Preparing Fock states in the micromaser Simon Brattke, Ben T. H. Varcoe, Herbert Walther Sektion Physik der Universit¨ at M¨ unchen and Max-Planck-Institut f¨ ur Quantenoptik 85748 Garching, Fed. Rep. of Germany [email protected]http://mste.laser.physik.uni-muenchen.de/ 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 lat- ter 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. c 2001 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.6570) Squeezed states References and links 1. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Exper- imental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett 77, 4281–4285 (1996). 2. D. Meschede, H. Walther, and G. M¨ uller, “The one-atom-maser,” Phys. Rev. Lett. 54, 551–554 (1985). 3. G. Rempe and H. Walther, “Sub-Poissonian atomic statistics in a micromaser,” Phys. Rev. A 42, 1650–1655 (1990). 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). 5. G. Raithel, O. Benson, and H. Walther, “Atomic interferometry with the micromaser,”Phys. Rev. Lett. 75, 3446–3449 (1995). 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). 7. B.-G. Englert, M. L¨ offler, O. Benson, B. Varcoe, M. Weidinger, and H. Walther, “Entangled atoms in micromaser physics,” Fortschr. Phys. 46, 897–926 (1998). 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). 10. P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,”Optics Lett. 13, 1078–1080 (1988). 11. M. Weidinger, B.T.H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,” Phys. Rev. Lett. 82, 3795–3798 (1999). 12. G. Antesberger, “Phasendiffusion und Linienbreite beim Ein-Atom-Maser,” PhD Thesis, Univer- sity 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). 15. P. J. Bardoff, E. Mayr, and W.P. Schleich, “Quantum state endoscopy: measurement of the quan- tum state in a cavity,”Phys. Rev. A 51, 4963–4966 (1995). 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). (C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 131 #29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
Transcript
Preparing Fock states in the micromaser
Simon Brattke, Ben T. H. Varcoe, Herbert Walther
Sektion Physik der Universitat Munchen andMax-Planck-Institut fur Quantenoptik85748 Garching, Fed. Rep. of Germany
References and links1. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Exper-
imental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett 77,4281–4285 (1996).
2. D. Meschede, H. Walther, and G. Muller, “The one-atom-maser,” Phys. Rev. Lett. 54, 551–554(1985).
3. G. Rempe and H. Walther, “Sub-Poissonian atomic statistics in a micromaser,” Phys. Rev. A 42,1650–1655 (1990).
4. G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atommaser,” Phys. Rev. Lett. 58, 353–356 (1987).
5. G. Raithel, O. Benson, and H. Walther, “Atomic interferometry with the micromaser,”Phys. Rev.Lett. 75, 3446–3449 (1995).
6. O. Benson, G. Raithel, and H. Walther, “Quantum jumps of the micromaser field – dynamicbehavior close to phase transition points,” Phys. Rev. Lett. 72, 3506–3509 (1994).
7. B.-G. Englert, M. Loffler, O. Benson, B. Varcoe, M. Weidinger, and H. Walther, “Entangled atomsin micromaser physics,” Fortschr. Phys. 46, 897–926 (1998).
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 asingle photon without destroying it,” Nature 400, 239–242 (1999).
10. P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,”OpticsLett. 13, 1078–1080 (1988).
11. M. Weidinger, B.T.H. Varcoe, R. Heerlein, and H. Walther, “Trapping states in the micromaser,”Phys. Rev. Lett. 82, 3795–3798 (1999).
12. G. Antesberger, “Phasendiffusion und Linienbreite beim Ein-Atom-Maser,” PhD Thesis, Univer-sity of Munich, 1999.
13. G. Raithel, et al., “The micromaser: a proving ground for quantum physics,” in Advances inAtomic, Molecular and Optical Physics, Supplement 2, pages 57–121, P. Berman, ed., (AcademicPress, New York, 1994).
14. J. Krause, M. O. Scully, and H. Walther, “State reduction and |n>-state preparation in a high-Qmicromaser,” Phys. Rev. A 36, 4547–4550 (1987).
15. P. J. Bardoff, E. Mayr, and W.P. Schleich, “Quantum state endoscopy: measurement of the quan-tum state in a cavity,”Phys. Rev. A 51, 4963–4966 (1995).
