Quantum Properties in Nonlinear Coupler with Raman Process.

In this work, we have examined the generation of squeezed states of light in a two-waveguide coupler in which Raman processes are active in one waveguide. Both waveguides are mutually linearly interacting through the field’s evanescent waves. We looked at a few interesting cases in which the system generated single mode squeezed states due to spontaneous and stimulated Raman processes. Under specific combinations of design parameters and phase mismatching conditions, squeezed states may take the form of collapses and revivals, or the second quadrature of the optical mode may oscillate completely below the short-noise limit of a coherent state.


Introduction
In contrast to the conventional electromagnetic wave theory, in quantum theory, light is not seen as a wave but rather as a continuous stream of photons.How these photons interact with atoms and molecules is the focus of quantum optics, which aims to put to the test many of the counterintuitive predictions of quantum mechanics, such as squeezed states of light, entanglement, Photon antibunching, and other quantum phenomena.A new quantum technology based on these phenomena is emerging and requires experimentation through fundamental research.For instance, coherent light (usually from a laser source) in its classical form is hindered by an inherent standard noise limit, making it unsuited for use in cutting-edge quantum applications that require performance below the standard quantum limit.Squeezed light, however, exhibits reduced noise in one of the two electric field quadrature components [1].This makes it suitable for use in high-precision measurements [2], such as the ultraprecise measurement of lengths for the detection of gravitational waves.Other potential applications of squeezed light include quantum teleportation (required for quantum computers) [3] and, communication [4].To generate squeezed light, several nonlinear systems have been proposed such as Bose-Einstein condensates [5], parametric down-conversion [6], and nonlinear couplers [7].These systems rely on the interaction of coherent light with a nonlinear material to generate squeezed light via various processes.Among these quantum systems, the nonlinear coupler has received a significant deal of interest due to its advantages as a simple-structured, experimentally realizable, and easily integrated system.Since the standard twowaveguide coupler has been proposed [8], several designs of nonlinear couplers have been suggested such as three-waveguide [9], four-waveguide nonlinear couplers [10], [11], cavity-assisted nonlinear couplers [12], [13], and nonlinear couplers with frequency mismatch [14].Also, nonlinear couplers based on various nonlinear processes such as Kerr nonlinear effect [15]- [18], second harmonic generation [19], and four-wave mixing have been proposed [20].
One of the most promising processes for producing nonclassical light is the Raman effect.Raman process is the third-order nonlinear scattering process that occurs when light interacts with solid material [21].In this interaction, light transfers some of its energy to the solid, and this results in two possibilities: Stokes Raman and anti-Stokes Raman scattering (See Fig. 1a).Stokes Raman scattering occurs when the emitted light has lower energy than the incident light.Inversely, anti-Stokes Raman scattering takes place when the energy of the emitted light is higher than the incident light.In addition, both of these processes will cause phonon vibrations in solids.The quantum description of various forms of Raman process in the nonlinear media has been a topic of interest among quantum opticians (see, e.g., [22] ).However, the quantum aspects of a nonlinear coupler operating with the Raman process are yet to be fully investigated [23], [24].Particularly, we believe that the single and compound mode squeezing in the current system is yet to be investigated.
Here we consider a two-waveguide coupler system with one nonlinear Raman-active waveguide coupled to another waveguide with just a linear process.Each waveguide is pumped with one fundamental coherent mode.The coherent fundamental mode pumped into the Raman-active waveguide produces Raman Stokes mode, anti-Stokes mode, and phonon mode (See Fig. 1b).The fundamental modes in both waveguides are also linearly

Mathematical Description of the System
Here we use the analytical perturbative (AP) method to study the current system.The AP method was introduced by Sen and Mandal [25] and subsequently used by others to investigate the quantum optical properties of various quantum systems (See, e.g.[26]).The AP method has shown improved results compared with the usual shortlength approximation (SLA) method used earlier in the literature (See, e.g., [27]).The quantum mechanical description of the current system can be expressed in terms of the following momentum operator.

