Motional Ground State Cooling Outside the Lamb-Dicke Regime
We report Raman sideband cooling of a single sodium atom to its three-dimensional motional ground state in an optical tweezer. Despite a large Lamb-Dicke parameter, high initial temperature, and large differential light shifts between the excited state and the ground state, we achieve a ground state population of % after ms of cooling. Our technique includes addressing high-order sidebands, where several motional quanta are removed by a single laser pulse, and fast modulation of the optical tweezer intensity. We demonstrate that Raman sideband cooling to the 3D motional ground state is possible, even without tight confinement and low initial temperature.
Trapped neutral atoms, assembled in an array of optical tweezers, are a promising platform to study quantum information and quantum simulations Schlosser et al. (2001); Weiss et al. (2004); Isenhower et al. (2010); Wilk et al. (2010); Kaufman et al. (2015); Labuhn et al. (2015); Murmann et al. (2015). The inherent single-particle detection and control, combined with tunable interactions, allow implementations of neutral-atom-based quantum logic gates Isenhower et al. (2010); Wilk et al. (2010), novel quantum phases Labuhn et al. (2015), and single-photon switches Dayan et al. (2008); Tiecke et al. (2014). Advances in real-time re-arrangement of optical tweezers enable rapid preparation of atoms in large and complex geometries Barredo et al. (2016); Endres et al. (2016). Quantum motional control of individual atoms Li et al. (2012); Kaufman et al. (2012); Thompson et al. (2013a); Liu et al. (2017); Robens et al. (2017) enable studies of the atomic Hong-Ou-Mandel effect Kaufman et al. (2014), high-fidelity single qubit gates Wang et al. (2016), and efficient coupling of single atoms to photonic crystal cavities Thompson et al. (2013b).
Extending optical tweezer arrays to include polar molecules will open a range of new applications that exploit long-lived molecular internal states and tunable long range interactions DeMille (2002); Gorshkov et al. (2011); Yan et al. (2013). Molecules could be assembled from atom pairs in optical tweezers Liu et al. (2017), or directly loaded from magneto-optical traps (MOTs) Barry et al. (2014); Truppe et al. (2017); Anderegg et al. (2017). For either approach, preparing the constituent atoms or molecules in the lowest motional quantum state is important to realize long coherence times in quantum applications.
Preparation of single atoms in the motional ground state has been achieved in optical tweezers Kaufman et al. (2012); Thompson et al. (2013a); Liu et al. (2017); Robens et al. (2017) using Raman sideband cooling (RSC) Monroe et al. (1995); Kerman et al. (2000); Han et al. (2000). However, RSC in these systems takes place in the Lamb-Dicke (LD) regime, where the spread of the initial wavefunctions is smaller than the reduced wavelength of light () used to address them. Systems with a large position spread (), either due to small mass or high initial temperature, fall outside of the LD regime and undergo strong recoil heating during RSC. In this letter we demonstrate cooling of single sodium atoms in optical tweezers, initially outside the LD regime (), to the motional ground state. We achieve a 3D ground state population of % by cooling via high-order sidebands in a carefully optimized cooling sequence. Our approach is general and applicable to other systems. Extending RSC beyond the LD regime opens the possibility for ground state cooling of systems such as light atoms and molecules that can be laser-cooled.
Our experiment has an overall repetition rate of Hz and begins by loading a single sodium atom into an optical tweezer from a MOT Hutzler et al. (2017). The tweezer is created by focussing a nm wavelength laser beam through an NA= objective to an elliptical waist with radii m. For mW of optical power, the trapping frequencies along the three axes are , where is the weakly confined axial direction, and and are the tightly confined radial directions. After each single atom loading attempt, a first image is taken with a ms exposure time to determine success, which occurs about 50% of the time due to the loading mechanism Schlosser et al. (2001). During imaging, the atom is cooled via polarization gradient cooling (PGC), which reduces the temperature of the single atom to K, corresponding to mean motional states along the three trap axes of , and therefore initial 3D ground state probability of %. The atom is then initialized into the stretched state via optical pumping (OP). Each experimental cycle ends with a second imaging sequence that measures the hyperfine state of the atom as or 111The final hyperfine state sensitive detection employs a strong beam that is resonant with the to cycling transition, to heat the population out of the trap. Therefore, any atoms which survive this heat out procedure are interpreted as having been in the state.. The data analysis includes only those experiments where the first image reveals successful atom loading. It should be noted that 4 % of atoms are lost from one image to the next due to recoil heating.
