Theory of high gain cavity-enhanced spontaneous parametric down-conversion
TTheory of high gain cavity-enhanced spontaneous parametric down-conversion
Joanna A. Zieli´nska ∗ and Morgan W. Mitchell , ICFO – Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain ICREA – Instituci´o Catalana de Recerca i Estudis Avan¸cats, 08015 Barcelona, Spain
We compute the output of multimode cavity-enhanced spontaneous parametric down-conversion(SPDC) for sub-threshold, but otherwise arbitrary, gain. We find analytic Bogoliubov transfor-mations that allow us to calculate arbitrary field correlation functions, including the second-orderintensity correlation function G (2) ( T ). The results show evidence of increased coherence due tostimulated SPDC. We extend an earlier model [Lu and Ou, Phys. Rev. A, , 033804 (2000)] toarbitrary gain and finesse, and show the extension gives accurate results in most scenarios. Theresults will allow simple, analytic description of cavity-based nonclassical light sources for quantumnetworking, quantum-enhanced sensing of atoms and generation of highly non-classical field states. I. INTRODUCTION
Cavity-enhanced spontaneous parametric down-conversion (CESPDC), in which a spontaneous para-metric down-conversion (SPDC) process is resonantlyenhanced by placing the χ (2) medium inside an opticalcavity, has been used to make highly efficient photon pairsources [1, 2] of interest for quantum networking withatomic quantum memories [3–6] and atomic quantummetrology [7, 8], applications that require both highspectral brightness and narrow line widths. SPDCsources in combination with coherent states have beenproposed as extremely bright photon pair sources [9],and as sources of entangled multi-photon states [10, 11].Many calculations of the fields emitted by CESPDCare based on techniques developed to calculate squeezingin parametric amplifiers [12, 13]. The cavity is describedin a modal expansion and quantum reservoir theory [14]is used to derive dynamical relationships between cavity,input, and output fields. When these are solved, the re-sulting Bogoliubov transformation expresses the outputfields as squeezed versions of the input fields [15, 16].Using this approach, Lu and Ou [13] computed G (2) ( T ),the second-order intensity correlation function for type-ICESPDC. Reflecting experimental conditions of the time,that calculation remained in the low-gain limit and ap-proximated the cavity line-shapes as Lorentzian, as ap-propriate to high finesse cavities.In contemporary applications, there is a trend towardlower-finesse cavities in CESPDC [17]. The availablesingle-pass gain has increased, due to periodically polednonlinear materials and more powerful pump lasers, andlowering the finesse allows higher escape efficiencies atthe same system gain level. At these lower finesses, the“tails” of the modes begin to overlap, and mode shapesdeviate from the simple Lorentzian. At the same time,higher-gain applications, for example in generation of“Schr¨odinger kitten” [18] states and other highly non-classical time-domain states [19–21] by photon subtrac-tion, are also becoming important. These higher-gain ∗ Corresponding author [email protected] processes necessarily involve stimulated SPDC [22], inwhich a photon or a pair of photons induces the produc-tion of more pairs. These developments motivate a newcalculation of CESPDC fields beyond the low gain, single-longitudinal-mode, and high-finesse approximations.Our method is similar to the classic works of Collettand Gardiner [15] and Gardiner