1.1 (a) A simple model of the interaction of light with a two-level atomic system. b) Diagram of energy levels of a two-level atomic system. Other parameters are the same as shown in Figure a) General energy level diagram for the M-system.
Theoretical background of light-matter interaction
Polarization and susceptibility of the medium
In the χ(2) medium, the three polarization components combine with the nine product components. The third-order sensitivity, χ(3), gives rise to important and interesting nonlinear phenomena such as third harmonic generation (THG), four-wave mixing (FWM), self-phase modulation (SPM), interphase modulation (XPM).
Atom-field interaction Hamiltonian
The last two terms in Eq. 1.23) represents the interaction between external EM field and atom. Next, we implement an important approximation known as the “electric dipole approximation,” which leads to significant simplification of the total Hamiltonian shown in Eq.
Density matrix formalism
In the quantum description, each atom has discrete energy levels, and each level is represented by a unique wave function, |ψ⟩. The evolution of |ψ⟩ corresponding to an energy state is described by the Schrödinger equation. The off-diagonal element ραα′ (α ̸=α′) is referred to as the connection between the modes|α⟩and|α′⟩. The coherence term basically represents the relative phase information between the coupled modes. we write the state vector as a superposition of basis states with explicit phases. then the coherence term ραα′ clearly shows the relative phase. 1.40) Some important properties of density matrix operator are listed below.
Interaction of light with a two-level atomic system
- Dynamics of population and coherence in a two-level system
- Effect of decoherence on dynamics of population and coherence 15
- Dynamics of population and coherence in a three-level system . 20
- Dressed state analysis of three-level Λ system
Energy level diagram of a three-level atomic system. is coupled to the ground states |1⟩ and |2⟩ by two monochromatic laser fields Ωp, Ωc. The rates of spontaneous decay from the excited state to the ground state are γ31 and γ32. 1.7 (a) Schematic diagram of coated states. b) The characteristics of the EIT window and the two absorption spots are shown by the dressed state analysis.
Basic propagation equation in a dielectric medium
Propagation equation for light pulse
For a light pulse, the transverse variation of the field is very small compared to the variation along the direction of propagation. We solve the pulse propagation Eq. 1.122) in a reference frame moving at the speed of light in vacuum c. 1.125) is the source term which determines how the light pulse propagates through the material medium.
Propagation equation for light beam
Solution of paraxial wave equation in free-space
1.8 (a) Intensity profile of a Gaussian beam in the transverse plane. b) Variation of the Gaussian beamwidth w(z) along the propagation axis. Next, we study some of these solutions of the paraxial wave equation in great detail. The phase profiles of the corresponding Laguerre-Gaussian LGlm modes can be found in Figs.
Orbital angular momentum of light
It can be seen from the equation. 1.150) that the angular quantity along the beam axis appears due to the non-zero ϕ-component, so that Lz =rpϕ. Torque ratio. The main feature of the vortex beam is the azimuthal phase dependence accounted for by the term eilϕ, where l is the OAM of the beam. In this chapter, we demonstrate the storage and retrieval of a shape-preserving FWM signal without compromising its intensity.
Theoretical Model
Model Configuration
The square root terms are the coupling strengths (Clebsch-Gordan coefficient) of the corresponding transitions. b) A simple illustration of the model system. In section 2.5, we derive the analytical expression of the nonlinear coherence under weak probe approximation to explain the FWM scheme. E⃗j(z, t) = ˆejE0j(z, t)ei(kjz−ωjt)+c.c., (2.1) whereE0j(z, t) is the space-time dependent amplitude,kj =ωj/c is the propagation constant along z - direction andeˆj is the polarization unit vector of the optical field.
Dynamical Equations
Due to the finite velocity, each atom experiences different Doppler shift in laser field tuning, i.e. Therefore, these effects can be incorporated into the equations of motion (5.11) by taking into account the average over a Maxwell velocity distribution.
Pulse Propagation Equations
We neglect the propagation of the control field because its intensity is much higher than the intensity of the probe and FWM signal. Therefore, the expressions inside the round brackets of Eq. (2.15) can be easily replaced by ∂/∂ξ in the frame of the moving coordinate system. Then, simultaneous solutions of Bloch's equations (5.11) and Maxwell's equations (2.15) in space-time coordinates explore the dynamical progression of optical fields within the medium.
