Slow light through Brillouin scattering in continuum quantum optomechanics

1 Introduction

Significant progress has been achieved in recent years in fabricating waveguides with cross-sections nearing nanoscale dimensions (Safavi-Naeini et al., 2019), opening new horizons for stimulated Brillouin scattering (SBS). A pivotal advance in SBS emerged with the identification of a dominant mechanism induced by radiation pressure, as has been theoretically predicted (Rakich et al., 2012; Van-Laer et al., 2016; Zoubi and Hammerer, 2016; Rakich and Marquardt, 2018) and experimentally realized (Shin et al., 2013; Beugnot et al., 2014; Van-Laer et al., 2015a; Van-Laer et al., 2015b; Kittlaus et al., 2016; Kittlaus et al., 2017). SBS in waveguides has found application across a broad spectrum of communication and information processing technologies. The substantial enhancement of SBS in waveguides facilitates the amplification of the Stokes field (Kittlaus et al., 2016; Kittlaus et al., 2017; Otterstrom et al., 2019), paving the path towards a Brillouin laser (Otterstrom et al., 2018a; Gundavarapu et al., 2019; Chauhan et al., 2021) and light storage (Zhu et al., 2007; Merklein et al., 2017). Various proposals for nanoscale waveguides (Safavi-Naeini et al., 2019) have emerged in the literature, where a waveguide’s mechanical quality factor—determining the sound wave lifetime (Eggleton et al., 2013)—significantly impacts the efficiency of each proposed device. Moreover, thermal phonons pose major challenges to efficient photon and phonon processes within waveguides (Kharel et al., 2016; Van-Laer et al., 2017; Behunin et al., 2018; Dallyn et al., 2022). To address these challenges, optomechanical cooling via sideband cooling in a continuous system has been demonstrated, using SBS to cool a continuum of traveling wave phonons in a waveguide by tens of kelvins (Otterstrom et al., 2018b). These achievements open the possibility of developing versatile light–matter interfaces (Hammerer et al., 2010) based on SBS achieving, for example, optomechanical entanglement (Zhu et al., 2024) or nonlinear photon interactions (Zoubi and Hammerer, 2017).

This study introduces a configuration to achieve slow photons using SBS within waveguides. By coupling a signal field to classical pump fields through Brillouin scattering mediated by acoustic waves, it is possible to achieve a low effective group velocity; however, the signal’s amplitude is significantly amplified when the pump frequency exceeds that of the signal and is considerably attenuated when the pump frequency is lower (Thevenaz, 2008). A stable signal amplitude can be maintained by employing two simultaneous pump fields with frequencies both above and below that of the signal. The Brillouin scattering from the higher pump field into the signal is balanced by the scattering from the signal field into the lower pump field. We account for the impact of thermal phonons in the waveguide medium and identify conditions under which thermal contributions to the signal amplitude are negligible. The real-space quantum Langevin equations of motion for the signal field are solved by assuming classical pump fields and adiabatically eliminating the phonon components. As a result, the signal field propagates through the waveguide without any gain or loss, with an effective group velocity significantly more reduced than the group velocity of light.

The ability to control the group velocity of light within waveguides opens new avenues for enhancing light–matter interactions, which are crucial for optical quantum information processing (O’Brien, 2007). Slowing the photons extends their interaction time with the medium, potentially increasing the efficiency of quantum gates and other processing elements (Zoubi and Hammerer, 2017; Zoubi, 2021; Zoubi, 2023). Furthermore, the stable propagation of slow light without gain or loss is essential for maintaining the coherence of quantum states necessary for quantum communication and computing.

The paper is structured as follows. Section 2 introduces a coupled system of photons and phonons via SBS within a waveguide. Section 3 describes two methodologies to achieve a slow propagating signal field utilizing SBS and a strong classical pump field. The first method employs a pump field with a frequency higher than that of the signal, leading to significant signal amplification. The second method utilizes a pump field with a frequency lower than the signal’s, resulting in considerable signal attenuation. The impact of thermal fluctuations is analyzed in both scenarios. Section 4 discusses the achievement of a slow signal at the single-photon level without gain or loss by implementing two pump fields with frequencies both above and below that of the signal while minimizing thermal contributions. Section 5 feature a discussion and conclusions. Detailed derivations of the equations of motion and their solutions are presented in the appendices.

