Laser trapping and cooling of atoms in a magneto-optical trap (MOT) has become a general technique in the experiments utilizing samples of cold atoms [1, 2, 3]. An atom temperature of a few hundred μK can be easily achieved with the MOT. Doppler broadening, thermal noise, collisional perturbation, and other temperature related nuisances are greatly reduced at such low temperature. Cold atoms are ideal for studies of quantum phenomena at long de Broglie wavelengths as well as scientific applications that require slow velocity or low kinetic energy of atoms. The MOT is the necessary experimental step in the studies of Bose-Einstein condensation (BEC) [4, 5], atom optics [6, 7], and cold collisions [8, 9], and gets more popular in experiments such as precision spectroscopy and measurement [10, 11, 12], slow light and storage of light [13, 14, 15, 16], etc. An atom cloud of large optical density (OD) is desirable in many experiments. For example, a large OD can enhance the signal-to-noise ratio of a weak transition in high-precision spectroscopy and significantly increase the efficiency of low-light-level nonlinear optics. Also, a light pulse can be entirely stored only in a medium with a large enough OD in the experiments of storage and retrieval of light.

Realization of BEC is a way to obtain a sample of cold atoms with a large OD, but the experimental setup and the procedure of achieving BEC are rather complicate and delicate. Inspired by the 2D MOT [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], we experimentally demonstrate a simple method to increase the OD of cold atoms in this paper. In a 2DMOT, there are typically four trapping beams and each beam has an elliptical intensity profile of large aspect ratio. This leads to confinement of the atoms in only two orthogonal directions, generating a steady atom flux. In our method, we achieve complete confinement by simply replacing a pair of the circular anti-Helmholtz coils of the MOT by a pair of rectangular ones. All the six trapping beams and other conditions of the conventional 3D MOT are kept the same. Such simple modification enables us to increase the optical density by about 2.7 folds. In the multiple scattering regime of the MOT [27, 28], as the trapped atom number (N) increases, the atomic density remains fixed and the size of the atom cloud increases with N ^1/3. Hence, a factor 2.7 increment in the OD would require a factor 20 increment in the atom number. In our method, the optical-density increment does not require to upgrade laser power and optics of the MOT or use some elaborate loading methods for trapping more atoms. It only needs to deform the shape of the atom cloud by using the rectangular coils.

We produced cold ⁸⁷Rb atoms in our vapor-cell MOT by two methods using different types of the coils. In the conventional method, a pair of the circular anti-Helmholtz coils was used to generate a “spherical quadrupole” magnetic field whose axial gradient is twice of the radial one. The resulting cloud of the trapped cold atoms was ellipsoid. We denote the setup with the circular anti-Helmholtz coils as the conventional MOT. In the new method to produce the cold atoms, a pair of the rectangular anti-Helmholtz coils was used. We set the separation of the two coils equal to the length of the rectangle’s short side such that the magnetic field generated by the coils has a nearly cylindrical quadrupole profile. Therefore, the cloud of the trapped cold atoms formed a cigar-like shape. We denote the setup with the rectangular coils as the cigar-type MOT. The magnetic-field gradient in the transverse plane is 13 folds of that along the longitudinal direction, which is parallel to the long side of the rectangular coils, in the cigar-type MOT.

We aligned the long side of the rectangular coils in a direction with an angle of 15° separated from the direction of one pair of the counter-propagating trapping beams as shown in Fig. 1. Such alignment has the following two advantages: Laser beams for science studies can easily access the longitudinal axis of the cigar-shaped atom cloud. The complication of merging the input laser beams for science studies into and separating the output beams from the optical path of the trapping beams is avoided. The second advantage is that attenuation of the trapping beams induced by the trapped atoms can be greatly reduced because the trapping beams do not propagate through the path of large optical density. The stability of the atom cloud is significantly improved and the fluctuation of the atom number is minimized. Figure 2(a) shows the representative fluorescence images of the atom cloud. Figure 2(b) show the fluorescence emitted by the trapped atoms as a function of time. The fluctuation or standard deviation of the steady-state fluorescence signal is about ±1.2%.

 

Fig. 1. Scheme of the experimental setup (drawing not to the scale). OP, the optical pumping beam; Coupling, the coupling beam; Probe, the probe beam; BS, beam splitter cube; PBS, polarizing beam splitter cube; λ/4, quarter-wave plate; L1-L6, lenses (f indicates the focal length in units of mm); CPL, optical fiber coupler, M, mirror; PD, photodetector; APD, avalanche photodetctor. The size of the rectangular coils is 155 mm by 55 mm. We use L2 to focus the probe beam, L1 and L3 to transform the optical pumping and coupling beams into plane waves in the region of the atom cloud, and L4, L5, L6, and CPL to collect entire laser light into the detectors.

