Abstract

In this paper we give a survey of our experiments performed with the micromaser on the generation of Fock states. Three methods can be used for this purpose: the trapping states leading to Fock states in a continuous wave operation; state reduction of a pulsed pumping beam and finally using a pulsed pumping beam to produce Fock states on demand where trapping states stabilize the photon number. The latter method is discussed in detail by means of Monte Carlo simulations of the maser system. The results of the simulations are presented in a video.

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References

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  1. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W. M. Itano, and D. J. Wineland, "Experimental determination of the motional quantum state of a trapped atom," Phys. Rev. Lett 77, 4281-4285 (1996).
    [CrossRef] [PubMed]
  2. D. Meschede, H. Walther, and G. M�ller, "The one-atom-maser," Phys. Rev. Lett. 54, 551-554 (1985).
    [CrossRef] [PubMed]
  3. G. Rempe and H. Walther, "Sub-Poissonian atomic statistics in a micromaser," Phys. Rev. A 42, 1650-1655 (1990).
    [CrossRef] [PubMed]
  4. G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353-356 (1987).
    [CrossRef] [PubMed]
  5. G. Raithel, O. Benson, and H. Walther, "Atomic interferometry with the micromaser," Phys. Rev. Lett. 75, 3446-3449 (1995).
    [CrossRef] [PubMed]
  6. O. Benson, G. Raithel, and H. Walther, "Quantum jumps of the micromaser field - dynamic behavior close to phase transition points," Phys. Rev. Lett. 72, 3506-3509 (1994).
    [CrossRef] [PubMed]
  7. B.-G. Englert, M. L�ffer, O. Benson, B. Varcoe, M. Weidinger, and H. Walther, "Entangled atoms in micromaser physics," Fortschr. Phys. 46, 897-926 (1998).
    [CrossRef]
  8. H. J. Kimble, O. Carnal, N. Georgiades, H. Mabuchi, E. S. Polzik, R. J. Thomson and Q. A. Turchettte,"Quantum optics with strong coupling," Atomic Physics 14, D. J. Wineland, C. E. Wieman, and S. J. Smith, eds., AIP Press, 314-335 (1995).
  9. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
    [CrossRef]
  10. P. Meystre, G. Rempe, and H. Walther, "Very-low temperature behaviour of a micromaser," Opt. Lett. 13, 1078-1080 (1988).
    [CrossRef]
  11. M. Weidinger, B.T.H. Varcoe, R. Heerlein, and H. Walther, "Trapping states in the micromaser," Phys. Rev. Lett. 82, 3795-3798 (1999).
    [CrossRef]
  12. G. Antesberger, "Phasendiffusion und Linienbreite beim Ein-Atom-Maser," PhD Thesis, University of Munich, 1999.
  13. G. Raithel, et al., "The micromaser: a proving ground for quantum physics," in Advances in Atomic, Molecular and Optical Physics, Supplement 2, pages 57-121, P. Berman, ed., (Academic Press, New York, 1994).
  14. J. Krause, M. O. Scully, and H. Walther, "State reduction and |n>-state preparation in a high-Q micromaser," Phys. Rev. A 36, 4547-4550 (1987).
    [CrossRef] [PubMed]
  15. P. J. Bardoff, E. Mayr, and W.P. Schleich, "Quantum state endoscopy: measurement of the quantum state in a cavity," Phys. Rev. A 51, 4963-4966 (1995).
    [CrossRef]
  16. B. T. H. Varcoe, S. Brattke, M. Weidinger, and H. Walther, "Preparing pure photon number states of the radiation field," Nature 403, 743-746 (2000).
    [CrossRef] [PubMed]
  17. S. Brattke, B.-G. Englert, B. T. H. Varcoe, and H. Walther, "Fock states in a cyclically pumped one-atom maser," J. Mod. Opt. (in print).
  18. S. Brattke, B. T. H. Varcoe, and H. Walther, manuscript in preparation.
  19. C.T. Bodendorf, G. Antesberger, M. S. Kim, and H. Walther, "Quantum-state reconstruction in the one-atom maser," Phys. Rev. A 57, 1371-1378 (1998).
    [CrossRef]
  20. M.S. Kim, G. Antesberger, C.T. Bodendorf, and H. Walther, "Scheme for direct observation of the Wigner characteristic function in cavity QED," Phys. Rev. A 58, R65-R69 (1998).
    [CrossRef]

Other (20)

D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W. M. Itano, and D. J. Wineland, "Experimental determination of the motional quantum state of a trapped atom," Phys. Rev. Lett 77, 4281-4285 (1996).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. M�ller, "The one-atom-maser," Phys. Rev. Lett. 54, 551-554 (1985).
[CrossRef] [PubMed]

