Degenerate Two-level System Research Articles

Unveiling the Impact of Degeneracy in an Isolated Two-Level System: An Undergraduate Problem

The study of isolated two-level systems (e.g. paramagnetic salt) is an integral part of every statistical mechanics coursework at undergraduate level. However, the thermodynamical properties of such a system are usually explained with very brief introduction to the degeneracy associated with the levels. Therefore, in this study, we revisited the macroscopic properties such as entropy, temperature and specific heat of a degenerate two-level system (both symmetric and asymmetric cases). We believe that this simple and analytic calculation may give very useful insight to undergraduate students.

Cavity-Induced Optical Nonreciprocity Based on Degenerate Two-Level Atoms.

We developed and experimentally realized a scheme of optical nonreciprocity (ONR) by using degenerate two-level atoms embedded in an optical ring cavity. For the degenerate transition Fg = 4 ↔ Fe = 3, we first studied the cavity-transmission property in different coupling field configurations and verified that under the strong-coupling regime, the single-dark-state peak formed by electromagnetically induced transparency (EIT) showed ONR. The stable ground-state Zeeman coherence for Λ-chains involved in the degenerate two-level system was found to be important in the formation of intracavity EIT. However, different from the three-level atom-cavity system, in the degenerate two-level system, the ONR effect based on intracavity EIT occurred only at a low probe intensity, because the cavity-atom coupling strength was weakened in the counter-propagating probe and coupling field configuration. Furthermore, ONR transmission with a high contrast and linewidth-narrowing was experimentally demonstrated.

Double Hanle resonance dependence on light polarization angle and transverse magnetic field direction

We present experimental and theoretical investigations of the dependence of double Hanle resonance spectrum on the light polarization angle and the direction of the applied transverse magnetic field (TMF). The experiments are done for Fg=2→Fe=1 transition of 87Rb D1 line using a rubidium vapor cell containing buffer gas. We show that a small light polarization component along the direction of TMF introduces asymmetry in the double Hanle resonance signal. Both the magnitude and sign of asymmetry in the signal are sensitive to the TMF orientation, suggesting a possible method for in-situ measurement of the direction of the magnetic fields generated by the coils. The physical origin of this asymmetry is explained by considering the redistribution of population among the ground-state Zeeman sublevels in the presence of TMF. In addition, we systematically vary both the polarization angle and TMF direction to study their effect on the line profile of Hanle resonances. We demonstrate that a double Hanle resonance changes to a dark Hanle resonance by rotating the light polarization vector irrespective of the TMF direction. We have developed a simple theoretical model based on a degenerate two-level system to explain our experimental observations.

Optical gain in a degenerate two-level system in the presence of a transverse magnetic field

Here we report a theoretical model that allows magnetically assisted narrowband probe beam amplification in a closed two-level system with triple Zeeman degeneracy. The gain mechanism can be understood in terms of coherent population transfer between the levels of the Zeeman ground state induced by the external magnetic field. The theoretical model we develop here using a simpler atomic transition Fg=1→Fe=0, in agreement with the experimental results already observed in recent literature. Additionally, we validate our theoretical predictions with the results obtained via density matrix numeric calculation and we believe our theoretical results could useful in communication protocols involving distant nodes.

Measurement of steep dispersion associated with magnetically assisted gain in Zeeman degenerate two-level system of cold atoms

We report on the measurements of magnetically assisted dispersion and gain in a Zeeman degenerate two-level system associated with the closed D 2 hyperfine transition Fg=3→Fe=2 cold cesium atoms. We use a Mach–Zehnder interferometer to measure the dispersive properties of this system in the presence of a magnetic field, transversal to the quantization direction, when the atomic system is excited by a coupling and probe beams with orthogonal polarizations. We have measured the spectrum of the refraction index for different magnitudes of the external magnetic field and different coupling beam intensities. A previously published theoretical model, using a simpler atomic transition Fg=1→Fe=0 , accounts reasonably for the experimentally observed gain and dispersive lineshapes.

Effects of neighboring transitions on electromagnetically induced absorption and transparency in 87Rb atoms with respect to varying hyperfine spacings

Effects of neighboring transitions through Doppler broadening on electromagnetically induced absorption (EIA) and electromagnetically induced transparency (EIT) of the degenerate two-level system at the F g =2→F e =1,2, and 3 transitions of the 87Rb D2 line have been investigated. Two different polarization configurations, such as linear parallel and linear perpendicular polarizations of coupling and probe lasers, have been studied. We study the significant variation of EIT and EIA due to the effect of the neighboring transitions via artificial variation of the hyperfine spacings of the 5P3/2 state. We have a new observation such as EIT instead of EIA due to the neighboring effect, even in the case of closed systems such as F g =2→F e =3 of 87Rb for the linear parallel configuration, for the first time to our best knowledge. The observed spectra match well with theoretically calculated spectra. EIAs for the open F g =2→F e =1 and 2 transitions at the linear perpendicular polarization are observed due to neighboring effects. This system may be applied to optical switching systems for future quantum processing application.

