Multimode analysis of the light emitted from a pulsed optical parametric oscillator
aa r X i v : . [ qu a n t - ph ] S e p APS/123-QED
Multimode analysis of the light emitted from a pulsed optical parametric oscillator
Anne E. B. Nielsen and Klaus Mølmer
Lundbeck Foundation Theoretical Center for Quantum System Research,Department of Physics and Astronomy, University of Aarhus, DK-8000 ˚Arhus C, Denmark (Dated: December 13, 2018)We present a multimode treatment of the optical parametric oscillator, which is valid for bothpulsed and continuous-wave pump fields. The two-time correlation functions of the output field arederived, and we apply the theory to analyze a scheme for heralded production of non-classical fieldstates that may be subsequently stored in an atomic quantum memory.
PACS numbers: 42.50.Dv, 03.65.Wj, 03.67.-a
I. INTRODUCTION
The process of parametric down conversion, in whichpump photons with frequency ω are sent through a non-linear crystal and converted into pairs of signal and idlerphotons with frequencies ω and ω = ω − ω , is a well-known and widely used phenomenon in quantum optics.Since the conversion efficiency is rather small, in the opti-cal parametric oscillator (OPO) the process is enhancedby placing the crystal inside an optical cavity. Due toits importance as a source of squeezed light when oper-ated below threshold, the OPO has been subject of muchinvestigation, in particular for the case of a time indepen-dent continuous-wave pump field [1, 2, 3]. The case ofa periodically modulated pump field has been studied in[4]. In the standard treatment of the OPO, the calcula-tions are performed in frequency space, the pump is takenas a time independent monochromatic beam, and it is of-ten assumed that the cavity allows only one or two modeswith well-defined frequencies, depending on whether de-generate or nondegenerate operation is considered. Thefield leaking out of the cavity is then determined frominput-output formalism.An OPO may also be pumped with a pulsed light field.This case is often discussed as if the OPO generates asingle mode pulse of squeezed light, but this is not quitethe case. The output consists of a number of independentmodes squeezed by different amounts as discussed for sin-gle pass down conversion in Refs. [5, 6]. In the presentpaper we use a completely different approach to charac-terize the output from a pulsed OPO. The treatment isa generalization of the discussion of the continuous-waveOPO in Ref. [7], and in contrast to Refs. [1, 2, 3, 5, 6] allcalculations are performed in the time domain.The time domain treatment has several advantages.Firstly, it is not necessary to assume the existence ofa single mode or a couple of independent modes in thecavity since all effects of cavity resonances appear natu-rally in the analysis from the specification of the lengthof the cavity via the cavity round trip time. Further-more, since we do not assume orthogonal cavity modes,our treatment is valid for all values of the transmission ofthe OPO output coupling mirror and not just for smalltransmissions. In particular, we may increase the mirror transmission to unity in our formulas and obtain the re-sults for single pass down conversion. Secondly, we do notassume any particular temporal shape of the pump field,and the analysis is thus suitable to investigate the transi-tion between the few mode situation for short pulses andthe highly multimode regime for continuous-wave fields.In the limit of a time independent pump field our re-sults reduce to those of Ref. [1], provided we invoke extraapproximations which guarantee the existence of only asingle cavity mode. Thirdly, the time domain analysis isconvenient if the light emitted from the OPO is detectedcontinuously in time and the back action of the measure-ments on the system has to be taken into account, see[7].Although the OPO output can be both squeezed andentangled, it belongs to the family of so-called Gaussianstates, and for a number of applications in quantum com-munication and computing, non-Gaussian states are nec-essary. The OPO output can be transformed into non-Gaussian states by measurements as shown schematicallyin Fig. 1: a small fraction of the light is extracted, pos-sibly frequency filtered, and detected by an avalanchephoto diode (APD). As, e.g., a squeezed vacuum state isa superposition of even photon number states, a photodetection in the APD heralds the generation of a statewith only odd photon number contributions such as sin-gle photon states and odd Schr¨odinger kitten states in theremaining beam as verified experimentally in [8, 9, 10].Ref. [11] provides a theoretical determination of the modeof the output state with the largest single photon fidelityfor the case of a continuous-wave nondegenerate OPO.We note that in the experimental verification by homo-dyne detection, one may use a constant amplitude localoscillator, measure the homodyne detector signal as afunction of time, and then at a later stage multiply therecorded data with the mode function, selected accordingto the instant of the photo detection by the APD [9].For several applications of the heralded non-Gaussianfield states it is, however, necessary to mode match thestate in real time. This is the case if one wants to interferedifferent pulses and if one wants to store the field state inan atomic medium and, e.g., perform quantum gates byoptical non-linearities in the medium [12]. Whether mak-ing use of electromagnetically induced transparency [13],a Raman transfer [14], or Faraday polarization rotation FIG. 1: Experimental setup to generate nonclassical states oflight. BS denotes a beam splitter with a small reflectance,and the curve to the right is an example of a possible modefunction that may contain a non-Gaussian state conditionedon a detector click in the APD. [15, 16], storage of light pulses in atomic media involvesa strong pump pulse that defines the temporal mode ofthe quantized field to be mapped onto the atomic col-lective degree of freedom [17]. This implies that storageof a field state heralded by an APD detection event re-quires a storage control field, which has to be producedat the appropriate time. Since the mode function oc-cupied by the non-Gaussian state may extend to timesboth before and after the trigger event [11], it may thusbe necessary to switch on the control field before the trig-ger event or, more realistically, to delay the arrival of thesignal to the storage medium. For demonstration exper-iments, it would be attractive to apply the storage pro-tocol irrespective of the APD output and subsequentlypost select the instances where APD detection actuallyoccurred in a suitable time window around the instantfor which the stored mode is optimal. In this paper wedo not model the storage process itself, but we identifythe mode functions that maximize the non-Gaussian fea-tures of the pulses to be stored, conditioned on realisticdetection events. In particular, we wish to identify modefunctions that are optimal independent of the precise de-tection time within a predefined time interval, which cannot be chosen arbitrarily small if one wants a satisfac-tory APD detection probability. We shall for concrete-ness take the optimal mode to be the one with the mostnegative value of the Wigner function at the origin.The paper is structured as follows. In Sec. II we de-velop a theoretical description of the OPO and derive thetwo-time correlation functions of the OPO output field.In Sec. III we show that a treatment similar to the oneapplied to the OPO can be used to express the outputfield from the filter in Fig. 1 in terms of the input field. InSec. IV we determine the mode of the conditioned statewith the most negative value of the Wigner function atthe origin, we compute the probability to obtain a triggerdetection event within a specified time interval, and wecompare the performance of continuous-wave and pulsedoperation of the OPO for preparation of pulses that canbe stored with predefined control pulses in an atomicmedium. Section V concludes the paper.
