Nowadays, computational applications are based on conventional electronic circuits, as extracted from the silicon technology. To meet the overwhelming demand for faster computers, the number of transistors confined in microprocessors has been doubling in capacity every two years,^[1] and this pace would possibly make the measures (of the number of required transistors) to be on the atomic scale in the future. However, transistors, being a classical electronic component, introduce difficulties in attaining circuit features on the atomic scale. Optical quantum computer,^[2] however, offers a way out with potential to deliver processing tasks much faster than the conventional ones. While squeezing cannot directly increase the storage capacity of the computer, within the context of phase estimation, quadrature noise squeezing remains advantageous, and the use of squeezed light would play an important role for future applications in optical computers.^[3] Also, the squeezed light remains useful in many other promising applications, such as secured optical communication systems,^[4] gravitational wave detectors,^[5] precision measurements,^[6] imaging,^[7] and quantum information.^[8] As such, in order to actualize squeezing for future applications, designing practical method of generating strong squeezing needs significant attention.

Rapid technological advancements in recent years allow strong squeezing to be generated in a number of nonlinear processes and quantum systems, e.g., parametric oscillators,^[9–12] second harmonic generation,^[13] atom–cavity couplings,^[14–16] semiconductor microcavities,^[17–19] optomechanical,^[20–22] and nonlinear directional couplers (NLDC).^[23–26] Systems having inherent fast nonlinear response, such as Kerr NLDC, however, possess the potential in generating strong squeezed light, especially when placed inside a cavity to resonantly amplify the nonlinearity. In addition, compared with other systems, Kerr NLDC is physically less complicated.^[27]

In general, squeezing can be enhanced by direct manipulation of interaction among the modes of Kerr NLDC. This may be achieved by introducing either multimode single coupler^[28–30] or multichannel single-mode configurations,^[31] where the adjacent waveguides interact only with the central channel. Another practical approach would be confining the NLDC inside an optical cavity to resonantly amplify the nonlinearity.^{[32, 33]}

Quantum statistical properties of the cavity-assisted single coupler utilizing nonlinear waveguides have been theoretically investigated from the viewpoint of energy coupling^[32] as well as the quantum dynamical effects.^[34–37] Such configuration remains essential to improve squeezing in other devices, such as the degenerate-parametric amplifier.^[37] In our previous work, we have reported the quantum properties of a multimode single coupler utilizing four-mode interaction with and without the aid of a cavity.^[38] In such model,^[38] the theory involves linear interaction between a single coherent mode in the first channel and three vacuum modes in the second channel.

The present paper aims to investigate the quantum dynamics of light modes (squeezing, in particular) propagating in a cavity-assisted multichannel NLDC, where four Kerr-like waveguides are engaged to form an eight-port coupler. The benefit of such configuration lies in achieving more versatile arrangement (of couplers) that allows easy control over the quantum correlations between modes. Moreover, the model presented here for the four-channel device gives a more adequate generalization of the traditional two-channel coupler system. In multimode waveguides, different modes propagate with different speeds, which makes it difficult to effectively estimate the generated squeezed states.^[29] On the other hand, the use of single-mode waveguides contributes better flexibility in controlling the interaction (linear and nonlinear) between the guided modes and an effective estimation of the produced squeezed states of light.

In the newly proposed multichannel system, the squeezed states of light are observed in both quadrature components compared with the previously reported two-channel system where the squeezed light was observed in one of the quadrature components only.^[26] The system under consideration appears to be a more efficient source of quantum light due to the evanescent field couplings of four interacting modes (instead of two, as in conventional two-waveguide coupler). To a significant extent, such quad-channel nonlinear coupler exhibits richer quantum effects due to the introduction of cavity, in addition to the involvement of the propagating fields in multimode interaction. Without accounting quantum fluctuations, the classical design of the system may find applications as essential elements in evanescent field optical sensing^[39] and electro-optic switches.^[40] For detailed design and other potential applications of multichannel directional coupler, see dissertation by McCosker.^[41]

