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cavity quantum electrodynamics ( cqed ) has continued to develop rapidly , driven both by recent experimental successes and by the promise of exciting new applications . advances in atom cooling techniques , as well as development of high - q optical cavities with large dipole coupling , have enabled testing of the strong - coupling regime of cqed  @xcite . single - atom experiments are now feasible  @xcite , and the possibility of trapping atoms in optical cavities is tantalisingly close  @xcite . potential applications include quantum logic gates @xcite . applications of cqed rely critically on the quantum effects , namely the entanglement between the field degree of freedom and the internal electronic state of the atom  @xcite . this entanglement is not only challenging to achieve , it is also difficult to probe . in the optical regime of cqed , photon coincidence spectroscopy ( pcs ) has been proposed as a feasible and unambiguous method for detecting genuine quantum effects in cqed . this technique employs a multichromatic driving field acting on the combined atom - cavity system and detects multiphoton decays by detecting photon coincidences in the cavity output field  @xcite . a difficulty arises in determining whether emitted photons are coincident or not . let us consider a single two - level atom ( 2la ) coupled to a single mode of an optical cavity , and  @xmath0 is the angular frequency of both the cavity mode and the 2la . multi - atom effects can be ignored provided that the atomic density is sufficiently small  @xcite . in the electric dipole and rotating - wave approximations , the hamiltonian is @xmath1 with  @xmath2 the position of the atom , @xmath3 the position - dependent dipole coupling strength , @xmath4 and @xmath5 the annihilation and creation operators for photons in the cavity field , @xmath6 , @xmath7 , and @xmath8 the 2la raising , lowering and inversion operators , respectively , and @xmath9 . the spectrum for this hamiltonian is depicted in fig . [ fig : ladder ] and is the well - known jaynes - cummings spectrum , or ` ladder '  @xcite . the ` dressed states ' of the combined atom - cavity system are designated by the lowest - energy state @xmath10 a positive integer , @xmath11 where  @xmath12 is the fock state of the cavity mode and @xmath13 is the ground ( excited ) state of the 2la . here we are concerned with two - photon coincidence spectroscopy ( 2pcs ) which proceeds , first by driving the atomic beam with a bichromatic field which causes two - photon excitation to the second couplet of the jc ladder , followed by two - photon decay from the atom - cavity system . the objective is to count photon pairs emitted from the cavity as the frequencies of the driving field are varied . when the sum frequency is @xmath14 , we expect to see a resonantly enhanced two - photon count rate ( 2pcr ) . of course , @xmath15 is a random variable due to beam fluctuations , and this leads to inhomogeneous broadening . despite these difficulties , 2pcs appears to be a feasible method for detecting the characteristic splitting in the jc model  @xcite . however , improvements in the procedure are important to ensure that the detection process is optimised . in the following analysis we investigate the appropriate window time for counting photon pairs . photon pairs are not emitted from the cavity simultaneously due to the randomness of photoemission from an optical cavity . the detection of a photon pair thus depends on identifying a window time  @xmath16 such that , for two photons detected with temporal separation  @xmath17 such that  @xmath18 , the two photons are deemed to be members of a pair , and , if  @xmath19 , are deemed to be independent single photons ( not members of a pair ) . here we determine the optimal window time @xmath20 which maximises the counting rate of genuine pairs relative to the rate of false pair counts . the hamiltonian ( [ jch ] ) for the combined atom - cavity system ignores the driving field emission of photons from the system . the hamiltonian is useful in so far as it yields the spectrum for the combined atom - cavity system , but the full quantum master equation is necessary to calculate the quantities relevant to experiments , namely the two - photon count rate ( 2pcr ) . the experiment proceeds by measuring the 2pcr as a function of the bichromatic driving