Abstract
Spin-orbit coupling (SOC) plays an essential role in many exotic and interesting phenomena in condensed matter physics. In neutral-atom-based quantum simulations, synthetic SOC constitutes a key enabling element. The strength of SOC realized so far is limited by various reasons or constraints. This work reports tunable SOC synthesized with a gradient magnetic field (GMF) for atoms in a harmonic trap. Nearly ten-fold enhancement is observed when the GMF is modulated near the harmonic-trap resonance in comparison with the free-space situation. A theory is developed that well explains the experimental results. Our work offers a clear physical insight into and analytical understanding of how to tune the strength of atomic SOC synthesized with GMF using harmonic trap resonance.
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Introduction
Resonance phenomena1 frequently occur in Nature. When the frequency of a time-periodic external drive matches a system’s resonance, the response is dramatic. A folklore wisdom warns that soldiers crossing a bridge should not march in unison to prevent its collapsing from accidentally stepping onto the resonance. Even at multiple resonant frequencies, such as parametric resonance when the driving frequency is twice the system’s characteristic frequency, the response can still be quite substantial. In quantum mechanics, resonances become ubiquitous as a result of quantization, where stationary states of a system feature definite eigenenergies. A transition between two eigenstates is resonantly enhanced when the frequency of an external coupling matches their energy difference2,3.
This article reports our experimental observation and theoretical vindication of an enhanced atomic spin-orbit coupling (SOC) synthesized with a modulated gradient magnetic field (GMF) applied to atoms in a harmonic trap. SOC, which couples a particle’s spin to its orbital motion, constitutes one of the most important interactions in condensed matter physics. In the strong coupling regime, SOC gives rise to nontrivial topological bands, which support many exotic states and phenomena, including topological band/Mott insulator, quantum-number fractionalization and magneto-electric effects4,5. In recent years, atomic quantum gases have emerged as powerful quantum simulators for condensed matter systems6,7. Strong atomic SOC often plays crucial roles in the increasing list of desired ingredients for artificial gauge fields8,9,10,11,12,13,14,15,16,17,18,19,20.
The breakthrough on synthetic SOC came in 201113, when Spielman’s group observed a special type of one-dimensional (1D) SOC: an equal-weighted sum of Rashba21 and Dresselhaus22 types of SOC created by the momentum-sensitive Raman coupling between two internal states of 87Rb atoms. Since then, the Raman scheme has become the prototype for studies involving 1D atomic SOC23,24,25,26,27. Recently, the observations of two-dimensional (2D) SOC relying on atom-photon interactions have been reported28,29,30. In the Raman scheme, the strength of synthesized SOC is limited by photon momentum transfer and constrained by Raman laser beam geometry. A protocol for tuning the strength of SOC including switching its sign through periodically modulating effective Rabi frequency31 in the Raman scheme has also been realized32.
An alternative method of creating synthetic atomic SOC is to use pulsed or time-periodic GMF33,34,35,36,37,38,39, which can be implemented free from atomic spontaneous emission. Its underlying mechanism is the Stern-Gerlach effect, whereby the periodic GMF imparts a spin-dependent momentum impulse to the atomic center-of-mass motion. This spin-dependent impulse can be described in terms of the same linear coupling between the spin (or pseudo-spin) with the atomic orbital (center-of-mass) motion as in the Raman scheme. Through concatenating GMF pulses along two orthogonal directions, genuine Rashba, Dresselhaus, or even arbitrary types of SOC in 2D can be synthesized for atoms with arbitrary hyperfine spins33,34,35. Very recently, essential features demonstrating 1D tunable SOC synthesized with a periodic modulated GMF have been reported38.
This work presents a different method for controlling the strength of atomic SOC synthesized with GMF by making use of harmonic trap resonance for atomic center-of-mass motion. It is beyond the straightforward scheme of tuning the strength of momentum impulse as demonstrated recently38. Furthermore, it differs from the reported tuning scheme32 based on the amplitude-modulated Raman coupling31, which can only decrease the strength of SOC. The harmonic trap resonance scheme we report here opens a different avenue for reaching the strong SOC regime.
