A Formalism Useful to Study Beyond Squeezing in Non-Linear Quantum

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arXiv:2005.04609v2 [quant-ph] 29 Jul 2020 oik Aibara Noriaki 6 1 eateto oilIfrainSuis tuaWmnsU Women’s Otsuma Studies, Information Social of Department aueadEvrnet aut fLbrlAt,TeOe Un Open The Arts, Liberal of Faculty Environment, and Nature 2 eateto nomto ein aut fAtadDesign and Art of Faculty Design, Information of Department omls sflt td eodSueigin Squeezing Beyond Study to Useful Formalism A 5 caoiuUiest,211Otua ukok,Tko11 Tokyo Bunkyo-ku, Ohtsuka, 2-1-1 University, Ochanomizu 4 oy ukoSuyCne,TeOe nvriyo aa (OUJ Japan of University Open The Center, Study Bunkyo Tokyo uiifnto,adteosralst xmn eodtes the beyond examine photons. produced to of of observables mechanism number the this averaged and the of function, Husimi estimating application reflec in Possible the discussed, ad as mirrors. is well the by as includes material effects propagator non-linear penetration the The via F effect is vertices. damping method and S-ma The or propagators amplitude stored. of transition is the expressing material one, non-linear graphical a which in ity, 3 iiino rsadSine,TeSho fGaut Studie Graduate of School The Sciences, and Arts of Division 1 h U oy ukoFedTer Collaboration Theory Field Bunkyo Tokyo OUJ The eea omls sgvni unu piswti igc ring a within optics quantum in given is formalism general A aaiFujimoto Naoaki , h pnUiest fJpn hb 6-56 Japan 261-8586, Chiba Japan, of University Open The o-ierQatmOptics Quantum Non-linear hyd-u oy 0-37 Japan 102-8357, Tokyo Chiyoda-ku, ahoi 9-34Japan 192-0394 Hachioji, sks Yumibayashi Tsukasa oy 1-02 Japan 112-0012, Tokyo 2 oKatagiri So , 6-56 Japan 261-8586, Abstract 1 3 koSugamoto Akio , 6 , iest,1 Sanban-cho, 12 niversity, 4 aaAtUniversity, Art Tama , vriyo aa,Chiba Japan, of iversity , 5 rxi terms in trix ocioYamaguchi Koichiro , formalism -60 Japan 2-8610, queezing photons, inand tion eynman ditional OCHA-PP-360 av- s, OUJ-FTC-5 ), 1 , 1 Introduction

In optics, laser technology has developed extensively and is now able to elucidate the quantum behavior of photons. That is, we can investigate quantum optics (QO) or quantum electrodynamics (QED) by using laser technologies. See for example [1]. It is amazing that there exist transparent materials, having non-linear dielectric constants, such as Silica, BBO (β-Barium Borate), KTP (Potassium Titanyl Phosphate), LN (Lithium Niobate), KDP (Potassium Dideuterium Phosphate) e.t.c.. For such non-linear materials the displacement vector (the electric ﬂux density) D is not linearly proportional to the electric ﬁeld E, but we have

(1) D = ε0(1 + χ )E + PNL, H = B/µ0, and (1) (P ) = ε χ(2) E E + χ(3) E E E + χ(4) E E E E + , (2) NL i 0 ijk j k ijkl j k l ijklm j k l m ··· i,j,k, X··· where PNL is the non-linear polarization induced by the electric ﬁeld, ε0 and µ0 are the dielectric constant and the magnetic permeability of the vacuum, respectively, and χ(n) (n =1, 2, ) are dielectric constants. ··· The tensor structure with indices (i, j, ) is important for an individual material, but here we ignore it for simplicity. ··· Then, the action of QO or QED in the material reads 1 S = dt d3x (E D H B) (3) QO 2 · − · Z 1 = dt d3x ε E2 + χ(1)E2 + χ(2)E3 + χ(3)E4 + χ(4)E5 + H B (4) 2 0 ··· − · Z which is non-linear in E. In this way the non-linear dielectric constant χ(n) introduces additional interactions to the QO (or QED) Lagrangian density, 1 (n) = ε χ(n)E(t, x)n+1. (5) LQO 2 0

The electric ﬁeld is composed of the dominant classical part EL, coming from the intense laser beam, and the quantum part Eˆq describing the creation and annihilation of photons:

E(t, x)= EL(t, x)+ Eˆq(t, x), where (6)

i(ωLt kL x) i(ωLt kL x) EL(t, x)= EL e− − · + EL† e − · , (7) ~ 3 i(ωt k x) i(ωt k x) Eˆ (t, x)= ∂ Aˆ (t, x)= d k (iω) aˆ(k)e− − · aˆ(k)†e − · , (8) q − t q (2π)3ε 2ω − Z s where Aˆq is the vector potential,a ˆ(k) anda ˆ(k)† are, respectively, annihilation and creation (1) 1 operators of photon with a wave vector k, and ε = ε0(1 + χ ). Here we assume a simple dispersion relation, ω = ω(k) = c′ k , with the light velocity c′ = 1/√εµ0 in the non-linear | | material. 1In reality, the vector potential, creation and annihilation operators have dependence on the direction of polarization vectors, but these tensor structure is also ignored here.

2 Now, we understand that the intense laser beam can create or annihilate photons by the interactions, (n) (n =2, 3, 4, ), proportional to χ(n), LQO ··· 1 (n) = (n + 1)ε χ(n)E (t, x)Eˆ (t, x)n. (9) LQO 2 0 L q At each interaction point, both energy (~ω) and momentum (~k) are compelled to conserve in the cavity. This is called the “phase matching condition” in QO, which implies that the interaction always satisfies the resonance condition in the cavity, and is amplified maximally. As an example, if the laser beam with (ωL, kL) create n photons with energy and momentum, (ω , k ), (ω , k ), , (ω , k ), then we have 1 1 1 1 ··· n n ω = ω + ω + + ω , and k = k + k + + k . (10) L 1 2 ··· n L 1 2 ··· n It is important to note that in the processes occurring in the “ring cavity”, all the photons, including the laser photons, propagate in one-direction, and the motion in the opposite direction is prohibited. The cavity called “optical parametric oscillator (OPO)” is an apparatus with one spacial dimension, such as a Fabry-Pérot resonator and a ring resonator. In Fabry-Pérot resonator, there are both forward and backward motions by mirror reflection, while in the ring resonator, the motion is restricted to the one-way round trip. Therefore, we will study this ring resonator in this paper, for simplicity. In such one-dimensional one-way motion, the energy-momentum conservation in Eq.(10) is reduced solely to the energy conservation, ω = ω + ω + + ω , (11) L 1 2 ··· n where the momentum conservation is automatically guaranteed, since the magnitude of a momentum is proportional to its energy, k = ω/c′. | | The other possible processes are ω +(ω + ω + + ω )= ω + + ω , (12) L 1 2 ··· m m+1 ··· n which implies that the laser beam and the photons with energy (ω ,ω , ,ω ) come into 1 2 ··· m the cavity and annihilate, and the photons with (ωm+1, ,ωn) are created and go out from the cavity. ··· Therefore, QO in one-dimensional apparatus can be analyzed without referring to the momentum or the space coordinate, if the apparatus is uniform in sufficiently long length scales in between mirrors. Now, it is reasonable to study QO in terms of the following non-linear quantum mechanics with time-dependent interactions, 1 1 1 1 L = x˙ 2 ω2x2 λ (t)x2 λ (t)x4, (13) 2 − 2 0 − 2! 2 − 4! 4 We can include more higher-order terms such as x6, x8, . Then, the quantum mechanics ··· (QM) has non-linear interactions via the couplings λn(t), given in the interaction picture as 1 L(n) = λ (t)ˆx(t)n, (14) −n! n where ~ iω0t iω0t xˆ(t)= aeˆ − +â†e (15) 2ω r 0 3 In comparison of Eq.(9) in QO with Eq.(14) in QM, we understand immediately that in case of one spacial dimension, these equations become equivalent under the following correspondence:

λ (t) ε χ(n)E (t), andx ˆ(t) Eˆ (t). (16) n ⇔ 0 L ⇔ q (2) The time dependent coupling λ2(t) in QM and the non-linear dielectric constant χ in QO equally generate the squeezing states, where the uncertainty for the “coordinate” is scaled up (down), while that of “momentum” is scaled down (up), keeping the product of them unchanged. In QO, the vector potential A(t, x) and the electric field E(t, x) form a canonical conjugate pair, playing the role of “coordinate” and “momentum”, respectively. It is remarkable that the squeezing is observed in OPO, in which the Fabry-Pérot resonator is implemented with a non-linear crystal of MgO:LiNbO3 stored inside the cavity [3]. The appearance of the squeezing states in the parametric down-conversion of laser photon γL into two degenerate photons γ1 and γ2 (ωL = ω1 + ω2 and ω1 = ω2), is confirmed using the homodyne interferometer. In this paper, we develop a simple formalism useful to study the phenomena, a more generally “beyond the squeezing mechanism”, for example to see beyond the squeezing in- (4) duced by the anharmonic coupling λ4(t) in QM or by the non-linear dielectric constant χ in QO. We consider a one-dimensional ring cavity, having a non-linear optical material and mirrors. See (Figure1). For this purpose, we develop a perturbation theory in which the Hamiltonian is separated into the quasi-free Hamiltonian and the real interaction Hamilto- nian. The former includes the effects of dispersion and absorption by the non-linear optical material and the mirrors. The intense laser light is prepared, which forms a classical light far above the threshold of laser oscillation. To formulate the perturbation theory consistently, L 2 2 ( ωk ωR) /2δ L the Gaussian averaging e− | |− is required over the laser energy ωk around the resonance energy ωR at k-th interaction point. This averaging tames the singularities (mass singularities), which are inevitable when both the conservations of energy and momentum are compelled to hold so that the phase matching conditions hold. A lot of works have been done so far on the non-linear quantum optics, based on the input-putout theory, the Fokker-Planck equation, and the QED in non-uniform dielectric matter [21, 23], [24], [27]. We prepare Appendices, A, B, and C corresponding to the three issues, in which we give comments on the relation of this paper to the issues. The more general case of beyond the squeezing, having the higher-order interactions, was studied separately in [4], by referring to the Virasoro algebra, in which a general formula is derived, which gives the number of particles produced, under a condition of the number of particles produced is large. Beyond the squeezing is also called “generalized squeezing” [5]. These generalized squeezed states are given by applying the following operator to the vacuum (k 3), ≥ k k Uˆk = exp zaˆ + z∗aˆ† (17) which are naturally conceivable from the analogy of coherent states (k = 1),

Uˆ1 = exp zaˆ + z∗aˆ† , (18) and the squeezed states (k = 2),

2 2 Uˆ2 = exp zaˆ + z∗aˆ† . (19) 4 It is, however, controversial, whether the vacuum expectation value of the generalized ˆ ˆ squeezed state 0 Uk†Uk 0 (k 3) is divergent or convergent. See Appendix D for the detail. h | | i ≥ In the next section, a ring resonator and its resonance condition are discussed. Transition amplitude for ring resonator is formulated in Section 3. In Section 4, the propagator and the vertex are explicitly given for QO inside ring resonator in the presence of NL-material, which are necessary items to estimate the transition amplitude. Transition amplitude is written using the propagators and the vertex functions in Section 5, so that our formalism can be applicable to higher-order calculations in QO in the cavity. The ﬁnal section is devoted to conclusion and discussion; where possible application of the formalism developed in this paper is discussed, such as in estimating averaged number of produced photons, Husimi function and the observables to examine beyond the squeezing of photons. Appendices A, B, C, and D survey , respectively, the input-output theory and observation of squeezed state in optical cavity, the Fokker-Planck equation (including the condition of laser light be classical), QED in non-uniform dielectric matter, and a controversial issue on generalized squeezing.