To reduce the temperature of the single atom and achieve high ground state population, we apply Raman sideband cooling Monroe et al. (1995); Kaufman et al. (2012). The energy levels, cooling sequence, and Raman beam geometries are shown in Fig. 1. RSC consists of two steps: driving a coherent Raman transition to remove motional quanta, followed by resetting the atom’s internal state via OP. These steps are repeated, until the motional ground state is populated with high probability.
Specifically, Raman transitions are driven in Na between the hyperfine states and in the presence of an G magnetic field. The external field is orthogonal to the effective magnetic field of the tweezer, to reduce vector light shifts Kaufman et al. (2012); Thompson et al. (2013a). Subsequently, an OP pulse, which consists of two frequencies both with -polarization, is applied to bring the atom out of the manifold and back to via spontaneous emission (Fig. 1A). It is important that the optical pumping step scatter as few photons as possible to minimize recoil heating. Therefore, to optically pump atoms from into while keeping the target state dark, we apply light resonant with Monroe et al. (1995); Gröbner et al. (2017) 222We find a reduction in the scattering rate by a factor of , as compared to using an OP resonant with , from which the state could always scatter a photon via the excited state..
During optical pumping, photon recoil can lead to significant heating, especially along the weakly-confined axial direction (). For an atom starting in an axial motional level , the probability to end in another axial state after absorbing and re-emitting one photon is approximated by an angular integral of the squared matrix element Wineland and Itano (1979). Here is the wave vector difference between absorbed and emitted photons, projected onto the -axis. As shown in Fig. 2A, the probability of gaining motional energy during OP grows with , and is an important effect for sideband-cooling outside the Lamb-Dicke regime.
Fortunately, the large LD parameters in our system also provide opportunities to overcome OP heating. The LD parameter for Raman transitions is defined as , where is the wave vector difference between the two beams that drive the Raman transition. Here is the ground state wavefunction spread ( is the atomic mass). The Raman transitions in our configuration have LD parameters of . To offset heating from OP initially when is large, higher-order Raman sidebands () remove several motional quanta in a single cooling pulse (see Fig. 2B). Because the coupling strengths of different orders do not reach minima for the same state of motion , using multiple orders of sidebands avoids accumulations of population near the coupling minima.
Taking the large motional-state changing heating and cooling sources into account, it is not immediately clear that ground-state cooling can be achieved. We therefore use a Monte-Carlo simulation to guide our search Dalibard et al. (1992). In particular, we find that alternating the cooling pulses (Fig. 1C) between two neighboring orders for the axial direction and and for the radial directions eliminates the accumulation of population in motional states that have zero Raman coupling, which would halt the cooling process. The simulation also indicates that setting the coupling strength of each sideband to drive a Rabi -pulse corresponding to the maximum matrix element motional state (i.e. the maxima in Fig. 2B) yields efficient cooling, initially. The efficiency of cooling on higher-order sidebands diminishes as the atom approaches the ground state, so the final cycles utilize only the sideband while alternating between the three axes.
Guided by the simulation results, we construct our axial cooling sequence by starting at the two highest observed sideband orders ( and ) and decreasing the orders to the next pair after every to cycles. This process is repeated until , and most of the population is in the first few excited states. We then switch to cooling only on with two different pulse lengths in order to efficiently cool atoms in the few remaining motional states. Radial cooling is performed similarly to axial cooling with the initial cooling of and and switching to only after to pulses. This sequence gives good initial cooling performance, which is then used to calibrate experimental parameters, including the Rabi rates of the Raman beams and optical pumping rates.
There are two additional important challenges in cooling single sodium atoms. First, the initial temperature populates high motional states, causing the atoms to sample the anharmonicity of the trap away from the center. Anharmonicity may be defined as for each trap axis , and calculated from the quartic term of the optical tweezers via perturbation theory. In the paraxial approximation, we find , where equals the beam radius for the radial directions and . Numerically, kHz. For thermal states, this broadens and shifts high-order sidebands due to the -dependence of the transitions. To mitigate this, we calibrate the frequency of each sideband order individually, and drive the sidebands with a large Rabi frequency and short pulses to Fourier broaden the spectrum.