and Savage [16], in thatwe use input-output relations for squeezing and cavityin/out-coupling to obtain equations relating input, out-put, and intra-cavity fields. In contrast to those works,we avoid quantum reservoir theory by posing the prob-lem directly in the time domain. As we describe below,narrow-band CESPDC is more naturally and transpar-ently described in this way. We find difference equationsdescribing the input, output, and cavity fields at con-secutive round-trip times. Eliminating the cavity fieldfrom these equations, we find the Bogoliubov transfor-mation expressing the output fields in terms of the inputfields. To study the time-domain structure, we calculatethe second-order intensity correlation function G (2) ( T )for a type-I OPO, including arbitrary finesse and gain.We find an envelope well approximated by a double ex-ponential with a gain-dependent decay constant, multi-plied by a comb structure with a period equal to thecavity round trip time. At low gain and high finesse thisagrees with the calculation of [13]. At higher gains wefind coherence beyond the cavity ring-down time due tostimulated SPDC. II. BOGOLIUBOV TRANSFORMATIONS
Let us consider a two-sided ring cavity as in Fig. 1with roundtrip time denoted as τ . We characterize thecavity amplitude transmission and reflection coefficientswith real numbers t i and r i , where a subscript i = 1 , t i,ce , the transmissionfrom the exterior to the interior of the cavity t i,ec , thereflection from inside the cavity r i,cc and the reflection a r X i v : . [ qu a n t - ph ] J a n from the outside the cavity r i,ee . These coefficients are re-lated by energy conservation: | t i,ce | + | r i,ee | = | t i,ec | + | r i,cc | = 1 and t i,ce r ∗ i,ec + t i,ec r ∗ i,cc = 0. We assume thatall t and r coefficients are real, and t i,ec = t i,ce ≡ t i , and r i,cc = − r i,ee ≡ r i . The intracavity field annihilation op-erator just before reaching the output coupler is denotedas a , while the input fields just before reaching the cav-ity are a in and b in . We denote the output field just afterexiting the cavity as a out .The field experiences three relevant transformationsduring a round-trip of the cavity. Interaction with theoutput coupler produces a OC → r a + t a in , (1)where a in is the input field. Other losses (here lumpedtogether in a single interaction) produce a loss → r a + t b in , (2)where b in is a bath mode assumed to be in vacuum.Finally there is the Bogoliubov transformation due tosqueezing on a single pass through the crystal a sq → a cosh( r ) + a † sinh( r ) , (3)where r is the squeezing amplitude.Applying these three transformations in sequence to a ( t − τ ) (understood to be the intra-cavity field at a lo-cation immediately before the output coupler), we have Af ekS a → r a + t a in (4) → r ( r a + t a in ) + t b in (5) → cosh( r )[ r ( r a + t a in ) + t b in ]+ sinh( r )[ r ( r a † + t a † in ) + t b † in ] . (6)Considering that a round-trip takes time τ and the field a ( t ) depends only on a ( t − τ ), which is true if we ne-glect the dispersion and finite bandwidth of the phase-matching (see below), we have a ( t ) = r r cosh( r ) a ( t − τ ) + r r sinh( r ) a † ( t − τ )+ t r cosh( r ) a in ( t − τ ) + t r sinh( r ) a † in ( t − τ )+ t cosh( r ) b in ( t − τ ) + t sinh( r ) b † in ( t − τ ) (7)with the hermitian conjugate: a † ( t ) = r r cosh( r ) a † ( t − τ ) + r r sinh( r ) a ( t − τ )+ t r cosh( r ) a † in ( t − τ ) + t r sinh( r ) a in ( t − τ )+ t cosh( r ) b † in ( t − τ ) + t sinh( r ) b in ( t − τ ) . (8)The output field is given by a out ( t ) = − r a in ( t ) + t a ( t ) . (9)Writing a ( t ) = 1 √ π (cid:90) ∞−∞ a ( ω ) e − iωt dωa † ( t ) = 1 √ π (cid:90) ∞−∞ a † ( ω ) e iωt dω ssq squeezing with amplitude r 𝑟 , 𝑡 𝑟 , 𝑡 𝑎 𝑖𝑛 𝑎 𝑜𝑢𝑡 𝑏 𝑖𝑛 𝑏 𝑜𝑢𝑡 𝑎 FIG. 1. (color online) An OPO scheme with input, outputand intracavity field operators for double-sided cavity with anonlinear crystal inside. and solving Eqs. (7),(8), (9) for a out , we find the Bogoli-ubov transformation a out ( ω ) = A ( ω ) a in ( ω ) + B ( ω ) a † in ( − ω )+ C ( ω ) b in ( ω ) + D ( ω ) b † in ( − ω ) (10)where A ( ω ) ≡ d ( ω ) t r [ e − iωτ cosh( r ) − r r ] − r (11) B ( ω ) ≡ d ( ω ) sinh( r ) t r e − iωτ (12) C ( ω ) ≡ d ( ω ) t t [ e − iωτ cosh( r ) − r r ] (13) D ( ω ) ≡ d ( ω ) sinh( r ) t t e − iωτ (14)and d ( ω ) ≡ e − iωτ − r r cosh( r )] − [ r r sinh( r )] . (15)Eqs. (10) to (15) constitute a full description of theoutput of the OPO, in the sense that any correlationfunction of interest can be calculated by taking expecta-tion values of products of a out and a † out . For example, thedegree of quadrature squeezing at a side-band frequencyof Ω can be computed as S (Ω) ≡ (cid:104) [ a out (Ω) + a † out ( − Ω)] (cid:105) , (16)where the expectation (cid:104)·(cid:105) is taken with respect to vacuumin both the a and b modes. S (Ω) is simply a polynomialin A (Ω) to D ( − Ω), so analytical results are available forany gain level.We have neglected dispersion in the cavity and the fi-nite phase-matching bandwidth of the crystal. In thiscase the emission spectrum of the source is not lim-ited by the phase matching profile, and depends only onthe cavity parameters. These approximations are jus-tified in typical narrow-band CESPDC scenarios [23],in which the phase matching bandwidth is several or-ders of magnitude larger than the free spectral range(FSR) of the cavity. Introducing a finite phase matchingbandwidth would modify the shape of the peaks com-posing the multimode G (2) ( T ), but at a time-scale be-yond the resolution of current electronics. As describedin [23] the KTP nonlinear crystal introduces a dispersionof dn/dλ = − . µ m − , which over a phase-matchingbandwidth of 100 GHz ( ≈ − , not shifting any of the resonances by morethan 10 − FSR. In contrast, broad-band CESPDC ex-periments are typically sensitive to the full output band-width of the SPDC process [24], and these approxima-tions would not be justified.
III. MULTIMODE G (2) ( T ) Time-domain correlation measurements on OPOs arean important diagnostic of the spectral content of theoutput [3–6], and are often used to demonstrate the quan-tum nature of the generated fields [1, 2]. In this sectionwe compute the intensity correlation function G (2) ( T ).As with the degree of squeezing, this can be computedanalytically for any sub-threshold gain level and includ-ing all modes.As described above, this correlation function is com-puted as a normally-ordered expectation value with re-spect to the vacuum state in both input modes: G (2) ( T ) ≡ (cid:104) a † out ( t ) a † out ( t + T ) a out ( t + T ) a out ( t ) (cid:105) (17)= (cid:90) d ω e − i ( ω + ω )( t + T ) e − i ( ω + ω ) t G (2) ( (cid:126)ω )(18)where d ω ≡ dω dω dω dω and G (2) ( (cid:126)ω ) ≡ (cid:104) a † out ( − ω ) a † out ( − ω ) a out ( ω ) a out ( ω ) (cid:105) . After the reduction of the operators using the com-mutation relation [ a ( ω ) , a † ( ω (cid:48) )] = δ ( ω − ω (cid:48) ) and know-ing that the coefficients A ( ω ), B ( ω ), C ( ω ) and D ( ω ) arehermitian