2 Ring Resonator for Optical Parametric Oscillation

We adopt a “ring resonator” as the apparatus to observe the beyond squeezing (generalized squeezing) by Optical Parametric Oscillation (OPO), which is depicted in (Figure 1).

Figure 1: A ring resonator as an apparatus to observe beyond the squeezing by Optical Parametric Oscillation (OPO).

The ring resonator, conﬁned in a cavity, consists of ﬂour mirrors, M1 M4. The M1 is the half-mirror, through which the input laser beam comes in from the left− to the cavity,

5 and M2 is the output half-mirror, through which the photons go out and are detected by the photo-diode (PD) located outside the cavity. The remaining M3 and M4 are ordinal mirrors. The non-linear (NL) crystal of length ℓ is placed in between M1 and M2. The path length of light from M1 to M2 is L′ and L is the length of a round trip in the resonator. The ring resonator extends in one spacial direction, along which the coordinate z is taken. Choosing z = 0 in the center of the crystal, any path length is measured by the coordinate z from the origin, clock-wisely along the direction of propagation of light. The crystal with non-linear (NL) dielectric constants is assumed to have crystal planes perpendicular to z. The planes are separated regularly by a separation of a lattice constant of the crystal, acrys. We consider the laser beam to be an intense “classical wave” in this paper. This assump- tion is not bad, since above the threshold of laser oscillation (the pumping rate dominates over the loss in the cavity), laser photons form a coherent state, so that the laser light can be considered as a classical electric field EL(t, z) with a well-defined phase without fluctua- tions [1]. The validity of treating laser beam as classical one at well above the threshold is discussed in Appendix B, using Fokker-Planck equation. The oscillation of this electric field of the laser in the perpendicular direction to z, induces the oscillation of electrons on the crystal planes in the same perpendicular direction, from which photons are generated via the non-linear interaction in Eq.(9). First, we will estimate the resonance condition of the cavity. We denote ri the reflection coefficient of the i-th mirror, for the amplitude. Then, the corresponding transmission 2 coefficient is 1 ri . The electric field− of light with angular frequency ω can be written as p iω(t z ) γ(z)z E(t, z)= E e− − c(z) − , (20) where the light velocity c(z) can depend on the position z to pass through; we choose c in the vacuum and c′ = c/ncrys in the crystal with the refraction coefficient ncrys, while γ(z) is the decay rate per unit length at z; we choose γ(z) = 0 in the vacuum and γ(z)= γ′ in the crystal. Consider the light wave starting from z at t. It undergoes round trips inside the cavity arbitrary times, being reflected by the mirrors, so that its electric field obtains an extra factor K(ωL/c), which includes both the phase change and the attenuation: 1 K(ωL/c)= , (21) 1 (r r r r )eiφ(ωL/c) γ′ℓ − 1 2 3 4 − where ω ω φ(ωL/c)= (L ℓ)+ ℓ. (22) c − c′ So, the strength of the electric field is proportional to K 2, which gives a resonance | | behavior: 1 K(ωL/c) 2 = , (23) | | 1 re γ′ℓ 2 +4re γ′ℓ sin2 1 φ(ωL/c) { − − } − 2 where r = r1r2r3r4.

6 If the resonance point at which K 2 takes the maximum value, is denoted by ω or its | | R frequency νR, we have the resonance condition as follows: φ(ω L/c) ω ℓ R = R 1+ (n 1) = integer, (24) 2π ω L crys − F SR where ωF SR =2πc/L is the angular frequency of a ﬁctitious wave with wave length L, which is called the “free streaming range”, since adjacent resonances are separated by ωF SR. The full width at half maximum (ωR/ωF SR)FWHM of the resonance is roughly γ′ℓ ω 1 re− R | − | 1, (25) ω ≈ √ γ′ℓ ≪ F SR FWHM re− since r 1. Therefore,≈ we can tune L very precisely so that the waves resonate and are enhanced in the cavity. The resonance is very sharp, and ωR can be precisely selected. It is important to note here that if ω1 satisﬁes the minimum resonance condition: φ(ω L/c) ω ℓ 1 = 1 1+ (n 1) =1, (26) 2π ω L crys − F SR then, 2ω , 3ω , 4ω , also satisfy the resonance condition Eq.(24) by choosing the integer 1 1 1 ··· as 2, 3, 4, , if the refraction index ncrys, (generally depends on the frequency ω), is constant for these··· waves. The FWHM in Eq.(25) depends on the waves, since it depends on the r and γ′ whose dependency on ω may not be ignored. We consider a crystal, having NL dielectric constants χ(2) and χ(4). Generalization to more general cases with χ(i) (i =3, 5, 6, ) is also possible.2 ··· Now, we choose the laser frequency to be ωL = ω0, then the down-converted photons with frequencies, ωL/4 = ω1 and ωL/2 = ω2 = 2ω1, can also resonate and survive in the cavity.3 In the cavity, the momenta of photons should always be parallel to the longitudinal direction. That is, if the photon in the cavity is scattered by the crystal and loses energy or alters the direction of momentum, then this photon can not exist in the cavity and disappears; the damping rate γ′ comes from these scatterings.

3 Transition amplitude for Ring Resonator

In this section, we will study the transition amplitude in QO, which describes the creation and annihilation of photons from a non-linear crystal, induced by the laser. In this QO inside 2The ring resonator which we are considering is especially suited for observing the parity violation effects from χ(i),i =odd integer. See for example [7], in which the ring resonator is examined to see the parity violation effect of neutrinos as dark matter, without QED background. On the other hand, the forward and backward motions in Fabry-Pérot cavity wash out the parity violation effects. 3To input the laser beam smoothly into the ring cavity and to extract the output photons effectively from it, some devise may be necessary. Referring to the gravitational wave detector [8], a way is to prepare two more cavities, the laser input cavity (LIC) and the signal extraction cavity (SEC). The former cavity (LIC) consists of the mirror M1 and mirrors in front and behind of M1 along the direction of laser beam, while the latter cavity (SEC) consists of the mirror M2 and mirrors in front and behind of M2 along the path to the photo-detector. The LIC can be designed almost transparent for ωL but very reflective for ω1 and ω2, while the SEC can be designed very reflective for ωL but almost transparent for ω1 and/or ω2. If necessary, the advanced technique such as modulation and demodulation of amplitude (AM) and of phase (PM) can be used.

7 the resonator, the Hamiltonian can be divided into two parts; the first part is denoted as Hˆ0, which is time dependent and not free but is “quasi-free”, including the effect of damping in the cavity and the reflection and penetration by the mirrors. The second part Hˆ1 gives the real interaction (creation and annihilation) of photons by the non-linear crystal. Thus, the Hamiltonian which we are going to study is time dependent, even in the “Shrödingier picture”:

Hˆ (t)= Hˆ0(t)+ Hˆ1(t). (27)

We will call Hˆ0 the “quasi-free Hamiltonian”, and Hˆ1 the “real interaction Hamiltonian”. We will mimic the process of picture changing from the “Schr¨odinger picture” to the “interaction picture”, by treating Hˆ0 as if it was the free Hamiltonian. If we introduce Uˆ0(t), U-matrix or the time evolution operator for the quasi-free Hamil- tonian Hˆ0(t), it satisﬁes ∂ i~ Uˆ (t)= Hˆ (t)Uˆ (t). (28) ∂t 0 0 0 Explicitly, we have

t ′ ′ R dt Hˆ0(t )/i~ Uˆ0(t)= T e 0 . (29)

In this “Schr¨odingier picture”, the equations of motion of wave function ψ(t)S and operator OˆS are given by ∂ ∂ i~ ψ(t) = Hˆ (t)ψ(t) , and i~ Oˆ =0, (30) ∂t S S ∂t S After moving to the “interaction picture” we have ∂ ∂ i~ Oˆ = [Oˆ , Hˆ (t) ], and i~ ψ(t) = Hˆ (t) ψ(t) , (31) ∂t I I 0 S ∂t I 1 I I where

ˆ 1 ˆ ˆ 1 ˆ ˆ ψ(t)I = U0− (t)ψ(t)S, and H1(t)I = U0(t)− H1(t)U0(t). (32)

Then, in the time evolution of the “interaction picture”, the quasi-free motion is subtracted from that of the “Schr¨odingier picture”. The motion of laser light and photons, before and after the real interaction in the cavity occurs, is quasi-free, so that in these periods the wave function in the “interaction picture” does not evolve. The transition amplitude of QO in the ring resonator can be expressed iteratively as usual, by solving the time evolution of wave function in the “interaction picture”. We will use the number representation, in which the sate is expanded in n having n photons. Then, | i the Schr¨odingier equation in the interaction picture, i~ ∂ n, t = Hˆ (t) n, t , can be solved, ∂t | i 1 I | i

