Controlled Correlation and Squeezing in Pr þ Y SiO to Yield ...7+28201729...01 ~ ω1 ρð3Þ 21ð SÞ (E ) and ρ ð0Þ 00 ~ ω1 ρð1Þ 20 ~ ωS ρð2Þ 10 ~ ω1 ρð3Þ 20ðASÞ

1 Controlled Correlation and Squeezing in Pr3þ∶Y2SiO5 to Yield Correlated Light Beams

2 Changbiao Li,1 Zihai Jiang,1 Yiqi Zhang,1 Zhaoyang Zhang,1 Feng Wen,1

3 Haixia Chen,1 Yanpeng Zhang,1,* and Min Xiao2,3,†

4 1Key Laboratory for Physical Electronics and Devices of the Ministry of Education and Shaanxi Key Lab5 of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China6 2Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA7 3National Laboratory of Solid State Microstructures and School of Physics,8 Nanjing University, Nanjing 210093, China9 (Received 26 March 2016; revised manuscript received 23 November 2016)

10 We report the generation of twin beams by the parametric amplification four-wave mixing process and11 triplet beams by the parametric amplification six-wave mixing (PA SWM) process associated with the12 multiorder fluorescence signals in a Pr3þ∶Y2SiO5 crystal. The intensity noise correlation and intensity-13 difference squeezing result from the nonlinear gain, which can be well controlled by the polarized dressing14 effect. The correlation value at the resonant position increases due to the double dressing effect; however,15 such correlation decreases if the triple dressing effect works. Specifically, correlation and squeezing16 between Stokes and anti-Stokes signals can be also switched by the relative nonlinear phase shift. The17 generated triplet beams from the PA SWM process have potential applications in three-mode all-optical18 information processing that can be used in on-chip photonic devices.

DOI:

19 I. INTRODUCTION

20 As a further step towards integrated quantum photonic21 devices, it is necessary to miniaturize external light sources22 to improve their performance, such as their portability,23 stability, and multifunctionality. To achieve such light24 sources, methods that are based on traditional nonlinear25 optical crystals, especially domain-engineered quadratic26 nonlinear photonic crystals [1,2], were developed.27 Recently, a certain valuable extension of the path-entangled28 states to high-dimensional entanglement from a domain-29 cascaded lithium niobate crystal was reported [3]. On the30 other hand, rare-earth-doped crystals such as Pr3þ∶Y2SiO5

31 exhibit unique features in comparison to nonlinear crystals.32 In these kinds of doped crystals, the “atomlike” properties33 of the dopant can be kept, in which the atomic coherence34 can be induced easily when interacting with multiple laser35 beams. As for the atomic coherence effects in solid-state36 materials, recent research includes electromagnetically37 induced transparency [4,5], light velocity reduction and38 coherent storage [6–8], optical quantum computing [9,10],39 all-optical routing based on optical storage [11], and40 enhanced four-wave mixing (FWM) based on atomic41 coherence [12], to name a few. It is worth mentioning that42 following the latest developments in intensity squeezed43 light with FWM in an atomic vapor, one can use the

44multiwave mixing processes in such rare-earth-doped45crystals to generate entangled lights.46In this paper, we report the generation of correlated light47beams from a parametric amplification four-wave-mixing48(PA FWM) process and a PA six-wave-mixing (PA SWM)49process in a Pr3þ∶Y2SiO5 crystal. Two nonlinear cascade50optical processes are controlled by adjustable dressing51fields, and the results show that the degrees of the three-52mode correlation and squeezing are induced by the dressed53nonlinear gain. The correlation and squeezing can be54controlled by the relative nonlinear phase shift caused55by the dressing fields. On the other hand, the correlation56and squeezing of PA FWM can be switched to anticorre-57lation and antisqueezing. Such controllable properties have58potential applications in all-optical communication and59optical information processing on photonic chips.

60II. THEORETICAL MODEL

61The sample is a 0.05% rare-earth Pr3þ-doped Y2SiO5

62(Pr3þ∶Y2SiO5) crystal, in which the triplet-energy-level633H4 and singlet-energy-level 1D2 are selected to couple64with each other. It is easy to identify the energy levels by65investigating the optical spectrum of Pr3þ. The degeneracy66of the Pr3þ energy levels is prohibited due to the crystal67field of Y2SiO5; the 3H4 and 1D2 states are split into nine68and five Stark components, respectively. The five relevant69energy levels are δ0 (j0i and j1i), γ0 (j2i), δ1 (j3i), γ�0 (j4i)70in a five-level atomic system as shown in Fig. 1(a). The71nature linewidths are about 2 kHz (site I) and 1 kHz (site II)72[13]. The input laser beams are along the [010] axis of the

*Corresponding [email protected]

†Corresponding [email protected]

12

3

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73 Y2SiO5 crystal, where the [010] axis is perpendicular to74 the optical axis. A strong pump beam E1 (frequency ω1,75 wave vector k1, Rabi frequencyG1, wavelength 605.97 nm,76 coupling transition j0i ↔ j2i with detuning Δi ¼77 ωmn − ωi, j1i ↔ j2i Δ0

1 ¼ ω21 − ω1, where ωmn denotes78 the corresponding atomic transition frequency, and ωi is the79 laser frequency), E2 (ω2, k2,G2, 609.27 nm, j3i ↔ j2i,80 Δ2), and E3 (ω3, k3, G3, 607.96 nm, j0i ↔ j4i, Δ3,81 j1i ↔ j4i, Δ0

3). The Pr3þ impurity ions occupy two82 nonequivalent cation sites (i.e., sites I and II) in the83 Y2SiO5 crystal. The energy levels for site I are labeled84 by a greek letter without an asterisk, while for site II by an85 asterisk, as shown in Fig. 1(a). Actually, with the induced86 dipole-dipole interaction considered, the coupling between87 the Pr3þ ions localized at different cation vacancies in the88 Y2SiO5 crystal can occur [14], so one can treat the two ions89 (at different sites) as a hetero-nuclear-like molecule.90 Therefore, one can establish an N-type four-level diagram91 (j3i ↔ j2i ↔ j1i ↔ j4i or j3i ↔ j2i ↔ j0i ↔ j4i),92 which is composed of one Λ-type (j3i ↔ j2i ↔ j0i or93 j3i ↔ j2i ↔ j1i) subsystem and one V-type (j2i ↔94 j0i ↔ j4i or j2i ↔ j1i ↔ j4i) subsystem. A two-photon95 coupling process can be implemented by the transition96 j2i ↔ j1i ↔ j4i.97 In the following theoretical section, we first introduce the98 nonlinear gain and then interpret that the intensity of such99 nonlinear gain can be controlled by the dressing effect.

