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A simple method to create a vortex in Bose–Einstein condensate of alkali atoms
Physica B 284}288 (2000) 17}18
A simple method to create a vortex in Bose}Einstein condensate of alkali atoms Mikio Nakahara *, Tomoya Isoshima, Kazushige Machida, Shin-ichiro Ogawa , Tetsuo Ohmi Department of Physics, Kinki University, Higashi-Osaka 577-8502, Japan Department of Physics, Okayama University, Okayama 700-8530, Japan Department of Physics, Kyoto University, Kyoto 601-8501, Japan
Abstract Bose}Einstein condensation in alkali atoms has materialized quite an interesting system, namely a condensate with a spin degree of freedom. In analogy with the A-phase of the super#uid He, numerous textures with nonvanishing vorticity have been proposed. In the present paper, interesting properties of such spin textures are analyzed. We propose a remarkably simple method to create a vortex state of a BEC in alkali atoms. 2000 Elsevier Science B.V. All rights reserved. Keywords: Bose}Einstein condensation; Texture; Vortex
The discovery of Bose}Einstein condensation (BEC) in alkali atoms has led to several exciting "elds to study. One of these developments is a BEC with a spin degree of freedom [1,2]. When the spin exchange interaction is ferromagnetic or the con"ning magnetic "eld is strong, the BEC behaves somewhat similarly to super#uid HeA, where topological objects called textures are known to exist. In the present paper, we investigate vortices in a BEC, which are analogous to the vortices and disgyrations in He-A. We "rst explain the order parameter of a BEC with a spin degree of freedom. Then the cross disgyration, expected to exist in the Io!e}Pritchard (IP) trap, is considered. Finally, we propose a simple method to create an ordinary vortex line in a BEC. Suppose an alkali atom has a hyper"ne-spin "F""1. Then the order parameter has three components W and W , which represent the amplitudes with ! F "$1,0, respectively. The basis vectors in this represX entation are +"$2, "02,. We introduce another set of basis vectors "x2, "y2 and "z2, which are de"ned by F "x2"F "y2"F "z2"0. These vectors are related V W X * Corresponding author. Fax: 81-6-6727-4301. E-mail address: [email protected] (M. Nakahara)
with the previous vectors as "$12"G(1/(2)("x2$i"y2) and "02""z2. When the z-axis is taken parallel to the uniform magnetic "eld, the order parameter of the weak "eld seeking state takes the form W "t and \ W "W "0, which is also written as W "iW " V W t/(2. Let us denote this state in a vectorial form as W"(t/(2)(x( !iy( ), where the common factor has been absorbed in the amplitude t. Suppose the magnetic "eld points to the direction BK "(sin b cos a,sin b sin a,cos b). Then the weak "eld seeking state takes the form W"(t/(2)e A(m( !in( ), where m( "(cos b cos a, cos b sin a, !sin b) and n( "(!sin a, cos a, 0). The unit vector lK "!m( ;n( is the direction of the spin polarization. The same amplitudes in the basis +"02,"$2, are [3] W "(t/2)(1!cos b)e\ ?> A, W "!(t/(2)sin be A, W "(t/2)(1#cos b)e ?> A. (1) \ The choice a"! , b"p/2 yields the cross disgyration shown in Fig. 1, where is the azimuthal angle. This texture has a nonvanishing vorticity n when c"n . It is expected that the cross disgyration is realized in the IP trap with B "0. X
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 9 5 2 - 3
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M. Nakahara et al. / Physica B 284}288 (2000) 17}18
This is nothing but the order parameter of a vortex with the winding number 2. The amplitude t(r) is determined by solving the Gross}Pitaevskii equation with appropriate boundary conditions. The supercurrent of the texture has the -component
Fig. 1. The lK -vector "eld of a cross disgyration. The circle shows the laser beam, where no atoms exist.