Second, the tweezers cause a large light shift (as large as MHz) in the excited state. We address this by strobing the trapping light at a rate of 3 MHz during the whole cooling sequence, similar to our loading and imaging process Hutzler et al. (2017), while leaving OP on with constant intensity. Due to the large light shift, the OP is effectively off whenever the trap light is on. Since the atom is still addressed by optical pumping light when the trap light is off, OP remains effective.
Our final cooling results are shown in Fig. 3 and 4. In total, cooling pulses (total duration ms) are applied along three axes with cooling beginning on the radial second order and axial fifth order. The full RSC pulse sequence is given in 333See supplemental material. To characterize the single atom thermal state before and after cooling, we perform Raman sideband thermometry Monroe et al. (1995); Meekhof et al. (1996). For the more tightly confined radial directions, we observe clear , , and sidebands before RSC, as is shown in Fig. 3A. After RSC, the and sidebands on both radial axes are strongly reduced. The formula that equates the ratio of and sideband heights to ( is the mean motional quantum number of the assumed thermal state) was derived in the LD regime Monroe et al. (1995). However, it is also valid outside the LD regime. The resulting and correspond to ground-state fractions of 98.1(5)% and 97.6(3)%, in agreement with fitted values of 98(1)% and 95(3)% from the Rabi flopping curves Meekhof et al. (1996) in Fig. 3B and 3C. The initial temperature of K before RSC is obtained from similar fits.
For the weak axial direction, cooling is challenging because the atom starts outside the LD regime. We observe up to 5th-order Raman cooling sidebands initially, which indicates population in highly-excited motional states. Nevertheless, our cooling sequence works efficiently as all the sidebands are reduced after RSC (Fig. 4A). Using the ratio of first-order sideband heights, we obtain , which corresponds to a ground state population of 97.6(5)%, in agreement with a ground state population of 95(4)% extracted from Rabi flopping when (Fig. 4B). For the sideband (Fig. 4C), we observe additional decoherence that is more pronounced due to the slower Rabi frequency. The decoherence rate is consistent with magnetic field fluctuations of mG measured independently in the lab, which would produce a Zeeman shift of kHz.
Combining the axial and radial cooling results, a single Na atom is in the 3D ground state with a probability of 93.5(7)% after RSC. The cooling efficiency is limited by spontaneous scattering rate (0.1-0.2 kHz) from the Raman beams, as well as spectral broadening from magnetic field fluctuations.
We measure a heating rate that corresponds to decreasing 3D ground state population at a rate of %/ms. The rate is consistent with off-resonant scattering of the trapping light Grimm et al. (2000), and is predominantly in the axial direction where the trapping beam propagates.
Monte-Carlo simulations show that the ground state probability after RSC could be enhanced by increasing the detuning of the Raman beams and implementing better control of the magnetic field. Another improvement could come from grey molasses cooling, to achieve a lower starting temperature before RSC Colzi et al. (2016).
We have shown that despite the difficulty in achieving a low optical cooling temperature of low mass sodium atoms, three dimensional cooling with significant ground state population can be achieved by using high-order Raman sidebands in an optimized cooling sequence. These techniques are well-suited for a large variety of systems and open up a route to ground state cooling for other species, including molecules and exotic atoms.
This work is supported by the Arnold and Mabel Beckman Foundation, the AFOSR Young Investigator Program, the NSF through the CUA, and the Alfred P. Sloan Foundation.
- Schlosser et al. (2001) N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, Nature 411, 1024 (2001).
- Weiss et al. (2004) D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren, U. Vazirani, and K. B. Whaley, Phys. Rev. A 70, 040302 (2004).
- Isenhower et al. (2010) L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, Phys. Rev. Lett. 104, 010503 (2010).
- Wilk et al. (2010) T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, Phys. Rev. Lett. 104, 010502 (2010).
- Kaufman et al. (2015) A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, and C. A. Regal, Nature 527, 208 (2015).
- Labuhn et al. (2015) H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Nature 534, 667 (2015).
