Qu.Entanglement.PRE2024 (1)

PHYSICAL REVIEW E110,065211(2024) Editors’ Suggestion Producing entangled photon pairs and quantum squeezed states in plasmas Kenan Quand Nathaniel J. Fisch Department of Astrophysical Sciences,Princeton University, Princeton, New Jersey 08544, USA (Received 16 August 2024; accepted 7 November 2024; published 20 December 2024) Plasma is capable of mediating the conversion of two pump photons into two different photons through a relativistic four-wave mixing nonlinearity. Spontaneously created photon pairs are emitted at symmetric angles with respect to the colinear pump direction, and the emission rate is largest if they have identical frequency. Thus, two orthogonally polarized pumps can produce polarization-entangled photon pairs through a millimeter-long homogeneous plasma. The noise from Raman scattering can be avoided if the pump detuning differs from twice the plasma frequency. However, pump detuning exactly equal to twice the plasma frequency can significantly enhance the interaction rate, which allows for the production of strong two-mode squeezed states. Remarkably, the amplified noise from Raman scattering are correlated and hence can be suppressed in one of the output quadratures, thereby maintaining the squeezing magnitude. DOI:10.1103/Phys Rev E.110.065211 I. INTRODUCTION Quantum entangled photon pairs and quantum squeezed states are two types of the most crucial resources for quantum information science. Entangled photon pairs have nonlocal correlations to enable a variety of applications like quan- tum computing and communication. Quantum squeezed states exhibit a lower noise level in one quadrature than the vac- uum state, offering a significant advantage in high-precision measurements. Notably, its application in advanced-LIGO de- tectors [1,2]hasdemonstrablyboosteddetectionratesbyover 60%. However, the advantages of utilizing quantum nonclas- sical light are constrained by low photon flux and narrow bandwidth. This limitation results in low frame rates, typically afractionofahertz,inquantumimaging[3]andquantum spectroscopy [4]experiments,duetotherestrictedphoton generation rates. Similarly, the SU(1,1) interferometer [5,6] has yet to surpass the conventional SU(2) interferometer due to limited squeezing performance. Production of entangled photons and squeezed light typically uses spontaneous parametric down-conversion in nonlinear crystals which, in a classical description [7], arises from the anharmonic potential of the crystal electrons in astrongdrivinglaserfield. Effortstoenhancethepho- ton flux and bandwidth of nonclassical light generation fall into two categories. First, the nonlinear optics commu- nity focuses on optimizing conventional nonlinear crystals through techniques like periodic poling [8,9]. This method effectively increases photon emission rates by achieving quasi-phase matching and minimizing phase drift. Sec- ond, researchers have explored alternative systems with higher nonlinear optical susceptibilities, including optical fibers [10,11], silicon waveguides [12–14], superconducting Josephson junctions [15], cold atoms [16–18], and optome- chanical systems [19–23]. However, all these approaches employ weak laser fields, fundamentally restricting the output photon flux. Recent advancements in attosecond physics have spurred investigations into high harmonic generation [24–27] and its potential for nonclassical light production using laser intensities of 10 12−14 Wcm −2 [24,27–29]. While the non- classical nature of high harmonic photons holds promise for testing quantum theory and studying electron interactions with strong quantum light [30], their practical applications in quantum optics remain unclear. Further increasing laser intensity, however, causes ther- mal damage to conventional nonlinear materials. Plasmas, however, can maintain optical properties above the ionization laser intensity. This high thermal damage threshold positions plasma as a potential candidate for delivering the next gen- eration of high-intensity laser sources [31–36]. Additionally, plasmas exhibit strong nonlinearity [37] at high intensities, allowing rapid amplification [32,34,38]andstorage[39–44] of light pulses, and merging laser energy of multiple k J [45]. Importantly, the plasma nonlinearity scales across a broad range of frequencies, enabling manipulation from microwaves to x rays. This paper investigates the use of the relativistic four-wave mixing (FWM) nonlinearity of plasmas to produce quantum entangled photon pairs and squeezed states. Several plasma experiments since the 1980s have demonstrated degenerate FWM [46–50]usingrelativelylowlaserpowers. However, these approaches employed a Brillouin grating generated through the laser ponderomotive force, which introduces clas- sical noise and hence is not suitable for directly producing quantum light. Recently, however, for the purposes of laser upconver- sion, all-optical parametric processes have been proposed at high power in under-dense plasma, in which the plasma is used for coupling electromagnetic