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Abstract
Multipartite entanglement has emerged as a valuable quantum resource for constructing large-scale quantum networks. However, the presence of non-Hermitian features induced by natural microscopic quantum systems significantly modifies the overall response of nonlinear parametric processes, thereby enabling direct manipulation of multipartite entanglement properties. In this study, we demonstrate the generation of multimode entanglement through atomic four-wave mixing (FWM) and analyze the properties of exceptional points (EP) under dressing control in non-Hermitian systems. By leveraging dressing-controlled atomic nonlinearity, we achieve versatile EPs and higher-order EPs by carefully tuning the atomic multi-parameter in the cascading FWM system. Additionally, we investigate the entanglement properties of various permutations of the output signal modes using the positive partial transpose (PPT) criterion. Notably, under non-Hermitian control, the application of single-, double-, and N-dressing splits leads to coherent multichannel control and further extends the scale of quantum entanglement. The outcomes of our research offer a novel approach to actively control non-Hermitian quantum phenomena without relying on artificial photonic structures. Furthermore, this paves the way for the realization of complex quantum information tasks by exploiting the non-Hermitian characteristics of the light-matter interaction.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Hermiticity is a fundamental and essential property observed in a wide range of physical systems, arising from the conservation of energy, and is characterized by the presence of real-valued eigenvalues. However, in realistic scenarios involving energy exchanges with the surrounding environment, the description of such open systems necessitates the use of a non-Hermitian Hamiltonian. Extensive investigation in this field has unveiled the remarkable ability of non-Hermitian processes to significantly transform system behaviors and give rise to counterintuitive phenomena when compared to their Hermitian counterparts [1–3]. In particular, the occurrence of degeneracy points, where the eigenvalues of the complex-valued non-Hermitian Hamiltonian coincide, provides a novel perspective for the study of physical properties in non-Hermitian systems. These special points, known as exceptional points (EPs), represent branch points where two eigenvalues and their corresponding eigenvectors simultaneously merge into a a value [4–6]. Furthermore, the intriguing phenomenon of higher-order EPs arises when more than two eigenvalues converge simultaneously [7–9].
Recently, exceptional points (EPs) have attracted considerable attention and have been the subject of extensive investigation within the field of optical systems. This research encompasses a diverse range of systems, including photonic crystals [10], optical microcavities [11–13], optical waveguides [14], and atomic systems [15,16]. Of particular significance are systems that exhibit parity-time (PT) symmetry, achieved through precise manipulation of gain or dissipation mechanisms [17–19]. Concurrently, there has been a growing interest in exploring EPs in other physical systems, such as plasmonics [20], acoustics [21,22], and optomechanics [23,24], among others. Moreover, variations in parameters within non-Hermitian systems can prompt broken-symmetry phase transitions, resulting in profound alterations in the sensitivity surrounding EPs. This exceptional characteristic of EPs finds widespread application in enhancing sensitivity measurements [9,13,25], pump or loss-induced lasers [26,27], laser mode selectivity [28,29], and mode switching [23]. However, previous studies on EPs have predominantly focused on fixed regimes, specifically the weak or strong coupling regime, within coupled artificial optical or photonic structures. As a consequence, the exploration of EPs in the context of light-matter coupling remains largely unexplored.
The realization of large-scale quantum networks relies heavily on multipartite entanglement [30–33]. Such networks have significant applications in the field of quantum information processing, spanning from quantum computing to quantum secure communications [34–36]. Despite the emergence of biphoton processes with nonlinear crystals as one of the most developed technologies for generating quantum entanglement, there remains a pressing need to discover scalable methods for constructing large-scale quantum states [30,37]. In recent years, the parametric amplifier four-wave mixing (PA-FWM) technique utilizing atoms has demonstrated promise as an effective approach for generating multimode entanglement [38–40]. Atomic media offer several inherent physical advantages such as strong nonlinear coefficients [41], narrow bandwidth [42,43], and long coherence time [44]. These properties make atomic systems highly suitable for advanced quantum technologies, including multi-photon generation [45,46], sources of multipartite quantum entanglement [47], and all-optical quantum information protocols [48–50].
