† Corresponding author. E-mail:

‡ Corresponding author. E-mail:

In this paper, after a brief review on the entangled squeezed states, we produce a new class of the continuous-variable-type entangled states, namely, deformed photon-added entangled squeezed states. These states are obtained via the iterated action of the *f*-deformed creation operator *A* = *f* (*n*)*a*^{†} on the entangled squeezed states. In the continuation, by studying the criteria such as the degree of entanglement, quantum polarization as well as sub-Poissonian photon statistics, the two-mode correlation function, one-mode and two-mode squeezing, we investigate the nonclassical behaviors of the introduced states in detail by choosing a particular *f*-deformation function. It is revealed that the above-mentioned physical properties can be affected and so may be tuned by justifying the excitation number, after choosing a nonlinearity function. Finally, to generate the introduced states, we propose a theoretical scheme using the nonlinear Jaynes–Cummings model.

In recent decades, entanglement^{[1]} is known as a favorite characteristic that distinguishes quantum mechanics from classical mechanics. The concept of entanglement plays an important role in the fields of quantum information processing including quantum teleportation,^{[2]} cryptography,^{[3]} and computation,^{[4]} both theoretically and experimentally. Recently, various types of entangled coherent states as one of the entangled continuous-variable-type states have been discussed and different schemes to generate such states have been introduced.^{[5–9]} In this regard, a new class of nonlinear excited entangled coherent states have been introduced and generated by us^{[10]} based on the *f*-deformed coherent states.^{[11]}

The increasing interest in achieving new quantum states with different nonclassical behaviors motivates researchers to introduce other types of entangled continuous-variable-type states, namely “entangled squeezed states”, and to propose many schemes to generate these states.^{[12,13]} The entangled squeezed states have been produced by using a squeezed state instead of a coherent state in “entangled coherent states”.^{[14–16]} The squeezed states were first formulated mathematically by Stoler^{[17,18]} and then have been observed experimentally by Slusher *et al.*^{[19]} In some of their applications one may refer to high quality optical communication,^{[20,21]} interferometric technique to detect very weak forces^{[22]} such as gravitational wave detection,^{[23]} optical wave-guide tap,^{[24]} optical sensors^{[25]}, computing,^{[26]} and so on.

In addition to entanglement, quantum polarization has also attracted a great deal of attention in recent years due to its key role in many experiments and demonstrations in quantum optics.^{[27–35]} In 1852, Stokes, in his seminal work, first discussed the concept of light polarization. The polarization of light is referred to as the property where the tip of the electric field vector describes an ellipse in the plane transverse to the propagating direction at a fixed space point.^{[36]} In this regard, a polarized light can be used as an excellent information carrier which retains coded information relatively intact on propagation.

On the other hand, Glauber coherent states were introduced in 1963,^{[37]} the photon-added coherent states were introduced by Agarwal and Tara in 1991^{[38]} and then they were generated experimentally by Zavatta *et al.* in 2005.^{[39]} In this regard, recently the deformed photon-added coherent states were introduced by one of us and their nonclassical properties were considered.^{[40]} Keeping in mind the three steps in the latter-mentioned research procedure, and the fact that we have recently introduced and generated the photon-added entangled squeezed states and studied their properties in Ref. [41], in this paper we aim to introduce a new class of entangled quantum states, namely the deformed photon-added entangled squeezed states. It is worth mentioning that for the squeezed states we will consider the vacuum and the one-photon squeezed states (see Eqs. (*m* times) of the *f*-deformed creation operator *A*^{†} = *f* (*n*)*a*^{†} on the first mode of the entangled squeezed states. Next, some of the nonclassical properties of the produced states such as entanglement, quantum polarization, sub-Poissonian photon statistics, correlation function and one-mode quadrature squeezing are studied by choosing a particular nonlinearity function, numerically. Moreover, a simple scheme based on the atom-field interaction using the generalized (nonlinear) Jaynes–Cummings model^{[42,43]} for the generation of the introduced entangled states is proposed. In particular, the investigation about the influence of the photon number excitation on the entanglement, polarization and other nonclassicality features of the introduced states motivates us to pay attention to the present contribution. At last, for more physical realization of this study we illustrate a simple theoretical implementation for the generation of the outlined states by using the nonlinear Jaynes–Cummings model.