100 Based on the above, we investigate the intensity noise101 correlation and intensity-difference squeezing in the dress-102 ing PA FWM.

103 A. Spontaneously parametric FWM

104 The SP FWM process involves a coupled Stokes channel105 and an anti-Stokes channel and produces twin photons.106 One can express such SP FWM process with the107 Hamiltonian [15]

H ¼ κ

vðaþbþ þ a bÞ; ð1Þ

108where aþðaÞ is the creation (-annihilation) operator that109acts on the electromagnetic excitation of the ES channel,110whereas bþðbÞ acts on the EAS channel. v is the group111velocity of light in the nonlinear medium, and κS;AS ¼112j − iϖS;ASχ

ð3ÞE21=2j is the pumping parameter of the SP

113FWM, which depends on the nonlinearity χð3Þ and the

114pump-field amplitude. ρð0Þ11 ~ω1

ρð1Þ21 ~ωAS

ρð2Þ01 ~ω1

ρð3Þ21ðSÞ and ρð0Þ00 ~

ω1

115ρð1Þ20 ~ωS

ρð2Þ10 ~ω1

ρð3Þ20ðASÞ are the central frequencies of the gen-

116erated Stokes and anti-Stokes signals. Different from the117case occurring in nonlinear crystals, χð3Þ is a function of118the density-matrix elements. A SP FWM process occurs in119the paraxial direction due to the so-called self-diffraction120phase-matching FWM process generated in a “double-121Λ-type” subsystem [j0i ↔ j1i ↔ j2i between two vertical122dashed lines in Fig. 1(a)]. The strong pumping field E1

123together with the generated Stokes (Es) and anti-Stokes124(EAS) fields satisfy the phase-matching conditions kS ¼1252k1 − kAS and kAS ¼ 2k1 − kS. Laser beams E2 and E3

126(counterpropagating with E1) can cause resonant absorp-127tion and are used only to dress such SP FWM. Thus, only128dressed SP FWM can be detected, and the generated fields129can be described by the perturbation chains [16,17]

130ρð0Þ11 ~ω1

ρð1Þ21 ~ωAS

ρð2Þ01 ~ω1

ρð3Þ21ðSÞ (ES) and ρð0Þ00 ~

ω1

ρð1Þ20 ~ωS

ρð2Þ10 ~ω1

ρð3Þ20ðASÞ

131(EAS). Therefore, one gets [18]

ρð3Þ21ðSÞ ¼ −iGAS

�G12=ðd21d01d021Þ; ð2Þ

132ρð3Þ20ðASÞ ¼ −iGS

�G12=ðd20d10d020Þ; ð3Þ

133where Gi ¼ μijEi=ℏ is the Rabi frequency of field Ei with134the electric dipole matrix elements μij, d20 ¼ Γ20 þ iΔ1,135d10 ¼ Γ10 þ iδ, d020 ¼ Γ20 þ iðΔ1 þ Δþ δÞ, d21 ¼ Γ21þ136iΔ0

1, d01 ¼ Γ01 − iδ, d021 ¼ Γ21 þ iðΔ1 − δÞ, and Γij is137the decay rate between energy levels jii and jji. The real138frequencies of the generated Stokes and anti-Stokes signals139can be expressed as ωS ¼ ϖS þ δ and ωAS ¼ ϖAS − δ140respectively, where the introduced symbol θ can be viewed141as the criterion for the linewidths of the generated signal.142The boson-creation (-annihilation) operator satisfies143the Heisenberg operator equation of motion in the dipole144approximation:

da=dz ¼ ½a; H�=iℏ ¼ κbþ; ð4Þ

145dbþ=dz ¼ ½bþ; H�=iℏ ¼ κa: ð5Þ

146147After some algebra, we get the photon numbers of the148Stokes and anti-Stokes field at the output site of the149medium [15,19]:

F1:1 FIG. 1. Experimental4 setup. (a) N-type four-level atomlikeF1:2 system in Pr3þ∶Y2SiO5. The width of ground state j0i representsF1:3 the broadened degenerate states. (b) Experimental setup scheme.F1:4 Flu, fluorescence; D, photomultiplier tube; L, lens. E2 and E3

F1:5 counterpropagate with E1.

CHANGBIAO LI et al. PHYS. REV. APPLIED XX, 000000 (XXXX)

2

haþoutaouti ¼1

2

�cos

�2L

ffiffiffiffiffiffiffiAB

psin

φ1 þ φ2

2

�

þ cosh

�2L

ffiffiffiffiffiffiffiAB

pcos

φ1 þ φ2

2

��AB; ð6Þ

150hbþoutbouti ¼

1

2

�cos

�2L

ffiffiffiffiffiffiffiAB

psin

φ1 þ φ2

2

�

þ cosh

�2L

ffiffiffiffiffiffiffiAB

pcos

φ1 þ φ2

2

��BA; ð7Þ

151 where L is the medium length, ρð3Þ21ðSÞ ¼ Aeiφ1 , ρð3Þ

20ðASÞ ¼152 Beiφ2 , and A (B) and φ1 (φ2) are the modulus and phase

153 angles of ρð3Þ21ðSÞ (ρ

ð3Þ20ðASÞ), respectively.