8 if we choose the initial condition n, = m , as follows | −∞i | i 1 t n, t = m + dt n n Hˆ (t ) m | i | i i~ 1 | ih | 1 1 I | i Z−∞ 1 2 t t1 + dt1 dt2 n n Hˆ1(t1)I m1 m1 Hˆ1(t2)I m + (33) i~ ∞ | ih | | ih | | i ··· −∞ Z t Z 1 1 = m + dt n n Uˆ (t)− n′ | i i~ 1 | ih | 0 | i Z−∞ n′ Uˆ (t, t ) m m Hˆ (t ) m m Uˆ (t , ) m′ m′ Uˆ ( ) m ×h | 0 1 | 1ih 1| 1 1 | 2ih 2| 0 1 −∞ | i×h | 0 −∞ | i 2 t t1 1 1 + dt1 dt2 n n Uˆ0(t)− n′ i~ ∞ | ih | | i Z−∞ Z n′ Uˆ (t, t ) m m Hˆ (t ) m m Uˆ (t , t ) m m Hˆ (t ) m m Uˆ (t , ) m′ ×h | 0 1 | 1ih 1| 1 1 | 2ih 2| 0 1 2 | 3ih 3| 1 2 | 4ih 4| 0 2 −∞ | i m′ Uˆ ( ) m + , (34) ×h | 0 −∞ | i ··· 1 where Uˆ0(t1, t2) Uˆ0(t1)Uˆ0(t2)− is a transition amplitude from t2 to t1 by the “quasi- ≡ 1 free Hamiltonian Hˆ0”. When t is ﬁnite, then the factor n Uˆ0(t)− n′ should be carefully estimated, but in our case, we will take t = , and considerh | the transition| i amplitude from −∞ t = to t = , that is the S-matrix, we may choose −∞ ∞ 1 n Uˆ ( )− n′ = δ ′ , m′ Uˆ ( ) m = δ ′ . (35) h | 0 ∞ | i nn h | 0 −∞ | i mm Next, we have to discuss an important point that QO is essentially a ﬁeld theory, having spacial degrees of freedom, x. Then, the Hamiltonian is given as the integral over x of the “Hamiltonian density (t, x)”, that is H Hˆ (t)= dx (t, x), and Hˆ (t)= dx (t, x). (36) 0 H0 1 H1 Z Z Thus, the transition amplitude in QO can be written as follows:

t 1 1 n, t = m + dt dx n n Uˆ (t)− n′ | i | i i~ 1 1 | ih | 0 | i Z−∞ Z n′ Uˆ (t, t ) m m (t , x ) m m Uˆ (t , ) m′ ×h | 0 1 | 1ih 1|H1 1 1 | 2ih 2| 0 1 −∞ | i m′ Uˆ ( ) m ×h | 0 −∞ | i 2 t t1 1 1 + dt1 dx1 dt2 dx2 n n Uˆ0(t)− n′ i~ ∞ | ih | | i Z−∞ Z Z Z n′ Uˆ (t, t ) m m (t , x ) m m Uˆ (t , t ) m ×h | 0 1 | 1ih 1|H1 1 1 | 2ih 2| 0 1 2 | 3i m H (t , x ) m m Uˆ (t , ) m′ m′ Uˆ ( ) m ×h 3| 1 2 2 | 4ih 4| 0 2 −∞ | i×h | 0 −∞ | i + . (37) ··· 4 Propagator and Vertex inside Ring Resonator con- taining Non-linear Material

We ﬁrst examine a simple case of quantum optics (QO) inside the uniformly distributed non-linear material, which was discussed a bit in Introduction. The Lagrangian density in

9 this case is Eq.(4), 1 = (E D H B)= + , (38) LQO 2 · − · L0 L1 where 1 1 1 = ε E2 B2 , and = ε χ(n)(E)n+1. (39) L0 2 − εµ L1 2 0 0 n 2 X≥ The quadratic terms are combined into a quasi-free Lagrangian with a new parameter ε = (1) ε0(1 + χ ), the dielectric constant of the material. In terms of vector potential A(t, x) and electric ﬁeld E = A˙ , the conjugate momentum of the former is given by Π = εE = εA˙ , and B = ∇ A, so− that the Hamiltonian density A − × reads H = + , (40) H H0 H1 where 1 1 1 = ε E2 + B2 , and = ε nχ(n)(E)n+1. (41) H0 2 εµ H1 2 0 0 n 2 X≥ The quasi-free Hamiltonian, in this case, is a free Hamiltonian in the medium, where c′ = c/n = c/√εµ0 (n: the refraction constant) is the light velocity in the medium. Now, we are ready to study QO inside the ring resonator.

4.1 Mode Expansion In QED, any mode having an arbitrary wave vector k have to be considered, but in QO, only a few modes, having speciﬁc values of wave vectors, have to be considered due to the resonance condition of the cavity. In this situation, it is better to formulate QO in terms of discretized coordinates and momenta. Let us denote discretized coordinates with equal spacing a by xn, and the discretized (angular frequency ω, wave vector k) with equal spacing by (ωm, km). The total number of them are equally N( 1), that is, (n, m)=1, 2, , N, and Na = L is the total length of the system. Then, the≫ spacing of the wave vector is··· 2π/L as usual. Corresponding to the mode expansion of E in Eq.(8), the mode expansion of the 4 vector potential A and its canonical conjugate ΠA, are as follows :

~ 3 i(ωt k x) i(ωt k x) Aˆ (t, x)= d k aˆ(k)e− − · +ˆa(k)†e − · , (42) q (2π)3ε 2ω Z s ~ 3 i(ωt k x) i(ωt k x) Πˆ (t, x)= ε d k ( iω) aˆ(k)e− − · aˆ(k)†e − · , (43) A,q (2π)3ε 2ω − − Z s giving the non-vanishing commutation relations,

(3) (3) [Aˆ (t, x), Πˆ (t, x′)] = i~ δ (x x′), or equivalently, [ˆa(k), aˆ(k′)†]= δ (k k′). (44) q A,q − − 4The reason why the dielectric constant ε appears in the mode expansion is that the conjugate momentum deﬁned by Πˆ A = ε∂tAˆ in the medium.

10 If we restrict the problem to QO in one-dimensional cavity, the discretized version of operators can be introduced by5

√a Aˆq(t, x)= Aˆ(t, xn), √a Πˆ A,q(t, x)= Πˆ A,q(t, xn), (45) 2π 2π aˆ(k)=ˆa(km), aˆ(k)† =ˆa(km)†. (46) r L r L Then, the commutation relations are expressed with Kroneker’s delta as follows:

[Aˆq(t, xn), Πˆ A,q(t, xn′ )] = i~ δnn′ , or equivalently, [ˆa(km), aˆ(km′ )†]= δmm′ . (47)

Thus, the discretized version of the vector potential and the electric ﬁeld are given by

N ~ i(ωmt kmzn) i(ωmt kmzn) Aˆ (t, z )= aˆ e− − +ˆa† e − , (48) q n Nε 2ω m m m=1 r m X N ~ i(ωmt kmzn) i(ωmt kmzn) Eˆ (t, z )= (iω ) aˆ e− − aˆ† e − . (49) q n Nε 2ω m m − m m=1 r m X These equations are valid for any spacial dimension, if we consider N to be the total number of discretized points in coordinate space and wave vector space. Similarly, the classical part of laser in Eq.(8), is renormalized, when using the discrete coordinates,

iωL(t z/c(z)) iωL(t z/c(z)) EL(t, z)= EL e− − + EL† e − , √aEL(t, z)= EL(t, zn) with (50) iωL(t zn/c(z)) iωL(t zn/c(z)) (EL(t, zn)= EL e− − + EL† e − .

In the above, the discrete sum over coordinates and that over wave vectors appear. These two discretizations are correlated, but it is convenient to convert, while keeping the discrete sum over modes, the discrete sum to the coordinate integration, by using n = dz/a. This is valid since a is small compared to the wave length. The discretization of modes appears R not from the correspondence between a and L, but from the resonanceP condition. In the following, we adopt this hybrid expression, that is, coordinate is continuous, while the wave vector is discrete. In our ring cavity, only two specific modes of photons, having angular frequencies, ω1 = ωL/4 and ω2 = ωL/2 are relevant. Therefore, in the hybrid expression, the classical part of electric field from laser is the same as Eq.(8), but the quantum part of the electric field is chosen as follows, namely

iωL(t z/c(z)) iωL(t z/c(z)) EL(t, z)= EL e− − + EL† e − , but (51) ~ ωi i(ω (t z/c(z)) iω (t z/c(z)) Eˆ (t, z)= i aˆ e− i − aˆ†e i − , (52) q 2Lε(z) i − i i=1, 2 s X where the system size L remains, and c(z), ε(z) are the light velocity and the dielectric constant at a position z, respectively. It is natural to choose L be a total path length of the ring resonator.

3 3 5 2 2π 2 In case of three spacial dimensions, the square roots should be replaced by a or L .

11 4.2 Propagator The “propagator” in the number representation is given as

1 n Uˆ (t, t′) n′ = n Uˆ (t)Uˆ (t′)− n′ . (53) h | 0 | i h | 0 0 | i

Here, Uˆ(t, t′) satisfies Eq.(31), so that it is true to give the phase change of wave in time, i ′ ′ Hˆ0(t t ) iω(t t ) n e− ~ − =(e− − ) for the n photon state, but that is not all. The propagation of laser beam and of produced photons move clock-wisely inside the cavity, inside which non-linear material and the mirrors exist, so that the wave changes position dependently. In our case with one spacial dimension, the position z is determined by the time t. Therefore, we will write the propagator as a time-ordered product, being defined piece-wisely. As an example, let us consider a photon with the frequency ω propagates first from (t0, z0) to (t1, z1) inside the medium with velocity c′, next propagates in the vacuum till (t2, z2), and it is reflected afterwards by a mirror with a reflection constant r. After that it is assumed to propagates in the vacuum, and finally arrives at (t3, z3). The corresponding propagator can be