2 Continuum quantum optomechanics in nanophotonic wires

We start by presenting a system of interacting light and sound waves within nanoscale waveguides via Brillouin scattering. The system consists of a waveguide composed of dielectric material placed in free space, characterized by a refractive index n greater than 1 (e.g., for silicon material, n ≈ 3.5), as depicted in Figure 1. The length of the waveguide, L, significantly exceeds its transverse dimension, d, with L ≫ d, and the light wavelength λ is comparable to the wire dimension, λ ≲ d. In our prior research (Zoubi and Hammerer, 2016), we formulated a microscopic quantum theory for the interaction between the light field and mechanical excitations in nanoscale waveguides, deriving a Brillouin-type Hamiltonian for the interplay of photons and phonons. This configuration allows photons and phonons to propagate freely along the waveguide while being confined in the transverse direction, leading to the emergence of photonic and phononic multi-mode branches. In Zoubi and Hammerer (2016), we derived the dispersion relations for photons and phonons and determined the photon–phonon coupling parameter by considering both electrostriction and radiation pressure mechanisms. In such an environment, the photon–phonon coupling via Brillouin scattering is significantly more intensified than conventional waveguides, a phenomenon corroborated by experimental findings (Rakich et al., 2012; Shin et al., 2013; Beugnot et al., 2014; Van-Laer et al., 2015a; Van-Laer et al., 2015b; Kharel et al., 2016; Kittlaus et al., 2016; Van-Laer et al., 2016; Zoubi and Hammerer, 2016; Kittlaus et al., 2017). This research broadens the scope of conventional quantum optomechanics, which typically focuses on localized modes of photons and phonons, to include continuum quantum optomechanics that encompass propagating modes (Aspelmeyer et al., 2014; Kharel et al., 2016; Zoubi and Hammerer, 2016; Safavi-Naeini et al., 2019).

The Hamiltonian for propagating photons within a waveguide is described bywhere a ̂ k α † and a ̂ k α represent the creation and annihilation operators for a photon of wavenumber k and branch α, respectively, with ω k α denoting the photon frequency. The Hamiltonian for propagating phonons within a waveguide is expressed aswhere b ̂ q μ † and b ̂ q μ are the creation and annihilation operators for a phonon of wavenumber q and branch μ, respectively, with Ω q μ indicating the phonon frequency. The Hamiltonian describing the photon–phonon interaction is given bywhere g k q , α β μ represents the photon–phonon coupling parameter. The first term describes the scattering of a photon from wavenumber k in branch α to wavenumber k − q in branch β through the emission of a phonon of wavenumber q in branch μ. Conversely, the Hermitean conjugate (h.c.) term accounts for the scattering of a photon from wavenumber k − q in branch β to wavenumber k in branch α via the absorption of a phonon of wavenumber q in branch μ. Owing to translational symmetry along the wire, these processes adhere to momentum conservation. Note that the momentum-space operators for photons and phonons, a ̂ k α and b ̂ q μ, are dimensionless.

H phot = ∑ k , α ℏ ω k α a ̂ k α † a ̂ k α , (1)

H phon = ∑ q , μ ℏ Ω q μ b ̂ q μ † b ̂ q μ , (2)

H phot - phon = ℏ ∑ k , q ∑ α , β , μ g k q , α β μ * b ̂ q μ † a ̂ k − q β † a ̂ k α + h . c . , (3)

In Zoubi and Hammerer (2016), we solved the equations of motion for the electromagnetic field and mechanical excitation to derive the photon and phonon dispersions analytically for the specific case of a cylindrical waveguide, obtaining the frequencies ω k α and Ω q μ. However, the scheme of the current paper can be implemented experimentally for nanoscale wires of any cross-section shape—for example, circular and rectangular (Rakich et al., 2012; Kittlaus et al., 2017; Safavi-Naeini et al., 2019). We focus here on a linear region of the dispersion and assume that the light injected into the waveguide possesses a finite bandwidth. For photons, we employ the linear dispersion relation ω k α = ω 0 α + v g α ( k − k 0 α ), where ω 0 α is the frequency at the center of the signal bandwidth for branch α. The wavenumber bandwidth is denoted by B 0 α k around k 0 α. The effective group velocity in the linear segment is v g α for branch α. A similar approach is applied to the phonon dispersion, where Ω q μ = Ω 0 μ + v s μ ( q − q 0 μ ). The wavenumber bandwidth B 0 μ q is centered around q 0 μ, with the sound velocity being v s μ for branch μ. For propagating both photons and phonons, the wavenumbers are determined by the periodic boundary condition in a wire of length L, where the wavenumber is quantized as k = 2 π L m with m being integers ( m = 0 , ± 1 , ± 2 , ⋯ ). We convert the Hamiltonian from momentum-space to real-space representation to accommodate the space–time dynamics of pulse light fields propagating through the waveguide. This transformation is achieved by defining the light field operator asand its inverse transformation byTranslational symmetry ensures the identities 1 L ∑ k e − i k ( z − z ′ ) = δ ( z − z ′ ) and 1 L ∫ 0 L d z e i ( k − k ′ ) z = δ k , k ′, allowing field operators to satisfy the boson commutation relations [ ψ ̂ α ( z ) , ψ ̂ α † ( z ′ ) ] = δ ( z − z ′ ). The real-space photon Hamiltonian is expressed asThis formulation allows for a nuanced treatment of the propagation dynamics of light pulses within the waveguide, encapsulating the effects of group velocity and phase shifts in real space.