pulses without affecting the resonance condition. In this way, the plasma can effi- ciently mediate the conversion of near-optical laser pulses to high-energy x rays in a cascaded manner [51–55]. In particular, the relativistic FWM in plasma was proposed for converting two pump photons into two output photons at different frequencies in under-dense plasma. The relativistic FWM process uses aχ (3) nonlinearity which couples four electromagnetic waves through anharmonic electron motion 2470-0045/2024/110(6)/065211(11)065211-1©2024 American Physical Society KENAN QU AND NATHANIEL J. FISCHPHYSICAL REVIEW E110,065211(2024) caused by the relativistic effects in the strong laser field. In fact, the similar FWM process in optical fibers [10,11]has been established as a key technique for generating broadband entangled photon pairs. However, in plasma, the tolerance to extreme laser intensities enables ultrahigh FWM interac- tion rate with a centimeter-scale growth length. Crucially, the nonlinear plasma dispersion relation can be utilized to create phase matched interaction through arranging the laser wave vectors and plasma density. Entangled photon pairs can then be produced in a submillimeter-long plasma and strong squeezing can be obtained in a longer plasma. The all-optical possibilities for FWM are crucial in realiz- ing entangled photons that survive plasma noise. The faster plasma Raman scattering process, which is a lower order parametric process, is subject to classical noise effects. While the thermal bath for unseeded FWM is the vacuum fluctuation, the thermal bath for spontaneous Raman scattering (SRS) is the random fluctuation of plasma density or thermal phonons. In fact, the fast growth rate of SRS is a significant challenge for many plasma photonic applications [56,57], particularly in high-density plasmas. This paper demonstrates two methods of suppressing the SRS noise. The first method takes advantage of the discrete emission spectrum of SRS, i.e., it emits to only the Stokes and anti Stokes side bands of the pump wave. The special phase matching condition, reported in Refs. [51,52], offers a route to tailor the FWM emission frequency by tuning the pump fre- quencies and the plasma density. Thus, SRS can be effectively suppressed by detuning the pumps away from the side bands of the output modes. Because detuning from plasma resonance reduces the FWM growth rate, this approach is particularly effective only for generating single-photon-level output. The second method exploits the correlation and cancellation of the SRS noise in both output modes. Specifically, when two pumps have equal amplitudes and their beat frequency matches twice the plasma frequency, the FWM growth rate is maximized, and the system functions as a two-mode squeezer and a phase-sensitive amplifier. While plasma wave phonons are created from scattering of the higher frequency pump, they are simultaneously annihilated by interacting with the lower frequency pump, thereby maintaining a fixed amplitude. Furthermore, the amplified plasma waves couple exclusively to one quadrature of the optical state, suppressing the SRS noise in the squeezed quadrature. The paper is organized as follows: In Sec. II,weanalyze two types of interactions between intense laser pulses and plasmas. Through quantization of the plasma wave, we obtain the interaction Hamiltonian of the laser-plasma system. In Sec. III,weinvestigatetheproductionofpolarizationen- tangled photon pairs using the relativistic FWM nonlinearity of plasmas. The condition for suppressing SRS is analyzed and the logarithmic negativity is obtained. In Sec. IV,we demonstrate how a quantum two-mode squeezed state can be produced using two colinear pumps with a frequency detun- ing equal to twice the plasma frequency. The solution to the quantum Langevin equations yields the squeezing magnitude and its degradation due to thermal phonons. We show that balanced pump strength can effectively reduce the noise in one quadrature. In Sec. V,wepresentourconclusionsand discussions. II. LASER PLASMA INTERACTIONS AND HAMILTONIAN Fully ionized plasmas, consisting of both electrons and ions (or positrons), can mediate laser interactions through a variety of processes. For the purpose of creating entangled photon pairs and quantum squeezed light, we focus on para- metric processes that have fast growth rates. For this purpose, we analyze the motion of electrons in a laser field which creates a polarization current Jthat drives the laser field E through the wave equation ! ∂ 2 ∂t 2 −c 2 ∂ 2 ∂z 2 " E= 1 ε 0 ∂ ∂t J,(1) wherecis the speed of light in vacuum andε 0 is the vac- uum permittivity. In the simplest “fluid” model of plasmas, we neglect electron kinetic effects. The acceleration of the polarization current can then be written as ∂ ∂t J= e 2 n e m e γ E,(2) whereeis the natural charge,n e is the electron number den- sity,m e is the electron rest mass, andγis the relativistic Lorentz factor. The explicit inclusion ofγreveals the first laser plasma interaction: If the laser field is sufficiently strong to drive the electron to relativistic speeds, then the electron mass increases by a factorγnear the laser antinodes when it reaches its maximum kinetic energy. Assuming the electrons start from rest, the value ofγcan be obtained using the conservation of canonical momentum,i.e.,γ(t)≈ 1+α 2 (t), where we introduced the parameterα=e A/(m e c 2 )anditsamplitude αto denote the normalized laser amplitude. 