U(t , t )= U(t , t , vac) r U(t , t ; vac) U(t , t ; medium), (54) 3 0 3 2 × × 2 1 × 1 0

iω (t1 t0) (z1 z0)/c where U(t1, t0; vac) = e− { − − − }, but in the medium, damping occurs with rate γ′ ′ ′ iω (t1 t0) (z1 z0)/c γ (z1 z0) per length, so that U(t1, t0; medium) = e− { − − − − − . The derivation of the propagator is the same both in the classical wave and the quantum wave.6 Now, we can understand that the propagator of n photons is

n n n Uˆ (t, t′) n′ δ ′ D (t, z; t′, z′), and D (t, z; t′, z′)= 1 Uˆ (t, t′) 1 D(t, z; t′, z′) , (55) h | 0 | i ≡ nn n n h | 0 | i ≡ where 1/n! appeared from the normalization of the state is cancelled by the degeneracy n! which comes from the ways of connecting n (initial and final) photons by propagators. The propagator of a single photon, D(t, z; t, z′), describes the transition amplitude of state by the quasi-free Hamiltonian. Its derivation is the same as that of the coefficient K(ωL/c)) in Eq.(2), including the round trips. Difference between K and the propagator D is that the starting point (t′, z′) and the arriving point (t′, z′) are equal in the former, but different in the latter. There are a number of propagators: 1) “propagator” from a crystal plane C′ to another crystal plane C:

iω (t t′ ) (z z′)/c′ γ′ z z′ ω e− { − − − }− | − | iω (t t′ ) (z z′)/c′ ω D ′ (t, z; t′, z′)= e− { − − − }K ′ (z, z′). (56) C;C 1 (r eiφ(ωL/c) γ′ℓ) ≡ C;C − ω −

Here, φ(ωL/c) is given in Eq.(22) in the previous section. This is applicable to ω = ω1, or ω n ω2(= 2ω1). If the n photons propagate, then the factor becomes (DC;C′ (t, z; t′, z′)) . 2) “propagator” for the laser light:

6If we wish to derive more rigorously the propagators, we can do it, by considering the Hilbert space of photons as a product of that inside the cavity (or that of the system) s and that outside the cavity (or H ′ that of the heat bath) b, namely, = s b. Since the attenuation caused by γ and r is the effect of escape of photons fromH the systemH to theH ⊗ heat H bath. If we trace out the heat bath and concentrate to the system, we have the effect of attenuation, caused by the crystal, and the reflection and penetration via mirrors. The rigorous result does not, however, differ from that given in the above. (See Appendix A and B for the details.)

12 The Uˆ0(t, t′) is purely quantum object, giving the transformation of operators from the Shr¨odingier picture to the interaction picture. Therefore, there is no such concept of “propagator” for the purely classical laser beam. However, it is reasonable to use the name “propagator”, Uˆ(t, t′) also for the classical electromagnetic ﬁeld, since the time evolution factor is the same as in the quantum theory. Then, the propagator of the classical laser beam is

′ ′ ′ iωL (t t ) (z z )/c ) ′ ′ ′ L e− { − − − } iωL (t t ) (z z )/c ) ωL D ′ (t, z; t′, z′)= e− { − − − }K (z, z′). (57) C;C 1 r (eiφ(ωL/c)) ≡ L − L Here, we have ignored the damping effect of γ′, since it should be cancelled by the strong pumping of the laser beam. 3) “propagator” from the final interaction point to the outside photo-detector (PD): This “propagator” has an additional effect to the propagator, which is caused by passing through the mirror M2, and by traveling from the final interaction point zf to the photo- detector located outside at a distance z L′/2 from the mirror M . − 2 We denote the final interaction plane be Cf , then we have

′ ′ ′ 1 r2 e iω (t tf ) (ℓ/2 zf )/c (z L /2)/c γ (ℓ/2 zf ) Dω (t, z; t , z )= 2 − { − − − − − }− − PD;Cf f f − iφ(ωL/c) γ′ℓ 1 rωe − p′ ′ − iω (t tf ) (ℓ/2 zf )/c (z L /2)/c ω e− { − − − − − }K (z, zf ). (58) ≡ PD;Cf The “propagators” include, other than the phase change, the additional factors Ks, which describes the damping eﬀect of photons in the cavity.

4.3 Vertex

The real interaction of our problem occurs at the crystal planes separated by acrys, so that the “vertex” is given in the vicinity of the crystal plane, giving

z+acrys/2 m Hˆ1(t)S m′ = dz′ m ˆ1(t, z′)S m′ , (59) h | | i z acrys/2 h |H | i Z − where we have assumed the local interaction at z. In the above expression we put the suffix S as ˆ1(t, z)S, to clarify the “Shrödingier picture”. We assume as usual that the operators in theH “Shrödingier picture” and the “interaction picture” coincide at t = 0. We will write the interaction Hamiltonian, after separating the dominant classical part of laser from the quantum part of photons, as follows:

z+acrys/2 m Hˆ1(t)S m′ = dz′ m ˆ1(t, z′)S m′ (60) h | | i z acrys/2 h |H | i Z − z+acrys/2 1 (n) n = ε n(n + 1)χ dz′ E (t, z′) m (Eˆ (t =0, z′)) m′ . (61) 2 0 L ×h | q | i n 2 z acrys/2 X≥ Z − In the following, we study a simple NL case of having only χ(2) and χ(4), and hence the real interaction Hamiltonian takes the following form:

1 (2) 2 (4) 4 ˆ′ (t, z) = ε E (t, z) 6χ (Eˆ (0, z)) + 20χ (Eˆ (0, z)) . (62) H1 S 2 0 L × q q n o

13 The real interaction occurs inside the non-linear crystal, where the light velocity is c′, so that the mode expansion used in writing the vertex function is7

′ ′ iωL(z/c ) iωL(z/c ) (classical part ) : EL(z)S = EL e + EL† e− , ~ ′ ′ (63) ωi iωiz/c iωiz/c (quantum part) : Eˆq(z)S = i aˆie aˆ†e− ,  i=1, 2 2Lε − i P q where all the classical and quantum photons are assumed to propagate in the positive (clock- wise) direction along the cavity, and the frequency is restricted to ω1 = ωL/4 and ω2 = ωL/2, by the resonance condition of the cavity. This is a good place to discuss the momentum conservation. From the position integral near the vertex point, if the space near the vertex point is uniform (ı.e. the relevant wave length λ of laser beam and photons satisﬁes λ acrys, then the momentum conservation holds, and the sum of all the wave vectors coming≪ in the vertex is equal to the sum of all the wave vectors coming out from the vertex. This property also holds in the usual old fashioned perturbation theory. As for the energy conservation, it does not hold in the old fashioned perturbation theory, since the transition amplitude becomes singular, if the energy conservation at each interaction point holds. This problem is solved in this paper, by averaging the frequency of laser beam around the resonance point. (See the details in the next section.) Now, the possible interactions can be listed as follows:

(4) : Laser emits 4γ s at a crystal surface, (ω =4ω 4ω ), 4 1 L 1 → 1 (4) 4 : 4γ1s are absorbed by Laser at the surface, (4ω1 ωL =4ω1),  − →  (4) (4)2 : Laser +1γ1 emitts 2γ2s +1γ1 at a crystal surface, χ :  (64) (ωL + ω1 ω1 +2ω2),  → (4) 2 : 2γ2s +1γ1 emit Laser + 1γ1 at a crystal surface, − (2ω + ω ω + ω ).  2 1 L 1  → (2) ; Laser emits 2γ s at a crystal surface, (ω =2ω 2ω ), χ(2) :  2 2 L 2 → 2 (65) ((2) 2 : 2γ2s are absorbed by Laser at the surface, (2ω2 2ω2 = ωL). − → where we denote the photons with energy ω1 and ω2 as γ1 and γ2, respectively. The diﬀerent types of interactions are labeled by (4) 4, (4) 2, and (2) 2 to make the individual processes ± ± ± clearer. It is noted that the other processes, such as (ω +1γ 3γ ) does not occur, since L 1 ↔ 1 the energy conservation is violated. Now, the various “vertices” can be written down explicitly as follows:

~ 2 ˆ (4) iωLt ω1 (4)4 : n1, n2 H1(t) n1 4, n2 = 10ε0χ ELe− acrys 2Lε (n1 3)(n1 2)(n1 1)n1, h | | − i (4) iω t ~ω 2 − − − (4) : n , n Hˆ (t) n +4, n = 10ε χ E† e L a 1 (n + 1)(n + 2)(n + 3)(n + 4),  4 1 2 1 1 2 0 L crys 2Lε p 1 1 1 1 − h | | i ~ ~ ˆ (4) iωLt ω1 ω2 (4)2 : n1, n2 H1(t) n1, n2 2 = 10ε0χ ELe− acrys (2n1 + 1) n2(n2 1),  h | | − i − 2Lεp 2Lε − (66)  (4) iω t ~ω1 ~ω2 (4) : n , n Hˆ (t) n , n +2 = 10ε χ E† e L a (2n + 1) (n + 1)(n + 2),  2 1 2 1 1 2 0 L crys 2Lε 2Lε 1 p 2 2  − h | | i − ~ (2) : n , n Hˆ (t) n , n 2 =3ε χ(2)E e iωLta ω2 (n 1)n , 2 1 2 1 1 2 0 L − crys 2Lε 2 2 p h | | − i (2) iω t ~ω −  ˆ † L 2 (2) 2 : n1, n2 H1(t) n1, n2 +2 =3ε0χ ELe acrys 2Lε (n2 + 2)(n2 + 1).  − h | | i p  7  We do put S for the classical laser beam as well as the quantum photons,p in order to indicate that all the time dependencies have been absorbed into the classical and quantum propagators.

