Nir Davidson's Cold Atoms Group

I. Introduction

The main effort of my group is in the filed of ultra-cold atoms and Bose-Einstein condensation (BEC). One major effort is to study the motion of ultra-cold atoms confined in dark optical traps shaped as optical billiards, and to study the interplay between their dynamics and spectrum. We believe that by combining the dark trapping approach which suppresses most of the perturbations to the trapped atoms, and our demonstration of optical billiards, where the small remaining perturbations are precisely controlled, may result in even longer atomic coherence times. Our long term goal is to establish ultra cold atoms as a tool to study the transition from classical to quantum nonlinear dynamics. Our second main effort is to study the spectrum, dynamics and decay of excitations  over a Bose-Einstein condensates. We start from simple conditions where linear perturbation theory is suitable and then study the effects of strong driving nonlinearities, confinement and inhomogeneity, strong interactions, wave mixing and periodic potential, where  new theories and concepts must be introduced.

Additional activities involve several fields in unconventional optics. These include: (1) electromagnetic induced transparency, where we concentrate on slowing and storing light beams with interesting spatial distribution such as topologically-stable helical beams and generalized images, (2) increasing the power and improving the beam quality of lasers by selecting pure high order modes and by phase locking and coherent combining of several lasers, and (3) developing new methods and elements for high numerical aperture laser focusing and microscopy, and diffuse light concentration at the thermodynamic limit, using new types of curved diffractive and reflecting optical elements, and by exploiting vectorial effects of the electromagnetic field which become important at large focusing angles. The expertise we achieved in these unconventional optical techniques were valuable also for manipulation of ultra cold atoms. Indeed, during the last years my group has used a variety of these techniques (fast scanning beams, holography and interference) to develop novel dark optical traps for ultra-cold atoms with better control over both internal degrees of freedom of the trapped atoms (related to spectroscopy) and their external degrees of freedom (related to dynamics).

Our experimental facility includes two main laboratories for atomic physics: one is devoted mainly to investigations of dark optical traps and optical billiards with laser cooled rubidium atom, and the second is devoted to research of Bose-Einstein condensation. Both labs contain ultra-high vacuum chambers, many tunable and frequency-stabilized lasers for cooling, trapping and manipulation the atoms, and various optical, mechanical and electrical equipment. A third laboratory serves our optical experiments.

Below our recent research activities in five main topics are reviewed: (1) dynamics and spectroscopy with ultra cold atoms in dark optical traps and billiards, (2) excitations in Bose-Einstein condensation, (3) spatial effects in electromagnetic induced transparency, (4) mode selection and coherent combining of lasers, and (5) optical focusing, concentration and microscopy with high numerical aperture systems. Finally, our future research plans are briefly outlined.

II. Research Activities

1. Classical and quantum dynamics in optical "billiards"

The strong suppression of Doppler broadening, and the possibility of long interaction times are obvious advantages of using free-falling ultra-cold atoms for precision spectroscopic measurements. Even longer interaction times are obtained for optically trapped atoms. To obtain long atomic coherence times in these traps, spontaneous scattering of photons and energy level perturbations caused by the trapping laser are reduced by increasing the laser detuning from resonance. To further reduce scattering, blue detuned optical traps, where repulsive light forces confine atoms mostly in the dark (“dark” traps), have been developed.

A main effort of my group, including my students N. Friedman, A. Kaplan, M. Andersen and T. Grunzweig was to develop new types of dark optical traps for ultra-cold atoms, where dephasing of the atomic coherence due to interaction with the trapping laser beams are largely suppressed, to enable long atomic coherence times. We investigated the properties of the trapped atoms using both spectroscopic and dynamical approaches. The research methology consisted of three main stages. First, to construct light distributions which are capable of trapping atoms “in the dark” using variety of unconventional optical techniques, e.g. rapidly scanning laser beams or diffractive optical elements. Second, to conduct precise and sensitive spectroscopic measurements to determine the amount of these perturbations, and third to study the remaining small perturbation by controlling the type of dynamics of atoms in the traps. Hence, different types of atomic dynamics (e.g. chaotic, regular or mixed dynamics) were demonstrated by changing the shape of the trap boundary. Such a system which we named an optical billiard, has many interesting aspects of its own.

