Review of Nonlinear Optics and its Applications

Executive Summary

Nonlinear optics is a fascinating field that explores the interaction of light with matter where the material's response depends nonlinearly on the electric field strength of the light. This report focuses on recent advancements and applications, particularly in the context of exciton-polariton microcavities and the generation of non-classical light states. A significant development highlighted is the extension of the Transfer Matrix Method (TMM) to model nonlinear and inhomogeneous optical systems, offering a computationally efficient alternative to traditional methods like FDTD and MoL. This extended TMM is shown to be particularly effective for simulating light propagation in polariton microcavities, where strong coupling between excitons and photons leads to prominent nonlinear effects such as self-focusing and spontaneous symmetry breaking.

Furthermore, the report delves into the generation and application of intense optical coherent state superpositions (CSS) in nonlinear optics. These non-classical light states, often referred to as optical Schrödinger's cats, are crucial for quantum technologies but have been limited by low photon numbers and decoherence. A novel decoherence-free approach is presented, enabling the generation of intense infrared CSS with significantly higher mean photon numbers. These intense CSS are then utilized to drive second harmonic generation in optical crystals, demonstrating that the non-classical nature of the input light is imprinted onto the generated second harmonic. This opens new avenues for quantum light engineering and applications in quantum information science, including quantum metrology, sensing, nonlinear spectroscopy, and imaging.

Table of Contents

1. Executive Summary
2. Introduction to Nonlinear Optics 2.1. Exciton-Polaritons in Semiconductor Microcavities 2.2. Challenges in Modeling Nonlinear Optical Media 2.3. The Transfer Matrix Method (TMM)
3. Extended Transfer Matrix Method for Inhomogeneous and Nonlinear Media 3.1. Derivation of the Wave Equation in Inhomogeneous Media 3.2. Fourier Transform and Convolution 3.3. Operator Form of the Helmholtz Equation 3.4. Transfer Matrix for a Thin Layer 3.5. Transfer Matrix for Exciton-Polariton Quantum Wells
4. Boundary Conditions and Field Propagation 4.1. Polarized Right-Propagating Input Field 4.2. Polarized Input Field
5. Numerical Simulations and Applications 5.1. Self-Focusing in Nonlinear Quantum Well 5.2. Exciton-Polariton Microcavity and Spontaneous Symmetry Breaking
6. Computational Efficiency of Extended TMM
7. Generation and Application of Intense Optical Coherent State Superpositions 7.1. Coherent State Superpositions (CSS) 7.2. Generating Intense Infrared CSS 7.3. Utilizing Intense CSS in Second Harmonic Generation 7.4. Characterization of Intense CSS 7.5. Quantum Light Engineering via Nonlinear Up-Conversion
8. Key Formulas and Theorems 8.1. Maxwell's Equations 8.2. Wave Equation in Inhomogeneous Media 8.3. Helmholtz Equation 8.4. Transfer Matrix for a Homogeneous Layer (TE Polarization) 8.5. Transfer Matrix for a Homogeneous Layer (TM Polarization) 8.6. Transfer Matrix for a Multilayer Structure 8.7. Transfer Matrix for a Thin Nonlinear Quantum Well 8.8. Reflection and Transmission Coefficients for a Quantum Well 8.9. Detuning Parameter 8.10. Relationship between Reflected and Incoming Fields 8.11. Relationship between Transmitted and Incoming Fields 8.12. Coherent State Superposition (CSS) 8.13. Wigner Function
9. Proofs of Theorems 9.1. Derivation of the Wave Equation in Inhomogeneous Media (Proof) 9.2. Derivation of the Helmholtz Equation (Proof) 9.3. Derivation of the Transfer Matrix for a Homogeneous Layer (TE Polarization) (Proof)
10. Conclusion
11. Sources

2. Introduction to Nonlinear Optics

Nonlinear optics is a branch of optics that describes the behavior of light in nonlinear media, where the polarization density P responds nonlinearly to the electric field E of the light. This nonlinear response can lead to a variety of interesting phenomena, such as harmonic generation, frequency mixing, optical parametric amplification, self-focusing, and optical bistability. These effects are typically observed at high light intensities, where the electric field strength is comparable to the interatomic electric fields within the material.

2.1. Exciton-Polaritons in Semiconductor Microcavities

A particularly active area of research in nonlinear optics involves exciton-polaritons in semiconductor microcavities [1, 2, 3, 4, 5, 6]. Exciton-polaritons [7] are hybrid light-matter quasiparticles that form when excitons (bound electron-hole pairs) strongly couple with photons confined within a microcavity. This strong coupling leads to unique properties, including the ability to exhibit nonlinear optical effects at very low power thresholds [8, 9, 10, 11, 12, 13, 14, 15]. The nonlinearities in these systems primarily arise from exciton-exciton interactions [16].

2.2. Challenges in Modeling Nonlinear Optical Media

Modeling the propagation of electromagnetic waves in complex, nonlinear optical media like microcavities presents significant challenges. Maxwell's equations, which govern electromagnetic field propagation, become more difficult to solve in materials with spatially dependent refractive indices and nonlinear responses [16]. While analytical approaches exist for simple cases, numerical simulations are often necessary for realistic structures.

2.3. The Transfer Matrix Method (TMM)

The Transfer Matrix Method (TMM) is a widely used technique for modeling the propagation of electromagnetic waves through stratified layered media [16]. In its original form, TMM is well-suited for linear, stratified problems where the refractive index is constant within each layer and independent of the field intensity [17]. However, extending TMM to handle spatially varying refractive indices or nonlinear responses, which cause spatial inhomogeneity under inhomogeneous excitation, requires modifications.

