Theory of biphoton generation in a single-resonant optical parametric oscillator far below threshold
aa r X i v : . [ qu a n t - ph ] F e b Theory of biphoton generation in a single-resonant optical parametric oscillatorfar below threshold
Ulrike Herzog, Matthias Scholz, and Oliver Benson
Nano-Optics, Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, D-10117 Berlin, Germany (Dated: October 26, 2018)We present a quantum-theoretical treatment of biphoton generation in single-resonant type-IIparametric down-conversion. The nonlinear medium is continuously pumped and is placed inside acavity which is resonant for the signal field, but nonresonant for the idler deflected by an intra-cavitypolarizing beam splitter. The intensity of the classical pump is assumed to be sufficiently low inorder to yield a biphoton production rate that is small compared to the cavity loss rate. Explicitexpressions are derived for the rate of biphoton generation and for the biphoton wave function. Theoutput spectra of the signal and idler field are determined, as well as the second-order signal-idlercross-correlation function which is shown to be asymmetric with respect to the time delay. Due tofrequency entanglement in the signal-idler photon pair, the idler spectrum is found to reveal thelongitudinal mode structure of the cavity, even though the idler field is not resonant.
PACS numbers: 03.67.-a,42.50.-p,42.50.Ar,42.65.Lm
I. INTRODUCTION
In parametric down-conversion, a pump photon of fre-quency ω p incident on a medium with a second-ordernonlinear susceptibility χ is split into two photons withlower frequency [1]. Spontaneous parametric down con-version produces photon pairs that can be entangled inmany degrees of freedom [2]. The resulting two-photonstate, consisting of a signal photon at frequency ω s andan idler photon at frequeny ω i = ω p − ω s , is often calleda biphoton. Upon post-selection on the idler photons,the process provides a source of heralded single photonswhich represent the principal resource in many quantuminformation processing protocols like quantum cryptog-raphy [3, 4, 5] or linear optics quantum computation [6].Quantum networks have been proposed [7, 8] that rely onstationary atoms or ions as information processing nodesand on single photons to transmit information via opticalfibers. First building blocks of this scheme have alreadybeen realized [9, 10]. For an efficient atom-photon cou-pling, the photon bandwidth has to match the linewidthof the atomic transition which is by orders of magnitudesmaller than the bandwidth of the photons emitted inspontaneous parametric down-conversion.In order to reduce the photon bandwidth, cavity-enhanced parametric down-conversion can be applied. Abright source of heralded narrow-band single photons wasexperimentally realized [11] using a double-resonant op-tical parametric oscillator (OPO). The nonlinear crystalwas placed inside a cavity resonant for both the signaland the idler field, and in the output a single narrow-band longitudinal signal mode was selected with an ex-ternal frequency filter. Clearly, to conditionally achievesingle-photon generation, the OPO has to be operated inthe regime far below threshold where the production rateof down-converted photons is small compared to the lossrate of the cavity. A number of preceding experiments forbiphoton generation in a double-resonant OPO far belowthreshold have been performed [12, 13, 14, 15, 16, 17], and the first theoretical description was given in Ref. [13].Recently, the theory of the double-resonant OPO hasbeen extended by analyzing the conditionally preparedsingle-photon state [18], and by providing a multi-modetreatment which is valid for both pulsed and stationarypump fields [19].To ensure reliable operation of a quantum network,continuous photon emission