The atomic four-wave mixing (FWM) process relies on third-order nonlinear effects to generate a pair of entangled beams [51]. To effectively modulate the nonlinear susceptibility, atomic coherence-assisted non-Hermitian processes are employed. These processes take advantage of the dressing effect, which arises from the constructive interference among different transition paths. Through this mechanism, the nonlinear susceptibility can be optimized to control the gains of the FWM [52], Rabi oscillations [53,54], and shape the correlation waveform of biphotons [55]. Furthermore, the dressing effects enable energy level splitting, facilitating the creation of multimode and multi-coherent channels in the FWM processes [52,56]. Consequently, this allows for the generation of high-dimensional time-energy entangled photon states [57] and enhances the degree of squeezing in the state [58–60].
However, coherently-prepared, multilevel atoms represent naturally occurring microscopic dissipative systems that are inherently endowed with EPs. In contrast to less controllable systems, the presence of atomic coherence within these dissipative atomic structures enables precise adjustment of the light-matter coupling between atomic states and external optical fields surrounding the EPs. This, in turn, facilitates active control over the quantum correlations generated within such non-Hermitian parametric processes.
This study explores the analysis of the dressing-controlled energy-level cascade four-wave mixing (ELC-FWM) process, a natural non-Hermitian nonlinear system, to demonstrate the presence of versatile exceptional points (EPs) and higher-order EPs, as well as multipartite entanglement within a single Rb cell. The coherent preparation of multilevel atoms offers distinct advantages in achieving flexible non-Hermitian control. We investigate the evolution of EPs in these non-Hermitian systems using three different doubly dressing structures, along with a single dressing structure. By adjusting the atomic multi-parameter, these systems can exhibit a wide range of EPs and higher-order EPs. Moreover, through the regulation of atomic coherence, the multipartite entanglement among the output modes can be modulated via multiple coherent channels originating from the degeneracy of the real and imaginary parts of the eigenvalues. Additionally, by utilizing generalized N-dressing fields in the interaction with the ELC-FWM process, non-Hermitian control can expand the quantum information capacity by generating a large-scale of multipartite entangled states with multiple modes. This research lays the foundation for studying the effects of quantum entanglement around EPs in non-Hermitian systems of light-matter coupling.
2. ELC-FWM process model
Using Four-wave mixing with the third-order nonlinear effect in rubidium atomic medium has been proven to be a promising candidate for generating two quantum correlated beams of light. Here we consider a multimode FWM system consisting of two pump fields E1 and E2 with one probe field ES2, as shown in Fig. 1(a), which depicts the cascade energy level of three modes scheme using a double-Λ FWM process in hot rubidium vapor. Here, we use two strong coherent pump beams cross in an 85Rb atomic vapor at a small angle, then another seed beam (weak coherent beam) is injected into medium vapor which has frequency detuning Δ1 to the D1 line of the 85Rb (5S1/2, F = 2→5P1/2). The strong pump E1 (frequency ω1, wave vector k1, and Rabi frequency Ω1) drives the atom transition from energy |1〉 to |3〉. When it decays to energy |2〉, the process will emit a photon with the mode ES1. Then the photon of E1 mode is excited from energy |2〉 to |3〉, from where it decays to the ground state, emitting a photon with the mode ES2 (ω2, k2, Ω2). The second strong beam E3 (ω3, k3, Ω3) is also excited from the ground state to energy |3〉, from where it decays to the energy |2〉, emitting a photon of the mode ES3.
Fig. 1. The scheme of the three modes is based on the energy-level-cascade four-wave mixing process. (a) The ELC-FWM processes in a single rubidium vapor. (b) Energy level diagram of the whole ELC-FWM system with two subsystems of FWM1 and FWM2. (c) The spatial structures of the two pump beams E1 and E2 with three output modes ES1, ES2 and ES3. (d) The diagram of the phase-matching conditions for the parametric amplified FWM1 and FWM2.