154 When three beams E1, E01, and E00

1 (from the same laser)155 are turned on [see Fig. 1(b)], a coherent FWM signal156 (kF ¼ k1 þ k00

1 − k01) is produced. If we inject this FWM

157 signal into the Stokes channel of the SP FWM [14], we158 obtain the PA FWM. Therefore, the gain of the Stokes159 channel is given by

g ¼ 1

2

�cos

�2L

ffiffiffiffiffiffiffiAB

psin

φ1 þ φ2

2

�

þ cosh

�2L

ffiffiffiffiffiffiffiAB

pcos

φ1 þ φ2

2

��: ð8Þ

160161 B. Dressed PA FWM

162 Taking into account the dressing effects of E2 and E3,163 Eqs. (2) and (3) can be rewritten as [16]

ρð3Þ21ðSÞ ¼ −iGAS

�G12=ðd21Dd01d021DÞ; ð9Þ

164ρð3Þ20ðASÞ ¼ −iGS

�G12=ðd20Dd10d020DÞ; ð10Þ

165 where d21D ¼ Γ21 þ iðΔ1 þ ΔÞ þ jG2j2=½Γ31 þ iðΔ1þ166 Δ − Δ2Þ þ jG1j2=Γ11�, d20D ¼ Γ20 þ iΔ1 þ jG2j2=½Γ30þ167 iðΔ1 − Δ2ÞþjG1j2=Γ00�, d021D ¼ Γ21 þ iðΔ1 − δÞ þ jG3j2=168 ðΓ41 þ iΔ3Þ, d020D¼Γ20þiðΔ1þΔþδÞþjG3j2=ðΓ40þiΔ3Þ.169 In addition, the polarization of the dressing field (E2 or170 E3) can be controlled by a quarter-wave plate (QWP) [see171 Fig. 1(b)]. We assume the P-polarization direction (along172 an optic axis of the crystal) to be the quantization axis, so173 the component perpendicular to it (S polarization) is174 decomposed into balanced left- and right-circularly-175 polarized parts, while the component parallel to it176 (P polarization) remains linearly polarized. So, the dressing177 terms in Eqs. (9) and (10) are replaced by C2

g2;linðcos4θ þ178 sin4θÞjG2j2 and C2

g2;cirð2cos2θsin2θÞjG2j2, respectively.

179 Here, Cg;lin and Cg;cir are Clebsch-Gordan (CG) coefficients180 for linear and circular polarization, respectively [20]. θ181 is the rotated angle between the QWP’s axis and182 P-polarization direction. Because of the relationship

183κ ∝ χð3Þi ∝ ρð3Þ, the output signals of this PA FWM process184can be manipulated by the dressing fields (E2 or E3), e.g.,185the power, frequency detuning, and polarization.

186C. Intensity noise correlation and187intensity-difference squeezing

188The output number of photons is given by haþoutaouti,189which is proportional to the intensity since I ∝ haþoutaouti.190The intensity fluctuation is δIðtÞ ¼ IðtÞ − hIðtÞi.191Therefore, the correlation between the intensity fluctua-192tions of the output Stokes and anti-Stokes PA FWMs fields193[19,21] can be obtained

Gð2ÞS;ASðτÞ ¼

h½δISðtSÞ�½δIASðtASÞ�iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih½δISðtSÞ�2ih½δIASðtASÞ�2i

q ¼ jΨS;ASj2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijCSj2jCASj2

p ;

ð11Þ

194where τ ¼ τAS − τs, τj ¼ tj − rj=c, rj is the optical path195from the output surface of the crystal to the jth detector196(j ¼ 1; 2), tj is the trigger time, h½δISðtSÞ�½δIASðtASÞ�i is the197cross-correlation function between the intensity fluctua-198tions of EAS and ES,CS ¼

ffiffiffiffiffiRs

p RdωSsinh2ðκLÞ, CAS ¼

199ffiffiffiffiffiffiffiffiRAS

p RdωASsinh2ðκLÞ, VQ is the quantization volume,

200and RS;AS ¼ V1=3Q E2

S;AS=ð2πνS;ASÞ are self-correlation func-201tions of ES and EAS, respectively, where ΨS;ASðτÞ ¼202iπE2

1ESEASω1

Re−iδτχð3ÞðδÞdδ.

203Considering the Boltzmann distribution with finite204temperature, the two-photon envelope function can be205obtained by performing a Fourier transform on the third-206order nonlinear susceptibility, Ψs−asðτÞ ¼ iπE2

1ESEASω1

207Re−iδτχð3ÞðδÞdδ¼ iπE2

1ESEASω1=½ðΓ21þ iΔ1 þ iΔÞðiΔ1þ208Γ21−Γ01Þ�½e−Γ−τ − e−iΔ1τe−Γþτ�. [22]. In addition, taking209into account the interaction between the sample and210coupling fields, the broadened linewidth of the measured211SP FWM signal is Γ� ¼ Γpop − Γð�δÞ þ Γion-spinþ212Γion-ion þ Γphonon, where Γpop ¼ ð2πT1Þ−1 depends on the213location of the energy level in phase space, with T1

214describing the population decay time, and the term215Γð�δÞ represents the location of the energy level, which216can be dressed by the coupling field. The last three terms217(Γion-spin þ Γion-ion þ Γphonon) are components of ð2πT∗

2Þ−1218(the reversible transverse relaxation time T∗

2 ). Γion-spin is219related to the ion-spin coupling effect of the individual ion.220Γion-ion is determined by the interaction among the rare-221earth ions and can be controlled by the power of the222external field and impurity concentration. Γphonon is related223to the sample temperature. To quantify Γion-ion, PðtÞ is224introduced PðtÞa ¼ exp½−cHPn¼6;8;10;12;13;14ðAnH=Rn

HÞ�225and PðtÞb ¼ exp½−cDPn¼5;6;7ðAnD=Rn

D�, where AnH=226AnD is the van der Waals attractive force coefficient,227Rn

H=RnD is the distance of the nucleus, cH and cD represent

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3

228 the population densities at the triplet energy level 3H4 and229 singlet energy level 1D2, respectively, and can be controlled230 by the pump power.

