## Preparing Fock states in the micromaser

Optics Express, Vol. 8, Issue 2, pp. 131-144 (2001)

http://dx.doi.org/10.1364/OE.8.000131

Acrobat PDF (659 KB)

### 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.

© Optical Society of America

## 1 Introduction

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. Müller, “The one-atom-maser,” Phys. Rev. Lett. **54**, 551–554 (1985). [CrossRef] [PubMed]

^{10}, leading to a photon lifetime in the cavity of 0.3s. The steady-state field generated in the cavity has already been the object of detailed studies of the sub-Poissonian statistical distribution of the field [3

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. Löffler, O. Benson, B. Varcoe, M. Weidinger, and H. Walther, “Entangled atoms in micromaser physics,” Fortschr. Phys. **46**, 897–926 (1998). [CrossRef]

^{-2}photons per mode.

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]

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]

^{3}

*He*-

^{4}

*He*dilution refrigerator houses the microwave cavity which is a closed superconducting niobium cavity. A rubidium oven provides two collimated atomic beams: a central one passing directly into the cryostat and a second one directed to an additional excitation region. The second beam was used as a frequency reference. A frequency doubled dye laser (λ=297 nm) was used to excite rubidium (

^{85}

*Rb*) atoms to the Rydberg 63 P

_{3/2}state from the 5 S

_{1/2}(F=3) ground state.

_{1/2}(F=3)-63 P

_{3/2}transition of the reference atomic beam excited under normal incidence. The reference transition was detuned by Stark shifting the resonance frequency using a stabilized power supply. This enabled the laser to be tuned while remaining locked to an atomic transition. The maser frequency corresponds to the transition between 63 P

_{3/2}and 61 D

_{5/2}. The Rydberg atoms are detected by field ionization in two detectors set at different voltages so that the upper and lower states can be detected separately.

10. P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,” Optics 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]

*t*

_{int}, are chosen such that in a cavity field with

*n*photons each atom undergoes an integer number,

*k*, of Rabi cycles. This is summarized by the condition,

*N*

_{ex}(pump rate per decay time of the cavity. The trapping state is therefore characterized by the photon number

*n*and the number of integer multiples of full Rabi cycles

*k*.

*-P*

_{g}*is plotted against interaction time and pump rate; P*

_{e}_{g(e)}is the probability of finding a ground (excited) state atom. At low atomic pump rates (low

*N*

_{ex}) the maser field cannot build up and the maser exhibits Rabi oscillations due to the interaction with the vacuum field. At the positions of the trapping states, the field increases until it reaches the trapping state condition. This manifests itself as a reduced emission probability and hence as a dip in the atomic inversion. Once in a trapping state the maser will remain there regardless of the pump rate. The trapping states therefore show up as valleys in the

*N*

_{ex}direction. Figure 3 shows the photon number distribution as the pump rate is increased for the special condition of the five photon trapping state. The photon distribution develops from a thermal distribution towards higher photon numbers until the pump rate is high enough for the atomic emission to be governed by the trapping state condition. As the pump rate is further increased, and in the limit of a low thermal photon number, the field continues to build up to a single trapped photon number and the cavity field approaches a Fock state.

*n*=0 to

*n*=5. The experimental realization requires a pump rate of

*N*

_{ex}=50, a temperature of less than 300mK, a high selectivity of atomic velocity[11

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]

**82**, 3795–3798 (1999). [CrossRef]

*≈1 the oscillatory behavior of the photon emission probability is given by*

_{ex}*N*

_{ex}=11. The results are shown for different interaction times. Whenever the photon number passes through a minium indicates that the trapping condition is fulfilled for this particular detuning. For the

*t*

_{int}=80

*µs*and

*t*

_{int}=70

*µs*results all the minima correspond to the vacuum trapping state. The minimum at detuning 0 for

*t*

_{int}=60

*µs*corresponds to the (1, 1) trapping state whereas the minima closest to the central maximum for

*t*

_{int}=90

*µs*corresponds to the (2, 1) trapping state.

**82**, 3795–3798 (1999). [CrossRef]

*n*=5 could be identified. The setup we use in the moment does not allow us to investigate the purity of the Fock states obtained under the trapping condition, however, the dynamical generation of Fock states described in the next chapter allows to perform such an experiment.

## 2 Dynamical preparation of |n〉-photon states in a cavity

*n*〉 then the interaction of an atom with the cavity leaves the cavity field in a superposition of the states |

*n*〉 and |

*n*+1〉 and the atom in a superposition of the internal atomic states |

*e*〉 and |

*g*〉. The entangled state can be described by:

*ϕ*is an arbitrary phase. The state selective field ionization measurement of the internal atomic state, reduces the field to one of the states |

*n*〉 or |

*n*+1〉. State reduction is independent of the interaction time, hence a ground state atom always projects the field onto the |

*n*+1〉 state independent of the time spent in the cavity. This results in an

*a priori*probability of the maser field being in a specific but unknown number state [14

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]

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]

*n*in a particular mode is not observed and the field always evolves into a mixture of such states. Therefore the inversion is generally given by

*P*

_{n}is the probability of finding

*n*photons in the mode.

