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Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 2 — Jan. 15, 2001
  • pp: 131–144
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Preparing Fock states in the micromaser

Simon Brattke, Ben T. H. Varcoe, and Herbert Walther  »View Author Affiliations


Optics Express, Vol. 8, Issue 2, pp. 131-144 (2001)
http://dx.doi.org/10.1364/OE.8.000131


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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

The quantum treatment of the radiation field uses the number of photons in a particular mode to characterize the quantum states. In the ideal case the modes are defined by the boundary conditions of a cavity giving a discrete set of eigen-frequencies. The ground state of the quantum field is represented by the vacuum state consisting of field fluctuations with no residual energy. The states with fixed photon number are usually called Fock or number states. They are usually used as a basis in which any general radiation field state can be expressed. Fock states thus represent the most basic quantum states and differ maximally from what one would call a classical field. Although Fock states of vibrational motion are routinely observed in ion traps [1

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]

], Fock states of the radiation field are very fragile and very difficult to produce and maintain. They are perfectly number-squeezed, extreme sub-Poissonian states in which intensity fluctuations vanish completely. In order to generate these states it is necessary that the mode considered has minimal losses and the thermal field, always present at finite temperatures, has to be eliminated to a large extent since it causes photon number fluctuations.

The one-atom maser or micromaser [2

2. D. Meschede, H. Walther, and G. Müller, “The one-atom-maser,” Phys. Rev. Lett. 54, 551–554 (1985). [CrossRef] [PubMed]

] is the ideal system to realize Fock states. In the micromaser highly excited Rydberg atoms interact with a single mode of a superconducting cavity which can have a quality factor as high as 3×1010, 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]

], the quantum dynamics of the atom-field photon exchange represented in the collapse and revivals of the Rabi nutation [4

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]

], atomic interference [5

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

], bistability and quantum jumps of the field [6

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]

], atom-field and atom-atom entanglement [7

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]

]. The cavity is operated at a temperature of 0.2 K leading to a thermal field of about 5×10-2 photons per mode.

There have been several experiments published in which the strong coupling between atoms and a single cavity mode is exploited (see e.g. Ref. [8

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).

]). The setup described here is the only one where maser action can be observed and the maser field investigated. The threshold for maser action is 1.5 atoms/s. This is a consequence of the high value of the quality factor of the cavity which is three orders of magnitude larger than that of other experiments with Rydberg atoms and cavities [9

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]

].

In this paper we present three methods of creating number states in the micromaser. The first is by way of the well known trapping states, which are generated in a c.w. operation of the pumping beam and lead to Fock states with high purity. We also present a second method using the entanglement between pumping atoms and cavity field. The field is prepared by state reduction and the purity of the states generated investigated by a probing atom. It turns out that the two methods of preparation of Fock states are in fact equivalent and lead to a similar result for the purity of the Fock states. The third method pumps the cavity with a pulsed beam using the trapping condition to stabilize the photon number, producing Fock states on demand.

The micromaser setup used for the experiments is shown in Fig. 1 and has been described in detail previously [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]

]. Briefly, in this experiment, a 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 P3/2 state from the 5 S1/2(F=3) ground state.

Fig. 1. The micromaser setup. For details see Ref. [11].

Velocity selection is provided by angling the excitation laser towards the main atomic beam at 11° to the normal. The dye laser was locked, using an external computer control, to the 5 S1/2(F=3)-63 P3/2 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 P3/2 and 61 D5/2. 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.

The trapping states are a continuous wave operation of the maser field peaked in a single photon number, they occur in the micromaser as a direct consequence of the quantization of the cavity field. At low cavity temperatures the number of blackbody photons in the cavity mode is reduced and trapping states begin to appear [10

10. P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,” Optics Lett. 13, 1078–1080 (1988). [CrossRef]

, 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]

]. They occur when the atom field coupling, Ω, and the interaction time, 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,

Ωtintn+1=kπ.
(1)

When Eq.1 is fulfilled the cavity photon number is left unchanged after the interaction of an atom and hence the photon number is “trapped”. This will occur regardless of the atomic pump rate 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.

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.

The build up of the cavity field can be seen in Fig.2, where the emerging atom inversion I=Pg-Pe is plotted against interaction time and pump rate; Pg(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.

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.

Under the condition that the pumping beam is very weak Nex≈1 the oscillatory behavior of the photon emission probability is given by

Pem(Δ,tint)=4Ω2Δ2+4Ω2sin2(12Δ2+4Ω2tint)
(2)

This oscillatory behavior results from Rabi flopping, however, since there is detuning the observed flopping frequency is higher than the one photon Rabi frequency, therefore many more periods are observed at finite detuning than at resonance.

If the flux is increased the average photon number in the cavity will increase since a steady state field will build up; nevertheless the oscillations are still visible as can be seen in Fig. 5 left column showing a simulation for 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=80µs and t int=70µs results all the minima correspond to the vacuum trapping state. The minimum at detuning 0 for t int=60µs corresponds to the (1, 1) trapping state whereas the minima closest to the central maximum for t int=90µs corresponds to the (2, 1) trapping state.

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

The corresponding experimental results are shown in the right column. For the experimental results the inversion is plotted which is experimentally determined. The agreement between experiment and theory is reasonable.

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

In the following we will describe an alternative method of generating number states. As mentioned above this method allows to analyze in an unambiguous way the purity of the states generated. For this purpose we use a pulsed excitation of the Rydberg atoms which pump the maser. We start the discussion of the method with some general remarks.

When the atoms leave the cavity in a micromaser experiment they are in an entangled state with the field. If the field is in an initial state |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:

Ψ=cos(ϕ)enisin(ϕ)gn+1
(3)

where 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]

]. If the initial state is the vacuum, |0〉, then a number state is created in the cavity being equal to the number of ground state atoms that were collected within a suitably small fraction of the cavity decay time. This is the essence of the method of preparing Fock states by state reduction proposed by Krause et al. [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]

].