Traditional methods for modeling complex optical systems include Finite-difference time-domain (FDTD) [24, 25, 26] and the Method of lines (MoL) [29]. While versatile and capable of handling complex geometries and inhomogeneous media, these methods can be computationally expensive, especially for large-scale systems or high-resolution simulations [23]. The finite element method (FEM) [27, 28] also offers flexibility but can suffer from increased computational overhead in multilayered systems. Sophisticated methods like the semianalytical method of lines and admittance transfer method provide stable and efficient solvers for linear eigenmode problems [29, 30].

The text highlights the need for an efficient method to model nonlinear optical phenomena in systems like polariton microcavities, motivating the extension of the TMM.

3. Extended Transfer Matrix Method for Inhomogeneous and Nonlinear Media

To address the limitations of the standard TMM for inhomogeneous and nonlinear systems, an extended framework is developed. This extension incorporates the effects of spatially varying permittivity and nonlinear responses.

3.1. Derivation of the Wave Equation in Inhomogeneous Media

The starting point is Maxwell's equations. For a medium with spatially varying permittivity $\epsilon(\vec{r})$, Faraday's law is given by: $\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$ (1) Taking the curl of both sides and using Ampère-Maxwell's law ($\nabla \times \vec{H} = \frac{\partial \vec{D}}{\partial t}$) with $\vec{D} = \epsilon(\vec{r})\vec{E}$ and $\vec{B} = \mu\vec{H}$, and assuming the electric field is transverse ($\nabla \cdot \vec{E} = 0$), the wave equation is derived as: $\nabla^2 \vec{E} = \frac{\epsilon(\vec{r})}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}$ (22) where $c = \frac{1}{\sqrt{\mu\epsilon_0}}$ is the speed of light in vacuum.

3.2. Fourier Transform and Convolution

For a monochromatic field with angular frequency $\omega$, substituting $\vec{E}(\vec{r},t) = \vec{E}(\vec{r})e^{i\omega t}$ into the wave equation yields the Helmholtz equation: $\nabla^2 \vec{E} + \frac{\omega^2}{c^2} \epsilon(\vec{r})\vec{E} = 0$ (23) To solve this for layered media, a Fourier transform is applied in the x and y directions, while keeping the z-direction in real space. The Fourier transform of the electric field is: $\tilde{\vec{E}}(k_x, k_y, z) = \frac{1}{2\pi} \int \vec{E}(x,y,z)e^{i(k_x x + k_y y)} dx dy$ (24) Applying the Fourier transform to the Helmholtz equation and using the convolution theorem, we get: $\frac{\partial^2 \tilde{\vec{E}}}{\partial z^2} - (k_x^2 + k_y^2)\tilde{\vec{E}} = -\frac{\omega^2}{c^2}(\tilde{\epsilon} \star \tilde{\vec{E}})$ (25) where $\tilde{\epsilon} \star \tilde{\vec{E}} = \int \tilde{\epsilon}(k'_x, k'_y, z)\tilde{\vec{E}}(k_x - k'_x, k_y - k'_y, z) dk'_x dk'_y$ (26) represents the convolution in momentum space, accounting for the spatial distribution of permittivity and nonlinear refractive index changes.

3.3. Operator Form of the Helmholtz Equation

Equation (25) can be written in a compact operator form: $\frac{\partial^2 \tilde{\vec{E}}}{\partial z^2} + K_\epsilon \tilde{\vec{E}} = 0$ (27) where $K_\epsilon(k_x, k_y, z) = -(k_x^2 + k_y^2) + k_0^2 \tilde{\epsilon}(k_x, k_y, z)\star$ and $k_0 = \omega/c$. In a discretized momentum space, the operator $K_\epsilon$ can be represented as a matrix with elements: $K_\epsilon[i,i',j,j'](z) = k_0^2 \tilde{\epsilon}(k_x[i] - k_x[i'], k_y[j] - k_y[j'], z) - \delta_{i,i'}\delta_{j,j'}(k_x[i]^2 + k_y[j]^2)$ (28)

3.4. Transfer Matrix for a Thin Layer

For a thin layer of thickness $\Delta z$, assuming $\tilde{\epsilon}(k_x, k_y, z) = \tilde{\epsilon}(k_x, k_y, z+\Delta z)$, the electric and magnetic fields can be expressed in terms of forward and backward propagating waves using the eigenvalues and eigenvectors of $K_\epsilon$. The transfer matrix $T_N(k_x, k_y)$ that propagates the fields from $z$ to $z+\Delta z$ is given by: $T_N(k_x, k_y) = \begin{bmatrix} \cos(\Delta z \sqrt{K_\epsilon}) & i k_0 \sin(\Delta z \sqrt{K_\epsilon}) \sqrt{K_\epsilon^{-1}} \\ \frac{i}{k_0} \sqrt{K_\epsilon} \sin(\Delta z \sqrt{K_\epsilon}) & \cos(\Delta z \sqrt{K_\epsilon}) \end{bmatrix}$ (32) For small $\Delta z$, this can be approximated as: $T_N(k_x, k_y) \approx \begin{bmatrix} 1 & i k_0 \Delta z \\ \frac{i}{k_0} \Delta z K_\epsilon & 1 \end{bmatrix}$ (33)

3.5. Transfer Matrix for Exciton-Polariton Quantum Wells

For a thin exciton-polariton quantum well (QW), the transfer matrix in the $[E \ cB]$ basis is given by [16]: $T_{QW} = \begin{bmatrix} 1 & \frac{2}{n} r_{QW} \\ \frac{2}{n} t_{QW} & 1 \end{bmatrix}$ (34) where $n$ is the refractive index, and $r_{QW}$ and $t_{QW}$ are the reflection and transmission coefficients of the QW in vacuum. These coefficients are nonlinear and depend on the exciton density $n_{ex}$, which is proportional to the local electric field intensity $|E(x,y,z=z_{QW})|^2$. The coefficients are given by [16]: $r_{QW}(x,y; \omega) = \frac{i\Gamma_0}{\omega_{QW} - \omega - g n_{ex} - i\Gamma_0}$ (35) $t_{QW}(x,y; \omega) = 1 + r_{QW}(x,y; \omega)$ (36) Here, $\omega_{QW}$ is the resonant frequency, $\Gamma_0$ is the exciton radiative broadening, and $g$ is the strength of exciton-exciton interaction. The detuning is defined as $\Delta = \omega_{QW} - \omega$ for linear systems and $\Delta = \omega_{QW} - \omega - g n_{ex}$ for nonlinear systems.