The paper is organized as follows. In the next section, we present the explicit form of entangled squeezed states. In Section 3, we introduce deformed photon-added entangled squeezed states. Next, we pay attention to the entanglement and quantum polarization of the introduced states, in Sections 4 and 5, respectively. Section 6 deals with a few other nonclassicality features in detail. A theoretical scheme for the generation of the introduced states has been proposed in Section 7 and finally, we present a summary and concluding remarks in Section 8.

At first, based on the introduced entangled coherent states in the literatures^{[9]} we have recently defined the two-mode entangled squeezed states generally in the following form^{[40]}

_{i,j}(

*ξ*) is the normalization coefficient. The subscripts e and o in Eq. (

^{[44]}

*â*and

*â*

^{†}are the bosonic annihilation and creation operators, respectively and

*ξ*=

*r*e

^{iθ}is a complex parameter. The normalization coefficients

*N*

_{e}(

*ξ*) and

*N*

_{o}(

*ξ*) may be obtained as:

*θ*and between the two components of the entangled squeezed states

*φ*. In this paper, by setting the phase differences

*θ*=

*π*and

*φ*= 0, we introduce three different entangled squeezed states (even–even, odd–odd, and even–odd) as follows:

*ξ*(

*r*,0)〉

_{e}and |

*ξ*(

*r*,

*π*)〉

_{e}(|

*ξ*(

*r*,0)〉

_{o}and |

*ξ*(

*r*,

*π*)〉

_{o}) have been introduced in Eq. (

_{e,e}(

*ξ*), 𝒩

_{o,o}(

*ξ*), and 𝒩

_{e,o}(

*ξ*) are appropriate normalization factors which can be easily determined.

It is worth mentioning that the first and second modes in the two-mode entangled squeezed states given by Eq. (*a* and *b*, respectively.

Now, we introduce the deformed photon-added entangled squeezed states that may be produced by the iterated action of the deformed creation operator *f* (*n*)*a*^{†} on the first mode of the two-mode entangled squeezed states in Eq. (

*θ*= 0,

*π*, as follows:

*d*(

_{e}*ξ*,

*f*,

*m*) and

*d*(

_{o}*ξ*,

*f*,

*m*) are determined as follows:

*N*(

_{e}*ξ*) and

*N*(

_{o}*ξ*) have been obtained in Eqs. (

By setting *f* (*n*) = 1 in Eqs. (^{[41]} Clearly, various classes of deformed photon-added entangled squeezed states can be achieved by choosing the well-known nonlinearity functions *f* (*n*) corresponding to different nonlinear quantum systems with various nonclassicality features and different physical properties.

In this contribution, as a physical realization of the presented formalism, we will choose the nonlinearity function ^{[40,45,46]} This function also arises in a natural way in the Hamiltonian illustrating the interaction with intensity-dependent coupling between a two-level atom and an electromagnetic radiation field.^{[46–48]} In particular, Fink *et al.* have explored a natural way in which this nonlinearity function will appear in physical systems.^{[49]} Henceforth, we are ready to produce the introduced entangled states corresponding to this particular nonlinearity function and investigate their physical behavior.

