1. Introduction

A few months after the Einstein-Podolsky-Rosen [ ¹] article was published in 1935, in the same year, articles were published in reaction by Erwin Schrödinger [²] and Niels Bohr [³]. Schrödinger coined the word "entanglement" to describe a condition in this famous EPR paper. The discussion of these three papers focuses on describing two entangled particle states and the characterization of the quantum correlations between them. The entanglement of two-qubits, two-qutrits or the two-quadrits is quantum-mechanically interesting. Correlation in an entangled system cannot be explained using classical correlations [⁴]. Quantum entanglement has been proposed as a proof of quantum correlations [²].

The study of quantum entanglement has advanced over the past years due to the further development of quantum physics [⁵ - ⁷]. Entangled states can be worked in different physical setups, such as massive particles like TRIs [⁵]. Entanglement between electronic levels of the TRI has been an active field of study in recent years [⁶]. The high-fidelity entanglement between a TRI and a photon is measured by quantum frequency conversion [⁷].

In this context, qudits are used as higher capacity quantum information sources, and could be of interest for quantum communication [⁸]. The higher dimension of a given Hilbert space, the more quantum correlation continuity is ensured. A quantum dit (qudit) is the unit of quantum information described by a superposition of "d" states, where the number of states is an integer greater than two. For example, qutrit or quadrit state can be called a qudit state. Pure and mixed qudits can be obtained due to the interaction between two laser beams and the three-levels TRI with specific initial state conditions [⁶, ⁹, ¹⁰]. It is fundamental within the probability interpretation as initiated by M. Born and pushed into a general form by P. A. M. Dirac, J. von Neumann, G. Birkhoff and many others [¹¹].

Fidelity (Transition Probability) for pairs of density matrices can be defined as a tool in the hierarchy of all quantum systems. Uhlmann and Jozsa’s papers are considered classic fidelity studies [¹²-¹⁴]. Fidelity, negativity, purity, and concurrence are popular methods for quantum entanglement [¹⁵-¹⁷]. A structure worthy of study is presented in Ref. [¹⁸] for multiplexed quantum repeaters that use local connectivity to improve accuracy in entanglement distribution.

The existence of atomic motion, which becomes most obvious for the high-dimensional Hilbert space of pure qudit states is shown in [¹⁹]. A two-level atom and a two-level ion, and a photon with spin up-spin down probability are associated with two-dimensional Hilbert space. In general, the n-level atom and the n-level ion are defined as the n-dimensional Hilbert space. Interaction of an atom or an ion with a photon is associated with a higher-dimensional Hilbert space in the tensorial product such as (H=H1⊗H2). We expand the physical model that has been improved before for studying Schrödinger cat states of the two phonons [20]. The n-dimensional Hilbert space in Ref. [²⁰] is restricted to 12 dimensions in this study. This limitation has led to easy analytical results. Characterization of the quantum entanglement can be explored using entropy [²¹] and fidelity [²²-²⁴], which apply to the bipartite systems of arbitrary dimensions in TRIs. Hence this situation fits our purposes. By means of quantum entropy, it is shown whether the entanglement lifetime is extended or not [²¹]. There are both contributions and compatibilities between Ref. [²⁵]’s fidelity calculations and this paper.

The outline of this study is as follows. The theoretical framework of the qutrit-quadrid in the physical system is introduced in Sec. 2. the quantum entropy and the fidelity are discussed with corresponding Figures in Sec. 3. Finally, we give the results of the quantum measure in Sec. 4.

2. Evaluation of qutrit-quadrid in the physical system

We focus a TRI in a harmonic potential and two coherent states (or two photons). Also, we develop a new physical system based on the quantum system mentioned in [²⁰]. The concept of the time evolution of the physical model is introduced using the density matrix, which affects the state necessary for quantum dynamics in [²⁶]. The harmonic trap frequency is designed to construct a linear trap so that the TRI’s center-of-mass (c.m.) motion is effectively one dimensional along the x axis. Therefore, we consider small vibrations of a TRI in the harmonic trap which can be defined as coherent displacements. The quantum state of the TRI’s c.m. motion is characterized by a coherent state |α〉 with α≪1 in this way. For the first order in α, this describes as a qubit of two phonons |0〉+α|1〉.

