Self-Pulsing in an Optical Ring Cavity Coupled with Three-Level Atoms

Resonant intracavity four-wave mixing in a composite atom–cavity system

Since real atomic systems have multiple Zeeman sublevels, one can use polarization states of the coupled laser beams to modulate the strength of the multi-wave mixing processes. By manipulating the dark-state or electromagnetically induced transparency (EIT) windows with the polarization states of the laser beams, the multi-wave mixing (MWM) processes can be modified and controlled. Such studies of controllable intermixing between differentorder nonlinear optical processes with phase or polarization states of the laver beams can be very important in high-precision measurements, coherent quantum control, and quantum information processing. In the following, some examples of phase or polarization controlled multi-wave mixing processes in multi-level atomic systems will be described. The degenerate Zeeman sublevels and their dressed-state effects are responsible for these observed phenomena. The relative intensities and polarization characteristics of fourwave mixing (FWM) signals in different laser polarization configurations and different level systems are experimentally investigated and compared. Also the results are theoretically explained by different transition path combinations. In the dressed-FWM processes, the dependence of dressing effect on the incident field’s polarization is shown. The FWM signal generated by a linearly-polarized pumping field is suppressed more by the dressing field than the one generated by a circularly-polarized pumping field. However, an opposite effect was observed when the probe field’s polarization is changed.

Summary form only given. We present our recent results in measuring the nonlinear Kerr index of refraction of multi-level rubidium atoms inside an optical ring cavity. By using a similar experimental arrangement as we used previously to demonstrate cavity linewidth narrowing due to EIT inside an optical ring cavity, we can measure the nonlinear coefficient χ/sup (3)/ through the asymmetric transmission profile of the ring cavity. By measuring the degree of asymmetry, we can obtain the Kerr nonlinear index of refraction directly, without worrying about contributions from linear absorption. Our preliminary results indicate a significant enhancement of Kerr nonlinearity due to atomic coherence at near-resonance conditions. We present our experimental results and their comparison with theoretical calculations. We discuss the implications of this enhanced nonlinear Kerr index of refraction to nonlinear optical processes in an optical cavity.

Cavity optomechanical systems have become a topic of great interest in recent years, and the coupled-cavity model is also a classic theoretical framework. This paper aims to construct a coupledcavity optomechanical system to study induced transparency, Fano resonance, and fast-slow light effects in such a system. By transferring phenomena typically studied in a single optical cavity to a coupled-cavity system, we analyze specific phenomena detected in optical and microwave cavities, such as transmission and absorption spectra, to investigate induced transparency. We also examine Fano resonance in the model by varying detuning, and study fast-slow light effects through group velocity. This paper first constructs the corresponding physical model, as shown in Figure 1. Based on the theoretical model, a reasonable Hamiltonian is proposed. By introducing appropriate dissipation and fluctuation noise terms, the Langevin equations of motion are derived. Next, the Langevin equations are linearized, and the resonant terms are retained to obtain <i>O</i><sub>+</sub> . The amplitude of the field modes is then derived using the input-output relations. Following the experimental data from referenced literature, a numerical simulation program is implemented in Mathematica. By substituting the relevant parameters and performing calculations, the results are obtained through simulation. For the first time, the interactions among photons, magnons, microwaves, and phonons— as well as the interplay between photons in the two cavities—are investigated in a coupled cavity optomagnomechanical system. Electromagnetically induced transparency (EIT), Fano resonance, and fast-slow light effects are studied in this coupled-cavity optomagnomechanical framework. Phenomena typically examined in a single optical cavity are extended to the coupled-cavity system, with specific observations analyzed separately in the optical and microwave cavities. When <i>δ</i>=<i>ω</i><i><sub>b</sub></i>, the absorption spectrum splits, and the absorption peak decreases from its maximum to its minimum. This phenomenon arises from the disruption of quantum interference effects. The resonance condition suppresses the generation of Fano resonance. At the resonant frequency <i>ω</i><i><sub>0</sub></i>, the group delay is greater than zero, indicating slow-light propagation, and this effect is enhanced with increasing coupling strength. Additionally, a group delay of τ is achieved. Meanwhile, on either side of the resonant frequency, the group delay peaks exhibit a decreasing positive value and an increasing negative value, respectively, signifying a gradual weakening of the slow-light effect and a corresponding enhancement of the fast-light effect. This paper investigates the MIT, MMIT, and OMIT windows in a coupled-cavity optomagnomechanical (OMM) system under a strong control field and weak probe field. The MMIT phenomenon is observed through nonlinear phonon-magnon interactions. Additionally, the photon-magnon interaction in the microwave cavity leads to MIT, while OMIT is achieved via the radiation pressure interaction between photons and nonlinear phonons in the optical cavity. The frequency of the probe field is tuned to interact with both the microwave and optical cavities. When the probe field couples with the microwave cavity, its absorption at the resonant frequency is significantly suppressed under optomechanical coupling, resulting in a pronounced optical switching effect on transmission. We analyze the asymmetric Fano resonance phenomenon, which reflects the existence of quantum interference mechanisms within the system and influences the fast- and slow-light conversion processes. Furthermore, by selecting appropriate coupling parameters, not only can the fast- and slow-light effects be enhanced, but dynamic switching between them can also be achieved.