over a long period of time isessential. For this purpose, an active stabilization of theOPO is necessary which proves to be a complicated taskin the double-resonant case while it is easier to achieve fora single-resonant cavity. Continuous biphoton generationin a single-resonant OPO far below threshold has recentlybeen demonstrated in our group [20] using a setup whereonly the signal mode experiences resonance enhancementin the cavity while the orthogonally polarized idler modeis nonresonant due to deflection by an intra-cavity polar-izing beam splitter. Additional passive filtering with thehelp of an external cavity can select a single longitudinalmode and will thus enable the generation of narrow-bandsingle photons.To our knowledge, a theoretical treatment of the single-resonant OPO far below threshold has not been per-formed so far. The present paper aims to fill this gap.Based on the concepts of the pioneering theoretical stud-ies of spontaneous parametric down-conversion [21, 22],we provide the theoretical background for our experi-mental results [20]. Different from the approaches usedfor the theoretical description of the double-resonantOPO [13, 18, 19], free-field quantization of the idler fieldis inevitable for our scheme.This paper is organized as follows: In Sec. II, thebasic equations for describing the nonlinear interactionbetween the quantized signal and idler fields are pro-vided. The biphoton production rate and the biphotonwave functions are derived in Sec. III by applying thestandard perturbative treatment in the Schr¨odinger pic-ture [1]. The results are used in Sec. IV to derive thespectral properties of the emitted radiation and to studythe the second-order signal-idler cross-correlation func-tion. Sec. V concludes the paper by establishing the con-nection to real experimental situations. II. BASIC EQUATIONSA. The interaction Hamiltonian
We consider type-II parametric down-conversion ina nonlinear crystal of length l that is pumped by amonochromatic linearly polarized classical field of fre-quency ω P . The crystal is assumed to be placed in-side a cavity which is resonant for the signal field, butnot for the idler field, polarized orthogonal to the signal.Fig. 1 shows a schematic picture of the correspondingexperimental setup [20]. The central frequencies ω S and nonlinearcrystal PBS w S coincidencecounter x0 - l w I w P PDPD
FIG. 1: (Color online) Scheme of the considered experiment.The cavity acts as a one-sided resonator for the signal field,while the idler field is non-resonant due to deflection at anintra-cavity polarizing beam splitter. The signal-idler cross-correlation function is determined by delayed coincidence de-tection. ω I of the signal and idler field depend on the propertiesof the birefringent nonlinear crystal and are determinedby energy and momentum conservation known as phase-matching [1] ω P = ω S + ω I , (2.1) ~k p ( ω P ) = ~k s ( ω S ) + ~k i ( ω I ) . (2.2)Here, ~k p , ~k s , and ~k i are the wave vectors of the pump,signal, and idler waves which are collinear in the consid-ered setup, with the signal leaving the cavity in positive x -direction. The standing-wave fields of the pump andthe signal inside the cavity are composed of two compo-nents, propagating in negative and positive x -direction,respectively. Because of the phase-matching conditions,only those components contribute to the parametric in-teraction that have the same propagation direction as theidler wave. Neglecting the vector notation of the fields,the positive-frequency part of the electric field in the rel-evant component of the classical pump inside the crystalcan be written as E P,cr ( x, t ) = E P e i [ k p ( ω P ) x − ω P t ] . (2.3)The weak signal and idler fields inside the crystal aredescribed by the operators E (+) S,cr = E ( − ) † S,cr and E (+) I,cr = E ( − ) † I,cr , respectively, denoting the positive-frequency partof their co-propagating field components. The interactionHamiltonian can be written in the simplified form H int = χ l Z − l dx ( E P,cr E ( − ) S,cr E ( − ) I,cr + E ∗ P,cr E (+) S,cr E (+) I,cr ) , (2.4)where the second-order nonlinear susceptibility χ isfrequency-dependent [1]. Before utilizing Eq. (2.4), weneed to find explicit expressions for the operators E (+) S,cr and E (+) I,cr . B. The free-field operators