In this section we evaluate the amount of entanglement of the deformed photon-added entangled squeezed states by studying two measures, i.e., ‘concurrence’ and ‘linear entropy’. In this way, we will consider the entanglement of the introduced entangled states to photon excitations for the particular nonlinearity function

To measure the degree of quantum entanglement of any two-mode entangled states we can use the concurrence which is generally defined for discrete-variable entangled states as:^{[50]} *C* = |〈*Ψ*|*σ*_{y} ⊗ *σ _{y}*|

*Ψ**〉|, where |

*Ψ**〉 is the complex conjugate of |

*Ψ*〉. The value of this criterion gets 1 for a maximally entangled state. As is clear from Eqs. (

*η*〉 and |

*ξ*〉 (|

*γ*〉 and |

*δ*〉) are normalized states of subsystem

*a*(subsystem

*b*) with complex numbers

*μ*and

*ν*. The normalization coefficient is given by:

N

^{2}= |

*μ*|

^{2}+ |

*ν*|

^{2}+ 2Re(

*μ**

*ν P*

^{(1)}

*P*

^{(2)}*) where

*P*

^{(1)}= 〈

*η*|

*ξ*〉 and

*P*

^{(2)}= 〈

*δ*|

*γ*〉.

By transforming the continuous-variable-type components to the discrete orthogonal bases and by making use of the Schmidt decomposition,^{[51,52]} the concurrence of the introduced entangled continuous-variable-type states in Eqs. (^{[50,53,54]}

*C*

_{e,e}(

*ξ*,

*f*,

*m*), i.e., the concurrence of the deformed photon-added even-even entangled squeezed states in Eq. (

*μ*= 1,

*ν*= 1,

*N*

_{e}(

*ξ*), 𝒩

_{e,e}(

*ξ*,

*f*,

*m*), and

*d*

_{e}(

*ξ*,

*f*,

*m*) have been introduced in Eqs. (

*C*

_{e,e}(

*ξ*,

*f*,

*m*).

After performing all the necessary calculations, we have plotted the concurrence of the deformed photon-added entangled squeezed states versus |*ξ* | for the nonlinearity function *m*.

It can be seen from Figs. *C*_{e,e}(*ξ*, *f*, *m*), *C*_{o,o}(*ξ*, *f*, *m*), and *C*_{e,o}(*ξ*, *f*, *m*), the concurrence of the even–even, odd–odd, and even–odd deformed photon-added entangled squeezed states corresponding to |*ψ*(*ξ*, *f*, *m*)〉_{e,e}, |*ψ*(*ξ*, *f*, *m*)〉_{o,o}, and |*ψ*(*ξ*, *f*, *m*)〉_{e,o}, have generally the same behavior. After starting from zero, they grow up by increasing |*ξ* | for all of the photon number excitations, *m*, and finally tend to 1 in the region of a strong field. Also, the rate of growing the concurrence by increasing the number of photon excitations, *m* is more rapid in the region of a weak field.

As another measure of entanglement, we calculate the linear entropy of the introduced entangled states which is defined for a bipartite state in the following form^{[55–57]}

*ρ*

_{a(b)}is the reduced density operator of the density operator

*ρ*

_{ab}= |

*ψ*〉

_{ab ab}〈

*ψ*| where |

*ψ*〉

_{ab}is one of the entangled squeezed states introduced in Eqs. (

*a*,

*b*= e,o. Finally, by using Eq. (

*S*(

_{ai,j}*ξ*,

*f*,

*m*) of the first mode,

*a*of the introduced states are obtained as:

*ξ*| in Fig.

*m*.

As is clear from this figure, the linear entropies *S*_{ai,j}(*ξ*, *f*, *m*) of the introduced states for each of the entangled states (even–even, odd–odd, and even–odd) have qualitatively the same behaviour as the concurrence.

In this section, we study the quantum polarization of the deformed photon-added entangled squeezed states by evaluating the Stokes operators. These operators are introduced in the following form^{[58]}

*a*

_{H}and

*a*

_{V}and

^{[59]}

To study the quantum polarization of the introduced entangled states in (

*H*and

*V*are referred to as horizontal and vertical components of the introduced states, respectively. By these explanations, we suppose here the first and second modes of the field respectively in the horizontal and vertical polarization.