PðAnH=RnHÞ and

PðAnD=RnDÞ re-

231 present the induced dipole-dipole interaction of states H-H232 and D-D, respectively.233 The degree of two-mode intensity-difference squeezing234 is given by [19,23,24]

sq ¼ log10hδ2ðIS − IASÞihδ2ðIS þ IASÞi

≈ log10hδ2ðNS − NASÞihδ2ðNS þ NASÞi

¼ −log10ð2g − 1Þ; ð12Þ

235 where hδ2ðNS − NASÞi is the mean square deviation of the236 intensity difference, and hδ2ðNS þ NASÞi is the mean237 square deviation of the intensity sum of the coherent laser238 beams.

239 D. Multiorder fluorescence

240 We adopt the perturbation theory to investigate the multi-241 order-fluorescence (MFL) signal [25,26]. When E1 and E2

242 are open, the MFL signal is generated in the Λ-type-level243 system of Pr3þ∶Y2SiO5. Taking into account the dressing

244 effects of E1 and E2 and the perturbation chains ρð0Þ00 ~ω1

245 ρð1Þ20 ~−ω1

ρð2Þ22 ~−ω2

ρð3Þ32 ~ω2

ρð4Þ22 , the intensity of MFL signal can be

246 described by the diagonal density matrix element ρð4Þ22 ,247 which is given by [14]

ρð4ÞMFL ¼ ρð4Þ22 ¼ jG2j2jG1j2=½d20DΓ00d32

× ðΓ22 þ jG2j2=d32 þ jG1j2=d20Þ�; ð13Þ

248 where d32 ¼ Γ32 þ iΔ2.

249 III. RESULTS AND DISCUSSION

250 A. System preparation

251 The sample (a 3-mm Pr3þ∶Y2SiO5 crystal) is held at 77 K252 in a cryostat (CFM-102). Three transform-limited tunable253 dye lasers (narrow scan with a 0.004-cm−1 linewidth)254 pumped by an injection-locked single-mode Nd: YAG laser255 (ContinuumPowerlite DLS 9010, 10-Hz repetition rate, 5-ns256 pulse width) are used to generate the pumping fields E1, E0

1,257 E00

1ðω1;Δ1Þ,E2 (ω2,Δ2), andE3 (ω3,Δ3). The diameter of all258 beams is 0.8 mm. The polarizations of the laser beams are259 controlled by inserting wave plates into the corresponding260 beam paths. In such multilevel atomlike system, one can261 obtain the PA FWMwith an injecting coherent FWM signal262 (kF¼k1 þ k00

1 − k01). Note that the generated signals (ES,

263 EAS) are quantum correlated [27,28]. Figure 1(b) shows the264 experimental arrangement taking into account the above265 phase-matching condition. The generated signals (ES, EAS)266 are associated with MFL signals and detected by photo-267 multiplier tubes. Such PA FWM process can be doubly

268dressed by applying dressing fields E1 and E2 (or E3) or269triply dressed by applying all three dressing fields E1, E2,270and E3 simultaneously.

271B. Two-mode experimental results

272Nonlinear processes can occur easily in the paraxial273direction of the incident laser beam. In the current experi-274ment, the PA FWM process occurs, which is well inter-275preted by a self-diffraction-type FWM process in which276a strong pumping field E1 is mixed with two weak277fields ES and EAS as shown in Fig. 1(a) (middle panel).278The parameter space to be searched for in our investigation279of the dressed PA FWM together with the MFL is large,280with the frequency detuningΔ1 of pumping fieldE1 and the281powers and the polarization states of both dressing fields E2

282and E3 as externally controlled variables. In addition, we283compare the dressing effects of two structures in Fig. 1(a),284i.e., E2 dressing in a Λ-type three-level subsystem of an285individual Pr3þ ion and E3 dressing in a V-type three-level286subsystem of a hetero-nuclear-like molecule.287Since the generated ES and EAS are twin beams and have288similar properties, we choose only ES and its MFL to show289the properties under various dressing conditions. First of all,290we analyze the accompanied MFL spectra [see Fig. 2(a)]291under various dressing conditions in the Λ-type subsystem.292The typical linewidths ofMFL signals are about 30GHz. On293the one hand, they are broadened by phonons (Γphonon) from294the lattice heat vibration and the power (Γion-ion) of the295pumping field. On the other hand, they are balanced by the296dressing effect [Γð�δÞ] [29]. At a relatively low power of297P1 ¼ 1 mW [Fig. 2(a1)], the MFL has a Lorentzian line298shape with a small Autler-Townes (AT) splitting [14],299which is induced by settingP2 ¼ 3 mW.Oneway to explain300these observed effects is by using the dressed-stated picture.

F2:1FIG. 2. Measurements of the PA FWM (ES) and MFL ex-F2:2citation spectra associated with their lifetimes under specifiedF2:3experimental parameters. (a)–(d) Dressed spectra and lifetimes.F2:4(a) The MFL spectra, (b) the PA FWM spectra ES, (c) temporalF2:5MFL, and (d) ES intensity. The parameters inside the parenthesesF2:6in (b1), (b2), and (b4) are E1 power, E2 power, and E2

F2:7polarization by QWP, where the linear and circular polarizationsF2:8are denoted as θ ¼ 0° and θ ¼ 45° of the QWP, respectively. InF2:9(b3), the parameters are E1 power, E2 power, and E3 power.

F2:10G1 ¼ 25 GHz at 3 mW.

