A novel two-mode squeezed light based on double-pump phase-matching
Xuan-Jian He Jun Jia Gao-Feng Jiao L. Q. Chen Weiping Zhang Chun-Hua Yuan






1State Key Laboratory of Precision Spectroscopy, Quantum Institute for Light and Atoms, Department of Physics, East China Normal University, Shanghai 200062, China

2School of Physics and Astronomy, and Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China

3Collaborative Innovation Center of Extreme Optics, Shanxi
University, Taiyuan, Shanxi 030006, China

4lqchen@phy.ecnu.edu.cn

*chyuan@phy.ecnu.edu.cn 





A novel two-mode non-degenerate squeezed light is generated based on a
four-wave mixing (4WM) process driven by two pump fields crossing at a small
angle. By exchanging the roles of the pump beams and the probe and conjugate
beams, we have demonstrated the frequency-degenerate two-mode squeezed light
with separated spatial patterns. Different from a 4WM process driven by one
pump field, the refractive index of the corresponding probe field n_p can
be converted to a value that is greater than 1 or less than 1 by an angle
adjustment. In the new region with n_p<1, the bandwidth of the gain is
relatively large due to the slow change in the refractive index with the
two-photon detuning. As the bandwidth is important for the practical
application of a quantum memory, the wide-bandwidth intensity-squeezed light
fields provide new prospects for quantum memories.






§ INTRODUCTION


Memory for quantum states of light is a necessary component for any future
quantum optical computer <cit.>. In order to extend the storage
procedure to squeezed states , we need squeezed light that is resonant to the
atomic medium we are using for storage, namely ^87Rb or ^85Rb. The
generation of squezeed light at atomic wavelengths have been obtained on the
rubidium D1 line <cit.> and D2 line <cit.>
. The squeezed vacuum state on the rubidium D1 line has been stored
<cit.>. Furthermore, the bandwidth is important for the practical
application of a quantum memory <cit.>. The generated wide-bandwidth
intensity-squeezed light fields at atomic wavelengths provides new prospects
for a quantum memory. Therefore, it is worth initiating a study on how to
generate a two-mode squeezed state of wide-bandwidth, especially frequency
degenerate two-mode squeezed state of wide-bandwidth.

The first experimental demonstration of squeezed states of light by Slusher
et al. <cit.> was based on four-wave mixing (4WM) in sodium
vapor. Since then, many techniques for producing different types of squeezing
have been explored, each with its own advantages and limitations for
particular applications <cit.>. Nondegenerate 4WM in a
double-Λ scheme <cit.> was identified as a possible scheme to
generate a squeezed state or squeezed twin beams, as described in Refs.
<cit.>.

The generated twin beams by the 4WM process in atomic system with higher
squeezing degree were firstly realized by McCormick et al.
<cit.> based on degenerate pump fields, as shown in
Fig. <ref>(a). A single linearly polarized pump beam, ν_pump, is crossed at a small angle with an orthogonally polarized, much weaker
probe beam, ν_probe. The 4WM process amplifies the probe and
generates a quantum-correlated conjugate beam, ν_conjugate, on
the other side of the pump (at a higher frequency), as shown in
Fig. <ref>(b). In this case, a pair of photons of the (single) pump is
transformed, via the 4WM process, into a photon in the probe beam and a photon
in the conjugate beam. By modulating the involved ground (excited) state with
one (two) laser beam (beams), the gain and squeezing degree can be enhanced
<cit.>. The best initial results for two-mode
intensity-difference squeezing at low frequencies seem to be ≈1.5 kHz
<cit.> to the recently reported ≈700 Hz <cit.> or even
≈10 Hz <cit.>. The generated entanglement between the probe and
conjugate beams can realize quantum imaging <cit.>. The
cascaded 4WM can generate the quantum correlated triple beams
<cit.> and can also be used to realized SU(1,1) interferometers
for highly sensitive phase measurements <cit.>. This 4WM process
also supports many spatial modes, making it possible to amplify complex
two-dimensional spatial patterns <cit.>.



Recently, as shown in Fig. <ref>(c), a new 4WM process driven by two pump
fields of the same frequency crossing at a small angle was realized
<cit.>. Instead of two superimposed rings centered around the
pump beams, we find that the output field is satisfied with a two-pump forward
phase matching geometry and is two-beam excited conical emission
<cit.>. That is, the light is emitted on the surface of a circular
cone centered on the bisector of the two pump beams. In this paper, we further
implement frequency-degenerate two-mode squeezed light based on a 4WM driven
by two pump fields crossing at a small angle through an optical phase locked
loop (OPLL) <cit.> and give theoretical explanations. By analyzing the
gain, we find that the phase matching condition can be achieved under the
conditions of n_p>1 and n_p<1 by an angle adjustment. The theoretical
range of angles for achieving different regions is given. In the new region
with n_p<1, the bandwidth of the gain is relatively large due to the slow
change in the refractive index with the two-photon detuning, which is
advantageous for realizing wide-bandwidth intensity-squeezed light.