16. B. T. H. Varcoe, S. Brattke, M. Weidinger, and H. Walther, “Preparing pure photon numberstates of the radiation field,” Nature 403, 743–746 (2000).
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 131#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
17. S. Brattke, B.-G. Englert, B. T. H. Varcoe, and H. Walther,“Fock states in a cyclically pumpedone-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).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).
1 Introduction
The quantum treatment of the radiation field uses the number of photons in a particularmode to characterize the quantum states. In the ideal case the modes are defined by theboundary conditions of a cavity giving a discrete set of eigen-frequencies. The groundstate of the quantum field is represented by the vacuum state consisting of field fluctu-ations with no residual energy. The states with fixed photon number are usually calledFock or number states. They are usually used as a basis in which any general radiationfield state can be expressed. Fock states thus represent the most basic quantum statesand differ maximally from what one would call a classical field. Although Fock states ofvibrational motion are routinely observed in ion traps [1], Fock states of the radiationfield are very fragile and very difficult to produce and maintain. They are perfectlynumber-squeezed, extreme sub-Poissonian states in which intensity fluctuations vanishcompletely. In order to generate these states it is necessary that the mode consideredhas minimal losses and the thermal field, always present at finite temperatures, has tobe 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 su-perconducting cavity which can have a quality factor as high as 3 × 1010, leading to aphoton lifetime in the cavity of 0.3s. The steady-state field generated in the cavity hasalready been the object of detailed studies of the sub-Poissonian statistical distributionof the field [3], the quantum dynamics of the atom-field photon exchange representedin the collapse and revivals of the Rabi nutation [4], atomic interference [5], bistabilityand quantum jumps of the field [6], atom-field and atom-atom entanglement [7]. Thecavity 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 hereis 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 valueof the quality factor of the cavity which is three orders of magnitude larger than thatof 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 alsopresent a second method using the entanglement between pumping atoms and cavityfield. The field is prepared by state reduction and the purity of the states generatedinvestigated by a probing atom. It turns out that the two methods of preparation ofFock states are in fact equivalent and lead to a similar result for the purity of the Fockstates. The third method pumps the cavity with a pulsed beam using the trappingcondition to stabilize the photon number, producing Fock states on demand.
The micromaser setup used for the experiments is shown in Fig. 1 and has beendescribed in detail previously [11]. Briefly, in this experiment, a 3He −4 He dilutionrefrigerator houses the microwave cavity which is a closed superconducting niobium
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 132#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
Fig. 1. The micromaser setup. For details see Ref. [11].
cavity. A rubidium oven provides two collimated atomic beams: a central one passingdirectly 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 (85Rb) atoms to the Rydberg 63 P3/2 statefrom the 5 S1/2 (F = 3) ground state.
Velocity selection is provided by angling the excitation laser towards the main atomicbeam at 11o to the normal. The dye laser was locked, using an external computercontrol, to the 5 S1/2 (F = 3)-63 P3/2 transition of the reference atomic beam excitedunder normal incidence. The reference transition was detuned by Stark shifting theresonance frequency using a stabilized power supply. This enabled the laser to be tunedwhile remaining locked to an atomic transition. The maser frequency corresponds tothe transition between 63 P3/2 and 61 D5/2. The Rydberg atoms are detected by fieldionization in two detectors set at different voltages so that the upper and lower statescan be detected separately.
The trapping states are a continuous wave operation of the maser field peaked ina single photon number, they occur in the micromaser as a direct consequence of thequantization of the cavity field. At low cavity temperatures the number of blackbodyphotons 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, tint, are chosensuch that in a cavity field with n photons each atom undergoes an integer number, k,of Rabi cycles. This is summarized by the condition,
Ωtint
√n + 1 = kπ. (1)
When Eq.1 is fulfilled the cavity photon number is left unchanged after the interactionof an atom and hence the photon number is “trapped”. This will occur regardless of theatomic pump rate Nex (pump rate per decay time of the cavity. The trapping state istherefore characterized by the photon number n and the number of integer multiples of
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 133#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
02
46
810
0 50 100 150 200 250 300
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0,0
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mic
Inve
rsio
n
N ex
Interaction time (µs)
−0,5
0
(2,1)
(1,1)
(0,3)
Interaction time (µs)
(0,1)(0,2)
Fig. 2. A theoretical plot, in which the trapping states can be seen as valleys in theNex direction. As the pump rate is increased, the formation of the trapped statesfrom the vacuum can be seen.