ψ ̂ α z = 1 L ∑ k ∈ B 0 α k a ̂ k α e i k − k 0 α z , (4)

a ̂ k α = 1 L ∫ 0 L d z ψ ̂ α z e − i k − k 0 α z . (5)

H phot = ∑ α ℏ ω 0 α ∫ d z ψ ̂ α † z ψ ̂ α z − i ℏ v g α ∫ d z ψ ̂ α † z ∂ ψ ̂ α z ∂ z . (6)

Similarly, we define the mechanical excitation field operator asand its inverse transformation bywhich satisfy the commutation relation [ Q ̂ μ ( z ) , Q ̂ μ † ( z ′ ) ] = δ ( z − z ′ ). The real-space phonon Hamiltonian is given byThe real-space photon and phonon field operators, having a dimension of 1 / length, represent slowly varying spatial amplitudes.

Q ̂ μ z = 1 L ∑ q ∈ B 0 μ q b ̂ q μ e i q − q 0 μ z , (7)

b ̂ q μ = 1 L ∫ 0 L d z Q ̂ μ z e − i q − q 0 μ z , (8)

H phon = ∑ μ ℏ Ω 0 μ ∫ d z Q ̂ μ † z Q ̂ μ z − i ℏ v s μ ∫ d z Q ̂ μ † z ∂ Q ̂ μ z ∂ z . (9)

The coupling parameter for photon–phonon interaction, g k q , α β μ, is considered constant across the photon and phonon bandwidths B 0 α k and B 0 μ q. Utilizing the local field approximation, the coupling parameter simplifies to g α β μ. Consequently, the real-space photon–phonon interaction Hamiltonian is expressed asThis formulation provides a compact description of the interaction between photons and phonons in the real-space framework, accommodating the direct and inverse scattering processes.

H phot - phon = ℏ L ∑ α , β , μ ∫ d z g α β μ * Q ̂ μ † z ψ ̂ β † z ψ ̂ α z + g α β μ ψ ̂ α † z ψ ̂ β z Q ̂ μ z . (10)

3 Slow light in Brillouin quantum optomechanics

We proceed on this basis to describe the phenomena of signal field amplification and attenuation by exploiting stimulated inter-modal Brillouin scattering of co-propagating photons that belong to distinct spatial optical modes (Kittlaus et al., 2017). We assume a signal field in branch ( s ) centered around frequency ω 0 s = ω s. Conversely, the pump field occupies a distinct branch ( p ) and is centered around frequency ω 0 p = ω p. Both branches are assumed to share identical slopes, leading to equal group velocities v g for the fields in each branch (Figure 2A). The pump field, being considerably stronger than the signal field, is treated as a classical quantity with a stationary (slowly varying) amplitude denoted by E = 〈 ψ ̂ p 〉. On the phonon side, we assume a non-dispersive single branch with a constant frequency Ω q μ = Ω 0 μ = Ω and negligible sound velocity v s μ (Figure 2B). Consequently, the photon Hamiltonian is formulated asBoth the signal and pump fields are assumed to propagate in the rightward direction. The rate of photon damping is considered negligible during their transit along the waveguide’s length L. We account for phonon dissipation by incorporating a damping rate Γ, and thermal fluctuations are represented through the Langevin force operators F ̂, adhering to the properties outlined in Gardiner and Zoller (2010)with n ̄ representing the average phonon count at frequency Ω. At low temperatures, the appearance of thermal photons is negligible, while thermal phonons are likely present and treated here as a heat reservoir in applying the Markovian approximation (Gardiner and Zoller, 2010).