- Murmann et al. (2015) S. Murmann, A. Bergschneider, V. M. Klinkhamer, G. Zürn, T. Lompe, and S. Jochim, Phys. Rev. Lett. 114, 080402 (2015).
- Dayan et al. (2008) B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, Science 319, 1062 (2008).
- Tiecke et al. (2014) T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, Nature 508, 241 (2014).
- Barredo et al. (2016) D. Barredo, S. de Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys, Science 354, 1021 (2016).
- Endres et al. (2016) M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic, M. Greiner, and M. D. Lukin, Science 354, 1024 (2016).
- Li et al. (2012) X. Li, T. A. Corcovilos, Y. Wang, and D. S. Weiss, Phys. Rev. Lett. 108, 103001 (2012).
- Kaufman et al. (2012) A. M. Kaufman, B. J. Lester, and C. A. Regal, Phys. Rev. X 2, 041014 (2012).
- Thompson et al. (2013a) J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, Phys. Rev. Lett. 110, 133001 (2013a).
- Liu et al. (2017) L. R. Liu, J. T. Zhang, Y. Y. Yu, N. R. Hutzler, Y. Liu, T. Rosenband, and K.-K. Ni, arXiv 1701.03121 (2017).
- Robens et al. (2017) C. Robens, J. Zopes, W. Alt, S. Brakhane, D. Meschede, and A. Alberti, Phys. Rev. Lett. 118, 065302 (2017).
- Kaufman et al. (2014) A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard, A. M. Rey, and C. A. Regal, Science 345, 306 (2014).
- Wang et al. (2016) Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, Science 352, 1562 (2016).
- Thompson et al. (2013b) J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Science 340, 1202 (2013b).
- DeMille (2002) D. DeMille, Phys. Rev. Lett. 88, 067901 (2002).
- Gorshkov et al. (2011) A. V. Gorshkov, S. R. Manmana, G. Chen, E. Demler, M. D. Lukin, and A. M. Rey, Phys. Rev. A 84, 033619 (2011).
- Yan et al. (2013) B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature 501, 521 (2013).
- Barry et al. (2014) J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, Nature 512, 286 (2014).
- Truppe et al. (2017) S. Truppe, H. J. Williams, M. Hambach, L. Caldwell, N. J. Fitch, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt, (2017), arXiv:1703.00580 .
- Anderegg et al. (2017) L. Anderegg, B. Augenbraun, E. Chae, B. Hemmerling, N. R. Hutzler, A. Ravi, A. Collopy, J. Ye, W. Ketterle, and J. Doyle, (2017), arXiv:1705.10288 .
- Monroe et al. (1995) C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev. Lett. 75, 4011 (1995).
- Kerman et al. (2000) A. J. Kerman, V. Vuletić, C. Chin, and S. Chu, Phys. Rev. Lett. 84, 439 (2000).
- Han et al. (2000) D.-J. Han, S. Wolf, S. Oliver, C. McCormick, M. T. DePue, and D. S. Weiss, Phys. Rev. Lett. 85, 724 (2000).
- Hutzler et al. (2017) N. R. Hutzler, L. R. Liu, Y. Yu, and K.-K. Ni, New J. Phys. 19, 023007 (2017).
- (30) The final hyperfine state sensitive detection employs a strong beam that is resonant with the to cycling transition, to heat the population out of the trap. Therefore, any atoms which survive this heat out procedure are interpreted as having been in the state.
- Kasevich and Chu (1992) M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992).
- Gröbner et al. (2017) M. Gröbner, P. Weinmann, E. Kirilov, and H.-C. Nägerl, Phys. Rev. A 95, 033412 (2017).
- (33) We find a reduction in the scattering rate by a factor of , as compared to using an OP resonant with , from which the state could always scatter a photon via the excited state.
- Wineland and Itano (1979) D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979).
- Dalibard et al. (1992) J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992).
- Meekhof et al. (1996) D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).
- (37) See supplemental material.
- Grimm et al. (2000) R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000).
- Colzi et al. (2016) G. Colzi, G. Durastante, E. Fava, S. Serafini, G. Lamporesi, and G. Ferrari, Phys. Rev. A 93, 023421 (2016).