Topological Nonlinear Optics with Spin-Orbit coupled Bose-Einstein Condensate in Cavity

TL;DR

This paper demonstrates topological nonlinear optical phenomena in a spin-orbit coupled Bose-Einstein condensate within a cavity, revealing new optical transparencies, topological features, and potential for quantum computation.

Contribution

It introduces a novel scheme combining spin-orbit coupling and cavity QED to realize topological nonlinear optics with unique optical and topological properties.

Findings

Discovery of inverted optical transparencies with atom-induced gain.

Emergence of gapless Dirac-like cones and topological edge states.

Enhanced control of probe light speed via spin-orbit and Zeeman effects.

Abstract

We report topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in a cavity. The cavity is driven by a pump laser and weak probe laser which excite Bose-Einstein condensate to an intermediate storage level, where the standard Raman process engineers spin-orbit coupling. We show that the nonlinear photonic interactions at the transitional pathways of dressed states result in new type of optical transparencies, which get completely inverted with atom induced gain. These nonlinear interactions also implant topological sort of features in probe transmission modes by inducing gapless Dirac-like cones, which become gaped in presence of Raman detuning. The topological features get interestingly enhanced in gain regime where the gapless topological edge-like states emerge among the probe modes, which can cause non-trivial phase transition. We show that spin-orbit coupling and Zeeman field effects also impressively revamp fast and slow probe light. The manipulation of dressed states for quantum nonlinear optics with topological characteristics in our findings could be a crucial step towards topological quantum computation.

Equations

$$\hat{\mathcal{H}}$$

$${\partial }_t\delta \hat{c}(t)=-$$

$${\partial }_t\delta {\hat{p}}_{\uparrow ,\downarrow }(t)$$

$${\partial }_t\delta {\hat{q}}_{\uparrow ,\downarrow }(t)$$

Full text

The quantum nonlinear optics, a stunning manipulation of nonlinear and strong photonic interactions at single photon level Ref1 ; Ref2 ; Ref3 , has the ability to practically demonstrate quantum computation Ref4 ; Ref5 ; Ref6 . The electromagnetically induced transparency (EIT) – quantum interference between two photon transitional paths excited from a single state Ref7 – is the finest example of such nonlinear interactions Ref8 ; Ref9 . These quantum interferences completely stop or slow-down Ref18 the probe light during the EIT interval which impressively appeals for the optical storage devices Ref4 ; Ref7 ; Ref91 . The coherent environment of optical cavities significantly enhances these optical nonlinear phenomena Ref10 , which yield in various significant advancements Ref11 ; Ref12 ; Ref13 ; Ref14 ; Ref15 ; Ref16 ; Ref30 . Further, the mechanical effects of light Ref18 ; Ref17 ; Ref19 ; Ref20 lead to optomechanically induced transparency (OMIT) Ref21 ; Ref22 ; Ref23 which, in coupling with other objects (like Bose-Einstein condensate (BEC)), result in multiple transparencies Ref24 ; Ref25 ; Ref26 ; Ref27 ; Ref28 ; Ref29 in a single cavity. However, it’s worth wondering that how nonlinear photonic interactions are significant to approach the topological defects in light.

