## Non-classical states of the electromagnetic field in cavity QED

Optics Express, Vol. 3, Issue 4, pp. 147-153 (1998)

http://dx.doi.org/10.1364/OE.3.000147

Acrobat PDF (500 KB)

### Abstract

We propose simple experiments in cavity quantum electrodynamics leading, for the first time, to the measurement of negative values of the Wigner function of an electromagnetic field. We also show that the realization of a controlled-not gate within the framework of cavity QED is a special case of our proposal, and is equivalent to the measurement of the Wigner function of a one-photon field at the origin of phase space.

© Optical Society of America

## 1. Introduction

1. R.J. Glauber, Phys. Rev. **131**, 2766 (1963). [CrossRef]

2. R.J. Glauber, Phys. Rev. Lett. **10**, 84 (1963). [CrossRef]

3. E.C.G. Sudarshan, Phys. Rev. Lett. **10**, 277 (1963). [CrossRef]

*c*-numbers, and the quantum averages by integrals over phase space, with the Glauber-Sudarshan function acting as a probability measure. States for which this probability measure is positive would then be considered as classical-like states, while those for which the Glauber-Sudarshan representation is negative or more singular than a tempered distribution would be non-classical. Examples of such non-classical states would be squeezed fields [4

4. D. Stoler, Phys. Rev. D **1**, 3217 (1970). [CrossRef]

5. D. Stoler, Phys. Rev. D **4**, 1925 (1971). [CrossRef]

6. H.P. Yuen, Phys. Lett. A **51**, 1 (1976). [CrossRef]

7. H.P. Yuen, Phys. Rev. A **13**, 2226 (1976). [CrossRef]

8. C.M. Caves, Phys. Rev. D **23**, 1693 (1981). [CrossRef]

9. E. Wigner, Phys. Rev. **40**, 749 (1932). [CrossRef]

*x*and momentum

*p*, with a state characterized by a density operator

*ρ*̂, it is defined by

*W*(

*x*,

*p*) is real, and that ∫

*W*(

*x*,

*p*)

*dxdp*= 1. This distribution has the remarkable property that its integrations with respect to

*x*and

*p*yield the momentum and the position probability distributions, respectively. For this reason, it is considered sometimes as the closest possible analog within quantum mechanics of a classical phase-space distribution. In terms of the raising and lowering operators

*a*̂

^{†}and

*a*̂, defined by

10. K.E. Cahill and R.J. Glauber, Phys. Rev. **177**, 1857 (1969). [CrossRef]

11. K.E. Cahill and R.J. Glauber, Phys. Rev. **177**, 1882 (1969). [CrossRef]

*D*̂(

*z*,

*z*

^{*}) = exp(

*z*

*a*̂

^{†}-

*z*

^{*}

*a*̂) is the displacement operator, and

*z*= (

*mω*/2ħ)

^{1/2}(

*x*+

*ip*/

*mω*). The operator

*D*̂(

*z*,

*z*

^{*}) yields the coherent state |

*z*〉 when applied to the harmonic oscillator ground state. Expression (2) differs from (1) by the multiplicative factor 2

*πħ*, chosen so that ∫

*W*(

*z*,

*z*

^{*})

*d*

^{2}

*z*/

*π*= 1, where

*d*

^{2}

*z*=

*d*(

*ℜez*)

*d*(

*ℑmz*). Expression (2) is also the Wigner function corresponding to a mode of the electromagnetic field, if

*a*̂ and

*a*̂

^{†}are interpreted as the photon annihilation and creation operators, respectively.

13. K. Vogel and H. Risken, Phys. Rev. A **40**, 2847 (1989). [CrossRef] [PubMed]

14. U. Leonhardt and H. Paul, Phys. Rev. Lett. **72**, 4086 (1994). [CrossRef] [PubMed]

15. G.M. D’Ariano, U. Leonhardt, and H. Paul, Phys. Rev. A **52**, R1801 (1995). [CrossRef]

16. D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. **70**,1244 (1993). [CrossRef] [PubMed]

17. G. Breitenbach, T. Müller, S. F. Pereira, J.-Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B **12**, 2304 (1995). [CrossRef]

18. S. Schiller, G. Breitenbach, S.F. Pereira, T. Müller, and J. Mlynek, Phys. Rev. Lett. **77**, 2933 (1996). [CrossRef] [PubMed]

19. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. **77**, 4281(1996). [CrossRef] [PubMed]

## 2. Measurement of generalized phase-space distributions

20. L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. **78**, 2547 (1997). [CrossRef]

*superconducting cavity*

**Q****C**is placed between two low-

*cavities (*

**Q****R**

_{1}and

**R**in Fig. 1). The cavities

_{2}**R**and

_{1}**R**are connected to the same microwave generator, the field in

_{2}**R**being dephased by

_{2}*η*with respect to the field in

**R**. Another microwave source is connected to

_{1}**C**, allowing the injection of a coherent state in this cavity, so that the density operator