Results
The experiment is inspired by the success of synthesizing atomic SOC from a time-periodic GMF34,38. Specifically, under a periodically modulated 1D GMF, atomic center-of-mass experiences a spin-dependent force, whose overall effect is simply to shift the momentum from px to for the atomic spin component mF, where denotes the minimum of the modified dispersion curve. The momentum of the equilibrium state is thus translated by a spin-dependent amount , which is equivalent to a synthetic SOC with strength . Experimentally, this effective strength of SOC is determined from the measured displacement of the atomic cloud.
A typical experiment starts with a 87Rb condensate of 1.2 × 105 atoms in the state confined inside a crossed dipole trap whose minimum potential region is approximately harmonic with frequencies (ωx, ωy, ωz) = 2π × (77, 136, 77) Hz along three orthogonal spatial directions x, y, and z. The 1D GMF is implemented by a combination of a 3D quadrupole magnetic field and a 5.7 Gauss bias field [Fig. 1(a)], whose linear and quadratic Zeeman shifts correspond to (2π)4 MHz and (2π)2.34 kHz, respectively. More details about the magnetic field control is as described in ref. 38. The amplitude for the GMF is sinusoidally modulated as B′(t) = , which translates into a 1D SOC strength , where gF denotes the Landé g-factor and μB the Bohr magneton, provided the modulation frequency ω is far away from trap resonance, as confirmed in a recent experiment38.
As shown in Fig. 1(b), the |mF = −1〉 condensate is loaded into the momentum-shifted equilibrium state by adiabatically ramping up the GMF modulation amplitude to a value corresponding to the SOC strength of kso = 1.25 μm−1 within 250 ms (or 25 modulation periods for ω < (2π)100 Hz), and then held on for another 50 ms (or 5 modulation periods for ω < (2π)100 Hz). At integer multiples of the modulation period τ = 2π/ω, the crossed dipole trap is turned off in less than 10 μs. Subsequently, the condensate expands for about 24 ms, during which different Zeeman components are Stern-Gerlach separated by an inhomogeneous magnetic field along the vertical direction. A bimodal fit to the atomic cloud density profile measured through the standard absorption imaging, as shown in Fig. 1(c), yields the shifted center-of-mass position for the condensate. The spatial displacement from that without SOC is used to derive the momentum shift , from which the scaled SOC strength ζ = /kso is computed.
A clear resonance behavior is observed for ζ, as shown in Fig. 1(c) and (d), for its dependence on the modulation frequency ω relative to the trap frequency ω0 = ωx = (2π)77 Hz. Above the trap frequency ω0, ζ increases with decreasing ω, from ζ = 1 for ω far above the resonance to a peak when ω approaches ω0. Below the trap frequency, ζ changes its sign, with its magnitude growing from ζ = 0 for ω far below the resonance to a peak around the resonance. The enhanced response on the opposite sides of the resonance is out of phase as a result of the π phase shift across a resonance. Limited by our present setup, we operate in the regime of small momentum impulse and observe nearly ten-fold enhancement for ζ at ω/ω0 = 1.03, where heating remains insignificant. The effect of heating-induced damping becomes noticeable in the immediate vicinity of resonance.
Discussion
The dramatic resonance enhancement of SOC due to the harmonic trap cannot be explained by the previous theory for atoms in free space34, which neglects the influence of the trapping potential on atomic motion. One might have naively concluded that an analogous calculation that incorporates the effect of the trap potential into the previously studied free-space model would find the agreement with the observed resonance. Unfortunately, this is easily said than done. To demonstrate it, we briefly recapitulate the basic idea of the previous theory34. For an atom of mass m in free space and in the presence of a sinusoidally modulated GMF along the x-direction, the effective 1D Hamiltonian is given by
where px is the momentum of an atom, Fx is the x-component of its spin F, and β(t) = ωsin(ωt) is the temporal profile for the coupling strength between the time-dependent GMF and atomic magnetic dipole moment measured in units of modulation amplitude ħkso. The Schrödinger equation for H0(t) of Eq. (1) can be more easily handled if we introduce a unitary transformation with , which corresponds to a momentum translation by the spin-dependent impulse from the GMF. The wave function in the rotating frame is then governed by the momentum shifted Hamiltonian , which commutes with itself at different times, . The corresponding time evolution operator takes a simple form . After one period of evolution τ = 2π/ω, we obtain Ax(τ) = 0, or R(τ) = 1. Hence, the wavefunctions in the two frames coincide, , and the effective Hamiltonian for the whole period is given by
where the first term describes the SOC of strength ħkso and the second term acts like a quadratic Zeeman shift.
