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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 2, Iss. 9 — Apr. 27, 1998
  • pp: 368–371
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Synthesis of arbitrary superposition of Zeeman states in an atom

C.K. Law and J.H. Eberly  »View Author Affiliations


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

In this paper, we show how Raman pulses can synthesize an arbitrary prescribed superposition of hyperfine magnetic states in an atom. The ability of controlling internal states of atoms could open doors for interesting applications [2

2. A.S. Parkins, P. Marte, P. Zoller, and H.J. Kimble, Phys. Rev. Lett.71, 3095 (1993).

, 10

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

, 11

11. C.K. Law and H.J. Kimble, J. Mod. Optics44, 2067 (1997).

]. Although the method of adiabatic passage is a powerful technique to transfer populations [12

12. See for example J. Martin, B.W. Shore, and K. Bergmann, Phys. Rev. A54, 1556 (1996) and references therein.

], the question how to achieve a superposition of hyperfine states with any desired values of complex amplitudes remains a challenge [13

13. 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).

]. In this paper we shall present a solution with the hyperfine ground state of sodium as an example.

To begin, let us review the key idea of our scheme of arbitrary state preparation. Consider an N’state system described by a complete basis {|n〉}, where n = 1,2, ..N. Our goal is to create a superposition state

|Ψ=n=1Ncn|n
(1)

with prescribed complex amplitudes cn. 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〉 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.,

|1=UV(λ1)UD(λ2)UV(λ3)UD(λN2)UV(λN1)|Ψ
(2)

where UV and UD, are the evolution operators for the vertical and diagonal flows respectively. For each evolution, λj characterizes the set of parameters (e.g. the duration, phases, amplitudes and polarizations of external fields) of the corresponding interaction. We choose λj in such a way that all the populations in the state |j + 1〉 are transferred to |j〉. Since Ua(λj) (a = V, D) also affects lower states |n < j〉, λj is determined by the history of the system. In Ref. [1

1. C.K. Law and J.H. Eberly, Phys. Rev. Lett.76, 1055 (1996).

] we have shown that a solution for {λj} always exists.

Fig. 1. Two types of interactions connecting different states. (a) Vertical (b) Diagonal

Quite remarkably, there are several interesting quantum systems that indeed allow the two types of interactions as depicted in Fig. 1. For example, a two-level atom in a quantized cavity field [1

1. C.K. Law and J.H. Eberly, Phys. Rev. Lett.76, 1055 (1996).

], a trapped ion [8

8. S.A. Gardiner, J.I. Cirac, and P. Zoller, Phys. Rev. A55, 1683 (1997).

], and the manifold of hyperfine levels. In the following we describe in detail the last case. Let us consider the hyperfine levels |F, mF〉 of 32 S 1/2 in sodium, where mF = -2, -1, 0,1, 2 for F = 2 and mF = -1, 0,1 for F = 1. We assume that the initial state is |F = 2, mF = -2) which can be prepared by optical pumping.

To generate an arbitrary superposition state of all hyperfine levels in the 32 S 1/2 manifold, we have to find a way to evolve the prescribed state back to |F = 2, mF = -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 (∆mF = 0, ∆F = ±1) are driven by two π-polarized pulses. The ‘diagonal’ transitions (∆mF = -1, ∆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.

Fig. 2. An 8-pulse sequence that force any given state (upper left) in the hyperfine manifold of 32 S 1/2 of sodium to evolve into |F = 2, mF = -2〉 (bottom right). The solid and empty circles represent occupied and unoccupied states respectively

|Ψ=12(|F=2,m=2+|F=2,m=2)
(3)

can be obtained without applying the pulses (3) and (4). In this simple case, all the required Raman pulses are π pulses, except that the pulse areas in the steps (7) and (8) should be adjusted according to the Clebsch-Gordan coefficients.

In conclusion, we have discussed a method to generate an arbitrary prescribed quantum state that is suitable for general systems that allow interactions defined in Fig. 1. As an example, we have described how a general superposition of hyperfine levels in sodium can be achieved by Raman pulses. The method can obviously be extended for other atoms with a similar structure of hyperfine states.

Acknowledgments

This research was supported by NSF grants No. PHY 94-08733 and PHY94-15583.

References

1.

C.K. Law and J.H. Eberly, Phys. Rev. Lett.76, 1055 (1996).

2.

A.S. Parkins, P. Marte, P. Zoller, and H.J. Kimble, Phys. Rev. Lett.71, 3095 (1993).

3.

A.S. Parkins, P. Marte, P. Zoller, O. Carnal, and H.J. Kimble, Phys. Rev. A51, 1578 (1995).

4.

K. Vogel, V.M. Akulin, and W.P. Schleich, Phys. Rev. Lett.71, 1816 (1993).

5.

B.M. Garraway, B. Sherman, H. Moya-Cessa, P.L. Knight, and G. Kurizki, Phys. Rev. A49, 535 (1994).

6.

A. Kozhekin, G. Kurizki, and B. Sherman, Phys. Rev. A54, 3535 (1996).

7.

D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett.76, 1796 (1996).

8.

S.A. Gardiner, J.I. Cirac, and P. Zoller, Phys. Rev. A55, 1683 (1997).

9.

K. Moler, D.S. Weiss, M. Kasevich, and S. Chu, Phys. Rev. A45, 342 (1992).

10.

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

11.

C.K. Law and H.J. Kimble, J. Mod. Optics44, 2067 (1997).

12.

See for example J. Martin, B.W. Shore, and K. Bergmann, Phys. Rev. A54, 1556 (1996) and references therein.

13.

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).

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

  1. C. K. Law and J. H. Eberly, Phys. Rev. Lett. 76, 1055 (1996).
  2. A. S. Parkins, P. Marte, P. Zoller, and H. J. Kimble, Phys. Rev. Lett. 71, 3095 (1993).
  3. A. S. Parkins, P. Marte, P. Zoller, O. Carnal, and H. J. Kimble, Phys. Rev. A 51, 1578 (1995).
  4. K. Vogel, V. M. Akulin, and W. P. Schleich, Phys. Rev. Lett. 71, 1816 (1993).
  5. B. M. Garraway, B. Sherman, H. Moya-Cessa, P. L. Knight, and G. Kurizki, Phys. Rev. A 49, 535 (1994).
  6. A. Kozhekin, G. Kurizki and B. Sherman, Phys. Rev. A 54, 3535 (1996).
  7. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).
  8. S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. A 55, 1683 (1997).
  9. K. Moler, D. S. Weiss, M. Kasevich and S. Chu, Phys. Rev. A 45, 342 (1992).
  10. H. J. Kimble and W. Lange (private communication).
  11. C. K. Law and H. J. Kimble, J. Mod. Optics 44, 2067 (1997).
  12. See for example J. Martin, B. W. Shore and K. Bergmann, Phys. Rev. A 54, 1556 (1996) and references therein.
  13. 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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