field s scanning field frequency . two - photon excitation is provided by driving the atom directly with a bichromatic field , characterised by the time - dependent variable @xmath21 the angular frequency  @xmath22 is fixed and resonantly excites the atom - cavity system from the ground state  @xmath23 to the excited state  @xmath24 for the subensemble  @xmath25 . that is , provided that @xmath26 the bichromatic driving field will resonantly excite the subensemble of atom - cavity systems for which @xmath25 . of course subensembles for which @xmath27 can also be excited , but these excitations are non - resonant and hence less significant . the second frequency , @xmath28 , is scanned over a particular frequency range . the purpose of the second component of the bichromatic field is to excite to one of the two states in the second couplet of the jaynes - cummings ladder , namely @xmath29 . thus , the range of scanning frequencies for @xmath28 must include the @xmath30 transition frequencies , @xmath31 respectively . the amplitudes of the two chromatic components must be large enough to ensure sufficient occupation of the excited state but not large enough that significant stark shifting or nonnegligible occupation of the higher - order states occurs . enhanced rates of photon pair detection are then sought as the scanning frequency  @xmath28 is varied . the enhanced 2pcr occurs at the resonances shown in fig . [ fig : ladder ] . in addition to a driving term , loss terms must also be included . the born - markov approximation is applied to both radiation reservoirs : the reservoir for the field leaving the cavity and the reservoir for direct fluorescence of the 2la from the sides of the cavity . the cavity damping rate is  @xmath32 , and the emission rate into free space is  @xmath33 , where  @xmath33 is the inhibited spontaneous emission rate due to the restriction of modes by the cavity . the master equation  @xcite can be expressed as @xmath34 for  @xmath35 the liouvillean superoperator . more specifically the liouvillean superoperator can be expressed as the sum of a time - independent liouvillean operator , a time - dependent liouvillean operator and a ` jump ' term . by defining @xmath36 , in the rotating picture of  @xmath37 the master equation is @xmath38 \rho(t;\delta , g)\ ] ] for @xmath39 a non - hermitian hamiltonian , the first term is    @xmath40,\ ] ]    the time - dependent term is    @xmath41,\ ] ]    and the jump term is @xmath42    the atom - field coupling strength  @xmath15 depends on the atomic position  @xmath43 . provided that the atoms move sufficiently slowly through the cavity @xcite , the atom can be treated as if it were at rest at some randomly located position  @xmath43 . the interaction of the atom with the cavity is then described by the master equation in the asymptotic large time ( @xmath44 ) . as the position  @xmath43 is a randomly varying quantity , the value of the coupling strength  @xmath15 itself is also random . hence , a coupling strength distribution  @xmath45 can be constructed  @xcite . the resultant density matrix is given by @xmath46 where @xmath47 is the coupling strength at a cavity node and @xmath48 is the effective lower bound cut - off for the coupling @xmath49 . the effect of averaging over  @xmath45 is an inhomogeneous spectral broadening . this broadening is due to atomic position variability . in fig . [ fig : p(g ) ] two typical distributions  @xmath45 are depicted , one for the case of a uniformly distributed atomic beam entering the cavity and the second for an atomic beam initially passing through a rectangular mask  @xcite . in both cases we assumed a single - mode cavity supporting a tem@xmath50 mode .    for a bichromatic driving field , the density matrix  ( [ density : matrix ] ) does not settle to a steady state value . the time - dependence of the density matrix in the long - time limit can be treated by making a bloch function expansion of the density matrix  @xcite . in the bloch function expansion , the density matrix is written as the sum @xmath51 with  @xmath52 time - dependent matrices . in the long - time limit , @xmath53 and @xmath54 can thus be regarded as time - independent . as the photocount integration time is expected to be long compared to the frequency  @xmath55 , it is reasonable to approximate @xmath56 by truncating the expansion  ( [ bloch ] ) . the two - photon count rate ( 2pcr ) can be obtained in more than one way . ideally one would have a perfectly efficient photodetector which detects all photons leaving one side of the cavity . the photodetector would then provide a complete record of photon emissions from the cavity as a function of  @xmath17 . a perfect coincidence would then arise as two simultaneously detected photons at some time  @xmath17 . however , there are two problems . one problem is that there does not exist a perfectly efficient photodetector . therefore , some pairs of photons are observed as single - photon emissions because one member of the pair escapes observation . in fact some pairs are missed altogether because both photons escape detection . the other problem concerns the detection of two simultaneously created photons . although created simultaneously , the emission from the cavity is not simultaneous due to the randomness of the emission time which is characterised by the cavity lifetime  @xmath57 .    a better and more accurate way to describe two - photon detections is to employ the 2pcr . to begin with , we consider two photons to be coincident provided that they arrive within a time interval  @xmath16 , the ` window time ' . the choice of window time is not obvious , and it is our aim here to determine what the window time should be . as the two simultaneous photons can be separated by a time of order @xmath58 , as discussed above , the window time  @xmath16 might be expected to be on the order of @xmath58 . however , our purpose here is to consider the choice of @xmath16 in detail and to identify the optimal choice of window time  @xmath16 which will produce the best measure of the 2pcr . the choice of optimal window time is further complicated by the method of detecting nearly simultaneous photons . in the ideal case discussed above of a perfect photodetector yielding a record of all photon emissions from the cavity , one can then define a two - photon event as taking place if a second photon arrives between times  @xmath59 and @xmath60 , _ conditioned _ on a photodetection at time  @xmath59 . we refer to this rate as the _ conditional _ 2pcr and define this rate to be @xmath61 the number operator in eq  ( [ 2pcr : con ] ) is defined as @xmath62 , and ` @xmath63 ' refers to normal ordering . the averaging is performed for the density matrix of eq  ( [ master ] ) . the conditional 2pcr for a window time @xmath64 was used in the quantum trajectory analysis of pcs in ref . @xcite . another natural way to measure the 2pcr is by counting all photon pairs defined as being separated by an interval less than  @xmath16 . this 2pcr is referred to as the _ unconditional _ 2pcr and does not rely on starting the count for the second photon conditioned on detecting the first photon . the definition of the unconditional 2pcr is @xmath65 as shown in appendix a , this expression can be simplified to read @xmath66    we solve analytically for two extreme cases in appendix a. the window time can be extremely long @xmath67 , yielding expression ( [ long : time ] ) , or extremely short @xmath68 , yielding expression ( [ short : time ] ) for both conditional and unconditional 2pcr . the short window time ( @xmath69 0 ) was the basis of the analysis of 2pcs in ref . @xcite . in this treatment both the conditional and unconditional 2pcr at time  @xmath17 is approximated by @xmath70 . in the long - time limit the 2pcr is dominated by poissonian statistics . the choice of optimal window time  @xmath20 depends on the technique for observing two - photon coincidences , but another factor must also be considered . the purpose of 2pcs is to observe two - photon decay resonances from the combined atom - cavity system . as explained in refs  @xcite , there are three peaks in the 2pcr as a function of the scanning frequency  @xmath55 . these peaks are shown in fig . [ adadaa ] as a function of the normalised scanning field frequency @xmath71 the choice of @xmath16 will depend on which peak is being observed . however , the best peak for observing a two - photon decay resonance occurs for @xmath72 this resonance corresponds to the transition @xmath73 . this peak does not occur in a semiclassical description of intensity correlations in the cavity output field and therefore serves as a signature of a genuine quantum field effect in cqed . moreover , this peak can be observed without the added complication of having to perform the experiment twice , once with a bichromatic field and once again with a monochromatic field , in order to perform the signal enhancement technique of background subtraction  @xcite . finally detection of this peak is the most feasible of the three dominant peaks in the two - photon spectrum . hence , we consider specifically @xmath16 for this peak at @xmath74 given by eq  ( [ delta : peak ] ) .    