Spinor condensate explained

Spinor condensates are degenerate Bose gases that have degrees of freedom arising from the internal spin of the constituent particles ^[1] .^[2] They are described by a multi-component (spinor) order parameter.Since their initial experimental realisation,a wealth of studies have appeared, bothexperimental and theoretical, focusingon the physical properties of spinor condensates, including their ground states, non-equilibrium dynamics, andvortices.

Early work

The study of spinor condensates was initiated in 1998 by experimental groups at JILA ^[3] and MIT.^[4] These experiments utilised²³Na and ⁸⁷Rb atoms, respectively.In contrast to most prior experiments on ultracold gases, these experiments utilised a purelyoptical trap, which is spin-insensitive. Shortly thereafter, theoretical work appeared^{[5] [6]} which described the possible mean-field phases of spin-one spinor condensates.

Underlying Hamiltonian

The Hamiltonian describing a spinor condensate is most frequently written using the language of second quantization. Here the field operator

\hat{\psi} ^{\dagger({\bf} _{m} r}) creates a boson in Zeeman level m at position {\bf r} . Theseoperators satisfy bosonic commutation relations:[\hat{\psi} _{m({\bf} r}), \hat{\psi} ^{\dagger({\bf} _{m'} r}')] = \delta({\bf r} - {\bf r}') \delta _{mm'} .

The free (non-interacting) part of the Hamiltonian is

H _{0} = \sum _{m} \int d ^{3} r \hat{\psi} ^{\dagger({\bf} _{m} r}) \left( - (\hbar ^{2})/(2 m) \nabla ^{2} + V _{\rm} ({\bf r}) \right) \hat{\psi} ^{\dagger({\bf} _{m} r}).

where

m denotes the mass of the constituent particles and V _{\rm} ({\bf r}) is an external potential.For a spin-one spinor condensate, the interaction Hamiltonian is^{[5] [6]} H _{\rm} = (1)/(2) \int d ^{3} r : \left( c _{0} \hat{\rho}({\bf r}) ^{2} + c _{1} (\hat{{\bf F}}({\bf r})) ^{2 \right):.}

In this expression,

\hat{\rho}({\bf r}) = \sum _{m} \hat{\psi} ^{\dagger({\bf} _{m} r}) \hat{\psi} _{m} ({\bf r}) is the operator corresponding to the density, \hat{{\bf F}}({\bf r}) = \sum _{mm'} \hat{\psi} ^{\dagger({\bf} _{m} r}) {\bf S} _{mm'} \hat{\psi} _{m'} ({\bf r}) is the local spin operator ({\bf S} _{mm'} is a vector composed of the spin-one matrices),and :: denotes normal ordering. The parameters c _{0,} c _{1} can be expressed in terms of the s-wave scattering lengths of the constituent particles.Higher spin versions of theinteraction Hamiltonian are slightly more involved, but can generally be expressed by using Clebsch–Gordan coefficients.

The full Hamiltonian then is

H = H _{0} + H _{\rm} .

Mean-field phases

In Gross-Pitaevskii mean field theory, one replaces the field operators with c-number functions:

\hat{\psi} _{m} ({\bf r}) → {\psi} _{m} ({\bf r}) . To find the mean-fieldground states, one then minimises the resulting energy with respect to these c-number functions.For a spatially uniform system spin-one system, there are two possible mean-field ground states. When c _{1} > 0 , the ground state is\psi _{\rm} = \sqrt{\bar{\rho}}( 0 , 1 , 0 ) while for c _{1<} 0 the ground state is \psi _{\rm} = \sqrt{\bar{\rho}}( 1 , 0 , 0 ). The former expression is referred to as the polar state while the latter is the ferromagnetic state.^[1] Both states are unique up to overall spin rotations. Importantly, \psi _{\rm} cannot be rotated into \psi _{\rm} .The Majorana stellar representation ^[7] provides a particularly insightful description of the mean-field phases of spinor condensates with larger spin.^[2]

Vortices

Due to being described by a multi-component order parameter, numerous types oftopological defects (vortices) can appear in spinor condensates.^[8] Homotopy theory provides a natural description of topological defects,^[9] and is regularly employed to understandvortices in spinor condensates.

Notes and References

1. Kawaguchi. Yuki. Ueda. Masahito. Spinor Bose–Einstein condensates. Physics Reports. 520. 5. 2012. 253–381. 0370-1573. 10.1016/j.physrep.2012.07.005. 2012PhR...520..253K. 1001.2072. 118642726 .
2. Stamper-Kurn. Dan M.. Ueda. Masahito. Spinor Bose gases: Symmetries, magnetism, and quantum dynamics. Reviews of Modern Physics. 85. 3. 2013. 1191–1244. 0034-6861. 10.1103/RevModPhys.85.1191. 2013RvMP...85.1191S.
3. Hall. D. S.. Matthews. M. R.. Ensher. J. R.. Wieman. C. E.. Cornell. E. A.. Dynamics of Component Separation in a Binary Mixture of Bose-Einstein Condensates. Physical Review Letters. 81. 8. 1998. 1539–1542. 0031-9007. 10.1103/PhysRevLett.81.1539. cond-mat/9804138. 1998PhRvL..81.1539H. 119447115 .
4. Stenger. J.. Inouye. S.. Stamper-Kurn. D. M.. Miesner. H.-J.. Chikkatur. A. P.. Ketterle. W.. Spin domains in ground-state Bose–Einstein condensates. Nature. 396. 6709. 1998. 345–348. 0028-0836. 10.1038/24567. cond-mat/9901072. 1998Natur.396..345S. 4406111 .
5. Ho. Tin-Lun. Spinor Bose Condensates in Optical Traps. Physical Review Letters. 81. 4. 1998. 742–745. 0031-9007. 10.1103/PhysRevLett.81.742. cond-mat/9803231. 1998PhRvL..81..742H. 10.1.1.242.852. 18956040 .
6. Ohmi. Tetsuo. Machida. Kazushige. Bose-Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases. Journal of the Physical Society of Japan. 67. 6. 1998. 1822–1825. 0031-9015. 10.1143/JPSJ.67.1822. cond-mat/9803160. 1998JPSJ...67.1822O. 119085696 .
7. Majorana. E. Nuovo Cimento. 1932. 2. 43 . Oriented atoms in a variable magnetic field. 9. 10.1007/BF02960953. 122738040.
8. Ueda. Masahito. Topological aspects in spinor Bose–Einstein condensates. Reports on Progress in Physics. 77. 12. 2014. 122401. 0034-4885. 10.1088/0034-4885/77/12/122401. 25429528. 2014RPPh...77l2401U. 46206348 .
9. Mermin. N. D.. The topological theory of defects in ordered media. Reviews of Modern Physics. 51. 3. 1979. 591–648. 0034-6861. 10.1103/RevModPhys.51.591. 1979RvMP...51..591M.