H phot = ℏ ω s ∫ d z ψ ̂ s † z ψ ̂ s z − i ℏ v g ∫ d z ψ ̂ s † z ∂ ψ ̂ s z ∂ z . (11)

〈 F ̂ z , t F ̂ z ′ , t ′ 〉 = 〈 F ̂ † z , t F ̂ † z ′ , t ′ 〉 = 0 , 〈 F ̂ † z , t F ̂ z ′ , t ′ 〉 = Γ n ̄ δ t − t ′ δ z − z ′ , 〈 F ̂ z , t F ̂ † z ′ , t ′ 〉 = Γ n ̄ + 1 δ t − t ′ δ z − z ′ , (12)

In our analysis, we explore two distinct scenarios based on the relationship between the pump and signal frequencies: 1) the pump frequency is higher than that of the signal, denoted as ω p > ω s, and 2) the pump frequency is lower than the signal frequency, indicated by ω p < ω s.

3.1 Slow light with signal amplification

In the scenario where ω p > ω s, a pump photon is scattered into a signal photon through the emission of a phonon, or conversely, a signal photon is converted into a pump photon by the absorption of a phonon (Figure 3). The amplitude of the pump field is represented by E u, with its frequency designated as ω u ≡ ω p. The phonon operator is expressed by Q ̂ u, and the associated Hamiltonian for the phonons is formulated asThe interaction Hamiltonian between photons and phonons is given by

H phon u = ℏ Ω ∫ d z Q ̂ u † z Q ̂ u z . (13)

H phot - phon u = ℏ L ∫ d z g u * E u Q ̂ u † z ψ ̂ s † z + h . c . . (14)

The Heisenberg–Langevin equations of motion for the photon and phonon field operators are formulated asThe resulting signal field operator is (see appendix A for details):where ψ ̂ s i n ( z − v g t ) represents the incoming signal field operator and Δ ω u = ω u − ω s − Ω denotes the detuning frequency (Figure 3B). The gain parameter is defined asand the shift in wavenumber is given bywith Δ u = 2 Δ ω u / Γ representing the scaled detuning and I u = L | E u | 2 denoting the dimensionless pump intensity.

∂ ∂ t + v g ∂ ∂ z ψ ̂ s z , t = − i ω s ψ ̂ s z , t − i L g u * E u Q ̂ u † z , t , ∂ ∂ t + Γ 2 Q ̂ u z , t = − i Ω Q ̂ u z , t − i L g u E u ψ ̂ s † z , t − F ̂ z , t . (15)

ψ ̂ s z , t = ψ ̂ s i n z − v g t e G u − i κ u z + i L g u * E u v g e − i Δ ω u t × ∫ 0 t d t ′ ∫ 0 z d z ′ F ̂ † z ′ , t ′ e − Γ 2 t − t ′ e G u − i κ u z − z ′ , (16)

G u = 2 | g u | 2 v g Γ I u 1 + Δ u 2 , (17)

κ u = 2 | g u | 2 Γ v g Δ u I u 1 + Δ u 2 , (18)

Using the relations (12), the average number of photons per unit length, or photon density, is calculated aswhere N u ( z , t ) represents the thermal contributionIn this formulation, correlations between the Langevin force operators and the initial signal operator are disregarded. This approach focuses on the significant impact of the gain and thermal noise on the evolution of the photon density within the medium, illustrating how amplification and thermal effects contribute to the overall behavior of the signal.

〈 ψ ̂ s † z , t ψ ̂ s z , t 〉 = 〈 ψ ̂ s i n † z − v g t ψ ̂ s i n z − v g t 〉 e 2 G u z + N u z , t , (19)

N u z , t = − | g u | 2 L | E u | 2 2 G u v g 2 n ̄ + 1 1 − e − Γ t 1 − e 2 G u z . (20)

The effective group velocity is defined byWe obtainThe rate of change of the gain G u with respect to the signal frequency is

1 v e u = 1 v g − ∂ κ u ∂ ω s . (21)

v e u v g = 1 + 4 | g u | 2 Γ 2 I u 1 − Δ u 2 1 + Δ u 2 2 − 1 . (22)

∂ G u ∂ ω s = 8 | g u | 2 v g Γ 2 I u Δ u 1 + Δ u 2 2 . (23)

