Cold Two-level Atoms Research Articles

Enhancing nonclassical correlations for light scattered by an ensemble of cold two-level atoms.

We report the enhancement of quantum correlations for biphotons generated via spontaneous four-wave mixing in an ensemble of cold two-level atoms. This enhancement is based on the filtering of the Rayleigh linear component of the spectrum of the two emitted photons, favoring the quantum-correlated sidebands reaching the detectors. We provide direct measurements of the unfiltered spectrum presenting its usual triplet structure, with Rayleigh central components accompanied by two peaks symmetrically located at the detuning of the excitation laser with respect to the atomic resonance. The filtering of the central component results in a violation of the Cauchy-Schwarz inequality to (4.8±1.0)≰1 for a detuning of 60 times the atomic linewidth, representing an enhancement by a factor of four compared with the unfiltered quantum correlations observed at the same conditions.

Observation of Nonclassical Correlations in Biphotons Generated from an Ensemble of Pure Two-Level Atoms.

We report the experimental verification of nonclassical correlations for an unfiltered spontaneous four-wave mixing process in an ensemble of cold two-level atoms, confirming theoretical predictions by Du etal. in 2007 for the violation of a Cauchy-Schwarz inequality in the system, and obtaining R=(1.98±0.03)≰1. Quantum correlations are observed in a nanoseconds timescale in the interference between the central exciting frequency and sidebands dislocated by the detuning to the atomic resonance. They prevail over the noise background coming from Rayleigh scattering from the same optical transition. These correlations are fragile with respect to processes that disturb the phase of the atomic excitation, but are robust to variations in number of atoms and to increasing light intensities.

Nonvolatile atomic memory in the spontaneous scattering of light from cold two-level atoms

The bunching of light resulting from the spontaneous emission of an atomic ensemble is a well known effect, shared with various sources of thermal fields. It is, on the other hand, also an integral part of recently developed quantum memories for various applications. Here we theoretically and experimentally investigate the statistical correlations in light spontaneously scattered from a cold ensemble of two level atoms. We demonstrate that the bunching of light in this situation is both long-lived and robust to various parameters of the excitation process. This memory comes from constructive interference of the emitted light for particular distributions of positions of the atoms, being washed out by their movement. We offer both a first-principles quantum theory for the process and a corresponding classical theory. We also introduce various methods to treat long-term fluctuations of the experimental data in order to be able to compare it to our simple theories. As a result, we obtain a detailed picture of such spontaneous scattering processes in cold atomic ensembles, which may result in better control of quantum memories heralded by these processes.

Kinetics of spontaneous decay in cold atomic systems

We discuss the spontaneous decay in a system of cold identical two-level atoms when, due to the strong dipole–dipole interaction, the collision-induced spontaneous decay plays a leading role in the process. We show that the time profile of the spontaneous transition is essentially non-exponential. Also, we argue that at a low initial temperature of the atomic system the spontaneous decay is accompanied by strong heating caused by the inelastic atom–atom collisions. We show that the spontaneous emission spectrum is asymmetric. In addition, the width of the emission spectrum is a function of time. When atoms decay, the emission spectrum becomes broader. The spectrum’s asymmetry and the atomic system’s heating have the same physical origin: coming from the peculiarities of the atoms’ distribution function.

Collective Shift in Resonant Light Scattering by a One-Dimensional Atomic Chain.

We experimentally study resonant light scattering by a one-dimensional randomly filled chain of cold two-level atoms. By a local measurement of the light scattered along the chain, we observe constructive interferences in light-induced dipole-dipole interactions between the atoms. They lead to a shift of the collective resonance despite the average interatomic distance being larger than the wavelength of the light. This result demonstrates that strong collective effects can be enhanced by structuring the geometrical arrangement of the ensemble. We also explore the high intensity regime where atoms cannot be described classically. We compare our measurement to a mean-field, nonlinear coupled-dipole model accounting for the saturation of the response of a single atom.

Light reflection and transmission in planar lattices of cold atoms.