In the limit $\Delta z \to 0$ for the QW, the term $\Delta z K_\epsilon$ converges to a finite value: $\Delta z K_\epsilon = -\Delta z (k_x^2 + k_y^2) + \Delta z k_0^2 \tilde{\epsilon}(k_x, k_y, z)\star \to k_0 \tilde{\epsilon}_{eff}(k_x, k_y, z)\star$ (37), (38) The transfer matrix for the QW in momentum space becomes: $T_{QW}(k_x, k_y) = \begin{bmatrix} 1 & i \tilde{\epsilon}_{eff}\star \\ 0 & 1 \end{bmatrix}$ (39) Comparing with Eq. (34), the effective permittivity of the nonlinear QW is $i \tilde{\epsilon}_{eff}(k_x, k_y) = 2n (\frac{r_{QW}}{t_{QW}})\tilde{}(k_x, k_y)$.

4. Boundary Conditions and Field Propagation

To analyze wave propagation through a multilayer structure, boundary conditions are applied at the interfaces between layers. The continuity of electric and magnetic fields is used to relate the fields in adjacent layers.

Consider an incoming right-propagating electric field $\vec{E}^{+}_{in}(x,y)$ incident from the left. Assuming no left-propagating field is incident from the right, the boundary condition at the right end of the structure is that the left-propagating field is zero.

4.1. Polarized Right-Propagating Input Field

For a right-propagating electric field in vacuum, the vector representation in momentum space is: $\tilde{\vec{E}}^{+}_{in}(k_x, k_y) = \begin{bmatrix} \tilde{E}_x \\ \tilde{E}_y \\ \frac{-k_x \tilde{E}_x - k_y \tilde{E}_y}{\sqrt{\frac{4\pi^2}{\lambda^2} - k_x^2 - k_y^2}} \end{bmatrix}$ (40) The TE and TM polarization basis vectors for the right-propagating field are: $\vec{e}_s^{+} = \frac{\vec{k} \times \vec{e}_z}{|\vec{k} \times \vec{e}_z|} = \frac{1}{\sqrt{k_x^2 + k_y^2}} \begin{bmatrix} k_y \\ -k_x \\ 0 \end{bmatrix}$ (41) $\vec{e}_p^{+} = \frac{\vec{k} \times (\vec{k} \times \vec{e}_z)}{|\vec{k} \times (\vec{k} \times \vec{e}_z)|} = \frac{1}{k_0 \sqrt{k_x^2 + k_y^2}} \begin{bmatrix} k_x k_z \\ k_y k_z \\ -k_x^2 - k_y^2 \end{bmatrix}$ (42) where $k_0 = \frac{2\pi}{\lambda}$ and $k_z$ is given by Eq. (16).

The transfer matrix for each layer is expressed in the $[\tilde{E}_{\tau s/p} \ c\tilde{B}_{\tau s/p}]$ basis. For a multilayer structure, the overall transfer matrix $M_{s/p}$ in the $[\tilde{E}^{+}_{s/p} \ \tilde{E}^{-}_{s/p}]$ basis is obtained by a change of basis: $M_{s/p} = C^{-1} T_{s/p} C$ (43) where $C$ is the change of basis matrix (Eq. 8) and $T_{s/p}$ is the transfer matrix for the layer (Eq. 13 or 14). For the entire structure, $T = T_N T_{N-1} \cdots T_1$ (Eq. 18). The boundary condition at the right end (no left-propagating wave) is expressed as: $M_{s/p} \begin{bmatrix} \tilde{E}^{+}_{s/p, in} \\ \tilde{E}^{-}_{s/p, in} \end{bmatrix} = \begin{bmatrix} \tilde{E}^{+}_{s/p, out} \\ 0 \end{bmatrix}$ (44) Solving this equation allows determination of the reflected field $\tilde{\vec{E}}^{-}$. The total electric field is the sum of contributions from all polarization components: $\vec{E} = \tilde{E}^{+}_s \vec{e}_s^{+} + \tilde{E}^{-}_s \vec{e}_s^{-} + \tilde{E}^{+}_p \vec{e}_p^{+} + \tilde{E}^{-}_p \vec{e}_p^{-}$ (45) where $\vec{e}_s^{-} = \vec{e}_s^{+}$ (46) and $\vec{e}_p^{-} = \frac{1}{k_0 \sqrt{k_x^2 + k_y^2}} \begin{bmatrix} -k_x k_z \\ -k_y k_z \\ -k_x^2 - k_y^2 \end{bmatrix}$ (47) are the polarization vectors for the left-propagating field.

