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Abstract
Quantum sensing with cold atoms is attracting more and more attention in the research frontier of quantum technologies. Isotropic laser cooling serves as a promising platform for this area and its many advantages include compactness, robustness, and not requiring fine-tuning of the optical alignment or magnetic field, which are particularly suitable for field and space applications. We have designed and demonstrated a special form of isotropic laser cooling system that is specifically tailored for the purpose of quantum sensing. In particular, the diffused cooling optical field is generated by hollow laser beam injection and the quality of detection process is improved. Moreover, the effective strength of diffused cooling laser is quantitatively deduced. We have carefully characterized the properties of our system, including the laser cooling performances with respect to different parameter settings, the influences of cooling laser polarization and repumping laser power, and radial atom density distribution. We anticipate that our progress will contribute to the future development of isotropic laser cooling for quantum sensing and quantum precision measurements.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum sensing [1] based on cold atomic clouds offers state-of-the-art performance across a variety of tasks, such as magnetometers [2,3], microwave sensors [4], and gravimeters [5–7]. A key advantage comes from the fact that the cold atomic clouds typically have much longer coherent time than atomic vapors at room temperature. Isotropic laser cooling (ILC) in the form of obtaining cold atoms via diffuse reflection light [8,9] serves as an ideal platform for critical applications in quantum sensing and quantum precision measurements [10–14] due to its unique characteristics of compactness and robustness. Unlike the case of optical molasses or magneto-optical trapping (MOT), its operation requires neither careful alignment of cooling lasers or presence nor magnetic gradient field. The implementation of ILC usually relies on an all-optical structure, which is preferable for magnetically sensitive applications. As a relatively simple method to obtain cold atoms with small footprint, it has been regarded an ideal cold atom source for miniature and portable devices. Moreover, space missions with cold atom systems have become a focus of many recent efforts [15–17], where ILC is deemed as a hopeful candidate.
Highly demanding applications in quantum sensing require various improvements of the ILC platform, especially in increasing the cold atom density along the detection axis and enhancing the signal-to-noise ratio (SNR) of the detection signal. Ever since the first demonstration of ILC in the form of integrating sphere [18], different configurations have been constructed over many years of development [13,19–21]. Moreover, such a robust form of cold atom platform can find applications in the research of quantum memory [22–24] if the cold atom density is further increased in the future. Meanwhile, hollow laser beam (HLB) has already been widely employed in many cutting-edge research areas including laser catheters, optical tweezers and manipulation of microscopic particles [25,26]. These progresses trigger us to think about the specific application of HLB under the theme of ILC. A direct consequence is the enhancement of the cold atom density around the detection region due to the axisymmetric ring light field structure of HLB. Moreover, external modulation is a well-known frequency stabilization technique that can work together with the saturated absorption spectroscopy (SAS). Equipping the probe laser lock system with external modulation instead of internal modulation avoids directly introducing the modulation sidebands into the ultimate detection signal of cold atoms. On the other hand, practical applications also require a systematic characterization of the ILC properties. In particular, we have developed a new method to quantitatively obtain the effective strength of diffused cooling laser received by the atoms. Furthermore, it has been observed experimentally that the cold atom temperature of ILC is significantly below the Doppler limit [11,27], but the details of sub-Doppler cooling mechanism in ILC still remains elusive and awaits a comprehensive discussion. These new efforts will not only enhance the relevant fundamental research in quantum optics but also inspire improvements in quantum sensing and quantum precision measurements [1,13].
In this article, we report our recent progress in the design, realization and characterization of a special form of isotropic laser cooling system for applications in quantum sensing. In particular, the diffused cooling optical field is generated by injection of HLB and the noises of probe laser system are effectively suppressed. We have carefully characterized the key properties of such a system, including the laser cooling performances with respect to different parameter settings, the influences of cooling laser polarization and repumping laser power, radial atom density distribution and so on.
2. Experimental setup
A sketch of the experimental setup is shown in Fig. 1(a). An ensemble of $^{87}$Rb atoms is cooled by diffuse reflection light. The frequency stability of the probe laser is realized by modulation transfer spectroscopy referenced to atomic transition, where the frequency modulation is generated via an EOM. A laser power controller (LPC) offers automated regulation and control of the probe laser power, which suppresses the amplitude noise. The polarizing laser and repumping laser A are combined to participate in the labeling process. An arrangement of polarizers (POL), polarizing beam splitters (PBS), and half wave plates (HWP) is used to adjust the powers and polarizations of the probe laser and the labeling lasers. A quarter-wave plate (QWP) turns the labeling lasers and the probe laser into circular polarization. A cylindrical cavity is employed to generate the diffuse reflection light for ILC, which is placed vertically. The probe laser and the labeling lasers propagate along the central axis of the cylindrical cavity.
