Perspective Chapter: Squeezing and Entanglement of Two-Modes Quantum X Waves

Abstract

Quantum theory of generalized $\mathrm{X}$ waves with orbital angular momentum in dispersive media, and the interaction of quantized $\mathrm{X}$ waves in quadratic nonlinear media were studied in (J. opt,20,065201 (2018)). We present a kind of phase matching, which is called velocity phase matching, and this phase matching can be used for determining the length of the nonlinear crystal or the interaction time in the experiment setup, to produce $\mathrm{X}$ waves with particular velocity $v$. Moreover, we introduce more analysis for the dependence of squeezing of $\mathrm{X}$ waves on its spectral order, and for spectral orders $j>0$, we predict the existence of a characteristic axicon aperture for maximal squeezing. Then, we find the quantum squeezed state of the down-converted state generated by the ${\chi }^2$-nonlinear process. Finally, we detect their entanglement using a criterion of separability.

1. Introduction

Based on the work of Tesla, German physicist Konstantin Meyl created a new unified field and particle theory. In Meyl’s theory [1], vortices are used to explain quantum and classical physics, mass, gravitation, the constant speed of light, neutrinos, waves, and particles. With the help of this model, the properties of subatomic particles may be precisely estimated. The unified equation can also be used to derive well-known equations. He offers resources for carrying out an experiment by Tesla that proves the existence of scalar waves. Simple energy vortices in the form of particles are what scalar waves are. The electromagnetic, eddy current, potential vortex, and special distributions are all covered by the unified field theory. According to Meyl, the field—which creates particles through decay or conversion—always comes first. Energy particles (i.e., potential vortices) were not included in the theory because they are not recognized by classical physics. Quantum physics attempted to explain everything in terms of vortices which is why it is incomplete. Gravitation is from the speed of light difference caused by proximity that, proportional to field strength, decreases the distance of everything for the field strength [2]. The spatially localized vortex structures include pseudo-nondiffracting (P-N) vortex beams and their superpositions and provide additional effects interesting for theoretical research, experiments, and applications [3]. As we see, this kind of vortex in its quantum form can connect to Meyl’s vortex to explain unified field and particle theory of Meyl. Also, it can connect to the localizability of Busch [4] and others for elementary particle physics to solve the theory of measurement in relativistic quantum mechanics problems as stated in Ref. [5]. Vortex solutions produced in some particle physics models like extended Abelian Higgs model [6], Galaxies and their dark-matter halos [7].

The primary characteristic that unifies numerous advancements in quantum physics is entanglement [8]. Entanglement is required, as a prime example, to distinguish between classical and quantum physics using Bell inequalities [9, 10]. Another way to think of entanglement is as the tool that permits real quantum protocols like teleportation [11] and Bell inequalities-based cryptography [12]. When a quantum method, like Shor’s algorithm, yields a meaningful performance advantage over a classical computer, large entanglement is anticipated to be present in quantum registers [13]. It is obvious that entanglement is essential to comprehending and using quantum physics. Therefore, it makes sense to examine the creation of entanglement at its most fundamental source, which are the particle physics theories of fundamental interactions. A Bell inequality would never be violated in nature if the quantum theory of electromagnetic, or QED, never produced electron entanglement. This suggests that at a fundamental level, quantum unitary development must produce entanglement. The gauge principle, which states that the physical laws are invariant despite internal local rotations for particular symmetry groups, has become widely accepted as a way to explain the fundamental interactions seen in nature. With regard to electroweak and strong interactions, the Standard Model uses a Lagrangian that is essentially constrained by gauge symmetry requirements, except from quantum gravity. It is only logical to continue looking for a principle that is even more basic. Previous research has looked at entanglement’s function in particle physics. Orthopositronium has been demonstrated in Ref. [14] to decay into 3-photon states that can be used in Bell-like experiments that reject classical physics more quickly than the conventional 2-particle Bell inequality. Bell inequalities have also been considered in regard to neutrino oscillations [15], kaon physics [16, 17], and their relationship with the characterization of T-symmetry violation [18]. The S-matrix formalism has been used to examine how entanglement varies in an elastic scattering process in Ref. [19]. A recent study on entanglement in deep elastic scattering is also worth mentioning [20]. Bell inequalities have also been discussed in regard to quantum correlations in CMB radiation [21].

Squeezed states, and two-mode-squeezed states, are foundational for describing the basis nature of quantum mechanical phenomena. Their quantum entanglement plays a central role in quantum physics, experiments of quantum optics, and quantum information science [22, 23, 24, 25, 26, 27]. Nonlinear quantum optics, in particular, parametric down-conversion (PDC), gives us ways to study issues in the foundations of quantum mechanics that are not readily available using other technologies [27, 28]. The quantum communication encode information into the polarization degrees of freedom of photons [12] or the orbital angular momentum (OAM) [29, 30, 31, 32, 33, 34, 35]. The important problem related to multi-modes quantum communications is the diffraction and dispersion of the electromagnetic waves [30]. Electromagnetic waves packet is usually subjected to diffraction and dispersion. Diffraction create a broadening in space during propagation of the wave and dispersion create a broadening in the time during the propagation. Ultimately, these effects are connected with the bounded nature of the wave spectrum and, therefore, to its finite energy content [36]. A great effort has been done in studying the effect of atmosphere turbulence in free-space communication [31, 37, 38, 39, 40]. Maxwell’s equations admit diffraction-free and dispersion-free solutions, the so-called localized waves [3]. In particular, such solutions in the monochromatic domain are the Bessel beams [41], and in the pulsed domain, the most renowned localized waves are the $\mathrm{X}$ waves, first introduced in acoustics in 1992 by Lu and GreenLeef [42]. Despite the great work in the literature concerning $\mathrm{X}$ waves, the investigations of their quantum properties are very few [43, 44]. Generalization of the traditional $\mathrm{X}$ waves to the case of OAM-carrying $\mathrm{X}$ wave and the coupling between angular momentum and the temporal degrees of freedom of ultrashort pulses have been investigated [45]. Very recently, quantum and squeezing of $\mathrm{X}$ waves with OAM in nonlinear dispersive media have been proposed [46, 47], and they can open a new direction for free-space quantum communication and different areas of physics. In this work, we state the propagation of a scalar electromagnetic field in a linear dispersive medium. We present a kind of phase matching called velocity phase matching and introduce a relation between the interaction time and the velocity of $\mathrm{X}$ wave as well as a relation between the length of the ${\chi }^{\left(2\right)}$-nonlinear crystal and the velocity of $\mathrm{X}$ wave generated by the spontaneous parametric down-conversion (SPDC) process in Section 2. We study the SPDC process in a quadratic medium, in particular the dependence of squeezing of the down-converted state generated by the ${\chi }^{\left(2\right)}$-nonlinear crystal on the spectral order of quantum $\mathrm{X}$ waves in Section 3. We introduce the quantum squeezed form of the state generated by the SPDC process and its entanglement in Section 4. Finally, the results are summarized in Section 5.

