Abstract

Using the symmetries of the three-dimensional Paul trap, we derive the solutions of the time-dependent Schrödinger equation for this system, in both Cartesian and cylindrical coordinates. Our symmetry calculations provide insights that are not always obvious from the conventional viewpoint.

© Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. J. H. Eberly and L. P. S. Singh, "Time operators, partial stationarity, and the energy-time uncertainty relation," Phys. Rev. D 7, 359-362 (1973).
    [CrossRef]
  2. P. H. Dawson, Quadrupole Mass Spectrometry and its Applications (Elsevier, Amsterdam, 1976), Chaps. I-IV. Reprinted by (AIP, Woodbury, NY, 1995).
  3. D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., "High-resolution spectroscopy of stored ions," Adv. Atomic Mol. Phys. 19, 135-186 (1983).
    [CrossRef]
  4. W. Paul, "Electromagnetic traps for charged and neutral particles," Rev. Mod. Phys. 62, 531-540 (1998).
    [CrossRef]
  5. M. Combescure, "A quantum particle in a quadrupole radio-frequency trap," Ann. Inst. Henri Poincare 44, 293-314 (1986).
  6. M. Feng, J. H. Wu, and K. L. Wang, "A Study of the characteristics of the wave packets of a Paul-trapped ion," Commun. Theoret. Phys. 29, 497-502 (1998).
  7. M. M. Nieto and D. R. Truax, "Coherent states sometimes look like squeezed states, and visa versa: The Paul trap," New J. Phys. 2, 18.1-18.9 (2000). Eprint quant-ph/0002050.
    [CrossRef]
  8. G. Schrade, V. I. Man'ko, W. P. Schleich, and R. J. Glauber, "Wigner functions in the Paul trap," Quantum Semiclass. Opt. 7, 307-325 (1995).
    [CrossRef]
  9. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).
  10. M. M. Nieto and D. R. Truax (in preparation).
  11. V. A. Kostelecky, V. I. Man'ko, M. M. Nieto, and D. R. Truax, "Supersymmetry and a time-dependent Landau system," Phys. Rev. A 48, 951-963 (1993).
    [CrossRef] [PubMed]
  12. J. R. Klauder, private communication.
  13. The phase factor can also be obtained [14, 15], by solving the eigenvalue equation 3 nz = (nz � ) nz, where 3 = {3 t � (Z z � ) - i/4 3z2} Then, solving the equation Jz-0=0 will yield the extremal state function up to a factor of (pi)^-1/4.
  14. D. R. Truax, "Symmetry of time-dependent Schr� odinger equations. II. Exact solutions for the equation {xx 2 t - 2 2(t)x2 - 2 1(t)x-2 0(t) = 0," J. Math. Phys. 23, 43-54 (1982).
    [CrossRef]
  15. M. M. Nieto and D. R. Truax, "Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras," J. Math. Phys. 38, 84-97 (1997).
    [CrossRef]
  16. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).
  17. V. A. Kostelecky, M. M. Nieto, and D. R. Truax, "Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions," Phys. Rev. D 32, 2627-2633 (1985). See. Eqs. (2.7) and (2.8).
    [CrossRef]
  18. M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

Other

J. H. Eberly and L. P. S. Singh, "Time operators, partial stationarity, and the energy-time uncertainty relation," Phys. Rev. D 7, 359-362 (1973).
[CrossRef]

P. H. Dawson, Quadrupole Mass Spectrometry and its Applications (Elsevier, Amsterdam, 1976), Chaps. I-IV. Reprinted by (AIP, Woodbury, NY, 1995).

D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., "High-resolution spectroscopy of stored ions," Adv. Atomic Mol. Phys. 19, 135-186 (1983).
[CrossRef]

W. Paul, "Electromagnetic traps for charged and neutral particles," Rev. Mod. Phys. 62, 531-540 (1998).
[CrossRef]

M. Combescure, "A quantum particle in a quadrupole radio-frequency trap," Ann. Inst. Henri Poincare 44, 293-314 (1986).

M. Feng, J. H. Wu, and K. L. Wang, "A Study of the characteristics of the wave packets of a Paul-trapped ion," Commun. Theoret. Phys. 29, 497-502 (1998).

M. M. Nieto and D. R. Truax, "Coherent states sometimes look like squeezed states, and visa versa: The Paul trap," New J. Phys. 2, 18.1-18.9 (2000). Eprint quant-ph/0002050.
[CrossRef]

G. Schrade, V. I. Man'ko, W. P. Schleich, and R. J. Glauber, "Wigner functions in the Paul trap," Quantum Semiclass. Opt. 7, 307-325 (1995).
[CrossRef]

W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).