*ρ*̂ of the field to be measured is transformed into

*ρ̂′*=

*D*̂(

*z*,

*z*

^{*})

*ρ̂*

*D*̂

^{-1}(

*z*,

*z*

^{*}) . This system is crossed by a velocity-selected atomic beam, such that an atomic transition

*e*↔

*g*is resonant with the fields in

**R**and

_{1}**R**, while another transition

_{2}*e*↔

*i*is quasi-resonant (detuning

*δ*) with the field in

**C**, so that the atom interacts dispersively with this field if it is in state

*e*, while no interaction takes place in

**C**if the atom is in state

*g*. The relevant level scheme is shown in Fig. 2. Just before

**R**, the atoms are promoted to the highly excited circular Rydberg state |

_{1}*e*〉 (typical principal quantum numbers of the order of 50, corresponding to lifetimes of the order of some milliseconds). As each atom crosses the low-

*cavities, it sees a*

**Q***π*/2 pulse, so that ||

*e*〉 → [|

*e*〉 + exp(

*iη*)|

*g*〉]/√2, and |

*g*〉 → [- exp(-

*iη*)|

*e*〉 + |

*g*〉]/√2, with

*η*= 0 in

**R**. If the atom is in state

_{1}*e*when crossing

**C**, there is an energy shift of the atom-field system (Stark shift), which dephases the field, after an effective interaction time

*t*

_{int}between the atom and the cavity mode. The one-photon phase shift is given by

*ϕ*= (Ω

^{2}/

*δ*)

*t*

_{int}, where the Rabi frequency Ω measures the coupling between the atom and the cavity mode. The atom is detected and the experiment is repeated many times, for each amplitude and phase of the injected field

*z*, starting from the same initial state of the field

*ρ*̂. In this way, the probabilities

*P*and

_{e}*P*of detecting the probe atom in states

_{g}*e*or

*g*are determined. It is easy to show that

*η*= 0 and

*ϕ*=

*π*, we can see from (2) that

*g*〉 does not interact with the field in

**C**, this expression differs from the one given in Ref. [20

20. L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. **78**, 2547 (1997). [CrossRef]

*et al*[21

21. L. Davidovich, A. Maali, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. **71**,2360 (1993). [CrossRef] [PubMed]

22. L. Davidovich, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A **53**, 1295(1996). [CrossRef] [PubMed]

*α*〉 ± | -

*α*〉) /

*N*

_{±}could be observed by measuring the conditional probability of detecting the first (the builder) and the second (the probe) atoms at given states. The atomic configuration considered in that reference coincides with the one adopted here. Since no field was injected into the cavity between the two atoms, it is clear now that the experiment proposed in Ref. [21

21. L. Davidovich, A. Maali, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. **71**,2360 (1993). [CrossRef] [PubMed]

22. L. Davidovich, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A **53**, 1295(1996). [CrossRef] [PubMed]

*α*|

^{2}), and increases again as dissipation takes place, bringing the field to the vacuum state. Following this proposal, the first observation of decoherence was realized by Brune

*et al*. [23

23. M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. **77**, 4887 (1996). [CrossRef] [PubMed]

23. M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. **77**, 4887 (1996). [CrossRef] [PubMed]

21. L. Davidovich, A. Maali, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. **71**,2360 (1993). [CrossRef] [PubMed]

22. L. Davidovich, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A **53**, 1295(1996). [CrossRef] [PubMed]

*e*〉 and |

*g*〉 lead to dephas-ings (in opposite directions) of the field in

**C**. In this case, it is easy to show that the Wigner function is again recovered, as long as the one-photon phase shift is

*ϕ*=

*π*/2, and a dephasing

*η*=

*π*/2 is applied to the second Ramsey zone. [20

20. L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. **78**, 2547 (1997). [CrossRef]

23. M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. **77**, 4887 (1996). [CrossRef] [PubMed]

*ϕ*was actually smaller than

*π*/2.