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301 The power of E2 splits the energy level j2i into jG2�i. From302 the Hamiltonian equation HjG2�i ¼ λ�jG2�i, we obtain303 λ� ¼ ½Δ2 � ðΔ2

2 þ 4jG2j2Þ1=2�=2. Therefore, the splitting304 distance between jG2�i is 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2

2 þ 4jG2j2p

, as shown in305 Fig. 2(a1). Next, by increasing P1 ¼ 3 mW [Fig. 2(a2)], the306 MFL is suppressed by the double-dressing effect of E1 and307 E2 which appears as a suppression dip [see Eq. (13)]. Here,308 the baseline comes from another fluorescence process309 excited byE2. Physically, this phenomenon can be attributed310 to the nest-type dressing term d20 in Eq. (13) [14,20]. The311 suppressing ability of E1 increases as the detuning Δ1 gets312 closer to resonance. However, this situation changes at the313 resonant region of Δ1 ¼ Δ2 ¼ 0, in which a peak of ES due314 to the high power ofE1 overlapswith the dip. Third, whenE2

315 andE3 are both turned on simultaneously, the triple-dressing316 effect must be considered. As demonstrated in Fig. 2(a3), the317 MFL is suppressed completely. If the beam E2 is set to be318 circularly polarized [Fig. 2(a4)] while all other parameters319 remain the same as those in Fig. 2(a2), one can easily see that320 the dressing effect in such case is stronger than that in the321 linear case in Fig. 2(a2). Since the CG coefficients may be322 different for different transitions between Zeeman sublevels,323 the Rabi frequency can vary with polarization even if the324 frequency and power of the laser field remain unchanged. For325 example, the CG coefficients, Cg;cir (circular polarization)

326 and Cg;lin (linear polarization) areffiffiffiffiffiffiffiffiffiffi4=35

pand

ffiffiffiffiffiffiffiffiffiffi1=35

p,

327 respectively with M ¼ 1=2. So, the dressing terms in328 Eq. (13) are replaced by C2

g2;linðcos4θ þ sin4θÞjG2j2 and

329 C2g2;cirð2cos2θsin2θÞjG2j2, respectively. Thus, the ratio

330 between dressing term of the circular and linear case is331 expressed as C2

g2;cir=C2g2;lin ¼ 4, with θ ¼ 45° and

332 M ¼ þ1=2, which indicates that the dressing effects in333 the circularly polarized subsystems are far greater than those334 in the linearly polarized subsystems.335 In addition, one can find that the dressed state, one of the336 most important consequences of the dressing effect of beam337 E2, makes the intensity curves in the time domain with338 Δ1 ¼ Δ2 ¼ 0 show double peaks of the AT splitting339 configuration. There exist two peaks, which correspond340 to the locations of the zero delay time and a longer delay341 time [30], as displayed in Figs. 2(c2)–2(c4). The obvious342 delay of the right peak in Fig. 2(c1) is caused by the343 residual particles in jG2þi transferring to jG2−i through344 phonon-assisted nonradiative transition, which is mainly345 determined by acoustic phonons at low temperature. As the346 power of either E1 or E2 increases, such space (time delay)347 between two peaks [Fig. 2(c)] gets larger, which can be348 attributed to the energy-level splitting by the dressing effect349 [corresponding to the AT splitting in the spectrum in350 Fig. 2(a)]. The curves in Figs. 2(d) and 2(d) are the dressed351 PA FWM spectra together with their lifetimes correspond-352 ing to each fluorescence case [30]. Based on the analysis of353 the above florescence signals, one can easily determine the354 physical mechanism behind each output ES as shown in

355Figs. 2(b) and 2(d). In other words, the dressing effect356makes the intensity of the PA FWM decrease and the357lifetime longer, whereas the dressing effect of the circularly358polarized field is stronger than that for the linear case [20].359However, since the PA FWM signal is insensitive to the360dressing field, there are no AT splittings in both the361spectrum and time domain, as shown in Figs. 2(b) and3622(d), respectively.363Now, we focus on the intensity noise correlation and364intensity-difference squeezing between the output beams365ES and EAS of the dressed PA FWM process as shown in366Fig. 3. Figure 3(a) shows the noise correlations under367various control conditions. First, the correlation levels at368the resonance and off-resonance positions are different—369the correlation is reduced at the suppression dip position370[shown in Fig. 2(a), satisfying suppression condition371Δ1 − Δ2 ¼ 0; Fig. 2(a1), Δ1 ¼ 0], while it is enlarged at372the enhanced-peak position [shown in Fig. 2(a), satisfying373the enhancement condition Δ1 þ λ� ¼ 0; Fig. 2(a2),374Δ1 ¼ 200 GHz]. To investigate in detail the role of the375dressed states on the correlation level, we plot a frequency376dependence of the maximum correlation value on the377discrete frequency detuning Δ1 presented as square points

378in Fig. 3(b1). The two-mode correlationGð2ÞS;ASðτÞ [Eq. (11)]

379and squeezing (sq) [Eq. (12)] are determined by the380nonlinear gains in Eq. (8), which can be modified by the381dressing effect in Eqs. (9), (10), and (13) [31]. The curve382includes three regions: the resonance, near resonance, and383far off resonance, respectively. At the far off resonance, the384nesting double-dressing configuration [d21D and d20D in385Eqs. (9) and (10)] reduces to zero, and the nonlinear gain386[Eq. (8)] gets the maximum value. At last, the maximum387correlation is obtained. As Δ1 is close to being near388resonance, the correlation value becomes smaller and389smaller due to the nonlinear gain (g) decrease, and the390correlation reaches its minimum value at Δ1 ¼ 100 GHz.391Similarly, the correlation value is minimized at Δ1 ¼392−100 GHz [the square points in Fig. 3(b1)]. However,393in the region jΔ1j ≤ 100 GHz, the dressing effect due to E2

394cannot be neglected. Therefore, at resonance, the nesting395double-dressing configuration (d20 and d21) makes the396correlation much smaller but not to its minimum value397due to the action of dressing field E2, as shown in the398square points in Fig. 3(b1). In addition, the modulated399correlation time has the opposite behavior depicted with400circular points in Fig. 3(b1). The single-dressing effect can401result in the increased lifetime of the PA FWM and the402decreased intensity of the PA FWM process [29]. The403lifetime of the PA FWM process is inversely proportional to404the decay rate of the PA FWM process, where the dressing405results between the lifetime of the PA FWMprocess and the

406correlation time of Gð2ÞS;ASðτÞ are coincident with each other

407because the lifetime of the PA FWM process [Fig. 2(d)] is408modulated in the sameway as shown in the circular curve in409Fig. 3(b1). As we discuss above, the correlation time and

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410 decay rate are still valid, as shown by the circular points in411 Figs. 3(b2), 3(b3), and 3(d2), and 3(d3), and the degree of412 correlation [square points in Fig. 3(b)] is inversely propor-413 tional to the correlation time [circular points in Fig. 3(b)].

