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Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 4 — Aug. 17, 1998
  • pp: 147–153
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Non-classical states of the electromagnetic field in cavity QED

L. G. Lutterbach and L. Davidovich  »View Author Affiliations


Optics Express, Vol. 3, Issue 4, pp. 147-153 (1998)
http://dx.doi.org/10.1364/OE.3.000147


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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

The concept of non-classical light has been closely tied to phase-space representations of the electromagnetic field. Most often, the Glauber-Sudarshan representation [1

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

, 2

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

, 3

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

] is used to define the non-classical character of the state. This representation arises naturally in the theory of photodetection, and it allows the calculation of quantum averages of normal-ordered functions of the photon annihilation and creation operators as if they were classical averages, the operators being replaced by 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

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

, 6

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

, 7

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

, 8

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

], for which the variance of a quadrature is smaller than the value corresponding to a coherent state.

A more strict characterization of the non-classical character of the field involves the Wigner function [9

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

], which allows the calculation of symmetrical products of operators as classical-like integrals. For a one-dimensional particle with position x and momentum p, with a state characterized by a density operator ρ̂, it is defined by

W(x,p)=12πħeipy/ħxy2ρ̂x+y2dxdp.
(1)

It is easy to show that 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 â and â, defined by x̂=ħ/2(â+â),p̂=iħ/2(ââ) one may write [10

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

, 11

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

]:

W(z,z*)=2Tr[ρ̂D̂(z,z*)eiπââD̂1(z,z*)],
(2)

where D̂(z, z *) = exp(z â - z * â) is the displacement operator, and z = (/2ħ)1/2(x + ip/). The operator D̂(z,z *) yields the coherent state |z〉 when applied to the harmonic oscillator ground state. Expression (2) differs from (1) by the multiplicative factor 2πħ, 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 â and â are interpreted as the photon annihilation and creation operators, respectively.

If one takes the Wigner distribution as a benchmark, a state would be non-classical only if its corresponding Wigner function becomes negative in some region of phase space. This criterion would exclude minimum-uncertainty squeezed states [4

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

, 5

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

, 6

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

, 7

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

, 8

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

] from the realm of non-classical states: indeed, the Wigner function associated with those states is a Gaussian, obtained by applying a scale transformation to the Gaussian Wigner distribution corresponding to a coherent state. Some authors [12

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

] actually claim that processes leading to electromagnetic fields characterized by positive Wigner functions may be described without evoking the concept of the photon.

Independently of the point-of-view one might adopt regarding the “classicality” of a state of the electromagnetic field, it is remarkable that, up to now, there has been no experiment leading to the measurement of negative values of the Wigner function of an electromagnetic field, notwithstanding the recent development of techniques for the experimental determination of the Wigner distribution. [13

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

, 14

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

, 15

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

] These techniques have allowed in particular the reconstruction of the Wigner representation of minimum-uncertainty squeezed states of light. [16

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

, 17

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

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

] The situation regarding the electromagnetic field contrasts with the one concerning material oscillators: indeed, the Wigner function corresponding to a harmonic oscillator eigenstate has been measured for a trapped ion, and a region of negative values was clearly displayed. [19

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]

]

In this paper, we propose an experiment which leads to the direct measurement of a negative value of the Wigner function, and show that this experiment admits a simple geometrical interpretation.

2. Measurement of generalized phase-space distributions

Figure 1. Experimental scheme

ΔP=PePg=e{eTr[D̂(z,z*)ρ̂D̂1(z,z*)eaa]}.
(3)

Setting η = 0 and ϕ = π, we can see from (2) that

ΔP=PePg=W(z,z*)/2.
(4)

Therefore, the difference between the two probabilities yields a direct measurement of the Wigner function (one should note that, due to the fact that here |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]

]).

Figure 2. Atomic level scheme. The transition ie is detuned by δ from the frequency ω of a mode of cavity C, while the transition eg is resonant with the fields in R1 and R2. State |g〉 is not affected by the field in C.

It is interesting to compare the procedure described above with the one suggested by Davidovich 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

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

], with the objective of observing the decoherence of a SchrÖdinger cat-like state. In that reference, it was proposed that the decoherence of the state |±〉 = (|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

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

] amounts to a measurement of the Wigner function at the origin, which is non zero for the pure state |±〉, vanishes after the decoherence time (shorter than the intensity decay time by the factor |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]

] There are however two main differences between the experiment reported in [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]

] and the one proposed in [21

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

, 22

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

]. In the experiment, both |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]

] This condition was not satisfied however in the experiment reported in Ref. [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]

]: due to experimental limitations, the angle ϕ was actually smaller than π/2.