Classical dynamics

We trapped ultra-cold atoms by arbitrary-shaped closed two-dimensional boundaries, known as optical billiards. These billiards were used to demonstrate different types of atomic motion, ranging from regular motion in circular and elliptical billiards, to chaotic motion in a Bunimovich-stadium billiard (see Fig. 1). The effects of decoherence on the stability of the motion in elliptical billiards was quantitatively investigated, and a generalized criteria was found. We also investigated the atomic dynamics in billiards with mixed phase-space, islands of stability and stickiness induced by wall softness and curved atomic trajectories in the presence of external force fields. In these experiments we established that we can precisely control the phase space structure, generate hierarchical structure of islands that leads to Levi-type dynamics, confirm mathematical predictions on smooth billiards and even predicted a new mechanism for the formation of islands, namely the smoothening of corners . Recently, together with my student A. Ridinger, we analyzed the effects of atomic motion in rapidly oscillating laser beams and discovered a surprising and strong dependence on initial phase slips even for arbitrarily high oscillation frequency.

Figure 1: Survival probability of atoms in billiards of various shapes, as a function of time after opening a hole in the boundary. (a) Elliptical billiard. The solid symbols denote the unperturbed case, in which the surviving fraction for the ellipse with the hole on the long side (solid squares) decays much faster than for the hole on the short side (solid circles). The open symbols show the case in which velocity randomizing molasses pulses are applied every 3 ms. The decay is now nearly identical for the two hole positions. (b) Decay of atoms from circular and stadium billiards. The decay from the stadium billiard (solid circles) shows a nearly pure exponential decay (dashed line). For the circle (solid diamonds) the decay curve flattens, indicating the existence of nearly stable trajectories. The solid lines represent numerical simulations, including all the experimental parameters, and no fitting parameters. The insets show CCD-camera images of the billiards’ cross sections at the beam focus.

Spectroscopy and quantum dynamics

While preserving long atomic coherence times, dark traps can provide tight confinement of trapped atoms to ensure good spatial overlap with a tightly focused excitation laser beam, yielding a further increase in sensitivity. We presented a new and ultra-sensitive method for measuring extremely weak optical transitions based on spin-shelving. Atoms that undergo the weak transition are “shelved” by spontaneous Raman transitions in a “dark” uncoupled level. After waiting long enough, a significant fraction of the atoms will be shelved. Similar to electron-shelving techniques, the detection benefits from multiply excited fluorescence from a strong cycling transition to yield a very large (up to 10⁷ fold) quantum amplification of the measured transition rate.

We also performed a series of experiments on the microwave spectroscopy of the ground-state hyperfine splitting of optically trapped atoms. We introduce a number of schemes to overcome the inherent perturbation of the atomic levels by the trapping potential. First, we developed an optimal dark trap, achieved by minimizing the surface to volume ratio of the trap and by choosing the minimal wall thickness which is allowed by diffraction, to yield atomic coherence times which are 200 longer than existing single-beam dark traps.

Next, we demonstrate a scheme for eliminating the trap-induced inhomogeneous broadening of the transition, by adding a weak ‘compensating’ laser, spatially mode matched with the trapping laser so as to exactly compensate for the trap-induced shifts. Despite being tuned close to resonance, this laser does not considerably increase the spontaneous scattering rate. We experimentally demonstrated the new scheme and obtained a 50-fold narrowing of the microwave spectrum (see Fig. 2a). With the suppression of inhomogeneous broadening, the atomic coherence time is now limited by the much smaller spontaneous scattering time. Whereas in a conventional optical trap the ac Stark shift of the line centre strongly depends on the temperature of the atoms, which may drift considerably, in the compensated trap the suppression of the line shift is equally effective for all temperatures. Hence, it provides a means of achieving a higher stability of the line centre than that achieved by simply stabilizing the trapping laser detuning and intensity.

Conversely, we presented a method in which a long microwave pulse is used to select a narrow energy band from the atomic ensemble, around any required central energy. The rest of the atoms are expelled from the trap using an on-resonant laser, without perturbing the selected atoms. With this method, the central energy can be chosen to maximize the number of selected atoms by selecting the energy with the highest density of populated states (see Fig. 2b).