Optical bistability and multistability are experimentally demonstrated in an optical ring cavity containing three-level (Λ-type) rubidium atoms. They are sensitive to the induced atomic coherence and can be controlled by coupling field and atomic density.

The nonlinear dynamical behavior of a micromechanical resonator acting as one of the mirrors in an optical resonance cavity is investigated. The mechanical motion is coupled to the optical power circulating inside the cavity both directly through the radiation pressure and indirectly through heating that gives rise to a frequency shift in the mechanical resonance and to thermal deformation. The energy stored in the optical cavity is assumed to follow the mirror displacement without any lag. In contrast, a finite thermal relaxation rate introduces retardation effects into the mechanical equation of motion through temperature dependent terms. Using a combined harmonic balance and averaging technique, slow envelope evolution equations are derived. In the limit of small mechanical vibrations, the micromechanical system can be described as a nonlinear Duffing-like oscillator. Coupling to the optical cavity is shown to introduce corrections to the linear dissipation, the nonlinear dissipation and the nonlinear elastic constants of the micromechanical mirror. The magnitude and the sign of these corrections depend on the exact position of the mirror and on the optical power incident on the cavity. In particular, the effective linear dissipation can become negative, causing self-excited mechanical oscillations to occur as a result of either a subcritical or supercritical Hopf bifurcation. The full slow envelope evolution equations are used to derive the amplitudes and the corresponding oscillation frequencies of different limit cycles, and the bifurcation behavior is analyzed in detail. Finally, the theoretical results are compared to numerical simulations using realistic values of various physical parameters, showing a very good correspondence.

The Kerr nonlinear index of refraction for rubidium atoms is measured by using an optical ring cavity with and without electromagnetically induced transparency (EIT). Significant enhancement and inhibition of the Kerr nonlinear index is observed near resonance with EIT. The nonlinear index of refraction is measured as functions of probe and coupling frequency detunings, respectively, with and without the presence of EIT. A simple theoretical calculation including Doppler broadening is presented and is found to be in good qualitative agreement with the experimentally measured results.

We present our experimental demonstration of controlling the cavity output intensity of one laser beam with the intensity of another laser beam in a composite system consisting of a collection of three-level ?-type rubidium atoms and an optical ring cavity. When the intensity of the controlling beam is modulated with a square waveform, the cavity output power switches on and off (with a distinction ratio better than 20:1) between two steady-state values. This all-optical switching effect is the result of combined absorption and enhanced Kerr nonlinearity near resonance in such three-level atomic systems because of atomic coherence and can find applications in optical communication and optical computation.

We present an investigation of electromagnetically induced transparency (EIT) without Doppler background due to a locked probe laser. The EIT is theoretically studied based on a multilevel ladder-type cesium atomic system 6${S}_{1/2}$--6${P}_{3/2}$--8${S}_{1/2}$ in a room-temperature vapor cell. The experimental results agree with the theoretical calculations. Compared with the traditional EIT spectra with a Doppler profile limiting the spectral resolution for keeping the coupling laser locked and scanning the probe laser, these EIT spectra with the probe laser locked and the coupling laser scanned have a flat background, which seem be of great benefit for applications such as the measurement of hyperfine intervals between excited states, the study of highly excited Rydberg states, laser-frequency stabilization, etc.