The revelation of topological states of light has emerged as a milestone in quantum optics Ref53 ; Ref54 ; Ref55 ; Ref56 ; Ref57 ; Ref59 ; Ref60 . The topological characteristics of photons have been studied in gyromagnetic photonic crystals Ref571 , quasicrystals Ref572 , optical waveguides Ref573 and many other systems Ref53 . Here the behavior of parity-time (PT)-symmetry plays a crucial role for quantum phase transition in photons Ref53 ; Refa0 ; Refa1 ; Refa2 ; Refa3 and has led to many stunning phenomena, like loss-induced transparency Refa4 , low power optical diodes Refa5 and single-mode lasers Refa6 ; Refa7 . Furthermore, the discovery of broken optical PT-symmetry in two coupled optomechanical oscillators Refa8 ; Refa9 ; Refa10 ; Refa11 ; Refa12 , by considering one in gain and other in lossy regime, has led to the demonstration of low threshold optical chaos Refa13 , inverted OMIT Refa14 and phonon laser Refa15 . However, to utilize topological light for quantum information, the manipulation of topological phases of light with nonlinear optics is highly desirable, which we intend to achieve with spin-orbit (SO)-coupled dressed states Ref31 ; Ref32 ; Ref33 ; Ref34 in a single cavity. As the SO-coupling is essential to study spin-Hall effect Ref35 ; Ref36 and topological insulators Ref37 ; Ref38 ; Ref39 ; Ref40 ; Ref41 ; Ref42 ; Ref43 ; Ref44 ; Ref441 ; Ref442 ; Ref443 ; Ref444 ; Ref445 ; Ref446 ; Ref447 ; Ref448 with atoms, so it is worthwhile to see its effects on transmitted light. The inclusion of SO-coupled BEC in optical (and optomechanical) cavities Ref45 ; Ref46 ; Ref47 ; Ref48 and the observation of topological edge state Ref58 as well as Weyl symmetries Ref61 in cavity transmission have further motivated us to use synthetic SO-coupling for topological-nonlinear optics.

In this Letter, we demonstrate topological characteristics of nonlinear optics mediated by SO-coupled BEC in a cavity. The SO-coupled dressed states excite to the intermediate state after interacting with strong pump and weak probe field, where a first EIT window appears. As the Raman process splits the BEC into hyperfine states, which than become momentum sensitive Ref31 ; Ref32 ; Ref33 ; Ref34 , so the quantum interference at these states opens another opportunity for the dark-state. We not only show that the photon interactions during the probe transition through the dressed states and the Zeeman shift generate EIT-like transparencies but also illustrate that the nonlinear interaction of dressed states induces gapless and gaped Dirac-like modes in probe transmission, preserved by the PT-symmetry, depending upon Raman detuning. However, in amplification case – where the atomic damping greater than cavity decay acts as a gain to the cavity field Refb – an edge-like topological mode appears between upper and lower band revealing the broken PT-symmetry. Furthermore, we show that the Zeeman field effects also dramatically alter the fast and slow dynamics of transmitted probe light.

The ${}^{87}Rb$ bosonic particles, with $N\approx 1.8\times 10^{5}$, are trapped inside a high-finesse optical cavity with length $L\approx 1.25\times 10^{-4}$m, Fig. 1(a). The external pump laser $\omega_{E}=\omega_{R}+\delta\omega_{R}=8.8\times 2\pi$ MHz and a weak probe laser $\omega_{p}<<\omega_{E}$ both driven the cavity and excite atomic mode to an intermediate storage state with electronic levels $5S_{1/2}$ Ref48 . Because of the high-Q factor, external pump field builds a strong cavity mode oscillating at $\omega_{c}=4\times 2\pi MHz$ with detuning $\Delta_{c}=\omega_{E}-\omega_{c}=\delta\omega_{R}$. The $10G$ bias longitudinal magnetic field $B_{0}$ produces a Zeeman splitting $\hbar\omega_{z}$ with $\omega_{z}\approx 4.8\times 2\pi$ kHz. To produce SO-coupling, two Raman transverse lasers ($\omega_{R}$ and $\omega_{R}+\delta\omega_{R}$), where $\delta\omega_{R}=\omega_{z}+\delta/\hbar\backsimeq 4.8\times 2\pi$ MHz with wavelength $\lambda=804.1nm$ and detuning $\delta=1.6E_{R}$ Ref34 , oppositely interact with atoms along $\hat{x}$-axis. This Raman process for SO-coupling yields in the coupling of two internal pseudo-spin states ($|\uparrow\rangle=|F=1,m_{F}=0\rangle$ and $|\downarrow\rangle=|F=1,m_{F}=-1\rangle$) at same electronic manifold $F=1$, see Fig. 1(b).