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The three pairs of trapping beams propagated along three orthogonal directions. Each trapping beam has a circular beam profile with a diameter of 20 mm which is set by the clear aperture of a quarter-wave plate. The total power of the six trapping beams is about 30 mW. The intensity of the trapping beams in the xz plane is about 26% higher than that in the y direction (the coordinate is illustrated in Fig. 1). The trapping field was red-detuned from the |5S _1/2, F = 2〉→|5P _3/2, F′ = 3〉 transition with a spectral width of 1.3Γ, where Γ = 2π × 5.9 MHz is the spontaneous decay rate of the excited state. A laser beam drove the |5S _1/2, F = 1〉→|5P _3/2, F′ = 2〉 transition resonantly and served to repump the population back to the |5S _1/2, F = 2〉 state. The repumping beam had a power of 4.4 mW and a e ⁻² full width of 24 mm.

We used the optical pumping method to measure the number of the trapped atoms in the MOT [29, 30]. An laser beam propagated nearly along the major axis of the cigar-shaped atom cloud and drove the |5S _1/2, F = 2〉→|5P _3/2, F′ = 2〉 non-cycling transition. This optical pumping beam had an e ⁻² full width of 7.9 mm and interacted with all atoms. On the average, each atom prepared in state |5S _1/2, F = 2〉 initially can be optically pumped to state |5S _1/2, F = 1〉 by absorbing two photons from the optical pumping beam. Therefore, measuring the absorption energy or the number of absorbed photons directly yields the atom number. The measured absorption energy neither depends on the detuning, intensity, and polarization of the laser field nor is affected by other system parameters. It is a robust and accurate method to determine the number of trapped atoms. Figure 3(a) shows the power of the optical pumping beam transmitted with (red line) and without (black line) the presence of the atoms versus time. When atoms are present, the pulse is partially absorbed in the beginning until all atoms are pumped into the ground state |5S _1/2, F = 1〉 and become transparent. The difference of both data sets in Fig. 3(a) yields the power absorbed by the atoms as shown by the blue line in Fig. 3(b). Details of the atom number measurement by the optical pumping method can be found in Refs. [29, 30].

 

Fig. 2. (a) Fluorescence images of the cigar-shaped atom cloud. The line of sight of the left image is along the z axis and that of the right image is along the x axis (the coordinate is illustrated in Fig. 1). (b) Fluorescence emitted by the trapped atoms versus time.

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The numbers of the trapped atoms in the conventional MOT and in the cigar-type MOT are 1.15 × 10⁹ and 7.2 × 10⁸ with time constants of 2.2 s and 1.9 s for loading the trap, respectively. We kept the power of the trapping beams and the Rb background pressure the same in both measurements. In the conventional MOT, the optimum axial magnetic field gradient and the trapping field detuning were 15 G/cm and 3.3Γ. The size of the atom cloud is about 2.7 × 2.9 × 2.7 mm³. In the cigar-type MOT, the optimum transverse magnetic field gradient and the trapping field detuning are 12 G/cm and 3.0Γ. The size of the cigar-shaped atom cloud is about 1.0 × 1.8 × 7.4 mm³. In both MOT’s, the weaker trapping beam intensity in the y direction results in that the atom cloud has a larger size in the y axis than in the x axis. The conventional MOT can trap more atoms than the cigar-type MOT. The shallow trapping potential along the longitudinal direction reduces the capture ability of the cigar-type MOT. Nevertheless, the decline of the trapped atom number due to the reduction of the capture ability is not severe.

In order to determine the OD of the atom cloud, we performed measurements on slow light and storage of light [31, 32, 33]. The group velocity delay time is inversely proportional to the OD. Hence, by measuring the delay time for a pulse passing through a cloud of cold atoms the OD of the system can be determined. In the system of slow light and storage of light, a weak probe field drives the transition between a ground state and an excited state and a strong coupling field drives the link transition between another ground state and the same excited state.

 

Fig. 3. (a) The power of the optical pumping beam transmitted with (red line) and without (black line) the presence of the atoms versus time. (b) The difference of the black and red lines in (a). The area below the blue line indicates the number of the trapped atoms in the cigar-type MOT is 7.2 × 10⁸.

