required to determine the atom number. The measured absorption energy is not affected by the measurement condition such as the intensity, detuning, and polarization of the probe beam, the magnetic field, etc. We have shown that atom numbers as low as a few thousands can be measured. The atom number is an important parameter in the studies of Bose condensates and its accuracy is greatly improved by this sensitive and robust method.
©2007 Optical Society of America
1. Introduction
Since the first realization of Bose-Einstein condensation (BEC) in 1995 [1, 2], it has been of huge experimental and theoretical interest. Research has been performed on, e.g., wave-particle duality [3], superfluid to Mott insulator transition [4], formation of molecular BEC from fermionic atoms [5, 6, 7], storage of light for quantum information [8, 9, 10], and vortices [11]. The mean-field energy plays a very important role in the properties and dynamics of a Bose condensate [12]. This energy is directly proportional to the atom number. Therefore, it is desirable to accurately determine the atom number of a Bose condensate in the BEC studies [13].
The conventional method to determine the number of atoms in a Bose condensate is taking an absorption image (AI). The transmission of an imaging laser beam through the condensate is measured, yielding the spatial distribution of the optical density, α(x,y), defined as
Here, n(x,y, z) is the particle density, σ is the absorption cross section, and z is the propagation direction of the imaging beam. If σ is constant, the integration of the measured optical density will give
and provide the information of the atom number, N. However, the absorption cross section given by [14]
is usually not constant. Here, λ is the wavelength, Cij and I0,ij are the Clebsch-Gordan coefficient and the saturation intensity of the transition from states |i〉 to |j〉,I(x,y, z) is the imaging beam intensity which decays along the propagation distance and depends on the position in the xy plane, Δ is the laser detuning, and Γ is the natural linewidth of the excited state. If the atoms are distributed amongst the Zeeman sublevels defined by the quantization axis in the measurement, the above absorption cross section should be summed over all allowed transitions weighted by the population distribution. By applying a uniform magnetic field during the measurement and aligning it appropriately with respect to the imaging beam, all population can in principle be prepared in a single Zeeman state. However, misalignment can result in having to sum over all allowed transitions also in this case. The laser linewidth can influence the absorption cross section and there is the run-to-run fluctuation of the laser detuning. Therefore, the absorption cross section is afflicted with a high uncertainty, as is the number of atoms measured.
In this paper we demonstrate a method using optical pumping (OP) to determine the number of atoms in a BEC. Unlike for AI, the derivation of the atom number from the measured signal is completely immune from the measurement condition. The method was already implemented in our group to accurately measure the number of atoms in a magneto-optical trap (MOT) [15]. Here now, a technical difficulty arises due the fact that the size of a BEC is much smaller than a cloud of ultracold atoms inside aMOT, leading to an increased background signal. The number of atoms in a Bose condensate is several orders of magnitude lower, reducing the signal-to-noise ratio even further. These differences do not make it straightforward to measure the number of atoms in a BEC by OP as implemented before in aMOT. Nevertheless, we show that the number of atoms in a Bose condensate can be successfully measured by OP. Multiple measurements for the same atom cloud can be performed allowing averaging of the data. This technique is easily implemented and can be applied when an accurate knowledge of the number of atoms in a Bose condensate is needed.
This paper is organized as follows: First we present the principle of measuring the atom number by OP. Then we introduce the experimental setup to reach BEC and measure the atom number. In the next section we present the experimental results and discuss them. The paper finishes with a summary.
2. Principle of Measurement
Consider a three-level system with two ground states |g 1〉 and |g 2〉. A probe laser beam drives the transition from state |g 1〉 to the excited state |e〉. The probabilities for spontaneously decaying from the excited state |e〉 to the ground states |g 1〉 and |g 2〉 are (1-p) and p, respectively. The expectation value 〈Np〉 for the number of photons that an atom absorbs to reach the ground state |g 2〉 is given by
The above expression is evaluated under the condition that either the atom number is large or many measurements are performed. At the minimum measured atom number in this work and p=1/2, it is valid to use the result of Eq. (4) to calculate the atom number. Each atom will absorb 1/p probe photons on the average, thereafter not interacting anymore with the probe field. Thus, if the number of photons absorbed by the atomic system is measured, i.e. the absorption energy of the probe field, the atom number can be calculated. All that has to be known is the expectation value 〈Np〉. Neither the intensity, polarization, detuning, and linewidth of the probe beam nor the magnetic field and other experimental conditions can affect the measured absorption energy as demonstrated in Ref. [15]. Therefore, this method is more accurate and robust to measure the atom number, compared to the method of AI.