§ FREQUENCY DEGENERATE SQUEEZED LIGHT


In our experiment <cit.>, the state |g,m⟩ (or state |1,2⟩)
involves the hyperfine levels |5S_1/2,F=2,3⟩, where the hyperfine
splitting of the ground state is ω_21=2π×3.035 GHz, and the
excited state |e⟩ (or state |3,4⟩) is |5P_1/2⟩ has
an excited state decay rate of γ=2π×5.75 MHz. The pump field is
blue-detuned approximately 1 GHz to the D1 line of Rb-85 5S_1/2
→5P_1/2. The powers of the pump fields E_P1 and E_P2 are
set to 350 mW, and their waists at the crossing point are 622 μ m and
596 μ m, respectively. The Rabi frequencies of Ω_P1 and
Ω_P2 are Ω_P1≃28γ and Ω_P2≃30γ
for a effective electric dipole d=1.47×10^-29 Cm <cit.>. The
atomic number density of Rb-85 at 125 ^∘C is approximately
N≃4.5×10^18 m^-3. As shown in Fig. <ref>(d), the pump
fields P_1 and P_2 have a certain angle θ_0 in one plane,
where the small angle θ_0 can vary within a certain range. The probe
field p is input at angles θ_1 and θ_2 relative to the
pump fields P_1 and P_2, respectively. The angle θ_0
determines the minimum of the sum of angles θ_1 and θ_2.
When only one pump field P_1 or P_2 exists, the conjugate field
c_1 or c_2 is generated under the respective phase matching condition
2𝐤_P_1-n_p𝐤_p-𝐤_c1=0 or 2𝐤
_P_2-n_p𝐤_p-𝐤_c2=0 <cit.>. When two
pump fields P_1 and P_2 exist at the same time and changing the angles
θ_1 and θ_2, under a certain condition a new conjugate
field c is generated due to the new phase matching condition 𝐤
_P_1+𝐤_P_2-n_p𝐤_p-𝐤_c=0 <cit.>,
and the conjugate fields c_1 and c_2 disappear due to mismatching. The
probe field p and the conjugate field c are at an angle of θ_3
relative to the z-axis, as shown in Fig. <ref>(d).

Now, we use this configuration to produce a frequency degenerate two-mode
squeezed light field. The approach to generating the frequency degenerate twin
beams is based on the idea of inverting the configuration <cit.>. Two
realtively strong beams is pumped the atomic system with the frequency of the
probe and conjugate beams, and along the direction of them, and a week beam
having the frequency and direction of the previous pump is also input.
However, this directly exchange of role of the pump and probe and conjugate
beams does not lead to the desired intensity difference squeezing and instead
we have found excess beams and noise due to the unexpected nonlinear process
as shown in Fig. <ref>(a). We found it necessary to tune the detuning of
Δ_1 and Δ_2 as approximately 1 GHz and -2 GHz to
suppress these extra processes, as shown in Fig. <ref>(b). This detuned
choice is beneficial to the acquisition of frequency degenerate two-mode
squeezed light. The reason is that the noise on the two sides of the atomic
line is asymmetrical as Davis et al. <cit.> pointed out. On the
other hand, we reduce the gain by adjusting the temperature from 125 ^∘C to 105 ^∘C, so that the unexpected nonlinear
process can be suppressed. Using this configuration we have investigated the
generation of the frequency degenerate and spatial nondegenerate twin beams.

We use the experimental setup with the two pump fields that are generated by a
Ti:Sapphire laser (Δ_1 ∼1 GHz) and a semiconductor laser
(Δ_2 ∼-2 GHz) to implement this scheme, and the frequency
difference of the two pumps is achieved by using an OPLL with the beat
frequency of 6.075 GHz as shown in Fig. <ref>(a). The probe beam is
generated by frequency-shift the light form Ti: Sapphire with double-passed
1.52 GHZ acousto-optics modulators (AOM). The AOM frequency shift, and hence
the two-photon detuning δ is adjusted to optimize performance and
change the scheme from frequency non-degenerate to degenerate twin beams. In
the experiment of degenerate four-wave mixing, two pump lights with a
frequency difference of 6 GHz or more are required to drive at the same
time. We use the amplifier lock scheme to generate pump light by frequency
shifting. In order to ensure a fixed phase difference between the two pumping
light fields, we use a beat frequency interlocking method-OPLL to lock a
semiconductor laser and a Ti:Sapphire laser to each other. Since the frequency
shift is as small as 0.1 Hz, the lock relative frequency difference is less
than 1 Hz.

The 6 GHz beat frequency signal after the frequency locking system is
stabilized is shown in Fig. <ref>(a). The modulating signal peaks
appearing in the frequency range of ±1 MHz around both sides of the peak,
which is caused by the feedback noise of the OPLL system itself. When the
probe and conjugate beams are near degenerate and frequency difference between
them ∼2.3 MHz, the intensity-difference noise is shown in Fig. <ref>
(b), and we find the OPLL feedback noise become the main limitation to get
better squeezing. When the beat signal is further reduced, as shown by the
arrow in Fig. <ref>(b), we obtain the intensity-difference noise with
frequency difference between probe and conjugate beams <1 Hz, as shown in
Fig. <ref>(c), which indicate the twin beams are totally
indistinguishable . The inset in Fig. <ref>(c) shows the process of
gradually reducing the beat signal to below 1 Hz, where the black and red
lines are phase locked within less than 2 KHz and 1 Hz, respectively.



