Abstract
We report the observation of synthesized spin-orbit coupling (SOC) for ultracold spin-1 87Rb atoms. Different from earlier experiments where a one dimensional (1D) atomic SOC of pseudo-spin-1/2 is synthesized with Raman laser fields, the scheme we demonstrate employs a gradient magnetic field (GMF) and ground-state atoms, thus is immune to atomic spontaneous emission. The strength of SOC we realize can be tuned by changing the modulation amplitude of the GMF and the effect of the SOC is confirmed through the studies of: 1) the collective dipole oscillation of an atomic condensate in a harmonic trap after the synthesized SOC is abruptly turned on; and 2) the minimum energy state at a finite adiabatically adjusted momentum when SOC strength is slowly ramped up. The condensate coherence is found to remain very good after driven by modulating GMFs. Our scheme presents an alternative means for studying interacting many-body systems with synthesized SOC.
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Introduction
Spin-orbit coupling (SOC), as is often referred to in condensed matter physics, couples the spin of a particle to its orbital degrees of freedom. It is believed that SOC constitutes an important ingredient for quantum simulation with ultracold atoms1,2,3,4. Research on SOC is an active area due to its ubiquitous appearance in condensed matter phenomena, such as topological insulator5,6, spin Hall effect7,8 and spintronics9. In contrast to solid-state materials, where SOC originates from the orbital motion of electrons inside a crystal’s intrinsic electric field, the coupling between atomic spin and its center of mass motion has to be engineered artificially. In recent years we have witnessed great successes of artificial atomic gauge fields10,11,12,13,14,15,16,17,18,19. A popular scheme employs Raman laser fields11 to couple two atomic ground states forming a pseudo-spin-1/2 system, leading to a SOC with equal Rashba20 and Dresselhaus21 contributions. This is routinely used nowadays for both bosonic11,14 and fermionic15,16 alkali atom species. More general forms of SOC are pursued actively in a variety of settings22,23,24,25, which together with the above well understood Raman scheme26,27,28,29,30, Our experiments synthesize an effective 1D SOC by applying a time-periodically modulating 1D GMF with zero average48,49, as illustrated in Fig. 1. The GMF provides a spin-dependent force for a spin F atom (with mass m). Here, μB is the Bohr magneton, gF is the Lande g-factor and Fx,y,z denotes the x-, y- and z-component of spin vector F. Although the net impulse over one period is zero, the accumulated distance an atom moves depends on its spin state, which implies the atom acquires a spin dependent group velocity. Thus atomic spin and its center-of-mass motion (orbital) is coupled by GMF pulses. The origin for the synthesized 1D SOC can be understood more straightforwardly when we consider the extreme case where each period contains a pair of two opposite GMF pulses of impulse ±kso cap** the ends of free evolution over time T. These two pulses enact a unitary transformation, which displaces the canonical momentum by a spin-dependent quantity, i.e., where , as shown in Fig. 1(b). Hence, the effective Hamiltonian results from a spin dependent momentum shift to the free particle Hamiltonian . For the sinusoidal modulating GMF used in our experiments, the atomic dynamics is governed by . Based on the Floquet theory, the effective time-independent Hamiltonian Heff is given by (more details in supplementary material) where , c1 = 1/2 and c2 = 3/8. q is the quadratic Zeeman shift (QZS) of the bias field used for selecting the 1D GMF from a 3D quadrupole magnetic field (see supplementary material). The c1 and c2 terms on the rhs of Eq. (1) describe the synthesized SOC and the effective QZS, respectively. The latter together with q can be further tuned to zero or negative by an off-resonant dressing microwave field56. The above physical picture remains approximately valid in the presence of an external harmonic trap if the modulation frequency ωmod = 2π/T is far greater than the trap frequency. Our experiments are performed in a single chamber BEC setup as described elsewhere57. We create a 87Rb BEC of 1.2 × 105 atoms in state in a crossed dipole trap with trap** frequencies , as illustrated in Fig. 1(c). The 1D GMF is implemented by a combination of a 3D quadrupole magnetic field and a 5.7 Gauss bias field , whose linear and quadratic Zeeman shifts are (2π) 4 MHz and (2π) 2.34 kHz, respectively (see supplementary material). The modulation frequency ωmod for the GMF is (2π) 1.0 kHz unless stated otherwise. To generate a SOC strength kso = 1 kL = 8.0554 μm−1, where kL = 2π/λL is the one photon recoil momentum at a laser wavelength λL = 