2. Propagation of a generalized $\mathrm{X}$ wave in dispersive media and its quantization

We consider the propagation of a scalar electromagnetic field in a linear dispersive medium with refractive index $n=n\left(\omega \right)$. Applying the paraxial and slowly varying envelope approximation to the envelope function $A\left(\mathbf{r},t\right)$ of an electric field

$$E\left(\mathbf{r},t\right)=\sqrt{\frac{2}{{\epsilon }_0{n}^2}}A\left(\mathbf{r},t\right)e^{i\left(\mathit{kz}-\mathit{wt}\right)},$$E1

where $A\left(\mathbf{r},t\right)$ varies slowly with $z$and $k=n\left(\omega \right)\omega /c,$ which are the propagation coordinate and the wave number, respectively.

The field envelope $A\left(\mathbf{r},t\right)$ satisfies the following eq. [48],

$$i\frac{\partial A}{\partial t}+i{\omega }^{\prime }\frac{\partial A}{\partial z}-\frac{{\omega }^{″}}{2}\frac{{\partial }^{2}A}{\partial z^2}+\frac{{\omega }^{\prime }}{2k}{\mathrm{\Delta }}_{\perp }^{2}A=0,$$E2

where ${\omega }^{\prime }=c^2\mathit{dk}/\mathit{d\omega }$ and ${\omega }^{″}=c^2{d}^{2}k/d{\omega }^2$ are the first and the second order dispersion, respectively.

The solution of Eq. (2) can be written as a superposition of the generalized $\mathrm{X}$ waves as follows [47]:

$$A\left(\mathbf{r},t\right)=\sum _{m,p}\int \mathit{dv}{C}_{\mathit{mp}}\left(v\right)e^{-i\frac{v^2}{2{{\omega }^{\prime }}^{\prime }}t}{\psi }_{m,p}^{\left(v\right)}\left(R,\zeta \right),$$E3

where ${\psi }_{m,p}^{\left(v\right)}\left(R,\zeta \right)$ is the generalized $\mathrm{X}$ wave of OAM number $m,$ spectral order $p$ and velocity $v$ defined as,

$${\psi }_{m,p}^{\left(v\right)}\left(R,\zeta \right)={\int }_0^{\infty }\mathit{d\alpha }{f}_p\left(\alpha \right)J_m\left(\sqrt{{{\omega }^{\prime }}^{\prime }k/{\omega }^{\prime }}\mathit{\alpha R}\right)e^{i\left(\alpha -v/{{\omega }^{\prime }}^{\prime }\right)\zeta }{e}^{\mathit{im\theta }},$$E4

with $R=\sqrt{x^2+y^2}$, $\zeta =z-\left({\omega }^{\prime }+v\right)t$ which is the co-moving reference frame associated to the $\mathrm{X}$ waves, and $f_p\left(\alpha \right)=\sqrt{k/{\pi }^2{\omega }^{\prime }\left(1+p\right)}\left(\mathrm{\Delta }\alpha \right)L_p^{\left(1\right)}\left(2\alpha \mathrm{\Delta }\right)e^{-\alpha \mathrm{\Delta }}$ which is the spectrum function, where $L_p^{\left(1\right)}\left(x\right)$ is the generalized Laguerre polynomials of the first kind of order $p$ [48], and $\mathrm{\Delta }$ is the reference length related to the spatial extension of the spectrum [47]. The quantized field of Eq. (3) can be written as [47],

$$\widehat{A}\left(\mathbf{r},t\right)=\sum _{m,p}^{\infty }\int \mathit{dv}{e}^{\frac{-i}{\mathrm{\hslash }}\left(\frac{{\mathit{Mv}}^2}{2}\right)t}\sqrt{\mathrm{\hslash }{\omega }_{m,p}\left(v\right)}{\psi }_{m,p}^{\left(v\right)}\left(R,\zeta \right){\widehat{a}}_{m,p}\left(v\right)+h.c.,$$E5

where $h.c.$ denotes to the Hermitian conjugate, ${\omega }_{m,p}\left(v\right)=v^2/\left(2{{\omega }^{\prime }}^{\prime }\right)$ and $M=\mathrm{\hslash }/{{\omega }^{\prime }}^{\prime }$ is the mass of $\mathrm{X}$ wave. The Hamiltonian operator can be written as follows [47]:

$$\widehat{H}=\sum _{m,p}\int \mathit{dv}\mathrm{\hslash }{\omega }_{m,p}\left(v\right)\left({\widehat{a}}_{m.p}^{†}\left(v\right){\widehat{a}}_{m,p}\left(v\right)+\frac{1}{2}\right),$$E6

with the following usual bosonic commutation relations of creation and annihilation operators of the field,

$$\begin{array}{lll}\left[{\widehat{a}}_{m,p}\left(v\right),{\widehat{a}}_{n,q}^{†}\left(u\right)\right]& =& {\delta }_{m,n}{\delta }_{p,q}\delta \left(u-v\right),\\ \left[{\widehat{a}}_{m,p}\left(v\right),{\widehat{a}}_{n,q}\left(u\right)\right]& =& 0=\left[{\widehat{a}}_{m,p}^{†}\left(v\right),{\widehat{a}}_{n,q}^{†}\left(u\right)\right].\end{array}$$E7

Now, we turn our attention to the quadratic nonlinear process involving $\mathrm{X}$ waves, particularly to the spontaneous parametric down-conversion (SPDC) [28]. The interaction Hamiltonian of this process, with a practical case $\rho =\sqrt{k_1{\omega }_2^{\prime }/k_2{\omega }_1^{\prime }}\simeq 1,$ can be written in two different forms [47]:

The first form is for the time-dependent interaction Hamiltonian which can be written as follows [47]:

$${\mathrm{\mathscr{H}}}_I=\chi ⟨A_1|A_2^{\ast }⟩+{\chi }^{\ast }⟨A_2|A_1^{\ast }⟩,$$E8

$$\chi ⟨A_1|A_2^{\ast }⟩=\chi \int d^{3}r{\widehat{A}}_1^{†}\left(\mathbf{r},t\right){\widehat{A}}_2^{†}\left(\mathbf{r},t\right)$$E9

$$\begin{array}{lll}\chi ⟨A_1|A_2^{\ast }⟩& =& \chi \sum _{m,p,n,q}\int \mathit{du}\mathit{dv}{e}^{\frac{i\left(u^2+v^2\right)t}{2{{\omega }^{\prime }}^{\prime }}}\sqrt{{\mathrm{\hslash }}^2{\omega }_{m,p}\left(u\right){\omega }_{n,q}\left(v\right)}\\ & & \times \int d^{3}r{\psi }_{m,p}^{\left(u\right)\ast }\left(R,{\zeta }_1\right){\psi }_{n,q}^{\left(v\right)\ast }\left(R,{\zeta }_2\right){\widehat{a}}_{m,p}^{†}\left(u\right){\widehat{b}}_{n,q}^{†}\left(v\right),\end{array}$$E10