1 Here,Ais the laser vector potential. We can thus expand 1/γ≈1−α 2 /2 forα<1. In a physical picture, the relativistic effect causes an anharmonic electron oscillation to lead to scatterings at different wavelengths. The second type of interaction is associated with the fluctu- ating plasma densityn e .Thedisplacementofanelectronfrom its stable position causes a electrostatic restoring force. If all the plasma electrons are driven by an external field, such as the laser ponderomotive force, then the electrostatic restoring force is amplified to drive the electrons into a longitudi- nal oscillation, called a plasma Langmuir wave. The plasma could thus exhibit a spatial density modulation which itself oscillates at the plasma frequencyω p = e 2 n e /(m e ε 0 ). The electron density ripple, functioning as a Bragg grating, scat- ters the incoming laser field. At the same time, the oscillation motion Doppler shifts the laser frequency to cause a Stokes side band and an anti-Stokes side band with a frequency detuning equal to±ω p .Thefrequency-detunedscatteredlight beats with the drive laser and reinforces the plasma oscillation. The positive feedback results in an instability, called Raman scattering. This scattering is further referred to as spontaneous Raman scattering and stimulated Raman scattering, depend- ing on whether a seed is used to initiate the instability. Here, we use the the acronym SRS for both scatterings. 1 It is usually denoted asa(ora 0 for its peak value) in the plasma community, but we reserve symbolafor quantum fields in this work. 065211-2 PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM ...PHYSICAL REVIEW E110,065211(2024) (a) (b) FIG. 1. (a) Wave-vector relations of FWM. Pump photons (k 1 andk 2 ) are directly converted into emitted photons (k 3 andk 4 ) if they satisfy the Manley-Rowe relation. (b) Wave-vector relation of SRS involving four electromagnetic waves and two plasma waves. Pump photons are converted into emitted photons through plasma waves (k p andk q ) ifω 1,2 ±ω 3,4 =ω p is also satisfied. Writingn e = ̄n e (1+δn), the wave equation (1)canbe transformed into ! ∂ 2 ∂t 2 −c 2 ∂ 2 ∂z 2 " α= ! 1+δn− α 2 2 " α.(3) The first-order Taylor expansion of 1/γshows a FWM cou- pling if the Manley-Rowe relations are satisfied,i.e.,ω 1 + ω 2 =ω 3 +ω 4 andk 1 +k 2 =k 3 +k 4 .Takingintoaccount the plasma dispersion relationω 2 =c 2 k 2 +ω 2 p ,theallowed wave vectors trace an ellipsoid as represented in 2D in Fig.1(a). The laser wave equation is accompanied by the plasma wave equation ! ∂ 2 ∂t 2 +ω 2 p " δn= c 2 2 ∇ 2 α 2 ,(4) which describes the driven motion of plasma density modu- lation by the laser ponderomotive force. Frequency matching condition shows that SRS takes place if two electromagnetic modes are detuned by the plasma frequencyω p ,whichisthe eigenfrequency of plasma oscillations and is independent of wave vector in a cold plasma. We illustrate the phase matching condition in Fig.1(b). To quantize the electromagnetic wave and the plasma wave (variablesαandδn)andtoobtaintheinteraction Hamilto- nian, we expand the wave equations to the first order using the slowly varying envelope approximation and the plasma dispersion relation. Including the plasma waves, the modes of interest can all be expanded as α= 4 $ i=1 α i e ik i r−iω i t +c.c.(5) δn=δn p e ik p ·r−iω p t +δn q e ik q ·r−iω p t +c.c.(6) Now we consider two pump modes with frequenciesω 1,2 and two output modes with frequenciesω 3,4 .Theirwave-vector relations are sketched in Fig.1.Then,weobtaintheequa- tions of motion after neglecting the fast rotating terms ! ∂ ∂t −v 3 ·∇ " α 3 = iω 2 p ω 3 (α 1 α 2 α ∗ 4 −α 1 δn ∗ p −α 2 δn q ),(7) ! ∂ ∂t −v 4 ·∇ " α 4 = iω 2 p ω 4 (α 1 α 2 α ∗ 3 −α 1 δn ∗ q −α 2 δn p ),(8) ∂ ∂t δn p = ic 2 ω p ∇ 2 (α 1 α ∗ 3 +α ∗ 2 α 4 ),(9) ∂ ∂t δn q = ic 2 ω p ∇ 2 (α 1 α ∗ 4 +α ∗ 2 α 3 ).(10) The pump fieldsα 1,2 have near relativistic intensities (10 16 –10 17 Wcm −2 ), so they can be treated classically. The variables to be quantized areα 3,4 andδn p,q .Thefieldα 3,4 can be quantized using the standard procedure [58]A= % k,s & ̄h 2ω k,s Vε 0 ˆa k,s ˆσ k,s e ik·r−iωt +H.c.,where ˆa k,s is the anni- hilation operator,Vis the normalization volume, and ˆσ i is the polarization vector. The plasma wave can be treated as a phonon which interacts with the laser by absorbing or emitting aphoton. Toquantizethephononmodeoftheplasmawave, we use the fact that the interaction Hamiltonian has the form H int =H FWM +H SRS ,(11) H FWM = ̄h)α 1 α 2 ˆa † 3 ˆa † 4 +H.c.,(12) H SRS = ̄hg[α ∗ 1 (ˆa 3 ˆp+ˆa 4 ˆq)+α ∗ 2 (ˆa 3 ˆq † +ˆa 4 ˆp † )]+H.c., (13) where)= ω 2 p 2 √ ω 3 ω 4 % i,j,k,l=1,2,3,4 (ˆσ i ·ˆσ j )(ˆσ k ·ˆσ l ). It thus leads to the relations|δn p | 2 ↔ ̄he 2 k 2 p 2Vε 0 ω 3 p m 2 e c 2 ˆp † ˆpandg= ck p 2 & ω p ω 3 .Be- causek p =k q andω 3 =ω 4 ,thenormalizationisthesame for mode ˆq. Adifferentnormalizationwouldchangethezero point fluctuation energy but would not alter the key result because the plasma wave amplitude would not grow exponen- tially as we will show. III. PRODUCING POLARIZATION ENTANGLED PHOTON PAIRS The interaction Hamiltonian (11)describestwoprocesses of creating photon pairs, including using FWM and us- ing phonon-mediated SRS. FWM is a parametric process and the electrons do not change their states after absorb- ing a pair of pump photons and emitting another pair of output photons. Thus, the combined property of the output photons, including their frequencies, wave vectors, polar- ization, and emission angles, will be identical to absorbed pump photon pair. The output states under the FWM inter- action can be expressed as|+⟩=exp[−( iz ̄hc )H FWM ]|0,0⟩= cosh −1 r % ∞ n=0 e inφ tanh n r|n,n⟩.Here,thequantumsqueezing parameter and phase is related to the interaction time via )α 1 α 2 z/c=re iφ .Thetwooutputmodeshaveanidentical photon number and are quantum correlated. For short plasma or weak pump fieldsr≪1, the output field becomes a sin- gle photon pair|1,1⟩.Variationofanyparameterofoneof the output photon will be correlated with the other photon. 065211-3 KENAN QU AND NATHANIEL J. FISCHPHYSICAL REVIEW E110,065211(2024) This process offers a mechanism to produce entangled photon pairs. However, SRS is an instability which can grow from scat- tering and amplifying a plasma wave. Because a plasma Langmuir wave is an eigenmode of the plasma medium, it could exist due to thermal effect and hence has a finite phonon number at finite temperatures. SRS of the pump wave is a fast process that creates photons that are not cor- related other photons and reduces the quantum purity of the output states. Considering degenerate frequency for the entangled photon pairsω 3 =ω 4 ,the FWMgrowthrateis α 1 α 2 )=α 1 α 2 3ω 2 p /(2ω 3 ), but the SRS growth rate isα 1,2 g≈ α 1,2 ω 3/2 p /(2ω 3 ). For plasma frequencyω p ≈ω 3 /10 and mod- erately intense laserα 1,2 ≈0.1, the SRS growth rate is larger than FWM by a factor of 10. Moreover, SRS of the pump could not be suppressed using techniques for plasma Raman amplifiers, such as a plasma density gradient, because fluc- tuations of the plasma density could broaden the scattering spectrum and shadow the entangled photon pairs. Therefore, the noise from SRS needs to be reduced by choosing the pump parameters such that the output photon frequency is detuned sufficiently far from the Stokes or anti- Stokes side band of each pump pulse. Such an arrangement spectral isolates the pump scattering from FWM and from SRS. The amount frequency detuning required depends on the linewidth of plasma resonance. For cold plasmas with Debye length longer than the plasma wavelength, the plasma density fluctuation spectrum is dominated [59,60]bythe collective dynamics and has a Lorentz shape. Its linewidth comes from Landau damping which is contributed by those few electrons in the tail of the Maxwell distribution whose velocity equals the phase velocity of the plasma oscillation. For plasma temperature of 10–100e V andk p ∼ω 1 /(10c), the Landau damping is negligible and hence SRS is negligible as long as the detuning exceeds the sum of plasma frequency and the laser linewidth. Although plasma waves are not resonantly excited in this regime, they nevertheless influence the FWM growth rate by inducing plasma oscillations (albeit not at the plasma fre- quency). It is pointed out in Ref. [51] that the FWM growth rate is to be changed to ) F = ω 2 p 2 √ ω 3 ω 4 (f 1,2 +f 1,−3 +f 1,−4 ),(14) f i,j = ' c 2 (k i +k j ) 2 (ω i +ω j ) 2 −ω 2 p −1 ( (ˆσ i ·ˆσ j )(ˆσ k ·ˆσ l ),(15) where we use the notationω −i =−ω i andk −i =−k i .Astwo mode detuningω i −ω j approaches the plasma frequencyω p , their beat drives plasma density oscillation which affects the FWM interaction rate. The polarization and wave vector of the emitted photon pairs are determined by the FWM interaction rate) F .Fortwo pumps with identical polarization, all three terms in Eq. (14) contribute to) F ,andbothoutputphotonshavethesamepo- larization. But if the two pumps have orthogonal polarization, then onlyf 1,−3 andf 1,−4 are nonzero, and they correspond to different quantum paths of photon generation:f 1,−3 is pro- portional to the probability that modes 1 and 3, and modes 2 FIG. 2. Two-color pump with orthogonal polarization creates po- larization entangled photon pairs at symmetric angles. The shades represent the most probable emission angles for given pump polarization. and 4, have the same polarization;f 1,−4 is proportional to the probability that modes 1 and 4, and modes 2 and 3, have the same polarization. A. Pumps with orthogonal polarization We first consider using two colinear pump pulses with or- thogonal polarization, as sketched in Fig.2.Theirinteraction in plasma produces photon pairs if their wave vectors satisfy the Manley-Rowe relations ω 1 +ω 2 =ω 3 +ω 4 ,(16) ω 3 = ) c 2 ! k 1 +k 2 2 +q " 2 +c 2 k 2 ⊥ +ω 2 p ,(17) ω 4 = ) c 2 ! k 1 +k 2 2 −q " 2 +c 2 k 2 ⊥ +ω 2 p ,(18) whereq=(k 3∥ −k 4∥ )/2. The possible combination of wave vectors traces an ellipse, as plotted in Fig.1(a).Photonpairs of the same frequency are emitted at angleαsuch thatk 3 =k 4 , cosα= c(k 1 +k 2 ) & (ω 1 +ω 2 ) 2 −4ω 2 p .