Manipulation of light using atoms plays a fundamental and important role in emerging technologies such as integrated photonics, information storage, and quantum sensors. Specifically, there have been intense theoretical efforts involving large samples of cold neutral atoms for coherent control of light. Here we present a theoretical scheme that enables efficient computation of collective optical responses of mono- and bi-layer planar square lattices of dense, cold two-level atoms using classical electrodynamics of coupled dipoles in the limit of low laser intensity. The steady-state transmissivity and reflectivity are obtained at a field point far away from the atomic lattices in the regime with no Bragg reflection. While our earlier method was based on exact solution of the electrodynamics for a small-scale lattice, here we calculate the dipole moments assuming that they are the same at all lattice sites, as for an infinite lattice. Atomic lattices with effectively over one hundred times more sites than in our earlier exact computations can then be simulated numerically with fewer computational resources. We have implemented an automatic selection of the number of sites under the given convergence criteria. We compare the numerical results from both computational schemes. We also find similarities and differences of a stack of two atomic lattices from a two-atom sample. Such aspects may be exploited to engineer a stack for potential applications.

Enhancement of long-distance Casimir-Polder interaction between an excited atom and a cavity made of metamaterials.

Within the framework of macroscopic quantum electrodynamics, we investigate both the radiation force and the potential of Casimir-Polder type acting on an excited cold two-level atom in a cavity made of left-handed materials and topological insulators. As the time-reversal symmetry is broken on the surface of the topological insulators, the spontaneous emission of the atom placed near the focus point(s) exhibits anisotropic properties. While the potential wells are normally shallow for topological trivial dielectric, they may be amplified in the presence of topological magnetoelectric effect. We find that when there exists only one focus point in the cavity, it is possible to boost the forces or the potential wells by up to one order of magnitude. Meanwhile, the lifetime of the atom could be prolonged owing to the focus effect of the left-handed materials, where the emitted photons can trace back to the atom and reabsorbed by itself. Our results indicate the possibility in forming long-lived potential wells, which may have potential applications in trapping and guiding cold atoms far away from the surface.

The Effect of Internal Dynamics on the Motion of Cold Atoms in 2D Optical Lattices with Interfering Laser Beams

We show theoretically and numerically that cold two-level atoms demonstrate unexpectedly complicated motion in an absolutely rigid two-dimensional optical lattice with interfering laser beams. The point-like atoms can move there in a chaotic way resembling motion of particles in a random potential. Chaos arises if the laser frequency is close to the atomic transition frequency where one should take into account the excitation of internal atomic states by the laser field. After loading cold atoms to the lattice, their induced electric dipole moments interact with a spatially inhomogeneous electric field of the two-dimensional standing laser wave. We find the range of the atom–field detuning where the synchronized (with the laser field) component of the atomic dipole moment changes in a random-like manner just after crossing the nodal lines of the standing wave where the electric field is zero. It, in turn, causes irregular changes in the momentum of atoms, leading eventually to chaotic walking of cold atoms in a deterministic potential. We show that adjusting the single control parameter, the atom–field detuning, it is possible to switch between regular and chaotic regimes of motion.

Coherent scattering of near-resonant light by a dense, microscopic cloud of cold two-level atoms: Experiment versus theory

We measure the coherent scattering of low-intensity, near-resonant light by a cloud of laser-cooled two-level rubidium atoms with a size comparable to the wavelength of light. We isolate a two-level atomic structure by applying a 300G magnetic field. We measure both the temporal and the steady-state coherent optical response of the cloud for various detunings of the laser and for atom numbers ranging from 5 to 100. We compare our results to a microscopic coupled-dipole model and to a multi-mode, paraxial Maxwell-Bloch model. In the low-intensity regime, both models are in excellent agreement, thus validating the Maxwell-Bloch model. Comparing to the data, the models are found in very good agreement for relatively low densities ($n/k^3\lesssim 0.1$), while significant deviations start to occur at higher density. This disagreement indicates that light scattering in dense, cold atomic ensembles is still not quantitatively understood, even in pristine experimental conditions.