In the presence of a 1D harmonic trap Vtrap = mω02x2/2, the Hamiltonian changes into
which in the rotating frame becomes
Unlike the case of a free atom discussed above, the two ’s in Eq. (4) at different times do not always commute due to the presence of Vtrap. The corresponding unitary evolution operator then takes a more general form , where denotes time ordering. This unitary evolution operator takes such a complicated form that it is difficult to derive the effective Hamiltonian in a straightforward manner. Therefore, we have to resort to other means for a compact solution capable of explaining the resonant behavior observed.
We note that Hamiltonian (3) also describes a sinusoidally driven harmonic oscillator, whose quantum-mechanical propagator can be obtained in the explicit analytic form. Hence, we can get the effective Hamiltonian of the system by making use of the propagator. For Hamiltonian (3), its propagator is given by (see Methods).
The effective Hamiltonian of the system should give the same propagator as Eq. (5). Without loss of generality, it is reasonable to infer that the effective Hamiltonian will not be much different from that in free space in Eq. (2). We therefore assume
where ζ and s denote modifications to the strength of SOC and the quadratic Zeeman shift, respectively, when the trap potential Vtrap is present. The corresponding propagator in this case is found to be (see Methods)
The equivalence between the two propagators of Eqs. (5) and (7) thus gives
and . In other words, the effective Hamiltonian for atoms in a harmonic trap driven by a sinusoidally modulated GMF is found to be given by Eq. (6), which is similar in form to the case of a free atom except for ζ and s in Eq. (8).
The factor ζ in Eq. (8) is plotted as the blue dotted curve in Fig. 1(d), which is found to agree well with the measured data except in the immediate vicinity of the trap resonance [shaded band region in Fig. 1(d)], where the finite lifetime of the condensate makes it difficult to reach equilibrium. It clearly shows that as the modulation frequency ω approaches the trap frequency ω0, the SOC becomes enhanced. More specifically, when approaching the resonance from above, the effective SOC is increasingly enhanced. Upon crossing the resonance ω0, the effective SOC reverses its sign, and the factor ζ gradually decreases and eventually tails off to zero at frequencies much smaller than ω0. This dependence of ζ on the modulation frequency highlights the tunability discussed in this work. In the immediate vicinity of the resonance, the amplitude of atomic micro-motion due to periodic modulation is so large that the Gaussian-shaped optical trap cannot be well approximated by a harmonic trap anymore, and as a result of the large amplitude oscillations the condensate collapses.
The observed enhancement of SOC is reminiscent of the resonance phenomenon in a driven harmonic oscillator. The factor ζ shown in Eq. (8) can be directly verified as well by comparing the equations of motion for the periodically-driven Hamiltonian (3) with the effective Hamiltonian form of (6) (see supplementary material). While the factor s cannot be derived in such a way. When the atomic spin is prepared into a superposition state, the seemingly classical driven equation of motion becomes spin dependent, a situation without a classical analog. For the 1D case considered here, the synthesized SOC can be gauged away by making the transformation with . However, when tracking the dynamics for the different atomic spin components, the accumulated phase from the SOC term, is real as confirmed in the recent experiment38. Thus, although the effective SOC we describe can be gauged away, the observation gives gauge dependent results which implicate the presence of synthetic SOC. Furthermore, in the presence of a uniform bias magnetic field, which gives rise to an interaction ∝ Fz, or when any other non-commuting interactions are present, the synthesized SOC discussed above persists and cannot be gauged away even in 1D system38,40.
In conclusion, for the atomic SOC synthesized from a time-periodic GMF, we observed a resonant behavior which highlights nearly ten-fold enhanced SOC when the modulation frequency is close to but higher than the trap frequency. This resonance is accompanied by a progression towards vanishing SOC on the lower modulation frequency side and a reduction to the value for a free atom in the higher modulation frequency side. We develop a theory that well explains the experimentally observed resonant behavior. Compared with atoms in free space under a sinusoidally modulated GMF, we find that an effective SOC Hamiltonian for atoms confined inside a harmonic trap takes an analogous form, except for a frequency-dependent prefactor. This prefactor reveals the resonant behaviour as the periodic drive hits the motional resonance of the harmonic trap.