The four-channel model of Kerr NLDC may be confined in an optical cavity by introducing partially reflecting mirrors (Fig. 1) at each end of the interacting waveguides;^[32–38] this serves as a linear resonator. The use of a Kerr-like dielectric medium as optical waveguides provides an effective medium for multimode optical interaction due to the intensity-dependent Kerr effect. Another closely related nonlinear model, following an identical approach, implements the four-well Bose–Hubbard model with two different tunneling rates.^[42] In Fig. 1, the guided mode in channel a₁ is prepared in the coherent state where the field E₁ is injected initially. Given that the neighboring waveguides are close enough to permit evanescent coupling, the energy from the coherent channel can be transferred to the neighboring vacuum waveguides a₂, a₃, and a₄. All the guided modes are assumed to be traveling in the same direction with constant phase and group velocities. Various physical quantities of the current system can be simulated by numerical means, as will be seen in the next section. The Hamiltonian describing the system is generally given by 1a where 1b 1c 1d 1e 1f In the system of Eqs. (1b)–(1f), and are the bosonic creation and annihilation ladder operators, respectively. Equation (1b) describes the system Hamiltonian for each individual mode inside the four waveguides without interaction; for example, in the first waveguide, the mode is described by . The coefficient ω is the operating frequency and ħ = h/2π is the reduced Planck constant. The Hamiltonian in Eq. (1c) represents the third-order Kerr effect which results from both self- and cross-action nonlinearities where g_a is the self-action nonlinear coupling constant, and g_b and g_c are the cross-action coupling constants with positive proportionality to the third-order susceptibility χ⁽³⁾. The Hamiltonian in Eq. (1d) represents the linear coupling between the modes in the adjacent waveguides due to the evanescent fields. The coefficient k is the linear coupling constant with the subscript 1 for the interaction between the mode a₁ with the other modes a₂, a₃, and a₄, and the subscript 2 for the interaction between a₂, a₃, and a₄, as shown in Fig. 1(b). The Hamiltonian in Eqs. (1e) and (1f) represents the inclusion of cavity in the system where H_pumping describes the pumping term derived from the classical driving field and H_damping describes the damping term due to the coupling with the heat bath. The cavity parameters comprised of Γ_ch1, Γ_ch2, Γ_ch3, and Γ_ch4 are the operators representing cavity losses in each channel, while , , , and are their respective conjugates. E denotes the coherent driving field.

Fig. 1.

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	Fig. 1. Schematic diagram of a cavity-assisted four-channel directional coupler. (a) Complete structure. (b) Cross section.

It should be noted that figure 1 shows one possible configuration of four-channel waveguides placed in a cavity. The current configuration exactly matches the coupling Hamiltonian in Eq. (1d), where a single waveguide (a₁) can be set at the center with three other vacuum waveguides (a₂, a₃, and a₄)in the surrounding. The coefficient k₁ refers to the interaction with the center waveguide, while k₂ gives the interaction among the surrounding ones. However, the arrangement of vacuum channels around the coherent channel is not unique. Nevertheless, regardless of which arrangement is followed, the separation distance among the vacuum waveguides remains almost identical. From Eq. (1), the master equation can be written as 2 with γ_j being the damping coefficient of modes. In Schrödinger picture, the time evolution equation for the reduced density matrix contains the heat bath coupling through the Liouvillian terms. To allow only the presence of quantum fluctuation, the coupling between laser modes and bath modes has been eliminated by assuming vanishing thermal fluctuation.