The complete system Hamiltonian, under rotating-wave approximation Ref34 ; Ref46 ; Ref47 ; Ref48 , can be expressed as,

[TABLE]

where $\boldsymbol{\hat{\psi}}=[\hat{\psi}_{\uparrow},\hat{\psi}_{\downarrow}]^{T}$ is the notation of bosonic field operators for atomic $|\uparrow\rangle$ and $|\downarrow\rangle$ states. $\hat{\mathcal{H}}_{SOC}=\hbar^{2}\boldsymbol{k}^{2}\sigma_{0}/2m_{a}+\tilde{\alpha}\boldsymbol{k_{x}}\sigma_{y}+\frac{\delta}{2}\sigma_{y}+\frac{\Omega_{z}}{2}\sigma_{z}$ is the Hamiltonian for a SO-coupled single particle, containing the strength of SO-coupling $\tilde{\alpha}=E_{R}/\boldsymbol{k_{L}}$ and the magnetic field effects $\delta=-g\mu_{B}B_{z}$ (Raman detuning) and $\Omega_{z}=-g\mu_{B}B_{y}$ (Raman coupling) along the $\hat{z}$ and $\hat{y}$ axis, respectively Ref34 ; Ref48 . The quasi-momentum will be one-dimensional $\boldsymbol{k}=[\boldsymbol{k_{x}},0,0]$ as the SO-coupling only occurs along $\hat{x}$-axis. $\sigma_{x,y,z}$ are the $2\times 2$ Pauli matrices with unit matrix $\sigma_{0}$. $\mathcal{V}=\hbar\hat{c}^{{\dagger}}\hat{c}U_{0}[cos^{2}(kx)+cos^{2}(ky)]$ is the two-dimensional optical lattice formed by the longitudinal and transverse fields Ref48 , where $\hat{c}$ ($\hat{c}^{\dagger}$) is an annihilation (creation) operator for the cavity field. $U_{0}=g_{0}^{2}/\Delta_{a}$ is the optical depth for the atoms defined by the Rabi oscillation $g_{0}$ and atomic-cavity detuning $\Delta_{a}$ Ref48 . The strength of atom-atom interactions is defined by $\mathcal{U}_{\sigma,\acute{\sigma}}=4\pi a_{\sigma,\acute{\sigma}}^{2}\hbar^{2}/m_{a}$, where $a_{\sigma,\acute{\sigma}}$ accounts for s-wave scattering. Last two terms define the relation of cavity field with external pump and probe field, with intensities $|\eta|=\sqrt{P\kappa/\hbar\omega_{E}}$ and $|\eta_{p}|=\sqrt{P_{p}\kappa/\hbar\omega_{p}}$, respectively.

We consider the strength of intra-species and inter-species interactions as $U_{\uparrow,\uparrow}=U_{\downarrow,\downarrow}=U$ and $U_{\uparrow,\downarrow}=U_{\downarrow,\uparrow}=\varepsilon U$, respectively, with laser configuration factor $\varepsilon$ Ref34 ; Ref48 . By applying plane-wave approximation for bosonic wave-function $\boldsymbol{\hat{\psi}}(r)=e^{ikr}\boldsymbol{\hat{\varphi}}$, with $\boldsymbol{\hat{\varphi}}=[\hat{\varphi}_{\uparrow},\hat{\varphi}_{\downarrow}]^{T}$, under normalization condition $|\hat{\varphi}_{\uparrow}|^{2}+|\hat{\varphi}_{\downarrow}|^{2}=N$, we derive the coupled Langevin equation equations for the system Ref48 ; Ref49 . We linearized Langevin equations to incorporate first order quantum fluctuations over their steady-state values $\mathcal{\hat{O}}(t)=\mathcal{O}_{s}+\mathcal{\delta O}(t)$, here $\mathcal{O}$ and $\mathcal{O}_{s}$ correspond to the associated subsystems and their steady-states, respectively. By considering equal amount of particles in each species under normalization conditions, which yield in $\hat{\varphi_{\uparrow}}^{\dagger}\hat{\varphi_{\uparrow}}=\hat{\varphi_{\downarrow}}^{\dagger}\hat{\varphi_{\downarrow}}=N/2$, we define dimensionless position and momentum quadratures for atomic subsystems, $\hat{q}_{\mathcal{N}}=\frac{1}{\sqrt{2}}(\hat{\mathcal{O}}+\hat{\mathcal{O}}^{{\dagger}})$ and $\hat{p}_{\mathcal{N}}=\frac{i}{\sqrt{2}}(\hat{\mathcal{O}}-\hat{\mathcal{O}}^{{\dagger}})$, respectively, where $\mathcal{N}\ni\{\uparrow,\downarrow\}$. Under this configuration, the linearized equations will read as,