Effects of neighboring transitions on electromagnetically induced absorption and transparency in 85Rb atoms based on the linear parallel polarization of coupling and probe beams

The effects of neighboring transitions through Doppler broadening on the electromagnetically induced absorption (EIA) and electromagnetically induced transparency (EIT) of the degenerate two-level system (DTLS) in 85Rb atoms with respect to the linear parallel polarization configuration of coupling and probe lasers have been investigated. The calculated coherent absorption profiles with and without neighboring effects have been compared to elucidate the neighboring effects between the DTLS and nearby states for the conversion between EIA and EIT. Even in the case of the closed system, i.e., Fg=3→Fe=4 of 85Rb, for the first time, EIT instead of EIA has been observed owing to the neighboring effect of dominant Fg=3→Fe=3 on the linear parallel polarization configuration case of 85Rb in the low power regime. All the transitions (Fg=3→Fe = 2 and 3 and Fg=2→Fe = 1, 2, and 3) of linear parallel polarization cases exhibit EIT owing to dominant neighboring EIT in the DTLS. In each polarization scheme, the observed spectral trends of the transitions are similar owing to the neighboring effect. The EIT spectral trends with the neighboring effect of Fg=2→Fe = 1, 2, and 3 are governed by the openness of pure DTLS calculation.

Effects of neighboring transitions on the mechanisms of electromagnetically induced absorption and transparency in an open degenerate multilevel system

In this study, optical Bloch equations with and without neighboring hyperfine states near the degenerate two-level system (DTLS) in the challenging case of ^{85}Rb D2 transition, which involves the Doppler broadening effect, are solved. The calculated spectra agree well with the experimental results obtained based on the coupling-probe scheme with orthogonal linear polarizations of the coupling and probe fields. The mechanisms of electromagnetically induced absorption (electromagnetically induced transparency) for the open F_g=3 rightarrow F_e=2 and 3 transitions (open F_g=2 rightarrow F_e=2 and 3 transitions) are determined to be the effect of the strong closed F_g=3 rightarrow F_e=4 transition line (strong closed F_g=2 rightarrow F_e=1 transition line); this finding is based on a comparison between the calculated absorption profiles of the DTLS without neighboring states and those of all levels with neighboring states, depending on the coupling and probe power ratios. Furthermore, based on the aforementioned comparison, the crucial factors that enhance or reduce the coherence effects and lead to the transformation between electromagnetically induced absorption and electromagnetically induced transparency, are (1) the power ratios between the coupling and probe beams, (2) the openness of the excited state, and (3) effects of the neighboring states due to Doppler broadening in a real atomic system.

Enhancing quantum annealing performance by a degenerate two-level system

Quantum annealing is an innovative idea and method for avoiding the increase of the calculation cost of the combinatorial optimization problem. Since the combinatorial optimization problems are ubiquitous, quantum annealing machine with high efficiency and scalability will give an immeasurable impact on many fields. However, the conventional quantum annealing machine may not have a high success probability for finding the solution because the energy gap closes exponentially as a function of the system size. To propose an idea for finding high success probability is one of the most important issues. Here we show that a degenerate two-level system provides the higher success probability than the conventional spin-1/2 model in a weak longitudinal magnetic field region. The physics behind this is that the quantum annealing in this model can be reduced into that in the spin-1/2 model, where the effective longitudinal magnetic field may open the energy gap, which suppresses the Landau–Zener tunneling providing leakage of the ground state. We also present the success probability of the Λ-type system, which may show the higher success probability than the conventional spin-1/2 model.

Koopman wavefunctions and classical-quantum correlation dynamics.

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman–von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical–quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect—the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical–quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

Quantum Phase Transition in Fully Connected Quantum Wajnflasz–Pick Model

We construct a quantum Wajnflasz-Pick model that is a generalized quantum Ising model, and investigate a nature of quantum phase transitions of the model with infinite-range interactions. Quantum phase transition phenomena have drawn attention in the field of quantum computing as well as condensed matter physics, since the phenomena are closely related to the performance of quantum annealing (QA) and adiabatic quantum computation (AQC). We add a quantum driver Hamiltonian to the Hamiltonian of classical Wajnflasz-Pick model. The classical Wajnflasz-Pick model consists of two-level systems as with the usual Ising model. Unlike the usual Ising spin, each of the upper and the lower levels of the system can be degenerate. The states in the upper level and the lower level are referred to as upper states and lower states, respectively. The quantum driver Hamiltonian we introduced causes spin flip between the upper and the lower states and state transitions within each of the upper and the lower states. Numerical analysis showed that the model undergoes first-order phase transitions whereas a corresponding quantum Ising model, quantum Curie-Weiss model, does not undergo first-order phase transitions. In particular, we observed an anomalous phenomenon that the system undergoes successive first-order phase transitions under certain conditions. The obtained results indicate that the performance of QA and AQC by using degenerate two-level systems can be controlled by the parameters in the systems.