Fig. 1. (a) Experimental setup. (b) Relevant energy levels and transitions of $^{87}$Rb D2 line for this experiment. (c) Details of the Zeeman sub-states for the nearly nondestructive detection method which relies on the polarization degrees of freedom. (d) A typical time sequence of the experiment.
Our setup comes from a thorough redesign based on previous versions, with the following major differences. The combination of cooling laser and repumping laser A passes through two axicons, and form the HLB for injection into the cylindrical cavity. A high vacuum is kept inside the cavity by an ion pump, and the coating on the surface of cavity generates the diffuse reflection light to cool $^{87}$Rb atoms. HLB with ring-shaped profile has several advantages for the implementation of ILC. (I) This method merely requires two colinear axicons instead of dividing cooling light into four or more beams as previous mechanisms [20], which simplifies the optical setup and reduces loss of laser power. (II) If the injection light directly shines on the cold atoms in the central region, it is known to significantly lower the cold atom density there. Since the optical detection axis usually travels through the center, injection via HLB avoids the troublesome consequence of reduction in signal strength. As stated earlier, the lock system of the probe laser is realized by the external modulation technology instead of the internal modulation, in order to achieve a better SNR. Meanwhile, the setup can also measure the temperature of cold atoms in situ with nearly nondestructive detection.
The relevant energy levels of $^{87}$Rb are shown in Fig. 1(b). The cooling laser is red detuned by 22 MHz ($\sim 3.6\Gamma$ , where $\Gamma$ denotes the natural line width) to the transition of $5^{2}S_{1/2}, F=2 \leftrightarrow 5^{2}P_{3/2}, F=3$. The probe laser and polarizing laser are tuned close to resonance with the transition of $5^{2}S_{1/2},F=2 \leftrightarrow 5^{2}P_{3/2},F=3$. The repumping laser is tuned to the transition of $5^{2}S_{1/2},F=1\leftrightarrow 5^{2}P_{3/2},F=2$, which is divided by a PBS into two beams. One beam (repumping laser A) becomes part of the diffuse reflection light in the cylindrical cavity to participate the cooling process. The other beam (repumping laser B) propagates along the central axis to participate the labeling process.
A typical time sequence for our experiment is shown in Fig. 1(d). Firstly, atoms are cooled down via ILC. Secondly, the short pumping phase transfers the atoms to the level of $5^{2}S_{1/2}, F=1$. Afterwards, labeling lasers populate the atoms to $|F=2, m_F=2\rangle$ of the ground level along the central axis. Finally, during the stage of detection, probe laser is applied to detect labeled atoms with nearly nondestructive performance. To ensure that all the signal comes from cold atoms generated by ILC process around the central axis, rather than any random entry of hot atoms or drift of cold atoms from outer regions, we enforce a labeling process via straightforward optical pumping method after the cold atoms are obtained. More specifically, via feeding the pumping laser as the intracavity diffuse reflection light, all the atoms are transferred to $F = 1$, and then labeling lasers shinning across the cavity pumps atoms along the optical axis to $|F=2, m_F=2\rangle$. A nearly nondestructive detection process can be realized through further manipulations with respect the polarization degrees of freedom as shown in Fig. 1(c).
3. Results and discussions
It turns out that the laser cooling process works effectively and we have obtained an ample number of cold atoms. We note that no magnetic field gradient is involved and therefore there is no trapping effects. The atom accumulation is the direct consequence of decelerating hot atoms due to the dissipative velocity dependent damping force of ILC. To evaluate the performance of the cooling process, we measure the obtained cold atom number as a function of the cooling time and the result is shown in Fig. 2. The power settings of cooling laser and repumping laser A are 120 mW and 8 mW, respectively. The width of HLB is about 1 mm, while the probe beam’s diameter is 2.6 mm and average intensity is 11.44 $\mu$W/cm$^{2}$. The pumping laser and repumping laser B are relatively weak, on the order of several tens of $\mu$W. For cooling time more than about one second, saturation behavior occurs where the capturing and loss rates are almost balanced.