$${\widehat{H}}_I\left(t\right)=\mathrm{\hslash }\sum _{m,p,q}\int \mathit{du}\mathit{dv}{\chi }_{m,p,q}\left(u+v\right)\sqrt{{\omega }_{m,p}\left(u\right){\omega }_{-m,p}\left(v\right)}{e}^{\mathit{\text{iF}}\left(u,v\right)t}{\widehat{a}}_{m,p}^{†}\left(u\right){\widehat{b}}_{-m,q}^{†}\left(v\right)+h.c.,$$E11

with

$$F\left(u,v\right)=\frac{1}{2{{\omega }^{\prime }}^{\prime }}\left[2\mathit{uv}+\left(v-u\right)\left({\omega }_1^{\prime }-{\omega }_2^{\prime }\right)\right],$$E12

and the interaction function is

$${\chi }_{m,p,q}\left(u+v\right)=\frac{{\left(-1\right)}^m\chi {\mathrm{\Delta }}^2}{{{{\omega }^{\prime }}^{\prime }}^2\sqrt{\left(1+p\right)\left(1+q\right)}}\left(u+v\right)L_p^{\left(1\right)}\left(\frac{\left(u+v\right)\mathrm{\Delta }}{2{{\omega }^{\prime }}^{\prime }}\right)L_q^{\left(1\right)}\left(\frac{\left(u+v\right)\mathrm{\Delta }}{2{{\omega }^{\prime }}^{\prime }}\right)e^{\frac{-\left(u+v\right)\mathrm{\Delta }}{2{{\omega }^{\prime }}^{\prime }}}\theta \left(u+v\right),$$E13

where $\theta \left(u+v\right)$, is the Heaviside step function [48]. The second form is for the finite length interaction Hamiltonian which can be given by Ref. [47]

$${\widehat{H}}_I\left(z\right)=\int \mathit{dt}\left[\chi ⟨A_1|A_2^{\ast }⟩+{\chi }^{\ast }⟨A_2|A_1^{\ast }⟩\right].$$E14

The final form of ${\widehat{H}}_I\left(z\right)$ can be written as follows:

$${\widehat{H}}_I\left(z\right)=\mathrm{\hslash }\sum _{m,p,q}\int \mathit{du}\mathit{dv}{\mathrm{\Xi }}_{m,p,q}\left(u,v\right)\sqrt{{\omega }_{m,p}\left(u\right){\omega }_{-m,q}\left(v\right)}{e}^{i\mathrm{\Lambda }\left(u,v\right)z}{\widehat{a}}_{m,p}^{†}\left(u\right){\widehat{b}}_{-m,q}^{†}\left(v\right)+h.c.,$$E15

where $\mathrm{\Lambda }\left(u,v\right)=2\mathit{uv}/{{\omega }^{\prime }}^{\prime }\left(u+v\right),$ and ${\mathrm{\Xi }}_{m,p,q}\left(u,v\right),$ which is the modified vertex function can be given by Ref. [47],

$${\mathrm{\Xi }}_{m,p,q}\left(u,v\right)=\frac{{\left(-1\right)}^{m}2\chi {\mathrm{\Delta }}^2}{{{{\omega }^{\prime }}^{\prime }}^2\sqrt{\left(1+p\right)\left(1+q\right)}}\frac{u^2+v^2}{u+v}{L}_p^{\left(1\right)}\left(\frac{\left(u^2+v^2\right)\mathrm{\Delta }}{\left(u+v\right){{\omega }^{\prime }}^{\prime }}\right)L_q^{\left(1\right)}\left(\frac{\left(u^2+v^2\right)\mathrm{\Delta }}{\left(u+v\right){{\omega }^{\prime }}^{\prime }}\right)e^{-\frac{\left(u^2+v^2\right)\mathrm{\Delta }}{\left(u+v\right){{\omega }^{\prime }}^{\prime }}}.$$E16

To calculate the state of the system after the interaction, we apply Schwing-Dyson expansion of the propagator $exp\left[-i{\widehat{H}}_I\left(t\right)/\mathrm{\hslash }\right]$, and consider only the first term of the expansion [49]. For the time-dependent form of the interaction Hamiltonian stated in Eq. (11), the final state after the ${\chi }^{\left(2\right)}$-nonlinear interaction can be expressed as follows [47]:

$$\mid {\psi }^{\left(1\right)}\left(t\right)⟩=\frac{-i}{\mathrm{\hslash }}{\int }_0^t\mathit{d\tau }{\widehat{H}}_I\left(\tau \right)\mid 0⟩.$$E17

Substituting Eq. (11) into the above equation and after some calculations, the final state can be written in the following form:

$$\mid {\psi }^{\left(1\right)}\left(t\right)⟩=\sum _{m,p,q}\int \mathit{du}\mathit{dv}{\mathcal{G}}_{m,p,q}\left(u,v\right)\mathrm{ℱ}\left(u,v,t\right)\mid m,p,u;-m,q,v⟩,$$E18

where $\mid m,p,u;-m,q,v⟩\equiv {\widehat{a}}_{m,p}^{†}\left(u\right){\widehat{b}}_{-m,q}^{†}\left(v\right)\mid 0,0⟩,$ the amplitude ${\mathcal{G}}_{m,p,q}\left(u,v\right)$ is defined as

$${\mathcal{G}}_{m,p,q}\left(u,v\right)=-i\sqrt{{\omega }_{m,p}\left(u\right){\omega }_{-m,q}\left(v\right)}\frac{{\chi }_{m,p,q}\left(u+v\right)}{F\left(u,v\right)},$$E19

and $\mathrm{ℱ}\left(u,v,t\right)=\left[e^{\mathit{\text{iF}}\left(u,v\right)t}-1\right].$ The above state represents the superposition of two particles corresponding to the modes ${\omega }_1$ and ${\omega }_2$ (generated by the nonlinear interaction) traveling with velocities $u$ and $v$, respectively. For the second form of the interaction Hamiltonian stated in Eq. (15), the final state of the system can be written as

$$\mid {\psi }^{\left(1\right)}\left(L\right)⟩=\frac{-i}{\mathrm{\hslash }}{\int }_0^L\mathit{dz}{\widehat{H}}_I\left(z\right)\mid 0⟩.$$E20

Substituting Eq. (15) into the above equation, the state of the field at the output face of the crystal can be written as follows:

$$\mid {\psi }^{\left(1\right)}\left(L\right)⟩=\sum _{m,p,q}\int \mathit{du}\mathit{dv}{\mathrm{ℒ}}_{m,p,q}\left(u,v\right)\mathrm{ℱ}\left(u,v,L\right)\mid m,p,u;-m,q,v⟩,$$E21

where $\mathrm{ℱ}\left(u,v,L\right)=\left[e^{i\mathrm{\Lambda }\left(u,v\right)L}-1\right],$ and

$${\mathrm{ℒ}}_{m,p,q}\left(u,v\right)=\frac{-i}{2}\sqrt{{\omega }_{m,p}\left(u\right){\omega }_{-m,q}\left(v\right)}\left[\frac{{{\omega }^{\prime }}^{\prime }\left(u+v\right)}{\mathit{uv}}\right]{\mathrm{\Xi }}_{m,p,q}\left(u,v\right).$$E22