(19) To find the probability of emission polarization, we next evaluate the value off 1,−3 +f 1,−4 .Letθ i be the polarization angle of modeiwith respect to the direction ˆ k 1 × ˆ k 3 .Using the law of cosine, we can write ˆσ 1 ·ˆσ −3 =cosθ 1 cosθ 3 +sinθ 1 sinθ 3 cosα = & cos 2 θ 1 +sin 2 θ 1 cos 2 αcos(θ 3 −φ 3 ) = & 1−sin 2 θ 1 sin 2 αcos(θ 3 −φ 3 ),(20) where tanφ 3 =cosαtanθ 1 .For ˆσ 2 ·ˆσ −4 ,weuseθ 2 =θ 1 − π/2, then ˆσ 2 ·ˆσ −4 =sinθ 1 cosθ 4 −cosθ 1 sinθ 4 cosα = & sin 2 θ 1 +cos 2 θ 1 cos 2 αcos(θ 4 −φ 4 ) = & 1−cos 2 θ 1 sin 2 αcos(θ 4 −φ 4 ),(21) 065211-4 PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM ...PHYSICAL REVIEW E110,065211(2024) FIG. 3. Values of ˆσ 1 ·ˆσ −3 and ˆσ 2 ·ˆσ −4 for cosα=0.8, and val- ues of ( ˆσ 1 ·ˆσ −3 )(ˆσ 2 ·ˆσ −4 )atθ 1 =π/4forcosα=0.8. where tanφ 4 =cosαtan(θ 1 + π 2 ). The values of ˆσ 1 ·ˆσ −3 and ˆσ 2 ·ˆσ −4 for differentαare plotted in Fig.3.Forα∼0, each term reaches its maximum value nearθ 1 andθ 1 −π/2, respec- tively. The product ( ˆσ 1 ·ˆσ −3 )(ˆσ 2 ·ˆσ −4 )reachesitsmaximum whenθ 1 = π 4 .Thesameresultcanbeobtainedforf 1,−4 .At this angle, the probability that mode 3 is polarized at angleθ 3 and mode 4 is polarized at angleθ 4 is proportional to f 1,−3 +f 1,−4 = ' c 2 (k 1 −k 3 ) 2 (ω 1 −ω 3 ) 2 −ω 2 p −1 ( ×(1+cos 2 α)cos(θ 3 −φ 30 )cos(θ 4 −φ 40 ), (22) tanφ 30 =cosα,tan * φ 40 + π 2 + = 1 cosα .(23) This probability function is plotted in Fig.3,showingthat the two modes have the maximum probability of polarizing at different angles tanφ 30 and tanφ 40 ,respectively. Therefore, polarization entangled photon pairs can be col- lected at the azimuthal angleαdefined in Eq. (19)andthe polar angleθ 1 =π/4. Now we define the horizontal polariza- tion at the angleθ 1 ,thenthebiphotonstatecanbewrittenas |+⟩= 1 √ 2 cos(φ 30 −φ 40 )(|H,V⟩+|V,H⟩) + 1 √ 2 sin(φ 30 −φ 40 )(|H,H⟩+|V,V⟩).(24) The photon pairs become entangled if either of the two coeffi- cients is significantly larger than the other. Quantitatively, the entanglement can be measured using logarithmic negativity, E N =log[2 cos(φ 30 −φ 40 )].(25) Thus, the emitted photon pair is in an entangled state if cos(φ 30 −φ 40 )>1/2, which limits the plasma frequency and the accompanying two-pump detuning. B. Pumps with the same polarization We next check the possibility of producing entangled pho- ton pairs using two colinear pump pulses with the same polarization. They produce output fields with the same polar- ization, which could be different from the pump polarization. All the terms, includingf 1,2 ,f 1,−3 ,andf 1,−4 , contribute to the production of photon pairs. Assume the pumps are horizon- tally polarized and letθ i be the polarization angle of modei with respect to the pump fields. The termf 1,2 allows creation of both horizontally or vertically polarized photon pairs, i.e., f 1,2 = ' c 2 (k 1 +k 2 ) 2 (ω 1 +ω 2 ) 2 −ω 2 p −1 ( ×[cosθ 3 cosθ 4 +sinθ 3 sinθ 4 cos(2α)].(26) However, becauseω 1 +ω 2 ≫ω p , the term in the first square bracket is strongly suppressed. The photon creation is domi- nated by the termsf 1,−3 andf 1,−4 ,whichequaltocosθ 3 cosθ 4 and become 0 for vertically polarized photon pairs. Therefore, it is impractical to produce polarization entangled photon pairs using two colinear pump pulses with the same polariza- tion. C. Rate of photon-pair emission The photon emission rate is determined by the product of the growth rate) F and the pump amplitudesα 1 α 2 .The value of) F increases asω 1 –ω 3 approachesω p .But,toonear the resonance point, strong spontaneous Raman scattering induces unwanted noise photons. As an example, consider the first pump with a wavelengthλ 1 =1μm, and plasma with a density of 3.5×10 15 cm −3 ,correspondingtoω p =0.1ω 2 .The second pump frequency is chosen well beyond the plasma res- onance frequency,ω 2 =ω 1 −4ω p .Inthisconfiguration,we find that the output photon frequency isω 3 =ω 4 =0.8ω 1 ,or λ 3 =λ 4 =1.25 μm, and they are separated by a 3.76 ◦ angle. For pump amplitudeα 1 α 2 =0.01, the FWM growth rate is 065211-5 KENAN QU AND NATHANIEL J. FISCHPHYSICAL REVIEW E110,065211(2024) ) F =0.0001ω 1 . Thus, a single photon pair (r≈0.15) could be produced in a 1.5mmlongplasma. Because the plasma frequency can be flexibly controlled through its density, a wide range of frequencies of the output entangled photon pairs are possible. If we instead consider a soft x-ray pump withλ=10 nm andα 1 α 2 =0.0001, then an entangled photon pair can be created in the same plasma length, if the plasma density is increased to 3.5×10 19 cm −3 . D. Effects of spontaneous Raman scattering In subpicosecond timescales, spontaneous Raman scatter- ing is the main process