Fig. 2. Atom number of the cold atom ensemble along the optical detection axis versus the cooling time. The atom numbers are normalized with respect to the highest data value in the figure, which corresponds to the averaged cold atom number density of $1.62 \times 10^{8}$ cm$^{-3}$ across the cavity with length 5.4 cm.
The measurement relies upon the cold atom ensemble’s absorption of the incident probe laser pulse. In general, if the probe laser power is not too strong, the concept of optical depth (OD) applies well for such processes. Namely, we have the relation $I=I_{0}e^{-OD}$ where $I$ and $I_{0}$ are emergent and incident intensities of the probe laser respectively, and there exists the straightforward relation with the average cold atom number density $n$ and cold atom ensemble length $L$ as $OD=\sigma n L$ where $\sigma$ is the cross section of the atom-light interaction. Therefore, according to detection signal and geometry of probe laser, the number of cold atoms along the cavity axis can be deduced. In the process of reaching the equilibrium cold atom number, there exists a competition between capturing and loss, where a simplified phenomenological model may be given as: $\frac {d}{dt}N=-\kappa N+\alpha$, where $N$ is the total cold atom number, $\kappa$ describes the loss and $\alpha$ describes replenishing rate of the newly captured cold atoms. The solution is $N(t)=\alpha /\kappa (1-e^{-\kappa (t-t_{0})})$ with equilibrium value $N(\infty )=\alpha /\kappa$. We include such a curve fitting based upon this observation in Fig. 2, from which we deduce that the loss rate is $\kappa =1.9 s^{-1}$.
Temperature of the cold atoms is another key parameter in quantum precision measurements and quantum sensing. Usually, the temperature measurement of cold atoms is instantiated by the method of time-of-flight (TOF) absorption imaging. It often begins with a sudden switching off of the trap potential or cooling laser such that the cold atoms subsequently expand freely under the influence of gravity. In order to make the intracavity optical field close to fully isotropic, the diffuse reflection coating material covers almost the entire cavity surface in ILC. Therefore it is difficult to incorporate a clear aperture to let all the atoms fall freely or a transparent window to image the atoms via fluorescent light, especially for a system tailored for field applications of quantum sensing. Nevertheless, we use a recently developed new method [27] to evaluate temperature of the cold atom ensemble, whose essence is to monitor the diffusion of cold atoms after labeling them through manipulating their internal states.
The basic mechanism of Zeeman sub-states and laser polarizations for the nearly nondestructive detection method is shown in Fig. 1(c). With the labeling process enforced by polarizing laser and repumping laser B, the cold atoms along the optical detection axis are optically pumped to $|F=2, m_F=2\rangle$ while the rest of the intracavity cold atoms stay in $F=1$. A right circularly polarized probe laser will then drive the cycling transition from $5S_{1/2}|F=2, m_F=2 \rangle$ to $5P_{3/2}|F=3, m_F=3\rangle$ of the cold atoms. Such type of detection process won’t destroy the prescribed labeling status and the internal atomic state remains nearly nondestructive. According to its basic principles of operations as discussed in Ref. [27], we develop a slightly upgraded form here for the purpose of better accuracy. More specifically, we use two short continuous probe pulses separated by milliseconds in the detection process. Under the geometry of this experiment, dynamics in the vertical dimension along the gravity direction does not have a significant impact on the signal, and therefore it suffices to consider the process in the transverse plane. Then it reduces to a 2D problem, and the velocity distribution is adequately described by the Maxwell-Boltzmann distribution $f(v)dv=(\frac {m}{2\pi k_{B}T})2\pi v$exp$(-\frac {mv^{2}}{2k_{B}T})dv$. Ultimately, the reduction in transmission signal of the probe pulses is proportional to:
We have implemented this method and constructed a quantitative analysis to interpret the data, as demonstrated in Fig. 3. The main purpose here is to examine the temperatures of cold atoms with respect to different cooling laser polarizations and repumping laser powers. For a very long time, it remains elusive whether the polarization of cooling laser injection and the power of repumping laser have a significant influence on the ultimate cold atom temperature in ILC. And this fact is further complicated by the sub-Doppler cooling properties of ILC. The purpose here is to experimentally examine this issue. According to fitting with 1, the temperatures of all these five cases shown in Fig. 3 are uniformly at about 19.4 $\mu$K with differences within 0.5 $\mu$K across the board, evaluated in the 2D considerations.