(
) On the right side of equation ( 1), h.c.means the Hermitian conjugate, and the first two terms (ℏ 1  ̂1 †  ̂1, ℏ 2  ̂2 †  ̂2) account for the pump modes in the first and second waveguides respectively.The third, fourth, and fifth terms (ℏ   ̂ † ̂, ℏ  ̂ †, ℏ   ̂ † ̂) accounts for the Stokes mode, the phonon mode, and the anti-Stokes mode, respectively.The coefficients kj are the wave vectors.The term ℏ ̂1 ̂ †̂ † + ℎ. . is responsible for the Stokes Raman scattering, where the annihilation of the pump mode creates the Stokes mode alongside the phonon mode.The coefficient  is the Stokes nonlinear coefficient that controls this process.Inversely, the term ℏ ̂1d ̂ † + ℎ. .indicates the anti-Stokes Raman scattering that creates the anti-Raman mode due to the annihilation of pump mode and phonon mode; the controlling parameter for this process is  (The anti-Stokes nonlinear coefficient).Finally, the last term in Equation (1) (ℏ ̂2 ̂1 † + ℎ. . ) expresses the evanescent linear coupling between the fundamental modes.The strength of this interaction is proportional to the evanescent coupling coefficient (); this coefficient is inversely proportional to the separation between the waveguides.
Using the Heisenberg picture, the evolution of an operator  ̂ be described through the spatial Heisenberg equation of motion in the form of where z is a spatial variable along the propagation direction [28].
By substituting the momentum operator from Eq. ( 1) into the Heisenberg equation of motion for each one of the five operators  ̂ = { ̂1,  ̂2,  ̂, ,  ̂ }, we arrive at the following system differential equation for the operator's evolution. ( These are the Heisenberg equations of motion that specifically describe the dynamics of each operator.In the analytical method, the solution to each operator in the previous system ( 2)-( 6) is assumed in the form of the Baker-Hausdorff (BH) formula as below. ( In the previous BH formula,  ̂(0) represents the operator at its initial state, i.e., at z = 0.The expansion is usually limited to only second-order terms on the assumption that the higher-order terms are insignificant and can be disregarded.Upon evaluating the commutator terms [ ̂(),  ̂(0)] and [ ̂(), [ ̂(),  ̂(0)]] in equation ( 7), closed-form analytical solutions to each operator could be obtained as follows.
( ) Solutions ( 8)-(12) describe the spatial evolution of the operators  ̂1(),  ̂2(),  ̂(), () and  ̂() in the zdirection expressed in terms of unknown spatial-dependent coefficients   (),   (),   (),   () and   ().These coefficients can be evaluated by substituting the operator solutions ( 8)-( 12) into each equation of motion in the set ( 2)-( 6) and collecting similar terms from both sides.This procedure produces a set of differential equations that define the spatial propagation of these unknown coefficients.This set is listed in Appendix A for completeness.As a result, even if the operator solutions in ( 8)-( 12) have an analytical closed form, we still need to numerically solve the set of equations (A1) -(A36) to acquire their coefficients.

Single-Mode Squeezing
In quantum optics, a quantized coherent electric field  ̂() ( ̂−  ̂ †) [29].Squeezing is the condition, where the noise in one of the quadrature components is reduced below the shot noise level (The noise level of coherent state) at the expense of the increased noise in the other quadrature.Meaning that, if the first quadrature component is squeezed, the second quadrature component is not, and vice versa.Mathematically, for a specific single-mode j to be squeezed, one of the quadratures has to fulfill one of these conditions.
where ⟨(Δ ̂) 2 ⟩ and ⟨(Δ ̂) 2 ⟩ are the variances of the first and the second quadrature respectively.The value '1/4' in the previous equation is the standard noise level in a coherent state (shot noise level), and single-mode squeezing occurs when the noise in a certain mode j reduces below this value.The noise fluctuation obeys the Heisenberg uncertainty relation which states that it is impossible for both conditions in equation ( 13) to hold at the same time, i.e., √〈 (∆ ̂) 2 〉 〈(∆ ̂) 2 〉 ≥ 0.25.In terms of creation and annihilation operators, the quadrature variances of some mode j can be written as.
In the previous equation ( 14), the bracket indicates the quantum expectation value of the operators in normal order.To obtain the quadrature variances for the first pump mode, we substitute  ̂1 and its complex conjugate  ̂1 † from equation ( 8) into equation ( 14).This yields the following equation. ( Similarly, we can develop an equation for possible squeezing exhibited by other modes.For example, if we substitute  ̂2 and its complex conjugate  ̂2 † from equation ( 9) into equation ( 14), we arrive at an expression for the quadrature variances for the second mode.However, here we will only look at squeezed states exhibited by the first pump mode, i.e., the mode propagates inside the Raman waveguide.We anticipate that the best possible squeezed states will be displayed by such a mode.In the previous equation, the variables   represent the initial complex amplitude of the operator  ̂, i.e., 〈 ̂( = 0)〉 =   and ⟨ ̂ † ( = 0)⟩ =  * where  * is the complex conjugate of α.

