March 29, 2009

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
^{87}Rb 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, m_{F} = -1) and
|2> (F = 2, m_{F} = +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 ω_{t} = 7.8
± 0.1 Hz. A two-photon microwave drives the transition between
|1> and |2>, which is allowed because $Δ m_{F} = 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 Ω_{e} 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 ω_{r}
= 100 Hz.

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

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 Ω_{e} = (Ω^{2} +
δ^{2})^{1/2}. The free Hamiltonian H_{0}
describes the dynamics of atoms in magnetic trap. H_{1}
describes the rotating laser beam. By coupling with H_{1}, 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).