Qu.Entanglement.Pre2024 (1)

Kenan Qu

Editors’ Suggestion

I. INTRODUCTION

and Nathaniel J. Fisch

DOI: 10.1103/PhysRevE.110.065211

PHYSICAL REVIEW E 110, 065211 (2024)

(Received 16 August 2024; accepted 7 November 2024; published 20 December 2024)

Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA

Producing entangled photon pairs and quantum squeezed states in plasmas

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.

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 a strong driving laser field. Efforts to enhance the pho- 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 including optical higher nonlinear optical susceptibilities, 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

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] has demonstrably boosted detection rates by over 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 a fraction of a hertz, in quantum imaging [3] and quantum spectroscopy [4] experiments, due to the restricted photon generation rates. Similarly, the SU(1,1) interferometer [5,6] has yet to surpass the conventional SU(2) interferometer due to limited squeezing performance.

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 a χ (3) nonlinearity which couples four electromagnetic waves through anharmonic electron motion

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] and storage [39-44] of light pulses, and merging laser energy of multiple kJ [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] using relatively low laser powers. 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.

intensities of 1012 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.

2470-0045/2024/110(6)/065211(11)

14 W cm− −

065211-1

(1)

∂ ∂t

ε0

c2 ∂ 2 ∂z2

∂ 2 ∂t 2 −

KENAN QU AND NATHANIEL J. FISCH

PHYSICAL REVIEW E 110, 065211 (2024)

II. LASER PLASMA INTERACTIONS AND HAMILTONIAN

” where c is the speed of light in vacuum and ε0 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

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 J that drives the laser field E through the wave equation

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.

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.

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 antiStokes 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 second type of interaction is associated with the ﬂuctu- ating plasma density ne. The displacement of an electron from 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 e2ne/(meε0). The oscillates at the plasma frequency ωp 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 ωp. The frequency-detuned scattered light detuning equal to 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.

The paper is organized as follows: In Sec. II, we analyze 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, we investigate the production of polarization en- 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, we present our conclusions and discussions.

The explicit inclusion of γ reveals 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 factor γ near the laser antinodes when it reaches its maximum kinetic energy. Assuming the electrons start from rest, the value of γ can be obtained using the conservation α2(t ), where we of canonical momentum, i.e.,γ (t ) eA/(mec2) and its amplitude introduced the parameter α

α to denote the normalized laser amplitude.1 Here, A is the α2/2 laser vector potential. We can thus expand 1/γ for α < 1. In a physical picture, the relativistic effect causes an anharmonic electron oscillation to lead to scatterings at different wavelengths.

where e is the natural charge, ne is the electron number den- sity, me is the electron rest mass, and γ is the relativistic Lorentz factor.

1It is usually denoted as a (or a0 for its peak value) in the plasma community, but we reserve symbol a for quantum fields in this work.

e2ne meγ

065211-2

∂ ∂t

(2)

≈

−

r ·

iωt

k,s

”

(a)

(b)

α4

α3

(9)

(8)

(7)

· ∇

δnq

(10)

α∗2 α3).

α∗2 α4),

α2δnq),

α2δnp),

ω3,4 =

¯h 2ωk,sV ε0

ˆak,s ˆσk,seik

α1δn∗q −

α1δn∗p −

ωp is also satisfied.

(α1α2α∗4 −

∂ ∂t − ∂ ∂t −

Writing ne

transformed into

PHYSICAL REVIEW E 110, 065211 (2024)

· ∇ ” ∂ δnp ∂t ∂ ∂t

PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM …

FIG. 1. (a) Wave-vector relations of FWM. Pump photons (k1 and k2) are directly converted into emitted photons (k3 and k4) 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 (kp and kq) if ω1,2 ±