The transition probability for the field to be in such a state after the time-dependent interaction with the ${\chi }^{\left(2\right)}$-nonlinear crystal is proportional with ${\left|\mathrm{ℱ}\left(u,v,t\right)\right|}^2$ as follows:

$$P\left(t\right)\propto {\left|\mathrm{ℱ}(u,v,t)\right|}^2,$$E23

similarly, the transition probability for the finite length interaction of the ${\chi }^{\left(2\right)}$-nonlinear crystal is proportional with ${\left|\mathrm{ℱ}\left(u,v,L\right)\right|}^2$ as follows [47]:

$$P\left(L\right)\propto {\left|\mathrm{ℱ}(u,v,L)\right|}^2.$$E24

Explicitly, ${\left|\mathrm{ℱ}\left(u,v,t\right)\right|}^2$ can be expressed as follows:

$${\left|\mathrm{ℱ}\left(u,v,t\right)\right|}^2=2\left[1-cos\left(F\left(u,v\right)t\right)\right].$$E25

From the above equation, we notice that the transition probability reaches its maximum, when ${\left|\mathrm{ℱ}\left(u,v,t\right)\right|}^2$ reaches its maximum, and that occurs when the condition $F\left(u,v\right)t=\left(2m+1\right)\pi$, with $m=\mathrm{0,1,2,3},\dots$, is achieved. The above condition is considered as a new phase matching condition, called velocity matching condition over the velocities of the two-modes $\mathrm{X}$ waves generated from SPDC process. This condition requires that

$$u=\frac{2k_m^{\prime }+v\mathrm{\Delta }{\omega }^{\prime }}{2v+\mathrm{\Delta }{\omega }^{\prime }},$$E26

where $\mathrm{\Delta }{\omega }^{\prime }={\omega }_1^{\prime }-{\omega }_2^{\prime },$ and $k_m^{\prime }=\left(2m+1\right)\pi {{\omega }^{\prime }}^{\prime }/t$. For the case, when $\mathrm{\Delta }{\omega }^{\prime }=0$, we get $\mathit{uv}=k_m^{\prime }=\left(2m+1\right)\pi {{\omega }^{\prime }}^{\prime }/t,$ and the interaction time can be given by $t=\left(2m+1\right)\pi {{\omega }^{\prime }}^{\prime }/\mathit{uv}$. When $u=v$, regardless of that $\mathrm{\Delta }{\omega }^{\prime }=0,$ we get $v^2=k_m^{\prime }=\left(2m+1\right)\pi {{\omega }^{\prime }}^{\prime }/t.$

Thus, the interaction time can be written in terms of the velocity of $\mathrm{X}$ waves modes as follows:

$$t=\frac{\left(2m+1\right)\pi {{\omega }^{\prime }}^{\prime }}{v^2}.$$E27

For the finite length of the nonlinear crystal, the velocity matching condition is given by Ref. [47],

$$u=\frac{k_{n}v}{v-k_n},$$E28

where $k_n=\left(n+\frac{1}{2}\right)\pi {{\omega }^{\prime }}^{\prime }/L$, with $n=\mathrm{0,1,2},\cdots .$ Therefore, the velocity matching condition fixes either the length of the crystal $L$ or the relative velocity of the two modes $\mathrm{X}$ wave involved in the SPDC process as, $\mathit{uv}/\left(u+v\right)=k_n=\left(n+1/2\right)\pi {{\omega }^{\prime }}^{\prime }/L$, and the length of the crystal $L$ can be given by,

$$L=\frac{\left(n+\frac{1}{2}\right)\pi {{\omega }^{\prime }}^{\prime }\left(u+v\right)}{\mathit{uv}}.$$E29

For $u=v$, the above equation becomes

$$L=\frac{\left(2n+1\right)\pi {{\omega }^{\prime }}^{\prime }}{v}.$$E30

From Eqs. (27) and (30), we note that the ordinary relation between the length and the time is achieved ($L=\mathit{vt}$). This result can be used for determining the length of the nonlinear crystal in the experimental setup, to produce $\mathrm{X}$ waves with velocity $v$. This is the first result of our work.

3. Squeezing of quantum $\mathrm{X}$ waves

Now, we can study the general case of the squeezing effect in the SPDC process, in particular, the dependence of squeezing of the down-converted state, generated by ${\chi }^{\left(2\right)}$-nonlinear media, on the spectral orders of quantum $\mathrm{X}$ waves, for the case, when the velocities of the two-modes $\mathrm{X}$ waves are equal $u=v$, and their spectral orders are equal as well $p=q=j.$ The interaction time Hamiltonian in Eq. (11) becomes

$${\widehat{H}}_I\left(t\right)=\mathrm{\hslash }\sum _{\mathit{mj}}\int \mathit{dv}{\chi }_{\mathit{mj}}\left(2v\right)\omega \left(v\right){\widehat{a}}_{\mathit{mj}}^{†}\left(v,t\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v,t\right)+h.c.,$$E31

where ${\widehat{a}}_{\mathit{mj}}^{†}\left(v,t\right)=e^{\mathit{\text{iF}}\left(v\right)t/2}{\widehat{a}}_{\mathit{mj}}^{†}\left(v\right),$ and ${\widehat{b}}_{-\mathit{mj}}^{†}\left(v,t\right)=e^{\mathit{\text{iF}}\left(v\right)t/2}{\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)$. Also, the interaction function in Eq. (13) becomes

$${\chi }_{\mathit{mj}}\left(2v\right)=\frac{2{\left(-1\right)}^m\chi {\mathrm{\Delta }}^{2}v}{{{{\omega }^{\prime }}^{\prime }}^2\left(1+j\right)}{L}_j^{\left(1\right)}\left(\frac{v\mathrm{\Delta }}{{{\omega }^{\prime }}^{\prime }}\right)L_j^{\left(1\right)}\left(\frac{v\mathrm{\Delta }}{{{\omega }^{\prime }}^{\prime }}\right)e^{\frac{-v\mathrm{\Delta }}{{{\omega }^{\prime }}^{\prime }}}.$$E32

And the modified vertex function in Eq. (16) can be rewritten as follows:

$${\mathrm{\Xi }}_{\mathit{mj}}\left(v\right)=\frac{2{\left(-1\right)}^m\chi {\mathrm{\Delta }}^{2}v}{{{{\omega }^{\prime }}^{\prime }}^2\left(1+j\right)}{L}_j^{\left(1\right)}\left(\frac{v\mathrm{\Delta }}{{{\omega }^{\prime }}^{\prime }}\right)L_j^{\left(1\right)}\left(\frac{v\mathrm{\Delta }}{{{\omega }^{\prime }}^{\prime }}\right)e^{\frac{-v\mathrm{\Delta }}{{{\omega }^{\prime }}^{\prime }}}.$$E33