Generation and control of FWM signal
However, the linewidth of the EIT window also depends on Ωc and can be expressed as (∆ω)EIT ∝ |Ωc|2/γ [119] which limits the efficiency of FWM. This figure precisely confirms that the generated FWM signal takes its shape from the probe field envelope and propagates as a shape-preserving pulse. However, the time resolution of the generated amplitude modulated pulse is higher compared to the probe pulse.
Storage and Retrieval of EM radiations
Decreasing (increasing) the intensity to zero (maximum) over time produces the turning off (on) of the control field as shown in the inset of Fig. The maximum intensity of the control field leads to the reception of the probe and the signal from the medium without loss of generality. Atomic coherence starts generating the copy of stored pulses after the control field is turned on.
Analysis and Discussion
Perturbative Analysis
The probe and signal pulses are stored inside the medium in the form of ground atomic coherence by turning off the control field. The Raman scattering between stored atomic coherence and control field intensity reproduces the stored signal. Therefore, the atomic coherence plays the main role behind the storage and retrieval of the probe as well as the FWM signal.
Nonlinear Susceptibility
At the same time, the probe pulse is absorbed and distorted due to the narrowness of the mean transmission window at low driving field intensity because (∆ω)EIT ∝ |Ωc|2/γ. However, the transmission and distortion of the probe pulse can be avoided by considering the appropriate spectral width of the probe pulse so that the spectrum of the probe pulse will be well contained within the transparency window of the medium. Therefore, the efficiency of generating a nonlinear signal can be increased with a suitable value of the control field strength.
Conclusion
A detailed study of the nonlinear atomic coherence governing such OAM conversion processes is required. The frequency of the generated FWM signal (ωg) depends on the frequency of the three fields and is given by ωg =ω1+ω2−ω3. The phase mismatch parameter,∆⃗k = (⃗k1+⃗k2)−(⃗kg+⃗k3) is essential to determine the efficiency of the nonlinear FWM process.
Theoretical Model
Perturbative analysis
In this section, we derive an analytical expression for the atomic coherence under Ω3 ≪ Ω1, Ω2 which states that the seed field (Ω3) will be treated as a perturbation in . The steady state value of these atomic coherences ρ34 and ρ41 can be expressed by the following expression.
Beam propagation equation
The first term on the right represents the phase-induced diffraction of the beam and the rotation of the wavefront during its propagation. The last term on the right represents the creation and dissipation of the medium. To generate the FWM signal, we use the corresponding spatially dependent transverse profile of the optical fields.
Vortex beam generation
Transfer OAM of Ω 1
Simultaneously, the blue dotted curve in Fig. 3.3 shows the transverse variation of the generated infrared intensity|Ω3/γ|2 along the propagation length. 5.4, the first row represents the phase structure of the input probe beam carrying different OAMs. Output phase and FWM signal intensity profile [(c) and (d)]. e) Comparison of the normalized intensity profile of the input control beam and the output FWM signal.
![Fig. 3.5 Transfer of control beam’s OAM (l 2 ) is demonstrated. Input phase and intensity profile of the control beam [(a) and (b)]](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10557424.0/86.892.172.733.339.966/transfer-control-demonstrated-input-phase-intensity-profile-control.webp)
Transfer OAM of Ω 2
Transfer OAM of Ω 1 and Ω 2
The phase structure and normalized intensity profile of the input probe and control beam [(a) and (b)].
Conclusion
In the above closed loop Λ system, population transfer is inevitable in the presence of a high power MW field as it acts on one of the populated states. The MW-induced population transfer between metastable states leads to imperfect transparency that limits many of the EIT-based applications [187] . In section 4.3, we first explain the line shape of the probe laser absorption as a function of probe detuning for various MW strengths and phases.
Theoretical formulation
Model Configuration
In the absence of control lasers, all magnetic sublevels of the Fg = 1 and Fg = 2 hyperfine states will be equally populated. The polarization of the probe laser is σ+ and is σ− for the two control lasers, Ω34 and Ω45 with respect to an applied magnetic field of 200 G along the z direction as shown in the figure. After exiting the glass cell, the probe laser is separated from the two control lasers using a λ/4 waveplate and PBS.
