Topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in cavity

We investigate topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in a cavity. The cavity is driven by a pump laser and a weak probe laser. Both lasers excite Bose-Einstein condensate, in the presence of standard Raman process for spin-orbit coupling, to an intermediate storage level. We theoretically show that the quantum interference at the transitional pathways of dressed atomic states results in different types of optical transparencies, which get completely inverted in atomic damping induced gain regime. The synthetic pseudo-spin states also implant different phases in the probe field forcing modes in probe transparencies to form gapless Dirac cones, which become gapped in presence of Raman detuning. These features get interestingly enhanced in gain regime where the amplified part of probe transparencies appear as gapless topological edge-like states between the probe bulk modes and cause non-trivial phase transition. We illustrate that the nonlinear interactions of the pseudo-spin states also enhance the slow light features in probe transmission. The manipulation of dressed states for topological optical transparencies in our findings could be a crucial step towards topological photonics and their application in quantum computation.

In this article, we theoretically demonstrate the topological characteristics in quantum nonlinear optical transparencies mediated by SO-coupled BEC in a cavity. The intra-cavity BEC excites to the storage state through two-photon excitation induced by pump and probe fields, where a single EIT window appears. In the case of SO-coupled BEC, the Raman process splits the BEC into two pseudo-spin states, which then become momentum sensitive [57][58][59][60] . We show that the quantum interference at these states provides an opportunity of two dark-states for EIT. Further, we illustrate that the interaction of pseudo-spin states also engineers different phase modes to the probe transparencies causing the formation of gapless and gapped Dirac-like states in probe bulk modes, depending upon Raman detuning. However, in amplification casewhere the atomic damping greater than cavity decay acts as a gain to the cavity field 85a gapless edge-like mode appears between the probe bulk modes revealing the topological phase transition. Moreover, we show that the Zeeman field effects also dramatically alter the fast and slow dynamics of the transmitted probe light.
and jη p j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P p κ=_ω p p are the intensities of pump and probe fields, respectively, where P and ω E (P p and ω p ) are power and frequency of pump (probe) field. κ is the cavity decay rate. With detuning Δ p = ω E − ω p , pump and probe fields excite atomic mode to an intermediate storage state within the spectrum of 5S 1/2 , see supplementary note 1 in supplemental information. Because of the high-Q factor, external pump field builds a strong cavity mode oscillating at ω c ≈ 1.9 × 2πGHz with detuning Δ c = ω E − ω c = κ, here ω c ≈ ω R + δω R only when κ ≈ 0. The 10G bias longitudinal magnetic field B 0 produces a Zeeman splitting ℏω z with condition |ω z /κ| > > 1 61 . To produce SO-coupling, two Raman transverse lasers (ω R and ω R + δω R ), with wavelength λ = 804.1nm and detuning δ = 1.6E R 60 , oppositely interact with atoms alonĝ x-axis. This yields in the coupling of two internal pseudo-spin states ( " Fig. 1(b). Here it should be noted that we choose these particular values from the experimental studies [86][87][88] available in literature in order to make this problem is experimentally feasible. However, the following results can be valid for a range of parameters. But their validation will critically depend upon the coupling strengths of SO-coupled BEC with the cavity.
The system Hamiltonian, under rotating-wave approximation 60,64-66 , can be expressed as, whereψ ¼ ½ψ " ;ψ # T corresponds to bosonic field operators for atomic " j i and # j i states.Ĥ SOC ¼ _ 2 k 2 σ 0 =2m a þαk x σ y þ δ 2 σ y þ Ωz 2 σ z is the Hamiltonian for SO-coupled single particle, containing SO-coupling strengthα ¼ E R =k L and the magnetic field effects δ = −gμ B B z (Raman detuning) and Ω z = −gμ B B y (Raman coupling) along theẑ andŷ-axis, respectively 60,66 . The quasi-momentum will be one-dimensional k = [k x , 0, 0] as the SO-coupling only occurs alongx-axis. σ x,y,z are the 2 × 2 Pauli matrices with unit matrix σ 0 . V ¼ _ĉ yĉ U 0 ½cos 2 ðkxÞ þ cos 2 ðkyÞ is the two-dimensional optical lattice formed by the longitudinal and transverse fields 66 , whereĉ (ĉ y ) is an annihilation (creation) operator for the cavity field. k is wave number associated with cavity field and it is same in both direction. U 0 ¼ g 2 0 =Δ a is the optical depth for the atoms defined by the Rabi oscillation g 0 and cavity-atomic detuning Δ a
We consider the strength of intra-species and inter-species interactions as U ↑,↑ = U ↓,↓ = U and U ↑,↓ = U ↓,↑ = εU, respectively, with laser configuration factor ε 60,66 . By applying plane-wave approximation for bosonic wave-functionψðrÞ ¼ e ikrφ , witĥ φ ¼ ½φ " ;φ # T , and by using normalization condition jφ " j 2 þ jφ # j 2 ¼ N, we derive the linearized quantum Langevin equations 66,89 , incorporating atomic damping rate γ (which we, for simplicity, consider equal for both atomic pseudo-spin states) and cavity decay rate κ. The details are given in the method section. In the cavity, the collective density excitations of each atomic species perform oscillations Ω = ℏk 2 /m bec under radiation pressure and thus, can be modeled as two atomic mirrors equally coupled to the cavity field, with effective SO-coupling α ¼ 4k xα . G ¼ ffiffi ffi 2 p g a jc s j and Δ = Δ a + g a N are the effective optomechanical coupling Fig. 1 Schematic system configuration. a Schematic setup for 87 Rb spin-orbit coupled Bose-Einstein condensate inside a high-Q cavity, driven by control pump field ω E and a probe field ω p . The bias magnetic field B 0 is applied along the cavity (ŷ-axis) while the Raman beams interact transversely (x-axis) with atoms. b The energy level excitation diagram, where left-side illustrates the probe-pump excitation of the atoms and right-side is the Raman excitation at storage level for spin-orbit coupling.