full Rabi cycles k.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 theprobability of finding a ground (excited) state atom. At low atomic pump rates (lowNex) the maser field cannot build up and the maser exhibits Rabi oscillations due tothe interaction with the vacuum field. At the positions of the trapping states, the fieldincreases until it reaches the trapping state condition. This manifests itself as a reducedemission probability and hence as a dip in the atomic inversion. Once in a trapping statethe maser will remain there regardless of the pump rate. The trapping states thereforeshow up as valleys in the Nex direction. Figure 3 shows the photon number distributionas 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 photonnumbers until the pump rate is high enough for the atomic emission to be governed bythe trapping state condition. As the pump rate is further increased, and in the limit of alow thermal photon number, the field continues to build up to a single trapped photonnumber and the cavity field approaches a Fock state.
Under the conditions we have in our present experiment the influence of thermalphotons is negligible, therefore the main reason that a deviation from a pure Fock statemay occur is dissipation. If a photon is lost due to dissipation the next incoming excitedatom will emit a photon with high probability so that the lost photon is replaced. Thephoton number in the cavity controls the Rabi flopping dynamics and therefore providesa stabilization process of the photon number. However, if a photon disappears it takes alittle 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
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 134#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
0 1 2 3 4 5 6Photon number
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ton
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ty
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0.5
0.0
0.5
1.0
Nex=0.1
Nex=2
Nex=5
Nex=15
Nex=50
Fig. 3. A numerical simulation of the photon number distribution as the atomicpump rate (Nex) is increased until the cavity field is in a Fock state with a highprobability.
4 shows micromaser simulations, for achievable experimental conditions, in which Fockstates with high purity are created from n = 0 to n = 5. The experimental realizationrequires a pump rate of Nex = 50, a temperature of less than 300mK, a high selectivityof 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 resultson trapping states we have obtained recently which underline our previous results. Themaser system has been improved so that the parameters such as mean velocity andvelocity 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, whichoccurs when the cavity frequency is tuned across the atomic resonance line.
Under the condition that the pumping beam is very weak Nex ≈ 1 the oscillatorybehavior of the photon emission probability is given by
Pem(∆, tint) =4Ω2
∆2 + 4Ω2sin2(
12
√∆2 + 4Ω2tint) (2)
This oscillatory behavior results from Rabi flopping, however, since there is detuningthe observed flopping frequency is higher than the one photon Rabi frequency, thereforemany more periods are observed at finite detuning than at resonance.
If the flux is increased the average photon number in the cavity will increase sincea steady state field will build up; nevertheless the oscillations are still visible as can beseen in Fig. 5 left column showing a simulation for Nex = 11. The results are shownfor different interaction times. Whenever the photon number passes through a miniumindicates that the trapping condition is fulfilled for this particular detuning. For thetint = 80µs and tint = 70µs results all the minima correspond to the vacuum trappingstate. The minimum at detuning 0 for tint = 60µs corresponds to the (1, 1) trapping
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 135#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
0 2 4 6 80 2 4 6 8
Photon number
0 2 4 6 8
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0 2 4 6 80.0
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P
hoto
n pr
obab
ility
(0,1) (1,1) (2,1)
(3,3) (4,3) (5,3)
Nex=25
Nex=50
Fig. 4. Purity of Fock states under the trapping condition for n=0 to n=5 (nth =10−4).
state whereas the minima closest to the central maximum for tint = 90µs correspondsto the (2, 1) trapping state.
The corresponding experimental results are shown in the right column. For the ex-perimental results the inversion is plotted which is experimentally determined. Theagreement 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 theFock states obtained under the trapping condition, however, the dynamical generationof 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 purityof the states generated. For this purpose we use a pulsed excitation of the Rydbergatoms which pump the maser. We start the discussion of the method with some generalremarks.
