2.1. Single-Dressing Effect
Figure 1a illustrates a streamlined experimental arrangement for the process under investigation, wherein the spontaneous six-wave mixing (SSWM) phenomenon takes place within a hot
85Rb atomic vapor. The rubidium vapor is contained inside a slender, elongated cylindrical cell of length. The corresponding atomic energy conservation is shown in
Figure 1b. In this setup, the atomic ensemble predominantly occupies the ground state |1>.
A pump beam
(frequency
, wave vector
, Rabi frequency
, wavelength 780 nm) with detuning
is employed to drive the atomic transition |1> to |3>. The detuning
refers to the deviation or difference between the frequency of the driving field (the laser frequency) and the intrinsic resonance frequency of the system resonant transition frequency). The detuning was defined as
,
is the resonant transition frequency, and
is the laser frequency of
.
(
,
,
, 780 nm, |2> to |4>) with detuning
and
(
,
,
, 776 nm, |1> to |4>) with detuning
are just like
. According to the principle of conservation of momentum,
. Regarding the energy level transition process, it can be represented in the form of a perturbation chain as follows:
The SSWM process satisfied the energy conservation principle
. Here,
(
= 1, 2, 3) denotes the frequency of the emitted photons. In the process illustrated in
Figure 1a, using single dressing as an example,
Figure 1c shows that a single channel emerges when the dressing Rabi frequency is dominant (represented by a dashed line), while two channels arise when the imaginary component dominates (represented by a wavy line). Thus, depending on the varying intensities of the dressing field and the dephasing rate, the number of coherent channels in the real and imaginary components manifests on either side of the exceptional point (EP). The solid lines in
Figure 1c,d indicate the pumping lasers, the dash line indicates the dressing Rabi frequency-induced photon channel, and the wavy line represents the dephasing rate-induced photon channel.
Figure 1c represents the coherent channels mainly induced by the strong dressing Rabi frequency Gi. The energy level splitting occurring under the strong dressing condition of Gi > Γij is denoted as the in-phase constructive dressing effect. The dash lines correspond to two splitting coherent channels induced by dressing Rabi frequency and the wavy line in the middle represents the decay in multi-photon coincidence counts resulting from the imaginary part of the eigenvalue. The diagrams in
Figure 1d exhibit the energy level splitting generated under the dominating dephasing rate with weak-dressing Rabi frequency, which is considered as the out-of-phase destructive dressing effect. And two splitting coherent channels can also be produced under the weak dressing condition of Gi < Γij. For this region, the coherent channels induced by dephasing rates are symbolized by the wavy lines while the dash line represents the decay produced by the real part of eigenvalue. As for the case of Gi = Γij, without further energy level splitting, only one unsplit photon channel can be produced. And the energy level distribution would exhibit high similarity to the initial configuration of the bare atom presented in
Figure 1c,d. Such a critical transition point between the strong dressing and weak dressing conditions can be denoted as the exceptional point of the atomic, natural, non-Hermitian system.
The
is to obtain the perturbation chain
.
Here,
,
,
,
,
, when considering the case of the single-dressing effect of
only, the formula can be written as follows:
. The expression for
can be simplified as follows:
. By tuning the ratio of the Rabi frequency
and the out-of-phase rate
of the corresponding dressing field, one can effectively manipulate the sign of the resulting expression. The solution is the eigenvalue
. When obtaining the pure real part without the imaginary part, ignore b
2 to solve a and ignore the imaginary part to solve the value of b. The solution of
are
, as well as the corresponding linewidth
. Therefore, the single-dressing effect can generate second-order EP. Similarly, the solution of
are
, as well as the corresponding linewidth
. It can be denoted by the second-order dressing-coupled Hamiltonian matrix (S3) and (S4) in the
Supplement material, respectively.
The main parameters are as follows: .