From Eqs. (32) and (33) we notice that the interaction function and the modified vertex function have the same form. In the interaction picture, we consider the time evolution controlled by ${\widehat{H}}_I\left(t\right)$. Thus, the equations of motion for ${\widehat{a}}_{\mathit{mj}}\left(v,t\right)$ and ${\widehat{b}}_{-\mathit{mj}}\left(v,t\right)$ are:

$$\frac{d{\widehat{a}}_{\mathit{mj}}\left(v,t\right)}{\mathit{dt}}=\frac{1}{i\mathrm{\hslash }}\left[{\widehat{a}}_{\mathit{mj}}\left(v,t\right),{\widehat{H}}_I\left(t\right)\right],$$E34

$$\frac{d{\widehat{b}}_{-\mathit{mj}}\left(v,t\right)}{\mathit{dt}}=\frac{1}{i\mathrm{\hslash }}\left[{\widehat{b}}_{-\mathit{mj}}\left(v,t\right),{\widehat{H}}_I\left(t\right)\right].$$E35

Substituting Eq. (31) into the above equations, we obtain

$$\frac{d{\widehat{a}}_{\mathit{mj}}\left(v,t\right)}{\mathit{dt}}=-2i\omega \left(v\right){\chi }_{\mathit{mj}}\left(2v\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v,t\right),$$E36

$$\frac{d{\widehat{b}}_{-\mathit{mj}}\left(v,t\right)}{\mathit{dt}}=-2i\omega \left(v\right){\chi }_{\mathit{mj}}\left(2v\right){\widehat{a}}_{\mathit{mj}}^{†}\left(v,t\right).$$E37

The general solutions of these equations are [50],

$${\widehat{a}}_{\mathit{mj}}\left(v,t\right)=cosh\left({\beta }_{\mathit{mj}}\left(v\right)t\right){\widehat{a}}_{\mathit{mj}}\left(v\right)+sinh\left({\beta }_{\mathit{mj}}\left(v\right)t\right)\frac{{\alpha }_{\mathit{mj}}}{{\beta }_{\mathit{mj}}\left(v\right)}{\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right),$$E38

$${\widehat{b}}_{-\mathit{mj}}\left(v,t\right)=cosh\left({\beta }_{\mathit{mj}}\left(v\right)t\right){\widehat{b}}_{-\mathit{mj}}\left(v\right)+sinh\left({\beta }_{\mathit{mj}}\left(v\right)t\right)\frac{{\alpha }_{\mathit{mj}}}{{\beta }_{\mathit{mj}}\left(v\right)}{\widehat{a}}_{\mathit{mj}}^{†}\left(v\right).$$E39

Here, we consider the expression ${\alpha }_{\mathit{mj}}=-2i\omega \left(v\right){\chi }_{\mathit{mj}}\left(2v\right)$, with $\omega \left(v\right)=v^2/2{{\omega }^{\prime }}^{\prime }.$ To find the squeezing parameter ${\xi }_{\mathit{mj}}\left(v\right)={\beta }_{\mathit{mj}}\left(v\right)t$, we use the relation ${\beta }_{\mathit{mj}}\left(v\right)=\sqrt{{\alpha }_{\mathit{mj}}{\alpha }_{\mathit{mj}}^{\ast }}$, then we can write the squeezing parameter as follows:

$${\xi }_{\mathit{mj}}\left(v\right)=\frac{2{\left(-1\right)}^m\chi t}{\left(1+j\right)\mathrm{\Delta }}\frac{{\tilde{v}}^3}{{{{\omega }^{\prime }}^{\prime }}^3}{\left(L_j^{\left(1\right)}\left(\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}\right)\right)}^2{e}^{-\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}},$$E40

with $\tilde{v}=v\mathrm{\Delta }.$ The above equation represents the squeezing parameter which depends on the OAM number $m$, the spectral order $j$, and the velocity of the two-modes $\mathrm{X}$ waves $v$. Now, we can show the normalized squeezing parameter modulus $\mid {\overline{\xi }}_{\mathit{mj}}\mid =\frac{\mathrm{\Delta }\mid {\xi }_{\mathit{mj}}\mid }{4\mathit{\chi t}}$ as a function of the normalized velocity $\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}$ for different values of $j$,($j=\mathrm{0,1,2,3,4,5}$), and fixing the time $t$ so that $\frac{4\mathit{\chi t}}{\mathrm{\Delta }}=1$, when $m=0$, as shown in Figure 1.

Figure 1.

Plot of the normalized squeezing parameter modulus $\mid {\overline{\xi }}_{\mathit{mj}}\mid =\frac{\mathrm{\Delta }\mid {\xi }_{\mathit{mj}}\mid }{\chi }$ in function of the normalized velocity $\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}$, for different values of the spectral order $j$ where, (a) for $j=0$, (b) for $j=1$, (c) for $j=2$, (d) for $j=3$, (e) for $j=4$, (f) for $j=5,$ with fixing the time $t$, so that $\frac{4\mathit{\chi t}}{\mathrm{\Delta }}=1,$ and $m=0$.

Figure 1 shows the dependence of the squeezing on the spectral order $j$ of $\mathrm{X}$ waves. This dependence can be explained in the following points:

1. The maximum value of the squeezing parameter $\mid {\overline{\xi }}_{\mathit{mj}}\mid$ and the corresponding normalized velocity $\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}$ increases as the spectral order $j$ increases.

2. The number of the peak values of squeezing depends on the spectral order $j$ and this number can be given by $j+1$, where the smallest peak is the first and the biggest one is the last.

3. The squeezing parameter depends on the velocity of $\mathrm{X}$ wave as shown in Eq. (40) where the amount of squeezing parameter produced by SPDC process will be maximized at an optimal velocity for every value of the spectral order $j$, and this velocity can be given by $v_{\mathit{opt}}=n_j{{\omega }^{\prime }}^{\prime }/\mathrm{\Delta }$, where $n_0=3$, $n_1=6$, $n_2=9.36$, $n_3=12.88$, $n_4=16.49$, $n_5=20.156$ for $j=\mathrm{0,1,2,3,4,5}$_, respectively as shown in Figure 1. Figure 1(a) shows that for $j=0$ as in Ref. [46], and the corresponding optimal axicon angles for different values of $j$ can be given by $cos{\theta }_j^{\mathit{opt}}=\mathrm{\Delta }/n_j\lambda .$ For example, given a nondiffracting pulse with a duration of $\mathrm{\Delta }t=8$ fs, a carrier wavelength of $\lambda =850$ nm and assuming that ${\chi }^2\simeq {10}^{-12}$ m/V, the optimal axicon angles ${\theta }_j^{\mathit{opt}}$ that maximize the squeezing for $j=\mathrm{0,1,2,3,4,5}$ will take the following values respectively, ${\theta }_0^{\mathit{opt}}\simeq {20}^{\circ }$, ${\theta }_1^{\mathit{opt}}\simeq {62}^{\circ }$, ${\theta }_2^{\mathit{opt}}\simeq {72}^{\circ }$, ${\theta }_3^{\mathit{opt}}\simeq {77}^{\circ }$, ${\theta }_4^{\mathit{opt}}\simeq {80}^{\circ }$, ${\theta }_5^{\mathit{opt}}\simeq {82}^{\circ }.$ Then, we can evaluate the corresponded maximal squeezing parameters to get the following values respectively, (${\xi }_m^{\left(0\right)}\simeq 100s^{-1}$, ${\xi }_m^{\left(1\right)}\simeq 280s^{-1}$, ${\xi }_m^{\left(2\right)}\simeq 520s^{-1}$, ${\xi }_m^{\left(3\right)}\simeq 820s^{-1}$, ${\xi }_m^{\left(4\right)}\simeq 1160s^{-1}$, ${\xi }_m^{\left(5\right)}\simeq 1550s^{-1}$).