When the atoms leave the cavity in a micromaser experiment they are in an entangledstate with the field. If the field is in an initial state |n〉 then the interaction of an atomwith 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 entangledstate can be described by:
Ψ = cos(φ)|e〉|n〉 − i sin(φ)|g〉|n + 1〉 (3)
where φ is an arbitrary phase. The state selective field ionization measurement of theinternal atomic state, reduces the field to one of the states |n〉 or |n+1〉. State reductionis independent of the interaction time, hence a ground state atom always projects thefield onto the |n + 1〉 state independent of the time spent in the cavity. This results inan 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 cavitybeing equal to the number of ground state atoms that were collected within a suitablysmall fraction of the cavity decay time. This is the essence of the method of preparing
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 136#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
Fig. 5. Maser resonance and trapping condition. The left column shows the resultsof a simulation. The oscillations are due to Rabi flopping. The right column showsthe corresponding experimental results. The atom flux is Nex = 11. The minimumat resonance for tint = 80µs corresponds to the vacuum trapping state. That fortint = 60µs is due to the (1, 1) trapping state. The minima at larger detunings aredue to Rabi flopping of the vacuum trapping state. For details see text.
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 therelative populations of the excited and ground states of the atom oscillate at a frequencyΩ√
n + 1. As mentioned above, experimentally the atomic inversion is investigated. Inthe presence of dissipation a fixed photon number n in a particular mode is not observedand the field always evolves into a mixture of such states. Therefore the inversion isgenerally given by,
I(n, tint) = −∑
n
Pn cos(2Ω√
n + 1tint) (4)
where Pn 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 meas-ured by a probing atom by studying the Rabi nutation. The signature that the quantumstate of interest has been prepared is simply the detection of a defined number of groundstate atoms. To verify that the correct quantum state has been projected onto the cavitya probe atom is sent into the cavity with a variable, but well defined, interaction time inorder to allow the measurement of the Rabi nutation. As the formation of the quantumstate is independent of the interaction time we need not change the relative velocityof the pump and probe atoms, thus reducing the complexity of the experiment. In thissense we are performing a reconstruction of a quantum state in the cavity using a similarmethod to that described by Bardoff et al. [15]. This experiment reveals the maximumamount of information that can be found relating to the cavity photon number. Wehave recently used this method to demonstrate the existence of Fock states up to n=2in the cavity [16].
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 137#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
As mentioned above, in contrast to previous experiments, pulsed excitation of theatoms is used so that the number of atoms passing through the cavity can be pre-determined. The probability of detecting an atom in either the excited or ground stateis 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 cavityand the detectors. To create and detect an n photon number state in the cavity, N = n+1atoms are required. That is n atoms to create the number state and the final atom asa probe of the state. However owing to the non-perfect detector efficiency and atomicdecay there are missed counts. By using a laser pulse of short duration the number ofexcited-state atoms entering the cavity per pulse is rather low. Hence we know that whenN atoms per pulse were detected the probability of having N + 1 atoms per pulse wasnegligibly small. This was achieved by modifying the UV excitation pulse such that themean number of atoms per pulse was between 0.2 and 0.8. With 40 % detector efficiencyand 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 anatom escapes detection. As the flux of atoms was variable, the pulse duration was alsovariable, a maximum sampling time of 3 ms for the n = 1 data and 5 ms for the n = 2data was imposed to limit the time delay between the pump and probe atoms. Actuallyin most cases the time delay was comparable to the excitation pulse duration. For themeasurement of an n-photon number state, the detection of the probe atom is triggeredby the detection of n ground state atoms within the length of the laser pulse. If too fewor 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, thereis a delay of 1.5 cavity decay times between the laser pulses. Hence the compromise thatthe Q value be lower than ultimately possible in our setup, since a higher Q would leadto an increase of the data collection time. Even with the reduced cavity life time of 25ms and large delay times between the laser pulses a cyclically steady state maser fieldcan build up in the cavity. The time delay between pulses was selected as a compromisebetween limiting the growth of the maser field and the length of the data collectiontime.
Fig. 6(a-c) displays three Rabi cycles obtained by measuring the inversion of a probeatom 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 datawas more difficult to collect than the other two measurements. The data collection timebecame substantially longer as the interaction time was increased and background effectshave a higher impact on the data. The fit to the n = 2 data includes an exponentiallydecreasing weight, so that measurements obtained for longer interaction times have lesssignificance than those at short times.