414 Now, we use the Cauchy-Schwarz inequalityGð2ÞS;ASðτ1Þ ≤ 1

415 to check our experimental result [Fig. 3(a2)]. The measured416 correlation at off resonance is about 1.5, which clearly417 violates of the Cauchy-Schwarz inequality, proving that418 there is a nonclassical correlation between ES and EAS.419 Next, we focus on the power and polarization depend-420 ences of the intensity noise correlation on E2. Figures 3(a1)421 and 3(a4) show the correlation curves at high power422 (P2 ¼ 5 mW) and low power (P2 ¼ 1 mW), respectively,423 when Δ1 ¼ 0 and θ2 ¼ 0° (linear polarization). As the424 power of the dressing field increases, the correlation425 increases at first due to the dominant gain by the terms426 d20 and d21, and then it becomes worse due to the dressing427 effect from the dressed term in Eq. (9), as shown by the428 square points in Fig. 3(b2). Figures 3(a1) and 3(a3) show the429 correlation curves at linear polarization (θ2 ¼ 0°) and cir-430 cular polarization (θ2 ¼ 45°), respectively, whenΔ1 ¼ 0 and431 P2 ¼ 5 mW.As site I of the Pr3þ ion is muchmore sensitive432 to the circular polarization, one finds a similar behavior [see433 the square points in Fig. 3(b3)]. The competition between the434 linear component C2

g2;linðcos4θ þ sin4θÞjG2j2 and circular

435 componentC2g2;cirð2cos2θsin2θÞjG2j2 can well interpret such

436 phenomena. Because of the relationship κ ∝ χð3Þi ∝ ρð3Þ, the437 output signals of this PA FWM process can be modified by438 the polarizations of the dressing fields (E2 or E3), mostly by439 the polarization of the dressing fields. Specifically, the440 intensity (jρð3Þj2Þ) of the PA FWM is suppressed when the

441polarization of the dressing field changes from linear to442circular due to the stronger dressing effect in the circular case.443Therefore, jΨs−asj2 in Eq. (11) becomes smaller, and this444leads to a lower correlation degree than that for the linearly445polarized case [Figs. 3(a1) and 3(a3)]. Similarly, according to446Eq. (12), the degree of squeezing decreases when the447dressing field is circularly polarized compared with the448linear case [Figs. 3(b1) and 3(b3)].449Figure 3(c) shows the intensity-difference squeezing450(lowest curve in each panel) in contrast to the total noise451(highest curve in each panel) or the shot-noise level [middle452curve in Fig. 3(c2)]. Specifically, the highest curve in each453panel is the total noise of the two output beams. The shot-454noise level (SNL) [dashed curves in Figs. 3(c) and 3(f)] for455the PA FWM process is defined as the intensity-difference456noise on a pair of equal power beams produced from a457coherent laser by a beam splitter with equal power as the458sum of ES and EAS [23]. As shown in Fig. 3(c2), the459intensity-difference squeezing measured here is −4.5 dB,460which is lower than the SNL. One finds that the double-461dressing results of squeezing [Eq. (12)] have the same462behavior as the correlation [Eq. (11)], which are determined463by the dressed nonlinear gain in Eq. (8), as shown by the464square points in each panel in Fig. 3(d). Therefore, the465degree of intensity-difference squeezing of the output466beams ES and EAS depends crucially on the correlation467functions [27,28].468When the dressing fields E2 and E3 are turned on469simultaneously, a triple-dressed PA FWM process can be470obtained [32]. Different from the case of the double-dressed471PA FWM, the dressed results in Eq. (12) can be switched at472Δ1 ¼ 0. For example, the correlation measured here is

F3:1 FIG. 3. Measured intensity noise correlation and squeezing. (a),(c) Measured intensity noise correlation and intensity-differenceF3:2 squeezing between output beams ES and EAS, where the PA FWM process is dressed by E2. All experimental conditions are identicalF3:3 except Δ1 [(a1), Δ1 ¼ 0] and [(a2), Δ1 ¼ 200 GHz], polarization state of the dressing field E2 (a1) θ2 ¼ 0° and (a3) θ2 ¼ 45°, and E2

F3:4 power (a1) P2 ¼ 5 mW, and (a4) P2 ¼ 1 mW, respectively. (b) Frequency detuning, power and polarization dependences of theF3:5 maximum correlation value Gð2Þð0Þ (squares), and correlation time Δτ (circles). (d) Dependences of the squeezing value at the analysisF3:6 frequency 1.5 MHz (squares) and decay rate (circles) of the output beam ES, respectively, corresponding to the cases in (b). Note that theF3:7 squares denote the case dressed by E3. (e),(f) Measured intensity noise correlation and intensity-difference squeezing between outputF3:8 beams ES and EAS at (1) Δ1 ¼ 200 GHz, (2) 60 GHz, and (3) 0, which are dressed by E2 and E3 simultaneously. (4) FrequencyF3:9 dependence of the maximum correlation and squeezing degree on the frequency detuning Δ1.