The photon random coincidence counting rate (Rcc) refers to the rate at which two or more detectors simultaneously register photon detection events within a defined temporal window. This metric is fundamental in quantum optics experiments for characterizing photon correlations, entanglement, and non-classical light properties. By analyzing coincidence counts, one can infer the degree of temporal or spatial correlation between photons, which is crucial for verifying quantum coherence and demonstrating phenomena such as photon bunching or antibunching. The photon coincidence counting rate thus serves as a key observation in the study of quantum states of light and their interactions. Tri-photons can be represented by the tri-photon amplitude
and the averaged tri-photon coincidence counting rate (Rcc), where
can be defined as follows:
The temporal correlation of the generated photons was examined through Rcc. Rcc is defined as follows:
where
is the third-order intensity correlation function of the tri-photon, which can be written as follows:
. Therefore, the temporal correlation characteristics can be studied via optical responses. We use the residue theorem to calculate
. The specific form is as follows:
.
is the derivative of the function
at
, reflecting the relationship between the higher-order derivative and the loop integral.
The eigenvalue splits when dressed by laser field . The real and imaginary parts have different phase transition points and opposite trends. The calculation result is obtained from dressing-coupled matrix (S1) and Equation (2). When the ratio of the dressing Rabi frequency to the corresponding energy level dephasing rate is considered on the right side of the exceptional point (EP), the root of Equation (2) becomes positive. In this case, the real part exhibits a splitting behavior dominated by , which leads to in-phase constructive dressing. Furthermore, a larger value of enhances the splitting of the real component.
Conversely, on the left side of the EP, when the ratio
is taken into account and the root of Equation (2) is negative, the imaginary part begins to split. This regime is dominated by the dephasing rate
(representing loss), resulting in out-of-phase destructive dressing. As shown in
Figure 2c, when analyzing the real part, the condition
causes the two branches of
to undergo simultaneous splitting. In contrast,
Figure 1d shows that when the imaginary component of
is dominant and
, phase transitions are more likely to occur. The roots are given by the following:
. As shown in
Figure 2d, the splitting effect caused by
is notable even at low field strengths, leading to a phase transition in the imaginary part of
, which can be expressed as follows:
. Additionally, in
Figure 2d, when the real part of
dominates and
, the observed four peaks result from phase splitting in the real component.
Due to the splitting of the atomic energy levels, the associated transition processes generate multiple coherent channels. The splitting is symmetrical and corresponds to a second-order process. The numerical results supporting this behavior are compiled in
Tables S1 and S2 in Supplementary Materials. When
, the dephasing rate dominates, the states with unsplit eigenvalues correspond to out-of-phase and dressing destructive quantization, and the noise is significant. It can be assumed that
and
. The dephasing rate and noise are challenging to control and tend to exhibit stochastic behavior, often resulting in out-of-phase interactions and destructive interference. In general, the case of eigenvalue’s real part splitting corresponds to the conditioning and detection of continuous signals, while eigenvalue’s imaginary part splitting corresponds to the conditioning and detection of impulse signals.
In the regime where the dressing Rabi frequency
significantly exceeds the dephasing rate
, coherent interaction dominates over dissipation. This leads to the emergence of split eigenvalues in the system’s real spectrum, associated with phase-matched, constructively interfered dressed states. Under approximately ideal conditions, the eigenvalue relation can be simplified using
, reflecting the inherent balance among the coupled energy levels. Due to the external controllability of
, in contrast to the intrinsic and fixed nature of
, the magnitude of real-part eigenvalue separation can be finely tuned by adjusting the intensity of the dressing field. This tunability allows for the creation of strongly coherent superposition states with high phase stability. As demonstrated in
Figure 2, once the real-part eigenvalues undergo splitting, they become distinct and non-degenerate. These non-degenerate solutions indicate that the system’s dressed states conform to an Abelian algebraic structure, suggesting commutativity in the adiabatic evolution path. Such behavior is a hallmark of spontaneous symmetry breaking in non-Hermitian systems and is often accompanied by nonlocal response features in the system’s spectral profile. Conversely, for the same parameter regime
, the imaginary components of the eigenvalues remain unchanged and degenerate. This degeneracy in the dissipative channel implies that no symmetry breaking occurs for the imaginary part, which continues to exhibit long-range coherence and nonlocal phase behavior. Moreover, because degeneracy is often a signature of non-Abelian quantum statistics, the imaginary-part eigenstates—despite being unaffected by field-induced splitting—can be associated with non-Abelian characteristics. These states are sensitive to the order of operations in cyclic evolution and may serve as a platform for realizing path-dependent geometric phases in open quantum systems.