[TABLE]

The $\partial_{t}$ corresponds to the first order time derivative. The parameter $M=\Omega/2+v-\gamma+UN(1-\epsilon)$, where $v=g_{a}n_{s}$ and $\Omega=\hbar\boldsymbol{k}^{2}/m_{a}$, while $G=\sqrt{2}g_{a}|c_{s}|$ and $\Delta=\Delta_{a}+g_{a}N$ are the effective atom-field coupling defined over $g_{a}$, and detuning of the system, respectively. The collective density excitations of each atomic species perform oscillations under cavity radiation pressure, mimicking as two atomic mirrors equally coupled to the cavity field. $g_{a}=\frac{\omega_{c}}{L}\sqrt{\hbar/m_{bec}\Omega}$ defines their optomechanical coupling strength, with mass $m_{bec}=\hslash\omega_{c}^{2}/(L^{2}U^{2}_{0}\Omega)$ Ref48 ; Ref49 . In order to determine and develop sufficient stability conditions, we use Routh-Hurwitz Stability Criterion Ref29 ; Ref48 ; Ref49 and strictly follow these conditions in numerical calculations.

We obtain cavity transmission by solving linearized Langevin equations, for details see Ref.Ref491 . The total probe transmission, from atom-cavity, can be obtained by taking the ratio of probe input and output probe components $\mathcal{E}_{p}(\omega_{p})=(\eta_{p}-\sqrt{2k}c_{-}(\Delta_{p}))/\eta_{p}=1-\sqrt{2k}c_{-}(\Delta_{p})/\eta_{p}$. Here amplitude $\varepsilon_{out}=\frac{\sqrt{2k}c_{-}(\Delta_{p})}{\eta_{p}}$, at $\omega_{p}$ and $\Delta_{p}=\omega_{e}-\omega_{p}$, determines the absorption (in-phase) and dispersion (out-of-phase) cavity transmission with its real and imaginary values. The transmission phase, i.e. $\Phi_{p}(\omega_{p})=arg(\mathcal{E}_{p})$ is crucially important to understand whole picture because, with its rapid dispersion, the dynamics of fast and slow probe transmission can be governed by computing transmission (or group) delay $\tau_{g}=\partial\Phi_{p}(\omega_{p})/\partial\omega_{p}$, as illustrated in Fig. 2.

In the empty cavity case, the weak probe field will simply pass the cavity without experiencing any quantum interference, as illustrated by the black curves in Fig. 2(a) and 2(b). However, when external fields resonantly interacts with intra-cavity atoms, they excite them to the intermediate energy level opening a way for quantum interference to generate the first EIT, as illustrated in the blue curves of $|\mathcal{E}_{p}|^{2}$ and $\Phi_{p}$ of Fig. 2, at $\Delta_{p}\approx 2.5\kappa$. This type of EITs have been extensively studied for the cavity containing cold/ultra-cold atoms Ref7 ; Ref20 ; Ref26 ; Ref27 ; Ref28 ; Ref29 , even for the single atom and ion Ref11 ; Ref12 .

The standard Raman process, in presence of magnetic field $B_{0}$, at the intermediate state further splits the atomic state into two hyperfine states, $|\uparrow\rangle$ and $|\downarrow\rangle$, by dressing their electronic manifolds, see Fig. 1(b). The SO-coupled atomic states acts as another double excitation forming at storage state, which eventually generates a second dark-state with quantum interference between transitional pathways of $|\uparrow\rangle$ and $|\downarrow\rangle$. These interactions result in new type of nonlinear photon interactions because of SO-coupling, as shown by the EIT window appearing in red curves of $|\mathcal{E}_{p}|^{2}$ and $\Phi_{p}$. Moreover, when we increase the SO-coupling, another interesting EIT-like structure appears in the transmission spectrum, as illustrated by green and magenta curves in Fig. 2(a) and 2(b), when $\alpha=1\Omega$ and $3\Omega$, respectively. The third EIT-like structure appears because at higher SO-coupling, the Zeeman shift, as shown by energy gap $\hbar\omega_{z}$ in Fig. 1(b), become more affective. These magnetic effects for photon interactions can be modeled as an analog of Magnetically induced transparency (MIT) Ref30 . The position of the third structure varies with applied $\alpha/\Omega$ but it can be adjust with $G/\Omega$ and $\epsilon$, see Ref. Ref491 . The inter-species interactions $\epsilon$ interestingly enhance the SO-coupling effects on probe transmission providing an agreement for increase in quantum interferences with SO-coupling.