*ϕ*=

*π*) be inferred from that measurement? This question can be answered by using the fact that the Wigner function belongs to a general class of phase-space distributions, parametrized by a complex parameter

*s*, and which can be written as [10

10. K.E. Cahill and R.J. Glauber, Phys. Rev. **177**, 1857 (1969). [CrossRef]

11. K.E. Cahill and R.J. Glauber, Phys. Rev. **177**, 1882 (1969). [CrossRef]

*W*(

*z*,

*z*

^{*},

*s*) is real when

*s*is real. For

*s*= 0, one obtains the Wigner distribution, while

*s*= - 1 and

*s*= 1 correspond respectively to the

*Q*and the Glauber-Sudarshan

*P*representations. [10

10. K.E. Cahill and R.J. Glauber, Phys. Rev. **177**, 1857 (1969). [CrossRef]

11. K.E. Cahill and R.J. Glauber, Phys. Rev. **177**, 1882 (1969). [CrossRef]

*s*= -

*i*cot

*ϕ*/2, (5) becomes [20

20. L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. **78**, 2547 (1997). [CrossRef]

*π*, one can see from (3) that by changing

*η*one may detect the real and the imaginary part of

*W*(

*z*,

*z*

^{*},

*ϕ*), given by (6). Therefore, one can measure phase space representations corresponding to imaginary values of

*s*. The connection between

*W*(

*z*,

*z*

^{*},

*ϕ*) and

*W*(

*z*,

*z*

^{*}) ≡

*W*(

*z*,

*z*

^{*},

*π*) can be obtained in the following way. It is easy to show from (5) that, setting

*τ*=

*is*and

*z*=

*x*+

*iy*,

*W*(

*x*,

*y*,

*τ*) obeys a free-particle SchrÖdinger equation, the parameter

*τ*playing the role of a time. As

*ϕ*changes from 2

*π*to 0,

*τ*changes correspondingly from

*τ*= -∞ to

*τ*= ∞. The behavior of the real part of

*W*(

*x*,

*y*,

*ϕ*) as

*ϕ*changes, for the state |-〉, is illustrated in the animation of Fig. 3. This behavior is easily understandable in terms of the development in time of a free wavepacket. In particular, the vanishing of

*W*(

*z*,

*ϕ*) when

*ϕ*= 0 (and therefore

*τ*= ∞) is a direct consequence of the wavepacket spreading. The interference fringes at the origin, displayed when

*ϕ*=

*π*(

*τ*= 0), may be thought as resulting from the collision of the two wavepackets counter-propagating along the

*x*axis and meeting at the origin of the phase space. Equation (7) also implies that

*W*(

*x*,

*y*,

*τ*) is connected to

*W*(

*x*,

*y*) through the free-particle propagator, which involves an integration of

*W*(

*x*,

*y*) over the whole phase space. Therefore, the reconstruction of the Wigner function from

*W*(

*z*,

*z*

^{*},

*ϕ*≠

*π*) is possible, but would require the knowledge of this generalized distribution for every value of the complex number

*z*. Furthermore, it is easy to see that a positive-definite Wigner function may give rise to negative values for the real and the imaginary parts of

*W*(

*z*,

*z*

^{*},

*ϕ*), and therefore no obvious characteristic sign of non-classicality can be found in

*W*(

*z*,

*z*

^{*},

*ϕ*).

## 3. Measuring a negative value of the Wigner function

*e*〉 is sent through the high-

*Q*cavity

**C**(vacuum state inside), its interaction time being such that the atom suffers a transition to state |

*g*〉 and one photon is left in the cavity. Such an experiment has been done recently by Maître

*et al*[24

24. X. Maître, E. Hagley, G. Nogues, and C. Wunderlich, Phys. Rev. Lett. **79**, 769 (1997). [CrossRef]

24. X. Maître, E. Hagley, G. Nogues, and C. Wunderlich, Phys. Rev. Lett. **79**, 769 (1997). [CrossRef]

*cavity, where it interacts dispersively with the field, producing a de-phasing equal to*

**Q***π*per photon if the atom is in level

*e*, and leaving the field unchanged if it is in level

*g*(our scheme is easily adapted to the case in which both atomic levels lead to a phase shift of the field in the cavity). We do not apply any displacement field, so that, according to (4), the quantity

*P*-

_{e}*P*yields the Wigner function at the origin of phase space. Since the Wigner function of a one-photon state is given by

_{g}*W*(

*z*,

*z*

^{*}) = 2

*e*

^{-|z|2}(2|

*z*|

^{2}- 1), it is clear that

*W*(0) = -2 < 0. This implies, according to (4), that Δ

*P*= 1, that is, the non-classical character of the state, associated to the negative value of the corresponding Wigner function at the origin of phase space, is translated, in this proposed experiment, into the fact that the probe atom comes out from the second Ramsey zone always in the upper state!

*e*〉 or in the |

*g*〉 state). This quantum-optics realization of a controlled-not gate was proposed in Ref. [25

25. L. Davidovich, N. Zagury, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A **50**,R895 (1994). [CrossRef]

*e*or in the

*g*state. Therefore, Δ

*P*= 0, corresponding to the well-known fact that the Wigner function of a coherent state with average number of photons much larger than one is practically equal to zero at the origin of phase space.