II. CORRELATION FUNCTIONS FOR THEOPTICAL PARAMETRIC OSCILLATOROUTPUT FIELD
A theoretical model of the OPO is based on the setupillustrated in Fig. 2. We have chosen a four-sided ringcavity for mathematical convenience, but this is not es-sential to the analysis. The field annihilation operatorsof the input and the output fields are denoted ˆ a ( t ) andˆ b ( t ), respectively, while ˆ c i ( t ), i = 1 ,
2, represents the fieldat different positions inside the cavity as shown, and ˆ v ( t )is the annihilation operator of a field in the vacuum state.The beam splitter BS couples the input and output fieldsto the intra cavity field, and the fictitious beam splitterBS models losses in the system. We model the entireloss as if it takes place between the crystal in the upperpart of the figure and the output coupling mirror, whichrepresents a worst case situation.The crystal is pumped by a classical pump field f ( t ) = | f ( t ) | e iφ f ( t ) , which is assumed to pass unhinderedthrough the cavity mirrors. The parametric process ina non-linear crystal is described in [5, 6], and leads, inthe time domain, to the following mapping of the fieldincident on the crystal to the field leaving the crystalˆ c ( t ) → cosh(2 χ | f ( t ) | )ˆ c ( t ) − ie iφ f ( t ) sinh(2 χ | f ( t ) | )ˆ c † ( t ) . (1)In (1) χ , which is taken to be real, is proportional tothe second order susceptibility of the crystal and to thelength of the crystal. We have assumed that the band-width of the down conversion is infinite and neglecteddepletion of the pump field and differences in the phasematching conditions for down conversion to different fre-quencies. This is likely to be a good approximation in theexperiment described in Ref. [9], since the crystal insidethe OPO is only few millimeters long, and thousands ofcavity modes are populated. Also, modes far from thecenter of the spectrum are irrelevant because they arefiltered out by other components in experiments (see thenext section).In order to express ˆ b ( t ) in terms of ˆ a ( t ) and ˆ v ( t ), weuse the beam splitter relationsˆ b ( t ) = t ˆ c ( t ) + ir ˆ a ( t ) (2)ˆ c ( t ) = ir ˆ c ( t ) + t ˆ a ( t ) (3)and express ˆ c ( t ) in terms of ˆ c ( t − τ )ˆ c ( t ) = − it (cid:0) cosh ( z ( t − τ )) ˆ c ( t − τ ) − ie iφ ( t − τ ) sinh ( z ( t − τ )) ˆ c † ( t − τ ) (cid:1) + ir ˆ v ( t ) . (4) t i ( r i ) is the field transmission (reflection) coefficient ofBS i , and τ is the round trip time in the cavity. In (4),we have assumed that each cavity mirror induces a phaseshift of π/
2, and we have defined the following quantities z ( t − τ ) ≡ χ | f ( t − τ + ξτ ) | (5) φ ( t − τ ) ≡ φ f ( t − τ + ξτ ) , (6) FIG. 2: Model of an optical parametric oscillator. Photonpairs are generated by parametric down conversion in thecrystal, which is pumped by the classical field f ( t ). The beamsplitter BS couples the intra cavity field to the input (vac-uum) field ˆ a ( t ) and the output field ˆ b ( t ), and the beam splitterBS is included to model intra cavity losses. where ξτ is the traveling time from BS to the crystal.Furthermore, for convenience, we have redefined ˆ v ( t ) ac-cording to ˆ v ( t − ζτ ) → ˆ v ( t ), where ζτ is the travelingtime from BS to BS . Isolating ˆ b ( t ) from Eqs. (2), (3),and (4) and assuming e iφ ( t ) = e iφ ( t − τ ) for all t , we findˆ b ( t ) = ir ˆ a ( t ) − it ∞ X n =0 r n t n +12 cosh n +1 X k =1 z ( t − kτ ) ! ˆ a ( t − ( n + 1) τ ) − ie iφ ( t ) sinh n +1 X k =1 z ( t − kτ ) ! ˆ a † ( t − ( n + 1) τ ) ! + it r ˆ v ( t ) + it r ∞ X n =0 ( r t ) n +1 cosh n +1 X k =1 z ( t − kτ ) ! ˆ v ( t − ( n + 1) τ ) − ie iφ ( t ) sinh n +1 X k =1 z ( t − kτ ) ! ˆ v † ( t − ( n + 1) τ ) ! . (7)The requirement e iφ ( t ) = e iφ ( t − τ ) , which is satisfied for φ ( t ) = − πN t/τ + φ , where N is an integer and φ is aconstant, means that the successive squeezing operationsadd up in phase, i.e., it is the same quadrature that issqueezed at times t , t ± τ , t ± τ , . . . .For the special case of a time independent continuous-wave pump field z ( t ) = z , below threshold r t e z <
