## 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.

©2001 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], 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] 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 3×10^{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], the quantum dynamics of the atom-field photon exchange represented in the collapse and revivals of the Rabi nutation [4], atomic interference [5], bistability and quantum jumps of the field [6], atom-field and atom-atom entanglement [7]. 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]). 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].

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]. 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 P_{3/2} state from the 5 S_{1/2}(F=3) ground state.

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 S_{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.

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, 11]. 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,

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*.

The build up of the cavity field can be seen in Fig.2, where the emerging atom inversion I=P* _{g}*-P

*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.

Under the conditions we have in our present experiment the influence of thermal photons is negligible, therefore the main reason that a deviation from a pure Fock state may occur is dissipation. If a photon is lost due to dissipation the next incoming excited atom will emit a photon with high probability so that the lost photon is replaced. The photon number in the cavity controls the Rabi flopping dynamics and therefore provides a stabilization process of the photon number. However, if a photon disappears it takes a little while until the next incoming excited atom can be used to replace the lost photon. Therefore smaller photon numbers show up besides the considered Fock state. Figure 4 shows micromaser simulations, for achievable experimental conditions, in which Fock states with high purity are created from *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] and very low mechanical noise of the system [12].

The first demonstration of trapping states in the maser field is described in Ref. [11]. We will not review these results here. However, we will show some new results on trapping states we have obtained recently which underline our previous results. The maser system has been improved so that the parameters such as mean velocity and velocity spread of the pumping atoms can be controlled more rigorously than before. As a consequence the trapping states can also be seen in the maser resonance, which occurs when the cavity frequency is tuned across the atomic resonance line.

Under the condition that the pumping beam is very weak N* _{ex}*≈1 the oscillatory behavior of the photon emission probability is given by

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}=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.

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.

In our previous experiments [11] trapping states up to *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

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:

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]. 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].

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 $\Omega \sqrt{n+1}$. 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

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]. 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].

As mentioned above, in contrast to previous experiments, pulsed excitation of the atoms is used so that the number of atoms passing through the cavity can be predetermined. The probability of detecting an atom in either the excited or ground state is about 40 % with a 3–5 % miscount rate. Owing to the finite lifetime of the atoms, additional atoms are lost through spontaneous emission in the flight between the cavity and the detectors. To create and detect an *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.

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

As dissipation is the most essential loss mechanism, it is interesting to compare the purity of the number states generated by the current method with that expected for trapping states (Fig. 7c). The agreement of the purity of the number states is striking. The trapping state photon distribution is generated in the steady-state, which means that whenever the loss of a photon occurs the next incoming atom will restore the old field with a high probability. The non-zero amplitudes of the states |0〉 in Fig. 7(c.2) and |1〉 in Fig. 7(c.3) are due to dissipative losses before restoration of a lost photon, which is not replaced immediately but after a time interval dependent on the atom flux. The atom rate used in these calculations was 25 atoms per cavity decay time, or an average delay of 1 ms. This can be compared to the delay between the preparation and probe atoms in the present experiment. In the steady state simulation loss due to cavity decay determines the purity of the number state, in the limit of zero loss the state measurement is perfect. It can therefore be concluded that dissipative loss due to cavity decay in the delay to a probe atom, largely determines the measured deviation from a pure number state. There is of course the question of the influence of the thermal field on the photon distribution. By the nature of the selection process in the current experiment we reduce the influence of the thermal field by only performing measurements of the field state following a trigger of ground state atoms. Hence the state of the field is well known. The simulations for the steady state case were therefore performed for a temperature of 100 mK, which makes the influence of the thermal field in the steady-state correspondingly low.

## 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.

We demonstrate the method by simulations which are shown in a video in Fig. 8. An average of four Rydberg atoms is sent into the cavity with a velocity corresponding to the trapping condition for the (1,1) trapping state. The atoms have Poissonian statistics. The results shown in the video are obtained by a Monte Carlo simulation. In each sequence there is a single emission event, producing a single lower state atom and leaving a single photon in the cavity. After that the following atoms perform single Rabi oscillations. In the case of the loss of a photon by dissipation, one of the next incoming excited state atoms will restore the single photon Fock state (e.g. pulse No 81 in the video). In this case two lower state atoms are produced. If the lost photon is not replaced during one sequence the resulting field is in the vacuum state. Further fluctuations can occur through thermal photons or through variations of the interaction time resulting from a velocity spread.

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 (0.01*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.

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,

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,

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/*P _{g}* 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.

For a large range of operating conditions, the production of Fock states of the field and single lower state atoms is remarkably robust against the influence of thermal photons, variations of the velocity of atoms and other influences such as mechanical vibrations of the cavity. Much more so than the steady state trapping states, for which highly stable conditions with low thermal photon numbers are required. Figure 9(c,d) shows the probability of finding exactly one atom in the lower state per pulse (*P*
^{(1)} for an extreme range of interaction time spread and increased thermal photons. This robustness results from the relatively short preparation pulse (≤0.1*τ*
_{cav}) which presents external influences from greatly affecting the generation of Fock states. In addition when fluctuations do occur they affect only a single experimental interaction after which the cavity is reset to the vacuum. It must be emphasized that the upper limit of fluctuations considered in Fig. 9(c, d) is well above that of a typical experiment and the routinely used experimental parameters of *n*
_{th}=0.03 (T=300 mK) and Δ*t*
_{int}/*t*
_{int}=2 %. Here we require high pumping rates (*N*
_{ex}≥40) and as a consequence the steady state operation of the micromaser would not exhibit trapping states, but even at these extreme conditions, the simulation shows that under pulse excitation the system still acts as an effective single photon Fock source. It is therefore possible that this Fock source is generalizable to a wide variety of systems including related systems for optical radiation.

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 ≥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

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]. The possibility to generate Fock states will allow us to perform the reconstruction of a single photon field or other Fock states [19, 20].

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