4.2. Polarized Input Field

A simplified approach assumes a specific polarization direction for the incoming field. The electric and magnetic fields are projected onto s and p polarizations using a projection matrix P: $\begin{bmatrix} \tilde{E}_{\tau s} \\ c\tilde{B}_{\tau s} \\ c\tilde{B}_{\tau p} \\ \tilde{E}_{\tau p} \end{bmatrix} = \begin{bmatrix} \vec{e}_E \cdot \vec{e}_{\tau s} & 0 \\ \vec{e}_B \cdot \vec{e}_{\tau s} & 0 \\ 0 & \vec{e}_B \cdot \vec{e}_{\tau p} \\ 0 & \vec{e}_E \cdot \vec{e}_{\tau p} \end{bmatrix} \begin{bmatrix} \tilde{E} \\ c\tilde{B} \end{bmatrix} = P \begin{bmatrix} \tilde{E} \\ c\tilde{B} \end{bmatrix}$ (48) The transfer matrix for the entire structure in the $[\tilde{E}_{\tau s}, c\tilde{B}_{\tau s}, c\tilde{B}_{\tau p}, \tilde{E}_{\tau p}]^T$ basis is a block diagonal matrix: $T = \begin{bmatrix} \prod_{j=1}^N T_{s,j}(k_x, k_y) & 0_{2\times2} \\ 0_{2\times2} & \prod_{j=1}^N T_{p,j}(k_x, k_y) \end{bmatrix}$ (49) The transfer matrix in the $[\tilde{E}^{+} \ \tilde{E}^{-}]$ basis is obtained by: $M = C^{-1} \cdot P^{-1} \cdot T \cdot P \cdot C$ (50) This results in a $2\times2$ matrix in momentum space: $M = \begin{bmatrix} m_{11}(k_x, k_y) & m_{12}(k_x, k_y) \\ m_{21}(k_x, k_y) & m_{22}(k_x, k_y) \end{bmatrix}$ (51) The boundary condition of no left-propagating field from the right gives: $\tilde{E}^{-}(k_x, k_y) = -m_{21}(k_x, k_y) m_{22}(k_x, k_y)^{-1} \tilde{E}^{+}_{in}(k_x, k_y)$ (52) $\tilde{E}^{+}_{out}(k_x, k_y) = (m_{11} - m_{12} m_{21} m_{22}^{-1})(k_x, k_y) \tilde{E}^{+}_{in}(k_x, k_y)$ (53)

5. Numerical Simulations and Applications

The extended TMM is applied to simulate nonlinear optical phenomena in exciton-polariton systems.

5.1. Self-Focusing in Nonlinear Quantum Well

Self-focusing, where the refractive index change induced by field intensity causes beam narrowing [33], is simulated for a Gaussian beam propagating through an air-QW-air structure. The QW is described by the nonlinear coefficients in Eqs. (35) and (36). Simulations show beam narrowing and interference patterns after transmission through the nonlinear QW (Fig. 2a in [1]). A close-up view confirms the continuity of the electric field at the boundary (Fig. 2b in [1]).

5.2. Exciton-Polariton Microcavity and Spontaneous Symmetry Breaking

Simulations of a polariton microcavity, consisting of Bragg mirrors with a nonlinear QW, demonstrate spontaneous symmetry breaking (SSB). The microcavity is formed by alternating layers with refractive indices $n_a = 3$ and $n_b = 3.5$, and thicknesses $d_a = 0.065 \ \mu\text{m}$ and $d_b = 0.0557 \ \mu\text{m}$, designed for resonance at $\lambda = 780 \ \text{nm}$ under normal incidence, satisfying $\lambda = 4 n_a d_a = 4 n_b d_b$ [16].

In the linear case, two symmetric Gaussian beams with opposite transverse wavevectors interfere, resulting in maximum intensity at the cavity center (Fig. 3b in [1]). With a linear QW, resonant reflection at small detuning causes significant field reflection (Fig. 3d in [1]).

Introducing a nonlinear QW leads to SSB. An initially symmetric right-propagating field results in unequal transmitted beam intensities after interacting with the nonlinear QW (Fig. 3e in [1]). Similar asymmetry is observed in the reflected field (Fig. 3f in [1]). The detuning in the nonlinear case dynamically depends on the local electric field within the QW, following $\Delta = -100\Gamma(1 - 10^3 |E_{QW}|^2)$, where $\Gamma = 100 \ \mu\text{eV}$ is the exciton radiative broadening. The self-consistent field distribution is found iteratively.

Fourier analysis shows that the transverse wavevectors of the incoming field remain unchanged, while the outgoing field intensities differ in the nonlinear regime due to SSB (Fig. 3g in [1]). The results for the linear microcavity without a QW show good agreement with simulations using the Photonic Laser Simulation Kit (PLaSK) [34], validating the numerical approach (Fig. 3b in [1]).

6. Computational Efficiency of Extended TMM

The extended TMM offers significant computational advantages compared to other methods. While methods like Beam Propagation Method (BPM) scale as $O(N^2)$, and FDTD and MoL scale as $O(N^3)$ with the number of modes $N$, the extended TMM, by leveraging the Fast Fourier Transform (FFT), achieves a time complexity of $O(N \log N)$ [1]. This makes it particularly efficient for systems requiring iterative calculations over transverse Fourier modes due to nonlinearities.

A comparison of execution time between the TMM-based method and PLaSK (based on admittance transfer method) shows the superior scalability of TMM with the number of modes (Fig. 4 in [1]). This highlights its potential for efficiently modeling large-scale optical systems with nonlinear effects.

7. Generation and Application of Intense Optical Coherent State Superpositions

Beyond the modeling of nonlinear systems, nonlinear optics plays a crucial role in generating and manipulating non-classical states of light, which are essential for quantum technologies.