1, aFourier transform of (7) leads to the following expressionfor the output field in frequency domain:ˆ b ( ω + ω ) = G ( ω + ω )ˆ a ( ω + ω )+ g ( ω − ω )ˆ a † ( ω − ω )+ G ( ω + ω )ˆ v ( ω + ω ) + g ( ω − ω )ˆ v † ( ω − ω ) , (8) where ω = N π/τ is half the frequency of the pump field, G ( ω + ω ) = ir − i t t G + ( ω + ω ) , (9) G ( ω + ω ) = it r + i t r t r G + ( ω + ω ) , (10) g ( ω − ω ) = − e iφ t t G − ( ω − ω ) , (11) g ( ω − ω ) = e iφ t r t r G − ( ω − ω ) , (12)and G ± ( ω ) = e z + iωτ − r t e z + iωτ ± e − z + iωτ − r t e − z + iωτ . (13)Assuming that the input field ˆ a ( t ) is in the vacuum state,the frequency correlation function h ˆ b † ( ω )ˆ b ( ω ′ ) i = t t (1 − r t ) sinh ( z ) δ ( ω − ω ′ ) (cid:14)(cid:16) r t + 4 r t cosh ( z ) + 2 r t (cid:0) r t cos(2 ωτ ) − r t ) cosh( z ) cos( ωτ ) (cid:1)(cid:17) (14)shows that resonances occur for ωτ = n π , n ∈ Z , asexpected. In particular, the degenerate frequency ω isonly a resonance frequency if N is even. To compare Eq.(8) to the results given in Ref. [1] for a cavity with a singleresonance frequency at ω , we assume N to be even and ωτ ≪ t ≪ r ≪
1, but tomaintain a finite width of the modes as in Ref. [1] wekeep γ ≡ t /τ and γ ≡ r /τ fixed. Since τ γ ≪ τ γ ≪ z must be correspondingly small,i.e., z must be proportional to τ . For | z | = | ǫ | τ we obtainEq. 46 of Ref. [1].Returning to the general case, we compute the two-time correlation functions of the output field for ˆ a ( t ) inthe vacuum state from Eq. (7): h ˆ b † ( t )ˆ b ( t ′ ) i = t t (1 − r t ) ∞ X q =1 ∞ X m =0 ( r t ) q +2 m sinh m +1 X k =1 z ( t − kτ ) ! sinh q + m +1 X k =1 z ( t ′ − kτ ) ! δ ( t − t ′ + qτ )+ ∞ X q =0 ∞ X m =0 ( r t ) q +2 m sinh q + m +1 X k =1 z ( t − kτ ) ! sinh m +1 X k =1 z ( t ′ − kτ ) ! δ ( t − t ′ − qτ ) ! (15)and h ˆ b ( t )ˆ b ( t ′ ) i = ie iφ ( t ) t t (1 − r t ) ∞ X q =1 ∞ X m =0 ( r t ) q +2 m sinh m +1 X k =1 z ( t − kτ ) ! cosh q + m +1 X k =1 z ( t ′ − kτ ) ! δ ( t − t ′ + qτ )+ ∞ X q =0 ∞ X m =0 ( r t ) q +2 m cosh q + m +1 X k =1 z ( t − kτ ) ! sinh m +1 X k =1 z ( t ′ − kτ ) ! δ ( t − t ′ − qτ ) ! . (16)These two expressions and h ˆ b ( t ) i = 0 are sufficient tocharacterize the output state completely, because it isGaussian.The correlations of an arbitrary single mode with modefunction h ( t ) and annihilation operatorˆ b = Z h ∗ ( t )ˆ b ( t ) dt (17)are easily calculated from Eqs. (15) and (16). Weshall only be concerned with the degenerate case be-low, and we thus choose h ( t ) = ˜ h ( t ) e iφ / iπ/ e − iω t =˜ h ( t ) e iφ ( t ) / iπ/ with ˜ h ( t ) real. The constant phase fac-tor is included to obtain a real value of h ˆ b i , which cor-responds to the case, where the axes of the squeezingellipse in phase space lie along the quadrature axes. Inthe following we take h ( t ) to be real and omit the factor ie iφ ( t ) in Eq. (16).The variances of ˆ x ≡ (ˆ b +ˆ b † ) / √ p ≡ − i (ˆ b − ˆ b † ) / √ / h ˆ b † ˆ b i + h ˆ b i and 1 / h ˆ b † ˆ b i − h ˆ b i , respectively,and the most efficiently squeezed mode is thus obtainedby choosing h ( t ) as the eigenfunction with the smallesteigenvalue λ of an integral equation with either the in-tegration kernel δ ( t − t ′ ) / h ˆ b † ( t )ˆ b ( t ′ ) i + h ˆ b ( t )ˆ b ( t ′ ) i or δ ( t − t ′ ) / h ˆ b † ( t )ˆ b ( t ′ ) i − h ˆ b ( t )ˆ b ( t ′ ) i , and λ is the corre-sponding variance. Since the output field at time t is onlycorrelated to the output field at times t + nτ , n ∈ Z , themode function of the most efficiently squeezed mode isin general a spike function that is only nonzero at times t + nτ , n = . . . , − , , , , . . . , for some t , but if the vari-ations of the pump field take place on a time scale τ ormore slowly, displaced spike functions, and therefore alsoa smooth mode function with the same envelope, will beonly slightly less squeezed.In the limit of a time independent pump field, Eqs. (15) and (16) reduce to h ˆ b † ( t )ˆ b ( t ′ ) i = t t ∞ X q = −∞ ( r t ) | q | (cid:18) − r t − r t e z e (2+ | q | ) z + 1 − r t − r t e − z e − (2+ | q | ) z − e | q | z − e −| q | z (cid:19) δ ( t − t ′ − qτ ) (18)and h ˆ b ( t )ˆ b ( t ′ ) i = ie iφ ( t ) t t ∞ X q = −∞ ( r t ) | q | (cid:18) − r t − r t e z e (2+ | q | ) z − − r t − r t e − z e − (2+ | q | ) z − e | q | z + e −| q | z (cid:19) δ ( t − t ′ − qτ ) . (19)Until now we have described the fields in terms of timedependent Heisenberg picture operators, but it is alsouseful to consider the OPO model from a Schr¨odingerpicture point of view. In particular, the latter approachis suitable for a numerical treatment of the pulsed OPO.To this end we divide the light beams into small segmentsof (infinitesimal) duration ∆ t and treat each segment asa single mode, i.e., we define g i ( t ) = (cid:26) / √ ∆ t for t i − ∆ t/ ≤ t < t i + ∆ t/