The fact that we do not measure pure number states is caused by dissipation in thetime interval between production and analysis of the cavity field. Our simulations whichare described in the following demonstrate that we are able to produce number stateswith a purity of 99 % for the n=1 state and 95 % for the n=2 state at the time ofgeneration which then is modified by dissipation between production and measurement.
For the simulations two idealizing assumptions were made: thermal photons are onlytaken into account for the long term build up of the cyclically steady state and Gaussianaveraging over velocity spread of atoms is considered to be about 3 %. Considered inthe calculations are the exponential decays for the cavity field during the pulse wheneither 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 Pois-sonian arrival times of the atoms. The details of this calculation have been discussed in
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 138#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
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ton
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abili
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(A)
(B)
(C)
(a)
(b)
(c)
Fig. 6. (A), (B) and (C): Three Rabi oscillations are presented, for the numberstate n=0, 1 and 2. (a), (b) and (c): plots display the coefficients Pn. The photondistribution Pn was calculated for each Rabi cycle by fitting Eq. 3 to each plot forthe set of photon numbers, n=0 to n=3. The relative phase of the Rabi frequencywas fixed since all the atoms enter in the excited state of the maser transition. Ineach fit the highest probability was obtained for the target number state. Unlikethe n=1 and n=2 Rabi cycles, the n=0 oscillation (Fig. 6(A)) was obtained in thesteady-state operation of the micromaser in a very low-flux regime. The fit to thiscurve was performed for Rabi cycles from n=0 to n=2. The low visibility of thiscurve 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 darkcounts become comparable to the real count rates and therefore contribute to alarge background. To improve the measurements for photon number of n=3 andhigher, 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 cavityphoton number changes periodically. At the maxima there is one photon more thanat the minima. The Rabi oscillation thus allow one to perform a non-destructiveand repeated measurement of the photon number. In connection with the discussionof trapping states, it is interesting to note that minima in the number state Rabioscillations correspond precisely to the trapping states conditions of the steady-statefield. Therefore the large possible storage times of single photons would allow oneto investigate the transition from a pulsed to a steady state experiment.
detail previously [17]. The results of these calculations are compared to the experimentalresults in Fig. 7a and Fig. 7b.
As dissipation is the most essential loss mechanism, it is interesting to compare thepurity of the number states generated by the current method with that expected fortrapping 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 meansthat whenever the loss of a photon occurs the next incoming atom will restore the oldfield 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, whichis not replaced immediately but after a time interval dependent on the atom flux. Theatom rate used in these calculations was 25 atoms per cavity decay time, or an averagedelay of 1 ms. This can be compared to the delay between the preparation and probeatoms in the present experiment. In the steady state simulation loss due to cavity decaydetermines the purity of the number state, in the limit of zero loss the state measurement
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 139#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
Fig. 7. Comparison between theory and experimental results on the purity of numberstates. The columns represent photon distributions obtained from; (a) a theoreticalsimulation of the current experiment; (b) the current experimental results; and (c)a theoretical model that extends the current experiment to the steady state at thepositions of the trapping states. The agreement between the two theoretical resultsand the experimental result is remarkable, indicating that dissipation is the mostlikely loss mechanism. Without dissipation, ie in the moment of generation thepurity of the states is 99 % for n=1 and 95 % for n=2.
is perfect. It can therefore be concluded that dissipative loss due to cavity decay in thedelay to a probe atom, largely determines the measured deviation from a pure numberstate. There is of course the question of the influence of the thermal field on the photondistribution. By the nature of the selection process in the current experiment we reducethe influence of the thermal field by only performing measurements of the field statefollowing a trigger of ground state atoms. Hence the state of the field is well known. Thesimulations for the steady state case were therefore performed for a temperature of 100mK, which makes the influence of the thermal field in the steady-state correspondinglylow.
3 Preparing Fock states on demand
In the following we would like to discuss a new method which allows us to produce pho-ton Fock states in the micromaser on demand. The method uses the trapping conditionin conjunction with a pumping of the maser cavity by a sequence of Rydberg atomswhich are sent into the cavity whenever a Fock state is required. Simultaneously alsoan atom in the lower state is populated so that the method also gives atoms in he lowerstate on demand.