(α1α2α∗3 − 2(α1α∗3 + 2(α1α∗4 + The pump fields α1,2 have near relativistic intensities (1016-1017 Wcm− 2), so they can be treated classically. The variables to be quantized are α3,4 and δnp,q. The field α3,4 can be quantized using the standard procedure [58] A

relations are sketched in Fig. 1. Then, we obtain the equa- tions of motion after neglecting the fast rotating terms iω2 p ω3 iω2 p ω4 ic2 ωp ∇ ic2 ωp ∇

The interaction Hamiltonian (11) describes two processes 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 ⟩ = | ⟩ = 0 einϕ tanhn r n, n . Here, the quantum squeezing cosh− ∞n ⟩

parameter and phase is related to the interaction time via reiϕ. The two output modes have an identical )α1α2z/c photon number and are quantum correlated. For short plasma or weak pump fields r 1, the output field becomes a sin- 1, 1 gle photon pair . Variation of any parameter of one of ⟩ the output photon will be correlated with the other photon.

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 frequency ωp, which is the eigenfrequency of plasma oscillations and is independent of wave vector in a cold plasma. We illustrate the phase matching condition in Fig. 1(b).

= H.c., where ˆak,s is the anni- + hilation operator, V is the normalization volume, and ˆσi is % the polarization vector. The plasma wave can be treated as a phonon which interacts with the laser by absorbing or emitting a photon. To quantize the phonon mode of the plasma wave, we use the fact that the interaction Hamiltonian has the form

. Be- to the relations 2V ϵ0ω3 ω4, the normalization is the same cause kp for mode ˆq. A different normalization would change the zero point ﬂuctuation energy but would not alter the key result because the plasma wave amplitude would not grow exponen- tially as we will show.

The first-order Taylor expansion of 1/γ shows a FWM cou- pling if the Manley-Rowe relations are satisfied, i.e.,ω1 + ω4 and k1 ω2 k4. Taking into account k3 the plasma dispersion relation ω2 c2k2 p, the allowed wave vectors trace an ellipsoid as represented in 2D in Fig. 1(a).

To quantize the electromagnetic wave and the plasma wave (variables α and δn) and to obtain the interaction 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

Now we consider two pump modes with frequencies ω1,2 and two output modes with frequencies ω3,4. Their wave-vector

III. PRODUCING POLARIZATION ENTANGLED PHOTON PAIRS

The laser wave equation is accompanied by the plasma

ˆσ j )( ˆσk · e c2 ˆp† ˆp and g

= 3 ˆa† ¯h)α1α2 ˆa†

δn), the wave equation (1) can be

1,2,3,4( ˆσi

¯he2k2 p pm2

δnp % | | kq and ω3

4 + α∗2 (ˆa3 ˆq†

ˆσl ). It thus leads

( iz ¯hc )HFWM]

∂ 2 ∂t 2 +

∂ 2 ∂t 2 −

c2 ∂ 2 ∂z2

wave equation

¯hg[α∗1 (ˆa3 ˆp

ω2 p 2sqrtω3ω4

c2 2 ∇

$

r ·

αieikir −

δnpeikp

δnqeikq

where )

ˆa4 ˆp†)]

065211-3

%

↔

HFWM

HSRS,

α2

HSRS

2α2,

H.c.,

ˆa4 ˆq)

¯ne(1

exp[

(11)

(12)

(13)

Hint

0, 0

ω2 p

ckp

c.c.

ωp ω3

i, j,k,l

(3)

(4)

(5)

(6)

ω2

1 r

ω3

≪

δn

iωpt

α.

iωit

−

”

r ·

−

≈

KENAN QU AND NATHANIEL J. FISCH

PHYSICAL REVIEW E 110, 065211 (2024)

p/(2ω3), but the SRS growth rate is α1,2g

This process offers a mechanism to produce entangled photon pairs.

and 4, have the same polarization; f1, 4 is proportional to the probability that modes 1 and 4, and modes 2 and 3, have the same polarization.

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.