The physical system is assumed to interact with two coherent states in a Λ scheme with the mass of the TRI m and the linear harmonic trap frequency v. A total Hamiltonian is determined in the Λ configuration for optical transitions. The total Hamiltonian of the TRI-coherent system is Htotal=Hion+He-g+He-r, and H _ion is called the Hamiltonian of the TRI. The H _ion is given as the following:

The assumptions and the parameters used in the TRI-coherent system are given as follows: δ1=ω1-ωeg and δ2=ω2-ωer. ωeg is the resonance frequency of e-g transition and ωer is the resonance frequency of e-r transition. The frequencies of the two photons are equal to ω = ω ₁ = ω ₂ (See Fig. 1). Here p _x is the momentum operator and x _ion is the x-component of the position operator using the Planck constant equal to one, (ℏ = 1). The TRI c.m. motion is described by the standard harmonic-oscillator quantization of H _ion [²⁰] via px=-i(1/2)mν(a†-a) and xion=(1/2mν)(a†+a). Bosonic operators a and a ^† according the usual Weyl-Heisenberg algebra are the annihilation and the creation of the phonons.

We consider the lower levels g and r to be degenerate. Therefore, the energy of the ground level is equal to the Raman energy ω _r = ω _g = 0, the excited energy and is ω _e . We focus on a nontrivial quantum case of a weakly detuned our system in which Δ=δ1-δ2=0 and δ1=-νη2. Δ = 0 can be true when lower levels of the ion are degenerate in Eq. (1). The Lamb-Dicke parameter (LDP) is η=k/2mν. The wavevectors k ₁ and k ₂ characterize the two photons with k = k ₁ = k ₂, and k = 2π/λ. The wavelength of both photon λ is red detuned from the upper level e by the same amount δ1=δ2=-νη2. If the frequencies and wavevectors of the two incoming photons do not take as equal (ω1≠ω2; k1≠k2), then Δ ≠ 0, that is, r and g levels can not be degenerate. So that, |r〉〈r| can be visible in Eq. (1) and Eq. (4). H _e-g and H _e-r are the interaction Hamiltonians between these levels e - g and e - r using ℏ = 1. The interaction Hamiltonians are defined as follows:

here, atomic levels of the TRI are given, such as |g〉→ TRI-ground level, |r〉→ Raman level and |e〉→ excited level (See Fig. 1). The Rabi frequencies of the dipole interactions between the photons and the TRI’s are given by Ω in Eq. (2) and Eq. (3). So that, the equality of Rabi frequencies is taken by Ω = Ω₁ = Ω₂. In this approach, the mathematical equations are obtained by δ1=δ2 and Ω₁ = Ω₂.

We now remove the Bose variables from the interaction part of the Hamiltonian in Eq. (2) and Eq. (3). Thus, the exponential expressions have some variations. Because of the degeneracy, Δ = 0 and the two exponential expressions are as follows: ei(k1xion-ωt)=eiη(a†+a) and ei(-k2xion-ωt)=e-iη(a†+a). Therefore, the total Hamiltonian is rewritten as

with respect to the base vectors as the following:

H~=U†HU is the transformed Hamiltonian. The Hamiltonian in Eq. (4) turns into the Hamiltonians of Eqs. (7-8) with the transformation process. In this transformation we used the most extended weak excitation regime. The weak excitation regime is applied by Ω = 2v. With this mathematical approach, this optical Λ configuration demonstrated to be equivalent to a cascade configuration for the phonon transition, under the unitary transformation [²⁰]. The TRI-coherent system evolves in the optical Λ configuration. The transformation matrix U is defined as [²⁰]:

Here, Glauber displacement operators are represented by B(±η)=e±(iη(a+a†)). The transformed Hamiltonian, H~ is described with cascade-type transitions of the two phonons, under a rotating wave approximation (RWA). The RWA is valid for η<1/α, where a characterizes the amplitude of coherent displacement of the phonons [²⁰]. The RWA and the transformation method in Eq. (6) allow for working effects of larger η, for example, the third value of η (See Figs. 2, 3 and 4). After this transformation operation, H~ is written as H~=H~0+V~, where

and

Under the unitary transformation method [?], the time evolution of an early position ψ(0) is defined as follows:

where K(t) is the propagator vector and e(-itH~0) is term of the interaction picture transformation. The two rightmost factors U†|ψ(0)〉=|ψK(t)〉 act as the early state for the cascade system, evolving K(t) into |ψK(t)〉. The rotating frame transformation is given by the prefactor matrix U0=exp(-iωt|e〉〈e|) [²⁰]. The propagator is obtained as

here ϵ=νη2,Λ=ϵ2a†a+1, G=(Λt)-1/Λ2 and S=(Λt)/Λ. The trap frequency of the system is taken by ν=104 Hz. Because the most suitable mathematical values for the systems are the trap frequencies between 10³ and 10⁶ Hz. Under all approaches and transformations, the new general formula of the system is given as