functions, e.g. A ( − ω ) = A ∗ ( ω ), we find theexpression under the Fourier transform G (2) ( (cid:126)ω ) = δ ( ω + ω ) δ ( ω + ω )Γ( ω , − ω )Γ( ω , − ω )+ δ ( ω + ω ) δ ( ω + ω )Υ( ω , − ω )Υ( ω , − ω )+ δ ( ω + ω ) δ ( ω + ω )Υ( ω , − ω )Υ( ω , − ω )(19) where Γ( ω, ω (cid:48) ) ≡ A ( ω ) B ( − ω (cid:48) ) + C ( ω ) D ( − ω (cid:48) ) (20)Υ( ω, ω (cid:48) ) ≡ B ( ω ) B ( − ω (cid:48) ) + D ( ω ) D ( − ω (cid:48) ) . (21)Performing one integral for each delta function, we arriveto an expression that is t -independent G (2) ( T ) = {F [Γ]( T ) } + {F [Υ]( T ) } + {F [Υ](0) } (22)where Γ( ω ) ≡ Γ( ω, ω ) and Υ( ω ) ≡ Υ( ω, ω ). Knowingthat r + t = 1 and r + t = 1, from Eqs. (11)–(14) wefind Γ( ω ) = d ( ω ) d ( − ω ) t sinh( r ) (cid:2) (1 + r r ) cosh( r ) − r r e iωτ − r r e − iωτ (cid:3) , (23)Υ( ω ) = d ( ω ) d ( − ω ) t sinh( r ) (1 − r r ) . (24) (cid:45) (cid:45)
10 0 10 200.00.20.40.60.81.0 Delay (cid:144) Τ g (cid:72) (cid:76) (cid:64) a . u . (cid:68) FIG. 2. (color online) Theoretical G (2) ( T ) calculated for cav-ity parameters as for the source presented in [23] with gainequal to 1% of the OPO threshold. The envelope of the G (2) ( T ) is calculated from Eqs. (22), (25) and (26), and nor-malized to unity at T = 0. For the purpose of plotting, thepeaks, which in the model are Dirac delta functions, havebeen replaced with finite-width Lorentzians. The necessary Fourier transforms are computed in theAppendix, see Eqs. (A19) and (A17), in terms of a func-tion F ( k ), defined in Eq. (A5). We find {F [Γ]( T ) } = t sinh( r ) ∞ (cid:88) k = −∞ δ ( T − kτ ) (25) × (cid:2) (1 + r r ) cosh( r ) F ( | k | ) − r r F ( | k | + 1) − r r F ( | k | − {F [Υ]( T ) } = t sinh( r ) (1 − r r ) (26) × ∞ (cid:88) k = −∞ δ ( T − kτ ) F ( | k | ) {F [Υ](0) } = t sinh( r ) (1 − r r ) F (0) , (27)the three terms necessary to calculate G (2) ( T ). As shownin Fig. 2, G (2) ( T ) of the multimode cavity output has anenvelope similar to the shape of double falling exponen-tial and peaks every cavity roundtrip time, resulting fromthe interference between the modes. In contrast, the sin-gle mode G (2) ( T ) would also have a double exponentialdecay, but without the comb structure [13]. IV. COMPARISON WITH EARLIER WORK
The G (2) ( T ) calculation of Lu and Ou [13] found themultimode G (2) ( T ) to be a comb of (approximate) Diracdelta functions spaced by the cavity round-trip time, mul-tiplied by an envelope given by the single-mode G (2) ( T ).This result has an appealing simplicity, and is intuitive inthe time-domain picture in which photon pairs are pro-duced simultaneously but may spend a different numberof round trips in the cavity before escaping. It is inter-esting to ask whether the same behaviour persists also athigher gains, i.e. in the presence of stimulated SPDC.We compare our G (2) ( T ), Eq. (22), against the natu-ral extension of the Lu and Ou model for arbitrary gain, (cid:45) (cid:45) (cid:45) (cid:45)
20 0 20 40 60 800.00.20.40.60.81.0 Delay (cid:144) Τ g (cid:72) (cid:76) (cid:64) a . u . (cid:68) FIG. 3. (color online) Three envelopes of multimode G (2) ( T ),computed from Eqs. (22), (25) and (26) and normalized tounity at T = 0. Curves show G (2) ( T ) for the gain r equalto 1% (blue), 50% (green, dashed), 90% (red, dotted) of thethreshold gain r th . Cavity parameters are as for the sourcepresented in [23]. but still within the high-finesse approximation. In thissection we follow the notation of Refs. [13] and [15], andwrite exp[ − γ i τ ] = r i to describe losses and 2 (cid:15) = r