Quantum simulations with photons in one dimensional nonlinear waveguides 3

Appendix A Derivation of Nonlinear Evolution Equation for Single-Species Photons A.1 Atomic Operators We start with the Hamiltonian H = − nz dz{δσcc + ∆0 σbb + ∆p σdd √ +[ 2π(gba σba + gdc σdc )(Eˆ+ eikQ z + Eˆ− e−ikQ z ) +(Ω+ eikC z + Ω− e−ikC z )σbc + h.c.]}, (A.1) which describes the quantum pulses Eˆ± and control lasers Ω± propagating along left- and right- directions in a medium with single-type atoms. With this Hamiltonian, the Maxwell-Bloch equation, which governs the 115 Appendix A. Derivation of Nonlinear Evolution Equation for Single-Species Photons evolution of the quantum light, can be written as 1∂ ∂ ± )Eˆ± (z, t) v ∂t ∂z √ nz = −i∆ω Eˆ± (z, t) + 2πi (gba σab,± (z, t) + gdc σcd,± (z, t)), (A.2) v ( and the evolution of atomic operator σab is given by ∂t σab = i[H, σab ] = −i[−∆0 σab + √ 2πgba (σbb − σaa )(Eˆ+ eikC z + Eˆ− e−ikC z ) −(Ω+ eikC z + Ω− e−ikC z )σac ]. (A.3) With the assumptions of σaa 1, σbb 0, (A.4) which means that the atoms are prepared in state |a at the initial time and the occupations to the excited states are small in the whole process, we have ∂t σab = i∆0 σab + √ 2πigba (Eˆ+ eikC z + Eˆ− e−ikC z ) + i(Ω+ eikC z + Ω− e−ikC z )σac . (A.5) For the slowly varying operators σab = eikC z σab,+ + e−ikC z σab,− , (A.6) we have a simplicit expression for the evolution of σab as ∂t σab,± = i∆0 σab,± + √ 2πigba Eˆ± + iΩ± σac , 116 (A.7) A.1. Atomic Operators which is without the presence of phase terms e±ikC z . As σac appears in the r.h.s. of the above evolution equation, next we write down the evolution equation for σac similarly ∂t σac = i[H, σac ] √ = iδσcc − 2πigba σbc (Eˆ+ eikC z + Eˆ− e−ikC z ) √ + 2πigdc (Eˆ+† e−ikC z + Eˆ−† eikC z )σad +i(Ω†+ e−ikC z + Ω†− eikC z )σab . (A.8) In the evolution process of the dark-state polaritons, the excitations to state |b is negligible due to a destructive interference effect of two excitation paths (the EIT effect). This gives us σbc ∂t σac = iδσac + and √ √ 2πigdc Eˆ+† σad,+ + 2πigdc Eˆ−† σad,− +iΩ†+ σab,+ + iΩ†− σab,− (A.9) with σad = eikC z σad,+ + e−ikC z σad,− (A.10) σab = eikC z σab,+ + e−ikC z σab,− . (A.11) and Eq. (A.9) implies that to obtain the expression for σac , we need to solve 117 Appendix A. Derivation of Nonlinear Evolution Equation for Single-Species Photons σad first. From the evolution equation of σad ∂t σad = i[H, σad ] √ 2πi(gba σbd − gdc σac )(Eˆ+ eikC z + Eˆ− e−ikC z ) √ ≈ i∆p σad + 2πigdc σac (Eˆ+ eikC z + Eˆ− e−ikC z ), (A.12) = i∆p σad − we straightforwardly write down the equation ∂t σad,± = i∆p σad,± + √ 2πigdc σac Eˆ± . (A.13) To solve the Maxwell-Bloch Eq. (A.2), we also need to know the expression for σcd . Similarly from the evolution equation of σcd ∂t σcd = i[H, σcd ] = −iδσcd + i∆p σcd + √ 2πigdc (σcc − σdd )(Eˆ+ eikC z + Eˆ− e−ikC z ) −i(Ω+ eikC z + Ω− e−ikC z )σbd √ ≈ i(∆p − δ)σcd − 2πigdc σcc (Eˆ+ eikC z + Eˆ− e−ikC z ), (A.14) we have ∂t σcd,± = i(∆p − δ)σcd,± + √ 2πigdc σcc Eˆ± . (A.15) The assumptions that σab , σad and σcd as slowly varing operators, i.e., ∂t σab 0, ∂t σad 118 0, ∂t σcd 0, A.1. Atomic Operators give us their expressions as σab,± σad,± σcd,± √ 2πgba ˆ Ω± = − σac , E± − ∆0 ∆0 √ 2πgdc σac Eˆ± = − , ∆p √ 2πgdc σcc Eˆ± = − . ∆p − δ (A.16) (A.17) (A.18) In Eq. (A.9), ∂t σac equals to a mixing part of σab , σad , and σcd . Then with Eqs. (A.16), (A.17), and (A.18), we have ∂t σac √ 2πigba † ˆ i(δ∆0 − Ω20 ) σac − (Ω+ E+ + Ω†− Eˆ− ) = ∆0 ∆0 2πigdc ˆ † − (E+ σac Eˆ+ + Eˆ−† σac Eˆ− ). ∆p (A.19) Here Ω20 = |Ω+ |2 + |Ω− |2 . With ∂t2 σac = and ∂t σac = √ 2πgba (Ω†+ ∂t Eˆ+ + Ω†− ∂t Eˆ− ) , (δ∆0 − Ω20 ) (A.20) the atomic operator σac is given by σac √ √ 2πgba (Ω†+ Eˆ+ + Ω†− Eˆ− ) gba ∆0 (Ω†+ ∂t Eˆ+ + Ω†− ∂t Eˆ− ) = − 2πi + (δ∆0 − Ω20 )2 δ∆0 − Ω20 2πgdc ∆0 + (Eˆ † σac Eˆ+ + Eˆ−† σac Eˆ− ). (A.21) ∆p (δ∆0 − Ω20 ) + In our scheme, we focus on the regime δ∆0 Ω20 , which indicates that the first and third terms in the r.h.s. of Eq. (A.21) are much smaller than the second term. Therefore, we substitute the second term into the r.h.s. of 119 Appendix A. Derivation of Nonlinear Evolution Equation for Single-Species Photons Eq. (A.21) and we have σac √ √ gba ∆0 (Ω†+ ∂t Eˆ+ + Ω†− ∂t Eˆ− ) 2πgba (Ω†+ Eˆ+ + Ω†− Eˆ− ) = − 2πi + (δ∆0 − Ω20 )2 δ∆0 − Ω20 √ 2π 2πgba ∆0 gdc + (Eˆ † Eˆ+ + Eˆ−† Eˆ− )(Ω†+ Eˆ+ + Ω†− Eˆ− ). (A.22) ∆p (δ∆0 − Ω20 )2 + Consequently, with the expression for σac , we can write down the expressions for σab and σcd as σab,± √ √ gba Ω± (Ω†+ ∂t Eˆ+ + Ω†− ∂t Eˆ− ) 2πgba ˆ E± + 2πi = − ∆0 (δ∆0 − Ω20 )2 √ 2πgba Ω± (Ω†+ Eˆ+ + Ω†− Eˆ− ) − ∆0 (δ∆0 − Ω20 ) √ 2π 2πgba gdc Ω± ˆ † ˆ (E+ E+ + Eˆ−† Eˆ− )(Ω†+ Eˆ+ + Ω†− Eˆ− )(A.23) − 2 ∆p (δ∆0 − Ω0 ) and σcd,± A.2 √ gdc (Ω+ Eˆ+† + Ω− Eˆ−† )(Ω†+ Eˆ+ + Ω†− Eˆ− )Eˆ± 2π 2πgba . =− (∆p − δ)(δ∆0 − Ω20 )2 (A.24) Quantum Light Evolution We define the polaritonic operator in the usual slow-light way: √ Ψ± = cos θEˆ± − sin θ 2πnz σca , cos θ = sin θ = By taking the limit sin θ Ω± , + 2πgba nz √ gba 2πnz . Ω2± + 2πgba nz Ω2± (A.25) which means that the excitations are mostly in the spin-wave form, we get a simplicit expression for the polaritonic 120 A.2. Quantum Light Evolution operator √ Ψ± = gba 2πnz Eˆ± /Ω± . (A.26) From the Maxwell-Bloch equation (A.2) and the expressions for σab and σcd , we know the evolution equation for polaritons as ( 1∂ ∂ 2πigba nz ± )Ψ± = −i∆ωΨ± − Ψ± v ∂t ∂z v∆0 2 2πgba nz Ω2 (∂t Ψ+ + ∂t Ψ− ) 2πigba nz Ω2 (Ψ+ + Ψ− ) − − v(δ∆0 − 2Ω2 )2 v∆0 (δ∆0 − 2Ω2 ) Ω4 (Ψ†+ Ψ+ + Ψ†− Ψ− )(Ψ+ + Ψ− ) 2πigdc − v∆p (δ∆0 − 2Ω2 )2 − 2πigdc Ω4 (Ψ†+ + Ψ†− )(Ψ+ + Ψ− )Ψ± . v(∆p − δ)(δ∆0 − 2Ω2 )2 (A.27) For a symmetric combination Ψ and an anti-symmetric combination A of Ψ+ and Ψ− : Ψ= Ψ+ + Ψ− Ψ+ − Ψ− ,A = , 2 (A.28) their evolutions are determined by 1∂ ∂ Ψ+ A v ∂t ∂z 2 4πgba nz Ω2 ∂t Ψ 2πigba nz 4πigba nz Ω2 Ψ Ψ− = −i∆ωΨ − − v∆0 v(δ∆0 − 2Ω2 )2 v∆0 (δ∆0 − 2Ω2 ) 2 8πigdc Ω4 (Ψ† Ψ + A† A)Ψ 8πigdc Ω4 Ψ† ΨΨ − − , (A.29) v∆p (δ∆0 − 2Ω2 )2 v(∆p − δ)(δ∆0 − 2Ω2 )2 2 1∂ ∂ 2πigba nz 8πigdc Ω4 Ψ† ΨA A + Ψ = −i∆ωA − A− . (A.30) v ∂t ∂z v∆0 v(∆p − δ)(δ∆0 − 2Ω2 )2 For the stationary polaritons, the phase matching mechanism allows us to adiabatically eliminate A, leading to the result A≈ iv∆0 ∂z Ψ v∆ω∆0 + 2πgba nz 121 (A.31) Appendix A. Derivation of Nonlinear Evolution Equation for Single-Species Photons from Eq. (A.30) and consequently a nonlinear evolution equation for Ψ 2πδgba nz vvg ∆0 ∂ 2Ψ )Ψ + 2 nz z v(δ∆0 − 2Ω ) v∆ω∆0 + 2πgba vg Ω4 (2∆p − δ) 8πgdc Ψ† Ψ2 + noise (A.32) + v(δ∆0 − 2Ω2 )2 ∆p (∆p − δ) i∂t Ψ = vg (∆ω + from Eq. (A.29). Here the group velocity of light in the nonlinear medium is given by vg nz ). Since the typical value of vg in a slow(vΩ2 )/(πgba light regime is around several tens to hundreds meters per second, we have employed the approximation v + vg (A.32). 