that destroys the entanglement. It converts a photon into a different frequency by either absorb- ing or creating a plasma phonon in the form of Langmuir wave. SRS introduces noise via two routes. First, the thermal phonon scatters one of the entangled photon pairs into a dif- ferent frequency and a different angle, causing direct loss of quantum correlation. But the scattering is negligible with low photon and phonon numbers. If we assume an equipartition theorem for plasma waves, and consider plasma phonon en- ergy to be ̄hω p ∼0.1e V (ω p equals to 1/10 of optical laser frequency), then a cold plasma at∼1e Vtemperatureonlyhas an average thermal phonon number of only ̄n th ≈10. Second, SRS can scatter the pump photons into the out- put modes. The created photons are not correlated with the entangled photon pairs, reducing the quantum purity of the output states. This process, however, could not be suppressed using techniques for plasma Raman amplifiers, such as a plasma density gradient, because fluctuations of the plasma density could broaden the scattering spectrum and shadow the entangled photon pairs. Suppression of SRS requires reducing the thermal photon number of plasma oscillation at frequency ω 1 –ω 3 . IV. PRODUCING QUANTUM SQUEEZED STATES AND SUPPRESSING STIMULATED RAMAN SCATTERING If multiple photon pairs could be produced, then the out- put becomes a quantum two-mode squeezed state|+⟩= cosh −1 r % ∞ n=0 e inφ tanh n r|n,n⟩.Thetwooutputmodeshave strong quantum correlation similar to the quantum entangled photon pairs but higher brightness than a single photon pair. What is remarkable about two-mode squeezed states is that covariance of their quadratures has below-shot-noise-level fluctuations. If the two output modes are linearly combined using a beam splitter, then the output becomes a continu- ous variable (CV) quantum entangled state. The squeezing operation can control the quantum noise which is encoded for quantum communication and manipulated for quantum computation. Because the angle for squeezed quadrature can be continuously tuned, there are unparalleled advantages com- pared to discrete variable (DV) quantum information, which uses entangled photon pairs. To achieve a significant squeezing magnitude, the system needs to have high photon emission rate and low noise input in the squeezed quadrature. To fulfill both requirements, we next consider pumping FWM using a bicolor pump with a detuning ω 1 −ω 2 =2ω p , which yields the maximum FWM interaction rate. This frequency configuration is avoided for producing entangled photon pairs, because it induces rapid scattering from thermal phonons. However, because quantum squeezing reduces noise only in one quadrature, the SRS noise might not degrade the squeezing magnitude if the they are induced in a correlated manner, i.e., the noise photon correlation could be made use of for noise cancellation. The enhancement of FWM growth rate)can significantly increase the photon emission rate. For example, if we only change the pump wavelength to satisfy|ω 1,2 −ω 3,4 |=ω p but keep other parameters as given in the last section, then) increases to a value similar tog∼0.015ω 1 , which is more than one order of magnitude higher than the value given in the last section. Under this condition, a 1 mm long plasma and 30 fs pump pulse could result in more than 10 9 photons. If the plasma length and laser duration is doubled, then the photon number would reach 10 17 ,whichcontainsenergyof0.01 J. Wth both FWM and SRS involved, the state evolu- tion is U=exp[− i ̄h , H int (t)dt]=D 3 (β 3 )D 4 (β 4 )S(ξ), where D i (β i )=exp(β i ˆa i −β ∗ i ˆa † i )isthedisplacementoperator, S(ξ)=exp(ξˆa † 3 ˆa † 4 −ξ ∗ ˆa 3 ˆa 4 )isthetwo-modesqueezingop- erator,ξ=−i) F β 1 β 2 ,β 3,4 =−i(α ∗ 1,2 g ̄p−α ∗ 2,1 g ̄q), and ̄pand ̄qare the average phonon amplitudes. The output state can then be expressed in the form of a two-mode squeezed coher- ent state|+ out ⟩=D 3 (β 3 )D 4 (β 4 )S(ξ)|0,0⟩.Interestingly,the displacement operators D 3 and D 4 shift the states with the same quadrature angle and amplitude if ̄p= ̄q. Thiscoher- ent component might, therefore, be canceled mutually if the modes are linearly combined. The bicolor pump configuration has been studied in opti- cal cavities involving oscillating a mechanical oscillator [20]. It shows that asymmetric pump produces strong quantum squeezing and symmetric pump produces phase sensitive amplification. The method of producing squeezing using asymmetric pumps, however, cannot be directly applied to the plasma medium, because it lacks the optical cavity for noise filtering. What could be used here is the combination of symmetric pumping and FWM. While a photon-phonon pair is produced by the high-frequency pump, the phonon is converted into another photon by the low-frequency pump. Thus, it creates correlated photon pairs. It also ensures that the phonon amplitude maintains a finite value, so that the me- diating plasma wave can maintain a finite amplitude without an exponential growth. However, the thermal noise is driven by plasma wave relaxation which cannot be analyzed through the Hamilto- nian itself. Instead, the photon emission process with thermal phonon noise taken into account is described by the quantum Langevin equations (QLE) ∂ ∂t "=L"+ √ 2κ" in ,(27) where we neglected the convection operator,"= (ˆa 3 ,ˆa † 4 ,ˆp † ,ˆq) T ," in =(0,0,ˆp † in ,ˆq in ) T ,and L= ⎛ ⎜ ⎜ ⎜ ⎝ 0iα 1 α 2 ) F −iα 1 g−iα 2 g −iα ∗ 1 α ∗ 2 ) F 0iα ∗ 2 giα ∗ 1 g iα ∗ 1 giα 2 g−κ0 −iα ∗ 2 g−iα 1 g0−κ ⎞ ⎟ ⎟ ⎟ ⎠ .