SO-coupled pseudo spin-states induced quantum nonlinear optics
The cavity transmission can be obtained by solving linearized Langevin equations, for details see Method section. The total probe transmission will be the ratio of probe input and output probe components determines the absorption (in-phase) and dispersion (out-of-phase) cavity transmission with its real and imaginary values. Here c − (Δ p ) defines the probe field components in cavity transmission. The transmission phase, i.e. Φ p ðω p Þ ¼ argðE p Þ is crucial 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 τ g = ∂Φ p (ω p )/∂ω p . In the empty cavity case, the weak probe field will be absorbed by the cavity without experiencing any quantum interference, as illustrated by the black curves in Fig. 2(a, b). However, when external fields resonantly interact with intra-cavity atoms, they excite twophoton transition for BEC opening a way for quantum interference to generate the single EIT, as illustrated in the blue curves of Fig. 2(a, b), at Δ p ≈ 2.5κ.
The Raman process at the intermediate state splits the atomic state into two pseudo-spin states, " j i and # j i, which then excite to excited states (having gap equivalent to ℏω z ) with pump laser, please see supplementary note 1. The SO-coupled atomic states act as two double excitation forming at storage state, which eventually generates two dark states with quantum interference between transitional pathways of " j i and # j i, with respect to g j i, see Fig. 1(b). These interactions result in another type of nonlinear photon interactions because of SO-coupling, as shown by the EIT window appearing in red curves of jE p j 2 and Φ p . Moreover, when we increase the SO-coupling, another interesting EIT-like structure appears in the probe transmission, as illustrated by green and magenta curves in Fig. 2(a, b), when α = 1Ω and 3Ω, respectively.
The third EIT-like structure appears because at higher SO-coupling, the Zeeman shift, as shown by energy gap ℏω z in Fig. 1(b), becomes more prominent and lead to another interference. The position of the third structure varies with applied α/Ω but it can be adjusted with G/Ω and ϵ, see supplementary note 2 and 3, and supplementary figs. 1 and 2.
The transparencies of probe field crucially depend on the effective loss and gain induced by the interacting passive and active medium [27][28][29][30][31]40,41 , respectively. As the dressed states in our system act like equally coupled oscillators getting damped after interacting with cavity field, so one can model these oscillators as active subsystems whose damping acts as gain for the probe transmission 31,85 . In our system, it is only possible when the condition κ/γ < 1 is fulfilled, where γ is damping rate of pseudospin states and κ is the cavity decay rate. At κ/γ = 1, the system appears to be in balance configuration, where the probe light will pass the cavity without facing any gain and loss, see blue curve in Fig. 2(c). In the gain configuration, the probe transmission, for both non-EIT as well as EIT case, gets completely inverted 31 because of the amplification (gain) in probe field, see dashed lines in Fig. 2(c). In uncoupled cavity case, jE p j 2 moves from absorption domain (i.e. jE p j 2 < 1) to amplification domain (i.e. jE p j 2 > 1) while for coupled cavity case EIT spectrum gets inverted. Here it should be noted that, at κ/γ > 1, the amplitudes of EIT-windows are partially appearing in amplification domain jE p j 2 > 1, and viseversa, which may be the reason behind the weak transport between bulk modes in lossy regime, as discussed later. Further, the rapid phase diffusion with α also prolongs the probe passage from the dressed states, as shown in Fig. 2(d) where τ g increases with the increase in α. It shows that the slow and fast light dynamics can also be controlled with ϵ.