3. Experimental Setup
The atomic system used to reach BEC is 87Rb. The experimental system consists of two separate MOT’s. In the collection MOT, 87Rb atoms are trapped from the background gas. The pressure inside the vacuum chamber of the collection MOT is about 2×10-9 torr and the lifetime of the trapped atoms is about 1 s. Typically, the collection MOT can trap about 1.3×10 9 atoms in the steady state. The vacuum chamber of the collection MOT is connected to that of the science MOT by a stainless steel transfer tube with an inner diameter of 10 mm and a length of 340 mm. This allows for differential pumping of both MOT’s, resulting in a lower pressure in the vacuum chamber of the science MOT of 2×10-11 torr. The lifetime of the trapped atoms in the science MOT is about 25 s. We transfer the atoms trapped in the collection MOT to the science MOT by 200 pulses of a pushing laser beam. The pushing beam is on resonance with the transition |5S 1/2,F=2〉→|5P 3/2,F ′=3〉. The pulse sequence is 40 s long with a pulse duration of 2 ms. A hexapole magnet outside along the transfer tube serves to guide the atoms. After the transfer process is finished, we trap about 1.5×109 atoms in the science MOT at a temperature of about 300 µK.
To cool the atoms in the science MOT to a temperature below the critical temperature for BEC, the following cooling procedure is used. During a period of 6 ms of polarization gradient cooling [16], the atoms are cooled down to 25 µK. We optically pump all the atoms into the |F=2,mF=2〉 state which can be trapped in a purely magnetic time-orbiting potential (TOP) trap [17]. We load the atoms into the TOP trap at a temperature of about 35 µK with an efficiency of about 70%. In the final cooling stage, evaporative cooling [18, 19] is performed. All the coils of the TOP trap are ramped up to the maximum values with the quadrupole field of an axial gradient of 342 G/cm and the rotating bias field of an amplitude 40 G at a frequency of 10 kHz. Next, we ramp down the bias field linearly to decrease the trap depth in two steps from 40 G to 12 G in 10 s and from 12 G to 5.0 G in 12 s. In the final step, the atoms are irradiated by rf radiation which is linearly ramped from 6.03 MHz to about 3.97 MHz in 8 s where the BEC is just reached. Nearly all the atoms become a pure condensate at the rf frequency of 3.84 MHz. Figure 1 demonstrates the transition of the Bose-Einstein condensation. A thermoelectric-cooled CCD camera (Princeton Instruments TE/CCD-576-EFT) is used to take the BEC image.
One way to measure the number of atoms in the condensate is to take an AI after the condensate has freely expanded as shown in Fig. 1. The other method that we present in this paper works as follows. A circular-polarized probe laser beam which is on resonance with the transition |F=2〉→ |F ′=2〉(see the energy levels in Fig. 2(a)) is passed through the condensate after a free expansion of 6.3 ms. The pulse duration of the probe beam is 5.0 µs and the intensity is about 7.0 mW/cm2. During the measurement of the atom number, we add a small magnetic field of about 1.4 G in the direction perpendicular to the propagation direction of the probe beam to prevent the population from being trapped in the dark state [15]. Figure 3 shows the setup of the detection system. Two biconvex lenses with focal lengths of 80 mm and 250 mm are used to create a magnified image of the condensate. We install a pinhole with a diameter of 300 µm in the image plane to improve the signal-to-background ratio by cutting off most of the background around the condensate. In the image plane of the pinhole, the magnification of the Bose condensate is 2.3. A biconvex lens of focal length 100 mm at a distance of 200 mm from the pinhole images the pinhole and condensate onto either a CMOS camera (Prosilica EC750) or an avalanche photodiode (APD, Hamamatsu C5460, photoelectric sensitivity 1.5×10 6 V/W, rise time 36 ns). We employ a mirror on a flip mount to change between detection by the CMOS camera and the APD. The CMOS camera serves to align the BEC and the pinhole and the APD temporally resolves the transmission of the probe pulse. The output of the APD is detected by an oscilloscope (Agilent MSO6054). To repeat the measurement several times for the same condensate, we apply a repumping laser pulse 2.5 µs after the probe pulse. It drives the population from the ground state |F=1〉 back into the ground state |F=2〉 via the excited state |F ′=2〉 which has a lifetime of about 27 ns. The duration of the repumping pulse is 7.5 µs with an intensity of about 20 mW/cm2. After applying ten cycles of the probe and repumping pulses to obtain the data with the presence of the atoms, we turn off the repumping field to obtain the data without the presence of the atoms. Figure 2(b) shows the timing diagram of the measurement sequence.