In the phase space, the operator equation is converted into a partial differential equation for the quasi probability function of the reduced density operator. In Wigner representation, the mean field is given by the expectation value of the symmetrically ordered ladder operator for single mode optical field basis . This expectation value corresponds to the average of the complex variables such that , where α₁ and are complex conjugate classical variables in the phase space. Applying a standard approach,^[43] the master equation (2) may be mapped onto its equivalent stochastic differential equation in Wigner representation for n = 4 to yield 3a 3b 3c 3d The quantum mechanical master equation (2) is converted to its equivalent classical system of Eq. (3) in two mathematical stages. First, the quantum-classical correspondence^[43] in Wigner representation is used to convert Eq. (2) to a Fokker–Planck equation. In the process of deriving the Fokker–Planck equation, higher-order derivatives above the second-order are usually truncated, and the resulting Fokker–Plank equation is a good approximation to the original problem. Second, the Ito calculus is used to convert the Fokker–Planck equation to the equivalent system of Langevin stochastic Eq. (3). The variables α_j in this case are the classical complex number representations to that of bosonic operators for modes 1 to 4 in the propagation direction τ.

In these equations, Δ_j(j = 1–4) represents detuning between the mode and cavity frequencies. The stochasticity of Eq. (3) is stipulated by the noise element η_j (τ) which, in turn, depends on the rate of photon losses γ_j. In order to obtain stable integration, equation (3) is scaled to the dimensionless form by introducing τ_s = k₁τ, Δ_js = Δ_j/k₁, γ_js = γ_j/k₁, k_1s = k₁/k₁, k_2s = k₂/k₁, g_(a,b,c)s = g_(a,b,c)/k₁, E_js = E_j/k₁, and η_js (τ) = η_j (τ)/h^1/2, where the subscript s refers to the scaled form of these coefficients and the parameter ζ is the controlling variable responsible in tuning the cross-action nonlinear coupling. An estimation of the input parameter can be found in our previous work.^[38] For realistic results, the input parameters used in the following section are adopted from the literature following Refs. [26], [28], [29], and [30] so that the scope remains consistent with the previous work in this area. The output fields may also be experimentally realized via standard detection schemes of direct photodetection and balanced homodyne detection.

Here, we have observed that the non-classical properties are sensitive to the cavity detuning. We, therefore, restrict the detuning to lower values (i.e., Δ_js = 1), high (i.e., Δ_js = 10), and the resonance case (i.e., Δ_js = 0). The classical driving force of E_1s = 1 is initialized in the first channel, and E_2s = E_3s = E_4s = 0 for the other channels. The first channel is sampled in the coherent state with the eigenvalue α₁ = 1 and the other in the vacuum states with α₂ = α₃ = α₄ = 0. The linear coupling coefficients are approximated at k_1s = 1, k_2s = 0.5, and the nonlinear self-action and cross-action coefficients are fixed at g_as = 0.01 and g_bs = g_cs = 2g_as with small damping γ_js = 0.001 and ζ = 1. The evanescent damping of coupling may lead to amplitude reduction of power transfer. Provided that the surface of waveguides is of high quality to prevent substantial energy loss, the self- and cross-action nonlinear coefficients may assume comparable spectral coupling.^[25]

3.1. Time evolution of the mean field

In Wigner representation, the operator product is calculated in symmetrical order so that the ensemble average of photon number ⟨|α|²⟩_W is given by (1/2) ⟨α^†α + αα^†⟩. Although certain approximation is used in the stochastic modeling for exact mapping of Fokker–Planck equation of the system, the results of Wigner representation remain feasible. At high detuning coefficient (Δ_js = 10), slight amplification in the mean amplitude of oscillation, as shown in Fig. 2(a), is observed. The overall energy distribution in the coherent channel is rather erratic; nevertheless, stable energy transfer across all channels is visible. During the evanescent coupling, most of the energy is transferred to other guides, leaving behind some in the coherent channel. The transferred energy oscillates across the multichannel structure with the same distribution on each channel.

Fig. 2.

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	Fig. 2. The evolution of mean photon number in the coherent ⟨n1⟩ and vacuum ⟨n2⟩ channels with the input parameters (a) Δjs = 10, (b) Δjs = 1, and (c) Δjs = 0.