todescribe gain. The single mode Bogoliubov transforma-tions from [15], without the low-gain approximation, are A single ( ω ) ≡ ( γ / − ( γ / − iω ) + | (cid:15) | ( γ / γ / − iω ) − | (cid:15) | (28) B single ( ω ) ≡ γ (cid:15) ( γ / γ / − iω ) − | (cid:15) | (29) C single ( ω ) ≡ √ γ γ ( γ / γ / − iω )( γ / γ / − iω ) − | (cid:15) | (30) D single ( ω ) ≡ √ γ γ (cid:15) ( γ / γ / − iω ) − | (cid:15) | . (31)We follow the same steps as from Eq. (17) to Eq. (22),to find G (2)single ( T ) = {F single [Γ]( T ) } + {F single [Υ]( T ) } + {F single [Υ](0) } (32)where {F single [Γ]( T ) } = π γ (cid:15) ( f − + f + ) (33) {F single [Υ]( T ) } = π γ (cid:15) ( f − − f + ) (34) f ± ≡ e − | T | ( γ + γ ± (cid:15) ) γ + γ ± (cid:15) (35)Finally, we multiply by a comb of (approximate) deltafunctions. Again following [13], for a multimode cavitywith 2 N + 1 modes we have: G (2)multi ( T ) ∝ G (2)single ( T ) sin [(2 N + 1) πT /τ ]sin [ πT /τ ] (36)lim N →∞ G (2)multi ( T ) ∝ G (2)single ( T ) ∞ (cid:88) n = −∞ δ ( T − nτ ) . (37) Eq. (32), computed by extension of [13], agrees veryclosely with our multimode result Eq. (22), shown in Fig.3. The only situation for which the two approaches givesignificantly different results is when the output couplerhas high transmission t . Even so, the difference betweenthe two calculations does not exceed 7 .
5% of the valueof G (2) ( T ), for r , r > . G (2) ( T ) enve-lope as a function of gain parameter r . This clearly showsa broadening of the correlations, along with a raising ofthe background level, which persists to arbitrarily large | T | . The background can be understood as a result of“accidental” coincidences, i.e. correlations among pho-tons that were not produced in the same SPDC event.The broadening is the time-domain manifestation of thenarrowing of the resonances with increasing r , visible e.g.in d ( ω ). Physically, it can be understood as the coherentamplification of SPDC photons already inside the cav-ity, i.e., stimulated SPDC. This change in photon tempo-ral distributions is of potential interest in wave-functionmatching for non-classical interference [25], matching toquantum memories [26], and detection of “Schr¨odingerkittens” and other time-localized non-classical fields [27]. V. CONCLUSION
We have computed the output of a multimode cavity-enhanced spontaneous parametric down-conversionsource, including realistic mode structure and sub-threshold but otherwise arbitrary gain. Using time-domain difference equations describing field operators atconsecutive roundtrips, we find multimode Bogoliubovtransformations that describe the output field. Thisanalytic solution provides a basis for calculations of anycorrelation function describing the multimode output.We compute the two-time intensity correlation function G (2) ( T ), and find increased temporal coherence due tostimulated SPDC in both single and multimode cases.We extend a calculation by Lu and Ou [13] to arbitrarygain, and find that it agrees well with our more exactcalculation. The results will be useful in describinghigh-gain spontaneous parametric down-conversion,in the context of quantum networking using atomicquantum memories [3–6] and studies of “Schr¨odingerkittens” and other exotic non-classical states [18–21]. VI. ACKNOWLEDGEMENTS
This work was supported by the Spanish MINECOproject MAGO (Ref. FIS2011-23520) and the EuropeanResearch Council project AQUMET, and Fundaci´o Pri-vada CELLEX. J. A. Z. was supported by the FI-DGRPhD-fellowship program of the Generalitat of Catalonia.