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.

In a system like the micromaser the spontaneous emission is reversible and an atom in the presence of a resonant quantum field undergoes Rabi oscillations. That is the relative populations of the excited and ground states of the atom oscillate at a frequency Ωn+1. As mentioned above, experimentally the atomic inversion is investigated. In the presence of dissipation a fixed photon number 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

I(n,tint)=nPncos(2Ωn+1tint)
(4)

where P n is the probability of finding n photons in the mode.

The method we are going to describe corresponds to a pump-probe experiment in which pump atoms prepare a quantum state in the cavity which is subsequently measured by a probing atom by studying the Rabi nutation. The signature that the quantum state of interest has been prepared is simply the detection of a defined number of ground state atoms. To verify that the correct quantum state has been projected onto the cavity a probe atom is sent into the cavity with a variable, but well defined, interaction time in order to allow the measurement of the Rabi nutation. As the formation of the quantum state is independent of the interaction time we need not change the relative velocity of the pump and probe atoms, thus reducing the complexity of the experiment. In this sense we are performing a reconstruction of a quantum state in the cavity using a similar method to that described by Bardoff et al. [15

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]

]. This experiment reveals the maximum amount of information that can be found relating to the cavity photon number. We have recently used this method to demonstrate the existence of Fock states up to 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]

].

To ensure that the cavity is in the vacuum state at the start of a measurement, there is a delay of 1.5 cavity decay times between the laser pulses. Hence the compromise that the 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.

Fig. 6(a–c) displays three Rabi cycles obtained by measuring the inversion of a probe atom that followed the detection of n=0, 1 or 2 ground state atoms respectively.

Because of the long waiting times for three atom events, the 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.

The fact that we do not measure pure number states is caused by dissipation in the time interval between production and analysis of the cavity field. Our simulations which are described in the following demonstrate that we are able to produce number states with a purity of 99 % for the 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.

For the simulations two idealizing assumptions were made: thermal photons are only taken into account for the long term build up of the cyclically steady state and Gaussian averaging over velocity spread of atoms is considered to be about 3 %. Considered in the calculations are the exponential decays for the cavity field during the pulse when either one photon (for 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

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).

]. The results of these calculations are compared to the experimental results in Fig. 7a and Fig. 7b.

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. 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.

3 Preparing Fock states on demand

In the following we would like to discuss a new method which allows us to produce photon Fock states in the micromaser on demand. The method uses the trapping condition in conjunction with a pumping of the maser cavity by a sequence of Rydberg atoms which are sent into the cavity whenever a Fock state is required. Simultaneously also an atom in the lower state is populated so that the method also gives atoms in he lower state on demand.

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 %).

The video shows the build-up of the probability distribution for the photon number in the cavity for 100 atom sequences. It follows that with an interaction time corresponding to the (1, 1) trapping state, both one photon in the cavity and a lower state atoms are produced with a 97 % probability. The same result is also valid for the population of lower state atoms. The duration of an atom sequence can be rather short (0.01τcavτ pulse≤0.1τ 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.

The variation of the time when an emission event occurs during an atom sequence is due to the variable time spacing between subsequent atoms as a consequence of Poissonian statistics and the stochasticity of the quantum process. The atomic rate therefore has to be high enough that there will be a sufficient number of excited atoms per sequence, in order to maintain the 97 % probability of an atom emitting. Figure 9(a,b) show the probability of a single Fock state creation as a function of the average number of atoms per pulse for the (1, 1) and (1, 2) trapping states. The (1 2) trapping state (Fig. 9(b)) shows a faster approach to the Fock state than for the (1, 1) trapping state. For a given cavity photon number the probability of emission into the cavity is given by,

Pg=sin2(n+1Ωtint)
(5)
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].

The faster rise time of the (1, 2) trapping state can therefore be attributed to the higher emission probability into the empty cavity (or vacuum) of 92.9 % as compared with the emission probability at the position of the (1, 1) trapping state being 63.3 %. The (1, 2) trapping state therefore appears to be the better position for single photon Fock source operation, but if the trapping condition is violated by thermal photon or other fluctuations, a higher stability is achieved when 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.

There is an upper bound to the probability of finding exactly one lower state atom per pulse, which is governed by the emission probability and the Poissonian distribution of atoms. This maximum probability is given by,

Pmax=1ePgNa.
(6)

where 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/Pg leading to a threshold of N Thr=3.16 for the (1, 1) trapping state and N Thr=2.15 for the (1, 2) trapping state.

To guarantee single-atom single-photon operation, the duration of the preparation pulses must be short in relation to the cavity decay time. For practical purposes, the pulse duration should be smaller than 0.1 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.

An obvious side effect of the production of a single photon in the mode is, as mentioned already, that a single atom in the lower state is produced. This atom is in a different state when it leaves the cavity and is therefore distinguishable from the pump atoms, hence under this operation, the micromaser also serves as a source of single atoms in a particular state.

Although the distribution of lower state atoms leaving the cavity will be maximally sub-Poissonian, the arrival time of an atom within the pumping pulse still shows a small uncertainty, the upper limit of which is determined by the pump pulse duration in the range of 0.01–0.1 τ cav for the parameters used in this paper. The separation of the pulses is ≥3τ 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

In this paper we gave a survey of the possibilities for generating Fock states in the micromaser. The generation of Fock states on demand has recently been experimentally confirmed and will be published elsewhere [18

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

]. The possibility to generate Fock states will allow us to perform the reconstruction of a single photon field or other Fock states [19

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

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 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]

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]

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 400, 239–242 (1999). [CrossRef]

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]

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]

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

  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]

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