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]

*n*=2 in the cavity [16

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]

*n*photon number state in the cavity, N=

*n*+1 atoms are required. That is

*n*atoms to create the number state and the final atom as a probe of the state. However owing to the non-perfect detector efficiency and atomic decay there are missed counts. By using a laser pulse of short duration the number of excited-state atoms entering the cavity per pulse is rather low. Hence we know that when

*N*atoms per pulse were detected the probability of having N+1 atoms per pulse was negligibly small. This was achieved by modifying the UV excitation pulse such that the mean number of atoms per pulse was between 0.2 and 0.8. With 40 % detector efficiency and the assumption that the probability of missing a count is statistically independent, there is a probability of about 1 % of the state preparation being incorrect because an atom escapes detection. As the flux of atoms was variable, the pulse duration was also variable, a maximum sampling time of 3 ms for the

*n*=1 data and 5 ms for the

*n*=2 data was imposed to limit the time delay between the pump and probe atoms. Actually in most cases the time delay was comparable to the excitation pulse duration. For the measurement of an

*n*-photon number state, the detection of the probe atom is triggered by the detection of

*n*ground state atoms within the length of the laser pulse. If too few or too many atoms (upper or lower state) are detected within the laser pulse duration, the measurement is rejected.

*Q*value be lower than ultimately possible in our setup, since a higher

*Q*would lead to an increase of the data collection time. Even with the reduced cavity life time of 25 ms and large delay times between the laser pulses a cyclically steady state maser field can build up in the cavity. The time delay between pulses was selected as a compromise between limiting the growth of the maser field and the length of the data collection time.

*n*=0, 1 or 2 ground state atoms respectively.

*n*=2 Rabi data was more difficult to collect than the other two measurements. The data collection time became substantially longer as the interaction time was increased and background effects have a higher impact on the data. The fit to the

*n*=2 data includes an exponentially decreasing weight, so that measurements obtained for longer interaction times have less significance than those at short times.

*n*=1 state and 95 % for the

*n*=2 state at the time of generation which then is modified by dissipation between production and measurement.

*n*=1) or two photons were deposited one by one (for

*n*=2) changing the photon number distribution. The simulations also average over the Poissonian arrival times of the atoms. The details of this calculation have been discussed in detail previously [17]. The results of these calculations are compared to the experimental results in Fig. 7a and Fig. 7b.

## 3 Preparing Fock states on demand

*τ*≤

_{cav}*τ*

_{pulse}≤0.1

*τ*

_{cav}) so there is little dissipation and the one photon state in the cavity following the pulse is very close to the probability of finding an atom in the lower state. Note that at no time in this process a detector event is required to project the field, the field evolves to the trapping state as a function of time automatically, when the suitable interaction time has been chosen.

*n*+1 emission probability is small. Thus although the (1, 2) trapping state is slightly more favorable for small average atom numbers, it is more unstable at higher average atom numbers and the (1, 1) trapping state reaches a higher total probability of single photon Fock state creation. The change of the emission probability as a function of the photon number

*n*by a single quantum thus has an appreciable effect on the evolution of the system. This discussion acquires more relevance when the creation of Fock states ≥2 is considered.

*N*

_{a}is the average number of atoms per pulse;

*N*

_{a}is the most important factor when comparing different operating conditions. A critical value of

*N*

_{a}can be defined that can be considered a threshold pump rate. We define the threshold pump rate to be

*N*

_{Thr}=2/

*P*leading to a threshold of

_{g}*N*

_{Thr}=3.16 for the (1, 1) trapping state and

*N*

_{Thr}=2.15 for the (1, 2) trapping state.

*τ*

_{cav}for dissipative losses to be less than 10 %. Apart from reducing the fidelity of the Fock state produced, losses increase the likelihood of a second emission event leading to a larger number of lower state atoms than photons in the field; whereby the 1:1 correspondence between both would be lost. Shorter atom pulses reduce the dissipative loss, however, the number of atoms per cavity decay time (usually labeled

*N*

_{ex}) must be larger than

*N*

_{Thr}to realize the Fock source with a significant fidelity. Since a minimum atom number is required to produce the desired state, care must also be taken to avoid atom beam densities violating the one-atom-at-a-time condition.

*τ*

_{cav}for the parameters used in this paper. The separation of the pulses is ≥3

*τ*

_{cav}leading to a small relative variation in the arrival times. If one would increase the pump rate still further, the pulse lengths could be further reduced and the arrival of an atom becomes even more predictable.

## 4 Conclusion

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]

## References and links

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 |

2. | D. Meschede, H. Walther, and G. Müller, “The one-atom-maser,” Phys. Rev. Lett. |

3. | G. Rempe and H. Walther, “Sub-Poissonian atomic statistics in a micromaser,” Phys. Rev. A |

4. | G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atom maser,” Phys. Rev. Lett. |

5. | G. Raithel, O. Benson, and H. Walther, “Atomic interferometry with the micromaser,”Phys. Rev. Lett. |

6. | O. Benson, G. Raithel, and H. Walther, “Quantum jumps of the micromaser field - dynamic behavior close to phase transition points,” Phys. Rev. Lett. |

7. | B.-G. Englert, M. Löffler, O. Benson, B. Varcoe, M. Weidinger, and H. Walther, “Entangled atoms in micromaser physics,” Fortschr. Phys. |

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 Physics14, 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 |

10. | P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,” Optics Lett. |

11. | M. Weidinger, B.T.H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. |

12. | G. Antesberger, “Phasendiffusion und Linienbreite beim Ein-Atom-Maser,” PhD Thesis, University of Munich, 1999. |

13. | G. Raithel, |

14. | J. Krause, M. O. Scully, and H. Walther, “State reduction and |n>-state preparation in a high-Q micromaser,” Phys. Rev. A |

15. | P. J. Bardoff, E. Mayr, and W.P. Schleich, “Quantum state endoscopy: measurement of the quantum state in a cavity,” Phys. Rev. A |

16. | B. T. H. Varcoe, S. Brattke, M. Weidinger, and H. Walther, “Preparing pure photon number states of the radiation field,” Nature |

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 |

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 |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 15, 2000

Published: January 15, 2001

**Citation**

Simon Brattke, Ben Varcoe, and Herbert Walther, "Preparing Fock states in the micromaser," Opt. Express **8**, 131-144 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-131

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### References

- 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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