Nonlinear probe transmission with topological effects
In previous section, we discuss that how quantum interference at pseudo-spin states level creates transparencies for probe light and how these transparencies get alter with gain and loss regimes. But the interesting thing arrives when the synthetic process of SO-coupling leaves spin signatures to the probe field 43,44,55,56 .
When the probe field travels through the pseudo spin-" j i and spin-# j i states, its phase gets split into two phases corresponding to spin states, yielding to two probe trajectories along the direction of SO-coupling. These topological features can be visualized by plotting the probe transparencies in cavity transmission parallel to the direction of SO-coupling (i.e. x-axis), as illustrated in Fig. 3. As the probe field appears to be the function of Δ p and k x is parallel to the wavevector of probe field along x-axis. Therefore, by plotting probe transmission jE p j 2 with respect to k x and Δ p /κ, one can illustrate the effects of pseudo-spin states on transmitting probe light and can see whether the different spin of the atomic states can spatially mold the probe light to form gapped and gapless Dirac cones like structures in cavity transmission. At weak Raman coupling Ω z = 0.1Ω, the probe transmission follow two side-modes like pattern, without forming any Dirac cone in the bulk modes, which can be seen at Δ p /κ > 0 and Δ p /κ < 0 in Fig. 3(a), corresponding to the incoherent creation and annihilation of quasi-particles, respectively. However, when we increase Ω z /Ω, the transparencies form two symmetrical Dirac cones at k x ≈ ±2π, as indicated with black arrows in Fig. 3(b, c). These Dirac cones result in a gapless Weyl-like points oriented along the x-axis in x − y-plane 56 . The gapless Dirac cones like features show evidence for the phase transitions in probe mode. It happens because the gap emerging between atomic spin-states 66 modifies the phase of probe field which then emerge as synthetic optical spin-modes yielding to bulk modes dispersion.
When we apply Raman detuning, the gap appears between Dirac cones with increase in δ/Ω 55,56 , as illustrated in Fig. 3(d-f). It is obvious because, in presence of δ/Ω, the energy spectrum of the pseudo-spin states became asymmetric which eventually cause the emergence of gap between probe bulk modes. A weak transport between bulk modes can be seen around k x ≈ 0, because of the amplified components of the probe field in lossy regime. However, these modes get significantly enhanced by considering optical gain regime κ/γ < 1, where atomic damping acts a gain for amplification for probe transmission. Here it should be noted that the location of these Dirac cones only depends on the parameters related to pseudo-spin states because they are forming because of the different spins of these atomic states. The Dirac cones do not depend upon the other system parameters. A topological transport between Dirac-like edge (or surface) state appears at k x ≈ 0 and k x ≈ ± 2π (illustrated by the increase in strength of transmission field around resonance Δ p = 0) which closes the gap between bulk modes of probe transmission, as illustrated in Fig. 3(g-i).
It is happening because of the interaction between amplified probe transmission and the edges of pseudo-spin states leading to the broken PT-symmetry 40 . Which means that the amplified photons with atomic gain can move from upper state of bulk mode to the lower state and vice versa. It can analogically be associated with optical Hall effect. The gapless edge modes trigger robust transport for non-trivial optical phase transition, especially to the quantum Hall phase with non-zero Chern number. Such topological transparenciesa stunning combination of quantum nonlinear optics and topological physics generating concepts of unconventional topological photonics in a single systemcould be a key step towards the implication of topological nonlinear optics for quantum information.