M. M. Nieto and D. R. Truax (in preparation).

V. A. Kostelecky, V. I. Man'ko, M. M. Nieto, and D. R. Truax, "Supersymmetry and a time-dependent Landau system," Phys. Rev. A 48, 951-963 (1993).
[CrossRef] [PubMed]

J. R. Klauder, private communication.

The phase factor can also be obtained [14, 15], by solving the eigenvalue equation 3 nz = (nz � ) nz, where 3 = {3 t � (Z z � ) - i/4 3z2} Then, solving the equation Jz-0=0 will yield the extremal state function up to a factor of (pi)^-1/4.

D. R. Truax, "Symmetry of time-dependent Schr� odinger equations. II. Exact solutions for the equation {xx 2 t - 2 2(t)x2 - 2 1(t)x-2 0(t) = 0," J. Math. Phys. 23, 43-54 (1982).
[CrossRef]

M. M. Nieto and D. R. Truax, "Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras," J. Math. Phys. 38, 84-97 (1997).
[CrossRef]

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).

V. A. Kostelecky, M. M. Nieto, and D. R. Truax, "Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions," Phys. Rev. D 32, 2627-2633 (1985). See. Eqs. (2.7) and (2.8).
[CrossRef]

M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1.
Fig. 1.

A plot of the allowed quantum numbers for the polar coordinates of the Paul trap. Shown are the (n,m) quantum numbers of Eqs. (54) and (55) as well as the (nr,ℓz ) quantum numbers of Eqs. (57) and (60).