Suppose a strong magnetic "eld B is applied to X a BEC, along with the quadrupole "eld. Atoms are assumed to be in the weak "eld seeking state so that they are con"ned in the trap. The "eld B is so strong comX pared to the quadrupole "eld that the order parameter is virtually W"(t/(2)(x( !iy( ). Clearly the vorticity of this texture vanishes. This con"guration is derived from Eq. (1) by putting b"0 and c"!a" . Then B is X adiabatically decreased, so that lK is always antiparallel to B, until B vanishes. The adiabatic condition is required X for atoms to remain in the weak "eld seeking state. Then the cross disgyration appears in the presence of the quadrupole "eld. In due process, the angle b increases from 0 to p/2 and c and !a are identi"ed with . Here, the trap must be plugged by laser beams along the axis, the top and the bottom of the trap so that the atoms do not escape from the trap. In the "nal step, the external "eld B is gradually increased in the opposite (!z) direction. X Then the lK -vector points up so that b"p. Substituting these angles into (1), one obtains W "W "0, W "te (. \
(2)
j "m"t"v , v "( /mr)(1!cos b), (3) ( ( ( where m is the atomic mass and v is the super#uid velocity. If the Na mass is substituted into Eq. (3), we obtain v K0.5/r cm/s for b"p, where r is measured in ( units of lm. In summary, we have proposed a simple method to create a vortex in a BEC of alkali atoms. A strong magnetic "eld is applied along the axis of the IP trap "eld, which is adiabatically varied toward a negative large value. Starting with a uniform order parameter "eld, we are eventually left with a vortex of the winding number 2. The initial state has no circulation while the "nal state does. This is because the external magnetic "eld transfers torque to the BEC while the spin vector is turned upside down. When the hyper"ne-spin is F in general, we will end up with a vortex with the winding number 2F since W and W have phases F(a#c) and \$ $ F(!a#c), respectively.
Acknowledgements We would like to thank M. Mitsunaga, Y. Takahashi and T. Yabusaki for discussions.
References [1] T. Ohmi, K. Machida, J. Phys. Soc. Japan 67 (1998) 1822. [2] T.-L. Ho, Phys. Rev. Lett. 81 (1998) 742. [3] K. Maki, T. Tsuneto, J. Low Temp. Phys. 27 (1977) 635.
A simple method to create a vortex in Bose}Einstein condensate of alkali atoms Mikio Nakahara *, Tomoya Isoshima, Kazushige Machida, Shin-ichiro Ogawa , Tetsuo Ohmi Department of Physics, Kinki University, Higashi-Osaka 577-8502, Japan Department of Physics, Okayama University, Okayama 700-8530, Japan Department of Physics, Kyoto University, Kyoto 601-8501, Japan
Abstract Bose}Einstein condensation in alkali atoms has materialized quite an interesting system, namely a condensate with a spin degree of freedom. In analogy with the A-phase of the super#uid He, numerous textures with nonvanishing vorticity have been proposed. In the present paper, interesting properties of such spin textures are analyzed. We propose a remarkably simple method to create a vortex state of a BEC in alkali atoms. 2000 Elsevier Science B.V. All rights reserved. Keywords: Bose}Einstein condensation; Texture; Vortex
The discovery of Bose}Einstein condensation (BEC) in alkali atoms has led to several exciting "elds to study. One of these developments is a BEC with a spin degree of freedom [1,2]. When the spin exchange interaction is ferromagnetic or the con"ning magnetic "eld is strong, the BEC behaves somewhat similarly to super#uid HeA, where topological objects called textures are known to exist. In the present paper, we investigate vortices in a BEC, which are analogous to the vortices and disgyrations in He-A. We "rst explain the order parameter of a BEC with a spin degree of freedom. Then the cross disgyration, expected to exist in the Io!e}Pritchard (IP) trap, is considered. Finally, we propose a simple method to create an ordinary vortex line in a BEC. Suppose an alkali atom has a hyper"ne-spin "F""1. Then the order parameter has three components W and W , which represent the amplitudes with ! F "$1,0, respectively. The basis vectors in this represX entation are +"$2, "02,. We introduce another set of basis vectors "x2, "y2 and "z2, which are de"ned by F "x2"F "y2"F "z2"0. These vectors are related V W X * Corresponding author. Fax: 81-6-6727-4301. E-mail address: [email protected] (M. Nakahara)