14 5 General Formula of Transition Amplitude or S-matrix in QO inside Cavity

To obtain the general formula of transition amplitude or S-matrix, we ﬁrst perform inte- grations over times (t1, t2, t3, ) of the product of vertex operators, which are arranged according to the time ordering.··· iωLt iωLt Time dependence of the vertex is ELe− or EL† e , and that of propagator D(t t′) iω(t t′) − is e± − , and hence the time integration is the same as in quantum mechanics. Here, we write the vertex with “energy” eigen-states as E′ V (t ) E , where E and E′ are energies h k| k | ki k k before and after the k-th interaction. Here we do not distinguish the “energy” and the “angular frequency”. Then, up to position dependent extra factors, Ks, the transition amplitude S(t, ), −∞ deﬁned by ψ(t) = S(t, ) ψ( ) , reads perturbatively, f −∞ −∞ i

N t t1 tN−1 ∞ L 1 i(ω +E1 E +iǫ)t1 S(t, ) =1+ dt dt dt e− 1 − f E Hˆ E −∞ |up to Ks i~ 1 2 ··· N h f | 1| 1i N=1 Z−∞ Z−∞ Z−∞ XL ′ L ′ i(ω +E2 E +iǫ)t2 i(ω +E E +iǫ)t e− 2 − 2 E′ Hˆ E e− N i− N N E′ Hˆ E × h 2| 1| 2i×···× h N | 1| ii N ∞ L L E′ Hˆ1 Ek /~ i(ω1 + +ωN +Ei Ef )t k =1+ e− ··· − L h | L | i , (67) (ω + + ω )+ E E′ + iǫ N=1 k N i k X kY=1 ··· − where Ei = EN and Ef = E1′ are initial and final “energies” of photons, respectively. The “energies” before and after each interaction can be expressed by using the number representation as (n ) , (n ) Hˆ (t ) (n ) (s ) , (n ) (s ) , where the suffix 1 and h k 1 k 2| 1 k | k 1 − k 1 k 2 − k 2i 2 indicate the particle numbers (nk)1,2 and their difference (sk)1,2 be for the two kinds of photons, γ1 and γ2, respectively. Then, we have

E′ = E +(s + s + + s ) ω +(s + s + + s ) ω , (68) k i k k+1 ··· N 1 1 k k+1 ··· N 2 2 where ω1 and ω2 are frequencies of photon γ1 and γ2, respectively. The formula obtained as Eq.(67) is nothing but the formula of old fashioned perturbation theory in QM. When we apply it to our QO, the phase matching condition tells that the energy and momentum conservation hold at each vertex, which leads to the vanishing of all the denominators in the formula, giving a singular (i. e. extremely dominant) behavior for the scattering amplitude. This singularity is called the “mass singularity”; it happens when the involved particles are massless like photons, and a particle decays into particles moving in the same direction to the parent particle. (See the reference [6]) A way to tame this singularity is to average the transition amplitude over the laser L frequency ωk at each interaction point k. This averaging is a kind of tuning of the laser beam at the resonance point. A candidate of the weight for this averaging is a Gaussian distribution, W ( ωL ), with a 8 | | center ωR and a width δ,

L 1 ( ωL ω )2/2δ2 W ( ω )= e− | k |− R . (69) | k | √2π δ

8Corresponding to absorption and emission of laser light, we consider ωL as positive and negative values, respectively.

15 L This averaging implies that the laser photon has a definite wave vector k = ωR/c′ at the interaction plane of crystal, but its energy is not exactly ω , but is distributed± as9 ± R ω δ < ωL <ω + δ. (70) R − | k | R Then, the averaging over the laser frequency becomes N ∞ ∞ S(t, ) =1+ dωL W ( ωL )S(t, ; ωL ) . (71) h −∞ |up to KsiωL k | k | −∞ { k } |up to Ks N=1 X Yk=1 Z−∞ Since the weight function vanishes at infinitely large frequency, W ( ωL ) 0, for ωL . (72) | k | → | k |→∞ we can perform the contour integration clock-wisely by the new variables with tilde, defined by ω˜L = ωL + ωL , ω˜L = ωL + ωL , , ω˜L = ωL , (73) 1 1 ··· N 2 2 ··· N ··· N N and pickup the pole residues. The choice of pole residues recovers the energy conservation L L at each vertex, ωk = Ek′ Ek, but ωk is not definitely fixed, but is distributed with a sharp peak around ω . Then,− we arrive at ± R N ∞ S(t, ) =1+ ( 2πi/~)W ( E′ E ) E′ Hˆ E , (74) h −∞ |up to KsiωL − | k − k| h k| 1| ki N=1 X kY=1 where

1 ′ 2 2 ( E Ek ωR) /2δ W ( E′ Ek )= e− | k− |− , (75) | k − | √2π δ i(ωL+ +ωL +E E )t where the pre-factor e− 1 ··· N i− f = 1 holds, and we can take t without problem.10 If the resonance peak is very sharp, we are allowed to approximate→∞ the Gaussian distribution at its peak value,

L 1 W (ω ) , with the condition E′ = Ek ωR. (76) k ≈ √2π δ k ±

Now, we are ready to include the factors K(z, z′)s, being ignored in the above, and to obtain the ﬁnal formula of transition amplitude in QO.

The formula for the transition amplitude S(t, ) ωL for t = , or the S-matrix S( , ) in QO is obtained as follows: h −∞ i ∞ h ∞ −∞ iωL N N L/2 ˆ ∞ 2πi ˆ Ef S Ei =1+ − dz1dz2 dzk K(z, z1)PD;C1 Ef H1(0) E1 h | | i ~√2π δ L/2 ··· h | | i N=1 k=1 Z− X L Y L K (z , z )K (z , z ) E′ Hˆ (0) E K (z , z )K (z , z ) × C1;C2 1 2 1 2 h 2| 1 | 2i C2;C3 2 3 2 3 ×··· L ˆ KCN−1;CN (zN 1, zN )K (zN 1, zN ) EN′ H1(0) Ei KCN ;C0 (zN , z0), (77) × − − h | | i 9At ﬁrst sight, this averaging seems to violate the “massless property” of photon, but is a standard way to tame the mass singularity. Physically, this δ can be the ambiguity to detect photon energy, which is inevitable. In QO, it can be the “ambiguity of tuning the laser frequency at the resonance point”. It can also be that the laser beam is not a plane wave but the wave packet. 10 L Here, we have to note that in doing the contour integral in the complex plane ofω ˜k , no other poles appear, even if the factor Ks are included, since the denominator of the factor KωL (z,z′) is 1 r eiφ(ωLL/c), L − L and it can not be zero, unless the mirror perfectly reﬂects the laser beam (rL < 1).

16 where the energy diﬀerence can be taken as E′ E = ω at each vertex, and δ/2 is the k − k ± R width of laser frequency distribution around the resonance point ωR. The factors Ks are given in Eqs.(56)-(58), and the vertices E′ Hˆ (0) E are in Eq.(65), being categorized into h | 1 | i six types, (4) 4, (4) 2 and (2) 2. ± ± ± Before ending this section, let us summarize the expressions for Ks and vertices: The damping factors Ks are

′ ′ ω e−γ |z−z | KC;C′ (z, z′)= iφ(ωL/c)−γ′ℓ , 1 rωe −( ) ω 1 K = KL (z, z′)= iφ(ωL/c) , (78) 1 rL(e )  − ′  √1 r2 e−γ (ℓ/2−zf )  ω − 2 K (z, zf )= iφ(ωL/c)−γ′ℓ , PD;Cf 1 rωe −   ω ω where rω, ωL =(r1r2r3r4)ω, ωL , and φ(ωL/c)= c (L ℓ)+ c′ ℓ (ℓ: the length of the non-linear crystal, L: the circumference of the ring resonator).− The vertices V s are

~ 2 ˆ (4) ωR (4)4 : n1, n2 H1(0) n1 4, n2 = 10ε0χ 8Lε acrys EL (n1 3)(n1 2)(n1 1)n1, h | | − i (4) ~ω 2 − − − (4) : n , n Hˆ (0) n +4, n = 10ε χ R a E† (n + 1)(n + 2)(n + 3)(n + 4),  4 1 2 1 1 2 0 8Lε crys Lp 1 1 1 1 − h | | i ~ 2  ˆ (4) ωR (4)2 : n1, n2 H1(0) n1, n2 2 = 10ε0χ acrys EL(2n1 + 1) n2(n2 1),  4Lε p (79)  h | | − i − (4) ~ω 2 − (4) : n , n Hˆ (0) n , n +2 = 10ε χ R a E† (2n + 1) (n + 2)(n + 1),  2 1 2 1 1 2 0 4Lε crys L 1 p 2 2  − h | | i − ~ (2) : n , n Hˆ (0) n , n 2 =3ε χ(2) ωR a E (n 1)n , 2 1 2 1 1 2 0 4Lε crys L 2 2 p h | | − i (2) ~ω − (2) : n , n Hˆ (0) n , n +2 =3ε χ R a E† (n + 2)(n + 1).  2 1 2 1 1 2 0 4Lε crys Lp 2 2  − h | | i  p  Now, the general formula has been obtained, which is useful in quantum optics (QO) to estimate the higher-order eﬀects inside the ring resonator with a non-linear material contained.

6 Conclusion and Discussion

We intend in this paper to derive a general formula to be used to estimate the higher-order effects in QO, especially in a ring resonator with a non-linear material contained. The obtained formula is expressed Feynman graphically using propagators and vertices, in which the propagators include the effects of damping in the non-linear material as well as the reflection and transmission by mirrors. See Appendices A, B and C for the relation of our method to the other works. We will discuss the possible observables being calculable in principle in the above- mentioned QO. (1) Average number of produced photons: We consider a simple case of NL-material with non-linear susceptibility, χ(2) and χ(4). L For the laser beam with a resonance energy ω , two photons γ1 and γ2 are relevant, having L L energies ω /4 and ω /2, respectively. We can estimate the averaged number of photons n1 m1 m2 h i and n2 for γ1 and γ2, as well as the correlations, (n1) (n2) , (m1, m2)=1, 2, . To do thish wei have to use the numerical simulations, andh to compare thei result so obtained··· with a theoretical study of the “generalized squeezing” in [4]. Also refer to Appendix D. (2) Husimi function:

17 Averaged number of the produced photons is directly connected to the Husimi function H(α) [9]. (Please refer to Wigner function [10].) It is a quantum version of the distribution function and is called “Q function” Q(α) in QO: H(t, α) Q(t, α)= α ρˆ(t) α , (80) ≡ h | | i whereρ ˆ(t) is a density matrix at time t, and α is a coherent state with a complex eigen-value | i α for the annihilation operatora ˆ. In our case, we have two sets of annihilation operators, so that the Husimi function can be written at t = in terns of S-matrices as follows: ∞ 2 2 ( α1 + α ) H(α ,α )= α ,α Sˆ 0 α ,α Sˆ 0 ∗ = e− | | | |2 1 2 h 1 2| | ih 1 2| | i ∞ 2¯n1 2m ¯ 1 2¯n2 2m ¯ 2 (α1∗) (α1) (α2∗) (α2) 2¯n1, 2¯n2 Sˆ 0 2¯m1, 2¯m2 Sˆ 0 ∗, (81) × (2¯n1)!(2m ¯ 1)!(2¯n2)!(2m ¯ 2) ×h | | ih | | i n¯1,2,m¯ 1,2 =1 { X } (3) Observable to measure beyond the squeezing: If we introduce a phase shifted wave, θ iθ iω (t z/c(z)) iθ iω (t z/c(z)) E (t, z)= e− E e− L − + e E† e L − , (82) then, it can accommodate the vector potential A(t, z) Eθ=π/2(t, z) and the electric ﬁeld E(t, z)= Eθ=0(t, z), which are canonical conjugate with∝ each other. Therefore, the correlation functions of the phase sifted waves described the squeezing. In our case, there exist various two-point and three-point correlation functions, since the decay of γ γ + γ is possible by ω =2ω , 2 → 1 1 2 1 θ θ θ θ θ θ ∆E1 (t)∆E1 (t′) PD = C11(t, t′), ∆E2 (t)∆E2 (t′) PD = C22(t, t′), for (t > t′), h θ θ i θ h θ i (83) T [∆E (t )E (t′ )∆E (t )] = C ′ (T [t , t′ , t ]), (h 1 1 1 1 2 2 iPD T [1,1 ,2] 1 1 2 the last three-point correlation function gives, by specifying the ordering of time, as