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The two fields form the three-level Λ-type configuration that gives rise to the electromagnetically induced transparency (EIT) effect. The large frequency dispersion due to the EIT effect results in an ultraslow group velocity of the probe pulse. Furthermore, the probe pulse can be stored in a medium by adiabatically switching off the coupling field and be subsequently released intact by the reverse process. This provides additional data for the determination of the OD. In the measurements on slow light and storage of light, the MOT as well as the repumping beam was turned off. The center frequency of the probe pulse was tuned to the resonance frequency of the |5S _1/2, F = 1〉→|5P _3/2, F′ = 2〉 transition and the coupling field drove the |5S _1/2, F = 2〉→|5P _3/2, F′ = 2〉 transition resonantly. The probe beam was focused to a e ⁻² full width of 0.27 mm with a peak Rabi frequency of about 0.1Γ. This probe Rabi frequency is weak enough that we can consider the probe field as a perturbation in the theoretical calculation. An avalanche photodiode (APD, Hamamatsu C5460, photoelectric sensitivity 1.5 × 10⁶ V/W, rise time 36 ns) detected the probe transmission. The coupling beam had a e ⁻² full width of 5.3 mm and interacted with all the trapped atoms. We determined the Rabi frequency of the coupling field by fitting the slow-light data with the theoretical prediction and will describe the procedure later. Both the probe and coupling fields propagated in nearly the same direction along the major axis of the cigar-shaped atom cloud. Other details of our experiment can be found in Refs. [15, 34, 35, 36].

Both coupling and probe fields were circularly polarized with right helicity (σ₊ polarization). Considering the degenerate Zeeman states, there are three sets of Λ-type EIT subsystems. According to Ref. [37], the three subsystems can be transformed into a simple three-state system. The Maxwell-Schrödinger equation of the probe pulse and the optical Bloch equation of the simple three-state EIT system are given by

In the equations above, Ω_p and Ω_c are the Rabi frequencies of the probe and coupling fields, ρ ₂₁ is the amplitude of the ground-state coherence and ρ ₃₁ is that of the optical coherence with respect to the probe transition, γ is the ground-state relaxation rate, α is the OD of the atoms, and L is the length of the atom cloud.

We can unambiguously and uniquely determine α, Ω_c, and γ by fitting the experimental data with the theoretical predictions calculated from Eqs. (1)–(3). Figure 4(a) shows the storage and retrieval of the probe pulse. The solid black and red lines are the experimental data and the best fit of the output probe pulse. The gap in the probe signal indicates the probe pulse is stored in the atoms when we momentarily turn off the coupling field (blue line). In the storing process, the probe pulse is converted into the ground-state coherence ρ ₂₁ of the atomic ensemble. During the storage, Eqs. (1) and (2) are idle and Eq. (3) becomes

The ground-state coherence ρ ₂₁ decays exponentially with a decay rate of γ. In the retrieving process, ρ ₂₁ is converted back to the probe pulse which is attenuated by the factor of exp(-2γt_s) where t_s is the storage time. The attenuation factor of the retrieved signal is determined by γ alone. Figure 4(b) shows the slow probe pulse under the constant presence of the coupling field. According to Ref. [36], the group velocity delay time is approximately equal to αΓ/Ω² _c and the broadening of the output pulse as well as the reduction of the pulse peak height is governed by 8αΓ²/(Ω⁴ _cτ²) where τ is the 1/e half width of the input probe pulse. Hence, only correct α (the OD of the atoms), Ω_c, and Γ can provide the best fits for the experimental data of slow light and storage of light altogether.

 

Fig. 4. The storage and retrieval of the probe pulse in (a) and the slow probe pulse under the constant presence of the coupling field in (b). Solid gray, black, and blue lines are the experimental data of the input and output probe pulses and the coupling field. The input probe pulse is plotted with the size reduced to one third. The data were measured with the cigar-shaped atom cloud of N = 7.2 × 10⁸. Dashed gray and blue lines in (a) are the functions of the input probe pulse and the coupling field used in the calculation. Solid red lines in (a) and (b) are the best fits calculated at (α,Ω_c, γ)=(32,0.330Γ,7.1 ×10⁻⁴Γ). Dotted blue lines in (b) have the same delay time as the solid red line, but are calculated at (38,0.365Γ,7.1 × 10-⁴Γ) and (26,0.295Γ,7.1 × 10⁻⁴Γ).