with the previous vectors as "$12"G(1/(2)("x2$i"y2) and "02""z2. When the z-axis is taken parallel to the uniform magnetic "eld, the order parameter of the weak "eld seeking state takes the form W "t and \ W "W "0, which is also written as W "iW " V W t/(2. Let us denote this state in a vectorial form as W"(t/(2)(x( !iy( ), where the common factor has been absorbed in the amplitude t. Suppose the magnetic "eld points to the direction BK "(sin b cos a,sin b sin a,cos b). Then the weak "eld seeking state takes the form W"(t/(2)e A(m( !in( ), where m( "(cos b cos a, cos b sin a, !sin b) and n( "(!sin a, cos a, 0). The unit vector lK "!m( ;n( is the direction of the spin polarization. The same amplitudes in the basis +"02,"$2, are [3] W "(t/2)(1!cos b)e\ ?> A, W "!(t/(2)sin be A, W "(t/2)(1#cos b)e ?> A. (1) \ The choice a"! , b"p/2 yields the cross disgyration shown in Fig. 1, where is the azimuthal angle. This texture has a nonvanishing vorticity n when c"n . It is expected that the cross disgyration is realized in the IP trap with B "0. X
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 9 5 2 - 3
18
M. Nakahara et al. / Physica B 284}288 (2000) 17}18
This is nothing but the order parameter of a vortex with the winding number 2. The amplitude t(r) is determined by solving the Gross}Pitaevskii equation with appropriate boundary conditions. The supercurrent of the texture has the -component
Fig. 1. The lK -vector "eld of a cross disgyration. The circle shows the laser beam, where no atoms exist.
Suppose a strong magnetic "eld B is applied to X a BEC, along with the quadrupole "eld. Atoms are assumed to be in the weak "eld seeking state so that they are con"ned in the trap. The "eld B is so strong comX pared to the quadrupole "eld that the order parameter is virtually W"(t/(2)(x( !iy( ). Clearly the vorticity of this texture vanishes. This con"guration is derived from Eq. (1) by putting b"0 and c"!a" . Then B is X adiabatically decreased, so that lK is always antiparallel to B, until B vanishes. The adiabatic condition is required X for atoms to remain in the weak "eld seeking state. Then the cross disgyration appears in the presence of the quadrupole "eld. In due process, the angle b increases from 0 to p/2 and c and !a are identi"ed with . Here, the trap must be plugged by laser beams along the axis, the top and the bottom of the trap so that the atoms do not escape from the trap. In the "nal step, the external "eld B is gradually increased in the opposite (!z) direction. X Then the lK -vector points up so that b"p. Substituting these angles into (1), one obtains W "W "0, W "te (. \
(2)
j "m"t"v , v "( /mr)(1!cos b), (3) ( ( ( where m is the atomic mass and v is the super#uid velocity. If the Na mass is substituted into Eq. (3), we obtain v K0.5/r cm/s for b"p, where r is measured in ( units of lm. In summary, we have proposed a simple method to create a vortex in a BEC of alkali atoms. A strong magnetic "eld is applied along the axis of the IP trap "eld, which is adiabatically varied toward a negative large value. Starting with a uniform order parameter "eld, we are eventually left with a vortex of the winding number 2. The initial state has no circulation while the "nal state does. This is because the external magnetic "eld transfers torque to the BEC while the spin vector is turned upside down. When the hyper"ne-spin is F in general, we will end up with a vortex with the winding number 2F since W and W have phases F(a#c) and \$ $ F(!a#c), respectively.
Acknowledgements We would like to thank M. Mitsunaga, Y. Takahashi and T. Yabusaki for discussions.
References [1] T. Ohmi, K. Machida, J. Phys. Soc. Japan 67 (1998) 1822. [2] T.-L. Ho, Phys. Rev. Lett. 81 (1998) 742. [3] K. Maki, T. Tsuneto, J. Low Temp. Phys. 27 (1977) 635.