§ THEORETICAL MODEL




In this section, we firstly describe the frequency non-degenerate squeezed
light based on non-collinear 4WM. As shown in Fig. <ref>(a), we assume
that the two pump fields E_P1 and E_P2 couple the transitions
|1⟩→|3⟩ and |2⟩→|4⟩,
respectively. The probe field couples the transition |2⟩→
|3⟩, and the conjugate field couples the transition |1⟩→|4⟩. The transitions |1⟩→|2⟩ and
|3⟩→|4⟩ are not dipole allowed transitions. Since the
two pump fields E_P1 and E_P2 have the same polarizations and
frequencies, the two pump fields E_P1 and E_P2 also couple the
transitions |2⟩→|4⟩ and |1⟩→
|3⟩ with probabilities of 1/2. For the second set of coupling
transitions, we just swap the pump field E_P1 and the pump field E_P2
in the result for the first set of coupling transitions. We add the two sets
of conclusions to obtain the final result.

Next, we describe the frequency-degenerate squeezed light based on exchanging
the roles of the pump beams and the probe and conjugate beams. As shown in
Fig. <ref>(b), the double-Λ four-level process is the
same as the non-degenerate process except that the magnitude of the detuning
is different. Therefore, the frequency-degenerate and non-degenerate squeezed
lights based on non-collinear 4WM can be described by a same set of equations.



In the dipole and rotating wave approximations, the Hamiltonian of the atoms
combined with the Hamiltonian of the light-atom interaction is given by

    Ĥ=Ĥ_atoms+Ĥ_I,

where

    Ĥ_atoms=ħω_41σ_44+ħω_31σ_33
    +ħω_21σ_22,

and

    Ĥ_I =-ħ(Ω_P1e^i(𝐤_P1·𝐫
    -ω_P1t)σ_31+Ω_P2e^i(𝐤_P2·𝐫
    -ω_P2t)σ_42
         +g_pℰ̂_pe^i(𝐤_p·𝐫-ω_p
    t)σ_32+g_cℰ̂_ce^i(𝐤_c·𝐫-ω_ct)σ_41)+H.c.

Here, ω_n1=ω_n-ω_1 (n=2,3,4), σ_nm
=|n⟩⟨ m| (n,m=1,2,3,4), 2Ω_P1=μ_31ℰ
_P1/ħ and 2Ω_P2=μ_42ℰ_P2/ħ are the Rabi
frequencies, g_n (n=p,c) are the atom-field coupling constants, and
ℰ̂_p and ℰ̂_̂ĉ are the slowly varying
envelope operators of the probe and conjugate field.



The equations for the atomic operators σ_nm (n,m=1,2,3,4) in the
Heisenberg picture are given in the Appendix. Using the atomic operators to
evaluate the linear and nonlinear components of the polarization at
ω_p and ω_c, the polarization of the atomic medium at a
particular frequency is given by P̂(  ω_p)
=Nd_23σ̃_23+H.c. and P̂(  ω
_c)  =Nd_14σ̃_14+H.c., where N is the
number density of the atomic medium. The polarizations of the medium at
frequency ω_n (n=p,c) are given by

    P̂_p(ω_p)       =√(ħω_p/2ϵ_0V
    )ϵ_0χ_pp(ω_p)ℰ̂_pe^i𝐤_p
    ·𝐫+√(ħω_c/2ϵ_0V)ϵ_0
    χ_pc(ω_p)ℰ̂_c^†e^i(𝐤
    _P1+𝐤_P2-𝐤_c)·𝐫+H.c.,
    P̂_c(ω_c)       =√(ħω_c/2ϵ_0V
    )ϵ_0χ_cc(ω_c)ℰ̂_ce^i𝐤_c
    ·𝐫+√(ħω_p/2ϵ_0V)ϵ_0
    χ_cp(ω_c)ℰ̂_p^†e^i(𝐤
    _P1+𝐤_P2-𝐤_p)·𝐫+H.c..

Here, the two coefficients χ_pp and χ_cc describe the effective
linear polarization processes for the probe and conjugate fields,
respectively, and unlike the usual linear coefficients, they depend
nonlinearly on the pump field. The other two coefficients χ_pc and
χ_cp are responsible for the 4WM process. A detailed calculation is
given in the Appendix.



Under the condition of the slowly varying amplitude approximation, considering
nearly co-propagating beams along the z axis, these field equations in the
co-moving frame are written as

    ∂/∂ zℰ̂_p     =ik_p/2
    [χ_pp(ω_p)ℰ̂_p+χ_pc^'(ω_p
    )ℰ̂_c^†e^iΔ k_zz],
    ∂/∂ zℰ̂_c     =ik_c/2
    [χ_cc(ω_c)ℰ̂_c+χ_cp^'(ω_c
    )ℰ̂_p^†e^iΔ k_zz],

where χ_pc^'=χ_pc√(ω_c/ω_p)≃χ
_pc, χ_cp^'=χ_cp√(ω_p/ω_c)≃χ_cp, and Δ k_z is the projection of the geometric phase
mismatch Δ𝐤=𝐤_P_1+𝐤_P_2
-𝐤_p-𝐤_c on the z axis. The solutions to the
propagation equations (<ref>) and (<ref>) with a medium of length
L are given by

    ℰ̂_p     =G_1ℰ̂_p(0)+g_1ℰ̂_c^†(0),
    ℰ̂_c^†     =[  G_2ℰ̂_c^†(0)+g_2ℰ̂_p(0)]  e^-iΔ k_zL,

where

    G_1     =e^δ aL[cosh(ξ L)+a/ξsinh(ξ L)],
    G_2=e^δ aL[cosh(ξ L)-a/ξsinh(ξ L)],
    
    g_1     =a_pc/ξe^δ aLsinh(ξ L),g_2
    =-a_cp/ξe^δ aLsinh(ξ L),

and

    a_pj     =ik_pχ_pj/2,a_cj=ik_cχ_cj^∗/2(j=p,c),
    
    a       =(a_pp+a_cc-iΔ k_z)/2,ξ=√(a^2-a_pca_cp
    ),δ a=(a_pp-a_cc+iΔ k_z)/2.