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With the atom number of 7.2 × 10⁸ in the cigar-type MOT, we achieved an OD of the atom cloud of about 32 for the |5S _1/2, F = 1〉→|5P _3/2, F′ = 2〉 transition of the probe field. The transition has the saturation intensity I ₀ of 5.8 mW/cm² [38], which is defined such that I/I ₀ = 2(Ω/Γ)² where I and Ω are the intensity and the Rabi frequency of the laser field. With the atom number of 1.16 × 10⁹ in the conventional MOT, the OD of the atom cloud was 12. There is a 2.7-fold increment of the OD by simply replacing the circular coils by the rectangular ones under similar trapping conditions. To achieve the same increase of OD for our system by increasing the atom number without changing the coils, the following estimation can be done. When the trapped atom number is far above 10⁸, the conventionalMOT is typically operated in the multiple scattering regime [27, 28]. In this regime, the atomic density reaches the maximum and remains fixed; as the atom number (N) increases the volume of the atom cloud expands with N or the cloud size enlarges with N ^1/3. Because the OD is proportional to the product of the atomic density and the atom cloud size, a factor 2.7 increment in OD requires a factor 20 increment in the atom number. The conventional MOT needs to trap more than 2 × 10¹⁰ atoms in order to achieve the same OD of the cigar-type MOT. For such large atom number in the MOT, it would be necessary to upgrade the trapping laser power as well as enlarge the trapping beam size or to use some elaborate loading methods.

 

Fig. 5. Transmission spectrum of the |6S _1/2, F = 4〉→|6P _3/2, F′ = 5〉 transition in lasercooled cigar-shaped Cs atom cloud. Black and red lines are the experimental data and the best fit. The fitting function is y = exp{-α/[1+4(x-x ₀)²]}, and α = 10⁵ and x ₀=-0.35Γ for the best fit.

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In another experimental setup, we trap Cs atoms with the cigar-type MOT. The trapping beams propagating in the xy plane (the coordinate is illustrated in Fig. 1) have an elliptical profile with the major and minor diameters of 40 and 25 mm. The total power of these four beams is 140 mW. The trapping beams propagating in the z axis have a circular profile with the diameter of 25 mm. The total power of these two beams is 33 mW. The trapping field has a detuning of 2.5Γ, where Γ = 2π×5.2 MHz is the spontaneous decay rate of the excited state in this experiment. The longitudinal and transverse magnetic field gradients are 9.0 and 1.5 G/cm, respectively. At the above condition, the number of the trapped atoms is 1.4 × 10⁹ with a time constant of 1.1 s for loading the trap and the size of the atom cloud is about 2.0 × 3.1 × 26 mm³. With this cigar-shaped atom cloud, we measured the absorption spectrum of the one-photon transition from the |6S _1/2, F = 4〉 state to the |6P _3/2, F′ = 5〉 state by sweeping the probe frequency. The transition has the saturation intensity of 2.7 mW/cm² [39]. Figure 5 shows the absorption spectrum of the transition. The probe transmission is given by

where Δ is the probe detuning and Γ′ is the spectral width of the transition in the low-OD limit. We set Γ′ = Γ and fit the experimental data with the above function. The OD of the transition at the resonance frequency is 105 in the cigar-type MOT. Such large OD has opened great opportunities for the experiments that perform measurements on weak transitions, study nonlinear optical processes, or require a large group velocity delay time in order to store a complete Gaussian probe pulse in the medium.

In conclusion, we have demonstrated that the optical density of a cold Rb atom cloud in the MOT is enhanced by 2.7 folds simply by replacing the circular anti-Helmholtz coils by the rectangular coils. The rectangular coils generate a magnetic field, which has a very weak longitudinal gradient, to make the cloud of the trapped atoms take a cigar-like shape. With the atom number of 7.2 × 10⁸ in the cigar-type MOT, we are able to achieve the optical densities of 32 for the transition with saturation intensity of 5.8 mW/cm². On the other hand, such large OD in the conventional MOT requires to trap 2 × 10¹⁰ atoms by using high-power and large-size setups or more elaborate loading methods. In the experimental setup of cold Cs atoms, we have achieved the OD of 105 in the cigar-type MOT for the transition with saturation intensity of 2.7 mW/cm². The cold atomic sample with large optical density is a desirable system in the studies of high-precision spectroscopy, low-light-level nonlinear optics, and the storage and retrieval of light pulses.

We thank Dr. Thorsten Peters for providing suggestions and proofreading the manuscript. This work was supported by the National Science Council of Taiwan under Grant Numbers 95-2112-M-007-039-MY3and 96-2112-M-001-006.