Fig. 3. Results of temperature measurement by dual-pulse detection method, in the nearly nondestructive form. The powers of the cooling laser, pumping laser, repumping laser B and polarizing laser are 120 mW, 74 $\mu$W, 60 $\mu$W, 1.2 $\mu$W, respectively. The cooling time is set as 602 ms for all these experiments. (a) and (b) show the effects of different polarizations with the injection HLB cooling laser, where (a) corresponds to linear polarization and (b) corresponds to circular polarization. Under the conditions of different repumping laser powers, the experimental data of temperature measurement are shown in (c), (d) and (e), with powers set at 6 mW, 16 mW, and 26 mW, respectively.
This temperature is clearly below the Doppler temperature $T_{D} \approx 145\mu K$ of $^{87}$Rb. These results confirm that in our system the polarization of injection laser does not have a significant impact on the sub-Doppler cooling process of ILC, which indicates that the polarization of intracavity optical field is close to ideally randomized, thanks to the diffuse reflective material coated on the outer surface. It also implies that the sub-Doppler cooling mechanism of ILC is closely connected with randomized polarizations of the diffuse cooling lasers, such that the concept of isotropic does extend to the polarization degrees of freedom in ILC. On the other hand, these results also confirm that as long as the power of repumping laser A is not too high, it does not have an essential effect on the ultimate temperature of cold atoms in ILC.
Next, we have measured the spatial distribution of the intracavity cold atoms by methods similar to that of Ref. [19]. More specifically, by a pair of anti-Helmholtz coils, we generate a magnetic gradient field to shift cold atoms out of resonance with the probe laser except for the point of interest. In this way, the absorption of probe laser reveals information about the atom density around that prescribed point. Then the spatial distribution can be measured by scanning the position of the on-resonance point and the result is shown in Fig. 4. It shows that the cold atom density is relatively higher in the middle, which indicates that the effectiveness of HLB injection in accumulating cold atoms of ILC around the optical detection axis. For applications in quantum sensing, this feature is helpful in enhancing SNR and reducing measurement time. Moreover, it will be interesting to try changing the injection method from HLB to vortex beams carrying non-zero orbital angular momentum [28–30] in the future.
Fig. 4. Measurement of spatial distribution of cold atoms. (a) Schematic of the experimental configuration to measure radial density of cold atoms. (b) Data of cold atom profile along the radial direction of the cylindrical cavity. The zero point of position corresponds to the middle of the cavity. For this measurement, the HLB with a diameter of 34 mm and a width of 1 mm is injected into the cavity to produce diffuse reflection light field.
As part of our efforts to study the influence of parameters on cooling properties, we investigate the relation between the number of accumulated cold atoms and the detuning of cooling laser. The experimental result is shown in Fig. 5, which indicates that the maximum cold atom number takes place at the red detuning of cooling laser around 20 MHz, about 3.3 $\Gamma$. We can perform a $\chi ^{2}$ fitting between the theory and data points to extract information of essential properties in the cooling process, where the theory values are computed according to numerical simulation of the underlying physical process. The result of our $\chi ^{2}$ fitting finds that $s_{0}=1.2$ with an effective detuning shift term $\delta =3.9$ MHz to account for the effects caused by repumping laser, multi-level structure, residual magnetic field and so on.
Fig. 5. Experimental data of normalized cold atom number versus the detuning of cooling laser $\delta$. The atom numbers are normalized with respect to the highest data value in the figure, which corresponds to the averaged cold atom number density of $1.04 \times 10^{8}$ cm$^{-3}$ across the cavity with length 5.4 cm. The cooling laser is set at 120 mW.