To illustrate the effect of the OAM on the squeezing more, we can introduce the quadrature operators ${\widehat{X}}_{\mathit{mj}}\left(v,t\right)={\widehat{a}}_{\mathit{mj}}\left(v,t\right)+{\widehat{a}}_{\mathit{mj}}^{†}\left(v,t\right),$ and ${\widehat{Y}}_{\mathit{mj}}\left(v,t\right)=i\left[{\widehat{a}}_{\mathit{mj}}^{†}\left(v,t\right)-{\widehat{a}}_{\mathit{mj}}\left(v,t\right)\right].$ For $e^{\mathit{i\varphi }}={\alpha }_{\mathit{mj}}/{\beta }_{\mathit{mj}}=1$ or $\varphi =0,$ we get ${\widehat{X}}_{\mathit{mj}}\left(v,t\right)=e^{{\xi }_{\mathit{mj}}\left(v\right)}{\widehat{X}}_{\mathit{mj}}\left(v,0\right),$ and ${\widehat{Y}}_{\mathit{mj}}\left(v,t\right)=e^{-{\xi }_{\mathit{mj}}\left(v\right)}{\widehat{Y}}_{\mathit{mj}}\left(v,0\right),$ and the variance of them is given by $\mathrm{\Delta }{\widehat{X}}_{\mathit{mj}}\left(v,t\right)=e^{{\xi }_{\mathit{mj}}\left(v\right)}{\widehat{Y}}_{\mathit{mj}}\left(v,0\right).$ This shows that the SPDC interaction Hamiltonian for the generalized $\mathrm{X}$ waves acts as a two-modes squeeze operator which will be illustrated more in the next section. Noticeably, the effect of the OAM number $m$ changes only the sign of the squeezing parameter ${\xi }_{\mathit{mj}}\left(v\right)$, that is, the squeezed quadrature changes depending on the parity of the orbital angular momentum number $m$. In particular, if $m$ is even number, ${\xi }_{\mathit{mj}}\left(v\right)>0$, the squeezing accrues on the $Y$ quadrature. On the other hand, if $m$ is an odd number, ${\xi }_{\mathit{mj}}\left(v\right)<0$ and the squeezing occurs on $X$ quadrature as in Ref. [46] for $j=0$, and for $j=\mathrm{1,2,3}.$ this effect can be shown as in Figure 2.

Figure 2.

Quadrature space representation of the squeezed down-converted state in the case of even [panel (a)] and odd [panel (b)] values of the OAM number $m$ with a normalized velocity $\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}=\mathrm{6,9.36,12.88}.$ for a spectral order $j=\mathrm{1,2,3},$ respectively.

4. The two-modes squeezed state of $\mathrm{X}$ wave and its entanglement

In analogy to traditional case [51, 52], we look for the eigenvalues of Hamiltonian operator, represented by Eq. (6). Let us introduce the $N$-particle states $\mid m,j,v;N⟩$ as states with $N$ field excitation with velocity $v$ in the traveling mode (i.e., the $\mathrm{X}$ wave) ${\psi }_{m,p}^{\left(v\right)}\left(\mathbf{r},t\right)$ with energy ${\mathit{Mv}}^2/2$ [47]. The eigenstates $\mid m,p,v,N⟩$ of the photon (particle) number operator $\widehat{N}\left(v\right)={\widehat{a}}_{\mathit{mp}}^{†}\left(v\right){\widehat{a}}_{\mathit{mp}}\left(v\right)$ are called Fock states, where ${\widehat{a}}_{\mathit{mp}}\left(v\right)$ is the annihilation operator of the single mode in Eq. (7). These states are orthogonal

$$⟨n,q,u,N|m,p,v,N⟩\equiv {\delta }_{\mathit{nm}}{\delta }_{\mathit{pq}}\delta \left(u-v\right),$$E41

and complete. The number state $\mid m,p,v,N⟩$ can be generated from the vacuum state (i.e., the eigenstate of $\widehat{N}\left(v\right)$ with the eigenvalue equal to zero) as

$$\mid m,p,v,N⟩=\frac{{\left[{\widehat{a}}_{\mathit{mp}}^{†}\left(v\right)\right]}^N}{\sqrt{N!}}\mid 0⟩.$$E42

The expectation value of the electric field operator is zero on the particle number operator eigenstates [47],

$$⟨m,p,v,N|\widehat{A}\left(r,t\right)|m,p,v,N⟩=0.$$E43

However, the Fock states $\mid m,p,v,N⟩$ are not normalizable [47], since their representation in configuration state $⟨\mathbf{r},t|m,p,v,N⟩={\psi }_{\mathit{mp}}^{\left(v\right)}\left(r,t\right)$ carries infinite energy. The particle number states of the two modes $\left(A,B\right)$ with particle numbers $\left(N_A,N_B\right)$ are defined as

$$\mid m,j,v;N_A⟩{}_A=\frac{{\left[{\widehat{a}}_{\mathit{mj}}^{†}\left(v\right)\right]}^{N_A}\mid 0⟩{}_A}{\sqrt{N_A!}},$$E44

and

$$\mid -m,j,v,N_B⟩{}_B=\frac{{\left[{\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)\right]}^{N_B}\mid 0⟩{}_B}{\sqrt{N_B!}}$$E45

Together with the conditions for the vacuums, ${\widehat{a}}_{\mathit{mj}}\left(v\right)\mid 0⟩{}_A=0$ and ${\widehat{b}}_{-\mathit{mj}}\left(v\right)\mid 0⟩{}_B=0$.