The whole system consists of dual-pump fields process by a cascade of energy levels, and two FWM processes will occur simultaneously (see Fig. 1(b)). Figure 1(c) that shows the spatial distribution amongst the correlated FWM output modes. In contrast to the single FWM process which annihilates two pump E1 photons to generate a single photon ES1 and an idler photon ES2, the new photon of ES3 will be generated by the second cascaded FWM interaction between pump E1 and E2. The two FWM processes are considered to be a whole process, and it also satisfies the phase-matching conditions ks1+ ks2 = 2k1 and ks2+ ks3= k1 +k3 (see Fig. 1(d)) with a cascade of energy levels in a single Rb vapor.
In the entire process, the interaction Hamiltonian of the entire system can be expressed as ${H_1} = i\hbar {\kappa _1}{\hat{a}^\dagger}_1{\hat{a}^\dagger}_2 + h.c$ and ${H_2} = i\hbar {\kappa _2}{\hat{a}^\dagger}_2{\hat{a}^\dagger}_3 + h.c$, where ${\hat{a}^\dagger}_i$(i = 1,2,3) are the boson creation operators of the three generated modes and the commutation relations can be expressed as $[{{{\hat{a}}_i},\hat{a}_j^\dagger} ]= {\delta _{ij}}$. Therefore, two pump parameters ${\kappa _1} = |{ - i{\mu_0}{\varpi_1}\chi_1^{(3)}E_1^2/2{k_s}} |$ and ${\kappa _2} = |{ - i{\mu_0}{\varpi_2}\chi_2^{(3)}{E_1}{E_3}/2{k_s}} |$ depend on the central frequency of Stokes and anti-Stokes signals ${\varpi _i}$(i = 1,2) as well as nonlinear susceptibility tensor $\chi _i^{(3)}$. Here, ${{\mathbf E}_i}$ is the pump optical field amplitude, and µ0 is the diploe matrix element of an atomic transition.
In the Heisenberg picture, the associated boson creation (annihilation) operators obey the time-evolution motion equation within the dipole approximation. The dynamic equations have the form
After the algebraic operation, the output of cascade FWM can be written as follows:
3. Non-Hermitian exceptional points under multi-dressing
In this section, let us consider the effect of the non-Hermitian system and the EPs under the control of the single- and doubly-dressing fields. Due to the third-order density matrix element $\rho _i^{(3)}$ being proportional to the nonlinear polarization χi(3), the study of its resonance modes can simplify the study evolution of exceptional points in the non-Hermitian system under the multi-dressing coherent control.
With the dipole and rotating wave approximation, the Hamiltonian of atomic interaction in the cascade energy level PA-FWM1 process can be expressed
3.1. Single dressing with non-Hermitian EP
With the strong coherent field E1 acting on the energy level |1〉, the energy level will be split into two dressed states. The density matrix elements of different orders can be modified by the dressing field. For convenience, without losing generality, here we consider the modified second-order density matrix element dressed by the pump field E1, as an example, to discuss atomic coherence effect within the coherent channel of PA-FWM1.
As shown in Fig. 2(a), with the dressing field E1, the dressed states|Ω1 + 〉 and |Ω1-〉 by splitting the ground energy |1〉 can be obtained, and the two eigenvalues define the two positions of the quasi-energy levels. Under the dressing effect of the E1 field, the density matrix element in Eq. (6a) can be expressed which can be written as
Fig. 2. Energy level diagram and EPs in the ELC-FWM processes under the single dressing field. (a) two dressed states splitting diagram with single dressing effect. The blue bidirectional dotted arrow indicates the dressing effect caused by the E1 field. (b) and (e) The eigenvalue surfaces of real and imaginary parts in the multi-coherent channels with the parameters Ω1/Γ31 and frequency detuning Δ1/Γ31. (c) and (f) The eigenvalues of the real and imaginary parts. The exceptional points occur at Ω1/Γ31 = 0.5 with Δ1 = 0. (d) and (g) The level repulsion of the eigenvalues in the real and imaginary parts with Δ1 = 0.15Γ31.
As the dressed states can be considered as the two-level system coupled with an electric field, the complex eigenvalues of the dressing system with the density matrix can be obtained
Figure 2(b), and 2(e) illustrate the eigenvalues surfaces of the real parts and imaginary parts around the exceptional point singularity in the parameter space. When the detuning Δ1 = 0, the two complex eigenvalues show repulsion at the EP with Ω1=|Γ21-Γ23|/2 in Fig. 2(c) and 2(f). Also, with no-zero valued detuning, the corresponding eigenvalues in the real part and imaginary part will occur level repulsion in Fig. 2(d) and 2(g).