θ θ θ θ ∆E1 (t1)E1 (t1′ )∆E2 (t2) PD = C112(t1, t1′ , t2) for (t1 > t1′ > t2), h θ θ θ i θ  ∆E1 (t1)∆E2 (t2)E1 (t1′ ) PD = C121(t1, t2, t1′ ) for (t1 > t2 > t1′ ), (84) h θ θ θ i θ  ∆E (t2)∆E (t1)∆E (t′ ) PD = C (t2, t1, t′ ) for (t2 > t1 > t′ ). h 2 1 1 1 i 211 1 1 Corresponding to the above correlation functions, we can introduce various spectra:

∞ θ iντ S11(ν, θ)= 0 dτ C11(t, t′ = t + τ)e− , S (ν, θ)= ∞ dτ Cθ (t, t = t + τ)e iντ ,  22 R0 22 ′ −  ∞ ∞ θ i(µτ+νσ) S112(µ,ν,θ)= 0 dτ 0 dσ C112(t1, t1′ = t1 + τ, t2 = t1′ + σ)e− , (85)  R S (µ,ν,θ)= ∞ dτ ∞ dσ Cθ (t , t = t + τ, t = t + σ)e i(µτ+νσ),  121 R0 R0 121 1 2 1 1′ 1 − S (µ,ν,θ)= ∞ dτ ∞ dσ Cθ (t , t = t + τ, t = t + σ)e i(µτ+νσ)  211 R0 R0 211 2 1 2 1′ 1 −   The second spectrumR S22(ν,R θ) was measured by L-A Wu et al.[3] using the OPO which contains the non-linear crystal with non-vanishing χ(2). This measurement shows the squeezing is realized in OPO. The spectrum S22(ν, θ) represents the down-conversion of the laser photon to two degenerate photons (ω 2ω ). L → 2 Therefore, it is interesting to examine the new spectra S112(µ,ν,θ), S121(µ,ν,θ) and (2) S211(µ,ν,θ), in addition to S11(ν, θ),S22(ν, θ), in case of NL-crystal with non-vanishing χ and χ(4). We hope to study these issues in the next paper of ours [11].

18 References

[1] “Introduction to Quantum Optics” by G. Grynberg, A. Aspect and C. Farbre, Cam- bridge University Press, Cambridge (2010). “Quantum Optics” (in Japanese) by E. Hanamura, Iwanami Pub. Co., Tokyo (1991).

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19 [16] V. V. Dodonov, “Nonclassical states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. of Optics B: Quantum and Semiclassical Optics 4.1 (2002): R1. [17] S. L. Braunstein and C. M. Caves. “Phase and homodyne statistics of generalized squeezed states,” Phys. Rev. A 42.7 (1990): 4115. [18] K. Banaszek and P. L. Knight. “Quantum interference in three-photon down- conversion,” Phys. Rev. A 55.3 (1997): 2368. [19] L. C. G. Govia, E. J. Pritchett, and F. K. Wilhelm. “Generating nonclassical states from classical radiation by subtraction measurements,” New J. of Phys. 16.4 (2014): 045011. [20] M. Cooper et al. “Experimental generation of multi-photon Fock states,” Optics express 21.5 (2013): 5309-5317. [21] C. W. S. Chang, et al. “Observation of Three-Photon Spontaneous Parametric Down- Conversion in a Superconducting Parametric Cavity,” Phys. Rev. X 10.1 (2020): 011011. [22] Walls, D. F., and G. J. Milburn. “Quantum Optics Springer-Verlag.” New York (1994). [23] Collett, M. J., and C. W. Gardiner. “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification.” Physical Review A 30.3 (1984): 1386. [24] M. Sargent III, M. O. Scully, W. E. Lamb, Jr.,“Laser Physics”, Additoin-Wesley, Pub- lishing (1974); E. Hanamura, “Quantum Optics” (in Japanese), Iwanami-Shoten Pub- lishing (1991). [25] see Appendix B: “Fokker-Planck equation and the operator formalism of thermodynam- ics”, in N. Aibara, N. Fujimoto, S. Katagiri, M. Saitou, A. Sugamoto, T. Yamamoto, and T. Yumibayashi, PTEP 2019, 073A02 (2019). [26] R. J. Glauber, Phys. Rev. Lett. 10 (1963) 83; Phys. Rev. 131 (1963) 2766; E. C. G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277. [27] L. Knöll, S. Scheel, D-G. Welsch, “QED in dispersive and absorptive medium”, arXiv: quant-ph 00061; P. D. Drummond and M. Hillery, “The quantum theory of nonlinear optics”, Cambridge University Press (2014).

Appendix A: -Input-output theory and observation of squeezed state in optical cavity-

In this Appendix, we review Input-Output Theory and squeezed state observation in optical cavity based on Corrett and Gardiner [22, 23], and compare it to our paper. Even if this Input-Output theory is popular in the nonlinear optics, it is not a rigorous theory, but is rather a phenomenological one. We will explain this, by following the theory. The input electric ﬁeld enters from outside into the optical cavity, interacts with it, and emits the output photon, which is observed by the photo-detector (PD). Suppose that a single mode Ω of photon remains in the optical cavity, where the creation and annihilation operators of the photon are denoted bya ˆ†, aˆ. The cavity (Fabry-P´erot type)

20 is assumed to have a perfect reﬂection on one side and a reﬂection mirror with transmittance on the other side. In this situation, the Hamiltonian is divided into three parts,

Hˆ = HˆCavity + HÊxternal + HÎnt, (86) where, HˆCavity is the Hamiltonian of the optical cavity, HÊxternal is that outside the cavity, and HÎnt is the interaction Hamiltonian between the inside and the outside of the cavity. We can give Hˆ Int as follows:

HˆInt = Γˆˆa†(t)+ Γˆ†aˆ(t), (87)

∞ Γ=ˆ i~ dωκ(ω)bω(t), (88) Z−∞ ˆ ˆ where, bω† (t), bω(t) are the creation-annihilation operators of photons in the external electric ﬁeld. The form of interaction shows this is a toy model, describing a transition of photon, γΩ γω, admitting the loss of energy (e.g. a phenomenological inelastic interaction, violating→ energy conservation). Then, Heisenberg equations are

˙ i ˆ ˆ aˆ(t)= ~ [â(t), HCavity] ∞ dω κ(ω)bω(t), − − −∞ (89) ˆ˙ i ˆ ˆ ( bω(t)= ~ [bω(t), HExternal]+ κ(ω)â(t), − R where, we suppose HÊxternal is free, then ˙ ˆb (t)= iωˆb (t)+ κ(ω)â(t). (90) ω − ω We are considering the Heisenberg picture, and should be careful about the time dependency, so that we have written the t-dependence explicitly at delicate places. We take the initial condition bω,0 = bω(t0) at t = t0 and final condition bω,1 = bω(t1) at t = t1. The solution to the above equations is

′ iω(t t0) t iω(t t ) ˆb (t)= e− − ˆb + κ (ω) e− − aˆ (t′) dt′ (t < t), ω ω,0 t0 0 ′ (91) iω(t t1) t1 iω(t t ) ˆb (t)= e− − ˆb κ (ω) e− − aˆ(t′)dt′ (t < t ) . ( ω ω,1 − Rt 1 Unfortunately, this equation is not rigorous,R but is valid only in the first order approximation. The reason can be understood from a well-known fact that if the Hamiltonian controlling ∂tˆbω is t-dependent as κ(ω)â(t), the rigorous solution is expressed in terms of R t dt′κ(ω)â(t′) R t dt′κ(ω)â(t′) T-ordered product of e t0 , or anti-T-ordered product of e t1 , We suppose κ(ω)2 = γ/2π, then (89) give two expressions,

˙ i ˆ γ aˆ = [â(t), HCavity]+ √γaÎN (t) aˆ(t), − ~ − 2 (92) ˙ i γ aˆ = [â(t), HˆCavity] √γaÔUT (t)+ aˆ(t), ( − ~ − 2 where aˆ (t) 1 ∞ dωe iω(t t0)b (ω) IN √2π − − 0 ≡ − −∞ (93) aˆ (t) 1 ∞ dωe iω(t t1)b (ω). ( OUT √2π R − − 1 ≡ −∞ R 21 Here,a ÎN anda ÔUT are called the Input and Output fields, respectively and have the rela- tionship:

aÔUT (t)+âIN (t)= √γaˆ(t). (94) To derive these equations we have to suppose that 2πκ(ω)2 = γ does not depends on ω. Therefore, the above equations on which the input-output theory is based, are phenomenological equations, derived with three assumptions, the phenomenological interaction, the first order perturbation theory and the ω independence of κ. From these, the two-time correlation functions of the input and output fields can be obtained from the correlation functions of the photons inside the optical cavity,

aˆ† (t), aˆ (t′) = γ aˆ†(t), aˆ(t′) , h OUT OUT i h i (95) aˆ (t), aˆ (t′) = γ aˆ(t), aˆ(t′) + γθ(t′ t) [â(t′), aˆ(t)] , ( h OUT OUT i h i − h i where U,ˆ Vˆ UˆVˆ Uˆ Vˆ . h i≡h i−h ih i The phase shifted operator Xˆθ is convenient to discuss the uncertainty relation between gauge field and its canonical conjugate, electric field: iθ iθ Xˆθ(t)= e− aˆ(t)+ e aˆ†(t), (96) The spectrum of the output photons is defined as a Fourier transform of the two-point correlation between the wave shifted fields,

OUT OUT OUT OUT iωt S (ω) 1 =: S (ω): dt : Xˆ (t), Xˆ (0) : e− (97) θ − θ ≡ h θ θ i Z iωt = γ dt T : X (t),X (0) : e− . (98) h θ θ i Z where : AˆBˆ : represents the normal order product and T is the time order product. Using this spectrum of the input-output theory, L.-A. Wu et al. [3] shows the squeezing by OPO.