Thick-medium model of transverse pattern formation in optically excited cold two-level atoms with a feedback mirror

We study a pattern forming instability in a laser driven optically thick cloud of cold two-level atoms with a planar feedback mirror. A theoretical model is developed, enabling a full analysis of transverse patterns in a medium with saturable nonlinearity, taking into account diffraction within the medium, and both the transmission and reflection gratings. Focus of the analysis is on combined treatment of nonlinear propagation in a diffractively- and optically-thick medium and the boundary condition given by feedback. We demonstrate explicitly how diffraction within the medium breaks the degeneracy of Talbot modes inherent in thin slice models. Existence of envelope curves bounding all possible pattern formation thresholds is predicted. The importance of envelope curves and their interaction with threshold curves is illustrated by experimental observation of a sudden transition between length scales as mirror displacement is varied.

Casimir-Polder force on a two-level atom in a structure containing metamaterials

We study the Casimir-Polder (CP) force on an excited cold two-level atom in structures containing metamaterials. We adopt two kinds of metamaterials: left-handed materials (LHMs) and zero-index materials (ZIMs). The CP force on an excited atom can be divided into two parts: the dispersive force that responds to all frequencies of the electromagnetic mode and the resonant force, which is determined by the frequency at the atomic transition. Left-handed materials and ZIMs can significantly modify the resonant part of the CP force due to their unique character. It is found here that the presence of LHMs can enhance the force on the atom far away from the surface due to its phase compensation, while the presence of ZIMs can lead to a force that is independent of dipole orientation. The Casimir effect within the combination of LHMs and ZIMs leads us to realize a potential well that is insensitive to the orientation of atomic dipole. Due to the spontaneous decay, the resonant part of the CP force disappears eventually; however, the decay at the position with maximum force is inhibited. Therefore, during the time evolution, there are special positions (focuses) at which the force is significant for a longer time. Our results show the trap effect that can work on an atom with arbitrary dipole orientation. This provides a method to either trap or reflect an atom in a position far away from surface.

Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms

The interplay between the superradiant emission of a cloud of cold two-level atoms and the radiation pressure force is discussed. Using a microscopic model of coupled atomic dipoles driven by an external laser, the radiation field and the average radiation pressure force are derived. A relation between the far-field scattered intensity and the force is derived, using the optical theorem. Finally, the scaling of the sample scattering cross-section with the parameters of the system is studied.

Spontaneous optomechanical pattern formation in cold atoms

Transverse pattern formation in an optical cavity containing a cloud of cold two-level atoms is discussed. We show that density modulation becomes the dominant mechanism as the atomic temperature is reduced. Indeed, for low but achievable temperatures the internal degrees of freedom of the atoms can be neglected, and the system is well described by treating them as mobile dielectric particles. A linear stability analysis predicts the instability threshold and the spatial scale of the emergent pattern. Numerical simulations in one and two transverse dimensions confirm the instability and predict honeycomb and hexagonal density structures, respectively, for the blue and red detuned cases.

Matter-wave propagation in optical lattices

Coherent propagation of atomic-matter waves in a one-dimensional optical lattice is studied. Wave packets of cold two-level atoms propagate simultaneously in two optical potentials in a dressed-state basis. Three regimes of the wave-packet propagation are specified by the quantity Δ2/ωD, where Δ and ωD are the dimensionless atom–laser detuning and the Doppler shift, respectively. At Δ2/ωD ≫ 1, the propagation is essentially adiabatic, at Δ2/ωD ≪ 1, it is (almost) resonant, and at Δ2 ≃ ωD, the wave packets propagate nonadiabatically, splitting at each node of the standing wave. The latter means that the atom makes a transition from one potential to the other one when crossing each node, and the probability of that transition is given by a Landau–Zener-like formula. All the regimes of propagation are studied with δ-like and Gaussian wave packets in the momentum and position spaces. Varying the control parameters, we can create wave packets trapped in a well of optical potentials and moving ballistically in a given direction in close analogy with point-like atoms. Within some range of the parameters, we force the atom to move in a pure quamtum-mechanical manner in such a way that a part of the packet is trapped in a well, and the other part propagates ballistically. The propagation modes are found to be characterized by different types of time evolution of the uncertainty product and the Wigner function.

The pump-probe approach to coherent backscattering of intense laser light from cold atoms

We present a new approach to near-resonant coherent backscattering of light from cold two-level atoms. In the dilute regime, where the distance between atoms is much larger than the laser wavelength, this approach is able to account for multi-photon scattering processes between many atoms through solutions of single-atom optical Bloch equations. We elaborate the method for double scattering from two atoms, and discuss the way of its extension for dilute, cold atomic clouds.