In general, the real component of the eigenvalues corresponds to in-phase, constructive quantum interference, whereas the imaginary component tends to exhibit out-of-phase destructive behavior. When the dressing strength satisfies , the real part displays localized characteristics, is spectrally non-degenerate, and follows Abelian dynamics with reduced symmetry. In contrast, the imaginary part remains degenerate, exhibits nonlocal features, and adheres to non-Abelian statistics, indicative of higher underlying symmetry.
The fifth-order nonlinear susceptibility
exhibits strong dependence on the single-dressing field, and its variation follows a trend qualitatively consistent with the eigenvalue behavior of the coupled system. In
Figure 3, two sets of three-dimensional plots display the amplitude distribution of
as functions of detuning variables
and
. These figures reveal multiple resonant peaks with varying intensities aligned along the respective coordinate axes, indicating a transition in the system’s optical response with respect to the control parameters. In
Figure 3(a2,b2) and (a3,b3), distinct patterns emerge along the
- and
-axes, with three and two peaks observed, respectively. These features correspond to the eigenvalue structures on either side of the exceptional point (EP). When
exceeds the EP threshold, the observed peak positions align well with the real-part eigenvalue splitting of
and
. In contrast, for
below the EP, the imaginary-part contributions become dominant, and the system exhibits features consistent with dissipative phase transitions.
A closer inspection of
Figure 3(a3) shows that under strong dressing conditions (
), the peak distribution resembles the spectral profile associated with real-part eigenvalue splitting. Conversely, in
Figure 3(b3), when the dressing field is weakened (
), the peak pattern shifts toward that of imaginary-part splitting, reflecting the dominance of dephasing in this regime. To quantify these effects, the corresponding eigenvalues are extracted as follows: Under real-part dominance,
,
,
,
. These values match the peak positions in the transverse slices of
Figure 3(a3,b3), and their origin traces back to the eigenvalue trends displayed in
Figure 2c,d. To evaluate the degree of asymmetry between real and imaginary contributions, a symmetry breaking ratio is introduced:
This indicates that the real-part splitting is over five times greater than the imaginary-part counterpart at the same EP-relative position. Such asymmetry is rooted in the eigenvalue expressions detailed in
Table S1–S3 in Supplementary Materials. Specifically, for the real-part splitting to dominate, the conditions Δ ≫ Γ, G ≥ Γ and Δ ≪ Γ, G ≤ Γ must be satisfied, where higher-order terms in Δ and G significantly influence the symmetry structure. Conversely, imaginary-part splitting prevails due to the predominance of the dissipative term Γ. Additionally, the midpoint between the paired peak positions—often corresponding to a spectral dip—represents the destructive interference or “dark state” configuration typically found near the EP. These transitions between bright and dark states further reinforce the interpretation of EP-induced symmetry breaking, offering a direct mapping between spectral features and eigenvalue dynamics in both the coherent and dissipative domains.
In
Figure 4, we can see that EP of single-dressing field G2 is
= 0.5 In
Figure 4(a1), where
= 0.8, through the Laplace transform, the two horizontal axes are
and
, respectively. The highest point is generally located at zero.
Figure 4(a2) shows the Rcc of eigenvalue
on the right side of the EP. When the eigenvalue of
is on the right side of the EP, the fifth-order nonlinear polarizability of
has 3 peaks, resulting in 3 cycles in the figure. Multi-period coupling generates complex cosine signals. In
Figure 4(a3),
has two distinct peaks. generating an oscillation period on the image so that it is a simple cosine periodic function with a decaying signal over time, which can be seen from the following formula:
.
In
Figure 4(b1), where
= 0.8, the Rcc of
on the left side of the EP exchanges resonant position and linewidth compared with the Rcc on the right side of the EP in the simulation diagram. In
Figure 4(b2), when the eigenvalue of
is on the left side of the EP, the gain and loss are interchanged, and there are still three different peaks of the fifth-order nonlinear polarizability, generating three oscillation periods, which presents as the complex cosine signal.
Figure 4(b3) shows the Rcc simulation diagram of
on the left side of the EP. The fifth-order nonlinear polarizability related to eigenvalue
has two peaks and only produces one oscillation period.