The dynamics of PT-symmetry plays an important role to describe quantum phase transition in photons Ref53 ; Refa0 ; Refa1 and crucially depends on the loss and gain of optical fields after interacting with passive and active medium Refa8 ; Refa9 ; Refa10 ; Refa11 ; Refa12 ; Refa13 ; Refa14 ; Refa15 , respectively. As, in our system, the dressed states acts as equally coupled oscillators inside the cavity which damp after interacting with cavity field, therefore, one can model these oscillators as active subsystems whose damping acts as gain for the probe transmission Refa14 ; Refb . It is possible under condition that the cavity decay rate is smaller than the atomic damping rate $\kappa<\gamma$. In the gain configuration, the probe transmission, for both non-EIT as well as EIT case, get completely inverted Refa14 because of the amplification (gain) in probe field, see Fig. 2(c). In empty cavity case, $|\mathcal{E}_{p}|^{2}$ moves from absorption domain (i.e. $|\mathcal{E}_{p}|^{2}<1$) to amplification domain (i.e. $|\mathcal{E}_{p}|^{2}>1$) while for coupled cavity case EIT spectrum get inverted. Here it should be noted that, at $\kappa/\gamma>1$, the amplitudes of EIT-windows are partially appearing in amplification domain $|\mathcal{E}_{p}|^{2}>1$, and vise-versa, which may be the reason behind the weak topological transport between bulk modes in lossy regime, as discussed later. Further, the rapid phase diffusion with $\alpha$ also prolongs the probe passage from the dressed states, as shown in Fig. 2(d) where $\tau_{g}$ increases with the increase in $\alpha$. It also shows the slow and fast light dynamics can also be controlled with $\epsilon$.

The interesting thing arrives when the synthetic process of SO-coupling leaves topological-like signatures to the probe field Ref53 ; Ref54 ; Ref58 ; Ref61 . These features can be visualized by plotting probe transmission components parallel to the direction of SO-coupling (i.e. $x$-axis), as illustrated in Fig. 3. As the probe field is appeared to be the function of $\Delta_{p}$ and $\mathbf{k_{x}}$ is parallel to the wavenumber of probe field along $x$-axis, therefore, the gaped and gapless modes can be obtained by plotting $|\mathcal{E}_{p}|^{2}$ verses $\mathbf{k_{x}}$ and $\Delta_{p}/\kappa$. At weak Raman coupling $\Omega_{z}=0.1\Omega$, no Dirac cone type (bulk states) structure appears in the transmission field resulting in trivial transmission. The two side modes can be seen at $\Delta_{p}/\kappa>0$ and $\Delta_{p}/\kappa<0$ corresponding to the incoherent creation and annihilation of quasi-particles, respectively, by the probe field. However, when we increase $\Omega_{z}/\Omega$, two symmetrical Dirac cones take form at $\mathbf{k_{x}}\approx\pm 2\pi$, as indicated with black arrows in Fig. 3(c) and (d). These Dirac comes result in a gap-less Weyl points oriented along the $x$-axis in $x-y$-plane Ref61 . The gap-less Dirac cones like features show agreement for the topological phase transitions of optical mode. It happens because the gap emerging between atomic spin-states Ref48 modifies the phase of probe field which then appears in new synthetic optical spin-states.