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473 about 1.4 [Fig. 3(e1)], which is reduced in comparison with474 that in Fig. 3(a1). However, there is also a nonclassical475 correlation betweenES andEAS, since the Cauchy-Schwarz476 inequality is violated clearly. On the other hand, due to the477 different polarization responses of site I and site II, the478 frequency dependence of the maximum correlation479 value on Δ1 presents an inverted Lorentzian line shape480 [see Fig. 3(e4)], while the double-dressed case is a normal481 Lorentzian line shape with a central peak [see Fig. 3(b1)].482 Similar behaviors can be found for the intensity-difference483 squeezing [Eq. (13)] as shown in Fig. 3(f4) (triple dressed)484 and Fig. 3(d1) (double dressed). Thus, the intensity-485 difference squeezing [Figs. 3(d) and 3(f4)] is proportional486 to the corresponding noise correlation [Figs. 3(b) and487 3(e4)] of the PA FWM process.488 In addition, we not only verify that the intensity-difference489 squeezing and intensity noise correlation can be easily490 manipulated by several parameters in Fig. 3, but we also491 show that the correlation and squeezing can be controlled by492 the relative nonlinear phase shift caused by the dressing493 beam. As shown in Fig. 4(a1), with P2 ¼ 6 mW, the494 amplitude of the correlation peak at delay time τ ¼ 0 is495 1.3. The Cauchy-Schwarz inequality is violated in Fig. 4(a1)496 and nonclassical correlation between ES and EAS is dem-497 onstrated. As P2 ¼ 8 mW, the amplitude is about 0, as498 shown in Fig. 4(a2), and there do not exist correlation and499 squeezing. In particular, the intensity fluctuation is switched500 from correlated to anticorrelated if P2 ¼ 12 mW, as shown501 in Fig. 4(a3). These results can be explained by the nonlinear502 refractive index of the Kerr medium. Since the dressing state503 created by E2 can modulate the nonlinear refractive index of

504the Kerr medium that results from the cross-phase modula-505tion (XPM), the relative nonlinear phase Δφ ¼ φS − φAS ¼5062ðkSnS2 − kASnAS2 ÞjE2j2e−r2z=n1 between the Stokes and507anti-Stokes signals is significantly modulated. Here, φS508(φAS) is the nonlinear phase induced on the Stokes (anti-509Stokes) signal. The Kerr effect is associated with nonlinear510refractive index n2 (corresponding to the third-order non-511linear response) and intensity of light I (n2I, n2 ¼512Reχð3Þ=ε0cn1). To be specific, by setting P2 ¼ 6, 8, and51312 mW, the relative nonlinear phaseΔφ changes toΔφ ¼ 0,514π=2, and π, respectively. Correspondingly, the correlation515[Eq. (12)] switches frompositive to zero and then to negative.516Moreover, by changing Δφ, Figs. 4(b1)–4(b3) show that517the intensity-difference signal [Eq. (13)] can be switched to518either higher or lower than the total noise signal, which519corresponds to squeezing or antisqueezing, respectively.520Therefore, the intensity fluctuation correlation and the521intensity-difference squeezing of the output beams ES and522EAS depend crucially on the relative nonlinear phase induced523by XPM.

524C. Triplet beams by the PA SWM process

525Furthermore, we investigate the three-mode correlation526and squeezing of the PA SWM with an injecting coherent527FWM. Different from the generated twin beams by the SP528FWM process, laser fields E2 and E3 both at large529detunings are set to be certain angles with E1 satisfying530the phase-matching condition. Thus, mutually correlated531triplet beams can be produced by a SP SWM process with532the phase-matching condition kS1 þ kS2 þ kS3 ¼ k1þ533k2 þ k3. Since such SP SWM process [14] can be534mimicked by cascading two involved closed-loop SP535FWM processes (kS1 þ kS2 ¼ k1 þ k2 and kS1 þ kS3 ¼536k1 þ k3) [Fig. 5(a)], one can write the Hamiltonian as537HI ¼ iℏκ1a

†S1a

†S2 þ iℏκ2a

†S1a

†S3 þ H:c:, where aþS1; a

þS2; a

þS3

538are the boson creation operators of the three generated

539fields, and κ1 ¼ −iϖS1χð3Þ1 E1E2=2 and κ2 ¼ −iϖS1χ

ð3Þ2

540E1E3=2 are the third-order nonlinear susceptibility of

F4:1 FIG. 4. (a) Intensity noise correlation curves between ES andF4:2 EAS versus delayed time τ with changing power of E2, with (a1)F4:3 6 mW, (a2) 8 mW, and (a3) 12 mW, respectively. (b1) P2 ¼F4:4 6 mW and ω ¼ 1.5 MHz. The lower curve and upper curve areF4:5 the noise level of intensity-difference squeezing and intensity-F4:6 sum squeezing versus analysis frequency Ω, respectively. (b2)F4:7 P2 ¼ 8 mW. The intensity-sum squeezing and intensity-F4:8 difference squeezing have the same noise level and are indis-F4:9 tinguishable. (b3) P2 ¼ 12 mW and ω ¼ 1.5 MHz. The lower

F4:10 curve and upper curve are the noise level of the intensity-sumF4:11 squeezing and intensity-difference squeezing, respectively.

F5:1FIG. 5. (a) N-type four-level atomlike system in Pr3þ∶Y2SiO5

F5:2for triple-photon generation by a SP SWM process composed ofF5:3Λ-type and V-type three-level subsystems. (b) New experimentalF5:4scheme for the SP SWM process with the phase-matchingF5:5condition kS1 þ kS2 þ kS3 ¼ k1 þ k2 þ k3.

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541 cascading two (χð3Þ1 and χð3Þ2 ) involved SP FWM processes.542 Therefore, one can easily get the dynamic equations as

daS2=dt ¼ κ1a†S1;

daS1=dt ¼ κ1a†S2 þ κ2a

†S3;

daS3=dt ¼ κ2a†S1: ð14Þ

543

544Similar to the PA FWM process, one can calculate the545output results from these equations. Here, we are concerned546about the properties of the output signals from this547PA SWM process with an injecting coherent FWM548(kF¼k1 þ k00

1 − k01) [15], and then we extend the Gð2ÞðτÞ

549function for the two-mode case to the three-mode correlation550function Gð3ÞðτÞ as

551552

Gð3Þ ¼ h½δIS1ðtS1Þ�½δIS2ðtS2Þ�½δIS3ðtS3Þ�iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih½δIS1ðtS1Þ�2ih½δIS2ðtS2Þ�2ih½δIS3ðtS3Þ�2i

q ¼ ðHS1S2S3 −HS3CS1S2 −HS2CS1S3 −HS1CS3S2 þ 2HS1HS2HS3ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijCS1j2ðjCS2j2 þ jC0

S2j2ÞjCS3j2p ;