(28) 065211-6 PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM ...PHYSICAL REVIEW E110,065211(2024) Here,κis the plasma wave relaxation rate mainly due to Landau damping at finite plasma temperature, andp in (t) andq in (t) are the phonon noise operator associated with the plasma waves. We assume their correlation functions have the similar form with those for Brownian motions ⟨p † in (t)p in (t ′ )⟩= ̄n p δ(t−t ′ ), ⟨p in (t)p † in (t ′ )⟩=( ̄n p +1)δ(t−t ′ ), ⟨q † in (t)q in (t ′ )⟩= ̄n q δ(t−t ′ ), ⟨q in (t)q † in (t ′ )⟩=( ̄n q +1)δ(t−t ′ ),(29) where ̄n p,q is the average thermal phonon number for modep andq. The thermal phonon noise is driven by plasma wave relax- ations, including Landau damping, collisional damping, and other decoherence processes. These relaxation processes are different from other systems like atoms and mechanical oscil- lators because the plasma electrons are not directly coupled to external reservoirs. Plasma waves decay mainly from phase mixing due to Landau damping or collision, i.e., by coupling to other plasmon modes. It also means that the thermal reser- voir is completely determined by the initial condition of the plasma system, in absence of laser interactions, which poten- tially leads to fewer thermally excited phonons compared to, e.g., a mechanical oscillating mirror. To satisfy ̄p= ̄q,weassumeequalpumpamplitudewith arealvalueα 1 =α 2 =α. Then, the QLEs have a simple solution ˆa 3 (t)=cosh() F α 2 t)ˆa 3 (0)+isinh() F α 2 t)ˆa † 4 (0) + 3 t 0 ' 1+i 2 4 ie −) F α 2 t ′ ) F α 2 −κ − e ) F α 2 t ′ ) F α 2 +κ 5 + ) F α 2 −iκ () F α 2 ) 2 −κ 2 e −κt ′ ( αg √ 2κ[p † in (t ′ )+q in (t ′ )]dt ′ , (30) ˆa 4 (t)=isinh() F α 2 t)ˆa † 3 (0)+cosh() F α 2 t)ˆa 4 (0) × 3 t 0 ' 1+i 2 4 ie −) F α 2 t ′ ) F α 2 −κ − e ) F α 2 t ′ ) F α 2 +κ 5 + ) F α 2 −iκ () F α 2 ) 2 −κ 2 e −κt ′ ( αg √ 2κ[p in (t ′ )+q † in (t ′ )]dt ′ . (31) The right-hand side of each expression has three terms. The first two terms denote the effect of FWM on the vacuum noise. In the square bracket of the integrand, the first term represents the noise amplification (and deamplification) due to SRS of the high-frequency and low-frequency pumps, and the last term describes the noise due to thermalization. The similar forms ofa 3 anda † 4 suggest the possibility of partial cancellation of the noise from phonons. Indeed, the correlation between the two modes is revealed in the X 3 and Y 4 quadratures, where X i =(ˆa i +ˆa † i )/ √ 2and Y i = (ˆa i −ˆa † i )/( √ 2i)fori=3, 4. Specifically, the correlation FIG. 4. The maximum squeezing magnitude (top) and the op- timal plasma length L=ct cr (bottom) for different plasma wave damping rateκand thermal phonon number ̄n. Otherparametersare given in the text. function V − =⟨(X 3 −Y 4 ) 2 ⟩=e −2) F α 2 t + 6 1−e −2κt 2κ + 1−e −2) F α 2 t 2) F α 2 − 2(1−e −() F α 2 +κ)t ) ) F α 2 +κ 7 × α 2 g 2 4κ (α 2 ) F −κ) 2 (2 ̄n p +1)(32) only includes terms proportional toe −2) F α 2 t ande −2κt but does not have any exponentially growing terms. The first term, representing squeezed vacuum noise, monotonically de- creases. The second term, representing thermal phonon noise, saturates at 2α 2 g 2 (2 ̄n+1)/[)α 2 ()α 2 +κ)](≫1). However, thermalization can be negligible in a short timeκt≪1 owing to the small Landau damping rate in cold plasmas. Hence,V − ≈2(g/α)) 2 (2 ̄n+1)(1−e −2κt )couldbelower than the vacuum level at a certain time. Exactly, the maximum squeezing magnitude is obtained att cr =(α 2 )−κ) −1 ln[1+ & α 2 )(α 2 )−κ) 2 2κα 2 g 2 (2 ̄n+1) ]. The maximum squeezing magnitude and the optimal plasma length are plotted in Fig.4.Itisseenthatthe squeezing magnitude crucially relies on a large FWM growth 065211-7 KENAN QU AND NATHANIEL J. FISCHPHYSICAL REVIEW E110,065211(2024) rateα 2 ),asmallplasmadecayrateκand a small thermal phonon number ̄n. The orthogonal correlation exhibits amplified thermal noise due to SRS V + =⟨(X 3 +Y 4 ) 2 ⟩=e 2) F α 2 t + ' 1−e −2κt 2κ − 1−e 2) F α 2 t 2) F α 2 t − 2(1−e () F α 2 −κ)t ) ) F α 2 −κ ( α 2 g 2 4κ (α 2 ) F +κ) 2 ×(2 ̄n p +1).