In Fig. 2(b), as a consequence of cavity amplification, the field amplitude is observed to be several times larger than its initial value. Such an effect is pronounced in the coherent channel where the mean field is scaled down to five times of its amplified value. The period of complete energy transfer is also longer than that simulated in the two-channel model,^[26] which indicates that the propagation of energy across waveguides is slower due to the multichannel structure. An identical feature of oscillations is observed in the other channels as well with the energy being shared approximately equally among the waveguides.

For the case of equal operating frequencies (i.e., vanishing detuning with Δ_js = 0), effects identical to that corresponding to the situation of Δ_js = 10 are observed (Fig. 2(c)); in this case, the oscillation of amplitude amplification appears in all channels. In general, the dynamics of intracavity mean fields show a classical damping behavior over their respective interaction length before reaching the steady-state oscillation. The total energy oscillates back-and-forth across the system in all considered cases. Such effective periodical exchange of photons provides potentials for the transfer of quantum effects from one channel to another.

3.2. Sub-Poissonian photon statistics

A convenient way of characterizing the statistical distribution property of multichannel intracavity fields is to use Mandel Q parameter^[44] 4 where (Δn′)² = (n′²) − (n′)² and . In Wigner representation, the symmetrical operator product of the number operator is given by 5 Thus, for x = y = 1,2, the number operator may be written as 6 7 yielding 8 A positive Q-value refers to the photon statistics as super-Poissonian following the classical interpretation, whereas a negative value of Q is associated with sub-Poissonian statistics of quantum phenomenon. In a photon-counting experiment, the intrinsic statistical nature of photons may be determined by analyzing the average photon arrival rate.

In Fig. 3(a), where the Q parameter is plotted against kτ (k = k₁) at Δ_js = 10, the evolution of Q parameter in the coherent channel reveals fluctuating statistics to be of both the super-Poissonian and sub-Poissonian kinds. Coherent light adherent Poissonian statistics (Q = 0) usually relates to photons arriving at the photodetector in clusters. Thus, both the super-Poissonian and sub-Poissonian statistics give the merit of comparison whether they have larger or smaller property of photo-count noise than that of the Poisson distribution, provided that those are of the same intensity.

Fig. 3.

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	Fig. 3. The evolution of Q parameter in the coherent channel αc1 versus kτ with the input parameters (a) Δjs = 10, (b) Δjs = 1, and (c) Δjs = 0.

At Δ_js = 10, photon statistical distribution of the system in the coherent channel can be either super-Poissonian or sub-Poissonian. It follows that for smaller detuning, such as Δ_js = 1 (Fig. 3(b)), fluctuation of the Q parameter can be higher than the coherent light for some interval of interaction, especially when the system is initially launched. The system may also exhibit the sub-Poissonian property. For example, there appeared an obvious reduction of the photo-count noise at kτ ≈ 7. The highest peak of super-Poissonian is observed at Δ_js = 0, as is shown in Fig. 3(c). However, a series of steep transitions to sub-Poissonian behavior occur at kτ ≈ 4 and kτ ≈ 10. It must be noted down that the negative value of Q may also relate to the existence of photon antibunching, provided that the time interval for detecting one arriving photon and another one at a later time is increasing.

3.3. Squeezing

In quantum optical systems, the optical fields obey the laws of quantum mechanics, and thus, have an inherent quantum indeterminacy that cannot be removed. In the framework of single-mode basis, the field quadrature components are given by and .^[45] The quadrature components X and Y obey the commutation relation [X, Y] = i. In accordance with the Heisenberg uncertainty principle, the quadrature variances (ΔX)² = (X²) − (X)² and (ΔY)² = (Y²) − (Y)² must satisfy ΔXΔY ≥ (1/2), so that the uncertainty principle is not violated. The squeezed light emerges with quadrature properties ⟨(ΔX)²⟩ < (1/4) or ⟨(ΔY)²⟩ < (1/4).^[46] For compound-mode squeezing, the uncertainty relation yields 9 where j takes the values 2, 3, and 4 for two-mode, three-mode, and four-mode, respectively. For a fair comparison, equation (9) may be rearranged to give squeezing expressions of the form^[28] 10a 10b with −S_jx being the quadrature squeezing in the X-component, whereas −S_jy being that in the Y-component. For a single-mode squeezing, j = 1, so that S_1x = 4⟨(ΔX₁)²⟩ − |C₁| with C₁ = 1. A similar definition may be given to the Y-quadrature component as well.

