Center for Theoretical Physics of the Polish Academy of Sciences

Gravity-induced resonanses in the rotating harmonic trap

Iwo Białynicki-Birula and Tomasz Sowiński

Center for Theoretical Physics of the Polish Academy of Sciences

tomsow@cft.edu.pl

Phys. Rev. A 71, 043610 (2005)

Abstract

It is shown that in an anisotropic harmonic trap that rotates with the properly chosen rotation rate, the force of gravity leads to a resonant behavior. Full analysis of the dynamics in an anisotropic, rotating trap in 3D is presented and several regions of stability are identified. On resonance, the oscillation amplitude grows linearly with time and all particles are expelled from the trap. The positions of the resonances depend only on the characteristic frequencies of the rotating trap and on the direction of the angular velocity of rotation.

Introduction

In this work we present a complete solution to the problem of the motion of a particle moving in a most general anisotropic rotating harmonic trap in 3D and in the presence of gravity.

A new significance of this problem brought about by experi- mental and theoretical studies of Bose-Einstein condensates and the accompanying thermal clouds in rotating traps.

All our results are valid not only for a single particle but also for the center of mass motion in many-body (classical or quantum) theory [3]. Therefore, a trap rotating at the resonant frequency will not hold the Bose-Einstein condensate.

Classical equations of motion

The best way to analyze the behavior of particles in a uni- formly rotating trap is to first perform the transformation to the rotating frame. In this frame the harmonic trap po- tential is frozen but the force of gravity is rotating with the angular velocity of the trap rotation. The Hamiltonian has the form

 



 



  

   



  



 









(1) The eigenvalues of the symmetric potential matrix

^

are the squared frequencies of the oscillations in the non-rotating trap. The angular velocity matrix

^{ }

is related to the com- ponents of the angular velocity vector through the formula

! #"

%$

&'"()&

. The vector of the gravitational acceleration





, as seen in the rotating frame, can be expressed in the form





( +*





-,

  

.

0/

 

132



.



e

54-6

 (2)

where

¹

denotes the direction and

^

denotes the length of the vector

^

. The parallel and the transverse compo- nents of the gravitational acceleration vector

^{7   98}

are defined as

^{-, 1  1: 7  }

and

^{. 7   1!  1: 7  }

, respec- tively. Note that the time-dependent part vanishes when the rotation axis is vertical.

The equations of motion determined by the Hamiltonian (1) have the following form

d

^{ }

d

^





(



;

  



<

(3a)

d

 ^

d

^

=

  

>?

  



(

  



@

(3b) In compact notation this equations have the form of one equation:

d

^{A (}

d

^



B     A



DC

,

EC

.

e

54-6

(4) where the physical trajectory in phase space is described by the real part of the complex vector

^{A }

and



B  



?



GFIH



I

  



;



!

<

C , 8

 

-,

!

< C . 8



 .

0/

 

132



 . 

!



Let us introduce a basis

^{J "}

of six eigenvectors of

^{B    }

:



B     J "

/LK

"    J " < M

"ON

HQP5P5P R

(5)

and expand

^{A }

and

^{C }

in this base as follows

A



R

X

"SN

HUT

"



e

WVYX[Z\4^]_6

J " <

C , R

X

"ON

Ha`

"

, J " < C . R

X

"ON

Hb`

"

. J " 

The equation of motion (4) can be rewritten now as a set of equations for the coefficient functions

T "



: d

T "

d

^

` "

,

e

^F

V X

Z\4^]_6

 ` "

.

e

cZ\4

F V X

Z\4^]d]_6

< M

"ON

HQP5P5P R

(6)

A mode amplitude

T "

(

will grow linearly in time – the sig- nature of a resonance – whenever the second term on the right hand side becomes time independent. It is so when the angular velocity of the trap rotation

^

satisfies the res- onance condition

^{ K "   }

and, of course,

` "

.fe

8

. We

must take into account that characteristic frequencies of the trap

^{K "}

depend on the frequency

^

. Therefore, the position of the resonance has to be determined selfconsis- tently. A full description of these resonances requires the knowledge of the behavior of

^{K "   }

’s as functions of

^

.

Regions of stability

The stability of motion for a harmonic oscillator is deter- mined by the values of its characteristic frequencies

^K

– eigenvalues of the matrix

^{B    }

. The characteristic poly- nomial is tri-quadratic:

g

ih> jhlk

nm

 

 < 



Qh

po

 

 < 



'h

0q

 

 < 

 

(7) where

^{h K}

and the coefficients A, B and C are well known [3]. Stable oscillations take place when all charac- teristic

^K

‘s are real. This means that all three roots of the polynomial

^{g rhs}

must be real and positive.

We shall exhibit this behavior by plotting the zero contour lines of

^{g rhs}

in the

^{ h}

plane [1, 2]. We assume that the trap potential and the direction of rotation are fixed and we treat the characteristic polynomial

^{g ih>}

as a function of

^

and

^h

only (Fig. 1).