7.1. Coherent State Superpositions (CSS)

Coherent state superpositions (CSS), also known as optical Schrödinger's cats, are superpositions of coherent light states. A common form is $|CSS\rangle_\pm = N_\pm (|\alpha\rangle \pm |-\alpha\rangle)$, where $|\alpha\rangle$ and $|-\alpha\rangle$ are coherent states with the same frequency, equal amplitude $|\alpha|$, and opposite phase, and $N_\pm$ are normalization factors [1]. Generalized CSS (GCSS) are superpositions of coherent states with the same frequency but different amplitudes, $|GCSS\rangle = A(|\alpha_1\rangle + \xi |\alpha_2\rangle)$. CSS are vital for quantum technologies, enabling fundamental tests of quantum theory and investigations in quantum information science [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Existing methods for generating CSS often result in low photon numbers, limiting their application in nonlinear optics and quantum technology [17, 18, 19, 20, 2, 21, 22, 23, 24, 25, 26, 27, 28]. Decoherence effects also pose a challenge, particularly for "large" CSS with $|\alpha| \ge 2$ [2, 29].

7.2. Generating Intense Infrared CSS

A novel decoherence-free approach is presented for generating intense femtosecond-duration infrared GCSS with mean photon numbers significantly higher than existing sources [1]. The scheme involves focusing an intense linearly polarized infrared laser pulse (≈25 fs duration, ≈800 nm central wavelength, ≈40 nm bandwidth, ∼10$^{14}$ W/cm$^2$ intensity) into Argon atoms, leading to high harmonic generation (HHG) [35, 36].

According to the fully quantized theory of HHG [21, 22, 23, 24, 25] (SM [34] Pt.1 in [1]), the interaction with atoms causes an amplitude shift in the fundamental and harmonic modes. By postselecting on events where HHG occurs, which corresponds to a depletion of the infrared state, an infrared GCSS is generated. This conditioning is achieved using a quantum spectrometer (QS) approach [39, 21, 22, 23, 24, 25, 40, 37, 38] (Fig. 1 in [1]). The QS uses shot-to-shot correlation measurements between the harmonic and infrared photocurrent signals to select events where energy conservation indicates IR depletion due to HHG. This effectively applies a projection operator $P_{HHG} = 1 - |\tilde{0}\rangle\langle\tilde{0}|$, where $|\tilde{0}\rangle$ is the state without HHG excitations, leading to the generation of the GCSS.

7.3. Utilizing Intense CSS in Second Harmonic Generation

The generated intense infrared GCSS are then used to drive the second harmonic generation (SHG) process in a BBO crystal. The infrared field, after passing through a Mach-Zehnder interferometer to introduce a time delay $\tau$ between two coherent states, is focused into the crystal. The state propagating towards the BBO crystal is given by: $|\psi(t,\tau)| = \frac{1}{2}(\alpha + \delta\alpha)[f_+ e^{i\omega(t+\tau/2)} + f_- e^{i\omega(t-\tau/2)}]$ (1) where $\alpha$ is the attenuated initial coherent state amplitude, $\delta\alpha$ is the depletion due to HHG, and $f_\pm = f(t\pm\tau/2)$ are the envelopes.

When the QS is "on" and conditioning on HHG is applied, the infrared state driving the SHG is a GCSS. The generated second harmonic signal $S_{2\omega}(\tau) \propto \int \langle\hat{I}_{2\omega}\rangle(t,\tau) dt$ is recorded as a function of $\tau$.

7.4. Characterization of Intense CSS

The non-classical nature of the intense infrared CSS is imprinted in the second-order autocorrelation (2-AC) traces of the generated second harmonic. Unlike the conventional 2-AC trace of a coherent state (Fig. 2a, d in [1]), the traces produced by the GCSS show prominent beating features around the center and tails (Fig. 2b, c, e, f in [1]). These features arise from quantum interference between the coherent states composing the GCSS. The Wigner function $W_\omega(x,p)$ in phase space, which visualizes the quantum character, shows a non-Gaussian distribution with negative values for GCSS (Fig. 2h, i in [1]), in contrast to the Gaussian distribution for a coherent state (Fig. 2g in [1]).

Quantitative characterization using $S_{2\omega}(\tau\approx0)$ and the modulation depth M of the second-order intensity autocorrelation (2-IAC) traces confirms the experimental results align with theoretical predictions for GCSS with high photon numbers (Fig. 3a in [1]). The method is applicable for even higher photon numbers, although the quantum interference effects in the AC traces become less pronounced (Fig. 3b in [1]).

7.5. Quantum Light Engineering via Nonlinear Up-Conversion

The theoretical analysis shows that the quantum features of the infrared GCSS are transferred to the generated second harmonic. The Wigner function of the second harmonic ($W_{2\omega}(x,p)$) produced by a GCSS exhibits non-classical features, which can be controlled by varying the depletion parameter $|\delta\alpha|$ (Fig. 4b, c in [1]). This demonstrates the potential for generating GCSS in different spectral regions through nonlinear up-conversion processes.

This capability opens new possibilities in quantum information science, including quantum sensing [7, 8, 9, 10, 42], nonlinear spectroscopy [43], and quantum imaging [44]. Intense, high-photon-number GCSS are a unique resource for advancing new quantum technologies and facilitating novel investigations in quantum information science, such as generating massively entangled states [8, 23, 25, 41] and applications in quantum metrology [42].

8. Key Formulas and Theorems

8.1. Maxwell's Equations

Faraday's Law: $\nabla \times \vec{E} = -i\omega\vec{B}$ (1) (in frequency domain) Ampère-Maxwell's Law: $\nabla \times \vec{H} = i\omega\vec{D}$ (2) (in frequency domain) Material relations: $\vec{D} = \epsilon\vec{E}$, $\vec{B} = \mu\vec{H}$. For homogeneous, isotropic medium with refractive index $n$, $\epsilon = n^2\epsilon_0$, $\mu = \mu_0$.