By comparing
Figure 4(a2) with
Figure 4(b2,a3) and with
Figure 4(b3), we can find that the real part has larger oscillation period, the imaginary part obtains a smaller oscillation period; the real part attenuates more slowly, and the imaginary part attenuates more greatly. This is because when the real part is dominant, dressing field strength G leads to oscillation and relaxation time Γ leads to attenuation. When the imaginary part is dominant, Γ leads to oscillation and G leads to attenuation. The splitting effect of the real part is stronger than the imaginary part.
Based on the Rcc expression, we perform numerical simulations of the signal amplitude, temporal periodicity, and coupling behavior under varying intensities of the dressing field. As shown in
Figure 1, the exceptional point (EP) associated with the single-dressing field
is located at
= 0.5. When the field strength is increased to
= 0.5, as depicted in
Figure 4(a1), the Laplace transform is applied to shift the analysis from the frequency domain to the time domain. The resulting three-dimensional plot uses
and
as its horizontal axes, with the global maximum typically appearing near the origin.
In
Figure 4(a2), corresponding to the right-hand side of the EP, the Rcc associated with eigenvalue
exhibits three distinct oscillation periods, reflecting the three-peak structure of its fifth-order nonlinear polarizability
. These multiple peaks result in a complex cosine-modulated signal profile, indicative of multi-period coupling dynamics. Meanwhile,
Figure 4(a3) presents the Rcc of eigenvalue
, which shows only two sharp peaks, producing a relatively simple cosine oscillation with clear exponential decay. The oscillation frequency can be approximated by the relation
, where
and
are the frequencies corresponding to the two peaks.
In contrast,
Figure 4(b1) shows the Rcc of
when
= 0.2, placing the system on the left-hand side of the EP. Compared with the configuration on the right-hand side of the EP, the resonance position and linewidth are interchanged due to the reversal of gain and loss contributions. In
Figure 4(b2), the nonlinear polarizability spectrum of δ1\delta_1δ1 continues to display three peaks, resulting in three periodic oscillations in time. The signal maintains a complex cosine structure even under gain–loss inversion. However, in
Figure 4(b3), the Rcc for
shows only two peaks in its nonlinear response and generates a single oscillation period, with a more rapid decay compared to its counterpart in (a3).
By comparing the Rcc in
Figure 4(a2,b2,a3,b3), several trends emerge. Signals associated with dominant real parts demonstrate longer oscillation periods and slower attenuation, while those governed by imaginary parts exhibit shorter periods and faster decay. This behavior arises because, under real-part dominance, the dressing field strength G determines the oscillation frequency, whereas the relaxation rate Γ controls attenuation. Conversely, when the imaginary component is dominant, Γ governs the oscillatory behavior and G contributes to the signal’s decay. These results illustrate that the eigenvalue splitting in the real part is more pronounced than that in the imaginary part, leading to more evident oscillatory features in the time domain.
The quantum information capacity can be characterized by the expression nm, where n denotes the number of entangled photons and mmm represents the number of coherent channels. In the context of this study, with three entangled photons considered, the information capacity increases with the number of available coherent channels. Introducing additional dressing fields enhances the number of these channels, thereby enabling exponential growth in the quantum information capacity.
2.2. Parallel Double-Dressing Effect
Considering the multi-dressing effect, the two types of multi-dressing effect that we are studying here are the parallel double-dressing effect and parallel triple-dressing effect. Firstly, the focus is on parallel double-dressing effect. The expression
for q can be written as follows:
In analyzing the influence of the parallel double-dressing configuration, it is important to emphasize that its mathematical structure is formally equivalent to the superposition of two independent single-dressing effects. Following a similar approach to the single-dressing analysis, the complete denominator of Equation (5) is set to zero to solve for the system’s eigenvalues. The corresponding results are systematically presented in
Tables S4 and S5 in Supplementary Materials.
Despite this formal equivalence, the physical implications of the double-dressing scenario are more intricate than those of single dressing. This increased complexity stems from the relative spatial and spectral arrangement of the two dressing fields, which introduces distinctions in their mutual priority, degree of coupling, and separability. It is worth emphasizing that in the parallel configuration, the dressing fields function independently, with no mutual coupling between their induced interactions. As a result, from a structural classification standpoint, both the single-dressing and parallel double-dressing scenarios can be interpreted within a consistent theoretical framework. Specifically, the single-dressing case corresponds to a single instance of second-order eigenvalue splitting. whereas the parallel double-dressing manifests as two isolated, second-order splitting—one from each dressing channel.
