{'fraction_non_alphanumeric': 0.03913454717427288, 'fraction_numerical': 0.017143532750059115, 'mean_word_length': 4.919174247725683, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Robustness of deformed catlike states under dissipative decoherence

The generation of coherent superposition of distinct physical systems and the construction of robust entangled states under decoherence are the most experimental challenges of quantum technologies. In this work, we investigate the behaviors of catlike states of a deformed harmonic oscillator under dissipative decoherence. Varying the deformation parameters, we obtain catlike states having more resistance against decoherence than catlike states of the ordinary harmonic oscillator. Furthermore, we study nonclassical properties and entanglement of different catlike states subjects to decoherence caused by a dissipative interaction with a large environment. Depending on different deformation parameters, we reveal that the nonclassical properties of catlike states under dissipative interaction can be more retarded and preserved in the time.


Introduction
Coherent states or quasi-classical states of the harmonic oscillator were first introduced by Schrödinger in order to make a connection between the classical and the quantum formalisms [1]. It has been shown that these states not only minimize the Heisenberg uncertainty inequality for position and momentum operators but also maintain maximum localizability during the time evolution of the harmonic oscillator [2,3]. In particular, their dispersions on the kinetic energy and on the potential energy are identical [2]. The importance and the physical applicability of these states, in quantum optics, have been investigated in many works including the papers of Glauber, Klauder and Sudarshan [4,5,6,7]. The concept of coherent states has not been restricted to only the harmonic oscillator and has been implemented to other physical systems [8,9,10]. In this way, these states are called nonlinear coherent states and have been constructed for certain physical systems [10,11,12]. Furthermore, it has been shown that coherent states can be built for any Lie symmetry [13,14,15]. Particularly, the su(1,1) coherent states related to the SU(1,1) group and the su(2) coherent states have been constructed and applied in quantum optics and in condensed matter physics [3,13,16]. It is worth noting that there are two different approaches to elaborate su(1,1) coherent states. The first one is called Perelomov approach which is based on the application of the displacement operator on the ground state [13]. The second one is associated with Barut-Girardello coherent states defined as eigenstates of the annihilation operator of the su(1,1) algebra [17]. Several mathematical generalizations of coherent states have been introduced [2,10,18,19,20,21,22,23,24,25]. In particular, the generalized Heisenberg algebra (GHA) nonlinear coherent states have been constructed [10] and studied for several physical systems including the square well potential [10], the Pöschl-Teller potential [11] and Morse potential [19].
Recently, a considerable attention is paid to the experimental observations of nonclassical properties of quantum systems such as squeezing [26], photon anti-bunching [27,28] and entanglement [29]. The latter is recognized as a key resource of quantum computing, quantum cryptography and quantum communications [30,31]. In contrast to classical systems whose states are always a mixture, quantum systems can be prepared in a superposition of quantum states. This property is an interesting quantum effect being very hard to be prepared and detected due to the decoherence occurring when the quantum system interacts with a relevant environment [32]. This interaction causes the loss of coherence of the quantum system generating classical mixture states. This mapping is called the decoherence [32]. The process of decoherence has been studied both theoretically and experimentally [33,34,35,36,37,38,39] for several physical systems such as the trapped ion in many works [40,41,42,43,44].
The main aim of this work is to contribute to these activities by investigating the generalized su(1,1) coherent states superposition for a four-parameter perturbed oscillator [45,46], known as catlike states or even and odd coherent states. Then, we study their nonclassical properties under decoherence in order to compare them with those of the generalized Heisenberg algebra catlike states. In particular, we discuss the robustness of generalized catlike states against decoherence caused by the interaction with an environment associated with an infinite collection of harmonic oscillators by using the fidelity. Concretely, we show that the resistance of generalized catlike states depends on the corresponding algebraic structure. Furthermore, we study the entanglement degree of generalized catlike states under decoherence. This paper is organized as follows. In section (2), we briefly recall the GHA and the generalized su(1,1) algebra and construct the generalized su(1,1) catlike states for a perturbed harmonic oscillator. Furthermore, in section (3) we study the resistance against decoherence of the constructed catlike states with that of the GHA catlike states. Moreover, in section (4) we study the effect of the decoherence on photon distribution function, statistical distribution and quantum entanglement of generalized catlike states in terms of various parameters of deformation. Finally, our conclusions are given in section (5).
It is recalled that the GHA is described by the generators H, A † and A satisfying and where H = H † is the dimensionless Hamiltonian of the physical system under consideration.
(A † ) † = A. f is an analytical function of H, called the characteristic function of the GHA. The associated Casimir operator reads as The irreducible representation of the GHA is given through an eigenvector |n of the Hamiltonian H, H |n = ε n |n such that ε n+1 = f (ε n ), for n = 0, 1, . . . .
The Fock space representation of GHA generators is then given in terms of the lowest energy eigenvalue ε 0 . The eigenvalue ε n = f n (ε 0 ) is just the n-iterate of ε 0 under the function f [47]. By