780 nm, the required peak magnetic field gradient . The centers of the gradient field coils and the optical trap are aligned within 50 μm to minimize short-term magnetic field fluctuation during modulation (see supplementary material). To confirm the effect of the synthesized SOC from the modulating GMF, we first excite the collective dipole mode of a single spin component atomic condensate in a harmonic trap by abruptly turning on the modulating GMF. By rewriting the effective Hamiltonian (1) as and interchanging the roles of the conjugate variable pair kz and z, it’s easy to find that this effective Hamiltonian is equivalent to that of a particle moving in a displaced harmonic trap. Here, we neglected the extra QZS which only causes an overall energy shift. A particle displaced from the center of a harmonic trap will oscillate back and forth periodically, which is indeed what we observe. Both the position and momentum of the condensate oscillate at the trap frequency ωz. Solving the Heisenberg equations of motion given by Heff, we obtain the averaged momentum , where for the component. The oscillation is around c1ksomF with a peak to peak amplitude . As shown in Fig. 2(a), in our experiments we abruptly turn on the modulating GMF, at a corresponding SOC strength of kso = 7.5 μm−1 and persist for variable hold time. After integer multiple periods of the modulation, the crossed dipole trap is turned off in less than 10 μs. Condensed atoms are expanded for about 24 ms, during which different Zeeman components are Stern-Gerlach separated by an inhomogeneous magnetic field along x-direction. For all three spin components, atomic center-of-mass momenta are derived from their shifted positions along z-direction with respect to their locations when SOC is absent. As shown in Fig. 2(c), the observed results are in good agreement with our theoretical predictions. As a second confirmation, we observe that the atom’s minimum energy state is adiabatically adjusted to a finite none-zero momentum kz = c1ksomF for the state when the modulation amplitude is slowly ramped up as shown in Fig. 3(a). In the presence of SOC, the minimum of the single-particle dispersion for the spin state is located at c1ksomF. According to the adiabatic theorem, if the ramp of kso is slow enough, atoms will follow the ramp and stay at the shifted minimum. In this set of experiments, atoms are prepared at the initial spin state of by applying a π/2 pulse to the state (0, 0, 1)T. The modulation amplitude is then ramped up to a final value within 200 ms. After turning off the optical trap, we measure the momentum for each spin component. We find good agreement with theoretical predictions as shown in Fig. 3(c). To confirm the adiabaticity, kso is ramped up from 0 to 4.9 μm−1 in 100 ms and then back to 0 in another 100 ms. We find atomic center-of-mass momentum returns to 0 without noticeable heating. We also check the dependence of c1 on the modulation frequency ωmod and find that c1 essentially remains a constant as long as . The results of Fig. 3 demonstrate tunability of the SOC we synthesize. For our protocol based on temporally modulating GMFs, a point of detrimental importance concerns heating which causes relaxation and atom loss from condensates. As mentioned before, one of the major heating mechanisms for the Raman scheme is photon scattering from the Raman laser. For different alkaline metal atoms, the situation varies significantly. A list of all alkaline metal atom data are shown in Table 1. For cesium and rubidium, the heating rate is very low, the Raman scheme works very well. For potassium, the heating will cause significant problem at the temperature scale of many quantum many body phenomenon. For sodium and lithium, the huge heating rate will quickly destroy the BEC or degenerate Fermi gas. As atomic spontaneous emission is absent in our experiments, the most likely heating mechanism comes from parametric processes associated with the temporal modulation. To minimize parametric heating, we modulate the GMF far away from the characteristic frequencies of our system. The typical trap** frequency is about (2π) 100 Hz and the mean field interaction energy is around (2π) 200 Hz. When modulated at ωmod = (2π) 1.0 kHz, heating is found to be moderate and acceptable for the reported experiments, based on the observed condensate life time. To measure the life time, we adiabatically increase the SOC strength to prepare atoms into an equilibrium state from a initial spin state and measure the fractions of remaining atoms as functions of time. The worst case occurs for condensates in the component, whose life time is found to be around 310 ms for kso = 4.9 μm−1 corresponding to a recoil energy of 1.3 kHz, as shown in Fig. 4(a). The life time of our system is comparable to