Prof. Robert B. Laughlin, Department of Physics, Stanford University

Vortex Creation in a Bose-Einstein Condensate

Kejie Fang
March 29, 2009

(Submitted as coursework for Physics 376, Stanford University, Winter 2009)

Fig. 1: (a) Schematic of the technique used to create a vortex. A rotating laser beam serves as a microwave drive to compensate the detuning δ. (b) A level diagram showing the transition from |1> to the vicinity of |2>. The energy shift of |2> comes from the laser's rotation.

Quantum vortices - quantized circular flow in superfluids and quantized magnetic flux in superconductors - have attracted interest for decades. The chief reason is that they are quantum topological defects - indeed the only examples of such defects that had been accessible in the laboratory until recently. However, in 1999, E. A. Cornell's group at JILA discovered vortices in a third quantum system, the atomic Bose-Einstein condensate (BEC). [1]

Matthew's et al. prepared the atoms 87Rb using the standard techniques of laser cooling and trapping, followed by evaporative cooling. [2] They also used standard techniques to observe the condensate behavior. [3] However, in between they excited a vortex in the condensate using a coherent process involving a transition between two internal spin states of the atoms. A detuned two-photon microwave field induces virtual transitions between the states. A laser beam whose spatial rotation frequency compensates the detuning then brings the system into resonance, in the process creating a vortex.

The details of the excitation method are illustrated in Fig. 1. States |1> (F = 1, mF = -1) and |2> (F = 2, mF = +1) are separated by ground-state hyperfine |splitting. The initial BEC of |1>, 54 microns in diameter, is confined in a trap with oscillation frequency of ωt = 7.8 ± 0.1 Hz. A two-photon microwave drives the transition between |1> and |2>, which is allowed because $Δ mF = 2. The power and oscillation frequency of the field are adjusted to obtain the desired effective Rabi frequency for the transition. The effective Rabi frequency Ωe is set to be 100 Hz by adjusting the detuning of the microwave frequency. The total detuning δ is about 94 Hz. To create a vortex in |2>, a 10 nW, 780 nm laser beam with a waist of 180 μm is rotated around the BEC core at ωr = 100 Hz.

The excitation process can be theoretically described by the Gross-Pitaevskii equation

Fig. 2: Dynamical evolution of a vortex. The coupling drive is turned on at time t = 0 and turned off at t = t_s. The top graph shows the fractional population of atoms in the |2> state. The small-amplitude rapid oscillations correspond to the cycling between internal levels due to off-resonant coupling (two-photon microwave). The gradual rise of this line is due to coupling of H1. The bottom graph shows the angular momentum of the |2> state. The rise and fall of this curve corresponds to a rapid cycling of |2> and |1> states. The graphs are obtained with parameters &omegar ≈ δ >> Ω.

where tensor product ⊗ is used to explicitly separate the spatial and spin operators. Ω is the resonant Rabi frequency (when δ = 0), and is related to the effective Rabi frequency by Ωe = (Ω2 + δ2)1/2. The free Hamiltonian H0 describes the dynamics of atoms in magnetic trap. H1 describes the rotating laser beam. By coupling with H1, the state |2> gains spatial angular momentum l = 1 and thus forms a vortex. The population of state |2> and its angular momentum are simulated by this equation and are plotted in Fig. 2.

After they formed the vortex, Matthews et al. used the overlap of the |1> and |2> fluids to image the phase profile of the vortex state via an interconversion interference technique. [3] In the presence of a near-resonant microwave filed, the two states interconvert at a rate sensitive to the local difference in the quantum phases of the two states. Thus the application of a resonant π/2 microwave pulse transforms the original two-fluid density distribution into a distribution that reflects the local phase difference. As a result, they read out the 2π phase winding of the vortex.

© 2009 Kejie Fang. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] M. R. Matthews et al., "Vortices in a Bose-Einstein Condensate", Phy. Rev. Lett. 83 ,2498 (1999).

[2] M. R. Matthews et al., Dynamical Response of a Bose-Einstein Condensate to a Discontinuous Change in Internal State", Phys. Rev. Lett. 81, 243 (1998).

[3] D. S. Hall et al., "Measurements of Relative Phase in Two-Component Bose-Einstein Condensates", Phys. Rev. Lett. 81, 1543 (1998).