20 otherwise (20)and replace the continuous field annihilation operatorˆ d ( t ) ( ˆ d ( t ) = ˆ a ( t ), ˆ b ( t ), ˆ c ( t ), or ˆ c ( t )) by the discreteannihilation operatorsˆ d i = Z g ∗ i ( t ) ˆ d ( t ) dt = ˆ d ( t i ) √ ∆ t (21)localized at time t i . In practice, we will have to deal witha finite ∆ t , and neglecting the field variation within each∆ t interval is equivalent to a cutoff in frequency, justi-fied by the experimentally relevant frequency regime. Athorough discussion of continuous and discrete operatordescriptions of light beams in both time and frequencyspace may be found in Ref. [18]. Since the output fieldand the cavity field are Gaussian for a vacuum inputfield, the state of all the small light beam segments isefficiently represented by a Wigner function. In general,the Wigner function of an n -mode Gaussian state withzero mean values is on the form W ( y ) = 1 π n p det( V ) e − y T V − y , (22)where y = ( x , p , x , p , . . . , x n , p n ) T is a column vec-tor of quadrature variables, and V is the covariancematrix of the n modes. V = h ˆ y ˆ y T i + h ˆ y ˆ y T i T , whereˆ y = (ˆ x , ˆ p , ˆ x , ˆ p , . . . , ˆ x n , ˆ p n ) T , ˆ x i = ( ˆ d i + ˆ d † i ) / √ p i = − i ( ˆ d i − ˆ d † i ) / √
2, and ˆ d i is the field annihilation op-erator of mode i . Before the pump pulse reaches thecrystal, all modes are in the vacuum state, and V is theidentity matrix. We include all the τ / ∆ t cavity modesand a sufficiently large number of ˆ a ( t ) and ˆ v ( t ) modes in(22). As time passes by modes are squeezed, when theyhit the crystal, input and cavity modes are transformedinto cavity and output modes, when they hit BS , andvacuum and cavity modes are transformed into cavityand lost modes, when they hit BS . For each of thesetransformations ˆ y is transformed according to ˆ y → S ˆ y ,where S is a matrix, which is easily determined from Eqs.(2), (3), and (4). The corresponding transformation of V is V → SV S T . At the end of the calculation all rows andcolumns of V that represent lost modes are erased, andthe result is a matrix, which contains the same informa-tion as Eqs. (15) and (16). We have used this method tocompute the results for the pulsed OPO in Sec. IV. III. FILTERING
The OPO produces pairwise quantum correlated fieldsin a large number of cavity field modes, and in experi-ments it is necessary to apply a frequency filter beforethe trigger detector, see Fig. 1, to ensure that the pho-ton is derived from a well defined sideband, and that thesignal field predominantly occupies a single mode. Wenow turn to a description of such a filter modeled by acavity with two beam splitters BS and BS and two per-fectly reflecting mirrors as depicted in Fig. 3. Copyingthe notation from Sec. II, ˆ a ( t ) is the input field, ˆ b ( t ) isthe output field, ˆ v ( t ) is a field in the vacuum state, ˆ c i ( t )is the field at different positions inside the cavity (see thefigure), t i ( r i ) is the field transmission (reflection) coef-ficient of BS i , and τ F is the round trip time in the filtercavity. The fields are related according to the equationsˆ c ( t ) = t ˆ a ( t ) + ir ˆ c ( t ) , (23)ˆ c ( t ) = ˆ c ( t − τ F / , (24)ˆ c ( t ) = t ˆ v ( t ) + ir ˆ c ( t ) , (25)ˆ c ( t ) = − ˆ c ( t − τ F / , (26)ˆ b ( t ) = t ˆ c ( t ) + ir ˆ v ( t ) , (27)from which we deriveˆ b ( t ) = ir ˆ v ( t ) − ir t ∞ X n =0 ( r r ) n ˆ v ( t − ( n + 1) τ F )+ t t ∞ X n =0 ( r r ) n ˆ a ( t − ( n + 1 / τ F ) . (28)Transforming the output field to frequency spaceˆ b ( ω ) = ir ˆ v ( ω ) − ir t e iωτ F − r r e iωτ F ˆ v ( ω )+ t t e iωτ F / − r r e iωτ F ˆ a ( ω ) , (29) FIG. 3: Filter cavity. The input field ˆ a ( t ) is partially trans-mitted into the cavity through the beam splitter BS . Thereflected component of the input field is lost as is the part ofthe intra cavity field that leaves the cavity through BS . Thepart of the intra cavity field that leaves the cavity throughBS contributes to the output field ˆ b ( t ). ˆ v ( t ) is a field in thevacuum state. we find h ˆ b † ( ω )ˆ b ( ω ) i = t t r r − r r cos( ωτ F ) h ˆ a † ( ω )ˆ a ( ω ) i , (30)and it is apparent that the frequencies transmitted mostefficiently through the filter are those which satisfy thecondition ωτ F = 2 πn , n ∈ Z . The free spectral range ofthe filter is thus 2 π/τ F , and for a given τ F the bandwidthis determined by t and t .In order to select just one of the frequency modesemerging from the OPO, the bandwidth of the filtershould be much smaller than the free spectral range ofthe OPO, but larger than the bandwidth of the OPO.Also, the free spectral range of the filter should be largecompared to the bandwidth of the parametric down con-version. Experimentally it can be difficult to build cavi-ties that satisfy the last condition, and one may use in-stead a sequence of different filter cavities [9]. Since weassumed an infinite bandwidth of the down conversion inSec. II, we must, for consistency, use an infinitely smallfilter cavity in the theoretical treatment. We thus assume( ω − ω ) τ F ≪ t ≪