In this paper, we present a rigorous investigation of the quantum phase transitions (QPTs) for a model which describes the degenerate collective regime of a three-level V-configuration atoms Bose-Einstein condensate (BEC). We consider that the three-level atoms of a BEC coupled to an optical resonator (cavity) with high finesse in the presence of a diamagnetic term, i. e., a system of N-identical three-level atoms interacting with a one-mode quantized electromagnetic cavity field. Also, the different components of this system are labeled by the different phase factors. By using the Glauber’s coherent states for the field, we calculate the free energy and study the finite-temperature phase transition for this model in the thermodynamic limit (N → ∞). Also, the potential energy surfaces constructed by taking the expectation value with respect to the direct product for a test state a direct product of coherent Heisenberg-Weyl HW(1)-states (for the electromagnetic field ), and U(3) (for the atomic field, i. e., matter contribution) coherent states. Moreover, the scaled ground-state energy is obtained by taking the expectation value of an effective Hamiltonian in the framework of the mean-field approach. In the thermodynamic limit, the energy surface takes a simple form for a direct description of the phase transitions. The relation between the QPTs and the symmetry physics are established and the QPTs for this model are investigated numerically. The properties of both stability and the equilibrium are calculated using the catastrophe theory. We notice that the second-order phase transitions (superradiant (SR)) with multi-level systems might be achievable in a wide range of physical systems, especially those where it is possible to engineer the spec-tra and oscillator strengths.

We experimentally demonstrate spatial interference between simultaneously generated four-wave mixing (FWM) and six-wave mixing (SWM) signals in a multilevel atomic system. By tuning the relative phase between the FWM and SWM processes the relative strengths between these two high-order nonlinear optical processes can be investigated.

We theoretically investigate a multimode quadratic coupling optomechanical system (OMS) assisted with N cold Λ-type three-level atoms. The coupling interaction between two mechanical membranes (MR1 and MR2) and the cavity is quadratic. The three-level atoms interact with an external controlling field and an internal cavity field. The optical cavity is driven by a strong coupling field and a weak probe field. In the resolved sideband regime, four transparency windows are obtained due to the atomic coherence and quadratic coupling in the absorption spectrum for the suitable choice of the controlling parameters. We analyze how to vary the number of the transparency window dips and to adjust their mutual spacing by changing the controlling parameters such as strength of the quadratic coupling between MR1 (MR2) and the cavity, the collective coupling strength between the atomic ensemble and the cavity field, the Rabi frequency associated with the coupling interaction between the controlling field and the three-level atom, power of the coupling field, different frequencies and decay rates of MR1 (MR2), decay rate of the cavity field, and the environmental temperature. Further, we also demonstrate the normal-mode splitting in the generated Stokes and anti-Stokes field in the OMS, where the changing collective coupling strength affects the splitting width of the output field at the Stokes (anti-Stokes) frequency.

We numerically investigate the dynamics of inward circular finite-energy Airy beams carrying different orbital angular momentum (OAM) numbers in a close-Λ three-level atomic vapor with the electromagnetically induced transparency (EIT) window. We report that due to the EIT induced by the microwave field, the transverse intensity distribution properties of Airy beam can be feasibly manipulated and modulated through adjusting OAM numbers l and the frequency detuning, as well as the propagation distance, in the multi-level atomic systems. What’s more, the rotation of the beam also can be observed with different positions in atomic ensembles. The investigation may provide a useful tool for studying particle manipulation, signal processing and propagation in graded-index (GRIN) fibers.