The operators ${\widehat{a}}_{\mathit{mj}}\left(v\right)$ and ${\widehat{b}}_{-\mathit{mj}}\left(v\right)$ obey the bosonic commutation relations in Eq. (7). Now, we can define the two-mode squeeze operator, which is a unitary operator, using the normalized squeezing parameter ${\overline{\xi }}_{\mathit{mj}}\left(v\right)=\frac{{\left(-1\right)}^m}{2\left(1+j\right)}\frac{{\tilde{v}}^3}{{{{\omega }^{\prime }}^{\prime }}^3}{\left(L_j^{\left(1\right)}\left(\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}\right)\right)}^2{e}^{-\frac{\tilde{v}}{{{\omega }^{\prime }}^{\prime }}},$ as

$$V_{{\overline{\xi }}_{\mathit{mj}}}=exp\left(e^{-\mathit{i\varphi }}{\overline{\xi }}_{\mathit{mj}}{\widehat{a}}_{\mathit{mj}}\left(v\right){\widehat{b}}_{-\mathit{mj}}\left(v\right)-e^{\mathit{i\varphi }}{\overline{\xi }}_{\mathit{mj}}{\widehat{a}}_{\mathit{mj}}^{†}\left(v\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)\right).$$E46

The two-mode squeezed vacuum TMSV state can be obtained from the vacuum state as [52],

$$\mid 0,0,{\overline{\xi }}_{\mathit{mj}}⟩{}_{\mathit{AB}}=V_{{\overline{\xi }}_{\mathit{mj}}}\mid 0,0⟩={\left[cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\right]}^{-1}\sum _{N=0}^{\infty }{\left(e^{\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right)\right)}^N\mid m,j,v,N;-m,j,v,N⟩.$$E47

Also, we can define the annihilation operators crossing over the separation of part $A$ and part $B$ as [53],

$$\begin{array}{lll}{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}& =& V_{{\overline{\xi }}_{\mathit{mj}}}{\widehat{a}}_{\mathit{mj}}\left(v\right)V_{{\overline{\xi }}_{\mathit{mj}}}^{†}=cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\left[{\widehat{a}}_{\mathit{mj}}\left(v\right)+e^{\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)\right]\\ {\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}& =& V_{{\overline{\xi }}_{\mathit{mj}}}{\widehat{b}}_{-\mathit{mj}}\left(v\right)V_{{\overline{\xi }}_{\mathit{mj}}}^{†}=cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\left[{\widehat{b}}_{-\mathit{mj}}\left(v\right)+e^{\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right){\widehat{a}}_{\mathit{mj}}^{†}\left(v\right)\right].\end{array}$$E48

The new field operators satisfy the bosonic commutation relations in Eq. (7)

$$\left[{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}},{\widehat{A}}_{{\overline{\xi }}_{m^{\prime }{j}^{\prime }}}^{†}\right]=\left[{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}},{\widehat{B}}_{{\overline{\xi }}_{m^{\prime }{j}^{\prime }}}^{†}\right]={\delta }_{m,m^{\prime }}{\delta }_{j,j^{\prime }}\delta \left(v-v^{\prime }\right),$$E49

and

$$\left[{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}},{\widehat{B}}_{{\overline{\xi }}_{m^{\prime }{j}^{\prime }}}\right]=\left[{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}},{\widehat{B}}_{{\overline{\xi }}_{m^{\prime }{j}^{\prime }}}^{†}\right]=0.$$E50

The TMSV state corresponds to the vacuum of the new field operators, as we have used ${\widehat{a}}_{\mathit{mj}}\left(v\right)\mid 0⟩=0$, ${\widehat{b}}_{-\mathit{mj}}\left(v\right)\mid 0⟩=0$, Eq. (47), Eq. (48), and the unitary property of the squeeze operator, we get ${\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}\mid 0,0,{\overline{\xi }}_{\mathit{mj}}⟩=0$, and ${\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}\mid 0,0,{\overline{\xi }}_{\mathit{mj}}⟩=0$.

The two-mode number state is defined by the number state of nonlocal modes as [53],

$$\mid N_A,N_B,{\overline{\xi }}_{\mathit{mj}}⟩=\frac{{\left[{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\right]}^{N_A}{\left[{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\right]}^{N_B}}{\sqrt{N_A!}\sqrt{N_B!}}\mid 0,0,{\xi }_{{\overline{\xi }}_{\mathit{mj}}}⟩.$$E51

In our case, we have $N_A=N_B=1$, so we obtain

$$\mid 1,1,{\overline{\xi }}_{\mathit{mj}}⟩={\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\mid 0,0,{\overline{\xi }}_{\mathit{mj}}⟩.$$E52

The state in the above equation represents a two-particles squeezed state of the two modes $\mathrm{X}$ wave. The superposition of the two particles corresponding to the two modes ${\omega }_1$ and ${\omega }_2$ see Eq. (18) (when $u=v$) can be written as

$$\mid {\psi }^{\left(1\right)}\left(t\right)⟩=\sum _{\mathit{mj}}\int \mathit{dv}{\mathcal{G}}_{\mathit{mj}}\left(v\right)\mathrm{ℱ}\left(v,t\right){\widehat{a}}_{\mathit{mj}}^{†}\left(v\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)\mid 0,0⟩,$$E53

where $\mathrm{ℱ}\left(v,t\right)=\left[e^{\mathit{\text{iF}}\left(u\right)t}-1\right]$, and ${\mathcal{G}}_{\mathit{mj}}\left(v\right)=-i\omega \left(v\right){\chi }_{\mathit{mj}}\left(2v\right)/F\left(v\right)$, with ${\chi }_{\mathit{mj}}\left(2v\right)$ as in Eq. (32). The superposition of the two particles squeezed state corresponds to the state in Eq. (53) is

$$\mid {\psi }_{{\overline{\xi }}_{\mathit{mj}}}^{\left(1\right)}\left(t\right)⟩=V_{{\overline{\xi }}_{\mathit{mj}}}\mid {\psi }^{\left(1\right)}\left(t\right)⟩=\sum _{\mathit{mj}}\int \mathit{dv}{\mathcal{G}}_{\mathit{mj}}\left(v\right)\mathrm{ℱ}\left(v,t\right)V_{{\overline{\xi }}_{\mathit{mj}}}{\widehat{a}}_{\mathit{mj}}^{†}\left(v\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)\mid 0,0⟩.$$E54

Using Eq. (48), the above state can be written in the form

$$\mid {\psi }_{{\overline{\xi }}_{\mathit{mj}}}^{\left(1\right)}\left(t\right)⟩=\sum _{\mathit{mj}}\int \mathit{dv}{\mathcal{G}}_{\mathit{mj}}\left(v\right)\mathrm{ℱ}\left(v,t\right){\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\mid 0,0,{\overline{\xi }}_{\mathit{mj}}⟩,$$E55

which is a superposition of the squeezed states $\mid 1,1,{\overline{\xi }}_{\mathit{mj}}⟩$ in Eq. (52).

Now, for calculating the entanglement of the state in Eq. (55), we consider the separable conditions based on the operators that form $\mathit{SU}\left(2\right)$ algebra and $\mathit{SU}\left(1,1\right)$ algebra [53]. The measurements of quadrature moments of the fourth order are sufficient to verify the entanglement of this state. Let us define the operators,

$$\begin{array}{lll}{J}_z& =& \frac{1}{2}\left(N_A-N_B\right),\\ K_x& =& \frac{1}{2}\left({\widehat{a}}_{\mathit{mj}}^{†}\left(v\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)+{\widehat{a}}_{\mathit{mj}}\left(v\right){\widehat{b}}_{-\mathit{mj}}\left(v\right)\right),\\ K_y& =& \frac{1}{2i}\left({\widehat{a}}_{\mathit{mj}}^{†}\left(v\right){\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)-{\widehat{a}}_{\mathit{mj}}\left(v\right){\widehat{b}}_{-\mathit{mj}}\left(v\right)\right)\end{array}$$E56

where ${\widehat{N}}_A={\widehat{a}}_{\mathit{mj}}^{†}\left(v\right){\widehat{a}}_{\mathit{mj}}\left(v\right)$ and ${\widehat{N}}_B={\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right){\widehat{b}}_{-\mathit{mj}}\left(v\right)$ are number operators of the local modes.