As such, the ELC-FWM systems exhibit all-optical controllable non-Hermitian characteristics with the dressing effect. As seen in Fig. 2(c) and 2(f), when we set Ω1/Γ31 < 0.5, the eigenvalues δ± of the real parts coalesce but the imaginary parts occur degenerate. In this case, the coupled system takes the same frequency mode with different dissipative rates, which, as a result, can induce a single absorption peak from the interference between the two coherent channels. When Ω1/Γ31 > 0.5, the real parts repel each other, but the imaginary part merges at the EP. With the increase of the strength of the dressing field, the interference between the two coherent channels become gradually disappears and it corresponds to different resonance frequencies and degenerate dissipation near the EP. Furthermore, such non-Hermiticity is involved within the parametric FWM processes, so it is directly related to bi-photons generation. The corresponding quantum entanglement property will be analyzed in Sec. 4. Note that, for the dressing effects and non-Hermitian control, in practice, it does not have to add extra dressing fields. One can directly modulate the coupling strength via changing the detuning and the power of the pump fields.
3.2. Multi-dressing with non-Hermitian higher-order EP
When the two strong pump fields exhibit the dressing effect on the energy level |1 > and |3>, the energy level split can be further induced [61], which can support higher-order EPs by tuning the system parameters. Due to the influence of the different split energy levels, the doubly dressing effect shows the three different structures in the ELC-FWM process, i.e., parallel-cascade scheme, sequential-cascade scheme, and nested-cascade scheme. These three structures will generate different effects for atomic coherence and parametric gain, as shown in Fig. 3.
Fig. 3. The doubly dressed states energy level diagram with two strong coherent fields E1 and E2. (a) The doubly dressed FWM of the parallel-cascade scheme with effecting on different density matrix element (b) The doubly dressing FWM of sequential-cascade scheme same as (a), but dressed same density matrix element. (c) The doubly dressed FWM of the nested-cascade scheme. (d) The entanglement scheme from multi-coherent channels of the parallel-cascade scheme (left), the other two schemes (right). (a)-(c) The blue and green bidirectional dotted arrow indicates the dressing effect by the fields E1 and E2.
According to the dressing effect acting on the different energy levels, it corresponds to the different density matrix elements in the ECL-FWM system. In the parallel-cascade scheme, the energy level |1〉 and |3〉 are dressed by two strong coherent fields independently in Fig. 3(a), while the sequential-cascade and nested-cascade scheme structural fields modify the level |3〉 at the same time, but with a different order, as seen in Fig. 3(b) and 3(c). So different dressing effect can be expressed by the corresponding density matrix element. Moreover, the multi-coherent channels via the doubly dressing control can further increase the scale of the produced states. As shown in Fig. 3(d), the parallel dressing regime and the other two dressing structures correspond to the four- and three-mode tripartite quantum states, respectively. The transitions of these particles between different splitting energy levels exhibit frequency modes expansion under the multi-dressing effect.
Firstly, we analyze the parallel-cascade doubly dressing (${\cal P}{\cal C}{\cal D}{\cal D}$) scheme which is the two strong coherent fields that modify the different density matrix elements and different energy levels. As shown in Fig. 3(a), the energy |1〉 is split into the two dressed states |Ω1 + 〉and |Ω1-〉 by the coherent field E1. Similarly, the energy |3〉 is split into the two dressed states |Ω3 + 〉 and |Ω3-〉 by the field E3, respectively. Thus, the corresponding output modes are regulated by the different dressing fields. Based on Eq. (7a), the third-order density matrix element of the anti-Stocks optical field ES2 with doubly dressing effect can be expressed by the perturbation chain $\rho _{11}^{(0 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{31}^{(1 )}\buildrel {{\omega _{S1}}} \over \longrightarrow \rho _{2D\Omega 1 \pm }^{(2 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{3D\Omega 3 \pm ({S2} )}^{(3 )}$, where the subscript of $\rho _{2D\Omega 1 \pm }^{(2 )}$ and $\rho _{3D\Omega 3 \pm ({S2} )}^{(3 )}$ indicates the two fields E1 and E3 dress the energy |1〉 and |3〉, respectively.