4. Experimental Results and Discussion
Figure 4(a) shows the probe power transmitted through the pinhole with (red line) and without (black line) the presence of the atoms from a pure condensate versus time as detected by the APD. The data has been averaged over nine measurements for the same condensate. When no atoms are present, the signal represents the temporal evolution of the probe pulse. When atoms are present, the probe pulse is partially absorbed in the beginning until all atoms are pumped into the ground state |F=1〉 and become transparent. The difference of both data sets yields the power absorbed by the atoms as shown by the blue line in the inset of Fig. 4. Because the power of the probe pulse can fluctuate, we multiply the data without the presence of the atoms by some factor such that the difference becomes zero when all the atoms are pumped out of the ground state |F=2〉 and the probe pulse is not absorbed any more. Fitting the data of the difference by an exponential decay function (black dashed line) and evaluating thereafter the area below the best fit, or directly integrating the area below the data, yields the energy Ep being absorbed by the atoms. As the probability for decaying from the state |F ′=2〉 to the states |F=1,2〉 is equal, namely 1/2, each atom absorbs 2 photons on the average as illustrated by Eq. (4) before it remains in state |F=1〉. The number N of the atoms in the Bose condensate is then given by N=Ep/(2h̄ω), where ω is the angular frequency of the probe beam. For the data shown in Fig. 4(a), the number of atoms in the Bose condensate obtained by this method is 4.9×10 4. Integration of the area below the best fit or the experimental data to obtain the absorbed energy Ep yields the same number of atoms within 2%. A probe pulse of higher intensity results in an absorption profile of larger amplitude and shorter time constant in the inset of Fig. 4. The absorption energy or the area below the absorption profile is the same however. Hence, the probe intensity does not influence the measured absorption energy, as do not the probe detuning and polarization (see the demonstrations in Figs. 2(a), 2(b), and 2(c) of Ref. [15]). We note that the data obtained from the first pulse of the probe pulse sequence had a smaller amplitude due to some leakage of the acousto-optic modulator that switched the probe pulse. Though this leaking radiation is very weak, it pumps a significant part of the population out of state |F=2〉 as the interval between switching on the shutter, which blocked the probe beam during the evaporative cooling process, and starting the measurement sequence is rather long, namely 3 ms. For the following probe pulses this leakage is negligible as they are separated by only 20 µs. Therefore, the first pulse was not used for averaging.
Fig. 1. The three images in the right column show the spatial distribution of the optical density for atom clouds of different temperatures. From top to bottom the images show thermal atoms above the transition temperature Tc, thermal atoms mixed with a condensate near Tc (≈250 nK) and a pure condensate below Tc, respectively, at a time-of-flight of 21 ms. The size of each image is 850×850 µm2. In the left column, each plot shows the optical density versus the horizontal axis intersecting the center of the image of the same row. The solid lines are the best Gaussian fits of the thermal (blue) and condensed (red) atoms.
Fig. 2. (a) Relevant energy levels in the OP measurement. The spontaneous decay rates from the excited state to the two ground states are both equal to 1/2. (b) The timing diagram of the measurement sequence.
Fig. 3. Experimental setup of the detection system. The image which is taken by the CMOS camera shows a condensate (the dark shadow in the bright spot) in the center of the pinhole (the bright spot in the image).
Fig. 4. (a) Power of the probe beam transmitted through the pinhole with (red line) and without (black line) the atoms in the optical path versus time. In the inset, blue line is the difference of the two signals and black dashed line is the best fit of the exponential decay function of y(x)=y0 exp(-x/τ). The data have been averaged over nine measurements for the same condensate. The product of y 0 and τ of the best fit indicates that the number of atoms is 4.9×104. We note that the spike on the rising edge of the pulse is the transient behavior due to the APD bandwidth. Its size is about 9% of the pulse amplitude. As the initial data points in the inset do not deviate from the exponential decay curve very much, the effect of the spike is negligible. (b) Transmitted power of the probe beam versus time for an atom number of 6×103. All the legends are the same as those in (a).