Figure 2: (a) Line centre (triangles) and RMS width (circles) of the Rabi spectrum for trapped atoms in a compensated tarp as a function of compensating beam power. The spectrum width is minimized to a Fourier-limited value, at a compensating beam intensity which also corresponds to a minimal shift from the free space resonance. (b) Rabi spectrum of energy-selected atoms [99]. Full dots: spectrum of trapped atoms with no pre selection. Empty dots: Rabi spectrum of atoms pre selected with a 20 ms pulse at two different frequencies. As seen, much narrower spectra at any chosen central frequency are obtained.

Next, we demonstrated that a macroscopic coherence, lost as a consequence of dephasing of the different populated vibrational levels, can be efficiently revived by stimulating an echo (by adding a microwave  pulse in between the two /2 pulses of the conventional Ramsey spectroscopy) if the trap detuning is large enough so the vibrational states of the optical potential felt by the two hyperfine states of the atoms have close to unity overlap for all thermally populated states. This suppression of dephasing due to perturbations induced by the trap yields a dramatic increase in the coherence time for trapped atoms long after the Ramsey fringes dephased (see Fig. 3). We further demonstrate that echo spectroscopy can be used to experimentally map the quantum dynamics of trapped atoms, showing a crossover between quantum and classical regimes.

Figure 3: (a) Echo signal P_↑ measured as a function of the time τ₂ between the π-pulse and the second π/2 pulse, for a fixed τ₁. A dip in P_↑ is seen, showing a ‘coherence echo’ at τ₂ = τ₁. (b) Ramsey fringe contrast (◦) and echo signal P_↑ (×) measured as a function of the time between the two π/2 pulses. For the echo signal the value 0 represents complete coherence, and the value 1/2 represents complete dephasing. A coherent echo (P_↑<< 1/2) persists long after the Ramsey fringe contrast has decayed.

We demonstrate that the dephasing in microwave spectroscopy of optical trapped atoms due to dynamical changes in the trap parameters, can be suppressed beyond the suppression offered by the simple echo, by using an improved pulse sequence containing up to 20 additional π pulses. The achieved coherence time is limited by increased mixing between transverse states and, to a lesser extent, the lifetime of the internal states of the atoms. The demonstrated pulse sequence can be used in precision spectroscopy of a periodic effect, where the π-pulses can be synchronized with the period of that effect.

Finally, we show that microwave spectroscopy of optically trapped atoms can serve as a tool to study the properties of the fidelity (or Loschmidt Echo) and its relation to the underlying classical dynamics of a quantum system. The main difficulty in experimental studies of these matters is the preparation of highly excited pure quantum states. This difficulty is overcome by using echo spectroscopy, which eliminates the need for pure quantum state preparation and allows the experimental observation of quantum fidelities even in a thermal ensemble and for extremely highly excited quantum states. Using a perturbative treatment we show that the coherence of the echo scheme is a function of the survival probability or fidelity of eigenstates of the atoms in the trap. The echo coherence and the survival probability display ‘system-specific’ features, even when the underlying classical dynamics is chaotic. We performed echo spectroscopy in chaotic and mixed atom–optics billiards, as a function of the perturbation strength, and observed two different regimes. First, a perturbative regime in which the decay of echo coherence is non-monotonic and partial revivals of coherence are observed at the Thoules time, even for traps where the dynamics is fully chaotic, as opposed to the prediction from random matrix theory (see Fig. 4). These revivals are more pronounced in traps with mixed dynamics. They again occur at the Thoules time, reflecting the non-generic nature of the perturbation applied, and the existence of low order periodic orbits motion of atoms in the trap. Next, for stronger perturbations, the decay becomes monotonic and independent of the strength of the perturbation. In this "perturbation independent" regime, no clear distinction can be made between chaotic traps and traps with mixed dynamics.

Figure 4: Echo signal for a light sheet wedge with chaotic classical dynamics, for different perturbation strengths (trap laser wavelength tuned from λ = 775.9 nm to 779.7 nm from bottom to top). For small perturbations a non-monotonic decay with revivals around the typical bouncing time is seen, whereas for large ones a monotonic decay is observed. Inset: CCD image of the trap laser at the focal plane, showing a fully chaotic wedge billiards with wedge half-angle α = 52.5◦ .