1, and t ≪ κ ≡ t /τ F and κ ≡ t /τ F , where ω = 2 πn/τ F , n ∈ Z , is a reso-nance frequency of the filter cavity, and expand Eq. (29)to lowest order:ˆ b ( ω ) = i (cid:18) − κ κ + κ − i ( ω − ω ) (cid:19) ˆ v ( ω )+ √ κ κ κ + κ − i ( ω − ω ) ˆ a ( ω ) . (31)In this limit the filter transmission function is aLorentzian h ˆ b † ( ω )ˆ b ( ω ) i = 4 κ κ ( κ + κ ) + 4( ω − ω ) h ˆ a † ( ω )ˆ a ( ω ) i . (32)We consider a single mode h ( t ) of the output field, anddefine h a ( t ) and h v ( t ) according to Z h ∗ ( t )ˆ b ( t ) dt = Z h ∗ a ( t )ˆ a ( t ) dt + Z h ∗ v ( t )ˆ v ( t ) dt. (33)Since the vacuum state does not contribute to normallyordered expectation values, we shall not need h v ( t ) in thefollowing, but from Eq. (28) h a ( t ) = Z ∞ t h ( t ′ ) √ κ κ e iω ( t ′ − t ) − ( κ + κ ) ( t ′ − t ) dt ′ , (34)i.e., the action of the filter is effectively to transform themode function h ( t ) into the (not properly normalized)mode function h a ( t ). Note that if the time dependentpart of the phase of h ( t ) is chosen as e − iω t , the time de-pendent part of the phase of h a ( t ) is also given as e − iω t .When we choose the resonance frequency of the filter toequal half the carrier frequency of the OPO pump beam,we may thus continue to use real mode functions and omitthe factor e iω ( t ′ − t ) in Eq. (34). For a given bandwidth κ + κ of the filter, Z | h a ( t ) | dt = 1 − Z | h v ( t ) | dt = Z | h ( ω ) | κ κ ( κ + κ ) + 4( ω − ω ) dω (35)is maximal for κ = κ , and we thus assume κ = κ = κ in the rest of the paper. IV. MODE FUNCTION OPTIMIZATION FORHERALDED GENERATION OF ANON-GAUSSIAN STATE OF LIGHT
With the necessary tools at hand we can now proceedto an analysis of the preparation of non-Gaussian lightstates that can be stored in atomic samples. Gaussianstates have Wigner functions that are positive for all ar-guments, and as a measure of non-Gaussian character weshall refer to negative values of the Wigner function oc-curring at the origin of phase space, e.g., for odd numberstates and odd Schr¨odinger cat states. In Ref. [11] wederived an expression for the value of the Wigner func-tion at the origin of the state of an arbitrary real modeof the multimode state generated when conditioning ona photo detection event W i (0 ,
0) = V V ( V + V − − V V − V V π ( V V ) / ( V + V − . (36) V jk are the elements of the Gaussian covariance matrixof the mode in which the APD detection takes place(quadrature variables 1 and 2) and the chosen mode ofthe output state (quadrature variables 3 and 4) beforeconditioning. V is computed from the definition givenjust below Eq. (22) by use of the mode functions of the two modes, the transformation (34) of the trigger modefunction due to the filter, Eq. (17) (with ˆ b ( t ) replaced bythe relevant linear combination of ˆ b ( t ) and the annihila-tion operator of the vacuum field entering into the systemat the beam splitter in Fig. 1), and the two-time correla-tion functions given in Eqs. (15) and (16). We take thetrigger mode function to be constant in a time intervalof duration ∆ t t positioned at time t i and zero otherwise,where ∆ t t is much shorter than all other time scales inthe system. In Appendix A we discuss the applied de-tector model and the choice of trigger mode function inmore detail.As explained in the Introduction the mode that willbe stored is determined by the shape and timing of thestrong storage pulse, and it is advantageous if one caninitiate the generation of the storage pulse before it isknown whether a trigger detection event will actuallytake place at the right time relative to the pulse stored.The protocol is thus probabilistic. We consider an at-tempted storage as successful if a trigger detection eventtakes place within a predefined time interval of duration T , whose position on the time axis is determined relativeto the pump pulse, if the OPO is pulsed, and relative tothe storage pulse, if the pump field is time independent.In practice, T is at least as large as the temporal resolu-tion of the APD detection system, which is of order 1 ns[10]. Since the shape of the storage pulse is independentof the actual trigger detection time, the quantity we op-timize is the mean value of the Wigner function at theorigin W (0 ,