using (1)-(3), one can show that
A |n = N n−1 |n − 1 , where The operators A † and A are interpreted as the creation and annihilation operators of GHA, respectively. It is noted that the vacuum state condition A |0 = 0 is verified. Taking f (H) = H + 1, where H is the dimensionless Hamiltonian of the harmonic oscillator, the GHA reduces to the ordinary Heisenberg algebra spanned by H and the bosonic ladder operators a and a + [47]. Similarly, considering f (H) = qH + 1, the relations (1)-(2) recover the q-harmonic oscillator algebra [47].
Furthermore, it has been shown that the GHA can be applied for physical systems involving known spectrum such as the square well potential [49], the Morse potential [50,51] and the Pöschl-Teller potential [11,52].
Furthermore, the q-deformed su(1,1) algebra has been also constructed and applied widely in many areas of physics [55,56,57]. Moreover, a generalization of the su(1,1) algebra can be constructed by following the same ideas developed for the GHA and the generalized su(2) introduced in [58,59].
The generalized su(1,1) algebra is defined by the Hamiltonian H and the ladder operators J + and J − satisfying where J † + = J − and f (H) is a given function of the Hamiltonian H. By considering f (H) = H + 1, this algebra becomes the standard su(1,1) algebra. The Casimir operator has now the following The representations of the generalized su(1,1) algebra generators can be easily deduced. Let ε n+1 = f (ε n ). Assuming that J − |0 = 0, we can show that where Similarly to the GHA, the generalized su(1,1) algebra involves the creation and annihilation operators J + and J − , respectively.

Construction of generalized su(1,1) coherent states superposition
Here, we would like to construct the catlike states associated with the generalized su(1,1) algebra.
For this purpose, we first construct the generalized su(1,1) nonlinear coherent state. We define this state as an eigenstate of the annihilation operator J − Since J − is a non hermitian operator, z is, in general, a complex number. The state |z is expanded in terms of energy eigenvectors |n as where c n are complex coefficients satisfying ∞ n=0 |c n | 2 = 1. The action of J − on such state, leads , the state |z can be rewritten as Imposing the normalization condition z|z = 1, we get The generalized su(1,1) catlike states can be easily obtained. They are given by where 2.4 Construction of generalized su(1,1) catlike states for a deformed harmonic oscillator As an application of the generalized su(1,1) algebra, we consider a perturbed harmonic oscillator having the dimensionless energy spectrum given by The function g(n) reads as where a, b, c, d are real parameters different from zero [45]. It is a perturbed function associated with the following conditions where r ∈ [0, 1]. Subject to such conditions, the spectrum (20) is strictly increasing. In this way, the associated generalized su(1,1) algebra can be constructed. For the simplicity reasons, we take c = 1.
By substituting (20) in (13), we find that and The normalization factor given in (17) can be easily computed. It can be written as ; |z| 2 is the generalized hypergeometric function. By using (29) and (32), the normalization factor (19) associated with the generalized su(1,1) catlike states can be given by the deformed harmonic oscillator. They are given by It is worth mentioning that the GHA catlike states can be obtained by replacing N n by N n in |z and in N ± (|z|). In the following, we only consider the catlike states having the form We will see that the construction of the generalized su(1,1) catlike states for physical systems is advantageous because it gives us the possibility to compare their nonclassical properties under decoherence with the catlike states associated with the GHA and select the ones whose properties are more preserved and retarded in the time. We will examine particularly the deformed harmonic oscillator having the spectrum of the form (20). It is noted that the deformed harmonic oscillator becomes the ordinary harmonic oscillator by considering a = 1 2 and e = d 2 .
In this case, the GHA catlike states become the ordinary catlike states given by where |z is now the ordinary Glauber coherent state given by 3 Dissipative decoherence of generalized su(1,1) deformed catlike states We now study the interaction between the perturbed oscillator presented above with a large environment composed by an infinite collection of harmonic oscillators. According to [60], the time evolution of the density operatorρ of the deformed harmonic oscillator is described by the following master equation where γ is the damping coefficient. a and a † are the annihilation and creation operators of the harmonic oscillator, respectively.
The density operatorρ(t) reads asρ where In the following, we will consider that at t = 0,ρ(0) = |ψ + ψ + |, where |ψ + is given in (35). In order to quantify the robustness of catlike states, we use the fidelity defined by This measures how statesρ(t) are evolved compared with the initial stateρ(0). By using (39), the fidelity (41) of the states (35) becomes where we have considered that z is a real number rather than complex. The fidelity of the gener- catlike states are always more resistant against decoherence than the GHA ones. In addition, we see that perturbed harmonic oscillators having a = 0.7 are more robust to decoherence than the ordinary harmonic oscillator associated with a = 0.5 and d = 2e. Furthermore, the figure (1c) shows that the robustness to decoherence of the perturbed oscillators depends also on the parameters d and e. Therefore, we can conclude that the catlike states' resistance to decoherence depends on the corresponding algebra from which they are constructed, i.e, the generalized su ( By using the results found in [61,62], the photon distribution function of generalized su(1,1) catlike states can be given by The photon distribution of GHA catlike states can be obtained by replacing N n by N n in (44).