The number operators of the probe beam and conjugate beam are defined as
N̂_p=ℰ̂_p^†ℰ̂_p and N̂_c=ℰ̂_c^†ℰ̂_c, respectively. From
the above result, we define the gain of the probe beam in the 4WM process as:

    G_p=⟨N̂_p⟩_out/⟨N̂_p⟩_in
    ≃| G_1| ^2,

where the initial condition is ⟨N̂_p⟩_in≫1 and
⟨N̂_c⟩_in=0. The 4WM generates a correlated probe and
conjugate beams, and the relative intensity fluctuations are reduced for the
amplification process. After the 4WM, the relative intensity fluctuation is
given by

    Δ^2(N̂_p-N̂_c)_out=(| G_1|
    ^2-| g_2| ^2)^2Δ^2(N̂_p
    )_in+| g_1^∗G_1-g_2^∗G_2| ^2
    [⟨N̂_p⟩_in+1].

Hence the beams have been amplified without increasing the relative intensity
noise, and they are relative intensity squeezed. The standard quantum limit
(SQL) is a differential measurement equal to the total optical power, that is

    ⟨N̂_p-N̂_c⟩_SQL≡⟨N̂_p+N̂_c⟩≃(| G_1| ^2+|
    g_2| ^2)⟨N̂_p⟩_in.

The noise figure of the process (or “degree of
squeezing”) is the ratio of the measured noise to the
corresponding shot-noise level for equal optical power. The typically the
noise figure is quoted as the noise in decibels relative to the SQL.



§ PHASE MATCHING




In this section, we describe the angles θ_1 and θ_2 between
the probe field and the pump fields by phase matching based on the different
refractive indices.

As shown in Fig. <ref>(a), when two pump fields E_P1 and E_P2 are
incident at an angle, the total projection of the wavevector of the pump
fields onto the z-axis is 2𝐤_PZ and becomes smaller; i.e.,
2|𝐤_PZ| <|𝐤_P1|
+|𝐤_P2|. The geometric phase matching
condition (GPMC) is given by

    Δ k_z     =|𝐤_P1|cos(θ
    _z1)+|𝐤_P2|cos(θ_z2)-(|𝐤_p| +|𝐤_c| )cosθ_3
         =2|𝐤_PZ| -(|𝐤
    _p| +|𝐤_c| )cosθ_3,

where θ_3 is the angle between the probe and the projected pump
field. In fact, if the 4WM is efficient, the GPMC of Eq. (<ref>) may not
be satisfied, but the effective phase matching condition (EPMC) must be met:

    𝐤_P1+𝐤_P2-n_p𝐤_p-n_c𝐤_c=0,

where the refractive index n_p=√(1+Re(χ_pp)), and
n_c=√(1+Re(χ_cc)). For the case of two pump
fields, the Eq. (<ref>) can be written as

    cosθ_3=ω_P1cos(θ_z1)+ω_P2cos(θ
    _z2)/n_pω_p+n_cω_c.

According to θ_3 and θ_z1 (θ_z2), we can determine
the angle θ_1 (θ_2) between the probe field and the pump
field P_1 (P_2).

In order to better explain phase matching, we first consider the
non-degenerate 4WM case. For the case of two degenerate pump fields, the
conservation of energy impose the condition ω_p+ω_c=ω
_P1+ω_P2=2ω_0, where ω_0 is the frequency of the
pump field. Considering θ_z1=θ_z2=θ_0/2, the Eq.
(<ref>) is written as

    cosθ_3=2ω_0cos(θ_0/2)/n_pω_p+ω
    _c,

where n_c≃1 due to the conjugate field with a large detuning. For a
given angle θ_0, when n_p=1, the EPMC of Eq. (<ref>) imposes
θ_3=θ_0/2. Under this condition, the GPMC Δ k_z=0 is
also satisfied, which is the phase matching condition in free space, where the
beams are required rigorously copropagating as shown in Fig. <ref>(b).

When n_p>1, the EPMC of Eq. (<ref>) is established to require that
θ_3>θ_0/2, which means that the GPMC of Eq. (<ref>) cannot
be satisfied and will occur Δ k_z>0, as shown in Fig. <ref>(c).
Considering θ_1=θ_2=θ, using the law of cosines we obtain
the angle requirement between the probe field and the pump fields:

    θ>cos^-1[  1+cosθ_0/2]  .



If n_p<1, similarly, the EPMC of Eq. (<ref>) requires that
θ_3<θ_0/2 and imposes Δ k_z<0, as shown in Fig.
<ref>(d). In addition, the generated probe and conjugate beams have
separate directions, which requires that the angle θ_3>0.
Furthermore, using the minimum value of the refractive index min(n_p
) according to Eq. (<ref>), we obtain

    0<(θ_3)_min<θ_3<θ_0/2,

where (θ_3)_min=cos^-1[  2ω_0cos(θ
_0/2)/(min(n_p)ω_p+ω_c)] and correspondingly,

    θ_0/2<θ_min<θ<cos^-1[  1+cosθ_0/2]  ,

where θ_min=cos^-1[  ω_0(1+cosθ_0)/(min
(n_p)ω_p+ω_c)]. Compared to the single pump field
case, this is a new region. Because for the single pump field case, the angle
θ_0 here is equivalent to 0, where the condition θ_3
<θ_0/2 cannot be satisfied because θ_3 cannot be less than
0. That is, when n_p<1, the EPMC cannot be satisfied for 4WM driven by a
single pump field.