In Fig. 4(a), single-mode quadrature evolution is presented at detuning Δ_js = 10. This figure shows squeezing with an interesting pattern of collapsed and revival in the coherent channel; similar features of oscillation can be obtained in the other channels as well. Squeezing occurs in both the X- and Y-quadrature components periodically, depending on the interaction length; it can be much lesser than the shot noise level, especially later in their evolution. In addition to squeezing in the system without cavity, the cavity-assisted system exhibits a swift squeezing oscillation. This, in turn, contributes to the enhancement of noise reduction in both the quadrature components (Fig. 4(b)). Since both the quadrature variances are oscillating in symmetrical pattern, the Y-quadrature is also subjected to the same impact.

Fig. 4.

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	Fig. 4. The evolution of quadrature variances in the coherent channel αc1 with the detuning fixed at Δjs = 10. (a) Both quadrature variances S1x and S1y, and (b) comparison between the first quadrature variances S1x with and without a cavity.

The numerical calculation of squeezing for detuning Δ_js = 1 is shown in Fig. 5. It is observed that, at small detuning coefficient, squeezing occurs in all channels. Unlike the cases at Δ_js = 10, squeezing at Δ_js = 1 exhibits lower frequency of oscillation between the two quadrature components. Although detuning the cavity to Δ_js = 1 eliminates swift oscillation pattern, the reduction of detuning remains advantageous in reducing the quantum noise. The comparison of both quadratures of the coherent channel between Δ_js = 10 and Δ_js = 1 (Fig. 5) shows maximal squeezing at Δ_js = 1. The system with smaller detuning is found to be more capable of generating longer squeezing where the highest value appears at a maximal distance before reaching the steady-state oscillation.

Fig. 5.

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	Fig. 5. Comparison of evolution for both quadrature variances S1x and S1y in the coherent channel αc1 at the detuning Δjs = 10 and Δjs = 1.

In the absence of mismatch between the operating frequencies (i.e., Δ_js = 0), it is found that stronger squeezing (than that obtained corresponding to Δ_js = 10 and Δ_js = 1) is possible. Figure 6 illustrates the comparison of squeezing features corresponding to the detuning coefficients Δ_js = 1 and Δ_js = 0. The case of zero detuning also results in the disappearance of the leaf-revival-collapse-like squeezing; much lower frequency of oscillation occurs in this case. Nevertheless, squeezing with better reduction of uncertainties product emerges periodically between the two oscillating quadrature components. While similar effect is also observed in the other channels, squeezing becomes maximum in the coherent channel.

Fig. 6.

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	Fig. 6. Comparison of evolution for both quadrature variances S1x and S1y in the coherent channel αc1 at the detuning Δjs = 1 and Δjs = 0.

To account for compound-mode squeezing, the interaction between the waveguides may be reduced so that the system interaction remains limited between the vacuum andcoherent channels only, while all the vacuum channels are isolated from each other. The mechanism is easily manifested by setting k_2s = 0 and ζ = 1/3 in Eq. (3). Also, we focus only on the highest compound-mode squeezing of the first-order basis where the effect is maximal by setting j = 4 in Eq. (10). Figure 7(a) gives the X-quadrature evolution of the coherent channel for both single-mode S_1x and four-mode S_4x quadrature. Launching the system at Δ_js = 0 results in quadrature squeezing with a strong oscillatory behavior of collapsed-and-revival pattern. It is to be noted that the maximal squeezing of single-mode may be improved following the previous arrangement where all the vacuum modes are allowed to couple with each other. With all the vacuum modes being isolated (as in the present case), the result predicted in four-mode basis shows a tendency of generating stronger squeezing limit than that exhibited in the single-mode.

Fig. 7.