Appendix A: Fourier transforms for Γ and Υ We first compute F [ d ( ω ) d ( − ω )]( T ), the Fourier trans-form of d ( ω ) d ( − ω ), where d is given in Eq. (15).We denote x ≡ (1 + r r e r ) / (2 r r e r ) and y ≡ (1 + r r e − r ) / (2 r r e − r ). In the below-thresholdregime we are considering, r < r th = − log( r r ) so that d ( ω ) is always finite. We find d ( ω ) d ( − ω ) = 14 r r x − cos( ωτ ) 1 y − cos( ωτ ) (A1)Since d ( ω ) d ( − ω ) is an even periodic function with a pe-riod of 2 π/τ we can write d ( ω ) d ( − ω ) = ∞ (cid:88) k =0 F ( k ) cos( kωτ ) (A2)Where F ( k ) = 2 π (cid:90) π d ( ω ) d ( − ω ) cos( kωτ ) dω (A3)The Fourier transform is then the sum of Dirac deltafunctions: F [ d ( ω ) d ( − ω )]( T ) = ∞ (cid:88) k = −∞ F ( | k | ) δ ( T − kτ ) (A4) The F ( k ) can be expressed in terms of hypergeometricfunctions F ( k ) = 24 r r x − y )(1 + x )(1 + y ) (A5) × (1 + x ) F (cid:16) { , , } , { − k, k } ; y (cid:17) Γ(1 − k )Γ(1 + k ) − (1 + y ) F (cid:16) { , , } , { − k, k } ; x (cid:17) Γ(1 − k )Γ(1 + k ) . It follows immediately that the Fourier transform of d ( ω ) d ( − ω ) e inωτ is F [ d ( ω ) d ( − ω ) e inωτ ]( T ) = ∞ (cid:88) k = −∞ F ( | k | + n ) δ ( T − kτ ) . (A6)Now in order to compute {F [Γ]( T ) } and {F [Υ]( T ) } , letus use the following trick. For a moment, let’s assumethat the bandwidth of the downconversion is finite, i.e.replace squeezing amplitude r by a function r rect( ω/ω bw )where rect(x) = (cid:40) , if | x | < / , otherwise (A7)later we will apply to the final expressions the limit ω bw → ∞ returning to the situation with the infi-nite bandwidth. In that case the functions Γ bw ( ω ) and Υ bw ( ω ) yield Γ bw ( ω ) = rect( ω/ω bw )Γ( ω ) (A8)Υ bw ( ω ) = rect( ω/ω bw )Υ( ω ) (A9)Therefore, if we write ∗ for convolution we find {F [Υ bw ]( T ) } = ω √ π {F [Υ]( T ) } ∗ sinc (cid:18) T ω bw π (cid:19) (A10)Knowing that {F [Υ]( T ) } = t sinh( r ) (1 − r r ) ∞ (cid:88) k = −∞ δ ( T − kτ ) F ( | k | )(A11)we arrive to {F [Υ bw ]( T ) } = t sinh( r ) (1 − r r ) (A12) × (cid:34) ∞ (cid:88) k = −∞ sinc (cid:18) (T − k τ ) ω bw π (cid:19) F( | k | ) (cid:35) . (A13)Now let’s notice that for k (cid:54) = l lim ω bw →∞ sinc (cid:18) (T − k τ ) ω bw π (cid:19) sinc (cid:18) (T − l τ ) ω bw π (cid:19) = 0(A14)and lim ω bw →∞ (cid:20) sinc (cid:18) T ω bw π (cid:19)(cid:21) = δ ( T ) (A15)in the sense of a weak limit, i.e.lim ω bw →∞ (cid:90) ∞∞ dT f ( T ) (cid:20) sinc (cid:18) T ω bw π (cid:19)(cid:21) = f (0) (A16)for any continuous function f with a compact support.It follows that: {F [Υ]( T ) } = t sinh( r ) (1 − r r ) (A17) × ∞ (cid:88) k = −∞ δ ( T − kτ ) F ( | k | ) (A18)An analogous argument leads to {F [Γ]( T ) } = t sinh( r ) ∞ (cid:88) k = −∞ δ ( T − kτ ) (A19) × (cid:2) (1 + r r ) cosh( r ) F ( | k | ) − r r F ( | k | + 1) − r r F ( | k | − . [1] M. Scholz, F. Wolfgramm, U. Herzog, and O. Benson,Appl. Phys. Lett. , 191104 (2007).