These bulk states are crucially protected by the $PT$ symmetry Ref53 which, in our case, can be modified with the Raman detuning, as illustrated in Fig. 3(d-f) where the gap appears between Dirac cones with increase in $\delta/\Omega$ Ref58 ; Ref61 . It is obvious because, in presence of $\delta/\Omega$, the energy spectrum of the dressed states became asymmetric which eventually cause the separation (or broke the $P$ type symmetry Ref53 ) of probe bulk and edge-like modes. A weak transport between bulk modes can be seen around $\mathbf{k_{x}}\approx 0$, because of the amplified components of the probe field in lossy regime, but this can be significantly enhanced by considering optical gain regime $\kappa/\gamma<1$. A topological transport between edge-like state appears at $\mathbf{k_{x}}\approx 0$ and $\mathbf{k_{x}}\approx\pm 2\pi$ (illustrated by the increase in strength of transmission field) which closes the gap between top and bottom interface (optical spin-states), as illustrated in Fig. 3(g-i). The reason behind the emergence of these edge modes can be the broken PT-symmetry, particularly T-symmetry. As discussed earlier, in the gain regime ($\kappa/\gamma<1$), the probe amplitude in output get categorically shifted from absorption to amplification domain interrupting PT-symmetry. The gapless edge modes trigger robust transport for non-trivial optical phase transition, especially to the quantum Hall phase with non-zero Chern number.

As $\alpha$ and $\Omega_{z}$ are interconnected in Raman process, so $\Omega_{z}$ will induce similar features in the probe transmission, as discussed in Ref. Ref491 . However, $\delta$ significantly effects the position and size of third transparency window, as illustrated by the blue and red curves of Fig. 4(a). The presence of $\delta$ alters the size of Zeeman gap which consequently appears as an asymmetric variations in eigen-energies spectrum of dressed states Ref48 . As the sign of $\delta$ is responsible for the $\pm\mathbf{k_{x}}$ direction of the asymmetric potential, therefore, it similarly shifts the third transparency over $\pm\Delta_{p}$, as can be seen in Fig. 4(a). The $\Omega_{z}$ also generates multiple Fano line-shapes in probe transmission which are significantly important for optical spectroscopy Ref50 ; Ref51 , for details see Ref. Ref491 . Furthermore, the increase in $\Omega_{z}$ is appear to be increasing the speed of probe light but the lossy ($\kappa/\gamma>1$) and gain ($\kappa/\gamma<1$) regime act as a switch between slow and fast probe transmission, as shown in Fig. 4(b), where $\gamma>\kappa$ significantly decreases $\tau_{g}$ to the fast light regime at certain $\Omega_{z}$. The behavior of slow and fast probe follow an interesting pattern against $\Omega_{z}$ and $\delta$, see Fig. 4(c), where both slow and fast light follow a semi-circular shape around $\Omega_{z}\approx 13\Omega$, $\Omega_{z}>13\Omega$ slow and $\Omega_{z}<13\Omega$ fast, over a specific interval of $\delta$. These semi-circles of slow and fast transmission dramatically meet each other $\delta\approx\pm 3.5\Omega$, at $\Omega_{z}\approx 13\Omega$, which is also the point where maximum slow as well as fast light occurs. However, these dynamics can be altered with the help of other parameters as discussed in Ref. Ref491 .

In conclusion, we propose a scheme to engineer topological nonlinear optics with SO-coupled BEC in an optical cavity. The quantum interferences during the probe transitions at dressed states and at Zeeman gap not only give birth to a new type of nonlinear optical transparencies which can be inverted in gain regime, but also imprint the gapless and gaped Dirac (bulk) modes in probe transmission, revealing the symmetric Weyl-like modes in gapless regime. We also illustrate the emergence of gapless edge-like topological modes in gain regime uncovering the broken PT-symmetry for non-trivial photonic phase transitions. Furthermore, we investigate the dynamics of fast and slow probe light under influence of dressed states and show that the speed of probe light can be increase or decrease with SO-coupling and Raman coupling. Interestingly, lossy and gain regimes act as a switch between the slow and fast light, respectively. Our findings provide new direction for quantum nonlinear optics with topological features, mediated by SO-coupled states, which could be crucial in order to bring topological physics to the quantum information science.