in fig . [ adadaa ] we observe that the 2pcr peak sits on a background 2pcr which is largely independent of  @xmath74 and occurs at  @xmath75 . let us characterise the quality of this 2pcr peak by the ratio of the peak height to the height of the background 2pcr . we can understand this ratio in terms of signal to noise , where signal is the 2pcr from the sought - for two - photon decay events , and the background noise corresponds to two - photon decays arising from unwanted off - resonance two - quantum excitations and decay events . the peak - to - valley ratio ( pvr ) is determined by the height of the peak to the height of the background ( or valley ) 2pcr . the optimal window time @xmath76 is defined such that the pvr for this 2pcr is maximal . that is , either a larger or a smaller choice of the window time would reduce the value of the pvr making the peak more difficult to detect .    there are other concerns besides the pvr in choosing the window time . for example , choosing a much shorter window time could improve the pvr but also lengthen the run time of the experiment in order to accumulate enough signal . that is , the absolute height of the peak is also a matter of concern in determining the feasibility of the experiment and is determined by the allowable timescale of the experiment . the minimum height would need to be on the order of  @xmath77 for  @xmath78 the timescale of the data collection . the pvr is obtained numerically . the matrix continued fraction method is used to solve the master equation to determine the peak height . the background , or valley , can be solved analytically though by treating the detuning of the scanning field as large . the details are provided in appendix b.    the 2pcr for large  @xmath74 is given by expressions  ( [ offres : c2pcs ] ) and  ( [ offres:2pcs ] ) . the peak - to - valley ratio 2pcr is thus @xmath79 where  @xmath80 . in fig . [ fig : surfpvr ] surface plots of the pvr _ vs _  @xmath15 and  @xmath16 reveals that the pvr increases as @xmath15 decreases . this is due to the background signal of two - photon coincidences for  @xmath74 large becoming negligible as shown in fig . [ adadaa ] . although the pvr improves as  @xmath15 decreases , the signal of two photon coincidences within the window time  @xmath16 decreases . this decrease is due to the fact that the resonant frequency for the transition @xmath81 is  @xmath82 , but we have constrained the pump field frequency by  ( [ gf ] ) . hence , as  @xmath15 decreases , the pump field drives the system more and more off resonance . the window time  @xmath16 for achieving the optimal pvr is an order of magnitude smaller than @xmath58 . the optimal time  @xmath20 exhibited in fig . [ fig : optvsg ] is a function of coupling strength and in fig . [ fig : resopt ] is a function of  @xmath83 ( averaged over  @xmath45 ) . we are concerned specifically with the 2pcr and the pvr for the system with a coupling constant distribution based on the tem@xmath50 mode  @xcite . the coupling strength distribution is depicted in fig . [ fig : p(g ) ] , and we treat the masked beam case which enhances the large - coupling effect .    in fig . [ fig : avgpvr ] we present the pvr for the density matrix  @xmath84 of expression  ( [ density : matrix ] ) , averaged over  @xmath45 . there is a peak of the pvr for each @xmath85 given by @xmath86 ( the optimal time window for observing the peak  ( [ delta : peak ] ) ) . these values of  @xmath20 are plotted in fig . [ fig : optvsg ] for @xmath87 for a range of values of  @xmath33 . of particular interest here is the very weak dependence of  @xmath20 on  @xmath33 where  @xmath33 is varied by a factor of  @xmath88 . moreover ,  @xmath20 is generally decreasing as  @xmath85 increases . in fig .  [ fig : avgpvr ]  we observe a maximum of the pvr for each of the assumed  @xmath45 in fig .  [ fig : p(g ) ] . this peak occurs at  @xmath89 for the conditional 2pcr and at  @xmath90 for the unconditional 2pcr . we can understand the location of these peaks by referring to fig . [ fig : optvsg ] .    