Effects of Raman detuning on probe transmission and fast and slow light
In order to further explore the quantum nonlinear interactions in our system, we study the effects of Raman detuning on probe transparencies in this section. As α and Ω z are interconnected in Raman process, so Ω z will induce similar features in the probe transmission, please see supplementary note 4 and figure 3. However, δ significantly affects the position and size of third transparency window, as illustrated in Fig. 4(a). The presence of δ alters the size of Zeeman gap which consequently appears as an asymmetric variations in eigen-energies spectrum of dressed states 66 . As the sign of δ is Fig. 3 Topological effects in probe transmission. The transmission amplitude jE p j 2 versus k x ( × π) and Δ p /κ. The first ((a), (d) and (g)), second ((b), (e) and (h)) and third ((c), (f) and (i)) columns illustrate the influence of Ω z = 0.1Ω, 4Ω and 8Ω, respectively, on the bulk modes. While the first ((a), (b) and (c)), second ((d), (e) and (f)) and third ((g), (h) and (i)) rows accounts for δ = 0Ω, 3Ω and 3Ω, respectively. The first and second rows correspond to the lossy regime (κ/γ > 1) while the third row is for gain regime (κ/γ < 1). The strength of SC-coupling is α = 2Ω in these results. Black arrows indicate the location of gapped and gapless transition in the bulk modes.
responsible for the ±k x direction of the asymmetric potential, therefore, it similarly shifts the phase of third transparency over ±Δ p , as can be seen in Fig. 4(a), enhancing the topological states discussed above. The Ω z also generates multiple Fano line-shapes in probe transmission which are crucial for optical spectroscopy 90,91 , please see supplementary note 6 and figure 5 for details. Furthermore, the increase in Ω z is appeared to be increasing the speed of probe light but κ/γ > 1 and κ/γ < 1 act as a switch between slow and fast probe transmission, as shown in Fig. 4(b). Here γ > κ significantly decreases τ g to the fast light regime at certain Ω z . The behavior of slow and fast probe light follows an interesting pattern against Ω z and δ, see Fig. 4(c), where both slow and fast light form a semi-circular shape around Ω z ≈ 13Ω (Ω z > 13Ω slow and Ω z < 13Ω fast) over the specific interval of δ. These semi-circles of slow and fast transmission dramatically meet each other at δ ≈ ±3.5Ω and Ω z ≈ 13Ω, which is also the point where maximum slow and fast light occurs. However, these dynamics can be altered with the help of other parameters, please see supplementary note 5 and figure 4 for details.

DISCUSSION
In conclusion, we propose a scheme to engineer topological nonlinear optics with SO-coupled BEC in an optical cavity. The quantum interferences during the two photon excitation at the pseudo-spin states not only give birth to a different type of nonlinear optical transparencies which can be inverted in gain regime, but these pseudo-spin states also imprint their signatures on the bulk modes of probe transmission. These signatures lead probe transparencies to form the gapless and gapped Dirac cones, revealing symmetric Weyl-like modes in gapless regime. We also illustrate the emergence of gapless edge-like topological modes in gain regime uncovering the non-trivial photonic phase transitions. This provides another way to demonstrate topological photonics different than the previous setups. 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 decreased with help of synthetic dressed state, which indeed is desirable for photonic storage devices. Interestingly, here lossy and gain regimes act as a switch between the slow and fast light.
Our findings provide a direction for quantum nonlinear optics with topological features which could be crucial in order to bring topological physics to the quantum information.

Cavity output field calculations
After considering the coupling of external pump field with cavity field larger than the probe field coupling (i.e. η > > η p ), we substitute system quadrature with δB ¼ P n!fþ;Àg B n e inωt , where B can be any associated subsystem. Here it should be noted that we have not incorporated the influences of associated quantum noises which is possible when the atomic mirrors oscillate at high frequency ℏΩ > > k B T. In order to extract the components of probe field from linearized Langevin equations, we compare the coefficient of exponential terms, which appear to be with exponent of probe detuning Δ p = ω E − ω p , and then solve them for intra-cavity field. To calculate output optical spectrum, we use standard input-output relation c out ¼ ffiffiffiffiffi 2κ p c À c in and extend the Fig. 4 Influence of Raman process on EIT spectrum as well as fast and slow light. a The absorption spectrum Re[ε out ] versus Δ p /κ under influence of δ/Ω. b The transmission delay τ g versus P/P p under influence of Ω z /Ω in lossy κ/γ > 1 and gain κ/γ < 1 regime. c The slow and fast dynamics of probe light (with τ g ) as a function of Ω z /Ω and δ/Ω when P = 1.5P p . α = 2Ω in these results.