0) = T/ ∆ t t X i =1 P i W i (0 , (cid:30) T/ ∆ t t X j =1 P j , (37)where the sum is over the T / ∆ t t trigger modes inside theacceptance interval T , P i is the probability to obtain adetection event in trigger mode i , which for infinitesimal∆ t t is equal to the expectation value of the number ofphotons in the i th trigger mode, and W i (0 ,
0) is the valueof the Wigner function at the origin of the chosen mode ofthe generated state conditioned on a detection in triggermode i given in Eq. (36). The total trigger probability P = T/ ∆ t t X i =1 P i (38)is small, and hence we neglect the possibility to have twoor more detection events within the time interval T .Under realistic experimental conditions there will besignificant losses, and these are included in the calcula-tions presented below. The chosen experimental parame-ters are given in Table I, and for the case of a pulsed OPOwe assume that the pump field has a Gaussian envelopeand write z ( t ) as z ( t ) = 2 sτ / π / T / exp (cid:18) − t T (cid:19) . (39) Quantity Symbol ValueTotal OPO cavity length L
81 cmRound trip time in the OPO cavity τ r t R κ . · s − Trigger channel efficiency η t η s W (0 , η t includes both propagation efficiency in thetrigger channel and the efficiency of the APD, while η s isthe propagation efficiency of the signal channel. The valuesare chosen in accordance with the numbers given in Refs.[9, 10, 19] Optimizing (37) for different values of T p , s , and T (or z and T for the case of a time independent pump field) byoptimizing the shape of the mode function of the modethat is to be stored, we obtain the values of W (0 , T hasbeen chosen to maximize the success probability. Also,the values of z for the OPO driven with a time indepen-dent pump field have been chosen in order to obtain atotal flux of photons in the degenerate mode of the out-put field from the OPO, which is of order 2 · s − asin Ref. [9]. This corresponds to a success probability of1 . · − for T /τ = 10. The values of s have been ad-justed to obtain comparable success probabilities in thecalculations for pulsed pump fields.Considering the trends in Fig 4b, we find that the suc-cess probability increases when z increases and when s increases for fixed T p as expected. The success probabil-ity is also an increasing function of T , because more termsare included in the sum in Eq. (38), when T increases.For a time independent pump field the increase is linear,because all the P i ’s are equal, while the success probabil-ity levels off to a constant value for a pulsed pump fieldwhen T increases beyond the temporal width of the in-tensity distribution of the generated output field, whichis roughly of order the width of the intensity distributionof the pump pulse plus the mean lifetime of a photonin the OPO cavity ( ≈ . τ ) plus the mean lifetime of aphoton in the filter cavity ( ≈ . τ ). The constant valueis thus reached faster for short pulses. It is apparent fromthe figure that a decrease in T p must be accompanied byan increase in s in order to keep the success probabilityunchanged. Both of these changes increase the requiredpeak value of z , i.e., a larger field strength or a largerconversion efficiency in the crystal is needed.To explain the trends in Fig. 4a we note that W (0 ,
0) ofa state with density matrix ρ = P ∞ n =0 P ∞ m =0 c n,m | n ih m | is P ∞ i =0 ( c i, i − c i +1 , i +1 ) /π , and thus an optimization of a) T / τ W ( , ) SP CW T p / τ = 8 T p / τ = 3 T p / τ = 1 b) −4 T / τ P SP FIG. 4: a) W (0 ,
0) and b) success probability as a functionof T . Solid lines: time independent continuous-wave (CW)pump field with z = 0 . . . . T p /τ = 8 and s = 0 .
03, 0 . .
05, and 0 .
06; dotted lines: pulsed pump field with T p /τ = 3and s = 0 .
04, 0 .
05, and 0 .
06; dash-dotted line: pulsed pumpfield with T p /τ = 1 and s = 0 .
07; dash-dotted line labeledSP: single pass (SP) operation for a pulsed pump field with T p /τ = 3 and s = 0 .