The third-order density matrix element of single dressing in Eq. (7) are modified by the two dressing fields which can be rewritten
Similarly, the complex eigenvalues with the dressing field E3 also can be obtained
Fig. 4. Exceptional points in the ELC-FWM processes under the doubly dressing ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme. (a) and (c) The eigenvalue surfaces of real and imaginary parts in the multi-coherent channels with the parameters Ω1/Γ31 and Ω3/Γ31; (b) and (d) the real parts and imaginary parts depending on the two parameters Ω1/Γ31 and Ω3/Γ31 with Ω1 = 0.5Γ31 and Ω3 = 0.5Γ31.
Secondly, the sequential-cascade doubly dressing (${\cal S}{\cal S}{\cal D}{\cal D}$) scheme has a different dressing level structure compared to the ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme. Here the energy level |3〉 is split by two strong coherent fields E1 and E3, and the corresponding second-order density matrix element is modified by the two dressing fields simultaneously. For the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme, the third-order density matrix element can be obtained by the perturbation chain of the ELC-FWM process $\rho _{11}^{(0 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{31}^{(1 )}\buildrel {{\omega _{S1}}} \over \longrightarrow \rho _{2D\Omega 1 \pm \Omega 3 \pm }^{(2 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{3({S2} )}^{(3 )}$, where the subscript $\rho _{2D\Omega 1 \pm \Omega 3 \pm }^{(2 )}$ indicates the dressing fields E1 and E3 dress the energy |3 > simultaneously. From the perturbation chain, the fields E1 and E3 drive the particle transition |2〉→|3〉→|2〉 and |1〉→|3〉→|1〉, respectively. According to Eq. (7), the third-order density matrix element can be rewritten as,
Via maximizing the third-order density matrix element, the three complex eigenvalues δ1, δ2, and δ3 can be obtained by analytically solving a cubic equation (more details seen in Supplement 1). Considering the higher-order EPs conditions, the three eigenvalues degenerate with,
Here, the higher-order EPs can be achieved when Eq. (13) takes positive real values (for dressing effect, Ω1 and Ω3 are taken as the strength of the Rabi frequency), and the corresponding parameters satisfy the constraint ${\Delta ^{\prime}_1} = {\Delta _3} = 0$ and ${\Gamma _{23}} \ne {\Gamma _{31}}$, which indicates the three eigenvalues can coalesce one value.
When changing the Rabi frequency of the dressing fields, the eigenvalue surfaces of real and imaginary parts are mutually relative cross together as shown in Fig. 5(a) and 5(e). In Fig. 5(b)-5(h), without considering the fields detuning, the real parts and imaginary parts of the three eigenvalues are plotted as a function with Ω1 = 0.1Γ31, 0.22Γ31, 0.35Γ31 in the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme. As seen in Fig. 5, the three eigenvalues δ1, δ2 and δ3 are represented by the red solid line, the blue and black dashed line, respectively. For Ω1 = 0.1Γ31, the two eigenvalues δ1 and δ2 coalesce one value at the Ω3 = 0.3Γ31, and the two eigenvalues δ1 and δ3 coalesce one value at the Ω3 = 0.46Γ31. In this case, the non-Hermitian system corresponds to two EPs. With increasing the Ω1 = 0.22Γ31, three eigenvalues δ1, δ2 and δ3 coalesce one value at the Ω3 = 0.55Γ31, corresponding to the higher-order EP in Fig. 5(c) and Fig. 5(g). Here, we find the two EPs can transform into the higher-order EP in the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme. If we continue to increase the Rabi frequency of the dressing field E1 to Ω1 = 0.35Γ31, it can be observed that the original degenerate EP has a level repulsion, and the eigenvalues of the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme are no longer degenerate, seen in Fig. 5(d) and Fig. 5(h). It is worth noting that, different from those in the ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme, here the two dressing fields co-modify the same item in the density matrix element in the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme. This makes the system appear higher-order non-Hermitian EPs. Therefore, higher-order EPs can be understood as a result of the coupling between multi-field and the atoms, while ordinary EPs are from the coupling between a single field and the atoms, similar to single dressing and ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme.