Perturbative Analysis
In order to ensure the correctness of the above approach, we plot in Fig.(4.2) the normalized absorption of the probe field (Im(ρ12)Γ2/Ω12) as a function of normalized probe detuning (δ12/Γ2) that is obtained. of analytical expression of ρ12 as given in Eq.(4.7) as well as the complete numerical solution of density matrix equation as stated in Eq.(4.4). The numerical solution of density matrix equation is calculated for the parameters given in the caption of Fig.
Results and Discussions
Lineshape of the probe absorption
The linewidth of EITAT is modulated by the three control field intensities Ω23, Ω34,Ω45 and the decay rate of the excited states |2⟩, |4⟩. The MW field, ΩM W35 splits or shifts the EITAT dip, depending on the phase ϕ as below as shown in Fig. ΩM W35 |=2π×0.3 MHz, there is further reduction of EITAT dip and division increases and visible as shown in Fig.
Effect of the MW power
Closed loop Λ vs M-system
Conclusion
We exploit this sensitive behavior of the Rydberg energy states to create a highly efficient and extremely tunable atomic waveguide. Second, the presence of buffer gas further manipulates the functions of the waveguide by expanding the transparency window and. Also, the improved contrast in refractive index focused the probe beam closely towards the center of the waveguide.
Theoretical model
Model Configuration
Therefore, the transmission of the weak diffraction-controlled probe beam at medium output increases from 10% to 40% in the presence of buffer gas, in contrast to the results based on the absorbing systems reported previously et al. Along with that, the waveguide has an exclusive and practical function where the absorption and refractive index profile of the waveguide is squeezed from both sides with the increase of the MW intensity, which makes this waveguide very effective in controlling the probe beam of arbitrary width. E⃗j(⃗r, t) = ˆejEj(⃗r)ei(kjz−ωjt)+c.c., (5.1) where Ej(⃗r), kj, ωj and ˆej are the slowly varying envelope curve, wave number, frequency and the unit polarization vector of EM respectively. fields. see probe, control and MW field.
Dynamical Equations
The following three coupled density matrix equations are sufficient to describe the dynamics of the active atoms in the buffer gas environment under weak probe approximation. The perturbative solution of the atomic correlation and population in the weak probe approximation limits can be defined as. We also incorporated the thermal stirring of the atom by performing velocity averaging of the atomic coherence.
Microwave field sensitivity
We next study how a weak MW field (Ωm=0.01γd) drastically changes the probe response in Rydberg system, which is different from normal system. These observations clearly confirm that Rydberg energy states strongly respond to the MW field unlike the normal atomic states [155] . We exploit this responsive behavior of MW field in Rydberg atomic system to create a highly tunable atomic waveguide.
Formation of atomic waveguide
The decreasing intensity of the control beam towards the wing region provides absorption by the cladding. Inset zoom figure shows Dicke's narrowing and amplification of the EIA peak due to Γc. In both cases, the core region of the atomic waveguide shows minimum absorption as shown in Fig.
Tunability of the waveguide
Furthermore, the transparent window of the waveguide becomes much wider and steeper in the presence of the buffer gas. This Dicke narrowing and VCC-induced enhancement of the EIA peak significantly facilitate the waveguide characteristics. 5.5 we draw the imaginary and real parts of the sensitivity of the probe along the transverse position for three different values of the MW field strength.
Beam propagation through the waveguide
Inset figure shows the normalized intensity profile of the probe beam in the presence and absence of MW LG20. The diffraction of the beam in the absence of MW LG20 beam and buffer gas is shown in Fig. The transverse structure of the MW beam and the buffer gas play the important role in steering the weak probe beam with narrow width and arbitrary modes.
Conclusion
5.7 (d) clearly shows that the undiffracted HG11 beam becomes tightly focused towards the center of the waveguide due to the strongly changing refractive index in the presence of the intervening gas medium. Intensity profile of the probe beam (HG11) in the presence of the MW LG20 beam after propagation at 5 Rayleigh lengths (z=5zp) through the atomic medium (c) without and (d) with a buffer gas environment. The solution of Ωp(ξ, τ) is possible because we know the input value of the impulse planeξ= 0.