Statistical properties of generalized catlike states
The statistical properties of generalized catlike states can be studied by using the Mandels param- where n is the average number of particles in the catlike state in question and (∆n) 2 = n 2 − n 2 is its variance. If Q > 0 (Q < 0), we say that the distribution is super-Poissonian (sub-Poissonian) and if Q = 0, the distribution is Poissonian. To calculate the Mandels parameter for generalized catlike states, we use the fact that nP n (t) and n 2 (t) = ∞ n=0 n 2 P n (t), where P n (t) is given in (43). In the figure (3)

Quantum entanglement
Protecting entanglement from decoherence is recognized as a considerable experimental challenge since the entanglement is a key resource for quantum technologies. Thence, several authors have studied the effect of decoherence on entanglement of catlike states for different physical systems.
Here, we aim to quantify the amount of entanglement of generalized su(1,1) catlike states constructed in section (2) for the deformed harmonic oscillator whose spectrum is given in (20) under decoherence caused by the interaction with a large environment composed by an infinite collection of harmonic oscillators. We will study the effect of deformation parameters and amplitude parameter |z| on the entanglement of generalized su(1,1) catlike states. For this purpose, we appeal to the von Neumann entropy which is a good quantifier of entanglement. The von Neumann entropy of the density operatorρ is given by where λ i are the eigenvalues of the reduced density matrixρ(t) of the generalized catlike states.
In the figure (4), we show the behavior of the time evolution of the von Neumann entropy. Immediately, we observe that the entropy increases quickly and reaches its maximum value. Then, it decreases and vanishes for large values of time. Thus, we can conclude that the decoherence affects on the correlations between the catlike states and the environment. The von Neumann entropy vanishes for long values of time t indicates that the state describing the system when time is significantly large, is pure and not entangled with the environment. Interestingly, the entanglement degree is shown to be large as the amplitude |z| is large. It depends also on different parameters of deformation. This is relevant because it gives us the possibility to find cases in which the entanglement loss is retarded in the time evolution. It is worth to mention that we have treated the von Neumann entropy for higher values of |z| and the entanglement is shown to be larger as |z| increases.
In the section (4.1), we have seen that the fidelity vanishes also for large values of the time for all generalized catlike states of the perturbed harmonic oscillator. This indicates that in this range of time, t >> 1, the quantum coherence (the interference property) of the state of the perturbed oscillator are lost due to the interaction with the environment. Furthermore, The Mandels param-eter Q and the photon distribution function tend to zero as the time is significantly large. This can be explained by the fact that the interaction of the system with the bath transfers all photons from the system to the environment, as P n (t) ≈ 0 when t → ∞, and leaves the system in a vacuum state which has a Poissonian distribution, i.e, Q ≈ 0 when t → ∞. This state is pure and not entangled.
Precisely, it can not be written as a coherent superposition of other quantum states. For such a reason, the fidelity and the von Neumann entropy vanish for large values of time.
Consequently, the interaction of the perturbed harmonic oscillators with a collection of harmonic oscillators decreases the fidelity, the amount of their entanglement with the environment, the super-Poissonian distribution and the photon distribution function. However, these properties can be preserved and retarded by varying the different parameters of deformation.

Conclusion
In this paper, we have built the catlike states associated with the generalized su(1,1) algebra for a four-parameter deformed oscillator. In terms of different deformation parameters, we have investigated the resistance of these states under decoherence caused by a dissipative interaction with an environment modeled by an infinite collection of harmonic oscillators. Subsequently, we have shown that the generalized su(1,1) catlike states are always more robust under decoherence than the GHA catlike states and that the robustness against the decoherence depends on different parameters of deformations. Among others, we have found that the different parameters of deformations give the possibility to find cases more resistant than the catlike states of the ordinary harmonic oscillator.
It has been revealed that the resistance of the catlike states depends on the algebraic structure from