We found the analytical expression for cross-Kerr nonlinear coefficient in a four-level Y-type atomic system. The analytical model is applied to 85Rb atoms and shown that under electromagnetically induced transparency, cross-Kerr nonlinear coefficient is enhanced by several orders of magnitude. At the same time, the amplitude and the sign of cross-Kerr nonlinear coefficient are controlled with respect to the intensity and the frequency of the coupling laser field. The analytical model can be useful to explain the experimental results and to study related effects in nonlinear optics. Full Text: PDF ReferencesC. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media", Phys. Rev. Lett. 90, 197902 (2003). CrossRef C. Zhu, G. Huang, "Giant Kerr Nonlinearity, Controlled Entangled Photons and Polarization Phase Gates in Coupled Quantum-Well Structures", Opt. Express 19, 23364 (2011). CrossRef C. Hang, G. Huang, "Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet", Opt. Express 18(3), 2952 (2010). CrossRef M. Fleischhauer, I. Mamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media", Rev. Mod. Phys. 77, 633 (2005). CrossRef H. Schmidt, And A. Imamogdlu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency", Opt. Lett., 21(23), 1936 (1996). CrossRef H. Kang And Y. Zhu, Phys. "Observation of Large Kerr Nonlinearity at Low Light Intensities", Rev. Lett., 91, 093601 (2003). CrossRef J. Kou, R. G. Wan, Z. H. Kang, H. H. Wang, L. Jiang, X. J. Zhang, Y. Jiang, and J. Y. Gao, "EIT-assisted large cross-Kerr nonlinearity in a four-level inverted-Y atomic system", J. Opt. Soc. Am. B. 27(10), 2035 (2010). CrossRef X. Yang, S. Li, C. Zhang, and H. Wang, "Enhanced cross-Kerr nonlinearity via electromagnetically induced transparency in a four-level tripod atomic system", J. Opt. Soc. Am. B. 26(7), 1423 (2009). CrossRef C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media", Phys. Rev. Lett. 90, 197902 (2003). CrossRef H. Sun, Y. Niu, S. Jin and S. Gong, "Phase control of cross-phase modulation with electromagnetically induced transparency", J. Phys. B: At. Mol. Opt. Phys. 40, 3037 (2007). CrossRef L.V. Doai, P.V. Trong, D.X. Khoa, and N.H. Bang, "Electromagnetically induced transparency in five-level cascade scheme of 85Rb atoms: An analytical approach", Optik, 125, 3666 (2014). CrossRef D. X. Khoa, P. V. Trong, L. V. Doai and N. H. Bang, "Electromagnetically induced transparency in a five-level cascade system under Doppler broadening: an analytical approach", Phys, Scr. 91, 035401 (2016). CrossRef D.X. Khoa, L.C. Trung, P.V. Thuan, L.V. Doai and N.H. Bang, "Measurement of dispersive profile of a multiwindow electromagnetically induced transparency spectrum in a Doppler-broadened atomic medium", J. Opt. Soc. Am. B 34 (6), 1255 (2017). CrossRef D. X. Khoa, L. V. Doai, D. H. Son, and N. H. Bang, "Enhancement of self-Kerr nonlinearity via electromagnetically induced transparency in a five-level cascade system: an analytical approach", J. Opt. Soc. Am. B., 31, 1330 (2014). CrossRef L.V. Doai, N.L.T. An, D.X. Khoa, V.N. Sau and N.H. Bang, "Manipulating giant cross-Kerr nonlinearity at multiple frequencies in an atomic gaseous medium", J. Opt. Soc. Am. B 36, 2856 (2019). CrossRef D. X. Khoa, L. V. Doai, L. N. M. Anh, L. C. Trung, P. V. Thuan, N. T. Dung, and N. H. Bang, "Optical bistability in a five-level cascade EIT medium: an analytical approach", J. Opt. Soc. Am. B, Vol. 33, 735 (2016). CrossRef N. T. Anh, L. V. Doai, and N. H. Bang, "Manipulating multi-frequency light in a five-level cascade-type atomic medium associated with giant self-Kerr nonlinearity", J. Opt. Soc. Am. B 35, 1233 (2018). CrossRef N. T. Anh, L. V. Doai, D. H. Son, and N. H. Bang, "Manipulating multi-frequency light in a five-level cascade EIT medium under Doppler broadening", Optik 171, 721 (2018). CrossRef D.A. Steck, Rb85 D Line Data: http://Steck.Us/Alkalidata/rubidium85numbers.pdf CrossRef

The aim of this work is to investigate the changing effects of the atom–cavity coupling constant on an atom–cavity system. A three-level atom in the Λ configuration with q-photon transition between levels 2 and 3 is confined in a single-mode Fabry–Pérot optical cavity. To solve the master equation of this system in the steady-state by using the appropriate physical quantities, the matrix continued fractions method for recurrence equations is applied. The behavior of physical observables including atom–field correlation, mean photon number, and second-order coherence function is discussed. The effect of altering the atom–cavity coupling constant for different transitions on these observables is fully considered. The results of calculations show that by increasing this coupling constant, the range of atom–cavity correlation becomes longer, the maximum value of the output mean photon number from the cavity remains almost constant, the broadening in the curves of the mean photon number increases and the lasing process is amplified in the system. Finally, the transformation of the three-level atom into a two-level one under several specific conditions in a four-photon transition case has been studied. The obtained results of the two-level atomic pattern are adequately confirmed by the simulations related to the three-level atom.