G. Rempe and H. Walther, "Sub-Poissonian atomic statistics in a micromaser," Phys. Rev. A 42, 1650-1655 (1990).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser," Phys. Rev. Lett. 58, 353-356 (1987).
[CrossRef] [PubMed]

G. Raithel, O. Benson, and H. Walther, "Atomic interferometry with the micromaser," Phys. Rev. Lett. 75, 3446-3449 (1995).
[CrossRef] [PubMed]

O. Benson, G. Raithel, and H. Walther, "Quantum jumps of the micromaser field - dynamic behavior close to phase transition points," Phys. Rev. Lett. 72, 3506-3509 (1994).
[CrossRef] [PubMed]

B.-G. Englert, M. L�ffer, O. Benson, B. Varcoe, M. Weidinger, and H. Walther, "Entangled atoms in micromaser physics," Fortschr. Phys. 46, 897-926 (1998).
[CrossRef]

H. J. Kimble, O. Carnal, N. Georgiades, H. Mabuchi, E. S. Polzik, R. J. Thomson and Q. A. Turchettte,"Quantum optics with strong coupling," Atomic Physics 14, D. J. Wineland, C. E. Wieman, and S. J. Smith, eds., AIP Press, 314-335 (1995).

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999).
[CrossRef]

P. Meystre, G. Rempe, and H. Walther, "Very-low temperature behaviour of a micromaser," Opt. Lett. 13, 1078-1080 (1988).
[CrossRef]

M. Weidinger, B.T.H. Varcoe, R. Heerlein, and H. Walther, "Trapping states in the micromaser," Phys. Rev. Lett. 82, 3795-3798 (1999).
[CrossRef]

G. Antesberger, "Phasendiffusion und Linienbreite beim Ein-Atom-Maser," PhD Thesis, University of Munich, 1999.

G. Raithel, et al., "The micromaser: a proving ground for quantum physics," in Advances in Atomic, Molecular and Optical Physics, Supplement 2, pages 57-121, P. Berman, ed., (Academic Press, New York, 1994).

J. Krause, M. O. Scully, and H. Walther, "State reduction and |n>-state preparation in a high-Q micromaser," Phys. Rev. A 36, 4547-4550 (1987).
[CrossRef] [PubMed]

P. J. Bardoff, E. Mayr, and W.P. Schleich, "Quantum state endoscopy: measurement of the quantum state in a cavity," Phys. Rev. A 51, 4963-4966 (1995).
[CrossRef]

B. T. H. Varcoe, S. Brattke, M. Weidinger, and H. Walther, "Preparing pure photon number states of the radiation field," Nature 403, 743-746 (2000).
[CrossRef] [PubMed]

S. Brattke, B.-G. Englert, B. T. H. Varcoe, and H. Walther, "Fock states in a cyclically pumped one-atom maser," J. Mod. Opt. (in print).

S. Brattke, B. T. H. Varcoe, and H. Walther, manuscript in preparation.

C.T. Bodendorf, G. Antesberger, M. S. Kim, and H. Walther, "Quantum-state reconstruction in the one-atom maser," Phys. Rev. A 57, 1371-1378 (1998).
[CrossRef]

M.S. Kim, G. Antesberger, C.T. Bodendorf, and H. Walther, "Scheme for direct observation of the Wigner characteristic function in cavity QED," Phys. Rev. A 58, R65-R69 (1998).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

The micromaser setup. For details see Ref. [11].

Fig. 2.
Fig. 2.

A theoretical plot, in which the trapping states can be seen as valleys in the N ex direction. As the pump rate is increased, the formation of the trapped states from the vacuum can be seen.

Fig. 3.
Fig. 3.

A numerical simulation of the photon number distribution as the atomic pump rate (N ex) is increased until the cavity field is in a Fock state with a high probability.

Fig. 4.
Fig. 4.

Purity of Fock states under the trapping condition for n=0 to n=5 (n th=10-4).

Fig. 5.
Fig. 5.

Maser resonance and trapping condition. The left column shows the results of a simulation. The oscillations are due to Rabi flopping. The right column shows the corresponding experimental results. The atom flux is N ex=11. The minimum at resonance for t int=80µs corresponds to the vacuum trapping state. That for t int=60µs is due to the (1, 1) trapping state. The minima at larger detunings are due to Rabi flopping of the vacuum trapping state. For details see text.

Fig. 6.
Fig. 6.