05. Within each series a smaller s or z corresponds to a more negative value of W (0 ,
0) and a smallersuccess probability. W (0 ,
0) is equivalent to a maximization of the odd pho-ton number components. An ideal single mode squeezedvacuum state is a superposition of even photon num-ber states, and if a photon is annihilated from the state,these are converted into odd photon number states. Thepresent experiment deviates from this ideal situation intwo respects. Firstly, the generated state is a multimodestate, since all the generated photon pairs do not belongto the same mode. The result is that the overlap betweenthe optimal mode and the modes of the generated pho-ton pairs is only partial, and this introduces even photonnumber components into the state of the optimal mode.The mechanism is particularly severe if photon pairs aregenerated in the outer regions of the optimal mode suchthat the overlap between the optimal mode and the modeof the photon pair is significantly smaller than unity, butalso significantly larger than zero. This suggests that ashort pump pulse, where the down conversion process isonly turned on in a short period, will lead to a morenegative value of W (0 , T . In this limit W (0 ,
0) approaches a con-stant value for pulsed pump fields, because P i is smallfor trigger modes far from the center of the intensity dis-tribution of the generated state and these modes do thusnot contribute significantly to the sums in Eqs. (37) and(38). For a time independent pump field, on the otherhand, W (0 ,
0) approaches the value of the unconditionalstate, because the success probability approaches unityfor very large T , and all storage attempts are acceptedas successful. The second deviation from ideal behavioris that losses degrade the odd photon number states intoboth odd and even photon number states, and since aphoton number state with a large number of photonsis more fragile than a single-photon state, it is to beexpected that a large intensity of down converted pho-tons will lead to a less negative value of W (0 , − , and this explains why we do notobtain more negative values of W (0 ,
0) than those pre-sented in Fig. 4a if the pulse duration is decreased below T p /τ = 1 but rather observe less negative values for veryshort pulses.In Fig. 4 we have included an example of single passdown conversion for T p /τ = 3, i.e., we have increased t to unity without changing the rest of the parameters inTable I. When the OPO cavity is absent, the temporalwidth of the light pulse is only broadened in the filtercavity, and this broadening is small since the mean life-time of a photon in this cavity is only 0 . τ . It is thuspossible to determine from the precise APD photo detec-tion time whether the photon pair that gives rise to theAPD detection was generated in the beginning, in themiddle, or in the end of the pulse. For small T this leadsto a negative value of W (0 ,
0) because we know ratherprecisely where the second photon in the pair is, if it hasnot been lost. On the other hand, if we do not discrim-inate between early and late trigger detection events asingle optimal compromise will only have a small overlapwith the actual modes, which are well localized in time.The resulting rapid increase towards positive values in W (0 ,
0) with increasing T can be avoided by increasingthe mean lifetime of a photon in the filter cavity, corre-sponding to a smaller frequency width of the filter, andthus the intensity of the light transmitted through thefilter will decrease, which leads to an even smaller suc-cess probability than in Fig. 4b. The success probabilitycan be increased by increasing s , but this will also resultin a less negative value of W (0 , a) −20 0 20 40 60 80−0.0500.050.10.150.20.250.30.35 t / τ h ( t ) τ ½ b) c) −30 10 50 90−0.050.050.150.250.35 t / τ h ( t ) τ ½ −30 10 50 90−0.050.050.150.250.35 t / τ h ( t ) τ ½ FIG. 5: Mode functions for the modes of the generated statewith the most negative values of W (0 ,
0) for a pulsed pumpfield with T p /τ = 8 and s = 0 . t = 0 is chosen to coincidewith the time where z is maximal. a) T /τ = 0 . T /τ = 2 .
4, and c)
T /τ = 10. ments it is preferable to apply pulsed fields and the OPOcavity in the setup to get the most efficient generation ofnon-Gaussian states for storage.The values of W (0 ,
0) presented in Fig. 4a are signifi-cantly above the theoretical minimum of − /π , and wenote that the results are all obtained for physical param-eters, which are already achieved in experiments. Morenegative values of W (0 ,
0) can be reached if it is possi-ble to reduce the losses. If, for instance, we increase thesignal channel transmission from 0 . W (0 ,
0) isdecreased to − .
23 for T p /τ = 3, s = 0 .
05, and large T , while the success probability is unchanged, and if wefurther assume zero loss in the cavity and reduce R to0 .
01, we find W (0 ,
0) = − .
27. An increase in the triggerchannel transmission or in the APD detector efficiencywill increase the success probability, and this increase canbe transformed into a more negative value of W (0 ,
0) bydecreasing z , s , T , or R .Considering the optimized mode functions for a pulsedpump field in Fig. 5, it is apparent that a fast variationin the mode function is present for small values of T , butthis variation is smoothed out for larger T . The inset inFig. 5a shows that the period of the fast variation is τ ,i.e., the round trip time in the OPO cavity. A photongenerated in one of the time localized modes in the OPOcavity can only leave the cavity at times separated by −20 0 20 40 60 80−0.0500.050.10.150.20.250.30.350.40.45 t / τ z ( t ) / ( s ) and h ( t ) τ ½ FIG. 6: Pump pulse (dash-dotted line) and mode functionof the mode of the generated state with the smallest valueof W (0 ,
0) (solid line) for T p /τ = 3, s = 0 .
05, and
T /τ =32 .