[2] F. Wolfgramm, A. Cer`e, and M. W. Mitchell, J. Opt.Soc. Am. B , A25 (2010).[3] E. Pomarico, B. Sanguinetti, N. Gisin, R. Thew,H. Zbinden, G. Schreiber, A. Thomas, and W. Sohler,New Journal of Physics , 113042 (2009).[4] C.-S. Chuu, G. Y. Yin, and S. E. Harris, Applied PhysicsLetters , 051108 (2012).[5] E. Pomarico, B. Sanguinetti, C. I. Osorio, H. Herrmann,and R. T. Thew, New Journal of Physics , 033008(2012).[6] J. Fekete, D. Riel¨ander, M. Cristiani, and H. de Ried-matten, Phys. Rev. Lett. , 220502 (2013).[7] F. Wolfgramm, A. Cer`e, F. A. Beduini, A. Predojevi´c,M. Koschorreck, and M. W. Mitchell, Phys. Rev. Lett. , 053601 (2010).[8] F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout,and M. W. Mitchell, Nat Photon , 28 (2013).[9] F. A. Beduini and M. W. Mitchell, Phys. Rev. Lett. ,143601 (2013).[10] I. Afek, O. Ambar, and Y. Silberberg, Science , 879(2010).[11] M. W. Mitchell and F. A. Beduini, New Journal ofPhysics , 073027 (2014).[12] Z. Y. Ou and Y. J. Lu, Phys. Rev. Lett. , 2556 (1999).[13] Y. Lu and Z. Ou, Physical Review A , 033804 (2000).[14] C. W. Gardiner and M. J. Collett, Phys. Rev. A , 3761(1985).[15] M. Collett and C. Gardiner, Physical Review A , 1386(1984). [16] C. Gardiner and C. Savage, Optics Communications ,173 (1984).[17] M. Mehmet, S. Ast, T. Eberle, S. Steinlechner,H. Vahlbruch, and R. Schnabel, Opt. Express , 25763(2011).[18] J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich,K. Mølmer, and E. S. Polzik, Phys. Rev. Lett. , 083604(2006).[19] U. L. Andersen and J. S. Neergaard-Nielsen, Phys. Rev.A , 022337 (2013).[20] O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre,and J. Laurat, Nat Photon , 570 (2014).[21] H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S.Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, NatPhoton , 564 (2014).[22] A. Lamas-Linares, J. C. Howell, and D. Bouwmeester,Nature , 887 (2001).[23] A. Predojevi´c, Z. Zhai, J. M. Caballero, and M. W.Mitchell, Phys. Rev. A , 063820 (2008).[24] Y. Jeronimo-Moreno, S. Rodriguez-Benavides, and A. B.U’Ren, Laser Physics , 1221 (2010).[25] M. Patel, J. B. Altepeter, Y.-P. Huang, N. N. Oza, andP. Kumar, New Journal of Physics , 043019 (2014).[26] M. R. Sprague, P. S. Michelberger, T. F. M. Champion,D. G. England, J. Nunn, X. M. Jin, W. S. Kolthammer,A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, NatPhoton , 287 (2014).[27] O. Morin, C. Fabre, and J. Laurat, Phys. Rev. Lett.111