Gravity induced resonanses

There are, in general, two resonant values of

^

. They occur when

^{h? }

. We may analyze this resonance condition graphically by superposing the parabola

^{hf }

on the plots of characteristic frequencies (Fig. 2).

Conclusions

We have analyzed the stability of motion in an anisotropic, rotating harmonic trap in 3D. We have found that, in gen- eral, there are three regions of stability. We have demon- strated the presence of resonances in a rotating harmonic trap subjected to the force of gravity whitch occur at two rotation rates but only when the rotation axis is tilted away from the vertical direction.

We hope that it should be possible to confirm experimentally the existence of the gravity-induced resonances for Bose- Einstein condensates in rotating traps.

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

tvu

tvw

x

t

(a) The contour lines in this figure represent the ze- roes of the characteristic polynomial (7) for

^y#zS{I|

,

y~}{

,

^y#€{-‚

, and

^{ƒ z} {„ƒ…}{„ƒc€{L|c†ˆ‡ ‚

. We

identify the lower region of instability (between blue lines) where one of the roots of (7) is negative, and also an upper region (between red lines) where there exists only one real root.

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

‰

Š

(b) The contour lines of the characteristic polyno- mial (7) for

^{y z {‹|}

,

^y~}S{U

,

^y#€Œ{‚

and

^{ƒ z {‡ ‚Ž†\}

,

ƒ…}'{

,

ƒc€Œ{‹|c†c

. In this case lower region of insta-

bility vanishes.

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

‘

’

(c) This plot is for the same trap as in Fig. (a) but for the rotation around the trap axis

^{ƒ z {}

,

^ƒ…}S{

,

ƒ € {“|

. In this degenerate case there exists only one (lower) region of instability, when one of the roots of (7) is negative.

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5

PSfrag replacements

” •

–

—˜d™

š

›

(d) Size of the upper region of instability as a function of direction of the angular velocity vector.

y z

{œ|

,

^{y } {L}

,

^{y € {‚}

, and

^{ƒ z {ž…Ÿ  ¢¡} £\¤d¥c¦¢#¡ §¨¤

,

ƒ } {L…Ÿ ˆ¡ £\¤Ž…Ÿ ˆ¡ §¨¤

,

^{ƒ € {L¥#¦¢#¡ £\¤}

. This size is de- fined as

^{©{ªQ«¬r­ ªQ«®}

(Fig. 1a).

Figure 1: Properties of characteristic frequencies for rotating trap.

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

¯

°

(a)

^{y z {^|}

,

^y~}S{U

,

^yc€Œ{‚

,

^{ƒ z {U…Ÿ ˆ¡ «²± ¤}

,

^ƒ~}S{

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

¯

°

(b)

^{y z {‹|}

,

^{y } {}

,

^{y € {‚}

,

^{ƒ z {U…Ÿ Ž¡ ±³ ¤}

,

^{ƒ } {}

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

´

µ

(c)

^{y z {‹|}

,

^{y } {}

,

^{y € {‚}

,

^{ƒ z {U…Ÿ Ž¡ ¶±· ¤}

,

^{ƒ } {}

0 1 2 3 4

1 2 3 4 5 6

PSfrag replacements

´

µ

(d)

^{y z {‹|}

,

^{y } {‹|cˆ†c‚}

,

^{y € {¸}

,

^{ƒ z {‹|}

,

^{ƒ } {ƒ € {}



Figure 2: Graphical representation of the resonace condition

^¹»º½¼

. The lower value always falls into the range of the first region of stable oscillations. The upper value may fall in the lower region of stability (a), in the lower region of instability (b), or in the higher region of stability (c). A degenerate case is also possible when the two resonance frequencies coincide (d). First region of instability is bounded by the blue lines.

X' Z'

(a) No rotation. In this case gravity just moved equilibrium position.

X' Z'

(b) (

ª{d¾#|#¸

) Slow rotation in first region of sta-

bility.

X' Z'

(c) (

^ª{

q

^¿À

) Gravity induced resonance.

X' Z'

(d) (

^ª{

) Fast rotation.

Figure 3: Trajectory of the particle moving in a rotating trap

^ÁOÂlºÃ

,

ÁOÄlºÆÅ

,

^ÇQȺÉÃ

,

^{Ç Â ºÊÇSĺÌË}

without (red line) and with (green line)

the gravitational field. Fig. (c) presents a resonant behavior.

References

[1] I. Bialynicki-Birula and T. Sowiński, Phys. Rev. A 71, 043610 (2005)

[2] T. Sowiński and I. Bialynicki-Birula, (quant-ph/0409070)

[3] I. Bialynicki-Birula and Z. Bialynicka-Birula, Phys. Rev. A 65, 063606 (2002)

Quantum Optics VI - „Quantum Engineering of Atoms and Photons” , Krynica, 13-18.VI.2005 This research was supported by a grant from the Polish Ministry of Scientific Research and Information Technology.