Methods
Propagator of a forced 1D harmonic oscillator
The quantum-mechanical propagator K(x″, t″; x′, t′) describes the transition amplitude from one space-time position (x′, t′) to another (x″, t″). For a time-dependent driven harmonic oscillator described by Hamiltonian
the propagator is given by ref. 41
where
For the example considered in the main text, we have J(t) = −β(t)ħksoFx and , and therefore we find
Thus, the propagator for a sinusoidally driven harmonic oscillator is given by Eq. (5) in the main text.
Propagator for the momentum-shifted 1D harmonic oscillator
To obtain the propagator for a momentum-shifted harmonic oscillator described by the effective Hamiltonian (6) in the main text, we first derive the propagator Kt for the unitary transformed Hamiltonian
with . Based on the propagator for the harmonic oscillator,
which is easily reproduced if we put in Eq. (10) and making use of the properties for the propagators, we find that
Therefore, by making inverse unitary transformation, we obtain the propagator for the effective Hamiltonian (6) as
or Eq. (7) in the main text.
Additional Information
How to cite this article: Wu, L.-N. et al. Harmonic trap resonance enhanced synthetic atomic spin-orbit coupling. Sci. Rep. 7, 46756; doi: 10.1038/srep46756 (2017).
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References
Landau, L. D. & Lifshitz, E. M. Mechanics, vol. 1 of Course of Theoretical Physics 3 ed. (Butterworth-Heinemann, 1976).
Rabi, I. I., Zacharias, J. R., Millman, S. & Kusch, P. A New Method of Measuring Nuclear Magnetic Moment. Phys. Rev. 53, 318–318 (1938).
Foot, C. J. Atomic Physics (Oxford University Press, Oxford, 2004).
Pesin, D. & Balents, L. Mott physics and band topology in materials with strong spin–orbit interaction. Nat. Phys. 6, 376–381 (2010).
Witczak-Krempa, W., Chen, G., Kim, Y. B. & Balents, L. Correlated Quantum Phenomena in the Strong Spin-Orbit Regime. Annual Review of Condensed Matter Physics 5, 57–82 (2014).
Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).
Lewenstein, M. et al. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys. 56, 243–379 (2007).
Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).
Galitski, V. & Spielman, I. B. Spin-orbit coupling in quantum gases. Nature 494, 49–54 (2013).
Cooper, N. Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008).
Fetter, A. L. Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys. 81, 647–691 (2009).
Lin, Y. J., Compton, R. L., Jiménez-García, K., Porto, J. V. & Spielman, I. B. Synthetic magnetic fields for ultracold neutral atoms. Nature 462, 628–632 (2009).
Lin, Y. J., Jiménez-García, K. & Spielman, I. B. Spin-orbit-coupled Bose-Einstein condensates. Nature 471, 83–86 (2011).
Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices. Phys. Rev. Lett. 111, 185301 (2013).
Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices. Phys. Rev. Lett. 111, 185302 (2013).
Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).
Struck, J. et al. Quantum Simulation of Frustrated Classical Magnetism in Triangular Optical Lattices. Science 333, 996–999 (2011).
Struck, J. et al. Tunable Gauge Potential for Neutral and Spinless Particles in Driven Optical Lattices. Phys. Rev. Lett. 108, 225304 (2012).
Parker, C. V., Ha, L. C. & Chin, C. Direct observation of effective ferromagnetic domains of cold atoms in a shaken optical lattice. Nat. Phys. 9, 769–774 (2013).
Jotzu, G. et al. Experimental realisation of the topological Haldane model. Nature 515, 237 (2014).
Bychkov, Y. A. & Rashba, E. I. Oscillatory effects and the magnetic susceptibility of carriers in inversion layers. J. Phys. C 17, 6039–6045 (1984).
Dresselhaus, G. Spin-Orbit Coupling Effects in Zinc Blende Structures. Phys. Rev. 100, 580–586 (1955).