ð15Þ

553 where HS1S2S3 is the cross-correlation function of intensity554 between ES1, ES2, and ES3, and HSi is the one-order555 correlation function of the intensity. CSiSj is the cross-556 correlation function between the intensity fluctuations of557 ESi and ESj (i, j ¼ 1; 2; 3, i ≠ j). CSi is the self-correlation558 function. The conditional intensity-difference squeezing for559 the three-mode case can be given by [24,31]

sq ¼ log10hδ2ðNS1 − NS2 − NS3Þihδ2ðNS1 þ NS2 þ NS3Þi

≈ 2log10g21 − g22 − g23g21 þ g22 þ g23

;

ð16Þ

560 where g1¼½κ22þκ21coshðΩLÞ�=ðκ22þκ21Þ, g2¼ κ1 sinhðΩLÞ=561

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiκ22þ κ21

p; and g3 ¼ κ1κ2ðcoshðΩLÞ − 1Þ=κ22 þ κ.

562 When beams E1, E2, and E3 are turned on simulta-563 neously, as shown in Fig. 5(a), a PA SWM process with the564 phase-matching condition kS1 þ kS2 þ kS3 ¼ k1 þ k2 þ565 k3 can be obtained in the N-type four-level system.566 Figure 5(b) shows the experimental arrangement for meet-567 ing the PA SWM phase-matching condition. To satisfy the568 requirement for such PA SWM process, two PA FWM569 processes share the pumping beam E1 and output beam ES1570 in both the frequency domain and spatial domain by using a571 specially designed experimental configuration (fields E2

572 and E3 both counterpropagating with E1 at a certain angle573 to meet the phase-matching condition). The major differ-574 ence between the current PA SWM experimental scheme575 [Fig. 5(a)] and that in Fig. 1(a) is the frequency detuning set576 for E1,E2, andE3. In addition, due to the angle between k2,577 k3, and k1, the generated SWM signal cannot be obtained578 in the same direction of the FWM when the detector is579 placed as shown in Fig. 1(a). Therefore, E2 and E3 are only580 dressing fields for the SP FWM in Fig. 1(a) while E1, E2,581 and E3 are all generating fields for the SP SWM.582 Figure 6 depicts the triple-beam correlation and inten-583 sity-difference squeezing results. Two sets of frequency584 detunings of E1 are set at off resonance [Δ1 ¼ 200 GHz,

585Fig. 6(a)] and on resonance [Δ1 ¼ 0, Fig. 6(b)], respec-586tively. Three-mode correlation [Eq. (15)] and squeezing587[Eq. (16)] are determined by the dressed nonlinear gains588[31]. By comparing the results of triple-beam correlation589and intensity-difference squeezing, one finds that the590maximum correlation value at off resonance is bigger than591the on-resonance case, which is similar to that shown for592the twin beams. However, the correlation level is weaker593than that for the case of triple-dressed twin beams since the594correlation condition is more stringent as shown in595Eq. (15). Similar behavior occurs with the intensity-596difference squeezing. In contrast to the twin beams, the597SNL for the PA SWM process becomes the intensity-sum598noise on three coherent laser beams with equal power as the599sum of ES1, ES2, and ES3. In Eq. (16), hδ2ðNS1 þ NS2 þ600NS3Þi is the total noise of the three coherent laser beams.

F6:1FIG. 6. Three-mode noise correlation and intensity-differenceF6:2squeezing of ES1, ES2, and ES3. Experimental conditions are setF6:3as follows: (a),(b) E1, E2, and E3 are linearly polarized, (c) E1 isF6:4linearly polarized, while E2 and E3 are circularly polarized.

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601 Based on the cascade processes [24], hδ2ðNS1 − NS2 −602 NS3Þi is the intensity noise difference of three output603 beams. Because of the more stringent condition of the604 three-mode direct correlation, the squeezing value is605 smaller than in the dressed twin-beam case. We get606 a -3.5-dB squeezing in Fig. 6(a3), but the same role played607 by the dressed state is still true for the squeezing results;608 i.e., it reduces the squeezing.609 Next, we focus on the case when both E2 and E3 are610 either linearly polarized [Fig. 6(b)] or circularly polarized611 [Fig. 6(c)] while E1 remains linearly polarized. Apparently,612 the correlation and squeezing with linear polarization are613 better than the case with circular polarization, which is614 similar to the twin-beam case with single dressing field E2

615 only (Fig. 3).

616 IV. CONLUSIONS

617 We investigate the generated twin beams by the PA618 FWM process and triplet beams by the PA SWM process619 associated with the MFL signals in a PR3þ∶Y2SiO5 crystal.620 First, the degrees of intensity noise correlation and inten-621 sity-difference squeezing between the output beams ES and622 EAS of the PA FWM process are determined by the623 nonlinear gain, which can be controlled by the dressing624 variables. The correlation at the resonant position can attain625 a higher value due to the double-dressing effect; however,626 such correlation is reduced in the case of the triple-dressing627 effect. Second, the correlation of the PA FWM can change628 to anticorrelation by the nonlinear phase shift, which is629 induced by the external dressing beam, and the noise level630 of the intensity-difference squeezing is switched synchro-631 nously. Finally, we investigate the intensity noise correla-632 tion and intensity-difference squeezing of the triplet beams633 generated by the PA SWM process, which is well inter-634 preted by two cascaded nonlinear processes. The correla-635 tion level is weaker than the case of twin beams since the636 three-mode correlation condition is more stringent.637 Meanwhile, the squeezing value is also smaller than the638 twin-beam case. Similarly, the polarized dark state still639 plays the same role in the results. Such results can find640 potential applications in engineering three-channel641 entangled imaging on chip.

642 ACKNOWLEDGMENTS

643 This work is supported by the 973 Program (Grant644 No. 2012CB921804), KSTIT of Shaanxi Province (Grant645 No. 2014KCT-10), the NSFC (Grant No. 11474228),646 China Postdoctoral Science Foundation (Grant No. 2016M647 590935), and NSFC of Shaanxi Province (Grant648 No. 2016JM6029).649

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