Absorption and quantum coherence of a degenerate two-level system in the presence of a transverse magnetic field in different directions

We investigate the absorption spectra for the ${F}_{g}=1\ensuremath{\rightarrow}{F}_{e}=0$ transition of the ${D}_{2}$ line of $^{87}\mathrm{Rb}$ excited by a linearly polarized light in the presence of a transverse magnetic field (TMF) in different directions. Using the theoretical methods in quantum optics, we obtain the analytical expressions for the atomic density-matrix elements and the absorption spectrum. We analyze the splitting proportional to the magnitude of the TMF in the coherent population trapping absorption spectrum and explain its physical origin. Then, the sensitivity of the resonance signal to the phase (in our case, to the angle between the TMF and the light polarization) is investigated. We show and explain the splitting of the bright resonance and find the simple dependence of the separation of this splitting on the magnitude and the direction of the TMF. Therefore, this system may provide a possible method for the measurement of the magnetic field vector. In addition, we study the coherence of the system based on an observable coherence measure, namely, robustness of coherence (ROC). We find that in the case where the absorption signal becomes difficult to detect, the change of the ROC spectrum becomes more obvious and the extreme points are well maintained. Therefore, this may provide a scheme for the measurement of the magnetic field which is complementary to that based on the absorption signal of the laser.

Magnetically assisted optical gain in Zeeman degenerate two-level systems of cold atoms

Abstract We report on the observation of magnetically assisted narrow band probe beam amplification in Zeeman degenerate closed two-level system associated with cold cesium atoms. In the experiment the closed cesium D 2 hyperfine transition F g = 3 → F e = 2 is coherently excited by a probe and an orthogonally polarized coupling beam in the presence of a transverse dc magnetic field. Single pass gain of 80% is observed for moderate values of the magnetic field and coupling beam power. The gain mechanism can be understood in terms of population transfer between the different Zeeman ground-state levels induced by the external magnetic field. We have developed a simple theoretical model using a simpler atomic transition F g = 1 → F e = 0 , which accounts reasonably for the main experimentally observed results.

Transverse magnetic field effect on the giant Goos–Hänchen shifts based on a degenerate two-level system

We study the effect of the Goos–Hänchen (GH) shifts through a cavity with degenerate two-level systems in the line of . For this purpose, we focus on the transverse magnetic field (TMF) in a Floquet frame to obtain the giant GH shifts. Physically, the collisional effects of TMF lead to increasing the population trapping in the ground state. However, we demonstrate that the population trapping generates the large negative or positive GH shifts and simultaneously switches from superluminal to subluminal (or vice versa). Also, we investigate the other optical properties such as the longitudinal magnetic field (LMF), which plays an important role in the control of the GH shifts and leads to the generation of new subsystems. In the next step, we evaluate the GH shifts beyond the multi-photon resonance condition by the control of TMF. Moreover, we compute the appearance of negative and positive GH shifts by setting the width of the incident Gaussian beams in the presence of a multi-photon resonance condition. Our results show that superluminal or subluminal light propagation can be simultaneously controlled by adjusting the rates of the TMF and LMF. The significant effects of these factors on the degenerate two-level systems provide different applications such as slow light, optical switches and quantum information storage.

Superluminal reflection and transmission of light pulses via resonant four-wave mixing in cesium vapor.

We report the experimental manipulation of the group velocities of reflected and transmitted light pulses in a degenerate two-level atomic system driven by a standing wave, which is created by two counter-propagating light beams of equal frequencies but variable amplitudes. It is shown that the light pulse is reflected with superluminal group velocity while the transmitted pulse propagates from subluminal to superluminal velocities via changing the power of the backward coupling field. We find that the simultaneous superluminal light reflection and transmission can be reached when the power of the backward field becomes closer or equal to the forward power, in this case the periodical absorption modulation for photonic structure is established in atoms. The theoretical discussion shows that the anomalous dispersion associated with a resonant absorption dip within the gain peak due to four-wave mixing leads to the superluminal reflection, while the varying dispersion from normal to anomalous at transparency, transparency within absorption, and electromagnetically induced absorption windows leads to the subluminal to superluminal transmission.