Quantum carpets woven by cold atoms in a laser field

Propagation of wave packets of cold two-level atoms in a standing-wave laser field can be interpreted in the dressed-state basis as motion in two optical potentials. The three distinct regimes of the wavepacket motion are specified by the ratio of the squared atom–laser detuning to the normalized Doppler shift. We calculate the momentum and position probability densities, which form patterns with minima and maxima of probability both in the momentum and the position spaces known as quantum carpets. At small and large detunings, the atomic motion is substantially adiabatic, and the quantum carpets have a simple form. At intermediate detunings, the wave packet moves nonadiabatically, splitting at each node of the standing wave, which causes a proliferation or branching of atomic trajectories with a single atom. Nonadiabatic transitions produce beautiful quantum carpets with a rich structure.

TWO-PHOTON QUANTUM-STATISTICAL PROPERTIES OF THE SINGLE-MODE CAVITY FIELD INTERACTING WITH A PAIR OF COLD ATOMS

The interaction between the pair of cold two-level atoms and the single-mode cavity field is investigated. The two-level atoms in the pair are supposed to be indistinguishable. This problem generalizes the two-photon Jaynes-Cummings model of a single two-level atom interacting with the squeezed vacuum. The model of the pair of indistinguishable two-level atoms is equivalent to the problem of the equidistant three-level radiator with equal dipole moment matrix transition elements between the adjacent energy levels. Supposing that at the initial moment the field is in the squeezed vacuum state we obtain the exact analytical solution for the atom-field state-vector. By using this solution the quantum-statistical and squeezing properties of the radiation field are investigated. The obtained results are compared with those for the single two-level atom system. We observe that in the model of the pair of cold two-level atoms the exact periodicity of the squeezing revivals is violated by the analogy with the single two-level atom one.

Cold three-level atom micromaser in the far-detuning regime

We analyze the quantum theory of the two-mode micromaser pumped by cold three-level atoms with a $\ensuremath{\Lambda}$-configuration under the two-photon resonance condition. We focus more specifically on the large detuning limit where the detuning between the two cavity mode frequencies and the two atomic transition frequencies is large in comparison with the atom-cavity coupling constants. We show that this regime can mimic a virtual two-level atom micromaser without spontaneous emission where the cavity acts for the atoms as a potential barrier or a potential well (depending on the sign of the detuning) and a zero potential but no more both as a potential barrier and a potential well as it is the case for the usual cold two-level atom micromaser [M. O. Scully, G. M. Meyer, and H. Walther, Phys. Rev. Lett. 76, 4144 (1996)]. This introduces interesting options for engineering the interaction between the cold atoms and the cavity. The elimination of the spontaneous emission solves a major issue of the conventional cold two-level atom micromaser as the transit time of cold atoms inside and in the vicinity of the cavity is usually much larger than the atomic level lifetimes. It is also shown that the cold atom regime is very sensitive to the sign of the detuning (in contrast to the hot atom regime) and that, according to this sign, the cavity may speed up or slow down the incident three-level atoms after their interaction with the field.

Coherent backscattering of light with nonlinear atomic scatterers

We study coherent backscattering of a monochromatic laser by a dilute gas of cold two-level atoms in the weakly nonlinear regime. The nonlinear response of the atoms results in a modification of both the average field propagation (nonlinear refractive index) and the scattering events. Using a perturbative approach, the nonlinear effects arise from inelastic two-photon scattering processes. We present a detailed diagrammatic derivation of the elastic and inelastic components of the backscattering signal for both scalar and vectorial photons. In particular, we show that the coherent backscattering phenomenon originates in some cases from the interference between three different scattering amplitudes. This is in marked contrast with the linear regime where it is due to the interference between two different scattering amplitudes. In particular we show that, if elastically scattered photons are filtered out from the photodetection signal, the nonlinear backscattering enhancement factor exceeds the linear barrier of 2, consistently with a three-amplitude interference effect.

Chaos and flights in the atom-photon interaction in cavity QED.

We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with <x(2)> approximately tau(1.13) at delta=1.2 to a superdiffusion with <x(2)> approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.