This classification is visually reinforced by
Figure 5a–d, which illustrates the splitting behavior of the eigenvalues under the G
2 and G
3 dressing fields. In these figures, the eigenvalues exhibit distinct bifurcation points and phase transitions, further supporting the interpretation that each dressing field independently modulates the system without mutual interference. In
Figure 5a, when the value of
is large, eigenvalue
is on the right-hand side of EP, and the real part of the eigenvalue
has a splitting effect.
Conversely, in
Figure 5b, when the value of
is small, the imaginary part of the eigenvalue
has a splitting effect.
.
In the case of parallel double-dressing, the number of coherent channels generated significantly exceeds that of the single-dressing scenario, reaching a total of eight. For the eigenvalue , which exhibits the highest multiplicity, four of these coherent channels are attributed to the influence of the dressing fields. When analyzing the evolution from energy level splitting governed by a strong dressing field to that driven by a weaker one, and interpreting the exceptional point (EP) as a quasi-superposition state exhibiting coherent channel-like properties, it can be approximated that the number of coherent channels remains constant throughout this transition—matching the EP count of four. This suggests that parallel dressing shares the same quantitative relationship between the number of coherent channels and the number of EPs as observed in the single-dressing scenario. This insight supports the interpretation that parallel double-dressing effectively behaves as a superposition of two independent single-dressing processes. Extending this reasoning, it is plausible to predict that a parallel multi-dressing configuration involving more dressing fields can likewise be viewed as a composite of multiple single-dressing effects.
Under the influence of the parallel double-dressing fields
and
, there are different peaks on the simulation images because of two independent dressing fields.
Figure 6(a1) shows that when the intensity of the dressing fields
=0.5 and
=0.7, the two eigenvalues are both on the right side of EP, while the fifth-order nonlinear polarizability of the eigenvalues has different number distributions and intensity distributions in the
and
directions. The peak pair corresponds to the values obtained by the eigenvalues in
Figure 5a,c, when
= 0.5 and
= 0.5. As for
Figure 6(a2), the four peaks in the figure are the result of the real parts of
undergoing energy level splitting; their values are
.
Figure 6(a3) has the two peaks of the symmetry breaking of
on the right side of the EP. Their values are
,
can be observed in
Figure 5c, corresponding to the horizontal coordinate of the peak values in
Figure 5a–c. In
Figure 6(b1), when the dressing field
is less than 0.4 and dressing field
is less than 0.3, and when
= 0.3 and
= 0.2, the imaginary part splits to produce multiple peaks. In
Figure 6(b2), the imaginary part of
dominates to produce four peaks, and the two peaks splitting from the same EPs are relatively close. Compared to
Figure 6(b3), when the dressing field
is 0.2 (at the left side of the EPs), the imaginary part’s peak bandwidth of
is different, with values of −0.91 GHz and −0.93 GHz. The symmetry breaking ratio
can be calculated by
Figure 7(b1) shows the simulation diagram of the Rcc when the imaginary part dominates and there is the exchange of resonance position and linewidth when the real part dominates. In
Figure 7(b2), when the dominant position is on the left side of the EP, the dispersion and loss are interchangeable, and the
still has four different peaks, generating six cycles, which is the complex cosine signal.
Figure 7(b3) shows the simulation diagram of the Rcc of
dominated on the left side of the EP. The fifth-order nonlinear polarizability has two peaks and only produces one period.
According to the law of conservation of energy , it can be seen that the maximum number of channels of the tri-photon with the parallel double-dressing effect is 8, and the corresponding eigenvalue is . Therefore, in this correspondence, in the simulation that conforms to the counting rate, we do not select the eigenvalue with the largest number of cycles for simulation, but according to the law of conservation of energy, we can judge that the largest number of cycles is 28.
2.3. Parallel Triple-Dressing Effect
The expression
for q can be written as follows:
where
Based on the computational results, it is evident that the presence of the dressing field leads to multi-channel splitting in both the real and imaginary components of the eigenvalue, and the splitting point is EP, which is the same for the real and imaginary parts. When the polarizability reaches the extreme value, the cubic term of the eigenvalue appears in the denominator. For this case, we obtain the corresponding eigenvalue simulation results from same method: we treat parallel triple-dressing effect as three independent single-dressing effects.
