## Symmetries and solutions of the three-dimensional Paul trap

Optics Express, Vol. 8, Issue 2, pp. 123-130 (2001)

http://dx.doi.org/10.1364/OE.8.000123

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### 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

## 1 Introduction

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 The Paul trap

4. W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. **62**, 531–540 (1998). [CrossRef]

4. W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. **62**, 531–540 (1998). [CrossRef]

*with an appropriate frequency*about the axis normal to the surface at the inflection point, the particle will be stably confined. The particle is oscillatory about the origin in both the x and y directions. But it’s oscillation in the

*z*direction is restricted to be bounded from below by some

*z*

_{0}>0.

*x, y*) directions but is different in the

*z*direction, since

*g*(

*t*) ≠

*g*

_{3}(

*t*).

## 3 The quantum-mechanical Paul trap

*ħ*=

*m*=1)

*z*coordinate, but not entirely [6]. Elsewhere [7

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]

## 4 Lie symmetries and separable coordinates

11. V. A. Kosteleck, 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]

*S*is one of the the Schrödinger operators we have discussed,

*L*is a generator of Lie symmetries, and λ is a function of the coordinates

*x, y, z*, and

*t*. An operator

*L*has the general form

*t*-dependent functions

*ϕ*(

*t*) and

*ϕ*3(

*t*) are given by

*ξ*(

*t*),

*ξ̄*(

*t*)} and {

*ξ*

_{3}(

*t*),

*ξ̄*

_{3}(

*t*)} are the complex solutions of the second-order, linear, differential equations in time

## 5 Cartesian symmetries

*z*coordinate, since formally the results are the same for the

*x*and

*y*solutions, with the exception that the

*ξ*3’s and ξ

*g*3’s, etc., lose the subscripts 3. [Elsewhere, we will discuss the symmetries of the Paul trap in much greater detail [10].]

*ω*

^{c}. This means that the operators generate a set of “number states” given by

*su*(1, 1) algebra, which we will not go into here. This is a generalization of the “squeeze algebra.”

*Z*

_{0}and “higher-order” states for

*f*must be a function of t.

*π*

*ϕ*3(

*t*)]

^{-1/4}. This would conserve the probability, as one would want for the time-development of a unitary Hamiltonian. But this turns out to be incorrect. Such a solution

*does not*satisfy the Schrödinger equation (16). Indeed, putting Eq. (33) into Eq. (16) yields a first order differential equation in

*t*for

*f*(

*t*). The resulting normalized extremal-state solution is

*ξ̄*3/ξ3)

^{1/4}there? It is there because, in this

*time-dependent*Schrödinger equation, it is the necessary generalization of the simple-harmonic oscillator ground-state energy exponential, exp[-

*iħω*/2]. (This follows since

*ξ*3(

*t*)→(2

*ω*)

^{-1/2}exp[

*iωt*].) This phase factor

*is necessary*[13, 14

14. D. R. Truax, “Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation {*∂*_{xx} +2*i∂*_{t} -2*g*2(*t*)*x*^{2}-2*g*1(*t*)*x*-2*g*0(*t*)}Ψ(*x*, *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]

## 6 Polar symmetries

11. V. A. Kosteleck, 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]

*I*, the operator

*I*,

*a*

_{±},

*c*

_{±},

*K,Lz*} forms a closed algebra. Further, if we make the transformations

*f*,

*a*

_{±},

*I*} and {

*d*,

*c*

_{±},

*I*}, which have only the identity operator,

*I*, in common. Therefore, the algebra is

*G*′

_{x,y}=

*os*

_{a}(1)+

*os*

_{c}(1), with the two Casimir operators

*os*(1) that are bounded below; namely, ↑-1/2+↑-1/2. Let Ω

_{n,m}be a member of the set of common eigenfunctions of

*f*and

*d*spanning the representation space. Then, for (

*n,m*)∈

*d*and

*f*be bounded below, we also have

*θ*dependence of the Ω

_{n,m}(

*r, θ, t*) can be given by

_{0,0}(

*r, θ, t*) is a function

*only*of

*r*or

*ρ*and

*t*. In this (

*n,m*)=(0, 0) case, both the equations in (51) have the same first-order differential form. So, we can solve for Ω

_{0,0}similarly to as was done for the Z

_{0}solution. The normalized solution to the Schrödinger equation (18) is found to be

17. V. A. Kosteleckŷ, 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). [CrossRef]

*n*

_{r}and

*ℓ*

_{z}cannot differ by an odd integer to allow a normalizable solution. [

*n*and

*m*quantum numbers, which reflect the fundamental symmetries, can be all non-negative integers. The

*n*

_{r}and

*ℓ*

_{z}quantum numbers reflect a rotation of the axes by 45°. (See figure 1.) So, reaching the allowed positions of quantum numbers along these diagonals, scaled by 1=√2, means all integer values of (

*n*

_{r},

*ℓ*

_{z}) are not allowed.

## Acknowledgements

## References and links

1. | J. H. Eberly and L. P. S. Singh, “Time operators, partial stationarity, and the energy-time uncertainty relation,” Phys. Rev. D |

2. | P. H. Dawson, |

3. | D. J. Wineland, W. M. Itano, and R. S. Van Dyck Jr., “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. |

4. | W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. |

5. | M. Combescure, “A quantum particle in a quadrupole radio-frequency trap,” Ann. Inst. Henri Poincare |

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. |

7. | M. M. Nieto and D. R. Truax, “Coherent states sometimes look like squeezed states, and visa versa: The Paul trap,” New J. Phys. |

8. | G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. |

9. | W. Miller Jr., |

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

11. | V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A |

12. | J. R. Klauder, private communication. |

13. | The phase factor can also be obtained [14, 15], by solving the eigenvalue equation |

14. | D. R. Truax, “Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation { |

15. | M. M. Nieto and D. R. Truax, “Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras,” J. Math. Phys. |

16. | W. Magnus, F. Oberhettinger, and R. P. Soni, |

17. | V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D |

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. |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(270.5570) Quantum optics : Quantum detectors

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 15, 2000

Published: January 15, 2001

**Citation**

Michael Nieto and D. Truax, "Symmetries and solutions of the three-dimensional Paul trap," Opt. Express **8**, 123-130 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-123

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### References

- 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.

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