After doing all the necessary calculations (not shown here), we have plotted the quantum polarization of the deformed photon-added entangled squeezed states versus |*ξ* | by choosing the nonlinearity function *m* in Fig.

As is observed from this figure, the even–even and odd–odd entangled squeezed states (*m* = 0) are complete unpolarized states, while the even–odd entangled squeezed state (*m* = 0) is a complete polarized state in the region of weak fields. Its polarization starts from 1 and suddenly decreases by increasing *r* and finally tends to 0.

The polarization of the even–even deformed photon-added entangled squeezed states (Fig. *m* starts from 1 and after two decrements in the medium region of fields remains fixed about *P* = 0.5. The amount of two decrements decreases by increasing the number of photon excitations, *m*.

In the cases of the odd-odd and even–odd deformed photon-added entangled squeezed states (Figs. *P* = 0.5. The initial nonzero values of polarization increase by increasing the number of photon excitations, *m*. Moreover, the initial values of polarization of the odd–odd deformed photon-added entangled squeezed states (Fig.

In this section, we intend to evaluate some other nonclassical features of the introduced states in Section 3. To achieve this purpose, we pay attention to a few important nonclassicality criteria such as “sub-Poissonian statistics”, “second-order correlation function”, “one-mode squeezing”, and “two-mode squeezing”.

To investigate the quantum statistics of any quantum state, as is usual, we consider the Mandel parameter^{[60]} which is defined for each mode as

*Q*> 0 and

_{i}*Q*< 0.

_{i}^{[61]}The negativity of

*Q*indicates the nonclassicality of the state.

^{[61]}We have plotted the Mandel parameters of the first mode,

*a*, of the deformed photon-added entangled squeezed states versus |

*ξ*| for a particular nonlinearity function

*m*. As is clear from this figure, the sub-Poissonian photon statistics appear in all three cases of the even–even, odd–odd, and even–odd deformed photon-added entangled squeezed states (|

*ψ*(

*ξ*,

*f*,

*m*)〉

_{e,e}, |

*ψ*(

*ξ*,

*f*,

*m*)〉

_{o,o}, and |

*ψ*(

*ξ*,

*f*,

*m*)〉

_{e,o}) for all of the numbers of photon excitations and over all of the regions of the field except the region of medium intensities of the field. This region becomes narrower by increasing the number of photon excitations. Also, only in the odd–odd entangled squeezed state, |

*ψ*(

*ξ*,0)〉

_{o,o}, the sub-Poissonian photon statistics are seen in the region of the weak field, while the even–even and even–odd entangled squeezed states, |

*ψ*(

*ξ*,0)〉

_{e,e}and |

*ψ*(

*ξ*,0)〉

_{e,o}, have no sub-Poissonian photon statistics.

In this subsection, we consider the second-order correlation degree of the two modes of the introduced states by calculating the second-order cross-correlation function which has been defined as^{[62]}

^{[63]}We have plotted the second-order correlation function between the two modes of the deformed photon-added entangled squeezed states versus |

*ξ*| for different numbers of photon excitations in Fig.

*m*> 1 show this nonclassicality sign in the region of the weak field.

In order to calculate the quadratures squeezing of the introduced entangled states, we consider two Hermitian operators *x*,*p*] = i. Accordingly, the following parameters for quadrature squeezing in *x* and *p* may be defined as *s _{xj}* = 2(Δ

*x*)

_{j}^{2}− 1, (

*x*=

_{j}*x*,

*p*) or equivalently

*x*=

_{j}*x*or

*p*if it satisfies the inequality −1 <

*s*< 0.

_{xj}In order to observe the squeezing in one of the field quadratures corresponding to one of the two modes of the deformed photon-added entangled squeezed states, we have plotted the *x*- and *p*-squeezing parameters for the first mode (*a*) of the introduced states versus |*ξ* |. By plotting the figures we observed that no squeezing occurs for the introduced entangled squeezed states for all of the photon number excitations (the figure not shown here).