For degenerate case, the form of Eqs. (<ref>-<ref>) is the same
except for the magnitude of the wave vectors. With two strong beams with the
frequency of the probe and conjugate beams, and along the direction of them,
and a week beam having the frequency and direction of the previous pump, we
can generate of the frequency degenerate and spatial nondegenerate twin beams
by changing the detunings of Δ_1 and Δ_2. Similar to
non-degenerate case, if n_p+n_c=2, then GPMC Δ k_z=0 is
satisfied. If n_p+n_c>2 or n_p+n_c<2, then the corresponding GPMC
Δ k_z>0 or Δ k_z<0 is also obtained.



§ DATA ANALYSIS


In this section, we numerically analyze the characteristics of the squeezed
light produced in this new region with n_p<1.

Fig. <ref>(a-b) show the direct susceptibilities χ_pp and
χ_cc for the probe and conjugate fields as a function of the two-photon
detuning δ/γ, and we obtain that χ_cc is far less than
χ_pp due to large detuning. In Fig. <ref>(a), when δ<0, the
real part of χ_pp is effectively responsible for the index of
refraction of the probe for the single pump field case
<cit.>. However, for our two pump fields input case, the
phase matching condition can also be satisfied on the other side δ>0.

According to Eq. (<ref>), we plot the probe gain G_p as a function of
the two-photon detuning δ and the geometric phase mismatch Δ
k_z in the presence of a single pump field and two pump fields, as shown in
Fig. <ref>. One can see that the maximum gains are obtained on the side
δ<0 with Δ k_z>0, for both cases. When Δ k_z<0, the
probe gain G_p does not exist for the single pump field input case and
occurs for two pump fields input case. The bandwidth of the probe gain is
relatively large due to the slow change in the refractive index with the
two-photon detuning.



The theoretical output probe gain G_p as a function of the two-photon
detuning δ/γ and the probe-pump angle θ with (a) a single
pump field and (b) two pump fields is shown in Fig. <ref>, where we
consider 𝐤_P1=𝐤_P2=𝐤_P and θ
_1=θ_2=θ. It can be seen from that for a single pump field, the
gain and 4WM process is on the δ<0 side as the angle θ increases
due to phase matching. For the case of two pump fields, the gain and 4WM
process can be achieved on the left side (Line L_2) or the right
side (line L_0or line L_1) by choosing the angle
θ between the probe field and the pump fields, for a given angle
θ_0.

The area intersecting the dashed line L_1in Fig. <ref>(b) is
the area selected by our experimental parameters, where θ_0
/2=0.615^o, and θ= 0.861^o. According to the
minimum value min(Re(χ_pp))=-1.609×10^-4 in Fig.
<ref>, using Eq. (<ref>), we obtain

    0.7630^o<θ<0.8697^o.



If we only choose the angle θ based on the bandwidth, we choose line
L_0[Fig. <ref>(b)] because it has the largest bandwidth.
However, in the experiment, the angle is finely adjusted according to the
degree of squeezing, and the optimum value of the angle θ is different.
If the angle θ is chosen as 1.2^o of line L_2in Fig. <ref>(b), the 4WM process driven by two pump fields will also be
observed on the δ<0 side due to the large gain. As shown in Fig.
<ref>, the strong conjugate field c and two weak conjugate fields
c_1 and c_2 may all occur because of their gains <cit.>.
However, in this region, the absorption is also large, which will affect the
degree of squeezing of the two generated beams.

Here, θ_0 has a fundamental effect on wavevector matching in the new
4WM process, thus opening up a region in which high-intensity-difference
squeezed light can be obtained over a wide bandwidth with low loss and
moderate gain. The gain curve in Fig. <ref>(b) shifts upward as the angle
θ_0 increases, because the minimum value of the angle θ is
greater than the angle θ_0/2.

The gain of the probe field and the squeezing as a function of the two-photon
detuning δ is shown in Fig. <ref>, where the square represents
experimental data and the solid line is a theoretical simulation. The
theoretical simulations and experimental data of the gain of the probe field
are in good agreement as shown in Fig. <ref>(a). The squeezing degree is
affected by the spatial mode mismatch, optical absorption by atomic system,
optical loss in the light path, and atomic decoherence. Fig. <ref>(b)
shows the theoretical simulations and experimental data of squeezing, where
the theoretical squeezing curve is reduced by 0.56 times and the agreement is
not very good because these effects are not included in our model in order to
clarify the physics picture concisely. On the new δ>0 side as shown in
Fig. <ref>(a), the bandwidth of the gain is relatively larger than that
for the single pump field case, which is advantageous for realizing
wide-bandwidth frequency-degenerate and nondegenerate intensity-squeezed
light. These light fields can be widely used in quantum information and other fields.