Our primary goal is to achieve a slow propagating signal, aiming for v e u v g ≪ 1 while also preferring the signal to propagate without significant gain, hence G u L ≪ 1. Additionally, it is crucial to minimize the impact of thermal phonons, ensuring that N u L ≪ 1. While the condition for slow light can be met, this comes at the cost of high signal amplification and increased thermal fluctuations. A specific physical example is in g u = 1 0 6 Hz, Γ = 1 0 8 Hz, I u = 1 4 × 1 0 8, L = 1 0 − 2 m, v g = 1 0 8 m/s, and Δ u = 1 2, we find v e u v g ≈ 2 × 1 0 − 4, and ∂ G u ∂ ω s ≈ 0.64 × 1 0 − 4 s/m. This results in slow light with a relatively large bandwidth, yet with a substantial gain factor of G u L ≈ 40. At a phonon frequency of Ω = 50 GHz, an average number of thermal quanta n ̄ ≈ 0.0224 is achievable at a temperature of T ≈ 0.1 K°. At the waveguide’s output ( z = L ) in the high gain limit of G u L ≫ 1, the thermal contribution becomes significant, leading to N out u L ≫ 1.

For the case of I u = 1 0 8, we plot in Figure 4 the gain factor G u L from Equation 50 as a function of Δ u. In Figure 5A, we plot the relative effective velocity v e u v g from Equation 22 as a function of Δ u. The rate of change of the gain factor with respect to the signal frequency, ∂ G u ∂ ω s from Equation 23, is plotted in Figure 5B as a function of Δ u. It is evident that the effective group velocity v e u is significantly smaller than the group velocity v g around zero detuning and that the rate of change of the gain factor is negligible in the same zone. However, the gain factor G u L is large in this interval, leading to significant amplification of the signal photons.

3.2 Slow light with signal attenuation

For the scenario where ω p < ω s, a pump photon is scattered into a signal photon by the absorption of a phonon, or conversely, a signal photon is scattered into a pump photon by the emission of a phonon (Figure 6). The amplitude of the pump field is represented by E l, with its frequency designated as ω l ≡ ω p. The phonon operator is denoted by Q ̂ l, and the associated Hamiltonian for the phonons is formulated asFurthermore, the interaction Hamiltonian between photons and phonons is described by

H phon l = ℏ Ω ∫ d z Q ̂ l † z Q ̂ l z . (24)

H phot - phon l = ℏ L ∫ d z g l ∗ E l Q ̂ l z ψ ̂ s † z + h . c . . (25)

The Heisenberg–Langevin equations of motion for the photon and phonon field operators are given byThe solution to the equations of motion, as provided in Appendix B, yieldswith the detuning frequency defined as Δ ω l = ω s − ω l − Ω (Figure 6B). The gain parameter isand the wavenumber shift iswhere Δ l = 2 Δ ω l / Γ represents the scaled detuning and I l = L | E l | 2 signifies the dimensionless pump intensity.

∂ ∂ t + v g ∂ ∂ z ψ ̂ s z , t = − i ω s ψ ̂ s z , t − i L g l ∗ E l Q ̂ l z , t , ∂ ∂ t + Γ 2 Q ̂ l z , t = − i Ω Q ̂ l z , t − i L g l E l ∗ ψ ̂ s z , t − F ̂ z , t . (26)

ψ ̂ s z , t = ψ ̂ s i n z − v g t e − G l + i κ l z + i L g l ∗ E l v g e i Δ ω l t × ∫ 0 t d t ′ ∫ 0 z d z ′ F ̂ z ′ , t ′ e − Γ 2 t − t ′ e − G l + i κ l z − z ′ , (27)

G l = 2 | g l | 2 v g Γ I l 1 + Δ l 2 , (28)

κ l = 2 | g l | 2 Γ v g Δ l I l 1 + Δ l 2 , (29)

The photon density, using relations (12), is given bywhere the thermal contribution is defined by

〈 ψ ̂ s † z , t ψ ̂ s z , t 〉 = 〈 ψ ̂ s i n † z − v g t ψ ̂ s i n z − v g t 〉 e − 2 G l z + N l z , t , (30)

N l z , t = | g l | 2 L | E l | 2 2 G l v g 2 n ̄ 1 − e − Γ t 1 − e − 2 G l z . (31)

The effective group velocity is given byThis leads toThe rate of change of the gain with respect to the signal frequency is calculated as

1 v e l = 1 v g − ∂ κ l ∂ ω s . (32)

v e l v g = 1 − 4 | g l | 2 Γ 2 I l 1 − Δ l 2 1 + Δ l 2 2 − 1 . (33)

∂ G l ∂ ω s = − 8 | g l | 2 v g Γ 2 I l Δ l 1 + Δ l 2 2 . (34)