8.2. Wave Equation in Inhomogeneous Media

$\nabla^2 \vec{E} = \frac{\epsilon(\vec{r})}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}$ (22)

8.3. Helmholtz Equation

$\nabla^2 \vec{E} + \frac{\omega^2}{c^2} \epsilon(\vec{r})\vec{E} = 0$ (23)

8.4. Transfer Matrix for a Homogeneous Layer (TE Polarization)

$T_s = \begin{bmatrix} \cos(k_z d) & \frac{i}{n \cos\theta} \sin(k_z d) \\ i n \cos\theta \sin(k_z d) & \cos(k_z d) \end{bmatrix}$ (13) where $k_z = n k_0 \cos\theta$ (4), $k_0 = \omega/c$, $d$ is the layer thickness, $n$ is the refractive index, and $\theta$ is the angle of incidence.

8.5. Transfer Matrix for a Homogeneous Layer (TM Polarization)

$T_p = \begin{bmatrix} \cos(k_z d) & \frac{i n}{\cos\theta} \sin(k_z d) \\ \frac{i \cos\theta}{n} \sin(k_z d) & \cos(k_z d) \end{bmatrix}$ (14)

8.6. Transfer Matrix for a Multilayer Structure

$T = T_N T_{N-1} \cdots T_2 T_1$ (18) where $T_i$ is the transfer matrix for the $i$-th layer.

8.7. Transfer Matrix for a Thin Nonlinear Quantum Well

In momentum space: $T_{QW}(k_x, k_y) = \begin{bmatrix} 1 & i \tilde{\epsilon}_{eff}\star \\ 0 & 1 \end{bmatrix}$ (39) where $i \tilde{\epsilon}_{eff}(k_x, k_y) = 2n (\frac{r_{QW}}{t_{QW}})\tilde{}(k_x, k_y)$.

8.8. Reflection and Transmission Coefficients for a Quantum Well

$r_{QW}(x,y; \omega) = \frac{i\Gamma_0}{\omega_{QW} - \omega - g n_{ex} - i\Gamma_0}$ (35) $t_{QW}(x,y; \omega) = 1 + r_{QW}(x,y; \omega)$ (36)

8.9. Detuning Parameter

Linear: $\Delta = \omega_{QW} - \omega$ Nonlinear: $\Delta = \omega_{QW} - \omega - g n_{ex}$

8.10. Relationship between Reflected and Incoming Fields

$\tilde{E}^{-}(k_x, k_y) = -m_{21}(k_x, k_y) m_{22}(k_x, k_y)^{-1} \tilde{E}^{+}_{in}(k_x, k_y)$ (52)

8.11. Relationship between Transmitted and Incoming Fields

$\tilde{E}^{+}_{out}(k_x, k_y) = (m_{11} - m_{12} m_{21} m_{22}^{-1})(k_x, k_y) \tilde{E}^{+}_{in}(k_x, k_y)$ (53)

8.12. Coherent State Superposition (CSS)

$|CSS\rangle_\pm = N_\pm (|\alpha\rangle \pm |-\alpha\rangle)$

8.13. Wigner Function

$W_\omega(x,p)$ is a quasi-probability distribution in phase space $(x,p)$ that characterizes the quantum state of a light field. $x = (\hat{a} + \hat{a}^\dagger)/\sqrt{2}$ and $p = (\hat{a} - \hat{a}^\dagger)/(i\sqrt{2})$ are the quadrature operators.

9. Proofs of Theorems

9.1. Derivation of the Wave Equation in Inhomogeneous Media (Proof)

Starting with Faraday's law: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (19) Take the curl of both sides: $\nabla \times (\nabla \times \vec{E}) = \nabla \times (-\frac{\partial \vec{B}}{\partial t}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$ (20) Using Ampère-Maxwell's law, $\nabla \times \vec{H} = \frac{\partial \vec{D}}{\partial t}$, and material relations $\vec{D} = \epsilon(\vec{r})\vec{E}$, $\vec{B} = \mu\vec{H}$: $\nabla \times \vec{B} = \nabla \times (\mu\vec{H}) = \mu \nabla \times \vec{H} = \mu \frac{\partial \vec{D}}{\partial t} = \mu \frac{\partial}{\partial t}(\epsilon(\vec{r})\vec{E})$ Substituting this into the curled Faraday's law: $\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\mu \frac{\partial}{\partial t}(\epsilon(\vec{r})\vec{E})) = -\mu \frac{\partial^2}{\partial t^2}(\epsilon(\vec{r})\vec{E})$ (21) Using the vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E}$. Assuming the electric field is transverse, $\nabla \cdot \vec{E} = 0$. So, $\nabla \times (\nabla \times \vec{E}) = -\nabla^2 \vec{E}$. Substituting this back into the equation: $-\nabla^2 \vec{E} = -\mu \frac{\partial^2}{\partial t^2}(\epsilon(\vec{r})\vec{E})$ $\nabla^2 \vec{E} = \mu \frac{\partial^2}{\partial t^2}(\epsilon(\vec{r})\vec{E})$ If we assume $\epsilon(\vec{r})$ is independent of time, and considering the case where the spatial variation of $\epsilon$ is slow compared to the wavelength such that $\nabla \cdot (\epsilon \vec{E}) \approx \epsilon (\nabla \cdot \vec{E}) + \vec{E} \cdot (\nabla \epsilon) \approx \epsilon (\nabla \cdot \vec{E})$, and if $\nabla \cdot \vec{E} = 0$, then $\nabla \cdot (\epsilon \vec{E}) \approx 0$. In this simplified scenario, the equation becomes: $\nabla^2 \vec{E} = \mu \epsilon(\vec{r}) \frac{\partial^2 \vec{E}}{\partial t^2}$ Using $c^2 = \frac{1}{\mu\epsilon_0}$, we can write $\mu = \frac{1}{\epsilon_0 c^2}$. $\nabla^2 \vec{E} = \frac{\epsilon(\vec{r})}{\epsilon_0 c^2} \frac{\partial^2 \vec{E}}{\partial t^2}$ The text provides a simplified form (Eq. 22) which assumes $\mu = \mu_0$ and incorporates $\epsilon_0$ into the definition of $c$. With $\epsilon = \epsilon_r \epsilon_0$, where $\epsilon_r$ is the relative permittivity, and $n^2 = \epsilon_r$, the equation becomes $\nabla^2 \vec{E} = \frac{n^2(\vec{r})}{c_0^2} \frac{\partial^2 \vec{E}}{\partial t^2}$, where $c_0$ is the speed of light in vacuum. The provided Eq. (22) uses $c$ as the speed of light in vacuum, implying $\epsilon(\vec{r})$ is the relative permittivity or that $\epsilon_0$ is absorbed into $\epsilon$. Assuming the latter for consistency with the provided equation: $\nabla^2 \vec{E} = \frac{\epsilon(\vec{r})}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}$ (22)