in fig . [ fig : optvsg](a ) the values of  @xmath91 for the conditional 2pcr are predominantly between  @xmath92 and  @xmath93 , but the value of  @xmath91 in the vicinity of @xmath94 is between  @xmath95 and  @xmath92 . due to the resonance condition  ( [ gf ] ) this region of the  @xmath91 _ vs _ @xmath85 curve is more significant . hence , the dependence of  @xmath20 on  @xmath33 as depicted in fig . [ fig : resopt ] is dominated by the  @xmath94 region of figs . [ fig : optvsg ] . similarly , we can estimate the precise value of  @xmath20 for the unconditional 2pcr from the dashed line of fig . [ fig : resopt ] in the context of the values of  @xmath20 in fig . [ fig : optvsg](b ) .    the linear dependence of  @xmath20 on  @xmath33 , as seen in fig . [ fig : resopt ] yields a correlation of 0.9983 for the conditional 2pcr , and a correlation of 0.9995 for the unconditional 2pcr . in the conditional case    @xmath96    and in the unconditional case @xmath97    the low values of the slopes are indicative of the weak dependence on  @xmath33 . formulae ( [ av : ctopt ] ) and ( [ av : topt ] ) can be used to fine - tune the choice of optimal window time , and the linear relationship simplifies the task of interpolating to obtain  @xmath20 . we have determined expressions for the optimal window time  @xmath20 for both conditional and unconditional 2pcr . these expression provide an optimal pvr for the 2pcr peak at  @xmath98 corresponding to the  @xmath99 transition . although we have determined  @xmath20 for certain parameters and for the coupling - strength distribution  @xmath45 ( solid line of fig . [ fig : p(g ) ] ) , the algorithm presented here is sufficiently general to allow calculation of  @xmath20 for other parameters and other coupling - strength distributions . in general  @xmath20 is smaller than  @xmath58 by an order of magnitude for both the conditional and unconditional 2pcr . there is some dependence on  @xmath33 , but this dependence is weak and is close to linear in the cases studied here . analyses of optimal window times are aided by studies of  @xmath20 for particular values of  @xmath15 , that is , for the coupling - strength distribution  @xmath45 corresponding to  @xmath100 for some  @xmath101 . these calculations provide good estimates of the optimal window time for general  @xmath45 . a longer window time may be desirable , however , if the timescale for collecting enough data is not experimentally feasible . a compromise between the two objectives of optimising the pvr and of accumulating sufficient data to produce a large peak height may be necessary . an important technique discussed in refs . @xcite was background subtraction . the principle behind this method is to remove the unwanted two - photon off - resonance excitation to the second couplet . background subtraction is particularly important to improve the pvr for 2pcr peaks . however , we choose to study detection of the most promising peak experimentally  ( [ delta : peak ] ) . in our simulations we determine the pvr for this peak both by performing background subtraction and without background subtraction , and we obtain plots in agreement with figs .  [ fig : surfpvr ] and  [ fig : avgpvr ] . these results confirm the assertion in ref . @xcite that `` the resonance frequency lies outside the inhomogeneous line and the resonance should be resolved even in the presence of the two - photon background '' . hence , background subtraction is not necessary to obtain optimal window times for resolving the peak  ( [ delta : peak ] ) . dr s.  m.  tan has provided matlab@xmath102 programs to us which was used initially to double check our simulations of the monochromatically driven jaynes - cummings system . we have benefited from valuable discussions with h.  j.  carmichael , j.  d.  cresser , z.  ficek , k .