4. The dotted line is the function in Eq. (40), while theapproximation (41) (dashed line) is almost indistinguishablefrom the optimum mode (solid line) in the figure. W (0 ,
0) = − . W (0 ,
0) = − . W (0 ,
0) = − . an integer number of τ ’s, and thus, in the output fieldfrom the OPO, the two photons in a pair are separatedby an integer number of τ ’s. The fast variation in theoptimal mode function can thus be explained from thefact that the mean lifetime of a photon in the filter cav-ity is smaller than τ , since this means that the observedphoton in a pair is typically delayed less than τ in thefilter cavity, and the detection time provides partial in-formation on the time of generation of the photon pair.The information is erased as T approaches τ , and thisleads to the steep rise in W (0 ,
0) towards less negativevalues observed for all curves to the very left in Fig. 4a.If the duration of the pump pulse is short compared to τ , the optimal mode function differs qualitatively fromthose presented in Fig. 5. In this case the optimal modefunction is a series of spikes separated by τ , and the widthof the spikes is determined by the width of the pumppulse provided the width of the pump pulse is also shortcompared to T . In the opposite limit of a time indepen-dent pump field, the optimal mode function for short T is qualitatively a function, which has a maximum close tothe time of the trigger detection and decays exponentiallywhen moving to the right or left from this maximum. Su-perimposed on this is a fast variation with period τ . Themode function is not completely symmetric around thetime of the trigger detection due to the filtering. Forlarger values of T the fast variation disappears, and themaximum becomes more rounded.In Fig. 6 we show the temporal shape of the pumpfield and the optimal mode function for T /τ = 32 . W (0 ,
0) and P are both independent of T for large T , and since we have to choose the same modefunction independent of the precise time for the APD de-tection, the optimal mode function is determined solelyfrom the generated output field when T is large, and itseems reasonable that the optimal mode function just fol-lows the square root of the intensity distribution of thegenerated state. Without applying the above theory, acrude guess for the optimal mode function is h ( t ) ∝ Z t −∞ z ( t ′ ) exp (cid:18) − t + r τ ( t − t ′ ) (cid:19) dt ′ , (40)and this function, which is also shown in the figure, isactually quite close to optimal. With the above theory athand, however, it is easy to calculate the actual intensitydistribution from Eq. (15), and this leads to the modefunction h ( t ) ∝ vuut ∞ X n =0 ( r t ) n sinh n +1 X k =1 z ( t − kτ ) ! , (41)which is even closer to the optimal mode function. Dueto its simplicity, Eq. (41) can be useful in a storage ex-periment, where T must be sufficiently large to obtain asatisfactory success probability. We note, however, thatneither Eq. (40) nor Eq. (41) is able to reproduce theoptimal mode function in Fig. 5a for T /τ = 0 . V. CONCLUSION
In conclusion, we have presented a multimode descrip-tion in time domain of the optical parametric oscilla-tor, which is valid for both pulsed and continuous-wavepump fields, and we have used the theory to analyzenon-Gaussian states that can be stored in atomic sam-ples. So far light storage has been demonstrated withGaussian states only, and the storage and retrieval of anon-Gaussian state is a hall mark in the demonstrationof atom-light quantum interfaces. Our analysis suggeststhat, with similar success probabilities, pulsed pumpfields lead to more negative values of the Wigner functionof the stored state than time independent continuous-wave pump fields.An essential ingredient in the experiment is the tem-poral spreading induced by the OPO and filter cavities,because it separates the two photons in a generated pho-ton pair temporally by an unknown amount and ensurethat it is impossible to infer the precise position of thesecond photon in a pair from the time at which a detec-tion took place in the APD. Without this spreading wedo not obtain large negative values of W (0 ,
0) for large T . After having thus characterized the system and iden-tified the optimum strategy for given setup and pulseparameters, one may now move a step further and try to0design filters and pump pulses in order to generate opti-mal mode functions which have high non-Gaussian statecontent and which are particularly easy to handle in thestorage part of an experiment.In the present treatment we have chosen to use the neg-ative value attained by the Wigner function as a measureof the non-classical character of the state generated, butwe note that only minor changes are required in order tooptimize other features as for example the single-photonor Schr¨odinger kitten state fidelity.The authors acknowledge discussions with Jonas S.Neergaard-Nielsen, Anders S. Sørensen, and Eugene S.Polzik. APPENDIX A: DISCUSSION OF THE APPLIEDDETECTOR MODEL
Various detector models are discussed in the literature.Considering first detection in a single mode, a detectormay for instance be a photon number resolving detectoror an on/off detector. The photon number resolving de-tector registers the number of photons in the detectedmode, and the detected mode is projected onto the de-tected photon number state, while the on/off detectoronly distinguishes between the outcomes ’vacuum’ and’not vacuum’, and, depending on the measurement re-sult, the detected mode is projected onto one of thesesubspaces. If a continuous beam of light is observed, athird detector model is often assumed, where a detectionevent at time t corresponds to an annihilation of a pho-ton in a mode of infinitesimal temporal width positionedat time t . Eq. (36) is derived under the assumption thata detection event is equivalent to application of the anni-hilation operator of the observed mode to the state of thesystem, but since the flux of trigger photons is small inthe experiment considered in Sec. IV and the possibilityto have two or more photons in a trigger mode can beneglected, all three detector models lead to practicallyidentical results.A real detector consists of a medium, which absorbsphotons, and typically several different microscopic tran-sitions in the medium are allowed. The detector is thenable to detect photons in more than a single mode. In an APD a photon absorption is accompanied by an excita-tion of an electron from a bound state to a free or a solidstate conduction band state, the signal is amplified, andthe resulting photo current is measured. The informa-tion concerning the precise microscopic transition is lostin the amplification process, and, when we condition onthe macroscopic detection event, we must average overall modes that could have led to the observed detection(see [20]). Since the filter in front of the detector se-lects a narrow band of frequencies, we assume that thedetector is equally sensitive to all frequencies transmit-ted by the filter, i.e., the detector bandwidth is assumedto be infinitely broad compared to the filter bandwidth,and all modes are observed by the detector. In this casewe should sum over a complete set of modes on the in-terval T in Eq. (37). The obtained values of W (0 , P are independent of the choice of basis which maybe seen from the following argument. 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