Fig. 5. Exceptional points in the ELC-FWM processes under the doubly dressing ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme. (a) and (e) The eigenvalue surfaces of the real and imaginary parts with the two dressing fields parameters Ω1/Γ31 and Ω3/Γ31. (b) and (f) The real and imaginary part of the eigenvalues with Ω1 = 0.1Γ31. (c) and (g) Ω1 = 0.22Γ31. (d) and (h) Ω1 = 0.35Γ31.
Thirdly, a different coupling between multi-field and the atoms, i.e., the nested-cascade doubly dressing (${\cal N}{\cal C}{\cal D}{\cal D}$) scheme, is considered. However, here the two dressing fields have hierarchy structural characteristics, i.e., the strong coherent field E1 is the primary dressing field and E3 is the secondary dressing field. If the coherent field E1 is equal to zero, there is no dressing effect regardless of the presence of coherent field E3. The third-order density matrix element can be obtained by the perturbation chain $\rho _{11}^{(0 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{31}^{(1 )}\buildrel {{\omega _{S1}}} \over \longrightarrow \rho _{2D\Omega 1 \pm{/}\Omega 3 \pm }^{(2 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{3({S2} )}^{(3 )}$, where the subscript $\rho _{2D\Omega 1 \pm{/}\Omega 3 \pm }^{(2 )}$ indicates the dressing field E1 and E3 acting on the same energy |3 > . So, also the atomic coherence effect is influenced in a hierarchy order by the two dressing fields E1 and E3. The third-order density matrix element can be rewritten as
The three complex eigenvalues δ1, δ2 and δ3 can also be obtained by solving the cubic equation. (more details seen in Supplement 1). The higher-order EP conditions, with the three degenerate eigenvalues, are obtained as
Based on Eq. (16), the relevant parameters become ${\Delta ^{\prime}_1} ={-} {\Delta _3}({\Delta _3} \ne 0)$ and ${\Gamma _{21}} = {\Gamma _{31}} - 3\sqrt 3 {\Delta _3}$.
In Fig. 6(a) and 6(e), we plot the eigenvalue surfaces of real and imaginary parts in the multi-coherent channels with the two dressing fields parameters Ω1/Γ31 and Ω3/Γ31. Considering the detuning of dressing fields, the real and imaginary parts of eigenvalues can be plotted with the different parameters Ω1 = 0.4Γ31, 0.5Γ31 and 0.6Γ31. The curves of the red solid line, blue and black dashed line represent the three complex eigenvalues of δ1, δ2 and δ3 in the ${\cal N}{\cal C}{\cal D}{\cal D}$ scheme. For Ω1 = 0.4Γ31, the three eigenvalues occur level repulsion in the real and imaginary parts, seen in Fig. 6(b) and Fig. 6(f). At the critical condition Ω1 = 0.5Γ31, the three complex eigenvalues coalesce one value at the Ω3 = 0.5Γ31, corresponding to the higher-order EP in Fig. 6(c) and Fig. 6(g). With increasing the Ω1 = 0.6Γ31, the three eigenvalues occur level repulsion again in Fig. 6(d) and Fig. 6(h). It is interesting to see, in contrast to the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme, when continuously altering the parameters, only the higher-order EP can exist, but the merger of the two eigenvalues, corresponding to ordinary EPs, does not. This exactly indicates the high-order dressing property in the ${\cal N}{\cal C}{\cal D}{\cal D}$ scheme. As here the two dressing fields E1 and E3 co-modify the atoms in a hierarchy order, and it cannot be reduced the ordinary EPs case, which is quite different from all the previous coupling regime. And, thus, we can think this corresponds to the genuine higher-order EP.