≈ α1,2ω3/2 p /(2ω3). For plasma frequency ωp ω3/10 and mod- 0.1, the SRS growth rate is larger erately intense laser α1,2 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] by the 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. ω1/(10c), the For plasma temperature of 10-100eV and kp Landau damping is negligible and hence SRS is negligible as long as the detuning exceeds the sum of plasma frequency and the laser linewidth.

The polarization and wave vector of the emitted photon pairs are determined by the FWM interaction rate )F . For two pumps with identical polarization, all three terms in Eq. (14) contribute to )F , and both output photons have the same po- larization. But if the two pumps have orthogonal polarization, then only f1, 4 are nonzero, and they correspond to different quantum paths of photon generation: f1, 3 is pro- portional to the probability that modes 1 and 3, and modes 2

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

To find the probability of emission polarization, we next 4. Let θi be the polarization f1, ˆk3. Using

where we use the notation ω ki. As two ω j approaches the plasma frequency ωp, mode detuning ωi their beat drives plasma density oscillation which affects the FWM interaction rate.

We first consider using two colinear pump pulses with or- thogonal polarization, as sketched in Fig. 2. Their interaction in plasma produces photon pairs if their wave vectors satisfy the Manley-Rowe relations

)/2. The possible combination of wave where q vectors traces an ellipse, as plotted in Fig. 1(a). Photon pairs of the same frequency are emitted at angle α such that k3 k4,

evaluate the value of f1,

− angle of mode i with respect to the direction ˆk1 the law of cosine, we can write

− cos2 θ1 cos2 α cos(θ4

sin2 θ1 cos2 α cos(θ3

A. Pumps with orthogonal polarization

where tan ϕ3 π /2, then

& cos α tan θ1. For ˆσ2

cos2 θ1 sin2 α cos(θ4

sin2 θ1 sin2 α cos(θ3

ω2 p 2sqrtω3ω4

cos θ1 sin θ4 cos α

sin θ1 sin θ3 cos α

k2 + 2 −

k2 + 2 +

4, we use θ2

cos θ1 cos θ3

sin θ1 cos θ4

3 and f1,

ω2)2

ωi and k

cos2 θ1

sin2 θ1

ω j )2

065211-4

ˆσ j )( ˆσk

c2(ki

(

= ’

= )

cos α

k j )2

4ω2 p

c2k2

( f1,2

= −

ϕ4),

ϕ3),

ω2 p,

p −

c(k1

ˆσl ),

(20)

(21)

(15)

(16)

(17)

(18)

(19)

(14)

⊥ +

ω4,

(ω1

ϕ4)

ϕ3)

∥ −

k2)

(ωi

(k3

4),

( ˆσi

fi, j

ω1

ω3

ω2

ω4

−

ω2

ˆσ1

ˆσ2

f1,

i −

θ1

∼

−

”

ˆσ

−

∥

)

⟩

−

⟩ =

(24)

V, V

H, V

V, H

⟩ + |

H, H

ϕ40)(

ϕ40)].

sin(ϕ30

cos(ϕ30

log[2 cos(ϕ30

sqrt2

PHYSICAL REVIEW E 110, 065211 (2024)

PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM …

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,

This probability function is plotted in Fig. 3, showing that the two modes have the maximum probability of polarizing at different angles tan ϕ30 and tan ϕ40, respectively.

Therefore, polarization entangled photon pairs can be col- lected at the azimuthal angle α defined in Eq. (19) and the polar angle θ1 π /4. Now we define the horizontal polariza- tion at the angle θ1, then the biphoton state can be written as

The photon emission rate is determined by the product of the growth rate )F and the pump amplitudes α1α2. The value of )F increases as ω1-ω3 approaches ωp. But, too near the resonance point, strong spontaneous Raman scattering induces unwanted noise photons. As an example, consider the first pump with a wavelength λ1 1 µm, and plasma with a density of 3.5 0.1ω2. The second pump frequency is chosen well beyond the plasma res- onance frequency, ω2 4ωp. In this configuration, we 0.8ω1, or find that the output photon frequency is ω3 λ3 1.25 µm, and they are separated by a 3.76◦ angle. For pump amplitude α1α2 0.01, the FWM growth rate is