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

V ( x , y , z , t ) = V x ( x , t ) + V y ( y , t ) + V z ( z , t ) ,
V x ( x , t ) = + e 2 r 0 2 V ( t ) x 2 g ( t ) x 2 ,
V y ( y , t ) = + e 2 r 0 2 V ( t ) y 2 g ( t ) y 2 ,
V z ( z , t ) = e r 0 2 V ( t ) z 2 g 3 ( t ) z 2 .
x ̈ c l = F x ( x , t ) = d V x ( x , t ) d x = 2 g ( t ) x c l ,
y ̈ cl = F y ( y , t ) = d V y ( y , t ) dy = 2 g ( t ) y c l ,
z ̈ cl = F z ( z , t ) = d V z ( z , t ) dz = 2 g 3 ( t ) z c l .
S Ψ ( x , y , z , t ) = { x x + y y + z z + 2 i t 2 g ( t ) ( x 2 + y 2 ) 2 g 3 ( t ) z 2 } Ψ ( x , y , z , t ) = 0 .
g ( t ) = + e 2 r 0 2 V ( t ) , g 3 ( t ) = e r 0 2 V ( t ) .
V ( t ) = V d c V a c cos ω ( t t 0 )
x = r cos θ , y = r sin θ , z = z ,
S c y l Φ ( r , θ , z , t ) = { r r + 1 r r + 1 r 2 θ θ + z z + 2 i t 2 g ( t ) r 2 2 g 3 ( t ) z 2 } Φ ( r , θ , z , t )
= 0 .
Ψ ( x , y , z , t ) = X ( x , t ) Y ( y , t ) Z ( z , t ) ,
S x X ( x , t ) = { x x + 2 i t 2 g ( t ) x 2 } X ( x , t ) = 0 ,
S y Y ( y , t ) = { yy + 2 i t 2 g ( t ) y 2 } Y ( y , t ) = 0 ,
S z Z ( z , t ) = { zz + 2 i t 2 g 3 ( t ) z 2 } Z ( z , t ) = 0 ,
Φ ( x , y , z , t ) = Ω ( r , θ , t ) Z ( z , t ) ,
S r θ Ω ( r , θ , t ) = { r r + 1 r r + 1 r 2 θ θ 2 g ( t ) r 2 } Ω ( r , θ , t ) = 0 .
[ S , L ] = ( x , y , z , t ) S .
L = C 0 t + C 1 x + C 2 y + C 3 z + C ,
x = x ϕ 1 / 2 ( t ) , y = y ϕ 1 / 2 ( t ) , z = z ϕ 3 1 / 2 ( t ) , t = t ,
ρ = x 2 + y 2 ϕ 1 / 2 ( t ) , θ = sin 1 ( y x 2 + y 2 ) , z = z ϕ 3 1 / 2 ( t ) , t = t .
ϕ = 2 ξ ξ ¯ ϕ 3 = 2 ξ 3 ξ ¯ 3 ,
γ ̈ + 2 g ( t ) γ = 0 , γ ̈ 3 + 2 g 3 ( t ) γ 3 = 0 ,
W ( ξ , ξ ¯ ) = W ( ξ 3 , ξ ¯ 3 ) = i .
J z = ξ 3 z i ξ ˙ 3 z = + 1 2 ( ξ ¯ 3 ξ 3 ) 1 / 2 [ z + z ( 1 i 2 ϕ ˙ 3 ) ] ,
J z + = ξ ¯ 3 z + i ξ ¯ ˙ 3 z = 1 2 ( ξ ¯ 3 ξ 3 ) 1 / 2 [ z z ( 1 + i 2 ϕ ˙ 3 ) ] .
[ J z , J z + ] = I ,
J z Z n z ( z , t ) = n z Z n z 1 ( z , t ) ,
J z + Z n z ( z , t ) = n z + 1 Z n z + 1 .
i t Z n z = H Z n z = i [ i t Z n z Z n z ] Z n z [ Const ] Z n z .
J z Z 0 ( z , t ) = 0 .
Z 0 ( z , t ) = f ( t ) exp { z 2 2 [ 1 i ϕ ˙ 3 2 ] } ,
Z 0 ( z , t ) = ( π ϕ 3 ) 1 / 4 ( ξ ¯ 3 ξ 3 ) 1 / 4 exp { z 2 2 [ 1 i ϕ ˙ 3 2 ] } .
Z n z ( z , t ) = [ n z ! ] 1 / 2 [ J z + ] n z Z 0 ( z , t ) .
Z n z ( z , t ) = ( 2 n z n z ! ) 1 / 2 ( π ϕ 3 ) 1 / 4 ( ξ ¯ 3 ξ 3 ) 1 / 2 ( n z + 1 2 )
H n z ( z ) exp { z 2 2 [ 1 i ϕ ˙ 3 2 ] } .
X n x ( x , t ) = Z n z n x ( z x , ξ 3 ξ , ϕ 3 ϕ , t ) ,
Y n y ( y , t ) = Z n z n y ( z y , ξ 3 ξ , ϕ 3 ϕ , t ) ,
Ψ n x , n y , n z ( x , y , z , t ) = X n x ( x , t ) Y n y ( y , t ) Z n z ( z , t ) .
a = a + = 1 2 ( J x + i J y ) = 1 2 [ ξ ( x + i y ) i ξ ˙ ( x + i y ) ]
= 1 2 ( ξ ξ ¯ ) 1 / 2 e i θ [ ρ + i ρ θ + ρ ( 1 i 2 ϕ ˙ ) ] ,
c = c + = 1 2 ( J x i J y ) = 1 2 [ ξ ( x i y ) i ξ ˙ ( x i y ) ]
= 1 2 ( ξ ξ ¯ ) 1 / 2 e i θ [ ρ i ρ θ + ρ ( 1 i 2 ϕ ˙ ) ] .
κ = J x + J x + J y + J y + 1 = a + a + c + c + 1 ,
z = i ( y x x y ) = i θ ,
f 1 2 ( κ z ) = a + a + 1 2 ,
d 1 2 ( κ + z ) = c + c + 1 2 ,
f Ω n , m = ( n + 1 2 ) Ω n , m , d Ω n , m = ( m + 1 2 ) Ω n , m ,
a Ω n , m = n Ω n 1 , m , c Ω n , m = m Ω n , m 1 ,
a + Ω n , m = n + 1 Ω n + 1 , m , c + Ω n , m = m + 1 Ω n , m + 1 ,
z Ω n , m = ( f d ) Ω n , m = ( m n ) Ω n , m ,
κ Ω n , m = ( d + f ) Ω n , m = ( n + m + 1 ) Ω n , m ,
a Ω 0 , 0 = 0 , c Ω 0 , 0 = 0 .
Ω n , m , ( r , θ , t ) = 𝓡 n , m ( r , t ) Θ n , m ( θ ) = 𝓡 n , m ( r , t ) exp [ i ( m n ) θ ] 2 π .
Ω 0 , 0 ( r , θ , t ) = 1 2 π [ 2 ϕ ] 1 / 2 ( ξ ¯ ξ ) 1 2 exp { ρ 2 2 [ 1 i ϕ ˙ 2 ] } = X 0 ( x , t ) Y 0 ( y , t ) .
Ω n , m ( ρ , θ , t ) = ( n ! m ! ) 1 / 2 a + n c + m Ω 0 , 0 ( ρ , θ , t ) .
Ω n , m ( r , θ , t ) = exp [ i ( m n ) θ ] 2 π ( ) k k ! ( n ! m ! ) 1 / 2 ( 2 ϕ ( t ) ) 1 / 2 ( ξ ¯ ξ ) 1 2 ( n + m + 1 )
ρ n m L k ( n m ) ( ρ 2 ) exp { ρ 2 2 [ 1 i ϕ ˙ 2 ] } ,
k 1 2 ( n + m m n ) .
= z , z = m n , n r = m + n 0 .
Ω n , m ( r , t ) R n r , ( r , t ) Θ z ( θ ) ,
Θ z ( θ ) = exp [ i z θ ] 2 π ,
R n r , ( r , t ) = ( 1 ) ( n r ) / 2 [ 2 [ n r 2 ] ! ϕ ( t ) [ n r + 2 ] ! ] 1 / 2 ( ξ ¯ ξ ) 1 2 ( n + m + 1 )
ρ L 1 2 ( n r ) ( ) ( ρ 2 ) exp { ρ 2 2 [ 1 i ϕ ˙ 2 ] } .
Φ n r , z , n z ( x , y , z , t ) = R n r , ( r , t ) Θ z ( θ ) Z n z ( z , t ) .

Metrics