The number of possible measurements for one condensate is limited by the time that the atom cloud expands out of the image area of the pinhole due to the heating by the laser excitation. Right before we apply the probe and repumping pulses, the 1/e diameters of the atom cloud in the vertical and horizontal directions are 41 and 56 µm, respectively. Less than 0.3% of the atoms is imaged outside the pinhole area. After ten cycles of the probe and repumping pulses, the 1/e diameters become 66 and 69 µm and about 3% of the atoms cannot be detected. The number of applicable probe pulses can be increased by using a pinhole of larger diameter. However, at the same time this will lead to an increase of the background level. The signal-to-noise ratio may not be improved due to the fluctuation of the background level. Besides, the detector can be saturated if the background level is too large. We choose a suitable pinhole diameter such that the signal-to-noise ratio is optimized and the APD is well below saturation.
To compare the results for the number of atoms, N, in the condensates obtained by OP and AI, we performed multiple measurements consecutively. By OP, the average atom number N in ten runs was 6.2×104 with a standard deviation of about ±5%. In the AI measurement, the circularly polarized imaging beam had a pulse width of 100 µs and drove the transition |F=2〉→ |F ′=3〉 resonantly. We used five different imaging-beam intensities of 0.076, 0.34, 1.0, 1.7, and 2.0 mW/cm2 which all fluctuated around ±4%. By AI, the average values of Nσ in Eq. (2) in ten runs were 1.8×104, 3.1×104, 3.9×104, 4.2×104, and 4.1×104 in units of 3λ 2/(2π) for the five intensities, respectively. The standard deviations in ten runs ranged from ±10% to ±20% and were larger for weaker intensities. To obtain the atom number N from Nσ, we calculated σ by using the input intensities of the imaging beam, the |mF=2〉→ |mF=3〉 saturation intensity of 1.6 mW/cm2, and Cij=1. This results in average atom numbers of N=1.9×104, 3.8×104, 6.3×104, 8.4×104, and 9.1×104 for the five intensities. The discrepancy of the atom numbers measured by AI is due to the following reasons. At weak imaging intensities, the background light level in the signal with the presence of the atoms becomes non-negligible. This decreases the measured atom number, because the optical density or Nσ is the natural logarithm of the ratio of the signals without and with the presence of the atoms. At large imaging intensities, the input intensity used in Eq. (3) to calculate σ is an over estimation, because the imaging beam decays along the propagation distance. This results in an increase of the measured atom number. The fluctuation of AI is larger than that of OP. This can be the consequence of frequency fluctuations of the imaging beam of about ±0.25Γ for our system. A detuning of 0.25Γ reduces the absorption cross section by 20% at the low-intensity limit according to Eq. (3).
In order to find the lowest number of atoms that can still be measured by the OP method, we decreased the atom number of the Bose condensate. By holding the condensate for 8.0 s before we turned off the trap and performed the OP measurement, the number of the atoms in the Bose condensate decayed in the trap due to three-body recombination. Figure 4(b) shows the absorbed power of the probe beam versus time (compare Fig. 4(a)) for the lowest number of atoms measured. The data are the average of nine measurements of the same condensate. Calculating the absorption energy as before indicates the number of the atoms in the Bose condensate is 6×103. According to the signal-to-noise ratio in the inset of Fig. 4(b), the atom numbers as low as two or three thousands could be measured.
5. Summary
We have presented a new method for measuring the number of atoms in a Bose condensate which can be easily implemented. It provides the number of atoms at a higher accuracy than the standard method of absorption imaging. A probe laser pulse optically pumps the population into a state where it cannot be absorbed anymore. By measuring the absorption energy of the probe pulse, we are able to determine the number of atoms in the condensate. No other parameters aside from the branching ratios of spontaneous decay in the system have to be known. By applying another laser pulse after the probe pulse to pump the population back to the initial ground state, about nine measurements for the same atom cloud can be done. The signal-to-noise ratio is greatly improved by averaging of these measurements. We have shown that the measurable atom number can be as low as a few thousand atoms. The method of optical pumping is robust and sensitive in the determination of the atom number. It will be a very useful tool in the BEC studies, as the mean-field energy is directly proportional to the atom number and plays a very important role in the properties and dynamics of a Bose condensate.
Acknowledgements
This work was supported by the National Science Council of Taiwan under Grant No. 95-2112-M-007-039-MY3.
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