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, including f1,2, f1, 4, contribute to the production of photon pairs. Assume the pumps are horizon- tally polarized and let θi be the polarization angle of mode i with respect to the pump fields. The term f1,2 allows creation of both horizontally or vertically polarized photon pairs, i.e.,

ˆσ where tan ϕ4 3 and − ˆσ2 0, each π /2, respec- term reaches its maximum value near θ1 and θ1 tively. The product ( ˆσ1 ˆσ 4) reaches its maximum · π when θ1 4 . The same result can be obtained for f1, 4. At − this angle, the probability that mode 3 is polarized at angle θ3 and mode 4 is polarized at angle θ4 is proportional to

However, because ω1 ωp, the term in the first square bracket is strongly suppressed. The photon creation is domi- 4, which equal to cos θ3 cos θ4 nated by the terms f1, − 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.

Thus, the emitted photon pair is in an entangled state if cos(ϕ30 ϕ40) > 1/2, which limits the plasma frequency and the accompanying two-pump detuning.

π 2 ). The values of ˆσ1 + 4 for different α are plotted in Fig. 3. For α

FIG. 3. Values of ˆσ1 · ˆσ

− − cos2 α) cos(θ3

= 3, corresponding to ωp

4 for cos α − π /4 for cos α

B. Pumps with the same polarization

− [cos θ3 cos θ4

C. Rate of photon-pair emission

ˆσ 4) at θ1 =

sin θ3 sin θ4 cos(2α)].

ues of ( ˆσ1 ·

3 and ˆσ2 ·

cos α tan(θ1

ϕ40), (22)

cos α

ϕ30) cos(θ4

1015 cm−

0.8, and val-

3, and f1,

3)( ˆσ2 ·

3 and f1,

ω2)2

− ω3)2

065211-5

tan ϕ30

c2(k1

cos α,

(

3)( ˆσ2

= ’

k2)2

k3)2

= 0.8.

p −

· ∼

(25)

(23)

(26)

(ω1

ϕ40

f1,2

tan

ω2

ω1

ω4

−

ω2

≫

λ4

f1,

−

ˆσ

−

i ¯h

∼

−

≈

| =

ω1,2

ω3,4

1019 cm−

0.15) could

10 nm and α1α2

KENAN QU AND NATHANIEL J. FISCH

PHYSICAL REVIEW E 110, 065211 (2024)

D. Effects of spontaneous Raman scattering

Wth both FWM and SRS involved,

0.0001ω1. Thus, a single photon pair (r

)F be produced in a 1.5 mm long plasma.

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 with λ 0.0001, then an entangled photon pair can be created in the same plasma length, if the plasma density is increased to 3.5

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 ωp but change the pump wavelength to satisfy keep other parameters as given in the last section, then ) 0.015ω1, which is more increases to a value similar to g 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 109 photons. If the plasma length and laser duration is doubled, then the photon number would reach 1017, which contains energy of 0.01 J.

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 ¯hωp 0.1 eV (ωp equals to 1/10 of optical laser ∼ frequency), then a cold plasma at 1 eV temperature only has an average thermal phonon number of only ¯nth

put becomes a quantum two-mode squeezed state ⟩ = cosh− . The two output modes have ⟩ 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.

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.

the state evolu- D3(β3)D4(β4)S(ξ ), where Hint(t )dt] tion is U exp[

− β∗i ˆa† exp(βi ˆai Di(βi ) is the displacement operator, i ) , − 3 ˆa† ξ ∗ ˆa3 ˆa4) is the two-mode squeezing op- S(ξ ) 4 −

erator, ξ α∗2,1g ¯q), and ¯p and i)F β1β2, β3,4 = − ¯q are the average phonon amplitudes. The output state can then be expressed in the form of a two-mode squeezed coher- ent state . Interestingly, the ⟩ displacement operators D3 and D4 shift the states with the same quadrature angle and amplitude if ¯p ¯q. This coher- ent component might, therefore, be canceled mutually if the modes are linearly combined.