As another indicator to analyze the quantum fluctuations of any two-mode entangled state, two hermitian parameters are defined as follows:^{[64]}

*U*

_{1},

*U*

_{2}] = i/2. The following parameters for two-mode squeezing in

*U*

_{1}and

*U*

_{2}can be defined as

After constructing and studying some of the nonclassical properties of the “deformed photon-added entangled squeezed states” in the previous sections, now we propose a theoretical scheme to generate such states in experiment. In what follows, we present a theoretical scheme based on a proposal for the interaction of a two-level atom with a two-mode quantized field.

The atom emits a photon by making a transition from the excited state |*e*〉 to the ground state |*g*〉. Based on the nonlinear Jaynes–Cummings model of atom-field interaction,^{[42,43,65–67]} the Hamiltonian of the atom–field system in the RWA and resonant condition is given by:^{[42]}

*g*is the atom–field coupling constant,

*σ*

_{+}= |

*e*〉〈

*g*| and

*σ*

_{−}= |

*g*〉〈

*e*| are the Pauli atomic operators and

*A*=

*af*(

*n*) and

*A*

^{†}=

*f*(

*n*)

*a*

^{†}are the deformed annihilation and creation operators, respectively. Clearly, it can be observed that we deal with a generalization of the Hamiltonian in the standard JCM Hamiltonian

*H*=

*g*(

*σ*

_{+}

*a*+

*σ*

_{−}

*a*

^{†}) which is related to the resonant interaction between a two-level atom with a single-mode quantized field in the dipole approximation and with the rotating wave approximation. By referring to the literature which is concerned with intensity-dependent (nonlinear) JCM,

^{[68–77]}it can be clearly found that in the above Hamiltonian (

*g*=

*gf*(

*n*)).

By assuming that the atom is initially in the excited state |*e*〉 and the field in each of the introduced entangled squeezed states in Eqs. (*t*, in the case of the weak coupling constant *g*, can be obtained as follows:

*gt*≪ 1 (recall that two generation schemes for the entangled squeezed states introduced in Eqs. (

*g*〉, the state of the field is collapsed to the deformed photon-added entangled squeezed states |

*Ψ*(

*ξ*,

*f*,1)〉

_{i,j}in Eqs. (

*m*= 1.

Using the above scheme, the introduced deformed photon-added entangled squeezed states in Eqs. (*m*(>1) successive atoms through the cavity and detecting all of them in the ground state |*g*〉. In another way, by supposing the multi-photon medium (for a two-photon medium see Ref. [78]) which has been introduced in the literature as:^{[66,69,79–83]}

*m*, can be generated. In a similar manner which we followed in Eq. (

*g*〉 in the interaction process expressed via Eq. (

In this paper, at first we had a brief review on general types of two-mode entangled quantum states namely “even–even, odd–odd, and even–odd entangled squeezed states” (|*ψ*(*ξ*)〉_{i,j}, *i*,*j* = e,o) which are constructed by replacing the coherent states in the previously introduced entangled coherent states^{[9]} with the (even and odd) squeezed states. Then, we introduced the corresponding “deformed photon-added entangled squeezed states” |*ψ*(*ξ*, *f*, *m*)〉_{i,j} via the iterated action of the deformed creation operator *f* (*n*)*a*^{†} on the first mode of each of the two-mode entangled squeezed states. In the continuation, after deriving the explicit forms of the introduced states in the number states, we studied the amount of entanglement and quantum polarization of the entangled states, numerically. Also, we discussed some of the nonclassicality criteria such as: the Mandel parameter, second-order correlation function, one-mode and two-mode quadrature squeezing. In particular, we investigated the influence of the photon number excitation on the considered criteria. Finally, we presented a theoretical scheme to generate the introduced states. The obtained numerical results are briefly shown as follows.

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