122 v to simplify the expression in Eq. Appendix B Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies B.1 Atomic Operators In the Schrödinger picture, the Hamiltonian describing transitions of twospecies four-level atoms driven by two pairs of quantum pulses with different frequencies and two pairs of control beams is H = H↑ + H↓ , where for each 123 Appendix B. Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies species s =↑, ↓, the Hamiltonian is Hs = − nsz dz{−(ωcc,s + δs )σc,s;c,s +(−ωQ,s + ∆s )σb,s;b,s √ +(−ωcc,s − ωQ,s + ∆ss )σd,s;d,s + [ 2π(gs σb,s;a,s + gss σd,s;c,s ) (Eˆs ,+ eikQ,s z e−iωQ,s t + Eˆs ,− e−ikQ,s z e−iωQ,s t ) × s +(Ωs,+ (t)eikC,s z e−iωC,s t + Ωs,− (t)e−ikC,s z e−iωC,s t )σc,s;b,s +h.c.]} (B.1) with the collective and continuous operators σi,s;j,s = |i, s j, s|. The quantum field is composed of two counter-propagating components Eˆs,± = k ak e±i(k−kQ,s )z e−i(ωk −ωQ,s )t . Similarly, the control field is expressed by Ωs,± (z, t) = k f e±i(k−kC,s )z e−i(ωk −ωC,s )t , where Ωs,± is a slowly varying op- erator of z and t. kQ,s and kC,s denote the wavevectors corresponding to central frequencies ωQ,s and ωC,s of Es,± and Ωs,± , respectively. nsz is the atomic density. ωcc,s denotes the level shifting of level |c, s . gs and gss are coupling strengths between the quantum fields and atoms, while δs is a two-photon detuning, and ∆s and ∆ss are one-photon detunings for corresponding transitions. Next by choosing nsz dz(ωcc,s σc,s;c,s + ωQ,s σb,s;b,s + (ωcc,s + ωQ,s )σd,s;d,s ) (B.2) H0 = s 124 Appendix B. Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies A, we next write down the evolution equations for atomic operators as ∂t σa,s;b,s = i[H, σa,s;b,s ] = −i[−∆s σa,s;b,s + √ 2πgs (σb,s;b,s − σa,s;a,s ) ×(Eˆs,+ eikQ,s z + Eˆs,− e−ikQ,s z ) √ + 2πgs (σb,s;b,s − σa,s;a,s ) ×(Eˆs,+ eikQ,s z + Eˆs,− e−ikQ,s z )ei(wQ,s −wQ,s )z −(Ωs,+ eikC,s z + Ωs,− e−ikC,s z )σa,s;c,s −(Ωs,+ eikC,s z + Ωs,− e−ikC,s z )ei(wC,s −wC,s )z σa,s;c,s ]. (B.6) With the assumptions of σa,s;a,s 1, σb,s;b,s 0, (B.7) which means that the atoms are prepared in state |a, s at the initial time and the occupations to the excited states are small in the whole process, we have √ ∂t σa,s;b,s,± = i∆s σa,s;b,s,± + 2πigs Eˆs,± √ + 2πigs Eˆs,± ei(kQ,s −kQ,s )z ei(wQ,s −wQ,s )z + iΩs,± σa,s;c,s +iΩs,± σa,s;c,s ei(kC,s −kC,s )z ei(wC,s −wC,s )z , (B.8) where we have introduced slowly varying operators σa,s;b,s = eikQ,s z σa,s;b,s,+ + e−ikQ,s z σa,s;b,s,− . (B.9) As σa,s;c,s appears in the r.h.s. of the evolution equation (B.8), we treat 126 B.1. Atomic Operators the evolution equation of σa,s;c,s similarly: ∂t σa,s;c,s = i[H, σa,s;c,s ] = iδs σc,s;c,s − √ gs σb,s;c,s ei(ωQ,s −ωQ,s )t 2πi s ×(Eˆs,+ eikQ,s z + Eˆs,− e−ikQ,s z ) √ † † + 2πi gs ei(ωQ,s −ωQ,s )t (Eˆs,+ e−ikQ,s z + Eˆs,− eikQ,s z )σa,s;d,s s † +i(Ωs,+ e−ikC,s z + Ω†s,− eikC,s z )σa,s;b,s + √ 2πigs ei(ωQ,s −ωQ,s )t † † ×(Eˆs,+ e−ikQ,s z + Eˆs,− eikQ,s z )σa,s;d,s . (B.10) In the evolution process of the dark-state polaritons, the excitations to state |b, s is negligible due to a destructive interference effect of two excitation paths (the EIT effect). This gives us σb,s;c,s and √ √ † † ∂t σa,s;c,s,± = iδs σc,s;c,s + 2πigs Eˆs,+ σa,s;d,s,+ + 2πigs Eˆs,− σa,s;d,s,− √ † σa,s;d,s,+ ei(kQ,s −kQ,s )z + 2πigs ei(ωQ,s −ωQ,s )t Eˆs,+ √ † + 2πigs ei(ωQ,s −ωQ,s )t Eˆs,− σa,s;d,s,− e−i(kQ,s −kQ,s )z +iΩ†s,+ σa,s;b,s,+ + iΩ†s,− σa,s;b,s,− (B.11) with σa,s;d,s = eikQ,s z σa,s;d,s,+ + e−ikQ,s z σa,s;d,s,− (B.12) σa,s;b,s = eikQ,s z σa,s;b,s,+ + e−ikQ,s z σa,s;b,s,− . (B.13) and 127 Appendix B. Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies Similarly, we have √ ∂t σa,s;d,s,± = i∆ss σa,s;d,s,± + 2πigs σa,s;c,s Eˆs,± √ + 2πigs σa,s;c,s Eˆs,± ei(ωQ,s −ωQ,s )t e±i(kQ,s −kQ,s )z (B.14) and √ ∂t σc,s;d,s,± = −iδs σc,s;d,s, + i∆ss σc,s;d,s,± + 2πigs σc,s;c,s Eˆs,± √ + 2πigs σc,s;c,s Eˆs,± ei(ωQ,s −ωQ,s )t e±i(kQ,s −kQ,s )z . (B.15) The assumptions that σa,s;b,s , σa,s;d,s and σc,s;d,s are slowly varying operators, i.e., ∂t σa,s;b,s 0, ∂t σa,s;d,s 0, ∂t σc,s;d,s give us their expressions as √ Ωs,± 2πgs ˆ Es,± − σa,s;c,s ∆s ∆s √ 2πgs − Eˆs,± ei(ωQ,s −ωQ,s )t e±i(kQ,s −kQ,s )z ∆s − ωQ,s + ωQ,s Ωs,± − σa,s;c,s ei(ωC,s −ωC,s )t e±i(kC,s −kC,s )z ,(B.16) ∆s − ωQ,s + ωQ,s σa,s;b,s,± = − σa,s;d,s,± √ 2πgs σa,s;c,s Eˆs,± = − ∆ss √ 2πgs σa,s;c,s Eˆs,± i(ωQ,s −ωQ,s )t ±i(kQ,s −kQ,s )z − e e , (B.17) ∆ss − ωQ,s + ωQ,s 128 B.1. Atomic Operators σc,s;d,s,± √ √ 2πgs σc,s;a,a σa,s;c,s Eˆs,± 2πgs σc,s;a,s σa,s;c,s Eˆs,± = − − ∆ss − δs ∆ss − δs − ωQ,s + ωQ,s i(ωQ,s −ωQ,s )t ±i(kQ,s −kQ,s )z ×e e . (B.18) We recall that in Eq. (B.11), ∂t σa,s;c,s equals to a mixing part of σa,s;b,s , σa,s;d,s , and σc,s;d,s . Replacing them by Eqs. (B.16), (B.17), and (B.18), Eq. (B.11) becomes ∂t σa,s;c,s √ i(δs ∆s − Ω2s ) 2πigs † ˆ = σa,s;c,s − (Ωs,+ Es,+ + Ω†s,− Eˆs,− ) ∆s ∆s √ 2πigs (Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) − ∆s − ωQ,s + ωQ,s ×ei(ωQ,s −ωQ,s )t e±i(kQ,s −kQ,s )z 2πigs2 ˆ † † − (Es,+ σa,s;c,s Eˆs,+ + Eˆs,− σa,s;c,s Eˆs,− ) ∆ss 2πigs2 − ∆ss − ωQ,s + ωQ,s † † ×(Eˆs,+ σa,s;c,s Eˆs,+ + Eˆs,− σa,s;c,s Eˆs,− ), (B.19) where we have introduced Ωs as Ω2s = |Ωs,+ |2 + |Ωs,− |2 . With ∂t2 σa,s;c,s = and ∂t σa,s;c,s = √ 2πgs (Ω†s,+ ∂t Eˆs,+ + Ω†s,− ∂t Eˆs,− ) , δs ∆s − Ω2s (B.20) we have σa,s;c,s √ ∆s gs (Ω†s,+ ∂t Eˆs,+ + Ω†s,− ∂t Eˆs,− ) = − 2πi (δs ∆s − Ω2s )2 √ 2πgs (Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) + δs ∆s − Ω2s 2π∆s gs2 † (Eˆ † σa,s;c,s Eˆs,+ + Eˆs,− σa,s;c,s Eˆs,− ) + ∆ss (δs ∆s − Ω2s ) s,+ 2π∆s gs2 + (∆ss − ωQ,s + ωQ,s )(δs ∆s − Ω2s ) † † ×(Eˆs,+ σa,s;c,s Eˆs,+ + Eˆs,− σa,s;c,s Eˆs,− ). (B.21) 129 Appendix B. Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies In our scheme, we focus on the regime δs ∆s Ω2s , which indicates that the first, third and fourth terms in the r.h.s. of Eq. (B.21) are much smaller than the second term. Therefore, we substitute the second term into the r.h.s. of Eq. (B.21) and we have √ ∆s gs (Ω†s,+ ∂t Eˆs,+ + Ω†s,− ∂t Eˆs,− ) σa,s;c,s = − 2πi (δs ∆s − Ω2s )2 2πgs (Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) + δs ∆s − Ω2s √ 2π 2π∆s gs3 † ˆ † ˆ + (Eˆs,+ Es,+ + Eˆs,− Es,− )(Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) 2 ∆ss (δs ∆s − Ωs ) √ 2π 2π∆s gs gs2 + (∆ss − ωQ,s + ωQ,s )(δs ∆s − Ω2s )2 † ˆ † ˆ ×(Eˆs,+ Es,+ + Eˆs,− Es,− )(Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ). (B.22) With σa,s;c,s , we can write down the expressions for σa,s;b,s and σc,s;d,s as σa,s;b,s,± √ √ gs Ωs,± (Ω†s,+ ∂t Eˆs,+ + Ω†s,− ∂t Eˆs,− ) 2πgs ˆ Es,± + 2πi = − ∆s (δs ∆s − Ω2s )2 √ 2πgs Ωs,± (Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) − ∆s (δs ∆s − Ω2s ) √ 2π 2πgs3 Ωs,± ˆ † ˆ † ˆ − (Es,+ Es,+ + Eˆs,− Es,− )(Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) 2 ∆ss (δs ∆s − Ωs ) √ 2π 2πgs gs2 