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 ω1-ω3.

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 ω1 2ωp, which yields the maximum FWM interaction rate. This frequency configuration is avoided for producing

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)

IV. PRODUCING QUANTUM SQUEEZED STATES AND SUPPRESSING STIMULATED RAMAN SCATTERING

If multiple photon pairs could be produced, then the out-

iα∗1 α∗2 )F iα∗1 g iα∗2 g

iα1α2)F

iα2g iα1g

iα2g − iα∗1 g

κ −

iα1g − iα∗2 g κ −

convection in, ˆqin)T , and

0 einϕ tanhn r ∞n

where we (ˆa3, ˆa†

D3(β3)D4(β4)S(ξ )

4, ˆp†, ˆq)T , “in

the (0, 0, ˆp†

= exp(ξ ˆa†

operator, ”

sqrt2κ”in,

i(α∗1,2g ¯p

neglected

065211-6

⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎠

∂ ∂t

= −

+out

L =

= L

(27)

(28)

0, 0

n, n

⟩ =

⎞ .

1 r

ω2

”

⎛

≈

∼

−

⟨

−

( ¯nq

( ¯np

t ′),

⟩ =

(29)

¯npδ(t

− 1)δ(t

¯nqδ(t

PHYSICAL REVIEW E 110, 065211 (2024)

PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM …

p† in(t )pin(t ′) pin(t )p† in(t ′) q† in(t )qin(t ′) ⟨ qin(t )q† in(t ′) ⟨

where ¯np,q is the average thermal phonon number for mode p and q.

Here, κ is the plasma wave relaxation rate mainly due to Landau damping at finite plasma temperature, and pin(t ) and qin(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

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.

2κt but only includes terms proportional to e− 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 2α2g2(2 ¯n 1)/[)α2()α2 1). However, ≫ + time κt

thermalization can be negligible in a short owing to the small Landau damping rate in cold plasmas. 2κt ) could be lower Hence, V than the vacuum level at a certain time. Exactly, the maximum squeezing magnitude is obtained at tcr

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.

4 suggest the possibility of partial cancellation of the noise from phonons. Indeed, the correlation between the two modes is revealed in the X3 and Y4 quadratures, where Xi (ˆai

κ )2

]. The maximum squeezing magnitude and the

optimal plasma length are plotted in Fig. 4. It is seen that the & squeezing magnitude crucially relies on a large FWM growth

FIG. 4. The maximum squeezing magnitude (top) and the op- timal plasma length L ctcr (bottom) for different plasma wave damping rate κ and thermal phonon number ¯n. Other parameters are given in the text.

¯q, we assume equal pump amplitude with α. Then, the QLEs have a simple α2

ˆa† i )/sqrt2 and Yi

= the correlation 3, 4. Specifically,

To satisfy ¯p a real value α1 solution

The similar forms of a3 and a†

)F α2 − ()F α2)2

ˆa† i )/(sqrt2i) for i

− 2)F α2 −

cosh()F α2t )ˆa4(0)

cosh()F α2t )ˆa3(0)

i sinh()F α2t )ˆa†

2)F α2t and e−

αgsqrt2κ[pin(t ′)

q† in(t ′)]dt ′.