6.1 Relation of our paper to this Appendix A The final result of the input-output theory [22, 23] is very simple, that is, the electromagnetic wave outside the cavity is damped from that inside the cavity by a damping factor γ = γ(input output). This damping is caused by the penetration of the wave through the last − mirror. If so, it is enough to take into account the damping effect of the wave by the mirror, phenomenologically. As was discussed in the above, the input-output theory is based on a number of assumptions, and is not a rigorous theory. We have a lot of sources of dispersion and damping; one is the scattering of the wave by the crystal surfaces of the non-linear optical material, and the others are the reflection by the mirrors and the penetration through them. In this paper we consider these dispersive and damping effects phenomenologically, by using the measured values of reflection coefficient r and transmission coefficient t = 1 r2 for i i − i the i-th mirror. The γ(input output) in the input-output theory for the final mirror is equal to − p the transmission coefficient tfinal mirror in our notation. − In our paper we separate this part of dispersion and damping as the “quasi-free Hamilto- nian”, and concentrate on the “real interaction Hamiltonian” of photons with the non-linear material, so that we can cope with the higher order calculations by many times round trips of photons inside the cavity.

22 Appendix B: -Fokker-Planck equation-

In quantum mechanics with loss, a standard method is to consider that the system is sur- rounded by a heat bath (or reservoir), so that the energy of the system can dissipate to the heat bath. In this situation, the master equation (q-number equation) satisﬁed by the density matrix is used, which is transformed by a coherent representation to the Fokker- Planck equation (c-number equation). The Fokker-Planck equation can be rewritten as the Langevin equation. [24, 25] The Hamiltonian (86) in such a system is divided into parts,

Hˆ = Hˆ + Hˆ + Hˆ Hˆ + Hˆ + Hˆ + Hˆ Hˆ + Hˆ (99) Cavity External Int ≡ Cavity,0 Cavity,I External Int ≡ 0 I

HˆI = HˆCavity,I + HˆInt, (100) where HˆCavity,0 is a free part of HˆCavity. The state can be written as the tensor product of the output photon state of b, and the photon state in the resonator of a, so that the total density operator is the tensorH product of the density operator for the heatH bath b and that for the system (a), ρ =ρ ˆ ρˆ. T otal b ⊗ Given the density matrix in the interaction picture ρT otal(t), the equation of motion is dρˆ (t) i T otal = [Hˆ (t), ρˆ (t)]. (101) dt −~ I T otal This master equation can be solved as a formal expansion as

n ∞ i t t1 tn−1 ρˆ (t) =ˆρ (0) + dt dt dt T otal T otal −~ 1 2··· n n=1 0 0 0 X Z Z Z [HÎnt(t1), [HÎnt(t2),..., [HÎ(tn), ρˆT otal(0)] ... ]] (102)

∞ 1+ Uˆ (t) (ˆρ(0)) Uˆ(ˆρ(0)). ≡ n ≡ n=1 ! X We do not observe the heat bath described by , so that we trace oﬀ the contribution Hb from the heat bath, giving the density operator of the system,

ρˆ(t) = Tr ρˆT otal(t), (103) Hb where we assume Tr HˆI (t)ˆρb =0. (104) Hb Unfortunately in the actual application, we have to restrict up to n = 2 for Un, and discard the higher order contributions. Then, the master equation forρ ˆ becomes

d 1 t ρˆ(t)= dt Tr Hˆ (t), [Hˆ (t ), ρˆ ρˆ(0)] . (105) ~2 1 b I I 1 b dt − 0 H ⊗ Z h i In the following, we will omit HˆCavity,I . Since Hˆ Int (of a toy model) is given as

∞ HˆInt = Γˆˆa† + Γˆ†a,ˆ Γ=ˆ i~ dω κ(ω)bω, (106) Z−∞

23 the master equation becomes d 1 ρˆ(t)= I (t)â†aˆ†ρˆ(0) + I (t)â†aˆρˆ(0) I (t)â†ρˆ(0)â† + I∗(t)â†ρˆ(0)â dt − ~2 1 3 − 1 4 + I (t)ˆρ aâˆ†ρˆ(0) + I (t)ˆρ aâˆρˆ(0) I∗aˆρˆ(0)â† + I aˆρˆ(0)â 4 b 2 b − 3 2 (107) I (t)â†ρˆ(0)â† I (t)âρˆ(0)â† + I (t)ˆρ(0)â†aˆ† + I∗(t)ˆρ(0)âaˆ† − 1 − 3 1 4 I (t)â†aˆρˆ(0) I (t)âρˆ(0)â + I∗(t)ˆρ(0)â†aˆ + I (t)ˆρ(0)âaˆ , − 4 − 2 3 2 where t ˆ ˆ iΩ(t+t1) I1 = 0 dt1 Γ(t)Γ(t1) e , t h i I = dt Γˆ (t)Γˆ (t ) e iΩ(t+t1),  2 R0 1 † † 1 − t h i (108)  I = dt Γ(ˆ t)Γˆ (t ) eiΩ(t t1),  3 R0 1 † 1 −  t h i I = dt Γˆ (t)Γ(ˆ t ) eiΩ( t+t1), 4 R0 1 † 1 −  h i  R  AˆBˆ Tr (AˆBˆρˆb). (109) h i ≡ Hb We suppose the form of the correlation function to be

ˆb ˆb 2πM(ω )δ(2Ω ω ω ), and ˆb† ˆb 2πN(ω )δ(ω ω ), (110) h ω1 ω2 i ≡ 1 − 1 − 2 h ω1 ω2 i ≡ 1 1 − 2 ˆ ˆ where N(ω1)= bω† 1 bω1 , then I1 become

t 2 ∞ i∆ωτ I1(t)=2π (i~) dτ d(∆ω) κ(Ω+∆ω)κ(Ω ∆ω)M(Ω+∆ω)e− , (111) 0 − Z Z−∞ where ∆ω = ω Ω. Here we assume− that κ(ω), M(ω) are a function that changes slowly around ω = Ω. Therefore we can take the Markov approximation and I1 becomes

t 2 ∞ 2 i∆ωτ I1 2π (i~) dτ d(∆ω) κ(Ω+∆ω) M(Ω+∆ω)e− . (112) ≈ 0 Z Z−∞ Here we assume that the integrals for frequency decay rapidly with respect to time, take the time integrals to inﬁnity, and interchange the order of the integrals, then we obtain

2 ∞ 2 1 I1 2π (i~) d(∆ω) κ (Ω+∆ω)M(Ω+∆ω)( i) + iπδ(∆ω) , ≈ − P ∆ω (113) Z−∞ 2(i~)2 γM(Ω) + i∆¯ ≡ where is the principal value of Cauchy and γ κ2(Ω). In theP following, we ignore ∆¯ as having a small≡ eﬀect. Similarly, we obtain

2 I1 2(i~) γM(Ω), ≈ 2 I 2(i~) γM ∗(Ω),  2 ≈ (114)  I 2(i~)2γ (N(Ω) + 1) ,  3  ≈ 2 I4 2(i~) γN(Ω),  ≈  where N(Ω) =a ˆ†aˆ is the number operator of photon in the cavity.

24 After all, including the part of HˆCavity,I , the master equation becomes d ρˆ(t) [Hˆ , ρˆ] dt ≈ Cavity,I γ γ + (N(Ω) + 1) 2âρâˆ† aˆ†aˆρˆ ρâˆ†aˆ + N(Ω) 2â†ρâˆ aâˆ†ρˆ ρââˆ† (115) 2 − − 2 − − γ γ + M(Ω) 2â†ρâˆ† aˆ†aˆ†ρˆ ρâˆ†aˆ† + M ∗(Ω) (2âρâˆ aâˆρˆ ρââˆ) . 2 − − 2 − − When the heat bath is characterized by a temperature T , we choose 1 M(Ω) = 0, N(Ω) = . (116) eβ~Ω 1 − In the following, we will consider the case of T = 0. Then, the master equation becomes simple and tractable, d γ ρˆ(t) =[Hˆ , ρˆ]+ aˆρâˆ† aˆ†aˆρˆ ρâˆ†aˆ . (117) dt Cavity,I 2 − − To estimate this master equation, it is known to be useful the Glauber-Sudarshan’s P representation [26] in terms of coherent state:

ρˆ(t)= P (α) α α d2α, (118) | ih | Z Then, the master equation becomes ∂ γ ∂ ∂ P (α)= H P (α)+ α∗ + α P (α) (119) ∂t Cavity 2 ∂α ∂α ∗

If we take the second-order squeezing term as a term in HCavity, we obtain the Fokker- Planck equation ∂ ∂ ∂ P (α)= λ∗(2Ω) α + λ(2Ω) α∗ ∂t − ∂α ∂α ∗ (120) γ ∂ ∂ 1 ∂2 ∂2 + α∗ + α + λ∗(2Ω) + λ(2Ω) P (α). 2 ∂α ∂α 2 ∂α 2 ∂α2 ∗ ∗ Generally, Fokker-Planck equation reads ∂ ∂ 1 ∂ ∂ P (t)= A α + B B +(c.c.) P (α) (121) ∂t ∂α ij j 2 ∂α ∂α ij ji i i i can be transformed into the Langevin equation dα i = A α + B ξj, (122) dt − ij j ij and its complex conjugate equation, where ξj is a Gaussian noise. When Aij is diagonalizable and its eigenvalue is real and positive, then we obtain steady- state solution t i (t t′)A k αSS(t)= e− − Bjkξ (123) ij Z−∞

25 So correlation function min(t,s) i Tj (t t′)A T (s t′)AT G (t s) α (t),α (s) = e− − BB e− − dt′ (124) SS − ≡h SS SS i Z−∞ is the Fourier transform of the spectrum matrix

1 ∞ iωτ 1 1 T T 1 S(ω)= e− G (τ)dτ = (A + iω)− BB A iω − . (125) 2π SS 2π − Z−∞ In case of Eq.(120),

γ 2 λ(2Ω) λ(2Ω) 0 A = − γ , B = (126) λ∗(2Ω) 0 λ∗(2Ω) − 2 OUT we can predict the spectrum Si , where i =1, 2 correspond to θ =0,π/2, respectively, as follows: 2γ λ(2Ω) 2 SOUT =1+ | | , (127) 1 γ 2 2 2 λ(2Ω) +(ω Ω) −| | 2 − OUT 2γ λ(2Ω) S2 =1 | 2 | . (128) − γ + λ(2Ω) +(ω Ω)2 2 | | − The same result was derived in the input-output theory [22, 23]. Therefore, the input-out-put theory is within the category of Fokker-Planck equation. If HˆCavity contains a higher nonlinear term, the higher-order derivatives with respect to α and α∗ arise, so that in this case the equation is beyond the Fokker-Planck equation which is the second order differential equation. From this we may understand that the Fokker-Planck equation is a powerful method for the dissipative system with quadratic Hamiltonian, but we will confront with a difficulty for the Hamiltonian with higher order terms than quadratic. The essential place which we have to apply the Fokker-Planck equation is the laser oscillation [24], where the infinite number of photons are involved; in the process of the laser pumping or the excitation of atoms, and the subsequent stimulated and spontaneous emissions of photons, under the various sources of absorption.