§ CONCLUSION


We have studied that a novel two-mode squeezed light is generated from a 4WM
process driven by two pump fields crossing a small angle, where the twin beams
are generated with a new phase matching condition. Different from 4WM realized
by a single pump field where the gain peak can only be achieved on the
δ<0 side, the new 4WM process is implemented from the δ<0 side
to the δ>0 side by an angle adjustment. The refractive index of the
corresponding probe field n_p can be converted from n_p>1 to n_p
<1, which can also be used to convert between slow light <cit.> and
fast light <cit.>. Based on slow light and fast light of the probe
field, two different time-order output pulses can be achieved. On the new
δ>0 side, the refractive index n_p changes slowly with the
two-photon detuning δ over a large range, which leads to a relatively
large gain bandwidth. With two strong beams with the frequency of the probe
and conjugate beams, and along the direction of them, and a week beam having
the frequency and direction of the previous pump, we have generated the
frequency degenerate and spatial nondegenerate twin beams with tuning the
detuning of Δ_1 and Δ_2. This type of twim beams can be
combined and interfered directly on the beam splitter. These wide-bandwidth
intensity-squeezed light fields can be applied in quantum information and
quantum metrology.



§ FUNDING


C.-H.Y. is supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 11474095 and the Fundamental Research Funds for the
Central Universities. L.-Q.C. is supported by the NSFC under Grant Nos.
11874152, 11604069, and 91536114 and the National Science Foundation of
Shanghai (No. 17ZR1442800). W.Z. is supported by the National Key Research and
Development Program of China under Grant No. 2016YFA0302001 and NSFC Grants
Nos. 11654005 and 11234003.






§ SUSCEPTIBILITIES WITH TWO PUMP FIELDS CROSSING A SMALL ANGLE INPUT


As shown in Fig. <ref>, we assume that the two pump fields E_P1 and
E_P2 couple the transitions |1⟩→|3⟩ and
|2⟩→|4⟩, respectively. The probe field couples the
transition |2⟩→|3⟩, and the conjugate field couples
the transition |1⟩→|4⟩. The transitions |1⟩→|2⟩ and |3⟩→|4⟩ are not dipole
allowed transitions.

Consequently, we obtain the following set of equations for the populations
σ_nn:

    ∂σ_11/∂ t=Γ_13σ_33+Γ
    _14σ_44+i(Ω_P1^∗e^-i𝐤_P1·𝐫
    σ̃_13+g_cℰ̂_c^†e^-i𝐤
    _c·𝐫σ̃_14-Ω_P1e^i𝐤_P1
    ·𝐫σ̃_31
         -g_cℰ̂_ce^i𝐤_c·𝐫σ̃_41),
       ∂σ_22/∂ t=Γ_23σ_33+Γ
    _24σ_44+i(Ω_P2^∗e^-i𝐤_P2·𝐫
    σ̃_24+g_pℰ̂_p^†e^-i𝐤
    _p·𝐫σ̃_23-Ω_P2e^i𝐤_P2
    ·𝐫σ̃_42
         -g_pℰ̂_pe^i𝐤_p·𝐫σ̃_32),
       ∂σ_33/∂ t=-Γ_3σ_33+i(Ω
    _P1e^i𝐤_P1·𝐫σ̃_31+g_p
    ℰ̂_pe^i𝐤_p·𝐫σ̃
    _32-Ω_P1^∗e^-i𝐤_P1·𝐫σ̃_13
         -g_pℰ̂_p^†e^-i𝐤_p·𝐫
    σ̃_23),
       ∂σ_44/∂ t=-Γ_4σ_44+i(Ω
    _P2e^i𝐤_P2·𝐫σ̃_42+g_c
    ℰ̂_ce^i𝐤_c·𝐫σ̃
    _41-Ω_P2^∗e^-i𝐤_P2·𝐫σ̃_24
         -g_cℰ̂_c^†e^-i𝐤_c·𝐫
    σ̃_14),

where Γ_mn is the population decay rate from the level n to level
m, and we introduce slowly varying matrix elements in time: σ
_13=σ̃_13e^-iω_p1t, σ_14=σ̃
_14e^-iω_ct, σ_24=σ̃_24e^-iω_p2t,
and σ_23=σ̃_23e^-iω_pt. In addition, the set
of equations for σ̃_nm (n≠ m) are given by

    ∂σ̃_42/∂ t     =[i(δ-Δ
    _2)-γ_42]σ̃_42-i[Ω_P2^∗e^-i𝐤
    _P2·𝐫σ_22,44-g_pℰ̂_p^†e^-i𝐤_p·𝐫σ̃_43+g_cℰ̂_c^†e^-i𝐤_c·𝐫σ̃
    _12],
       
    ∂σ̃_41/∂ t     =(-iΔ_2-γ
    _41)σ̃_41+i[Ω_P1^∗e^-i𝐤_P1
    ·𝐫σ̃_43-g_cℰ̂_c^†e^-i𝐤_c·𝐫σ_11,44-Ω_P2^∗e^-i𝐤_P2·𝐫σ̃_12],
    ∂σ̃_32/∂ t     =[i(δ-Δ
    _1)-γ_32]σ̃_32+i[Ω_P2^∗e^-i𝐤
    _P2·𝐫σ̃_34-g_pℰ̂_p^†e^-i𝐤_p·𝐫σ_22,33-Ω_P1^∗e^-i𝐤_P1·𝐫σ̃_12],
       