the reported values for the Raman scheme with Rb atoms11. It can be further improved by increasing the modulation frequency. Figure 4(b) displays the dependence of the atomic cloud size after 24 ms of TOF expansion on modulation frequency at a fixed kso = 4.9 μm−1. The cloud radius is found to decrease with increasing modulation frequency, which confirms the expected heating suppression with increased modulation frequency. Thus enhanced performances of the GMF scheme is expected if our experiments can be carried out with atomic chip based setups, which routinely provide higher GMFs and faster modulations58,59. As demonstrated in our experiments, the SOC synthesized from GMF enacts spin-dependent momentum shifts to the single-particle dispersion curves, leading to curve crossings between different spin states. Inspired by the idea of ref. 50, we find that these crossings can be tuned into avoided crossings when spin flip mechanism is introduced as elaborated in more detail in the supplementary material. In conclusion, we experimentally demonstrate a tunable SOC synthesized by a modulating GMF for a spin-1 87Rb BEC. We tune the SOC strength by changing the momentum impulse from the GMF. The observed coherence time is reasonably long compared to the Raman scheme for rubidium and expected to be much better than the Raman scheme for sodium and lithium, pointing to promising experimental opportunities. The scheme we realized relies on spin-dependent Zeeman interactions, thus is naturally extendable to high-spin atomic states, like the spin-1 case we demonstrate here. It adds to the recent report of spin-1 SOC. A condensate in state with 1.2 × 105 atoms is produced every 40 seconds. To minimize heating from the near-resonant driving of the modulating GMF, we further ramp down ωz to 2π × 31.8 Hz in 500 ms after BEC production, which reduces ωx and ωy respectively to 2π × 74.6 Hz and 2π × 67.5 Hz. Three pairs of Helmholtz coils are used to control the homogeneous bias magnetic field. While transferring atoms from the hybrid trap into the crossed dipole trap, a 0.7 Gauss bias field along z-direction is simultaneously turned on in order to maintain atoms in state. In the last 1.5 seconds of evaporative cooling, we ramp up the bias field to 5.7 Gauss and hold on to this value. The Larmor frequency of the bias field is calibrated by RF driven Rabi oscillations between Zeeman sublevels. The residual magnetic field gradient is compensated to below 2 mGauss/cm by a pair of anti-Helmholtz coils along z-direction. A pair of small anti-Helmholtz coils is used to modulate the GMF with a modulation amplitude up to 100 Gauss/cm at a frequency of (2π) 1.0 kHz. The radius for the gradient coils is 15 mm. The two coils are separated at an inner distance of 36 mm. Each coil consists of 12 turns of winding and produces an inductance of about 10 μH. The gradient coil size is much larger than the 120 μm beam waist of our dipole trap, which produces a homogeneous gradient field inside the crossed dipole trap. The coils are small enough to ensure fast and strong GMF modulation. The current for the gradient coils is regulated by a home made fast (10 μs rise time) and precise (100 ppm) linear bipolar current controller, whose phase compensation is carefully analysed and tuned for stable running with the inductive load. The gradient coil is mounted on a 3D low magnetic translation stage for precise alignment of the gradient coils. The center of the gradient field is aligned within 50 μm with the BEC, which is found to be crucial for minimizing short term bias field fluctuations during GMF modulation. How to cite this article: Luo, X. et al. Tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field. Sci. Rep. 6, 18983; doi: 10.1038/srep18983 (2016).Results
Dipole oscillations
SOC shifted minimum energy state
Discussion
Magnetic field control
Additional Information
References
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Acknowledgements
This work is supported by MOST 2013CB922002 and 2013CB922004 of the National Key Basic Research Program of China and by NSFC (No. 91121005, No. 11374176, No. 11404184, No. 11474347 and No. 11574100).
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X.L. and Q.G. performed the experiment. X.L. and L.W. analysed the data. X.L., K.G., Q.G. and R.W. built the experimental setup. L.W., Z.-F.X., J.C. and L.Y. developed the theory. L.W. performed the numerical calculation. X.L., L.W., Z.-F.X., R.W. and L.Y. wrote the manuscript. L.Y. and R.W. supervised the whole research project.
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Luo, X., Wu, L., Chen, J. et al. Tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field. Sci Rep 6, 18983 (2016). https://doi.org/10.1038/srep18983
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