## 4. Conclusions

_{q}(Conselho Nacional de De-senvolvimento Científico e Tecnológico), CAPES (Coordenaça͂o de Aperfeiçoamento de Pessoal de Ensino Superior), PRONEX (Programa de Apoio a Núcleos de Excelência), and FUJB (Fundaça͂o Universitária José Bonifácio).

## References

1. | R.J. Glauber, Phys. Rev. |

2. | R.J. Glauber, Phys. Rev. Lett. |

3. | E.C.G. Sudarshan, Phys. Rev. Lett. |

4. | D. Stoler, Phys. Rev. D |

5. | D. Stoler, Phys. Rev. D |

6. | H.P. Yuen, Phys. Lett. A |

7. | H.P. Yuen, Phys. Rev. A |

8. | C.M. Caves, Phys. Rev. D |

9. | E. Wigner, Phys. Rev. |

10. | K.E. Cahill and R.J. Glauber, Phys. Rev. |

11. | K.E. Cahill and R.J. Glauber, Phys. Rev. |

12. | T.W. Marshall and E. Santos, quant-ph/9711046 (1997). |

13. | K. Vogel and H. Risken, Phys. Rev. A |

14. | U. Leonhardt and H. Paul, Phys. Rev. Lett. |

15. | G.M. D’Ariano, U. Leonhardt, and H. Paul, Phys. Rev. A |

16. | D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani, Phys. Rev. Lett. |

17. | G. Breitenbach, T. Müller, S. F. Pereira, J.-Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B |

18. | S. Schiller, G. Breitenbach, S.F. Pereira, T. Müller, and J. Mlynek, Phys. Rev. Lett. |

19. | D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. |

20. | L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. |

21. | L. Davidovich, A. Maali, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. |

22. | L. Davidovich, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A |

23. | M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. |

24. | X. Maître, E. Hagley, G. Nogues, and C. Wunderlich, Phys. Rev. Lett. |

25. | L. Davidovich, N. Zagury, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5290) Quantum optics : Photon statistics

(270.5580) Quantum optics : Quantum electrodynamics

**ToC Category:**

Focus Issue: Quantum noise reduction in optical systems

**History**

Original Manuscript: April 21, 1998

Published: August 17, 1998

**Citation**

L. Lutterbach and Luiz Davidovich, "Non-classical states of the electromagnetic field in cavity QED," Opt. Express **3**, 147-153 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-4-147

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### References

- R. J. Glauber, Phys. Rev. 131, 2766 (1963). [CrossRef]
- R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963). [CrossRef]
- E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). [CrossRef]
- D. Stoler, Phys. Rev. D 1, 3217 (1970). [CrossRef]
- D. Stoler, Phys. Rev. D 4, 1925 (1971). [CrossRef]
- H. P. Yuen, Phys. Lett. A 51, 1 (1976). [CrossRef]
- H. P. Yuen, Phys. Rev. A 13, 2226 (1976). [CrossRef]
- C. M. Caves, Phys. Rev. D 23, 1693 (1981). [CrossRef]
- E. Wigner, Phys. Rev. 40, 749 (1932). [CrossRef]
- K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969). [CrossRef]
- K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969). [CrossRef]
- T. W. Marshall and E. Santos, quant-ph/9711046 (1997).
- K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989). [CrossRef] [PubMed]
- U. Leonhardt and H. Paul, Phys. Rev. Lett. 72, 4086 (1994). [CrossRef] [PubMed]
- G. M. D'Ariano, U. Leonhardt and H. Paul, Phys. Rev. A 52, R1801 (1995). [CrossRef]
- D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Phys. Rev. Lett. 70,1244 (1993). [CrossRef] [PubMed]
- G. Breitenbach, T. M" uller, S. F. Pereira, J.-Ph. Poizat, S. Schiller and J. Mlynek, J. Opt. Soc. Am. B 12, 2304 (1995). [CrossRef]
- S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller and J. Mlynek, Phys. Rev. Lett. 77, 2933 (1996). [CrossRef] [PubMed]
- D. Leibfried D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano and D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996). [CrossRef] [PubMed]
- L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997). [CrossRef]
- L. Davidovich, A. Maali, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. Lett. 71, 2360 (1993). [CrossRef] [PubMed]
- L. Davidovich, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 53, 1295 (1996). [CrossRef] [PubMed]
- M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 77, 4887 (1996). [CrossRef] [PubMed]
- X. Maitre, E. Hagley, G. Nogues and C. Wunderlich, Phys. Rev. Lett. 79, 769 (1997). [CrossRef]
- L. Davidovich, N. Zagury, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 50, R895 (1994). [CrossRef]

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