Ωs,± − (∆ss − ωQ,s + ωQ,s )(δs ∆s − Ω2s )2 † ˆ † ˆ ×(Eˆs,+ Es,+ + Eˆs,− Es,− )(Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− ) (B.23) 130 B.2. Quantum Light Evolution and 2πgs σc,s;a,s σa,s;c,s Eˆs,± ∆ss − δs 2πgs σc,s;a,s σa,s;c,s Eˆs,± i(ωQ,s −ωQ,s )t ±i(kQ,s −kQ,s )z e − e ∆ss − δs − ωQ,s + ωQ,s √ 2π 2πgs3 † † = − (Ωs,+ Eˆs,+ + Ωs,− Eˆs,− ) 2 (∆ss − δs )(δs ∆s − Ωs ) ×(Ω†s,+ Eˆs,+ + Ω†s,− Eˆs,− )Eˆs,± . (B.24) σc,s;d,s,± = − B.2 Quantum Light Evolution We define the polaritonic operator as Ψs,± = gs 2πnsz Eˆs,± /Ωs,± (B.25) in the limit that the excitations are mostly in the spin-wave form. By employing the Maxwell-Bloch Eq. (B.5) and the expressions for σa,s;b,s and σc,s;d,s , we know the evolution equation for polaritons as ∂ ∂ ± )Ψs,± ν ∂t ∂z 2πigs2 nsz = −i∆ωs Ψs,± − Ψs,± v∆s 2πgs2 nsz Ω2s (∂t Ψs,+ + ∂t Ψs,− ) 2πigs2 nsz Ω2s (Ψs,+ + Ψs,− ) − − v(δs ∆s − 2Ω2s )2 v∆s (δs ∆s − 2Ω2s ) ( 2πigs2 Ω2s (Ψ†s,+ Ψs,+ + Ψ†s,− Ψs,− )(Ψs,+ + Ψs,− ) − v∆ss (δs ∆s − 2Ω2s )2 2πigs2 Ω2s (Ψ†s,+ + Ψ†s,− )(Ψs,+ + Ψs,− )Ψs,± − v(∆ss − δs )(δs ∆s − 2Ω2s )2 − † 2πigs3 Ω2s (Ψs,+ Ψs,+ + Ψ†s,− Ψs,− )(Ψs,+ + Ψs,− ) . gs v(∆ss − ωQ,s + ωQ,s )(δs ∆s − 2Ω2s )2 131 (B.26) Appendix B. Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies For a symmetric combination Ψs and an anti-symmetric combination As of Ψs,+ and Ψs,− : Ψs = Ψs,+ + Ψs,− Ψs,+ − Ψs,− , As = , 2 (B.27) their evolutions are determined by ∂ ∂ Ψs + As ν ∂t ∂z 2πgs2 nsz Ω2s ∂t Ψs 2πigs2 nsz 4πigs2 nsz Ω2s Ψs Ψs − = −i∆ωs Ψs − − v∆s v(δs ∆s − 2Ω2s )2 v∆s (δs ∆s − 2Ω2s ) 8πigs2 Ω4s (Ψ†s Ψs + A†s As )Ψs 8πigs2 Ω4s Ψ†s Ψs Ψs − − v∆ss (δs ∆s − 2Ω2s )2 v(∆ss − δs )(δs ∆s − 2Ω2s )2 − 8πigs3 Ω4s (Ψs† Ψs + As† As )Ψs gs v(∆ss − ωQ,s + ωQ,s )(δs ∆s − 2Ω2s )2 (B.28) and ∂ ∂ 2πigs2 nsz 2πigs2 Ω2s Ψ†s Ψs As As − Ψs = −i∆ωs As − . As − ν ∂t ∂z v∆s v(∆ss − δs )(δs ∆s − 2Ω2s )2 (B.29) For the stationary polaritons, the phase matching mechanism allows us to adiabatically eliminate As , leading to the following nonlinear evolution equation for Ψs : i∂t Ψs = ∆s ννs 2πgs2 νs † 2πgs3 νs † (∂ Ψ ) + Ψ Ψ + Ψ Ψ Ψs Ψs + noise. (B.30) s s s 2πgs2 nsz z ν∆ss s νgs ∆ss s 132 Appendix C Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations C.1 Aomtic operators The Hamiltonian for the system where single species of multi-level atoms interact with two pairs of differently polarized quantum lights and two pairs of control beams reads H = − H∆ = nz dz(H∆ + HQ + HC ), where ∆s σb,s;b,s + s HQ = ∆ss σd,s,s ;d,s,s , gss σd,s,s ;c,s )Es + h.c., (gs σb,s;a + s HC = (C.1) s,s (C.2) s σc,s;b,s Ωs + Ω0 σc,↑;c,↓ + h.c (C.3) s Here, the quantum filed is composed of two counter-propagating components, Es = Es,+ (z, t)eikQ,s z + Es,− (z, t)e−ikQ,s z . Similarly, the control field 133 Appendix C. Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations is given by Ωs = Ωs,+ (z, t)eikC,s z +Ωs,− (z, t)e−ikC,s z . kQ,s and kC,s denote the wavevectors corresponding to central frequencies ωQ,s and ωC,s of Es,± and Ωs,± , respectively. The collective and continuous operators σµ;ν ≡ σµ;ν (z, t) describes the average of the flip operators |µ ν| over atoms in a small region around z. nz is the atomic density. As a Raman transition, we have the laser strength Ω0 Ωs . gs and gss are coupling strengths between the quantum fields and atoms, while ∆s and ∆ss are one-photon detunings for corresponding transitions. For simplicity, we assume that gs = gss = g. For the quantum light Es,± = ∂t Es,± ± v∂z Es,± = k ak e±i(k−kQ,s )z e−i(ω−ωQ,s )t , we have (∂t ak )e±i(k−kQ,s )z e−i(ω−ωQ,s )t k = i[H, Es,± ], (C.4) where v = ωQ,s /kQ,s is the light speed in an empty medium. With σc,s ;d,s,s )e∓ikQ,s z , [H, Es,± ] = nz g(σa;b,s + (C.5) s the Maxwell-Bloch equation becomes (∂t ± v∂z )Es,± = inz g(σa;b,s,± + σc,s ;d,s,s ,± ). (C.6) s Similar to the case of solving one-species polaritons in Appendix A, we 134 C.1. Aomtic operators write down the evolution equations for atomic operators as ∂t σa;b,s = i[H, σa;b,s ] = i∆s σa;b,s + igEs + iΩs σa;c,s , (C.7) ∂t σc,s;d,s,s = i[H, σc,s;d,s,s ] = i∆ss σc,s;d,s,s + igσc,s;c,s Es , (C.8) ∂t σc,s;d,s,s = i[H, σc,s;d,s,s ] = i∆ss σc,s;d,s,s +igσc,s;c,s Es + igσc,s;c,s Es , (C.9) ∂t σa;c,s = i[H, σa;c,s ] = igEs† σa;d,s,s + igEs† σa;d,s,s +iΩs σa;b,s + iΩ0 σa;c,s , ∂t σa;d,s,s = i[H, σa;d,s,s ] = i∆ss σa;d,s,s + igσa;c,s Es , (C.10) (C.11) ∂t σa;d,s,s = i[H, σa;d,s,s ] = i∆ss σa;d,s,s +igσa;c,s Es + igσa;c,s Es . (C.12) By introducing the slowly varying operators σµ;ν = σµ;ν,+ eikQ,s z + σµ;ν,− e−ikQ,s z (C.13) ∂t σµ;ν,± = 0, (C.14) and considering we have the expressions for the atomic operators: gEs,± Ωs,± − σa;c,s , ∆s ∆s g − σc,s;a σa;c,s Es,± , ∆ss g g − σc,s;a σa;c,s Es,± − σc,s;a σa;c,s Es,± , ∆ss ∆ss g − σa;c,s Es,± , σa;d,s,s,± ∆ss g g − σa;c,s Es,± − σa;c,s Es,± . ∆ss ∆ss σa;b,s,± = − σc,s;d,s,s,± = σc,s;d,s,s,± = σa;d,s,s,± = = 135 (C.15) (C.16) (C.17) (C.18) Appendix C. Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations Replacing the atomic operators appearing in Eq. (C.10) by the above expressions, Eq. (C.10) becomes g2 † † = −i (Es,+ σa;c,s Es,+ + Es,− σa;c,s Es,− ) ∆ss ig † † − (E σa;c,s Es,+ + Es,− σa;c,s Es,− ) ∆ss s,+ ig † † (E σa;c,s Es,+ + Es,− − σa;c,s Es,− ) ∆ss s,+ ig − (Ωs,+ Es,+ + Ωs,− Es,− ) ∆s ∂t σa;c,s − 2iΩs σa;c,s + iΩ0 σa;c,s , ∆s where a new quantity Ωs with Ωs = 2Ω2s,+ + 2Ω2s,− (C.19) has been introduced. With ∂t2 σa;c,s = and ∂t σa;c,s = − g (Ωs,+ ∂t Es,+ 2Ωs + Ωs,− ∂t Es,− ), (C.20) we have σa;c,s = − i∆s g 2 (Ωs,+ ∂t Es,+ + Ωs,− ∂t Es,− ) 2Ωs 2Ωs g ∆s † † + (Es,+ Es,+ + Es,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− ) ∆ 4Ω ss s + + − g3 2 4Ωs Ωs g 4Ωs ∆s † † (E Es,+ + Es,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− ) ∆ss s,+ ∆s † † (E Es,+ + Es,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− ) ∆ss s,+ g (Ωs,+ Es,+ + Ωs,− Es,− ) 2Ωs g∆s Ω0 − 2 (Ωs,+ Es,+ + Ωs,− Es,− ). 4Ωs Ωs (C.21) Since we have the expression for σa;c,s , we can write down the expressions 136 C.1. Aomtic operators for σa;b,s , σc,s;d,s,s , and σc,s;d,s,s as gEs,± igΩs,± + (Ωs,+ ∂t Es,+ + Ωs,− ∂t Es,− ) ∆s 4Ωs g Ωs,± † † − (Es,+ Es,+ + Es,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− ) 4∆ss Ωs g Ωs,± † † − 2 (Es,+ Es,+ + Es,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− ) 4∆ss Ωs Ωs g Ωs,± † † − (Es,+ Es,+ + Es,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− ) 4∆ss Ωs gΩs,± + (Ωs,+ Es,+ + Ωs,− Es,− ) 2∆s Ωs gΩs,± Ω0 + (C.22) 2 (Ωs,+ Es,+ + Ωs,− Es,− ), 4Ωs Ωs σa;b,s,± = − σc,s;d,s,s,± = − g3 † (Ωs,+ Es,+ 4∆ss Ωs † + Ωs,− Es,− ) ×(Ωs,+ Es,+ + Ωs,− Es,− )Es,± , (C.23) and σc,s;d,s,s,± = − − g3 † (Ωs,+ Es,+ 4∆ss Ωs g † + Ωs,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− )Es,± † 2 (Ωs,+ Es,+ 4∆ss Ωs Ωs † + Ωs,− Es,− ) ×(Ωs,+ Es,+ + Ωs,− Es,− )Es,± . 137 (C.24) Appendix C. Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations C.2 Quantum Light Evolution We define the polariton operators as [43, 127] Ψs,+ √ √ nz g nz g = Es,+ , Ψs,− = Es,− . Ωs,+ Ωs,− (C.25) By employing the Maxwell-Bloch Eq. (C.6) and the expressions for σa;b,s , σc,s;d,s,s , and σc,s;d,s,s , we know the evolution equation for polaritons as (∂t ± v∂z )Ψs,± nz g 2 inz g 2 Ψs,± − (Ωs,+ ∂t Ψs,+ + Ωs,− ∂t Ψs,− ) ∆s 4Ωs ig † † 2 2 − (Ωs,+ Ψs,+ Ψs,+ + Ωs,− Ψs,− Ψs,− )(Ωs,+ Ψs,+ + Ωs,− Ψs,− ) 4∆ss Ωs ig † † − 2 (Ωs,+ Ωs,+ Ψs,+ Ψs,+ + Ωs,− Ωs,− Ψs,− Ψs,− ) 4∆ss Ωs Ωs ×(Ω2s,+ Ψs,+ + Ω2s,− Ψs,− ) = − ig † † 2 2 (Ωs,+ Ψs,+ Ψs,+ + Ωs,− Ψs,− Ψs,− )(Ωs,+ Ψs,+ + Ωs,− Ψs,− ) 4∆ss Ωs inz g inz g Ω0 2 2 + (Ω Ψ + Ω Ψ ) + s,+ s,+ s,− s,− 2 (Ωs,+ Ψs,+ + Ωs,− Ψs,− ) 2∆s Ωs 4Ωs Ωs ig † † 2 2 − (Ωs,+ Ψs,+ + Ωs,− Ψs,− )(Ωs,+ Ψs,+ + Ωs,− Ψs,− )Ψs,± 4∆ss Ωs ig † † 2 2 − (Ωs,+ Ψs,+ + Ωs,− Ψs,− )(Ωs,+ Ψs,+ + Ωs,− Ψs,− )Ψs,± 4∆ss Ωs ig Ωs,± − (Ω2s,+ Ψ†s,+ + Ω2s,− Ψ†s,− ) 2 4∆ss Ωs Ωs Ωs,± ×(Ω2s,+ Ψs,+ + Ω2s,− Ψs,− )Ψs,± . (C.26) − Next, we introduce a symmetric combination Ψs and an anti-symmetric 138 C.2. Quantum Light Evolution combination As of Ψs,+ and Ψs,− as: Ψs = αs,+ Ψs,+ + αs,− Ψs,− , As = Ψs,+ − Ψs,− , (C.27) where αs,+ = Ω2s,+ Ω2s,− , α = s,− Ω2s,+ + Ω2s,− Ω2s,+ + Ω2s,− (C.28) have been introduced to characterize the imbalance between Ωs,+ and Ωs,− . We note here that their commutation relation still holds: [Ψ† (z), A(z )] = [Ω+ E+† (z) + Ω− E−† (z), E+ (z ) E− (z ) − ] = 0, Ω+ Ω− (C.29) and they obey the equations ∂t Ψs + v∂z (αs,+ Ψs − αs,− Ψs + 2αs,+ αs,− As ) tan2 θs tan2 θs 2g † ∂t Ψs + i Ω0 Ψ_s − i Ψ Ψs Ψs 2 ∆ss s g2 −i _ [2 + cos(ϕ_s − ϕs )]Ψ†_s Ψ_s Ψs , ∆s s ∂t As + v∂z (2Ψs + αs,− As − αs,+ As ) − nz g ig † ig As − Ψs Ψs As − _ Ψ†_s Ψ_s As ∆s ∆ss ∆s s _ ig Ω s ,+ † Ψ_ Ψs A_s . − _ ∆s s Ωs,+ s (C.30) −i (C.31) In the above two equations, we have introduced two angular parameters θs and ϕs as tan2 ϕs = Ω2s,− /Ω2s,+ , tan2 θs = g nz /Ωs . (C.32) The phase matching condition allows us to adiabatically eliminate As , lead- 139 Appendix C. Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations ing to the result As i∆s v∂z (2Ψs + αs,− As − αs,+ As ) nz g (C.33) from Eq. (C.31), and consequently a nonlinear evolution equation for Ψs : i ∂t Ψs = sin2 (2ϕs )vs2 ∆s Ωs ∂z2 Ψs − 2i vs cos 2ϕs ∂z Ψs Ωs † + Ω0 Ψ s + ΨΨ ∆ss nz s s _ Ωs [2 + cos(ϕ_s − ϕs )] † _ + Ψ_s Ψ s Ψs + noise ∆s_s nz from Eq. (C.30). 140 (C.34) C.2. Quantum Light Evolution 141 [...]... Ψ_ Ψs A_ − _ s ∆s s Ωs,+ s (C .30 ) −i (C .31 ) In the above two equations, we have introduced two angular parameters θs and ϕs as 2 tan2 ϕs = Ω2 /Ω2 , tan2 θs = g 2 nz /Ωs s,+ s,− (C .32 ) The phase matching condition allows us to adiabatically eliminate As , lead- 139 Appendix C Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations ing to the result As i∆s v∂z (2Ψs... σc,↑;c,↓ + h.c (C .3) s Here, the quantum filed is composed of two counter-propagating components, Es = Es,+ (z, t)eikQ,s z + Es,− (z, t)e−ikQ,s z Similarly, the control field 133 Appendix C Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations is given by Ωs = Ωs,+ (z, t)eikC,s z +Ωs,− (z, t)e−ikC,s z kQ,s and kC,s denote the wavevectors corresponding to central... describes the propagation of quantum lights in the nonlinear medium, can be written as 1 ∂ ∂ ˆ ± )Es,± (z, t) ν ∂t ∂z √ 2πigs ns z ˆ = −i∆ωs Es,± (z, t) + v ×(σa,s;b,s,+ (z, t) + σc,s;d,s,+ (z, t)) ( (B.5) Similar to the case of solving single-component polaritons in Appendix 125 Appendix B Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies A, we next write down... σc,s;d,s,s,± = − g3 † 4 (Ωs,+ Es,+ 4∆ss Ωs † + Ωs,− Es,− ) ×(Ωs,+ Es,+ + Ωs,− Es,− )Es,± , (C. 23) and σc,s;d,s,s,± = − − g3 † 4 (Ωs,+ Es,+ 4∆ss Ωs 3 g † + Ωs,− Es,− )(Ωs,+ Es,+ + Ωs,− Es,− )Es,± † 2 2 (Ωs,+ Es,+ 4∆ss Ωs Ωs † + Ωs,− Es,− ) ×(Ωs,+ Es,+ + Ωs,− Es,− )Es,± 137 (C.24) Appendix C Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations C.2 Quantum Light... 2πgs νs † 2πgs νs † (∂z Ψs ) + Ψs Ψs Ψs + Ψ Ψs Ψs + noise (B .30 ) 2 2πgs ns ν∆ss νgs ∆ss s z 132 Appendix C Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations C.1 Aomtic operators The Hamiltonian for the system where single species of multi-level atoms interact with two pairs of differently polarized quantum lights and two pairs of control beams reads H = − H∆... 2Ω2 )2 s − † 3 2πigs Ω2 (Ψs,+ Ψs,+ + Ψ† Ψs,− )(Ψs,+ + Ψs,− ) s s,− gs v(∆ss − ωQ,s + ωQ,s )(δs ∆s − 2Ω2 )2 s 131 (B.26) Appendix B Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies For a symmetric combination Ψs and an anti-symmetric combination As of Ψs,+ and Ψs,− : Ψs = Ψs,+ + Ψs,− Ψs,+ − Ψs,− , As = , 2 2 (B.27) their evolutions are determined by 1 ∂ ∂... ing to the result As i∆s v∂z (2Ψs + αs,− As − αs,+ As ) nz g 2 (C .33 ) from Eq (C .31 ), and consequently a nonlinear evolution equation for Ψs : i ∂t Ψs = 2 2 sin2 (2ϕs )vs ∆s 2 Ωs 2 ∂z Ψs − 2i vs cos 2ϕs ∂z Ψs 2 4 Ωs † 2 + Ω0 Ψ s + ΨΨ ∆ss nz s s _ 2 2 Ωs [2 + cos(ϕ_ − ϕs )] † _ s + Ψ_ Ψ s Ψs + noise s ∆s_ nz s from Eq (C .30 ) 140 (C .34 ) C.2 Quantum Light Evolution 141 ... σa;d,s,s,± ∆ss g g − σa;c,s Es,± − σa;c,s Es,± ∆ss ∆ss σa;b,s,± = − σc,s;d,s,s,± = σc,s;d,s,s,± = σa;d,s,s,± = = 135 (C.15) (C.16) (C.17) (C.18) Appendix C Derivation of Nonlinear Evolution Equation for Two-species Photons with Different Polarizations Replacing the atomic operators appearing in Eq (C.10) by the above expressions, Eq (C.10) becomes g2 † † = −i (Es,+ σa;c,s Es,+ + Es,− σa;c,s Es,− ) ∆ss... Es,− ) ˆ ˆ ˆ ×(Es,+ (B.21) s,− 129 Appendix B Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies In our scheme, we focus on the regime δs ∆s Ω2 , which indicates that the s first, third and fourth terms in the r.h.s of Eq (B.21) are much smaller than the second term Therefore, we substitute the second term into the r.h.s of Eq (B.21) and we have ˆ ˆ √ ∆s gs (Ω†... s s − † † 3 8πigs Ω4 (Ψs Ψs + As As )Ψs s gs v(∆ss − ωQ,s + ωQ,s )(δs ∆s − 2Ω2 )2 s (B.28) and 2 2 1 ∂ ∂ 2πigs ns 2πigs Ω2 Ψ† Ψs As z s s As − Ψs = −i∆ωs As − As − ν ∂t ∂z v∆s v(∆ss − δs )(δs ∆s − 2Ω2 )2 s (B.29) For the stationary polaritons, the phase matching mechanism allows us to adiabatically eliminate As , leading to the following nonlinear evolution equation for Ψs : i∂t Ψs = 2 3 ∆s ννs 2 . simplify the expression in Eq. (A .32 ). 122 Appendix B Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies B.1 Atomic Operators In the Schrödinger picture, the. (B.5) Similar to the case of solving single-component polaritons in Appendix 125 Appendix B. Derivation of Nonlinear Evolution Equation for Two-Species Photons with Different Frequencies A, we. shifting of level |c, s. g s and g ss are coupling strengths between the quantum fields and atoms, while δ s is a two-photon detuning, and ∆ s and ∆ ss are one- photon detunings for corresponding