Electromagnetically induced absorption due to transfer of coherence and coherence population oscillation for the [formula omitted] transition in 85Rb atoms

Lineshapes for the electromagnetically induced absorption (EIA) of thermal 85Rb atoms in a degenerate two-level system have been investigated using matching (σ∥σ,π∥π) and orthogonal (σ⊥σ,π⊥π) polarization configurations of coupling and probe beams. EIA signals, which result from coherence population oscillation and transfer of coherence of the excited state, are obtained in detail theoretically and experimentally. The observed EIA linewidths, which are limited due to the decoherence rate between the magnetic sublevels in the ground state from transit-time relaxation, match well with the calculated ones. Decompositions of the absorption signals analyzed with respect to magnetic sublevels of the ground state show that enhanced or decreased absorption signals for each component of magnetic sublevels in the ground state depend on several factors. These factors include the decay rates and transition strengths, which determine the overall absorption spectral profile.

Electromagnetically induced absorption and transparency in degenerate two level systems of metastable Kr atoms and measurement of Landé g-factor

We report electromagnetically induced absorption (EIA) and transparency (EIT) resonances of sub-natural linewidth in degenerate two level systems (DTLSs) of metastable 84Kr (84Kr⁎) and 83Kr (83Kr⁎) atoms. Using the spectrally narrow EIA signals obtained corresponding to the closed hyperfine transition 4p55s[3/2]2(F=13/2) to 4p55p[5/2]3(F′=15/2) in 83Kr⁎ atom, we have measured the Landé g-factor (gF) for the lower hyperfine level involved in this transition by application of small values of magnetic field of few Gauss.

On the efficiency of 1D atom localisation via EIT in a degenerate two-level atomic system

We analyse one-dimensional (1D) subwavelength atom localisation in a cold atomic medium under the action of two optical fields, the standing-wave and travelling probe fields, in the presence of a magnetic field. Optical Bloch equations are solved numerically for the hyperfine atomic transition of the 87Rb D1 line. All Zeeman sublevels are included in the calculations. This atomic scheme allows electromagnetically induced transparency (EIT) if the applied magnetic field is zero or small. The results for the position-dependent probe absorption are presented for two configurations, depending on the orientation of the magnetic field with respect to the optical fields’ polarisations. The efficiency of the atom localisation is analysed for a large range of field intensities and applied magnetic fields. The observed behaviour of the probe absorption is analysed through the effects of EIT induced by two fields of various strengths and its dependence on the applied magnetic fields.

The manipulation of light pulse from subluminal to superluminal propagation in a degenerate two-level Cs atomic system

We experimentally demonstrate the propagation of light pulse from subluminal to superluminal light based on quantum coherence in a degenerate two-level atomic system in a Cs vapor cell. It is shown that the group velocity of light pulse can be switched from subluminal to superluminal propagation via changing the coupling field from a traveling wave to a standing wave, while can also be continuously manipulated by varying the intensity of two waves superposed to form a standing wave. The observed maximum delay and advance times are about 0.45 and 0.54 μs, corresponding to the group velocity of Vs = 168 km/s and vg =−138 km/s, respectively. This investigation may have the practical applications of devices for optical tunable delay lines, optical switching and optical buffering.

Cascade splitting of two atomic energy levels due to multiphoton absorption

We have theoretically and experimentally studied the spectroscopic properties of dressed levels in a strong monochromatic field, and propose a model of cascade splitting of two atomic energy levels. In this model two related dressed levels can be split into four levels, and transitions connecting four new levels will constitute spectroscopic structures. Two types of proof-in-principle experiments are performed to verify the model. One experiment measures the probe absorption spectra of a degenerate two-level atomic system with two strong monochromatic coupling fields. The system consists of ${5}^{2}{S}_{1/2},F=2$ and ${5}^{2}{P}_{3/2},{F}^{\ensuremath{'}}=3$ states of $^{87}\mathrm{Rb}$ atoms in a magneto-optical trap (MOT) as well as the cooling beams and an additional coupling field. New spectral features are observed and proven to be due to the transitions of new levels generated by splitting of the dressed levels. The other experiment measures the pump-probe spectra in a degenerate two-level atomic system with one strong monochromatic coupling field. The system consists of ${5}^{2}{S}_{1/2},F=2$ and ${5}^{2}{P}_{3/2},{F}^{\ensuremath{'}}=3$ states of the $^{87}\mathrm{Rb}$ atom in a magneto-optical trap and one coupling field. We have observed spectral features that obviously differ from the prediction that comes from the two-level dressed-atom approach. They cannot be explained by existing theories. The model of cascade splitting of two atomic energy levels is employed to explain the observations in these two types of experiments.