The initial state of the TRI-coherent system is defined as

here, the matrix representations of the coherent states are 〈0|=(1,0), and 〈1|=(0,1). The generalized coherent state is Fn(α)=e-|α|2/2(αn/n!) in Eq. (11). n = 0 and n = 1, F ₀ = b and F ₁ = a are the amplitudes of the Fock number states of two phonons. The probability amplitudes of the coherent states are given as b=1 and α=0.005. The TRI normalization condition is exactly 1/32+1/32+1/32=1, and two phonons normalization condition is approximately ∥b∥2+∥α∥2 =|1|2+|0.005|2≃1. We have shown the TRI-coherent system as l⊗l' in Eqs. (11-12). Considering Eq. (12), the dimensionality of the Hilbert space is defined as: The Hilbert space dimensions are l=4 for the two-phonons and l’ = 3 for the full TRI. The TRI-coherent system is in Hilbert 12-space. For the two rightmost terms of Eq. (9), the TRI-coherent state is transformed into an early case of the cascade

Because of the transformation matrix, Eq. (12) is produced by ∑σ,m‍Mσ,m(t)|σ,m〉 in the system. The twelve probability amplitudes belong to the TRI-coherent system are given as

and the three amplitudes are zero: Me3(t)=Mr2(t)=Mr3(t)=0. In Eqs. (14)-(22), the first index σ is located in the atomic states (e, r, g), the second index m is located in the vibrational quantum numbers (0,1,2,3). In the above equations, t is dimensionless time scaled by vη. In Figs. 2, 3, and 4, a scaled time of 6 is equal to 2000 microseconds for LDP=0.3. The calculation is as follows: If η = 0.3 and v = 1 × 10⁴ Hz, then νη=3×103, and the scaled time of 6 is (6/νη)=2×10-3=2000 microseconds. Under the unitary transformation and the interaction picture, we calculated the new general existing state vector as the following:

where An(t),Bn(t),Cn(t) are these amplitudes of the state vector for the TRI-coherent system. Then, the new detailed existing state vector (Eq. 23) is given by

We denoted the total density operator by ρ _ion-phonon for a composite quantum state with finite-dimensional Hilbert space. Therefore, we limit the Hilbert space so that the quantum state is easily examined through Eqs. (23-24). The total density operator for three-levels of the TRI interacting with two photons has the form

For the new detailed state vector and the density operator of the TRI-coherent system, we use the new notation below:

Here e1=e,0, e2=e,1, e3=e,2, e4=e,3, e5=r,0, e6=r,1, e7=r,2,  e8=r,3, e9=g,0, e10=g,1, e11=g,2,  e12=g,3 for both j and l. After long calculations, these coefficients of the new detailed state vector are defined as follows:

where, ω _eg is the resonance frequency of e-g levels, ω=ωeg-νη2. Four of the 12 coefficients (Eq. 29) are plotted for η = 0.3 in Fig. 2. The number of peaks in all plots of Fig. 2 are more numerous than in the remaining eight probability amplitudes.

We showed two the measures of entangled states in Figs. 3, 4 and 5. By quantum mechanical tracing method, we obtained a reduced density operator ρion=Trphonon(ρion-photon) using Eq. (39). From Eq. (25), the analytic solution of the total density operator becomes

Qudit entangled states can be made observable by measuring quantum entropy [⁶, ²⁴]. The operator ρ _ion-phonon is represented by a 12×12 matrix. Taking trace over the phonon system, 3×3 reduced density operator, ρ _ion can be written as

where diagonal terms, |e〉〈e|, |r〉〈r|, and |g〉〈g| are these 4×4 matrices. The matrix representation of first diagonal term using basis vectors is defined as:

One of the nine traces in Eq. (39) is defined as: Tr|e〉〈e|=A0A0*+A1A1*+A2A2*+A3A3*. Similarly, other diagonal trace terms can be calculated.