Fig. 6. Exceptional points in the ELC-FWM processes under the doubly dressing ${\cal N}{\cal C}{\cal D}{\cal D}$ scheme. (a) and (e) The eigenvalue surfaces of real and imaginary parts with the two dressing fields parameters Ω1/Γ31 and Ω3/Γ31. (b) and (f) The real and imaginary part of the eigenvalues with Ω1 = 0.4Γ31. (c) and (g) Ω1 = 0.5Γ31, (d) and (h) Ω1 = 0.6Γ31
4. Coherent dressing control of entanglement
The presence of multiple coexisting FWM processes within the ELC process can be attributed to an increase in coherent channels via the dressing effect. This dressing effect leads to the emergence of multiple resonance peaks and the subsequent appearance of multiple frequency modes in the original output mode ES1, ES2, and ES3. Consequently, it is possible to modify the nonlinear susceptibility of the ELC-FWM process to exert control over the multipartite entanglement within the non-Hermitian system. As shown in Fig. 7, the normalized real parts of the third-order density matrix are plotted to reflect the influence of the dressing effect on the field ES2. Figure 7(a) illustrates the third-order density matrix influenced by the single dressing effect and depicts the presence of two resonance peaks as the deviation δ is varied from -40 MHz to 40 MHz. The resonance positions of the two coherent channels are (I) ${\delta _{RE( - )}} = ({\Delta _1} - \sqrt {\Delta _1^2 + 4{\Gamma _{21}}{\Gamma _{23}} + 4\Omega _1^2} )/2$ and (II)${\delta _{RE( + )}} = ({\Delta _1} + \sqrt {\Delta _1^2 + 4{\Gamma _{21}}{\Gamma _{23}} + 4\Omega _1^2} )/2$.
Fig. 7. The normalized the real parts of the third-order density matrix intensity under the single and doubly dressing effect. (a) The condition of the existing single-dressing effect with two resonance peaks (I) and (II). (b) The condition of the doubly-dressing effect of the ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme with four resonance peaks (I), (II), (III), and (IV). (c) The condition of the doubly-dressing effect of the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme and (d) ${\cal N}{\cal C}{\cal D}{\cal D}$ scheme with three resonance peaks (I), (II) and (III).
Figure 7(b) represents four resonance peaks (I), (II), (III) and (IV) of the third-order density matrix under the doubly-dressing effect in the ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme. Additionally, Fig. 7(c) and 7(d) exhibit the three resonance peaks (I), (II) and (III) under the doubly-dressing effect in the ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme and the ${\cal N}{\cal C}{\cal D}{\cal D}$ scheme, respectively. It is worth noting that analogous outcomes to those observed in the real part can also be identified when examining the imaginary part of the eigenvalues across multiple coherent channels. Attributed to the multi-dressing effect, which causes additional splitting of energy levels, it generates multiple coexisting FWM channels within the ELC-FWM scheme. Consequently, the system exhibits multiple frequency modes for each output signal. These additional frequency modes not only construct multimode entanglement but also enhance the system's capacity for quantum information transmission [62].
We use the PPT criterion to describe the entanglement properties of the doubly-dressing effect and theoretically investigate the entanglement of the tripartite in the ELC-FWM system. Here, we consider the multipartite entanglement of two partitions in the type of ${\hat{a}_i}|{\hat{a}_j}$ and ${\hat{a}_i}|({\hat{a}_j},{\hat{a}_k})$(i, j, k = 1, 2, 3). We set gain G2 = 1.3, and vary the deviation δ to trace out the entanglement of the partition of three output modes.
In Fig. 8, the PPT values below 0 indicate the existence of entanglement among the output modes of the ELC-FWM process. Each of the rectangle pink regions in Fig. 8 corresponds to the coherent channels in Fig. 7, where the parametric gain is maximized. According to Eq. (2), the dressing effect can directly modulate the parametric gain of the ELC-FWM process. Consequently, the entanglement characteristics of the FWM system can be actively controlled in the preparation of entangled sources. Figure 8(a) and 8(b) illustrate the modulation of the bipartite and tripartite entanglement under the single dressing effect with corresponding coherent channels. Similarly, Figs. 8(c)-8(h) demonstrates the variation trend of the multipartite entanglement under the doubly-dressing effect. The absence of entanglement between the type of ${\hat{a}_1}|{\hat{a}_3}$ is not given in the three doubly-dressing schemes.