We demonstrate the method by simulations which are shown in a video in Fig. 8. Anaverage of four Rydberg atoms is sent into the cavity with a velocity corresponding to thetrapping 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 eachsequence there is a single emission event, producing a single lower state atom andleaving a single photon in the cavity. After that the following atoms perform singleRabi oscillations. In the case of the loss of a photon by dissipation, one of the nextincoming excited state atoms will restore the single photon Fock state (e.g. pulse No81 in the video). In this case two lower state atoms are produced. If the lost photon
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 140#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
Photonemission
Photon Number
Pulse Count
99
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 atomswith the cavity. A sequence of four Rydberg atoms on the average is interactingwith 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 witha Poissonian statistics. The photon emission events in the cavity are recorded andsummed up in the box on the lower part of the figure. The accumulated result showsthe deviations from single photon emission. It is shown that 100 pulses lead to 98single photon emissions, one with no emission and one with the emission of twophotons. In the video only the result of every third sequence of atoms is shown inorder to reduce the length of the video, however, the real outcome of the situationis incorporated in the emitted photon number. Simultaneously with the emissionof a photon one lower state atom is produced. The video shows that besides singlephotons,single lower state atoms can be generated on demand with high probability(also 97 %).
is not replaced during one sequence the resulting field is in the vacuum state. Furtherfluctuations can occur through thermal photons or through variations of the interactiontime resulting from a velocity spread.
The video shows the build-up of the probability distribution for the photon number inthe cavity for 100 atom sequences. It follows that with an interaction time correspondingto the (1, 1) trapping state, both one photon in the cavity and a lower state atoms areproduced with a 97 % probability. The same result is also valid for the population oflower state atoms. The duration of an atom sequence can be rather short (0.01τcav ≤τpulse ≤ 0.1τcav) so there is little dissipation and the one photon state in the cavityfollowing 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, thefield evolves to the trapping state as a function of time automatically, when the suitableinteraction time has been chosen.
The variation of the time when an emission event occurs during an atom sequenceis due to the variable time spacing between subsequent atoms as a consequence ofPoissonian statistics and the stochasticity of the quantum process. The atomic ratetherefore has to be high enough that there will be a sufficient number of excited atomsper sequence, in order to maintain the 97 % probability of an atom emitting. Figure9(a,b) show the probability of a single Fock state creation as a function of the averagenumber of atoms per pulse for the (1, 1) and (1, 2) trapping states. The (1 2) trappingstate (Fig. 9(b)) shows a faster approach to the Fock state than for the (1, 1) trappingstate. For a given cavity photon number the probability of emission into the cavity is
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 141#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
0 2 4 6 8 10 12 14Avg. number of atoms per pulse Na
0.0
0.2
0.4
0.6
0.8
1.0
p(1)
nth=0.001nth=0.03nth=0.1
0.0
0.2
0.4
0.6
0.8
1.0
p(1)
∆tint/tint=0%∆tint/tint=5%∆tint/tint=10%
(c)
(d)
NThr
Fig. 9. Figures 9 (a) and (b) show a comparison for one photon Fock state genera-tion under the conditions of the (1, 1) and (1, 2) trapping states. Higher emissionprobability into the vacuum for the (1, 2) trapping state means a faster approach tothe operation of an unconditional single photon Fock source. However, violation ofthe trapping conditions by a thermal photon causes higher emission at high pumprates, 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 con-ditions for this simulation are τcav = 100 ms, τpulse = 2 ms, nth = 10−4. Figures(c) and (d) demonstrate the robustness of the unconditional Fock source. Presentedhere is the probability of finding exactly one atom per pulse (P (1)) for a range ofexperimental conditions. Figure (c) shows the robustness of the Fock source againstinteraction time averaging. Figure (d) shows the robustness of the Fock source asa function of temperature. It should be emphasized that the upper level of vibra-tions and thermal photons considered in this figure are extreme conditions andvery much higher than those of a typical experiment. Experimental parameters of(nth = 0.03, ∆tint/tint = 0.02) are well within these limits. The threshold, NThr,for Fock state operation (dotted vertical line) and the pump rate, Na, attained inour present experiment (broken vertical line) are both indicated on the figure (seeRef. [18].