(A), (B) and (C): Three Rabi oscillations are presented, for the number state n=0, 1 and 2. (a), (b) and (c): plots display the coefficients P n. The photon distribution P n was calculated for each Rabi cycle by fitting Eq. 3 to each plot for the set of photon numbers, n=0 to n=3. The relative phase of the Rabi frequency was fixed since all the atoms enter in the excited state of the maser transition. In each fit the highest probability was obtained for the target number state. Unlike the n=1 and n=2 Rabi cycles, the n=0 oscillation (Fig. 6(A)) was obtained in the steady-state operation of the micromaser in a very low-flux regime. The fit to this curve was performed for Rabi cycles from n=0 to n=2. The low visibility of this curve was due to the low flux (one atom/s) which was required to reduce the steady-state operation of the micromaser to below-threshold behavior hence detector dark counts become comparable to the real count rates and therefore contribute to a large background. To improve the measurements for photon number of n=3 and higher, the range of interaction times would have to be extended beyond 120 µs, this is not possible with the current apparatus. During the Rabi cycle the cavity photon number changes periodically. At the maxima there is one photon more than at the minima. The Rabi oscillation thus allow one to perform a non-destructive and repeated measurement of the photon number. In connection with the discussion of trapping states, it is interesting to note that minima in the number state Rabi oscillations correspond precisely to the trapping states conditions of the steady-state field. Therefore the large possible storage times of single photons would allow one to investigate the transition from a pulsed to a steady state experiment.

Fig. 7.
Fig. 7.

Comparison between theory and experimental results on the purity of number states. The columns represent photon distributions obtained from; (a) a theoretical simulation of the current experiment; (b) the current experimental results; and (c) a theoretical model that extends the current experiment to the steady state at the positions of the trapping states. The agreement between the two theoretical results and the experimental result is remarkable, indicating that dissipation is the most likely loss mechanism. Without dissipation, ie in the moment of generation the purity of the states is 99 % for n=1 and 95 % for n=2.

Fig. 8.
Fig. 8.

(974 kb) Video to demonstrate the generation of Fock states on demand. Shown is a Monte Carlo computer simulation of the interaction of Rydberg atoms with the cavity. A sequence of four Rydberg atoms on the average is interacting with the maser cavity. After the cavity atoms in the upper maser state are in red; those in the lower state are indicated by black squares. They enter the cavity with a Poissonian statistics. The photon emission events in the cavity are recorded and summed up in the box on the lower part of the figure. The accumulated result shows the deviations from single photon emission. It is shown that 100 pulses lead to 98 single photon emissions, one with no emission and one with the emission of two photons. In the video only the result of every third sequence of atoms is shown in order to reduce the length of the video, however, the real outcome of the situation is incorporated in the emitted photon number. Simultaneously with the emission of a photon one lower state atom is produced. The video shows that besides single photons,single lower state atoms can be generated on demand with high probability (also 97 %).

Fig. 9.
Fig. 9.

Figures 9 (a) and (b) show a comparison for one photon Fock state generation under the conditions of the (1, 1) and (1, 2) trapping states. Higher emission probability into the vacuum for the (1,2) trapping state means a faster approach to the operation of an unconditional single photon Fock source. However, violation of the trapping conditions by a thermal photon causes higher emission at high pump rates, which means that the (1, 2) tapping state is more vulnerable. The (1, 1) condition therefore reaches a higher final Fock state creation probability. The conditions for this simulation are τ cav=100 ms, τ pulse=2 ms, n th=10-4. Figures (c) and (d) demonstrate the robustness of the unconditional Fock source. Presented here is the probability of finding exactly one atom per pulse (p (1)) for a range of experimental conditions. Figure (c) shows the robustness of the Fock source against interaction time averaging. Figure (d) shows the robustness of the Fock source as a function of temperature. It should be emphasized that the upper level of vibrations and thermal photons considered in this figure are extreme conditions and very much higher than those of a typical experiment. Experimental parameters of (n th=0.03, Δt int/t int=0.02) are well within these limits. The threshold, N Thr, for Fock state operation (dotted vertical line) and the pump rate, N a, attained in our present experiment (broken vertical line) are both indicated on the figure (see Ref. [18].

Equations (6)

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Ω t int n + 1 = k π .
P em ( Δ , t int ) = 4 Ω 2 Δ 2 + 4 Ω 2 sin 2 ( 1 2 Δ 2 + 4 Ω 2 t int )
Ψ = cos ( ϕ ) e n i sin ( ϕ ) g n + 1
I ( n , t int ) = n P n cos ( 2 Ω n + 1 t int )
P g = sin 2 ( n + 1 Ω t int )
P max = 1 e P g N a .

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