The dephasing rate Γij in this paper is similar to the gain/loss in microcavity/waveguide coupling system. In next part, by using the process of six-wave mixing, the theoretical calculations and simulations of the eigenvalues, polarization rates, and conformal count rates of the PT symmetry using the single-dressing effect, the parallel double-dressing effect, as well as the parallel triple-dressing effect have been carried out to analyze the differences and connections between them. It is also found that the maximum number of multi-channels generated increases with the increase in the number of photons and the number of applied parallel dressing fields. Furthermore, the multi-photon correlation can be expanded from the regions of the strong dressing (small dephasing rate) to weak dressing (large dephasing rate).
Compared to the parallel double-dressing fields and single-dressing effect, the parallel triple-dressing setup introduces an additional perturbation field. However, all parallel triple-dressing fields are mutually independent and do not exhibit any sequential or internal–external relationships. In contrast, when compared to the parallel double-dressing fields, two of parallel double-dressing fields simultaneously influence the same eigenvalue,
. Under the influence of these two perturbation fields,
experiences an energy level splitting. The relative magnitudes of the splitting depend on the interplay between the dressing Rabi frequencies and the dephasing rate, as reflected in the varying degree of level splitting shown in
Figure 8a,b,d,e. Additionally, the dressing field G
3 acts independently on the eigenvalue
, yielding results consistent with the single-dressing effect case. The parallel triple-dressing effect adds an additional dimension at the dressing field level, thereby enhancing the quantum information capacity. This suggests that as the number of dressing fields increases, the level splitting brings about more resonant positions, which in turn leads to an increase in the number of coherent channels as the number of dressing fields rises.
A closer inspection of
Figure 9(a3) shows that under strong dressing conditions
, the peak distribution resembles the spectral profile associated with real-part eigenvalue splitting. Conversely, in
Figure 9(b3), when the dressing field is weakened (
), the peak pattern shifts toward that of imaginary-part splitting, reflecting the dominance of dephasing in this regime. To quantify these effects, the corresponding eigenvalues are extracted as follows: under real-part dominance,
,
,
,
,
,
. These values match the peak positions in the transverse slices of
Figure 3(a3,b3), and their origin traces back to the eigenvalue trends displayed in
Figure 2c,d. To evaluate the degree of asymmetry between real and imaginary contributions, a symmetry breaking ratio is introduced as follows:
Figure 10(a2) shows the Rcc of δ
1 on the right side of the EP. When positioned to the right of the EP, the eigenvalue δ
1 exhibits four peaks, resulting in six cycles. The multi-period coupling generates a complex cosine signal. In
Figure 10(a3), δ
3 displays two distinct peaks, leading to a single cycle, which is a simple cosine periodic function, with its signal decaying over time.
Figure 10(b1) shows the Rcc of δ
1 on the left side of the EP, along with the simulated exchange of resonance positions and linewidths compared to the case on the right side of the EP. In
Figure 10(b2), when positioned to the left of the EP, dispersion and loss are exchanged. There are still four distinct peaks, resulting in six cycles, corresponding to a complex cosine signal.
Figure 10(b3) shows the Rcc simulation of δ
3 on the left side of the EP. The eigenvalue δ
3 exhibits two peaks, generating only one cycle. According to the energy conservation law, it can be deduced that the maximum number of channels in the tri-photon is 8, with the corresponding eigenvalue being δ
2. Based on this, and following the energy conservation law, the maximum number of cycles is determined to be 28.
In summary, as the number of dressing fields increases, the quantum information capacity grows exponentially. Simulations of single-dressing, parallel double-dressing, and parallel triple-dressing schemes demonstrate that the number of exceptional points (EPs) increases accordingly. This results in a doubling of energy level splitting and a corresponding rise in the number of coherent channels. Consequently, the number of peaks in the nonlinear polarization spectra increases, and this enhancement is reflected in the Rcc simulations as a further increase in the number of oscillation periods. These findings highlight the scalability of coherent control and signal complexity in multi-field dressed quantum systems.