One is led therefore to a natural question: can the Wigner function (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

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

]:

W(z,z*,s)=ezξ*z*ξesξ2/2Tr[ρ̂D̂(ξ,ξ*)]π1d2ξ.
(5)

Note that 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

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

] Setting s = -icotϕ/2, (5) becomes [20

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

]:

W(z,z*ϕ)=2ie/2sinϕ2Tr[D̂(z,z*)ρ̂D̂(z,z*)eââ].
(6)

For ϕ = π (s = 0), we recover Eq. (2).

If the phase shift is different from 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,

iW(x,y,τ)τ=18(2x2+2y2)W(x,y,τ),
(7)

so that W(x, y, τ) obeys a free-particle SchrÖdinger equation, the parameter τ playing the role of a time. As ϕ changes from 2π 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 * ,ϕπ) 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).

Figure 3. Animation showing the real part of the Wigner function for the “odd cat” (|α〉 - | - α〉)/N -, as ϕ changes from 0 to π. The frame above corresponds to ϕ = Pi/2. [Media 1]

3. Measuring a negative value of the Wigner function

We propose now a simple experiment which leads to the measurement of negative values for the Wigner function associated with a quantum state of the electromagnetic field in a cavity. The experiment is realized in two steps. In the first step, the two Ramsey zones are inactive, and a resonant atom in state |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]

]. One should note that it is relatively easy to tune the atom in and out of resonance with the cavity mode, by Stark shifting the atomic transition with the help of electric fields applied across the gap between

the mirrors [24

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

]. Having thus constructed the one-photon Fock state, one proceeds as described above, sending a second atom which crosses the two Ramsey zones and the high-Q cavity, where it interacts dispersively with the field, producing a de-phasing equal to 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 Pe - Pg yields the Wigner function at the origin of phase space. Since the Wigner function of a one-photon state is given by W(z,z *) = 2e -|z|2 (2|z|2 - 1), it is clear that W(0) = -2 < 0. This implies, according to (4), that ΔP = 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!

This result has a simple geometrical interpretation, in terms of the Bloch vector associated with the two-level atom, as shown in Fig. 4: the final position of the Bloch vector depends on whether the number of photons in the cavity is odd or even, and this is at the origin of the manifestation of the non-classical behavior of the field. It follows immediately that the Wigner functions corrresponding to the states |-〉 and |+〉 have negative and positive values at the origin, respectively. One should note that the dependence on the photon number parity of the final atomic state leads to a controlled-not logic gate: the atomic state does not change if there is one photon in the cavity, while the absence of photons in the cavity leads to a change of atomic state. We have therefore the situation in which the state of one of the bits (0 or 1 photon in the cavity) controls what happens to the other bit (atom in the |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]

] as a basic ingredient for the teleportation of atomic states. We see that, in the present context, it is associated to the measurement of a striking non-classical manifestation of a Fock state: the negative value of the corresponding Wigner function at the origin of phase space. In the classical limit, attained for coherent states with an average number of photons much larger than one, odd and even photon populations come up with approximately the same weight, implying that there is an equal probability that the atom comes up in the e or in the g state. Therefore, ΔP = 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.

Figure 4. Evolution of the Bloch vector (green) of an atom that crosses the proposed setup, interacting with the eletromagnetic field (red) in the two Ramsey zones, and having a dispersive interaction with the field in the superconducting cavity C, when the number of photons of the field in C has a well defined parity. The atom is initially in the state e. As the atom crosses the first Ramsey zone, its Bloch vector is rotated by π/2 around the vector representing the electromagnetic field along the real polarization axis, as shown in (a) and (d). As the atom crosses the cavity C, the Bloch vector rotates around the population axis. If the number of photons in the cavity is odd (b), the Bloch vector ends up pointing towards the opposite direction, and the rotation in the second zone leads the atom back to state |e〉 (c). On the other hand, if the number of photons in C is even, the Bloch vector turns by an integer multiple of 2π, so its direction does not change (d). The second Ramsey zone then brings the atom to |g〉 (f).

4. Conclusions

This research was partially supported by CNPq (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. 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]

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]

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]

12.

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

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]

20.

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

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]

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]

24.

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

25.

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

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

  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]
  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]
  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]
  12. T. W. Marshall and E. Santos, quant-ph/9711046 (1997).
  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. M" uller, 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. Muller 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]
  20. L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997). [CrossRef]
  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]
  23. 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]
  24. X. Maitre, E. Hagley, G. Nogues and C. Wunderlich, Phys. Rev. Lett. 79, 769 (1997). [CrossRef]
  25. L. Davidovich, N. Zagury, M. Brune, J. M. Raimond and S. Haroche, Phys. Rev. A 50, R895 (1994). [CrossRef]

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