    ∂σ̃_31/∂ t     =-(iΔ_1+γ
    _31)σ̃_31-i[Ω_P1^∗e^-i𝐤_P1
    ·𝐫σ_11,33-g_cℰ̂_c^†e^-i𝐤_c·𝐫σ̃_34+g_pℰ̂_p^†e^-i𝐤_p·𝐫σ̃_21],
    ∂σ̃_43/∂ t     =[i(Δ_1-Δ
    _2)-γ_43]σ̃_43+i[Ω_P1e^i𝐤_P1
    ·𝐫σ̃_41+g_pℰ̂_pe^i𝐤
    _p·𝐫σ̃_42-Ω_P2^∗e^-i𝐤
    _P2·𝐫σ̃_23
         -g_cℰ̂_c^†e^-i𝐤_c·𝐫
    σ̃_13],
    ∂σ̃_21/∂ t     =-(iδ+γ
    _21)σ̃_21+i[Ω_P1^∗e^-i𝐤_P1
    ·𝐫σ̃_23+g_cℰ̂_c^†e^-i𝐤_c·𝐫σ̃_24-Ω_P2
    e^i𝐤_P2·𝐫σ̃_41
         -g_pℰ̂_pe^i𝐤_p·𝐫σ̃_31],

where σ_22,44=σ_22-σ_44, σ_11,44=σ
_11-σ_44, σ_22,33=σ_22-σ_33, and
σ_11,33=σ_11-σ_33, the single-photon detunings are
Δ_1=ω_P1-ω_31 and Δ_2=ω_P2-ω
_42+δ, the two-photon detuning is δ=ω_P1-ω
_p-ω_21, and the slowly varying matrix elements are σ
_43=σ̃_43e^i(ω_p2-ω_p)t and σ
_12=σ̃_12e^-i(ω_c-ω_p2)t. γ_nm gives
the dephasing rate of the σ_nm coherence, and γ_nm
=(Γ_m+Γ_n)/2+γ_nm^c, where Γ_n is the total
decay rate out of level n and γ_nm^c is the dephasing rate due to
any other source of decoherence.

Now we are in a position to solve the properties of the system. For
convenience, we let Γ_3=Γ_4≡γ and Γ
_13=Γ_14=Γ_23=Γ_24=γ/2, and the complex decay
rates are

    ξ_42     =-γ/2+i(δ-Δ_2),ξ_41=-(
    γ/2+iΔ_2)  ,
    ξ_32     =-γ/2+i(δ-Δ_1),ξ_31=-(
    γ/2+iΔ_1)  ,
    ξ_43     =-γ+i(Δ_1-Δ_2),ξ_21=-(
    γ_21+iδ)  .

In order to obtain analytical expressions, we assume that the pump fields
propagate without depletion, and the steady-state expectation values for the
zeroth-order atomic operators σ_33 and σ_44 are equal to

    ⟨σ_33⟩=⟨σ_44⟩=|Ω_P1| ^2|Ω_P2| ^2/|Ω_P2| ^2|ξ_31| ^2+|Ω_P1| ^2|ξ_42| ^2+4|Ω_P1| ^2|Ω_P2| ^2.

Then, the population differences are given by

    ⟨σ_11,33⟩     =⟨σ_11,44⟩=|ξ_31| ^2|Ω_P2| ^2/|Ω_P2| ^2|ξ_31| ^2+|Ω_P1| ^2|ξ_42| ^2+4|Ω_P1| ^2|Ω_P2| ^2,
    ⟨σ_22,33⟩     =⟨σ_22,44⟩=|ξ_42|^2|Ω_P1| ^2/|ξ_31|
    ^2|Ω_P2| ^2+|ξ_42|
    ^2|Ω_P1| ^2+4|Ω_P1Ω
    _P2| ^2,



We also assume that the probe and conjugate fields are weak fields, such that
we only keep terms to first order in Ω_p and Ω_c. The
steady-state density matrix elements σ̃_23 and σ̃_14 are given by

    σ̃_23     =iξ_41/D×{g_pℰ̂
    _pe^i𝐤_p·𝐫[σ_11,33(  ξ_43
    /ξ_31+|Ω_P1P2| ^2/ξ_31ξ_41
    )  |Ω_P1| ^2-(ξ_43|Ω_P2| ^2+ξ_21|Ω_P1| ^2
    /ξ_41
         +ξ_21ξ_43)σ_22,33+|Ω_P2|
    ^2(  ξ_21/ξ_42-|Ω_P1P2
    | ^2/ξ_42ξ_41)  σ_22,44]+Ω_P1
    Ω_P2[ξ_21+ξ_43/ξ_41σ_11,44
         +(  ξ_43/ξ_42^∗+|Ω
    _P1P2| ^2/ξ_42^∗ξ_41)  σ
    _22,44+(  ξ_21/ξ_31^∗-|Ω_P1P2| ^2/ξ_31^∗ξ_41)  σ
    _11,33]e^iΔ𝐤_p·𝐫g_cℰ̂
    _c^†},