(33) A. Single-mode squeezed state The two-mode squeezed output can be converted into a single-mode squeezed state using a beam splitter. The output state ˆccan be written as ˆc=(ˆa 3 −iˆa 4 )/ √ 2, which obeys the commutation relation [ˆc(t),ˆc † (t ′ )]=δ(t−t ′ ). The spectra of its quadratures X c =(ˆc+ˆc † )/ √ 2and Y c =(ˆc−ˆc † )/( √ 2i) can be found S Xc = 8 X 2 c 9 = 1 2 e −2) F α 2 t + ' 1−e −2κt 2κ + 1−e −2) F α 2 t 2) F α 2 − 2(1−e −() F α 2 +κ)t ) ) F α 2 +κ ( × α 2 g 2 2κ (α 2 ) F −κ) 2 (2 ̄n p +1),(34) S Yc = 8 Y 2 c 9 = 1 2 e 2) F α 2 t ' 1−e −2κt 2κ − 1−e 2) F α 2 t 2) F α 2 t − 2(1−e () F α 2 −κ)t ) ) F α 2 −κ ( α 2 g 2 2κ (α 2 ) F +κ) 2 (2 ̄n p +1). (35) It is seen that the spectrum of Xquadrature is squeezed for certain timet. Thespectrumof Yquadrature, however, shows antisqueezed noise fluctuation. Although squeezing is obtained, excessive noise degrades the state purity Trρ 2 =1/ √ detσ.Itcanbeobtainedusing the covariance matrixσ=( 2⟨X 2 c ⟩⟨X c Y c +Y c X c ⟩ ⟨X c Y c +Y c X c ⟩2⟨Y 2 c ⟩ ), where ⟨X c Y c ⟩=⟨Y c X c ⟩=0andweused⟨X c ⟩=⟨Y c ⟩=0. Thus, the state purity is Trρ 2 =4S Xc S Yc ,whichdecreasesathigher thermal phonon number ̄n p and larger plasma wave relaxation rateκ. V. CONCLUSIONS In conclusion, we investigated the use of ionized plasmas and ultraintense laser pulses to generate quantum states of light with high photon flux and broad bandwidth. Our model demonstrates the ability of the relativistic FWM nonlinear susceptibility to convert two pump photons into two output photons at distinct frequencies and angles. The all-optical parametric nature of FWM ensures that the properties of the output photon pairs are solely determined by the pump photon pairs, independent of plasma resonances, though plasma den- sity influences the photon emission rate. To mitigate classical noise from SRS, the pump frequencies are tailored to ensure substantial detuning of the output frequencies from the Stokes and anti Stokes sidebands. Employing orthogonally polarized dual-color pumps then enables the generation of polarization- entangled photon pairs. Setting the pump detuning to twice the plasma frequency enhances both FWM and SRS interaction rates. While this increases SRS noise in both output modes, the quantum cor- relation of this noise allows for its suppression in one of the quadratures. This configuration facilitates the production of quantum two-mode squeezed states. It should be noted that the result is obtained by assum- ing a Brownian-noise-like correlation function [Eqs. (29)] for the plasma thermal phonons. Plasmas follow the same fluctuation-dissipation theorem of statistical dynamics near equilibrium [61]thatdescribestheemissionof Brownianpar- ticles. However, note several distinctive features of plasma fluctuations. First, while the Brownian motion of particles is captured in the frequency spectrum of the particle energy distributions, the fluctuation of plasma waves is dependent on both frequency and wave vector. This leads to the second difference between particle fluctuations and wave fluctuations in that the plasma wave relaxation is caused by dephasing of different wave vectors, known as Landau damping, but Brownian motion decays mainly due to Stokes dragging. Third, plasma couples to electromagnetic waves via its dis- crete particle distributions, but Brownian particles radiate due to random acceleration by thermal fluctuations. Nevertheless, the two different models should be equivalent in the limit that the plasma wavelength is shorter than the Debye length and the radiating elements are purely collective plasma wave phonons. The calculation also neglects the large mismatch of the group velocities between the laser pulses and the plasma waves. A more accurate analysis needs to conduct quantum numerical simulations of the laser-plasma interaction pro- cesses to fully characterize the produced quantum states. The simulations should include more modes that we have to ne- glect in analytical calculations. The most important modes are the series of Stokes and anti Stokes sidebands of the strong squeezed mode. As the output amplitude grows, SRS is shown to be capable of broadening the spectrum into a frequency comb [62]withspacingofω p .Thecombmodes,whichare produced through cascaded SRS, could show unique quantum features. These processes, as well as other plasma instabilities, will be investigated using particle-in-cell simulations in future works. The advantage of utilizing plasmas lies in their capac- ity for high photon flux and broad bandwidth generation. Plasma excels in mediating short-wavelength light, like x rays, compared to conventional materials. For example, consider a soft x-ray pump withλ=10 nm. Even if the pumps have an amplitude ofα 1 ≈α 2 =0.00001, the same output photon numbers could be produced in the same plasma length if the plasma density is increased to 3.5×10 19 cm −3 .Withmildly relativistic laser intensities, each millimeter-long plasma can produce nine orders of magnitude growth of photon fluxes before the pump is depleted. 065211-8 PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM ...PHYSICAL REVIEW E110,065211(2024) The plasma-based methods for generating ultrastrong and broadband quantum light open new avenues for di- verse applications in science and technology, including enhanced x-ray imaging, quantum metrology, and x-ray nuclear spectroscopy [63–67]. A notable example is its po- tential application in quantum lithography [68–73], a concept proposed to surpass the Rayleigh diffraction limit using quantum light sources. 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