Fig. 8. Entanglement criteria in the single and doubly-dressing effect with three output modes. (a) and (b) The PPT value of the single dressing. (c) and (d) The ${\cal P}{\cal C}{\cal D}{\cal D}$ scheme under the doubly-dressing effect. (e) and (f) The ${\cal S}{\cal S}{\cal D}{\cal D}$ scheme. (g) and (h) The ${\cal N}{\cal C}{\cal D}{\cal D}$ scheme. The gray region shows the separability state, and below it is the entanglement state. The rectangle pink region (I), (II), (III), and (IV) indicates the change in the PPT criterion for each coherent channel.
5. Generalized N-dressing effect
We illustrate the N-dressing fields interact with the cascade energy level of the atomic system in Fig. 9(a). Each of the dressing fields Edn couples the ground state |1〉 and excited state |3〉. Under the approximation condition, the N-dressing PA-FWM process can be considered as a superposition of the multiple dressing effects. As a result, the generalized Hamiltonian describing the atomic interaction in the N-dressing cascade energy level system can be expressed
Fig. 9. The energy level diagram and theoretically calculated normalized the real parts of the third-order density matrix with the multi-dressing fields in the ELC-FWM processes. (a) the energy level diagram of interaction with the multilevel atomic system under the N-dressing fields. (b)-(e) the multiple coherent channels with a different number of multi-dressing fields, i.e., the number of dressing fields is N = 3, N = 4, N = 5, and N = 6.
The subscripts $2D\sum\limits_{n = 1}^N {\Omega n \pm }$ denote the second-order density matrix element $\rho _{21}^{(2)}$ dressed by the N-dressing fields. Multiple coherent fields acting on the energy level |3〉 simultaneously constitute the cascade multi-dressing structure. Subsequently, the third-order density matrix element can be expressed
Under the interaction of the multi-dressing effect, the atomic coherence induces the multi-channel nonlinear optical processes in the multilevel system. As a result, the theoretically calculated normalized real parts of the third-order density matrix show the rich multiple resonance peaks structure in Fig. 9(b)-9(e). Here, we consider the cases of the multi-dressing fields N = 3, N = 4, N = 5, and N = 6. Furthermore, the coherent channels can be extended by simply adding the number of control fields without altering the overall quantum system structure.
In Fig. 10, the triple partitions of the entanglement criterion of the ECL-FWM process can be further optimized through the utilization of multiple dressing fields (ie., N = 3, 4, 5, 6). The dotted line indicates the position corresponding to the resonance peaks as in Fig. 9. Therefore, the multipartite entanglement of the ELC-FWM process can be effectively modulated by increasing the number of dressing fields. The atomic coherence control using multi-dressing fields can further extend the entanglement modes, which provides an effective and scalable method to improve the quantum information capacity in non-Hermitian system.
Fig. 10. Entanglement criterion in the multi-dressing coherent control with three output modes (a) the positivity under transposition (PPT) criterion under the number of N = 3 dressing fields. (b)-(d) N = 4,5,6 dressing fields. The dotted lines indicate the position of the multichannel regulation.
6. Conclusion
In summary, our study demonstrates the capability of the ELC-FWM process to achieve versatile EPs and higher-order EPs, as well as to establish scalable and all-optically controlled multimode entanglement, based on the multi-dressing effects of atomic coherence. Remarkably, these achievements are obtained without the need for modifying the experimental optics architecture or introducing artificial photonic structures in the non-Hermitian systems. Furthermore, we employ the PPT criterion to characterize the multipartite entanglement under single- and doubly-dressing as well the generalized N-dressing fields. The results reveal that the properties of multipartite entanglement can be actively, flexibly, and continuously controlled around the EPs. This research opens up new avenues for exploring non-Hermitian physics through the integration of light-matter coupling, nonlinear optics, and entanglement generation, thereby paving the way for non-Hermiticity-enhanced quantum information processing.
Funding
National Natural Science Foundation of China (11904279, 12074303, 12074306, 12174302, 61975159, 62022066); Natural Science Foundation of Shandong Province (ZR2022LLZ004); Shaanxi Key Science and Technology Innovation Team Project (2021TD-56).
Disclosures
The authors declare no competing interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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