9.2. Derivation of the Helmholtz Equation (Proof)

Starting with the wave equation for a monochromatic field $\vec{E}(\vec{r},t) = \vec{E}(\vec{r})e^{i\omega t}$: $\nabla^2 \vec{E}(\vec{r},t) = \frac{\epsilon(\vec{r})}{c^2} \frac{\partial^2 \vec{E}(\vec{r},t)}{\partial t^2}$ (22) Substitute the time dependence: $\nabla^2 (\vec{E}(\vec{r})e^{i\omega t}) = \frac{\epsilon(\vec{r})}{c^2} \frac{\partial^2}{\partial t^2} (\vec{E}(\vec{r})e^{i\omega t})$ $\nabla^2 \vec{E}(\vec{r}) e^{i\omega t} = \frac{\epsilon(\vec{r})}{c^2} \vec{E}(\vec{r}) \frac{\partial^2}{\partial t^2} (e^{i\omega t})$ $\nabla^2 \vec{E}(\vec{r}) e^{i\omega t} = \frac{\epsilon(\vec{r})}{c^2} \vec{E}(\vec{r}) (i\omega)^2 e^{i\omega t}$ $\nabla^2 \vec{E}(\vec{r}) e^{i\omega t} = \frac{\epsilon(\vec{r})}{c^2} \vec{E}(\vec{r}) (-\omega^2) e^{i\omega t}$ Divide by $e^{i\omega t}$ (assuming $e^{i\omega t} \neq 0$): $\nabla^2 \vec{E}(\vec{r}) = -\frac{\omega^2}{c^2} \epsilon(\vec{r}) \vec{E}(\vec{r})$ Rearranging the terms, we get the Helmholtz equation: $\nabla^2 \vec{E}(\vec{r}) + \frac{\omega^2}{c^2} \epsilon(\vec{r}) \vec{E}(\vec{r}) = 0$ (23)

9.3. Derivation of the Transfer Matrix for a Homogeneous Layer (TE Polarization) (Proof)

For a homogeneous, isotropic medium, the wave equation for the transverse-electric (TE) polarized component $E_\tau^s$ in the z-direction simplifies to: $\frac{\partial^2 E_\tau^s}{\partial z^2} + k_z^2 E_\tau^s = 0$ (5) where $k_z = n k_0 \cos\theta$. The general solution is a superposition of forward and backward propagating waves: $E_\tau^s(z) = E_0^+ e^{ik_z z} + E_0^- e^{-ik_z z}$ (6) The corresponding magnetic field component $B_\tau^s$ is related by Maxwell's equations. From Faraday's law (Eq. 1), $\nabla \times \vec{E} = -i\omega\vec{B}$. For TE polarization with $\vec{E} = E_\tau^s \vec{e}_\tau$, where $\vec{e}_\tau$ is in the transverse plane, and propagation along z, the relevant component of the curl is $(\nabla \times \vec{E})_\tau = \frac{\partial E_\tau^s}{\partial z}$. So, $\frac{\partial E_\tau^s}{\partial z} = -i\omega B_\tau^s$. $B_\tau^s(z) = \frac{i}{\omega} \frac{\partial E_\tau^s}{\partial z} = \frac{i}{\omega} (ik_z E_0^+ e^{ik_z z} - ik_z E_0^- e^{-ik_z z}) = \frac{-k_z}{\omega} (E_0^+ e^{ik_z z} - E_0^- e^{-ik_z z})$ Using $k_z = n k_0 \cos\theta$ and $k_0 = \omega/c$, we have $\frac{k_z}{\omega} = \frac{n k_0 \cos\theta}{\omega} = \frac{n (\omega/c) \cos\theta}{\omega} = \frac{n \cos\theta}{c}$. So, $B_\tau^s(z) = -\frac{n \cos\theta}{c} (E_0^+ e^{ik_z z} - E_0^- e^{-ik_z z})$. Or, $c B_\tau^s(z) = -n \cos\theta (E_0^+ e^{ik_z z} - E_0^- e^{-ik_z z})$. The text provides a slightly different form for $c B_\tau^s(z)$ in Eq. (7): $c B_\tau^s(z) = \frac{k_z}{k_0} (E_0^+ e^{ik_z z} - E_0^- e^{-ik_z z})$ (7) Using $k_z/k_0 = n \cos\theta$, this matches our derivation except for a sign. The sign difference might arise from the definition of the transverse magnetic field component or the direction of propagation. Assuming the provided Eq. (7) is correct for the chosen basis: $E_\tau^s(z) = E_0^+ e^{ik_z z} + E_0^- e^{-ik_z z}$ $c B_\tau^s(z) = n \cos\theta (E_0^+ e^{ik_z z} - E_0^- e^{-ik_z z})$ (using $k_z/k_0 = n \cos\theta$)