For later use, we start by providing the operators forthe signal and idler fields in free space. The positive-frequency electric field operator of a wave with transversecross-section A propagating freely in x -direction is givenby E (+) ( x, t ) = lim L →∞ P ∞ j =0 q ~ ω j ǫ LA a j e iω j ( xc − t ) with ω j = 2 πjc/L where L is the quantization length and a j denotes the photon annihilation operator of mode j . Letus consider the signal field and introduce the frequencydifference Ω j = ω j − ω S . Using ∆Ω = 2 πc/L , the transi-tion to the continuum limit is performed via the replace-ment P j a j . . . → (∆Ω) − / R ∞− ω S d Ω a ( ω S + Ω) . . . wherethe continuous field operators a ( ω ) have the dimensions / . Since the bandwidth of the signal is small comparedto its central frequency ω S , the integration interval canbe extended to −∞ , and we arrive at the approximateoperator representation E (+) S ( x, t ) = r ~ ω S ǫ cA Z ∞−∞ d Ω √ π a ( ω S + Ω)e i ( ω S +Ω)( xc − t ) , (2.5)where [23] [ a ( ω ) , a † ( ω )] = δ ( ω − ω ) . (2.6)Similarly, the corresponding operator for the idler fieldin free space is given by E (+) I ( x, t ) = r ~ ω I ǫ cA Z ∞−∞ d Ω √ π b ( ω I + Ω)e i ( ω I +Ω)( xc − t ) , (2.7)where [ b ( ω ) , b † ( ω )] = δ ( ω − ω ) . (2.8)Since in type-II parametric down-conversion signal andidler photons are polarized orthogonally, we have[ a ( ω ) , b † ( ω )] = 0 . (2.9) C. The field operators inside the crystal
We now turn to the fields inside the crystal. If the non-linear interaction is small, we do not need to consider thecomplicated problem of field quantization in a nonlinearmedium, but we can represent the field operators insidethe crystal by adapting the corresponding expressions forthe free-field operators in order to account for the pres-ence of a lossless dispersive medium [1]. In accordancewith Ref. [24], the operator for the positive-frequencypart of the idler field inside the crystal then takes theapproximate form E (+) I,cr ( x, t ) = (2.10) r ~ ω I ǫ cAn I Z ∞−∞ d Ω √ π b ( ω I + Ω)e i [ k I (Ω) x − ( ω I +Ω) t ] , where we introduced the wave vector at frequency ω I +Ω, k I (Ω) = ω I + Ω c n i ( ω I + Ω) . (2.11)Here, the replacements ǫ → ǫ n i and c → c/n i havebeen performed where n i is the refractive index of theidler wave and n I = n i ( ω I ).To describe the signal field, we have to take the pres-ence of the resonator into account. Let us first assumea lossless resonator completely filled with the nonlinearmedium. The quantization length of the field is thenequal to the crystal length l . When n s denotes the re-fractive index of the signal, the adapted resonator eigen-frequencies characterizing the longitudinal modes can bewritten as ω m = ( m + m ) πcn s ( ω m ) l with m = ω S n S lπc , (2.12)where n S = n s ( ω S ), m = 0 , ± , ± , . . . and m ≫ | m | .Since the frequency difference between adjacent modes issmall compared to the total spectral width of the signal,as will become obvious in Sec. IV, we can assume withoutlack of generality that ω S coincides with the frequency ofa longitudinal mode and m is an integer, i.e. that thecavity is tuned to resonance. Using the Taylor expan-sion n s ( ω m ) = n S + ( ω m − ω S ) ∂n s ∂ω | ω = ω S , we find fromEq. (2.12) after minor algebra that ω m ≈ ω S + m π v g,S l ≡ ω S + m ∆ ω c , (2.13)where v g,S = cn S + ω S ∂n s ∂ω | ω = ω S (2.14)is the group velocity of the signal at frequency ω S . Byadapting the empty-cavity field operator [23] with thereplacements ǫ → ǫ n s and c → c/n s , the part ofthe standing-wave signal field operator inside the crys-tal that corresponds to a component traveling in positive x -direction in the lossless resonator is found to be E (+) S,cr ( x, t ) = r ~ ω S ∆ ω c ǫ n S cAπ ∞ X m = −∞ a m e iω m [ xc n s ( ω m ) − t )] . (2.15) Here, we replaced the quantization length under thesquare-root sign by the expression l = π v g,S / ∆ ω c ≈ π c/ ( n S ∆ ω c ), following Eqs. (2.13) and (2.14). Moreover,the summation has been extended to m = −∞ , in anal-ogy to the expanded integration range in Eq. (2.5). Thephoton annihilation and creation operators for mode m obey the usual commutation relation [ a m , a † m ′ ] = δ m,m ′ .When resonator losses are incorporated, the modesturn into quasi-modes and the annihilation operators a m in Eq. (2.15) become time-dependent. According to theinput-output formalism [23, 25] for a one-sided cavitywith loss constant γ , the damping of mode m is describedby ˙ a m ( t ) = − γ a m ( t ) + √ γa INm ( t ) . (2.16)The operators a INm ( t ) and a OUTm ( t ) characterize the ingo-ing and outgoing photon flux at frequency ω m and havethe dimension s − / . They are related by the boundarycondition a INm ( t ) = √ γa m ( t ) − a OUTm ( t ) . (2.17)In order to determine a m ( t ), we use the representation a OUTm ( t ) = 1 √ π Z ∞−∞ d Ω a ( ω m + Ω)e − i Ω t , (2.18)where, in analogy to Eq. (2.6),[ a ( ω m + Ω) , a † ( ω m ′ + Ω ′ )] = δ m,m ′ δ (Ω − Ω ′ ) . (2.19)Eq. (2.19) implies that the quasi-modes do not overlapwhich is justified in the good-cavity limit γ ≪ ∆ ω c . (2.20)After inserting Eq. (2.17) into Eq. (2.16), we obtain byFourier transformation the solution a m ( t ) = 1 √ π Z ∞−∞ d Ω a ( ω m + Ω) √ γ γ + i Ω e − i Ω t (2.21)which has to be applied to Eq. (2.15). The operator forthe relevant field component of the signal in the lossycavity can then be written as E (+) S,cr ( x, t ) = r ~ ω S ǫ n S cA √ γ ∆ ω c π (2.22) × ∞ X m = −∞ Z ∞−∞ d Ω a ( ω m + Ω) γ + i Ω e i [ k S,m (Ω) x − ( ω m +Ω) t ] . Here, k S,m (Ω) = ω m + Ω c n s ( ω m + Ω) (2.23)is the wave vector corresponding to a traveling-wave com-ponent of frequency ω m + Ω. The denominator in theintegral in Eq. (2.22) describes radiation suppression forfrequencies ω with | ω − ω m | = | Ω | ≫ γ while resonanceenhancement occurs for ω ≈ ω m .So far, we have assumed a resonator length L r thatcoincides with the crystal length l . If L r > l , a rigor-ous quantization of the signal field has to account for theexact position of the crystal inside the resonator, but isbeyond the scope of the present paper. For the purposesof our approximative treatment, however, it is sufficientto describe the signal field inside the crystal by Eq. (2.22)with ∆ ω c = ∆ ω and ω m = ω S + m ∆ ω where ∆ ω is the ef-fective free spectral range. The latter can be representedas ∆ ω = 2 πT with T = 2 lv g,S + 2( L r − l ) c , (2.24)where T is the effective cavity round-trip time of a signalphoton. III. THE RATE OF BIPHOTON GENERATIONAND THE BIPHOTON WAVE FUNCTION
With the expressions for the operators of the signal andidler field at hand, we are now in the position to specifythe interaction Hamiltonian and to derive a perturba-tive solution of the Schr¨odinger equation. Making use ofEqs. (2.1), (2.3), (2.10) and (2.22), as well as Eq. (2.13)with ∆ ω c → ∆ ω and taking the frequency-dependenceof the nonlinear susceptibility into account, we find fromEq. (2.4) H int = i ~ α ∞ X m = −∞ Z ∞−∞ d Ω √ γ γ − i Ω Z ∞−∞ d Ω ′ F m (Ω , Ω ′ ) × a † ( ω m + Ω) b † ( ω I + Ω ′ )e i ( m ∆ ω +Ω+Ω ′ ) t + H.A., (3.1)where we introduced the function F m (Ω , Ω ′ ) = (3.2) χ ( ω P ; ω m + Ω , ω I + Ω ′ ) χ ( ω P ; ω S , ω I ) l Z − l dx e i [ k P − k S,m (Ω) − k I (Ω ′ ] and defined the constant α = − iE P πǫ cA r ω S ω I n S n I χ ( ω P ; ω S , ω I ) √ ∆ ω. (3.3)We are interested in the regime far below threshold wherethe biphoton production rate κ is much smaller than thecavity damping rate, κ ≪ γ, (3.4)and the mean photon number in the resonator there-fore close to zero. Since the mean time interval betweenbiphoton emission events is large compared to the cav-ity damping time, the resonator can be assumed to beempty before each emission event. For this case, we can perform a perturbative treatment of the nonlinear inter-action, assuming that at the initial time t = 0, the com-bined signal-idler field is in the vacuum state, describedby | i = | i S ⊗ | i I . In the following we rely on theideas developed for the theory of spontaneous paramet-ric down-conversion [1, 21, 22]. By expanding the formalsolution of the time-dependent Schr¨odinger equation, upto a normalization factor, the state vector describing thecombined signal-idler field at a time t = ∆ t ≪ κ − isfound to be | ψ (∆ t ) i ∝ | i + | ˜ ψ (∆ t ) i with | ˜ ψ (∆ t ) i = 1 i ~ Z ∆ t dt H int ( t ) | i . (3.5)After substituting Eq. (3.1) into Eq. (3.5), integrationwith respect to t yields | ˜ ψ (∆ t ) i = ∞ X m = −∞ Z ∞−∞ d Ω α √ γ γ − i Ω Z ∞−∞ d Ω ′ F m (Ω , Ω ′ ) × ∆ t sinc (cid:20)
12 ( m ∆ ω + Ω + Ω ′ )∆ t (cid:21) e i ( m ∆ ω +Ω+Ω ′ )∆ t × a † ( ω m + Ω) b † ( ω I + Ω ′ ) | i , (3.6)where we used the sinc-function defined as sinc( z ) = sin zz with sinc(0) = 1.The nonnormalized vector | ˜ ψ (∆ t ) i refers to the stateof the radiation field on the condition that a signal-idlerphoton pair has been produced during the time inter-val ∆ t . Since this condition applies with the probability h ˜ ψ (∆ t ) | ˜ ψ (∆ t ) i , the biphoton production rate is given by κ = 1∆ t h ˜ ψ (∆ t ) | ˜ ψ (∆ t ) i . (3.7)Applying the commutation relations Eqs. (2.8) and(2.19), we get from Eq. (3.6) h ˜ ψ (∆ t ) | ˜ ψ (∆ t ) i = (∆ t ) ∞ X m = −∞ Z ∞−∞ d Ω | α | γ ( γ ) + Ω (3.8) × Z ∞−∞ d Ω ′ | F m (Ω , Ω ′ ) | sinc (cid:20)
12 ( m ∆ ω + Ω + Ω ′ )∆ t (cid:21) . Because of the properties of the sinc-function, the inte-gral is dominated by the region m ∆ ω +Ω+Ω ′ ≤ π/ ∆ t . If∆ t is sufficiently large and F m (Ω , Ω ′ ) is a slowly varyingfunction of Ω ′ in this region, the latter function can bereplaced by its value at Ω ′ = − m ∆ ω − Ω [1]. Using therelation R ∞−∞ sinc ( z ) dz = π , the integration with respectto Ω ′ is then readily performed, and we obtain h ˜ ψ (∆ t ) | ˜ ψ (∆ t ) i = (3.9)2 π | α | ∆ t ∞ X m = −∞ Z ∞−∞ d Ω γ ( γ ) + Ω | Φ m (Ω) | , where Φ m (Ω) = F m (Ω , − m ∆ ω − Ω) . (3.10)For further evaluation, we need to specify the func-tion Φ m (Ω). Using Eqs. (2.2), (2.11), and (2.23) with ω m = ω S + m ∆ ω , Taylor expansion around the centralfrequencies ω S and ω I yields k P − k S,m (Ω) − k I ( − m ∆ ω − Ω) ≈ ( m ∆ ω +Ω) τ l , (3.11)where we introduced the time constant τ = lc n I + ω I ∂n i ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω I − n S − ω S ∂n s ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω S ! . (3.12)The latter is equivalent to τ = lv g,I − lv g,S (3.13)and describes the difference between the transit timesof a signal and idler photon through a crystal of length l , originating from the difference in the signal and idlergroup velocities, v g,S and v g,I , defined by Eq. (2.14) andby the corresponding equation for the idler wave, respec-tively. Since in any real experiment | v g,S − v g,I | ≪ v g,S ,it follows that | τ | ≪ (∆ ω ) − where we used Eq. (2.24).Hence, we are considering a parameter range in this prob-lem that is characterized by the combined inequality κ ≪ γ ≪ ∆ ω ≪ | τ | − , (3.14)where Eqs. (2.20) and (3.4) have been incorporated [27].Assuming a constant nonlinear susceptibility within thebandwidth given by | τ | − , we find from Eqs. (3.10),(3.11), and (3.2) thatΦ m (Ω) ≈ l Z − l dx e i ( m ∆ ω +Ω) τ l x . (3.15)In order to determine the rate of biphoton generation,we have to insert Eq. (3.15) into Eq. (3.9) where the in-tegration with respect to Ω is effectively restricted to thecavity bandwidth γ with γ ≪ ∆ ω . Hence, we neglectΩ compared to m ∆ ω in Eq. (3.15) for m = 0. More-over, Eq. (3.14) implies | Ω τ | ≪ | Ω | . γ and therefore Φ (Ω) ≈