6.2 Relation of our paper to this Appendix B Our paper uses the solution of Fokker-Planck equation applied for the ﬂuctuation of phase θ of photons, deﬁned by α = α eiθ. The Fokker-Planck equation| | of this problem is [24], ∂ 1 ∂ ∂2 P (α, t)= (A C B α 2)αP (α, t) +(c.c.) + A P (α, t). (129) ∂t −2 ∂α − − | | ∂α∂α ∗ Let b, a denote the upper level and the lower level of atoms, between which the emission and absorption of light occurs. Atoms at the ground state c are pumped to the higher level b, but the atom at b and a can loose energy and be relaxed to the ground c by rates γb and γa, respectively. The various constants are given by N rg C c A = a , B = , C = T + α(ω)c, (130) 2γb β 2L

26 where Na is the number of atoms, r is the rate of pumping per time, c is light velocity, L and V are length and volume of the Fabry-Pérot cavity, T is transmission coefficient of the mirror, α(ω) is absorption rate per length by the matter inside cavity. The g and β are given by the dipole emission rate er 2 of photon from the atom, | ab| e2ω r 2 2γ γ g =2π ba| ab| , β = a b . (131) 3ǫ0V ~(γa + γb) (γa + γb)g In terms of polar coordinate, α = α eiθ, we have | | ∂ 1 ∂ P ( α ,θ,t)= α 2(A C B α 2)P ( α ,θ,t) ∂t | | −2 α ∂ α | | − − | | | | | | | | A ∂2 A ∂ ∂ + + α P ( α ,θ,t). (132) 4 α 2 ∂θ2 4 α ∂ α | |∂ α | | | | | | | | | | Assume the solution is P ( α ) P (θ, t), then we have two equations, | | × 1 ∂ 2 2 A ∂ ∂ 2 α ∂ α [ α (A C B α )P ( α )]+ 4 α ∂ α α ∂ α P ( α )= λP ( α ), − | | | | | | − − | | | | | | | | | | | | | | | | (133)  ∂ A ∂2 ∂t P (θ, t)= 4 α 2 ∂θ2 λ P (θ, t).  | | − The first equation gives the stationary distribution for the photon number, n = α 2, while | | the second equation describes the temporal development of the phase θ. For simplicity we take λ = 0. Then the solution of the first equation is

2 B α 4+ A−C α 2 B ( α 2 A−C ) P ( α ) e− 2A | | A | | e− 2A | | − B (134) | | ∝ ∝ This Gaussian distribution gives the averaged number of photon and its standard deviation as A C A n = − , σn = . (135) h i B rB The laser is above the threshold, when the pumping rate A is much larger than the cavity 2 loss C, A C, leading to n = σn, that is the laser light above threshold obeys the Poisson distribution,≫ and the two pointh i correlation function becomes unity: E2E2 g = h 1 2 i =1. (136) 12 E2 E2 h 1 ih 2 i This means that the laser light becomes classical light. As for the second equation, if choosing α 2 as the average number of photons n , it | | h i becomes a diﬀusion equation with a diﬀusion constant D = A/2 n , (Einstein’s equation for Brownian motion): Then, we have h i ∂ A ∂2 P (θ, t)= P (θ, t). (137) ∂t 4 n ∂θ2 h i The solution is

2 n hni(θ−θ0) P (θ, t)= h i e− At , (138) rπAt ×

27 for which the standard deviation squared increases linearly in time, but can be zero: A B σ2(t)= t = Dt t 1, (139) θ 2 n ≈ 2 ≪ h i for A C above threshold. If an actual laser gives (A, C) = O(106)/[s], and B 0.1/[s], 2 ≫ 9 ∼ σθ (t) 10− 1, when the diffusion time 10 [s] is compared with the typical time of laser ∼ ≪ 8 operation, t =2L/c = 10− [s] for the cavity with length L =1.5 [m]. Now, we can understand that if the laser is tuned sufficiently above the threshold A C, the laser light becomes the classical one, having a Poisson distribution for the number,≫ and a common phase which is constant without caused by the diffusion. This is the starting line of our paper, by assuming the classical electric field for the input laser light.

Appendix C: QED in non-uniform dielectric matter

QED in non-uniform dielectric matter is studied by L. Knöll, S. Scheel, D-G. Welsch [27]. For the recent developments, refer to the book by P. D. Drummond and M. Hillery [27]. Considering a system of doped atoms and radiation field (photons) in the dielectric matter. The dielectric matter is assumed to consist of a finite number of dipoles Xi with discrete spectra ω (i =1 N), and the infinite number of dipoles X with continuum spectra i − ω ω (0 ω < ). This is the microscopic Hopfield model of dielectrics. The Lagrangian is ≤ ∞ = + + + , (140) L Latom Lrad Lmatters Lint where

= 1 ε (E2 c2B2) , = ma x˙ 2, Lrad 2 0 − Latom 2 a N µ ˙ 2 2 2 µ ˙ 2 2 2 ˙ matters = Xi ωi Xi + dω Xω ω Xω dω Xωvi(ω)Xi, (141)  L i=1 2 − 2 − −  N ˙ ∇  int = eaP(x˙ aA(xa) φ(xa)) R αi X iA(Xi) φ(Xi)R Xi . L − − − i=1 −  Putting aside the Lagrangian of theP doped atom, the electric field E couples to the discrete dipoles Xi, but not to the continuum dipoles Xω, from which we understand in the first order perturbation theory, the effective action for photons becomes,

N 3 1 2 2 2 3 3 1 2 S = d r ε E c B + d r d r′ E(r) α X (r)X (r′) E(r′) eff 2 0 − 2 i h i i i Z Z Z i=1 N X 3 3 + d r d r′ dω A (r) α X˙ (r)X˙ (r′) v (ω)X (r′) (142) ω ih i i i i ω i=1 Z Z Z X 3 1 2 2 2 3 = d r ε(r) E(r) c′(r) B + d r A (r)j (r), (143) 2 − ω ω Z Z where X (r)X (r′) is a two point Green function or the Feynman propagator, and h i i i N (0) ε = ǫ (1 + χ ), and j (r)= α X˙ (r)X˙ (r′) v (ω)X (r′). (144) 0 ω ih i i i i ω i=1 X

28 In the above discussion, the current jω(r) is induced by the continuum modes of the dipoles, Xω(r), which is considered in [27], the noise of electric current coming from the reservoir, being understood by the Fokker-Planck equation. The eﬀective action gives the diversion and the absorption of photon, since ε is a complex number having real part and imaginary part, so that the refraction constant n = √ε is also complex number. In this second order eﬀective action of photon, the propagator of photon in two dimensional space-time reads

n n i(ω(t t′) R ω z z′ ) I ω z z′ G (r, t; r′, t) e− − − c | − | e− c | − |. (145) ω ∝ As a microscopic model of dielectrics, this simple model of dipoles [27] is further developed, using multi-polar expansion and the polariton, considering the levels of N dielectric N atoms as the levels of a higher spin 2 , and so on. For the non-linear QED, a delicate problem seems to be discussed in the process of quantization [27]. To consider this problem, we have to think about carefully, the difference of the original action and the non-linear effective action derived from it, without doubling the quantization procedures. For example, we are assumed to start from an action S(0), perform the quantization (or path integration) over the higher energy modes with E>µ, and to obtain an non-linear effective action S(µ) at energy scale µ. Then, the further quantization (or path integration) is allowed only for the lower energy modes of E<µ.

6.3 Relation of our paper to this Appendix C Our paper does not consider the microscopic structure of non-linear material, but the expression of dielectric permeability of it phenomenologically, using the values χ(n) obtained by measurement. The purpose of out paper is to examine how the perturbation theory works well even in the presence of non-uniform dielectric matter and mirrors. Therefore, the quantization is performed for the quasi-free Hamiltonian (the quadratic part in the non- linear QED), and the real interaction Hamiltonian is considered as a perturbation. This is a kind of “interaction picture”, in which the creation and annihilation operator is deﬁned for the quasi-free Hamiltonian, and the interaction Hamiltonian is expressed in terms of the operators. Therefore, the numerator of our propagators in Sec.4.2 coincides with Eq.(145) in [27]. As far as the perturbation theory is applied as ours, no delicate problem discussed in [27] appears.

Appendix D: -Generalized squeezing-

On the normalizablity of the generalized squeezing state, Fisher, Nieto and Sandberg [12] shows that the vacuum expectation value of the generalized squeezed state diverges and the perturbative calculations are broken. Elyutin and Klyshko [13] indicates that the state diverges in a ﬁnite time and behaves very diﬀerently from the normal squeezing state. To get around this problem, k-photon generalized boson operator was studied [14, 15] which uses the following operators [16]:

1/2 nˆ (ˆn k)! k Aˆ = − aˆ† , and Sˆ = exp zAˆ† z∗Aˆ , (146) (k) k nˆ! (k) (k) − (k)

29 where

nˆ =ˆa†a,ˆ [Aˆ , Aˆ† ]=1, [ˆn, Aˆ ]= kAˆ , (147) (k) (k) (k) − (k) nˆ nˆ and k denotes the maximal integer less than or equal to k . On the other hand, Braunstein and McLachlan [5] argues that the divergence of the vacuum expectation value is a mere mathematical artifact, and compute it numerically using the Pade approximation. Later, Braunstein and Caves [17] calculates statistics on Aˆk by homodyne and heterodyne measurements. Banaszek and Knight [18] computes the Wigner function about k = 3 and shows that there is a negative region in its value. Govia, Pritchett and Wilhelm [19] proposes a method to generate generalized squeezed states from coherent states using Josephson photomultiplier (JPM). Experimentally, Cooper et al. [20] generated a multiphoton Fock state. Recently, Chang et al. [21] observed the distribution of star states, which have been expected to be observed for k = 3.