## Synthesis of arbitrary superposition of Zeeman states in an atom

Optics Express, Vol. 2, Issue 9, pp. 368-371 (1998)

http://dx.doi.org/10.1364/OE.2.000368

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### Abstract

We present a general strategy of quantum state engineering. We describe how an arbitarily prescribed superposition of internal Zeeman levels of an atom can be prepared by Raman pulses.

© Optical Society of America

*N*’state system described by a complete basis {|

*n*〉}, where

*n*= 1,2, ..

*N*. Our goal is to create a superposition state

*c*. We assume that the initial state is a simplest state |1〉 which can be easily prepared. Since the quantum evolution is time reversible, the state preparation problem is equivalent to finding a way to evolve |Ψ〉 back to |1〉 unitarily. Quite generally, if the system allows the two types of external interaction indicated in Fig. 1, we can always find a solution. In Fig. 1 we label the states |

_{n}*n*〉 in a two-row format, and the two types of interaction are classified by the vertical (Fig. 1a) and diagonal (Fig. 1b) flows. For each type of interaction, each state couples with one state only. Such a two-state interaction structure is the key because populations can always be transferred completely from one state to another. Therefore by applying the two interactions alternatively, we can sweep all the populations down to |1〉, i.e.,

*U*and

_{V}*U*, are the evolution operators for the vertical and diagonal flows respectively. For each evolution,

_{D}*λ*characterizes the set of parameters (e.g. the duration, phases, amplitudes and polarizations of external fields) of the corresponding interaction. We choose

_{j}*λ*in such a way that all the populations in the state |

_{j}*j*+ 1〉 are transferred to |

*j*〉. Since

*U*(

_{a}*λ*) (

_{j}*a*=

*V, D*) also affects lower states |

*n*<

*j*〉,

*λ*is determined by the history of the system. In Ref. [1] we have shown that a solution for {

_{j}*λ*} always exists.

_{j}*F, m*〉 of 3

_{F}^{2}

*S*

_{1/2}in sodium, where

*m*= -2, -1, 0,1, 2 for

_{F}*F*= 2 and

*m*= -1, 0,1 for

_{F}*F*= 1. We assume that the initial state is |

*F*= 2,

*m*= -2) which can be prepared by optical pumping.

_{F}^{2}

*S*

_{1/2}manifold, we have to find a way to evolve the prescribed state back to |

*F*= 2,

*m*= -2). This can be done by a Raman pulse sequence as shown in Fig. 2. There are two types of Raman transition. The ‘vertical’ transitions (∆

_{F}*m*= 0, ∆

_{F}*F*= ±1) are driven by two

*π*-polarized pulses. The ‘diagonal’ transitions (∆

*m*= -1, ∆

_{F}*F*= ±1) are driven by a

*π*-polarized pulse and a circularly polarized pulse with + or - helicity. We assume that all Raman pulses are far-detuned from the

*P*–states so that effects of spontaneous decay of P states can be suppressed. In addition we assume that Raman pulses satisfy the resonance condition, ∆

*ω*=

*ω*

_{0}, where ∆

*ω*is the frequency difference in each pulse pair and

*ω*

_{0}is the frequency difference between the

*F*= 2 and

*F*= 1 levels. There could be intensity-dependent level shifts during the Raman process. If these level shifts become significant (i.e., comparable with the effective Rabi-frequency), ∆

*ω*should be adjusted in order to maintain the resonance condition. We should point out that at each step resonance is needed only for the two levels between which we want to transfer population completely from one level to another.

*π*pulses, except that the pulse areas in the steps (7) and (8) should be adjusted according to the Clebsch-Gordan coefficients.

## Acknowledgments

## References

1. | C.K. Law and J.H. Eberly, |

2. | A.S. Parkins, P. Marte, P. Zoller, and H.J. Kimble, |

3. | A.S. Parkins, P. Marte, P. Zoller, O. Carnal, and H.J. Kimble, |

4. | K. Vogel, V.M. Akulin, and W.P. Schleich, |

5. | B.M. Garraway, B. Sherman, H. Moya-Cessa, P.L. Knight, and G. Kurizki, |

6. | A. Kozhekin, G. Kurizki, and B. Sherman, |

7. | D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, and D.J. Wineland, |

8. | S.A. Gardiner, J.I. Cirac, and P. Zoller, Phys. Rev. A |

9. | K. Moler, D.S. Weiss, M. Kasevich, and S. Chu, |

10. | H.J. Kimble and W. Lange (private communication). |

11. | C.K. Law and H.J. Kimble, |

12. |
See for example J. Martin, B.W. Shore, and K. Bergmann, |

13. |
Measurement of atomic angular momentum states can be made by Stern-Gerlach measurements proposed by
R.G. Newton and B. Young, |

**OCIS Codes**

(020.2930) Atomic and molecular physics : Hyperfine structure

(020.7490) Atomic and molecular physics : Zeeman effect

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Focus Issue: Control of loss and decoherence in quantum systems

**History**

Original Manuscript: November 14, 1997

Published: April 27, 1998

**Citation**

C. Law and J. Eberly, "Synthesis of arbitrary superposition of Zeeman states in an atom," Opt. Express **2**, 368-371 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-9-368

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### References

- C. K. Law and J. H. Eberly, Phys. Rev. Lett. 76, 1055 (1996).
- A. S. Parkins, P. Marte, P. Zoller, and H. J. Kimble, Phys. Rev. Lett. 71, 3095 (1993).
- A. S. Parkins, P. Marte, P. Zoller, O. Carnal, and H. J. Kimble, Phys. Rev. A 51, 1578 (1995).
- K. Vogel, V. M. Akulin, and W. P. Schleich, Phys. Rev. Lett. 71, 1816 (1993).
- B. M. Garraway, B. Sherman, H. Moya-Cessa, P. L. Knight, and G. Kurizki, Phys. Rev. A 49, 535 (1994).
- A. Kozhekin, G. Kurizki and B. Sherman, Phys. Rev. A 54, 3535 (1996).
- D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).
- S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. A 55, 1683 (1997).
- K. Moler, D. S. Weiss, M. Kasevich and S. Chu, Phys. Rev. A 45, 342 (1992).
- H. J. Kimble and W. Lange (private communication).
- C. K. Law and H. J. Kimble, J. Mod. Optics 44, 2067 (1997).
- See for example J. Martin, B. W. Shore and K. Bergmann, Phys. Rev. A 54, 1556 (1996) and references therein.
- Measurement of atomic angular momentum states can be made by Stern-Gerlach measurements proposed by R. G. Newton and B. Young, Ann. Phys. (N.Y) 49, 393 (1968).

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