given by,
Pg = sin2(√
n + 1Ωtint) (5)
The faster rise time of the (1, 2) trapping state can therefore be attributed to thehigher emission probability into the empty cavity (or vacuum) of 92.9 % as comparedwith 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 photonFock source operation, but if the trapping condition is violated by thermal photon orother 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 atomnumbers, it is more unstable at higher average atom numbers and the (1, 1) trappingstate reaches a higher total probability of single photon Fock state creation. The changeof the emission probability as a function of the photon number n by a single quantumthus has an appreciable effect on the evolution of the system. This discussion acquiresmore 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
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 142#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
per pulse, which is governed by the emission probability and the Poissonian distributionof atoms. This maximum probability is given by,
Pmax = 1 − e−PgNa . (6)
where Na is the average number of atoms per pulse; Na is the most important factorwhen comparing different operating conditions. A critical value of Na can be definedthat can be considered a threshold pump rate. We define the threshold pump rate tobe NThr ≡ 2/Pg leading to a threshold of NThr = 3.16 for the (1, 1) trapping state andNThr = 2.15 for the (1, 2) trapping state.
To guarantee single-atom single-photon operation, the duration of the preparationpulses must be short in relation to the cavity decay time. For practical purposes, thepulse duration should be smaller than 0.1 τcav for dissipative losses to be less than10 %. Apart from reducing the fidelity of the Fock state produced, losses increase thelikelihood of a second emission event leading to a larger number of lower state atomsthan 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 percavity decay time (usually labeled Nex) must be larger than NThr to realize the Focksource with a significant fidelity. Since a minimum atom number is required to producethe desired state, care must also be taken to avoid atom beam densities violating theone-atom-at-a-time condition.
For a large range of operating conditions, the production of Fock states of the fieldand single lower state atoms is remarkably robust against the influence of thermalphotons, variations of the velocity of atoms and other influences such as mechanicalvibrations of the cavity. Much more so than the steady state trapping states, for whichhighly 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. Thisrobustness results from the relatively short preparation pulse (≤ 0.1τcav) which presentsexternal influences from greatly affecting the generation of Fock states. In addition whenfluctuations do occur they affect only a single experimental interaction after which thecavity is reset to the vacuum. It must be emphasized that the upper limit of fluctuationsconsidered in Fig. 9(c, d) is well above that of a typical experiment and the routinelyused experimental parameters of nth = 0.03 (T = 300 mK) and ∆tint/tint = 2 %.Here we require high pumping rates (Nex ≥ 40) and as a consequence the steady stateoperation of the micromaser would not exhibit trapping states, but even at these extremeconditions, the simulation shows that under pulse excitation the system still acts asan effective single photon Fock source. It is therefore possible that this Fock source isgeneralizable 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 men-tioned already, that a single atom in the lower state is produced. This atom is in adifferent state when it leaves the cavity and is therefore distinguishable from the pumpatoms, hence under this operation, the micromaser also serves as a source of singleatoms in a particular state.
Although the distribution of lower state atoms leaving the cavity will be maximallysub-Poissonian, the arrival time of an atom within the pumping pulse still shows a smalluncertainty, the upper limit of which is determined by the pump pulse duration in therange of 0.01 - 0.1 τcav for the parameters used in this paper. The separation of thepulses is ≥ 3τcav leading to a small relative variation in the arrival times. If one wouldincrease the pump rate still further, the pulse lengths could be further reduced and thearrival of an atom becomes even more predictable.
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 143#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
4 Conclusion
In this paper we gave a survey of the possibilities for generating Fock states in themicromaser. The generation of Fock states on demand has recently been experimentallyconfirmed and will be published elsewhere [18]. The possibility to generate Fock stateswill allow us to perform the reconstruction of a single photon field or other Fock states[19, 20].
(C) 2001 OSA 15 January 2001 / Vol. 8, No. 2 / OPTICS EXPRESS 144#29368 - $15.00 US Received November 15, 2000; Revised December 01, 2000