The relations of the annihilation and creation operators ${\widehat{a}}_{\mathit{mj}}\left(v\right)$, ${\widehat{a}}_{\mathit{mj}}^{†}\left(v\right)$, ${\widehat{b}}_{-\mathit{mj}}\left(v\right)$, and ${\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)$ with the new field operators in Eq. (48) can be represented by

$$\begin{array}{lll}{\widehat{a}}_{\mathit{mj}}\left(v\right)& =& cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\left({\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}-e^{\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right){\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\right)\\ {\widehat{a}}_{\mathit{mj}}^{†}\left(v\right)& =& cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\left({\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}-e^{-\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right){\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}\right)\\ {\widehat{b}}_{-\mathit{mj}}\left(v\right)& =& cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\left({\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}-e^{\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right){\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\right)\\ {\widehat{b}}_{-\mathit{mj}}^{†}\left(v\right)& =& cosh\left({\overline{\xi }}_{\mathit{mj}}\right)\left({\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}-e^{-\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right){\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}\right)\end{array}$$E57

The separable condition is [53],

$$\left[⟨{\mathrm{\Delta }}^2{K}_y⟩-\frac{1}{4}\right]⟨{\mathrm{\Delta }}^2{J}_z⟩\ge \frac{1}{4}{\left|⟨K_x⟩\right|}^2$$E58

where $⟨\widehat{O}⟩=\mathit{tr}\left(\widehat{O}\rho \right)$ denotes the expectation value of the observable $\widehat{O}$ with respect to state $\rho$, $\mathrm{\Delta }\widehat{O}=\widehat{O}-⟨\widehat{O}⟩$, and $⟨{\mathrm{\Delta }}^2\widehat{O}⟩=⟨{\widehat{O}}^2⟩-{⟨\widehat{O}⟩}^2$. The state we use in Eq. (58) is the superposition of simultaneous eigenstates of the number operators ${\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}$ and ${\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}$. From the definition of $J_z$ in Eq. (56), we have

$${\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}-{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}=N_A-N_B=2J_z$$E59

This implies that the two-mode squeezed (TMS) number states are eigenstates of $J_z$ and $⟨{\mathrm{\Delta }}^2{J}_z⟩=0$.

Also, we can write

$$K_x=\frac{1}{2}{cosh}^2\left({\overline{\xi }}_{\mathit{mj}}\right)\left[\left({\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}-{\eta }^{\ast }{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}\right)\left({\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}-{\eta }^{\ast }{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}\right)+\left({\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}-\eta {\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\right)\left({\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}-\eta {\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}\right)\right],$$E60

with $\eta =e^{\mathit{i\varphi }}tanh\left({\overline{\xi }}_{\mathit{mj}}\right)$.

Then, the expectation value of $K_x$ for the TMS number state of Eq. (55) can be given by

$$⟨K_x⟩=-\sum _{\mathit{mj}}\int \frac{\mathit{dv}}{2}{cosh}^2\left({\overline{\xi }}_{\mathit{mj}}\right){\left|{\mathcal{G}}_{\mathit{mj}}\left(v\right)\right|}^2{\left|\mathrm{ℱ}(v,t)\right|}^2\left({\eta }^{\ast }+\eta \right)⟨{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}+{{\widehat{B}}^{†}}_{{\overline{\xi }}_{\mathit{mj}}}{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}+1⟩$$E61

with $⟨{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}^{†}{\widehat{A}}_{{\overline{\xi }}_{\mathit{mj}}}+{{\widehat{B}}^{†}}_{{\overline{\xi }}_{\mathit{mj}}}{\widehat{B}}_{{\overline{\xi }}_{\mathit{mj}}}+1⟩=3.$ From Eq. (25) we get the maximum value ${\left|\mathrm{ℱ}\left(v,t\right)\right|}^2=4$, thus

$$⟨K_x⟩=-6\sum _{\mathit{mj}}\int \mathit{dv}{cosh}^2\left({\overline{\xi }}_{\mathit{mj}}\right){\left|{\mathcal{G}}_{\mathit{mj}}\left(v\right)\right|}^2\left({\eta }^{\ast }+\eta \right)$$E62

where, $\left({\eta }^{\ast }+\eta \right)=2cos\left(\varphi \right)tanh\left({\overline{\xi }}_{\mathit{mj}}\right)$, which implies that $⟨K_x⟩=0$, if $\varphi =\frac{\pi }{2}$, this problem can be avoided by applying a local phase shift [53], we obtain

$$\begin{array}{lll}⟨K_x⟩& =& -12cos\left(\varphi \right)\sum _{\mathit{mj}}\int \mathit{dv}{\left|{\mathcal{G}}_{\mathit{mj}}\left(v\right)\right|}^{2}cosh\left({\overline{\xi }}_{\mathit{mj}}\right)sinh\left({\overline{\xi }}_{\mathit{mj}}\right)\\ & =& -6cos\left(\varphi \right)\sum _{\mathit{mj}}\int \mathit{dv}{\left|{\mathcal{G}}_{\mathit{mj}}\left(v\right)\right|}^{2}sinh\left(2{\overline{\xi }}_{\mathit{mj}}\right).\end{array}$$E63

If we take $\varphi =\left(2n+1\right)\pi ,n=\mathrm{0,1,2,3},\cdots$, then $⟨K_x⟩$ will be maximized. This implies that $\mid ⟨K_x⟩\mid >0$, for $\varphi =\mathit{n\pi },n=\mathrm{0,1,2,3},\cdots$. Therefore, for the state in Eq. (55), the L.H.S. of Eq. (58) is equal to zero and its R.H.S. is positive. Hence, we can confirm the inseparability of the state in Eq. (55) from the violation of the condition in Eq. (58), means that we can take it as an indicator of quantifying the amount of the entanglement of squeezed states which can be detected by increasing $\mid ⟨K_x⟩\mid$. In other words, we have found that $\mid ⟨K_x⟩\mid$ increases as the spectral order $j$ increases, which indicates that the amount of entanglement of the state in Eq. (55) increases.

5. Summary and comments

To summarize, we obtain the velocity phase matching, which can be used for determining the length of nonlinear crystal in the experiment setup to produce $\mathrm{X}$ a wave with velocity $v$. We have introduced the relation between the squeezing parameter and the spectral order of $\mathrm{X}$ waves which shows that the maximal squeezing increases as the spectral order increases, and there exists an optimal velocity (i.e., axicon angle) that maximizes the amount of squeezing generated for each value of the spectral order. Moreover, we have constructed the squeeze operator of the two modes $\mathrm{X}$ wave, then we act by it on the two-particle state generated by SPDC process to obtain the squeezed form of it. Finally, we have used the criterion of the quadrature moment of the fourth order to verify the entanglement of the two modes squeezed state of $\mathrm{X}$ wave. We have observed that the entanglement of the state increases as the spectral order increases.