1. After integration withrespect to x , we get the approximationΦ m (Ω) ≈ Φ m (0) = sinc (cid:16) m ∆ ω τ (cid:17) e − im ∆ ω τ (3.16)which can be used in connection with Eq. (3.9), (3.7),and (3.3) to determine the rate of biphoton generation κ = (cid:18) χ | E p | ǫ cA (cid:19) ω S ω I n S n I ∆ ω ∞ X m = −∞ sinc (cid:16) m ∆ ω τ (cid:17) . (3.17)Further simplification is possible if we transform the suminto an integral with respect to z = m ∆ ωτ /
2, introduc-ing the positive increment dz = ∆ ω | τ | / dz ≪ κ ≈ (cid:18) χ | E p | ǫ cA (cid:19) ω S ω I n S n I π | τ | (3.18) that does not depend on the properties of the resonatorbecause the mean photon number in our cavity is approx-imately zero and, in addition, an associated idler modeexists for each signal mode which meets the requirementfor energy conservation.The presented perturbative treatment relies on thecondition | τ | ≪ ∆ t ≪ κ − so that the wave function | ˜ ψ (∆ t ) i in Eq. (3.6) and the approximation leading toEq. (3.9) are valid simultaneously. The normalized vector | ψ i = ( κ ∆ t ) − / | ˜ ψ (∆ t ) i can be denoted as the biphotonwave function since it represents the state of the radia-tion field on the condition that exactly one signal-idlerphoton pair is present.A more convenient representation of the biphoton wavefunction is obtained if the sinc-function in Eq. (3.6) isreplaced by 2 πδ ( m ∆ ω +Ω+Ω ′ ) according to the standardprocedure [22]. Mathematically, this corresponds to thelimit ∆ t → ∞ , implying κ →
0. Then, the integrationwith respect to Ω ′ can be performed immediately. Inanalogy to the expression given in Ref. [22], we obtainthe biphoton wave function | ψ i = N ∞ X m = −∞ Z ∞−∞ d Ω Φ m (Ω) γ − i Ω × a † ( ω S + m ∆ ω + Ω) b † ( ω I − m ∆Ω − Ω) | i , (3.19)where Φ m (Ω) is given by Eq. (3.15) and N is a normaliza-tion constant. The explicit value of N [26] is not impor-tant as long as only normalized quantities characterizingthe radiation field are considered. Eq. (3.19) clearly re-veals the frequency-entanglement between the signal andidler photon and will serve as our basic equation to de-termine the properties of the emitted radiation. IV. PROPERTIES OF THE EMITTEDRADIATIONA. Output spectra of the signal and idler field
First we investigate the output spectra S S ( ω ) and S I ( ω ) of the signal and idler field, defined as S S/I ( ω ) = 12 π Z ∞−∞ dτ G (1) S/I ( τ ) e iωτ , (4.1)where G (1) S ( τ ) and G (1) I ( τ ) are the first-order temporalcorrelation functions of the respective fields outside theresonator. In the following, it is convenient to use theHeisenberg picture and to start from the expression G (1) S/I ( τ ) = h ψ | E ( − ) S/I ( x, t ) E (+) S/I ( x, t + τ ) | ψ i . (4.2)Here, | ψ i is the time-independent biphoton wave func-tion given by Eq. (3.19) and E (+) S = E ( − ) † S and E (+) I = E ( − ) † I denote the positive-frequency parts of the time-dependent operators of the electric fields in free space, | S i gna l /I d l e r S pe c t r u m S S /I () / S S /I ( S /I ) Frequency -