3. Fidelity, quantum entropy and discussion

It is significant to measure whether there is entanglement in the TRI-coherent system with entropy and fidelity. We investigate the entanglement of the system for values of the LDP between 0.0 and 1.0, according to entropy and fidelity. The internal states of the TRI subsystem are associated with a three-dimensional Hilbert space H _i of a qutrit spanned by the basis states H _i . The coherent state subsystem is described by a four-dimensional Hilbert space H _p . In particular, the Hilbert space of the TRI-coherent system H is the twelve-dimensional, such as H=Hi⊗Hp=C3⊗C2⊗C2=C3⊗C4=C12. The dimensions of Hilbert space of the system are guides for understanding the density operators. From the new detailed state vector |ψlower(t)〉 (See Eq. 24), the density operator of the TRI-coherent system is shown by ρion-phonon=|ψ(t)〉〈ψ(t)| in Eq. (38). The quantum entropy (S) of the TRI-coherent system is defined as in [²⁷]

where ρion=Trphonon(ρion-phonon) is the reduced density operator in Eq. (39). Entropy can be calculated with three eigenvalues (Schmidt coefficients) of the reduced density operator [²¹]. The normalization condition of the three eigenvalues is exactly ∥λ1∥+∥λ2∥+∥λ3∥=1.

In quantum mechanics, especially in quantum information science, fidelity is used to evaluate the transformation between the changing states and the fixed states. With the help of the basic formulas in [²⁵], we obtained the fidelity of the system as follow: [²⁸, ²⁹]

where |ψ(t)〉 is given by Eq. (24) and |ψ(0)〉 is given by Eq. (12).

We focus on the quantum dynamics and the quantum correlations of two quantum measures with figures. The entropy in Fig. 3 and the fidelity in Fig. 4 are obtained for three different LDPs. These three values of LDP are within the deep LDP. This is called the deep LD regime is qualified by a small η = 0.01 or η = 0.3; on the other hand, beyond the LD regime is η = 0.4 [²⁰]. The fidelity is certainly recorded F = 0.34 at the peak for η = 0.1 in Fig. 4. The scaled optimum time (scaled time=33, 11.000 ns) was found with this research, by evaluating in Figs. 3 and 4. These findings are consistent with the findings obtained in previous studies [¹⁹, ²¹, ²⁵]. We can see in Fig. 3 that the entanglement values increase with the rise of LDP. If Fig. 5 and these results are evaluated together, the experimental limit for fidelity is compatible with LDP = 1.0 via Eq. (42) [²⁸].

Figure 5 appears that the entropy (left panel) rather than fidelity (right panel) allows more entanglement lifetime. We demonstrated quantum entanglement with entropy in the deep LD regime distinctively in Ref. [²¹]. In this graph, as LDP grows the entropy increases, which means that the entangled lifetime is raising. For example, the state of the TRI-coherent system is disentangled at scaled time 18, t = 6.000 μs for η = 0.3. The system reaches a local minimum which corresponds to a separable quantum state

Similarly the state of the TRI-coherent system is maximally entangled at scaled time 33, t = 11.000 μs for η = 0.3. We get a local maximum which corresponds to a partially quantum state

The optimum time to reach an arbitrary entangled state is 11.000 ns (scaled time 33); see Fig. 5. In line with of the measures and the results reported above, we observed that the maximally entangled states do not collapse in the TRI-coherent system. We examined with S that the measurement degrees have a sudden appearance of the entangled state in parallel with raising η, which is in agreement with previous observations [²¹, ²⁸ - ³⁰]. The article in Ref. [³¹] characterized the application of Schmidt mode analysis for pure states.

4. Concluding remarks

In this paper, we examined whether the full TRI-coherent system with a harmonic trap frequency of 10⁴ Hz is entangled or not. We analyzed the quantum correlations of the system via three coupling parameters (η, v and Ω). After that, we discovered that for η = 0.3 it survives longer than for η = 0.1 and η = 0.01. This observation demonstrates that entangled states are directly connected with the LDP. We show a high degree of the quantum entanglement of the full TRI-coherent system using two elaborated measures of the family consisting of S and F. The entropy behavior is similar to the fidelity behavior when we change the LDP. Comparing three graphs in the Fig. 3, we determined that entangled states are stored in the TRI-coherent system. Besides, the amount of entanglement has two maximum values for S = 0.68 and F = 0.35 in η = 0.3. Also, the first graph of Fig. 3 shows that the system’s entropy is described by a Gaussian-like profile as a function of scaled time.

Our results can be helpful in engineering the science of entanglement. The quantum correlations yielded the following significant developments in this study.

In summary, the S measurement indicates the length of life of entanglement, while F indicates the maximally entangled state. The long-lived and maximally entangled states presented in this study can serve as a guide to researchers trying to add innovative aspects to the science of entanglement.