Our main goal is to achieve a slow propagating signal, aiming for v e l v g ≪ 1. It is essential for the signal to propagate without loss along the wire, requiring G l L ≪ 1. Additionally, minimizing the influence of thermal phonons is crucial, ensuring N l L ≪ 1. Although achieving slow light is possible, it comes at the cost of high signal attenuation. Using the previously mentioned physical values with I l = 1 0 8 and Δ l = 2, we find v e l v g ≈ 2 × 1 0 − 4 and ∂ G l ∂ ω s ≈ − 0.64 × 1 0 − 4 s/m. This scenario yields slow light with a relatively large bandwidth but incurs a significant loss factor of G l L ≈ 40. At the waveguide output—that is, at z = L — and under the condition of high loss G l L ≫ 1, the thermal contribution becomes negligible, where N out l L ≪ 1.

In Figure 4, we plot the gain factor G l L from Equation 60 as a function of Δ l. In Figure 7A, the relative effective velocity v e l v g from Equation 33 is plotted as a function of Δ l. The rate of change of the gain factor with respect to the signal frequency, ∂ G l ∂ ω s from Equation 34, is depicted in Figure 7B as a function of Δ l. The plots demonstrate that the effective group velocity v e l is significantly smaller than the group velocity v g around zero detuning. Meanwhile, the rate of change of the gain factor is negligible in the same region, but the loss factor G l L is substantial in this interval, leading to significant attenuation of the signal photons.

4 Slow light without gain and loss

Based on this discussion, we conclude that achieving a slow signal within a waveguide while maintaining a constant signal amplitude using SBS with a single pump field is unattainable. Our primary interest lies in slowing down the signal field to the level of single photons. Our objective is to attain a propagating signal with an effective group velocity significantly lower than that in free space while also ensuring a constant average number of quanta. Additionally, it is crucial to minimize the impact of thermal fluctuations, preventing them from significantly affecting the propagating signal. Therefore, our goal is to introduce a configuration that enables the realization of slow signals at the single-photon level without inducing gain or loss.

To address the challenges previously discussed, we propose a unique configuration in which the signal field is coupled through SBS to two pump fields, involving a dispersion-less vibration mode. This approach aims to demonstrate that by merging the two above scenarios, a slow signal can be achieved without gain or loss, where the processes of signal amplification and attenuation counterbalance each other. Specifically, a signal with frequency ω s and group velocity v g is coupled to two classical pump fields with amplitudes E l and E u, and frequencies ω l and ω u respectively, where ω u > ω s > ω l (Figures 8, 9). The involved dispersion-less vibration mode operates at frequency Ω. The SBS process adheres to the phase matching condition for coupling with both the upper and lower pump fields. The photon–phonon coupling parameter is considered to be real, local (i.e., wavenumber independent), and identical for both interactions, with g = g l = g u. Additionally, the lower and upper detuning frequencies are defined as Δ ω l = ω s − ω l − Ω and Δ ω u = ω u − ω s − Ω, respectively, as schematically illustrated in Figure 9B. Both the upper and lower SBS processes involve phonons at the same frequency Ω but with distinct wavenumbers. The phonon damping rate is denoted by Γ, and the Langevin force operator F ̂ is considered identical for both Brillouin scattering processes.

The photon Hamiltonian is specified in (11), and the phonon Hamiltonian combines both upper and lower phonon contributions, H phon = H phon u + H phon l, utilizing Hamiltonians (13) and (24). Correspondingly, the photon–phonon interaction Hamiltonian merges the two interaction scenarios H phot - phon = H phot - phon u + H phot - phon l, conferring Eqs 14, 25.

In the interaction picture, the equation of motion for the photon operator is expressed asand the phonon equations of motion follow from Equations 15 and 26. Adopting a similar approach to that in Appendices A and B for solving these equations, we arrive atwhere G = G u − G l and κ = κ u + κ l, integrating G u , κ u from, 50, 51 and G l , κ l from 60, 61. The gain G and phase shift κ are given byandThe key control parameters remain the scaled detunings Δ u = 2 Δ ω u / Γ and Δ l = 2 Δ ω l / Γ, alongside the dimensionless pump intensities I u = L | E u | 2 and I l = L | E l | 2.