S/I -2 / | | FIG. 2: (Color online) Schematic plot of the normalized sig-nal and idler output spectra. The width of the Lorentzianpeaks is determined by the cavity damping rate γ , and theyare separated by the free spectral range ∆ ω = 2 π/T . Theenvelope, described by the sinc-function in Eq. (4.6), yieldsthe total spectral width 2 π/ | τ | . given by Eqs. (2.5) and (2.7). Let us first determine thespectrum of the idler field. Considering b ( ω I + Ω ′ ) b † ( ω I − m ∆ ω − Ω) | i = δ (Ω ′ + m ∆ ω + Ω) | i , (4.3)due to Eq. (2.8), we obtain from Eqs. (2.7) and (3.19) E (+) I ( x, t ) | ψ i ∝ ∞ X m = −∞ Z ∞−∞ d Ω Φ m (Ω) γ − i Ω (4.4) × e i ( ω I − m ∆ ω − Ω) ( xc − t ) a † ( ω S + m ∆ ω + Ω) | i . By taking the inner product of E (+) I ( x, t + τ ) | ψ i and E (+) I ( x, t ) | ψ i and applying the commutation relationEq. (2.19), we find G (1) I ( τ ) ∝ ∞ X m = −∞ Z ∞−∞ d Ω | Φ m (Ω) | (cid:0) γ (cid:1) + Ω e − i ( ω I − m ∆ ω − Ω) τ . (4.5)In analogy to the derivation of Eq. (3.17), the dependenceof Φ m on Ω can be neglected within the relevant band-width determined by γ . After inserting Eq. (3.16) intoEq. (4.5), the Fourier transform yielding the idler spec-trum according to Eq. (4.1) is readily performed. Thesignal spectrum can be determined in a completely anal-ogous way, and we finally arrive at the relation S S/I ( ω ) ∝ ∞ X m = −∞ sinc (cid:0) m ∆ ω τ (cid:1)(cid:0) γ (cid:1) + ( ω S/I − m ∆ ω − ω ) . (4.6)Eq. (4.6) indicates that both the signal and idler spec-trum are composed of Lorentzians of halfwidth γ centered at frequencies ω S/I − m ∆ ω with m = 0 , ± , . . . wherethe respective spectral envelopes are determined by thesinc-function in the nominator. From Fig. 2 it becomesobviouos that the frequency bandwidth of the signal andidler photons is characterized by | τ | − . Their tempo-ral uncertainty is thus equal to the modulus of the timeconstant τ introduced in Eq. (3.12) and resulting fromthe phase-matching conditions. Even though the idlerwave is not resonant, the longitudinal mode structure ofthe resonator is revealed in the idler spectrum due tothe frequency entanglement between the signal and idlerphoton which arises from the interaction underlying thebiphoton generation process. B. Signal-idler cross-correlations
The coincidence rate for detecting an idler photon attime t and a signal photon at time t + τ , both at equaldistance from the end facet of the crystal, is proportionalto the temporal correlation function G (2) IS ( τ ) = (4.7) h ψ | E ( − ) I ( x, t ) E ( − ) S ( x, t + τ ) E (+) S ( x, t + τ ) E (+) I ( x, t ) | ψ i , where again E (+) S and E (+) I are the free-field operatorsdefined by Eqs. (2.5) and (2.7). With the explicit ex-pression for the biphoton wave function | ψ i , given byEq. (3.19), we get E (+) S ( x, t + τ ) E (+) I ( x, t ) | ψ i ∝ e i [ ( ω S + ω I ) ( xc − t ) − τ ω S ] × ∞ X m = −∞ e − im ∆ ωτ Z ∞−∞ d Ω Φ m (Ω) γ − i Ω e − i Ω τ | i , (4.8)where Eq. (4.3) and the corresponding relation for thesignal modes a ( ω S + Ω ′ ) a † ( ω S + m ∆ ω + Ω) | i = δ (Ω ′ − m ∆ ω − Ω) | i , (4.9)following from Eq. (2.6), have been applied. Accordingto Eq. (4.7), the correlation function G (2) IS ( τ ) is propor-tional to the squared norm of the Hilbert vector on theright-hand side of Eq. (4.8). It can be determined fromthe explicit expression for Φ m (Ω), given by Eq. (3.15),together with the integral identities − π Z ∞−∞ d Ω e − i Ω tγ − i Ω = t <
01 if t = 02e − γ t if t > . (4.10)First, it is important to observe that G (2) IS ( τ ) = 0 if τ + τ < − | τ | (4.11)which is equivalent to τ < − τ for τ > τ < τ <
0, respectively. Mathematically, Eq. (4.11) is due tothe expression Ω t = Ω( τ − xl τ ) which results from insert-ing Eq. (3.15) into (4.8) and is negative in the given case -2 0 2 4 60.00.20.40.60.81.01.2 N o r m a li z ed C r o ss - C o rr e l a t i on F un c t i on Time Delay / T /T FIG. 3: Schematic representation of the normalized second-order signal-idler cross-correlation function G (2) IS ( τ ) accordingto Eq. (4.13) for τ > γ/ ∆ ω = 0 .