Results and Discussions
It is easier to obtain numerical solutions by putting the relevant equations in dimensionless form.The system of equations ( A1) -(A36) can be converted into dimensionless form by dividing both sides of each equation with  1 .
The evolution of single-mode quadrature variances in the case of the spontaneous Raman process is shown in Figure 2. Observe the oscillation of both quadrature variances of the first pump mode below the shot noise level (y = ¼ line) of the coherent state.In particular, at certain distances, the second quadrature ΔY evolves totally below the shot-noise level.Here, both quadrature variances oscillate with 180° out of phase, which means, when the first quadrature variance reaches the maximum point, the second quadrature variance will be at its minimum point, and vice versa.In addition, the maximal squeezing amplitude of the second quadrature component ΔY is always lower than the first quadrature component ΔX, and this makes sense could be inferred from Equation (15).For the stimulated Raman process, all modes are initially set in coherent states [25].In Figure 3, the graphs show the evolution of squeezed states as a function of scaled distance phase mismatching, i.e., when the wave numbers of both pumping modes are not equal,  ̃1≠  ̃2, which means their frequencies are not equal.In Fig. 3a, we choose the input pump fields to have scaled wavenumbers of  ̃1 = 10 and  ̃2 = 12 respectively.Under this condition, the squeezing in the first pump mode shows a collapses and revivals pattern.This behavior may result from the superposition of two modes with distinct phases.In this case, the second quadrature component ΔY oscillates almost below the shot noise level of a coherent state, which is an intriguing property.This is an excellent level of squeezed state with minimal noise.When the frequency discrepancy between the two pumping modes increases, as depicted in Fig. 3b, the collapses and revivals pattern begin to vanish and the quadrature variances of the first pump mode evolve with irregular amplitudes.As the mismatching increases further (Figure 3c), the amplitude of both quadrature variances increases progressively as a function of distance.This does not seem reasonable, and we anticipate that the AP method may produce inaccurate results for certain combinations of high initial input parameters and greater evolution distances.However, the AP method performs well with comparatively low initial input parameters, especially for early evolution distances.
We have finally examined the effect of linear coupling between the two pump modes on the generated squeezed states.Results are shown in Fig. 4a and Figure 4.b. for the case of linear coupling ̃ = 0.2.̃ = 0.7 respectively.Figure 4a also depicts the evolution of the quadrature variances as a function of scaled distance.The oscillatory nature of the quadrature variances below the shot-noise level is evident.However, as observed in Fig. 3c, the amplitude of quadrature variances steadily increases which seems unreasonable.Nevertheless, the general feature of the oscillatory nature of the quadrature variances around the shot-noise limit remains valid, especially at short evolution distances.An intriguing aspect of Figure 4b is that the first quadrature ΔX does not oscillate symmetrically about the shot-noise level and never oscillates completely above it.Nonetheless, at some distance ranges, the second quadrature variance ΔY oscillates above the shot noise level, indicating the absence of squeezing, whereas at other distance ranges, it oscillates completely below the shot noise level, indicating complete squeezing.In this case, this type of squeezing is only appropriate for short-distance signal transmission and not for long-distance optical communications.

Conclusion
To conclude, the AP method was used to study the generation of squeezed states in an asymmetric nonlinear coupler comprised of a Raman-active waveguide and a linear waveguide.In the AP method, analytical solutions of the operators are obtained assuming weak nonlinearity and expressions of quadrature variances of single mode squeezing can be derived.we have investigated the production of single-mode squeezed states of light in the Raman-active waveguide in cases of spontaneous and stimulated Raman processes.spontaneous Raman requires preparing the pump modes in coherent states and the Raman modes in vacuum states, while stimulated Raman requires preparing all modes in coherent states.during the spontaneous Raman process, the quadrature variances of the first waveguide mode oscillate around the shot noise level of the coherent state.The second quadrature evolves completely below the shot-noise level at specific distances.In the stimulated Raman process, all modes are initially set in coherent states.Under reasonable phase mismatching, the field shows squeezed states in the form of collapses and revivals patterns.This behavior may result from the superposition of two modes with distinct phases.In this case, the second quadrature component oscillates almost below the shot noise level of a coherent state as well.This is an excellent level of squeezed state with minimal noise.When the frequency mismatching between the two pumping modes increases, the collapses and revivals pattern begin to vanish and the quadrature variances of the first pump mode evolve with irregular amplitudes.We have finally examined the effect of linear coupling between the two pump modes on the generated squeezed states.Both field quadratures exhibit squeezed states of oscillatory nature with asymmetry about the shot-noise level.Nonetheless, at some distance ranges, the second quadrature variance oscillates above the shot noise level, indicating the absence of squeezing, whereas at other distance ranges, it oscillates completely below the shot noise level, indicating complete squeezing.In this case, this type of squeezing is appropriate for short-distance signal transmission in optical communications.This calculation paves the way for the future use of squeezed states in optical communications and precise measurements.

6th
Photonics Meeting 2023 Journal of Physics: Conference Series 2627 (2023) 012005IOP Publishing doi:10.1088/1742-6596/2627/1/012005 2 coupled (cross-action coupling) via evanescent waves.We will examine the generation of single and compound mode squeezed states in the current system due to spontaneous Raman under phase mismatching conditions.

Figure 1a .
Figure 1a.Diagram for Raman process Figure 1b: Basic diagram of the twowaveguide asymmetric nonlinear coupler

6th 5 Figure 2 :
Figure 2: Evolution of the quadrature variances of the electric field of the pump mode  ̂1 as a function of spatial distance showing fluctuation of squeezed states below the shot-noise level (Thin black line-y = 0.25) under spontaneous Raman process, i.e., when the pump modes are prepared in a coherent state, while other modes are in vacuum states, i.e.,  1 ≠ 0,  2 ≠ 0 and   =   =   = 0. Other parameters used in this simulation are fixed at  ̃1 =  ̃2 = 10,  ̃ = 10,  ̃ = 0.00001,  ̃ = 10,  ̃=  ̃= 0.0001 and ̃= 0.8.