This type of measurement can yield further information about the essential properties of ILC. Generally, a direct measurement of the effective saturation parameter of intracavity diffuse cooling light does not seem easy for ILC. This is different from the scenario of six-beam optical molasses, where saturation parameter may be readily obtained from a straightforward measurement of cooling laser intensity. Nevertheless, we try an alternative approach to extract the information from the data of Fig. 5. And this is part of the motivation to carry out a numerical simulation for the relation between atom number and cooling laser detuning. As a starting point, according to a recent theoretical analysis in Ref. [31], the mean force of ILC for a two-level atom model may be expressed as:
The capturing process can be understood as one hot atom is laser cooled to near zero velocity within the finite range determined by the actual physical system. Based on the force $F(v)$ given by the effective two-level atom model as of 2, if we postulate a value for $s_{0}$, we may compute the capture velocity as a function of the cooling laser detuning according to the geometry of the cavity. Then according to the Maxwell-Boltzmann distribution of $^{87}$Rb atoms at room temperature, the atom number may be readily obtained via the capture velocity. With there preparations, we may enforce a $\chi ^{2}$ fitting process between the simulation results and experimental data, in order to retrieve an optimal value of $s_{0}$ and relevant two-level atom model parameters. For our case, $\chi ^{2}$ is a reasonable choice to evaluate the deviation between the theory and experiment points. The outcome of $\chi ^{2}$ fitting is also shown in Fig. 5.
Furthermore, we also measure the relation between the number of accumulated cold atoms and the power of cooling laser, with the cooling time consistently set at 602 ms. The experimental result is presented in Fig. 6, which clearly demonstrates that the linear relation holds well when the cooling laser power is at a relatively low level. In general, if the cooling laser power is not too strong, we have the relation $s_{0}= \alpha _s \times I_{cooling}$ where $\alpha _s$ and $I_{cooling}$ denote the proportional constant and the cooling laser power respectively. Again, by the force $F(v)$ given by 2, we can compute the capture velocity as a function of the saturation parameter $s_0$, and then convert the capture velocity into accumulated cold atom numbers according to the Maxwell-Boltzmann distribution of $^{87}$Rb atoms. Through this type of analysis similar to that of Fig. 5, we perform a $\chi ^{2}$ fitting process as shown in Fig. 6 to obtain that $\alpha _s=11.8$ W$^{-1}$. In addition, the fitting result of Fig. 6 indicates that $s_{0}=1$ when the power of cooling laser is 85 mW, which is within a reasonable range with the fitting result of the cold atom number versus detuning as in Fig. 5. In fact, when the cooling laser power is at 120 mW as in Fig. 5, the saturation behavior takes place and the relation with the accumulated cold atom number deviates from the linear case.
Fig. 6. Relation between the normalized cold atom number and the power of cooling laser. The atom numbers are normalized with respect to the highest data value in the figure, which corresponds to the averaged cold atom number density of $5.8 \times 10^{7}$ cm$^{-3}$ across the cavity with length 5.4 cm. The cooling time is set at 602 ms and the detuning of cooling laser is set at 22 MHz. A theory fitting curve is also included in the figure, which is based numerical simulations of an effective two-level atom model, similar to that of Fig. 5.
Along the way of analyzing Fig. 5 and Fig. 6, we also observe that the injection method of HLB works particularly well. While the nature of ILC force remains the same, the changes caused by different cooling laser injection methods can be significant for the cooling process and outcome. From a geometry point of view, HLB is almost equivalent to increasing the number of individually injected narrow cooling laser beams. According to the measurement of cold atoms properties, It helps to provide a more uniform diffuse reflection light field to generate ILC. In fact, compared with many of the previous configurations for ILC, the injection method of HLB has a clear advantage in accumulating more cold atoms in the central region along the optical detection axis. From the quantitatively obtained values of $s_0$ and $\alpha _s$, the effective strength of intracavity cooling laser field is sufficient for our system, indicating that the power efficiency of HLB is reasonably well.
4. Conclusion
In conclusion, we have designed and implemented a special physical form of ILC for future applications in quantum sensing. The intracavity cooling laser field is realized by the diffuse reflection light with HLB injection, and the SNR of optical detection is carefully improved. Moreover, we have systematically characterized the properties of ILC and examined the influence of essential parameters. From our experiment and simulation, the underlying physical process of ILC in our system is thoroughly described and understood. In particular, we have quantitatively deduced the strength of intracavity cooling laser field. We hope that our efforts will promote the applications of ILC in quantum sensing and quantum precision measurements.
Funding
National Natural Science Foundation of China (12074391, 92165107); National Key Research and Development Program of China (2016YFA0301504); Chinese Academy of Sciences; China Manned Space Engineering Office.
Acknowledgments
We thank Xiao Zhang, Wenli Wang, Yanling Meng and Jinyin Wan for technical assistance.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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