At $z=0$: $E_\tau^s(0) = E_0^+ + E_0^-$ (9) $c B_\tau^s(0) = n \cos\theta (E_0^+ - E_0^-)$ (using $k_z/k_0 = n \cos\theta$)

At $z=d$: $E_\tau^s(d) = E_0^+ e^{ik_z d} + E_0^- e^{-ik_z d}$ (11) $c B_\tau^s(d) = n \cos\theta (E_0^+ e^{ik_z d} - E_0^- e^{-ik_z d})$ (using $k_z/k_0 = n \cos\theta$)

We want to relate the fields at $z=d$ to the fields at $z=0$. From the equations at $z=0$: $E_0^+ + E_0^- = E_\tau^s(0)$ $E_0^+ - E_0^- = \frac{c B_\tau^s(0)}{n \cos\theta}$ Adding the two equations: $2 E_0^+ = E_\tau^s(0) + \frac{c B_\tau^s(0)}{n \cos\theta} \implies E_0^+ = \frac{1}{2} (E_\tau^s(0) + \frac{c B_\tau^s(0)}{n \cos\theta})$ Subtracting the second from the first: $2 E_0^- = E_\tau^s(0) - \frac{c B_\tau^s(0)}{n \cos\theta} \implies E_0^- = \frac{1}{2} (E_\tau^s(0) - \frac{c B_\tau^s(0)}{n \cos\theta})$

Now substitute these into the equations at $z=d$: $E_\tau^s(d) = \frac{1}{2} (E_\tau^s(0) + \frac{c B_\tau^s(0)}{n \cos\theta}) e^{ik_z d} + \frac{1}{2} (E_\tau^s(0) - \frac{c B_\tau^s(0)}{n \cos\theta}) e^{-ik_z d}$ $E_\tau^s(d) = \frac{1}{2} E_\tau^s(0) (e^{ik_z d} + e^{-ik_z d}) + \frac{1}{2 n \cos\theta} c B_\tau^s(0) (e^{ik_z d} - e^{-ik_z d})$ Using Euler's formulas, $\cos(k_z d) = \frac{e^{ik_z d} + e^{-ik_z d}}{2}$ and $\sin(k_z d) = \frac{e^{ik_z d} - e^{-ik_z d}}{2i}$: $E_\tau^s(d) = E_\tau^s(0) \cos(k_z d) + \frac{i}{n \cos\theta} c B_\tau^s(0) \sin(k_z d)$

$c B_\tau^s(d) = n \cos\theta [\frac{1}{2} (E_\tau^s(0) + \frac{c B_\tau^s(0)}{n \cos\theta}) e^{ik_z d} - \frac{1}{2} (E_\tau^s(0) - \frac{c B_\tau^s(0)}{n \cos\theta}) e^{-ik_z d}]$ $c B_\tau^s(d) = \frac{n \cos\theta}{2} E_\tau^s(0) (e^{ik_z d} - e^{-ik_z d}) + \frac{n \cos\theta}{2 n \cos\theta} c B_\tau^s(0) (e^{ik_z d} + e^{-ik_z d})$ $c B_\tau^s(d) = i n \cos\theta E_\tau^s(0) \sin(k_z d) + c B_\tau^s(0) \cos(k_z d)$

In matrix form, relating $[E_\tau^s(d) \ c B_\tau^s(d)]^T$ to $[E_\tau^s(0) \ c B_\tau^s(0)]^T$: $\begin{bmatrix} E_\tau^s(d) \\ c B_\tau^s(d) \end{bmatrix} = \begin{bmatrix} \cos(k_z d) & \frac{i}{n \cos\theta} \sin(k_z d) \\ i n \cos\theta \sin(k_z d) & \cos(k_z d) \end{bmatrix} \begin{bmatrix} E_\tau^s(0) \\ c B_\tau^s(0) \end{bmatrix}$ (13) This matches the provided transfer matrix for TE polarization.

10. Conclusion

This report has provided a detailed review of recent advancements in nonlinear optics, focusing on the modeling of complex systems and the generation of non-classical light states. The extension of the Transfer Matrix Method (TMM) to incorporate nonlinearities and inhomogeneities, particularly relevant for exciton-polariton microcavities, represents a significant step towards computationally efficient simulations of light-matter interactions in these systems. The $O(N \log N)$ time complexity of the extended TMM, achieved by leveraging FFT, offers a substantial advantage over traditional methods like FDTD and MoL, which scale as $O(N^3)$. Numerical simulations using this extended TMM successfully capture key nonlinear phenomena such as self-focusing and spontaneous symmetry breaking in polariton microcavities, validating its accuracy and potential for designing future low-power nonlinear optical devices and photonic circuits.

Furthermore, the report highlighted a breakthrough in generating intense optical coherent state superpositions (CSS) using a decoherence-free approach based on high harmonic generation and quantum spectroscopy. The ability to produce CSS with significantly higher photon numbers overcomes a major limitation for their application in nonlinear optics. The experimental and theoretical demonstration that the non-classical nature of these intense CSS is transferred to the generated second harmonic opens exciting new avenues for quantum light engineering across different spectral regions. These intense non-classical light states are poised to become a powerful resource for advancing quantum technologies, including quantum metrology, sensing, nonlinear spectroscopy, and imaging, by enabling novel investigations into the interplay between the quantum properties of light and matter on ultrafast timescales.

11. Sources

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