Zhang, J. Y. et al. Collective Dipole Oscillations of a Spin-Orbit Coupled Bose-Einstein Condensate. Phys. Rev. Lett. 109, 115301 (2012). Ji, S. C. et al. Experimental determination of the finitetemperature phase diagram of a spin–orbit coupled Bose gas. Nat. Phys. 10, 314–320 (2014).
Olson, A. J. et al. Tunable Landau-Zener transitions in a spin-orbit-coupled Bose-Einstein condensate. Phys. Rev. A 90, 013616 (2014).
Hamner, C., Zhang, Y., Khamehchi, M. A., Davis, M. J. & Engels, P. Spin-Orbit-Coupled Bose-Einstein Condensates in a One-Dimensional Optical Lattice. Phys. Rev. Lett. 114, 070401 (2015).
Cheuk, L. W. et al. Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas. Phys. Rev. Lett. 109, 095302 (2012).
Wang, P. et al. Spin-Orbit Coupled Degenerate Fermi Gases. Phys. Rev. Lett. 109, 095301 (2012). Fu, Z. et al. Production of Feshbach molecules induced by spin-orbit coupling in Fermi gases. Nat. Phys. 10, 110–115 (2014).
Huang, L. et al. Experimental realization of a two-dimensional synthetic spin-orbit coupling in ultracold Fermi gases. Nat. Phys. 115, 073002 (2015).
Meng, Z. et al. Experimental observation of topological band gap opening in ultracold Fermi gases with two-dimensional spin-orbit coupling. arXiv: 1511.08492 (2015).
Wu, Z. et al. Realization of Two-Dimensional Spin-orbit Coupling for Bose-Einstein Condensates. arXiv:1511.08170 (2015).
Zhang, Y., Chen, G. & Zhang, C. Tunable spin-orbit coupling and quantum phase transition in a trapped Bose-Einstein condensate. Sci. Rep. 3, 1937 (2013).
Jiménez-García, K. et al. Tunable Spin-Orbit Coupling via Strong Driving in Ultracold-Atom Systems. Phys. Rev. Lett. 114, 125301 (2015).
Xu, Z. F., You, L. & Ueda, M. Atomic spin-orbit coupling synthesized with magnetic-field-gradient pulses. Phys. Rev. A 87, 063634 (2013).
Anderson, B. M., Spielman, I. B. & Juzeliunas, G. Magnetically Generated Spin-Orbit Coupling for Ultracold Atoms. Phys. Rev. Lett. 111, 125301 (2013).
Goldman, N. & Dalibard, J. Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields. Phys. Rev. X 4, 031027 (2014).
Eckardt, A. & Anisimovas, E. High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective. New. J. Phys 17, 093039 (2015).
Struck, J., Simonet, J. & Sengstock, K. Spin-orbit coupling in periodically driven optical lattices. Phys. Rev. A 90, 031601 (2014).
Luo, X. et al. Tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field. Sci. Rep. 6, 18983 (2016).
Jotzu, G. et al. Creating State-Dependent Lattices for Ultracold Fermions by Magnetic Gradient Modulation. Phys. Rev. Lett. 115, 073002 (2015).
Yu, J., Xu, Z. F., Lü, R. & You, L. Dynamical Generation of Topological Magnetic Lattices for Ultracold Atoms. Phys. Rev. Lett. 116, 143003 (2016).
Grosche, C. & Steiner, F. Handbook of Feynman path integrals, vol. 1, (Springer, Berlin, 1998).
Acknowledgements
We thank Qifeng Xie for the assistance of the experiment and Jinlong Yu for helpful discussions. This work is supported by MOST 2013CB922002 and 2013CB922004 of the National Key Basic Research Program of China, and by NSFC (Nos 11404184, 11574100, and 11474347).
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X.Y.L. performed the experiment. L.N.W. and X.Y.L. analysed the data. L.N.W. performed the numerical calculation. Z.-F. Xu, M.U. and L.Y. developed the theory. All the authors wrote the manuscript. L.Y. and R.W. supervised the whole research project.
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Wu, LN., Luo, XY., Xu, ZF. et al. Harmonic trap resonance enhanced synthetic atomic spin-orbit coupling. Sci Rep 7, 46756 (2017). https://doi.org/10.1038/srep46756
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DOI: https://doi.org/10.1038/srep46756
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