05. If the time delay τ is equal to zero or multiples of the cavity round-trip time T = 2 π/ ∆ ω , the function exhibits pronounced peaks whichdecay with the cavity damping time γ − . for any x inside the crystal, i.e. for − l ≤ x ≤
0. There-fore, the upper line of Eq. (4.10) applies. Physically, thecorrelation function vanishes because the arrival time ofthe signal photon in a photon pair can precede the arrivaltime of the corresponding idler photon at most by a timeinterval that is within the temporal uncertainty interval | τ | inherent in the biphoton generation process.On the other hand, the presence of the resonator al-lows a signal photon to be detected considerably laterthan the associated idler photon since the signal photonmay bounce back and forth between the resonator mir-rors repeatedly before leaving the resonator. For timedelays τ outside the limits of Eq. (4.11), we approxi-mate Φ m (Ω) by Eq. (3.16) where the integration withrespect to x has been performed and the dependenceon Ω has been neglected. After inserting Eq. (3.16)into Eq. (4.8), the damping of the cavity gives rise toa factor proportional to e − γ τ for G IS ( τ ) = 0 whichfollows from the third line of Eq. (4.10) [28]. Since G (2) IS ( τ ) = k E (+) S ( x, t + τ ) E (+) I ( x, t ) | ψ ik , we finally getfrom Eq. (4.8) the approximate result G (2) IS ( τ ) ∝ e − γτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X m = −∞ sinc (cid:16) m ∆ ω τ (cid:17) e − im ∆ ω ( τ + τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) if τ + τ ≥ − | τ | . (4.12)According to Eq. (3.13), the sign of τ is determined bythe relation between the signal and idler group velocities,i.e. τ > v g,S > v g,I and τ < v g,S < v g,I . Anumerical evaluation reveals that Eq. (4.12) describes adecaying periodic function with peaks of width | τ | cen-tered at τ = j T − τ / j = 0 , , . . . and T = 2 π/ ∆ ω is the cavity round-trip time. The time shift − τ / τ > τ / τ < z = m ∆ ωτ / dz = ∆ ωτ /
2, we find that G (2) IS ( τ ) ∝ e − γτ I with I = R ∞−∞ sinc( z ) cos( az ) dz sincesinc( z ) is an even function of z . Because of the period-icity of the cos-function, the parameter a can be writtenas a = 1 + 2 (cid:2) τ − (cid:0)(cid:2) τT (cid:3) + 1 (cid:1) T (cid:3) /τ where (cid:2) τT (cid:3) denotes thelargest integer that does not exceed τ /T . Since I = π for | a | < I = 0 for | a | >
1, Eq. (4.12) takes a simpleform that can be combined with Eq. (4.11) to yield thecompact approximate representation G (2) IS ( τ ) ∝ ∞ X j =0 (cid:26) e − γjT if (cid:12)(cid:12) τ − j T + τ (cid:12)(cid:12) ≤ | τ | j = (cid:12)(cid:12)(cid:2) τT (cid:3)(cid:12)(cid:12) (see Fig. 3). Here, we took into accountthat due to Eq. (3.14) cavity damping is negligible duringthe time interval | τ | , i.e. exp( − γτ ) ≈ V. DISCUSSION AND CONCLUSIONS
In a real experiment, the sharp peaks of the function G (2) IS ( τ ) will be broadened due to the finite resolutiontime of the detector setup. When the latter is taken intoaccount by performing the convolution with respect toa Gaussian function, Eq. (4.13) yields the time-averaged -2 0 2 4 60.00.20.40.60.81.01.2 N o r m a li z ed and A v e r aged C r o ss - C o rr e l a t i on F un c t i on Time Delay / T
FIG. 4: Time-averaged second-order signal-idler cross-corre-lation function G IS (2) ( τ ) for γ/ ∆ ω = 0 .
05. The ratio be-tween the resolution time ∆ T and the cavity round-trip time T = 2 π/ ∆ ω is assumed as ∆ T /T = 0 .
02 (dashed line) and∆
T /T = 1 (solid line), respectively. cross-correlation function G IS (2) ( τ ) ∝ ∞ X j =0 exp (cid:20) − γjT − jT − τ ) (∆ T ) (cid:21) , (5.1)where ∆ T characterizes the effective resolution time andwhere we have assumed ∆ T ≫ | τ | . The resulting aver-aged function is plotted in Fig. 4 for two different valuesof ∆ T . A second-order cross-correlation function show-ing the behavior of the solid line in Fig. 4 has recentlybeen measured in our group [20], and the results havebeen found to be in excellent agreement with the predic-tions derived from Eqs. (4.11) and (4.12).We still note that for spontaneous parametric down-conversion in a double-resonant cavity the signal-idlercross-correlation function has also been found to exhibita comb-like structure which is, however, symmetric withrespect to the time delay [14, 15, 16]. The effect hasbeen explained by applying the concept of mode-lockingto the frequency-entangled biphoton state, pointing outthat due to the large coherence time of the pump, pho- ton pairs with different frequencies have a common phaseand form a coherent superposition [14].To summarize, we performed a theoretical investiga-tion of biphoton generation by spontaneous paramet-ric down-conversion in a single-resonant OPO far belowthreshold. We derived analytical expressions for the rateof biphoton generation, for the output spectra of the sig-nal and idler fields, as well as for the second order signal-idler cross correlation function. Our investigations pro-vide the theoretical background for explaining the resultsof a recent experiment [20], where stable continuous op-eration of a single-resonant OPO far below threshold hasbeen demonstrated. Acknowledgments
This work was supported by Deutsche Forschungsge-meinschaft DFG, grant BE 2224/5. M. Scholz acknowl-edges funding by Deutsche Telekom Stiftung. [1] L. Mandel and E. Wolf,
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10 MHz, ∆ ω ∼ | τ | − ∼∼