∂ ∂ t + v g ∂ ∂ z ψ ̂ s z , t = − i L g E u e − i Δ ω u t Q ̂ u † z , t − i L g E l e i Δ ω l t Q ̂ l z , t , (35)

ψ ̂ s z , t = ψ ̂ s i n z − v g t e G − i κ z + i L g v g ∫ 0 t d t ′ ∫ 0 z d z ′ e G − i κ z − z ′ e − Γ 2 t − t ′ × E l F ̂ z ′ , t ′ e i Δ ω l t + E u F ̂ † z ′ , t ′ e − i Δ ω u t

G = 2 g 2 v g Γ I u 1 + Δ u 2 − I l 1 + Δ l 2 , (36)

κ = 2 g 2 v g Γ Δ u I u 1 + Δ u 2 + Δ l I l 1 + Δ l 2 . (37)

For the photon density, we obtainwhere the thermal fluctuation contribution is given byUtilizing relations (12) for both the upper and lower processes, correlations among the Langevin force operators corresponding to the upper and lower processes are neglected.

〈 ψ ̂ s † z , t ψ ̂ s z , t 〉 = 〈 ψ ̂ s i n † z − v g t ψ ̂ s i n z − v g t 〉 e 2Gz + N z , t , (38)

N z , t = − g 2 2 G v g 2 I l n ̄ + I u n ̄ + 1 1 − e − Γ t 1 − e 2Gz . (39)

The effective group velocity is defined byWe haveThe rate of change of gain with respect to the signal frequency is expressed asThe objective is to achieve a slow propagating signal, where v e v g ≪ 1. Additionally, it is essential for the signal to propagate without gain or loss along the wire, indicated by G L ≪ 1. Concurrently, we aim to minimize the influence of thermal fluctuations, ensuring that N l ≪ 1. Our goal is to determine the conditions necessary to satisfy these three requirements.

1 v e = 1 v g − ∂ κ ∂ ω s . (40)

v e v g = 1 + 4 g 2 Γ 2 I u 1 − Δ u 2 1 + Δ u 2 2 − I l 1 − Δ l 2 1 + Δ l 2 2 − 1 . (41)

∂ G ∂ ω s = 8 g 2 v g Γ 2 I u Δ u 1 + Δ u 2 2 + I l Δ l 1 + Δ l 2 2 , (42)

We aim to achieve the propagation of light without gain or loss, which is possible when G u ≈ G l, leading to G L ≈ 0. This condition can be satisfied by ensuring thatAdditionally, the thermal fluctuation contribution to the signal needs to be significantly less than 1. At the waveguide output, at z = L in the limit G L ≪ 1 and under the condition Γ L / v g ≪ 1, the thermal contribution is given byThe contribution of thermal fluctuations to the average number of signal photons at the waveguide output should also be much smaller than 1: N out ≪ 1.

I u I l ≈ 1 + Δ u 2 1 + Δ l 2 . (43)

N out ≈ g 2 Γ L 2 v g 3 I l n ̄ + I u n ̄ + 1 . (44)

For further analysis of the result, we define the ratios a = I u I l and b = Δ u Δ l. We use I l = I, then I u = a I, and Δ l = Δ, then Δ u = b Δ. The requirement (43) is written as Δ 2 = 1 − a a − b 2. Note that 1 < a < b 2 or b 2 < a < 1. For example, we use the previous physical values, with I = 1 0 8. We choose a = b = 1 4, then Δ = 2. We obtain v e v g ≈ 1 0 − 4 and ∂ G ∂ ω s ≈ 1.28 × 1 0 − 4 s/m. We obtain a slow light with relatively large bandwidth without gain or loss. For the thermal contribution, we obtain N out ≈ 2.8 × 1 0 − 3.

For the case of zero gain G = 0, the relative effective velocity v e v g from Equation 41 is plotted in Figure 10A as a function of Δ u / Δ l for I u / I l = 1 / 4 and in Figure 10B as a function of I u / I l for Δ u / Δ l = 1 / 4. The rate of change of the gain factor with respect to the signal frequency ∂ G l ∂ ω s from Equation 42 is plotted in Figure 11A as a function of Δ u / Δ l for I u / I l = 1 / 4 and in Figure 11B as a function of I u / I l for Δ u / Δ l = 1 / 4. The effective group velocity v e is significantly smaller than the group velocity v g, where v e v g ≈ 1 0 − 4 for detunings up to Δ u / Δ l < 1 / 3. Note that the rate of change of the gain factor is negligible in the same zone, allowing the propagation of a wide-band signal without gain or loss and with negligible thermal contribution.