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	Fig. 7. Comparison of evolution between the single-mode quadrature variance S1x, S1y and four-mode quadrature variance S4x, S4y in the coherent channel αc1 at the detuning (a) Δjs = 10, (b) Δjs = 1, and (c) Δjs = 0.

In Fig. 7(b), we compare the dynamics of both the quadrature variances in the coherent channel for single-mode S_1x, S_1y and four-mode S_4x, S_4y at Δ_js = 1. Within the considered interaction length, the quantum noise between the two quadrature components of the single-mode field is uniformly distributed, and squeezing is generated alternatingly between the quadrature obeying the uncertainty rule. The maximal squeezing is also found to be generated at the maximal distance. As one would hope to achieve for the multichannel system, the compound-mode squeezing at Δ_js = 1 exemplifies even stronger squeezing. Identical to the single-mode squeezing, symmetrical noise distribution between the two quadrature components also appears in the compound mode. However, the maximal squeezing does not necessarily depend on the interaction length, as illustrated in Fig. 7(b); a sudden peak of noise reduction may appear, for example at kτ = 1, 2.5.

We further investigate the possible existence of single-mode S_1x, S_1y, and four-mode S_4x, S_4y squeezing by simulating the system equation at zero detuning. A comparison between quadrature evolutions in the coherent channel is depicted in Fig. 7(c). As shown, the perfect matching between the two operating frequencies contributes to the generation of enhanced squeezing. In the single-mode basis, the quadrature fluctuation in the X-component is found to be completely below the shot noise level for a certain range, without violating the uncertainty principle. At Δ_js = 1 squeezing does not need to alternatingly appear between the two quadrature components to complete the period of oscillation. The outcome is essential in a sense that the practical application of squeezed light requires exact prediction of when the squeezing will appear. In such a case, one can be certain that the X-component will exhibit squeezing at specific time interval corresponding to Δ_js = 0. On the other hand, the result corresponding to four-mode squeezing suggests that quadrature variances may exhibit squeezing in both the quadrature components and the quadrature transition happens at every half-period of their oscillation with equal noise distribution. Finally, the squeezing spectrum corresponding to the four-mode basis shows a stronger squeezing than that in the case of single-mode one where the maximal noise reduction appears at the longest interaction length.

It should also be noted that a higher number of waveguide channels (following similar configuration) can be employed to extend the non-classical effects. Nevertheless, in the present approach, the mathematical dimension of the system is proportional to the number of interacting modes between the waveguides. A large number of modes results in a higher number of system equations. This requires integration of a larger set of linear equations with noise sources over many trajectories. Here, for example, a minimum of 10⁴ trajectories was employed to generate the current results. This requires complex measures and time-consuming computational operations. For this reason, we have limited the number of interacting modes to n = 4 in the Hamiltonian given by Eq. (1).

Switching properties, photon statistics, and squeezing profiles of the single- and compound-mode intracavity four-channel Kerr NLDC are studied within the approximated solution of Wigner phase-space representation. The abrupt coupling of mean field dynamics is simulated, and the effective transfer of energy across waveguides is discussed under different choices of detuning. In all cases, the mean fields show a strong transition of photon property between the super-Poissonian and sub-Poissonian statistics. The cavity setup, in addition to the multichannel arrangement, allows different types of squeezing to be generated, and enhanced squeezing is obtained for all the considered detuning values. In terms of single-mode squeezing, close to cavity resonance (Δ_js = 0), a stronger leaf-revival-collapsed-like squeezing appears as opposed to the system without cavity. Detuning the cavity to Δ_js = 1 eliminates the collapsed-and-revival pattern to yield better noise reduction with larger periodic oscillations. The maximum squeezing is found at Δ_js = 0. For the case where all vacuum channels are without interaction, the case of Δ_js = 0 exhibits maximum squeezing. The enhanced squeezing noticed in the compound-mode for all detuning values indicates that the multichannel system, with the aid of a cavity setup, may be used as a viable source of squeezed light.