- marzlin , and s.  m.  tan . this research has been supported by australian research council large and small grants and a macquarie university research grant . in the long - time limit , the conditional two - photon count rate ( 2pcr ) is given by @xmath103 if the time window @xmath16 is large , compared to @xmath58 ( the cavity lifetime ) , the two photons are highly decorrelated , and we can approximate @xmath104 thus , @xmath105 this count rate reflects the poissonian nature of the count statistics for long window times . on the other hand , for @xmath106 , the correlation between photon pairs can not be neglected . hence , the count rate reduces to @xmath107 which is the approximation employed in ref  @xcite . similarly , in the long - time limit , the unconditional 2pcr is    @xmath108    this expression can be simplified as we show below . first we make the substitution  @xmath109 . we also introduce the notation  @xmath110 and let  @xmath111 be the union of the two regions  @xmath112 and  @xmath113    this substitution transforms the above double integral into the sum of two double integrals : @xmath114 the advantage of this expression is that the two - time photon number correlation depends on only one term in the double integral instead of both terms in the double integral . greater simplification is possible and desirable for studying the short and long window time  @xmath16 . substituting @xmath115 transforms eq  ( [ two : arrange ] ) to @xmath116           \nonumber \\ & &      \left\langle : \hat{n}(0 ) \hat{n } ( w_+ ) : \right\rangle ( \delta , g)\end{aligned}\ ] ] which reduces to @xmath117    for large ( @xmath118 ) and small ( @xmath119 ) window times  @xmath120 reduces identically to  @xmath121 as shown in equations  ( [ long : time ] ) and ( [ short : time ] ) . for the scanning field far off resonance ( @xmath55 large ) , the time - dependent component of the liouvillean  ( [ l(t ) ] ) can be ignored . in the interaction picture , the master equation can be written as  @xmath122 with @xmath35 time - independent . if @xmath84 is expressed as a vector , then  @xmath35 can be expressed as a complex matrix with  @xmath123 the set of eigenvalues for  @xmath124 the number of levels in the jaynes - cummings ladder retained after truncation . the density matrix can be approximated by the sum @xmath125 for  @xmath126 a set of time - independent @xmath127 matrices . thus , the conditional 2pcr  ( [ 2pcr : con ] ) can be written as @xmath128 }       \nonumber \\      & = & \ ! c_0(g ) \ !      \ !   + \ ! \sum_{n=1}^{n^2 } \ ! \frac { c_n ( g ) } { \mu_n(g ) } \ ! \left \ { \ ! 1- e^ { -\mu_n ( g ) \ ! } \right \ } \ ! \ ! .\end{aligned}\ ] ] with @xmath129 scalar constants  @xmath130 and @xmath129 constants  @xmath131 where @xmath132 as well . in the long window time limit , we equate @xmath133 expansion  ( [ offres : c2pcs ] ) provides a useful method for calculating @xmath134 . the function @xmath135 is monotonically increasing because @xmath136 if @xmath137 . thus , @xmath138 as the function approaches the limit given by ( [ limit : delta ] ) .    in the same way , the unconditional 2pcr can be obtained : @xmath139 }       = c_0(g ) \nonumber \\         & & + \ ! 2 \ ! \sum_{n=1}^{n^2 } \frac { c_n ( g ) } { \mu_n ( g ) } \ ! \left\ { \ ! \frac { e^ { -\mu_n ( g ) } - 1 } { \mu_n ( g ) } + 1 \ ! \right\}\ ! . \end{aligned}\ ] ]      r.  j.  thompson , q.  a.  turchette , o.  carnal and h.  j.  kimble , phys . a , * 57 * , 3084 ( 1998 ) . c.  j.  hood , m.  s.  chapman , t.  w.  lynn and h.  j.  kimble , phys . * 80 * , 4157 ( 1998 ) . h.  mabuchi , j.  ye and h.  j.  kimble , j.  appl . b unpublished ( 1998 ) . a.  s.  parkins and h.  j.  kimble unpublished . q.  a.  turchette , c.  j.  hood , w.  lange , h.  mabuchi and h.  j.  kimble , phys . * 75 * , 4710 ( 1995 ) . h.  j.  carmichael , p.  kochan , and b.  c.  sanders , phys . * 77 * , 631 ( 1996 ) . m.  brune , f.  schmidt - kaler , a.  maali , j.  dreyer , e.  hagley , j.  m.  raimond , and s.  haroche , phys .  rev .  lett .  * 76 * , 1800 ( 1996 ) . b.  c.  sanders , h.  j.  carmichael , b.  f.   wielinga , phys . a * 55 * , 1358 ( 1997 ) . l.  horvath , b.  c.  sanders and b.  f.  wielinga , j.  opt . b : quant .  and semiclassical opt . 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photon coincidence spectroscopy relies on detecting multiphoton emissions from the combined atom - cavity system in atomic beam cavity quantum electrodynamics experiments . 
 these multiphoton emissions from the cavity are nearly simultaneous approximately on the cavity lifetime scale . 
 we determine the optimal time for the detection window of photon pairs in two - photon coincidence spectroscopy . 
 if the window time is too short , some photon pairs will not be detected ; if the window time is too long , too many nearly coincident independent single photons will be falsely interpreted as being a photon pair .