5 Discussion and conclusion

Optical quantum information processing is currently a leading candidate for the development of quantum computers. Generally, the components used in quantum information processing differ from those used in communication, which implies a need for interfaces between devices with varying physical properties. Such interfacing can significantly affect the coherence of quantum information. Nanophotonic structures involving photons can serve purposes in both quantum communication and quantum computing. This setup marks a crucial step toward an all-optical on-chip platform, using the same photons for quantum communication and computing, thereby avoiding the decoherence effects associated with interfacing. Interactions among photons are critical for developing optical quantum logic gates. One of the primary obstacles to fabricating efficient photon-based quantum logic gates is the rapid propagation of optical fields within extensive nanophotonic structures. The high speed of light in these structures limits the accumulation of the nonlinear phases necessary for operating quantum logic gates.

In this study, we introduce a configuration that enables slow signal propagation at the single-photon level by exploiting stimulated Brillouin scattering (SBS) within waveguides. The signal field can be significantly slowed via Brillouin scattering, which involves a classical pump field and propagating phonons. When the pump frequency exceeds that of the signal, it results in a substantial amplification of the signal amplitude; conversely, a pump frequency lower than that of the signal causes notable attenuation. To achieve a slow signal field without gain or loss, we propose a novel configuration that utilizes two pump fields with frequencies both above and below that of the signal. This arrangement allows the effects of amplification and attenuation to counterbalance each other, thus enabling the signal to propagate at a constant amplitude with an effective group velocity significantly more reduced than that in free space. Additionally, this configuration can accommodate slow signals over wide bandwidths, extending up to tens of megahertz. We also consider the effects of thermal fluctuations by calculating the scattering of the pump fields off thermal phonons into and out of the signal field and establish conditions under which thermal contributions are negligible.

Slow light has been realized in a free-space medium containing an atomic ensemble (Tey et al., 2008; Hammerer et al., 2010). The control over light propagation in an optical medium can be achieved through electromagnetic induced transparency (EIT), which enables the generation of both fast and slow light. In this process, coherent destructive interference prevents excitation within the optical medium (Lukin et al., 2001; Fleischhauer et al., 2005; Chang et al., 2014). EIT inherently satisfies the phase-matching requirement due to the presence of atomic components. To illustrate EIT, we examine a three-level atom configured in a lambda scheme with two lower metastable states, | g 〉 and | s 〉, and a higher excited state | e 〉, where the transition between the lower states is dipole-forbidden. A probe field near resonance with the dipole-allowed transition | g 〉 ↔ | e 〉 is affected by a strong control field close to resonance with the transition | s 〉 ↔ | e 〉. The control field induces a superposition of the probe field and a coherent mix of the lower atomic states, mapping the photon onto a collective state of the atomic ensemble. This configuration creates a transparent window with an extremely narrow transparency band for the probe field in an otherwise opaque atomic medium, significantly reducing the probe field’s effective group velocity.

EIT has been demonstrated in cavity optomechanics via coupling between vibrational modes and photon modes through radiation pressure (Safavi-Naeini et al., 2011), where photons and phonons are localized within the resonator and phase-matching occurs naturally (Weis et al., 2010). Brillouin scattering induced transparency was shown by utilizing long-lived propagating light and phonons in a silica resonator under the required phase-matching conditions (Kim et al., 2015). Moreover, higher-order side-band induced transparency in optomechanical systems (Xiong et al., 2012), and optomechanical group delays in spinning resonator (Zhang and Shen, 2024) have been demonstrated. The approach introduced in the current paper allows for the propagation of signals across a broader bandwidth than achievable with the EIT scheme. Here, the phonon component serves a role analogous to the atomic component in EIT, ensuring phase-matching for the Brillouin scattering between the signal and pump fields.

The generation of slow photons is important for fundamental physics, such as for quantum nonlinear optics at the level of single photons, which rely on the derivation of effective photon–photon interactions (Zoubi and Hammerer, 2017). Additionally, the formation of photon bound states has been explored (Zoubi, 2021). Slow photons in waveguides provide a test system for studying quantum phases of a gas of interacting photons. Moreover, slow photons have practical applications in nanophotonics for physical implementation in quantum information and quantum communication. The time delay achieved by slowing photons inside waveguides can serve as a memory device, a critical component for quantum computing with photons. A time delay in the order of microseconds can be achieved once the effective group velocity approaches the velocity of sound waves inside a waveguide.

Author contributions

HZ: Writing–review and editing, Writing–original draft. KH: Writing–review and editing, Writing–original draft.

Publisher’s note

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frqst.2024.1437933/full#supplementary-material

Supplementary material Supplementary material at the source