2(g/α))2(2 ¯n

i + 2 4

κ )t ) + κ

e− − )F α2

ie− )F α2

αgsqrt2κ[p†

α2)(α2) − 2κα2g2(2 ¯n

qin(t ′)]dt ′,

2)F α2t

)F α2t ′

iκ κ 2

function

e− 2κ

α2g24κ

065211-7

e)F α2t ′

(α2)F

=

1 ln[1

)F α2

in(t ′)

− = ⟨

(α2)

ˆa4(t )

ˆa3(t )

Y4)2

4(0)

3(0)

(2 ¯np

κ 5

2)F α2t

κ −

1)(1

(31)

(30)

(32)

− ≈

κ )](

κ )−

0 ’

()F α2

⟩ =

(X3

2(1

κ )2

(ˆai

e−

≪

−

2κt

(

κt ′

(

2κt

−

( ˆc

δ(t

1).

κ )2

2(1

(X3

⟩ =

SX c

(33)

(2 ¯np

Y4)2

’
= ⟨

(α2)F

e2)F α2t

α2g24κ

e2)F α2t

e− 2κ

− )F α2

− 2)F α2t

noise due to SRS

A. Single-mode squeezed state

e()F α2 κ )t ) − κ

KENAN QU AND NATHANIEL J. FISCH

PHYSICAL REVIEW E 110, 065211 (2024)

The orthogonal correlation exhibits amplified thermal

rate α2), a small plasma decay rate κ and a small thermal phonon number ¯n.

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.

The two-mode squeezed output can be converted into a single-mode squeezed state using a beam splitter. The output i ˆa4)/sqrt2, which obeys the state ˆc can be written as ˆc (ˆa3 −

commutation relation [ ˆc(t ), ˆc†(t ′)] t ′). The spectra of −

ˆc†)/(sqrt2i) ˆc†)/sqrt2 and Yc its quadratures Xc can be found

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 antiStokes sidebands. Employing orthogonally polarized dual-color pumps then enables the generation of polarization- entangled photon pairs.

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] that describes the emission of Brownian par- 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 antiStokes 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] with spacing of ωp. The comb modes, which are 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 with λ 10 nm. Even if the pumps have an amplitude of α1 0.00001, the same output photon α2 numbers could be produced in the same plasma length if the 3. With mildly plasma density is increased to 3.5 relativistic laser intensities, each millimeter-long plasma can produce nine orders of magnitude growth of photon ﬂuxes before the pump is depleted.

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

Although squeezing is obtained, excessive noise degrades the state purity Trρ2 1/sqrtdet σ . It can be obtained using

XcYc the covariance matrix σ ), where ⟨

Yc YcXc XcYc 0. Thus, the ⟨ state purity is Trρ2 4SX cSY c, which decreases at higher thermal phonon number ¯np and larger plasma wave relaxation rate κ.

It is seen that the spectrum of X quadrature is squeezed for certain time t. The spectrum of Y quadrature, however, shows antisqueezed noise ﬂuctuation.

X 2

c ⟩ ⟨ XcYc YcXc

⟨ + Xc 0 and we used ⟨

e()F α2 κ )t ) − κ

− 2)F α2 −

YcXc + Y 2 c ⟩ ⟨ ⟩ =

V. CONCLUSIONS

κ )t ) + κ

e− − )F α2

− 2)F α2t

1019 cm−

⟩ ⟩ = ⟨

− )F α2

e− 2κ

e2)F α2t

α2g22κ

e2)F α2t

065211-8

(α2)F

=

⟩ = ⟨

(2 ¯np

2)F α2t

(35)

(34)

()F α2

X 2 c

Y 2 c

⟩ =

SY c

2(1

κ )2

1),

1).

e−

−

≈

2κt

(

’

(

⟩

ACKNOWLEDGMENT

PHYSICAL REVIEW E 110, 065211 (2024)

Tutorial, J. Opt. Soc. Am. B 40, 3073 (2023).

and by NNSA Grant No. DE-NA0004167.

[3] C. Moodley and A. Forbes, All-digital quantum ghost imaging:

PRODUCING ENTANGLED PHOTON PAIRS AND QUANTUM …

This work was supported by NSF Grant No. PHY-2206691

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