    σ̃_14=iξ_32/D^∗×{g_cℰ̂_ce^i𝐤_c·𝐫[σ_11,33(
    ξ_21^∗/ξ_31+|Ω_P1P2|
    ^2/ξ_31ξ_32)  |Ω_P1| ^2
    -(ξ_21^∗|Ω_P2| ^2+ξ_43^∗|Ω_P1| ^2/ξ_32
         +ξ_21^∗ξ_43^∗)σ_11,44+|Ω
    _P2| ^2(  ξ_43^∗/ξ_42-|Ω_P1P2| ^2/ξ_42ξ_32)  σ_22,44
    ]+Ω_P1Ω_P2[ξ_43^∗+ξ_21^∗/ξ_32
    σ_22,33
         +(  ξ_21^∗/ξ_42^∗+|Ω_P1P2| ^2/ξ_42^∗ξ_32)  σ
    _22,44+(  ξ_43^∗/ξ_31^∗-|Ω_P1P2| ^2/ξ_31^∗ξ_32)  σ
    _11,33]e^iΔ𝐤_c·𝐫g_pℰ̂
    _p^†},

where

    D=ξ_41ξ_21ξ_43ξ_32^∗+(ξ_41ξ_43+ξ_21
    ξ_32^∗)|Ω_P1| ^2+(ξ_41ξ_21
    +ξ_43ξ_32^∗)|Ω_P2| ^2+|Ω_P1P2| ^4,
       |Ω_P1P2| ^2=|Ω_P1|
    ^2-|Ω_P2| ^2,Δ𝐤_p
    =𝐤_P1+𝐤_P2-𝐤_c,Δ𝐤
    _c=𝐤_P1+𝐤_P2-𝐤_p.



Using the atomic operators to evaluate the linear and nonlinear components of
the polarization at ω_p and ω_c, the polarization of the
atomic medium at a particular frequency is given by P̂(  ω
_p)  =Nd_23σ̃_23+H.c. and P̂(  ω
_c)  =Nd_14σ̃_14+H.c.. The polarization of the medium
at frequency ω_i can be divided into two different terms: one that is
proportional to the field at frequency ω_i and one that is
proportional to the field at frequency ω_P1+ω_P2-ω_i,
such that

    P̂_p(ω_p)       =√(ħω_p/2ϵ_0V
    )ϵ_0χ_pp(ω_p)ℰ̂_pe^i𝐤_p
    ·𝐫+√(ħω_c/2ϵ_0V)ϵ_0
    χ_pc(ω_p)ℰ̂_c^†e^iΔ𝐤_p
    ·𝐫+H.c.,
    P̂_c(ω_c)       =√(ħω_c/2ϵ_0V
    )ϵ_0χ_cc(ω_c)ℰ̂_ce^i𝐤_c
    ·𝐫+√(ħω_p/2ϵ_0V)ϵ_0
    χ_cp(ω_c)ℰ̂_p^†e^iΔ𝐤_c
    ·𝐫+H.c.,

where the two coefficients χ_pp, and χ_cc are given as follows:

    χ_pp     =iN| d_23| ^2ξ_41/ϵ
    _0ħ D[|Ω_P1| ^2(  ξ_43/ξ_31+|Ω_P1P2| ^2/ξ_31ξ_41
    )  σ_11,33-(ξ_43|Ω_P2|
    ^2+ξ_21|Ω_P1| ^2/ξ_41
         +ξ_21ξ_43)σ_22,33+|Ω_P2|
    ^2(  ξ_21/ξ_42-|Ω_P1P2
    | ^2/ξ_42ξ_41)  σ_22,44],


    χ_cc     =iN| d_14| ^2ξ_32/ϵ
    _0ħ D^∗[|Ω_P1| ^2(  ξ_21^∗/ξ_31+|Ω_P1P2| ^2
    /ξ_31ξ_32)  σ_11,33-(ξ_21^∗|Ω_P2| ^2+ξ_43^∗|Ω_P1|
    ^2/ξ_32
         +ξ_21^∗ξ_43^∗)σ_11,44+|Ω
    _P2| ^2(  ξ_43^∗/ξ_42-|Ω_P1P2| ^2/ξ_42ξ_32)  σ_22,44],

The two coefficients χ_pc and χ_cp are given by

    χ_pc     =iNd_23d_14ξ_41Ω_P1Ω_P2/ε_0ħ D[ξ_21+ξ_43/ξ_41σ
    _11,44+(ξ_43/ξ_42^∗+|Ω
    _P1P2| ^2/ξ_42^∗ξ_41)σ_22,44
         +(  ξ_21/ξ_31^∗-|Ω
    _P1P2| ^2/ξ_31^∗ξ_41)  σ_11,33],


    χ_cp     =iNd_23d_14ξ_32Ω_P1Ω_P2/ε_0ħ D^∗[ξ_43^∗+ξ_21^∗/ξ_32σ_22,33+(ξ_21^∗/ξ_42^∗
    +|Ω_P1P2| ^2/ξ_42^∗ξ_32
    )σ_22,44
         +(  ξ_43^∗/ξ_31^∗-|Ω_P1P2| ^2/ξ_31^∗ξ_32)  σ
    _11,33].

The two coefficients χ_pp and χ_cc describe the effective linear
polarization processes for the probe and conjugate fields, respectively, and
unlike the usual linear coefficients, they depend nonlinearly on the pump
field. The other two coefficients χ_pc and χ_cp are responsible
for the 4WM process.

Since the two pump fields have the same polarizations and frequencies, the two
pump fields E_P1 and E_P2 also couple the transitions |2⟩→|4⟩ and |1⟩→|3⟩ with probabilities
of 1/2. Due to coupling another set of transitions with the same
effective electric dipole, we just swap the Rabi frequencies Ω_P1 and
Ω_P2 in the above 4 coefficients χ_pp, χ_cc, χ
_pc and χ_cp to obtain a new set of transitions.

99                                                                                               


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