More recently, we extended our studies of the time dependence of the Loschmidt echo to open dynamical systems. By placing a hole in the base of the wedge trap of Fig. 4. we allowed the atoms in the chaotic sea to escape the wedge, and thus observed the evolution of the Loschmidt echo from that of mixed phase space (with nearly monotonic decay) to that of a semi regular phase space, with strong revivals occurring at the Thoules time. We thus showed that the Loschmidt echo is sensitive to dynamical changes in type of motion in the trap.

We also considered a broader class of so-called "generic" perturbations using an additional perturbation laser field, while minimizing the trap related dephasing using our compensating technique. First, we studied the case of complete spatial symmetry breaking of the perturbation and trap, by adding a weak spackle perturbation beam. The spackle beam was detuned to half the distance between the two hyperfine levels such that its associated dipole potential is time averaged over the whole echo sequence to zero, while its mixing effect was comparable to the uncompensated trap. The time scale associated with the spackle beam was smaller then all other dynamical timescales. Unlike the non-generic perturbation of the trap itself (blue circles in Fig. 5), the spackle field caused a monotonically increasing dephasing (red crosses in Fig. 5) reflecting the lack of a well defined timescale emerging from the perturbation. Furthermore, when the spackle generic perturbation strength was increased, the dephasing rate increased, contrary to the non-generic perturbation case, where a perturbation-independent regime was observed.

Figure 5: Echo coherence as a function of time. For wedge billiard with classically mixed phase space dynamics shows coherence revival at time scale associated with trap parameters (blue circles), while for a compensated wedge in the presence of perturbing spackle pattern revival is suppressed (red cross). The vertical axis normalized signal is proportional to the amount of dephasing. The insets show the billiard potential without (top) and with (bottom) the speckle field perturbation. [T. Grunzweig et. al., in preparation]. 

Finally, we studied the decay of echo coherence due to a "point" perturbation in a mixed phase-space wedge billiard. Here by controlling the spot position of a perturbation laser, having the same detuning as in the speckle case, we apply a localized perturbation to either the island of stability or to the chaotic sea and confirmed that different parts of phase space have a qualitatively different echo signal. When the point is localized on an island, the time scale of the low-order periodic-orbit trajectories is clearly expressed as a partial revival of the echo coherence. Contrary, when the perturbation point misses the island a nearly monotonic decay is observed, in agreement with the predictions of random matrix theory. Phase space dependent perturbation allows us to study and verify experimentally the relation between stability of quantum eigenstates, and stability of classical trajectories.

Some of the ideas presented here have been found useful by other groups. A compensating’ technique has been used to compensate the Stark shift in an ion trap quantum processor. Two groups working on quantum information processing with single ions and atoms have reported the use of coherence echoes. The combined possibilities offered by our experimental system (control of dynamical phase-space and spectroscopy) can have impact on other fields of study: first, the effect of interactions between the particles trapped in a billiard is an interesting question in the context of semiconductor quantum dots. In atom–optic billiards, the interactions can be tuned in a controlled way, Next, open systems, which are characterized by a scattering matrix (and not by eigenstates), can be investigated, and related to results in quantum dots, such as universal conductance fluctuations. Finally, chaotic optical billiards, which their unique ability to move from the quantum to the classical regime, can be used for experimental studies of parametrically-dependent Hamiltonians and driven quantum systems.

2. Excitations in Bose-Einstein condensation (BEC)

The ability to tune experimental parameters over a broad range is one of the exciting feature of dilute-gas Bose-Einstein condensates. Together with my postdoc J. Steinhauer, and my students R. Ozery, N. Katz, E. Rowen and N. Bar-gill, we used dynamically controlled optical lattice potentials to trap and Bragg excite condensates in many different physical regimes of interest. Initially, by using relatively weak Bragg pulses, we excited perturbative Bogoliubov quasi particles and studied their spectra, dynamics and decay. Within this framework excitations of controlled momenta varying from very low (in units of the inverse healing length) to high momenta were studied. The pulse duration was then extended to study higher resolution spectra and the dynamics of the time-of-flight process were analyzed for both high and low momentum excitations. By strengthening the Bragg pulse, we entered the regime of strong excitations, where the condensate is depleted and perturbative theories no longer apply. This regime was studied both in the time and spectral domains. The main results of this research are:

·       Bogoliubov excitation spectrum has been confirmed in atomic vapor BEC to better than 1% accuracy by using perturbative Bragg spectroscopy (see Fig. 6). In particular, the linear slope of the phonon branch was accurately measured, yielding a precise value for the sound velocity, which is also the critical super fluid velocity for dilute gas condensates. The static structure factor was also measure (with lower accuracy) and its strong suppression for low momenta was confirmed. This was the first time that a momentum dependent experiment directly measured the excitation energy, and the Feynman-Bijl relation was confirmed.

Figure 6: The measured excitation spectrum w(k) of a trapped BEC. The solid line is the Bogoliubov spectrum with no free parameters, in the local density approximation for chemical potential =1.91 kHz. The dashed line is the parabolic free-particle spectrum. The vertical line at ^–1 shows the separation between collective (phonon) and single particle regimes.

·       Using the cylindrical symmetry of our excited condensates we applied computerized tomography to ontain the full three dimensional atomic distribution from the time of flight images to measure the excitation energy distribution in space after condensate expansion. This method is useful for quantifying nontrivial density distributions that evolve due to excitation of the condensate (including the decay products of such excitations). It was used to study the dynamics of the Bragg-excited condensate, and coherent interactions between the condensate and the emerging excitations was measured, revealing a surprising Mossbauer-like effect.

·       By extending the duration of our Bragg pulses (and thus enhancing their spectral resolution), we observed radial modes in the BEC, and explained them by use of the inhomogeneous Bogoliubov equations. The cigar shape of the condensate (due to the tight radial confinement) leads to the observable quantization of the excitations in this dimension. The observation of these modes, which are coupled to the Bragg lattice by the nonlinearity, has lead to a set of theory papers, both in our group and in others studying excitations in such cylindrical condensates in various regimes.

·       The technique of momentum echo was developed to suppress the inhomogeneous mean field broadening of Bragg spectroscopy, and then implemented experimentally. We showed that by reflecting Bogoliubov excitations using an additional Bragg grating, narrower spectral lines are measured, approaching the natural line width .

·       For very low momentum excitations, a clear matter-wave interference fringe pattern was observed in the time-of-flight images (see Fig. 7). We developed tools for analyzing these fringes and to use them to measure remarkably weak excitations, approaching the single excitation limit. The process by which such excitations evolve during time-of-flight has proven to be governed by nontrivial effects (the so-called quantum evaporation and an effective non-hermitian evolution due to the expanding condensate).

Figure 7: Radial and axial density profiles of the released, excited condensate after 38 msec of flight. The profiles are generated by computerized tomography of the absorption images. The excitations’ momenta correspond to (a) k=0.67 and (c) k=0.1. For the lowest momentum value (c), the released excitations form fringes in the released condensate. For (a) the condensate and the released excitations form two fairly distinct clouds. (b) and (d) are the result of a full Gross-Pitaevskii equation simulation with the same experimental parameters as in (a) and (c), respectively.

·       The Beliaev damping of excitations was measured by carefully (using computerized tomography) comparing the number of observed atoms in the excitations as compared to the momentum they carry. Remarkably, Beliaev theory was never applied explicitly to the problem at hand (all the theory papers contained implicit integral expressions). By solving exactly, for the first time to our knowledge, the energy-momentum conserving collision problem of Bogoliubov modes, we were able to analytically calculate exactly the decay rate for all momenta and to compare with experiment. Beliaev theory was found to agree with experiment within an overall factor, where both confirm the strong suppression of decay at low momenta. This was the first time that such a test was conducted for Beliaev damping in BEC for bulk excitations.

·       We then used a post selection technique to measure Beliaev damping in an elongated BEC. We find a dependence of the damping rate on both the excitation energy and the momentum of the decayed excitations. We measure a shift downward in the excitation energy of the undamped excitation for high momentum, while for small momentum we measure no such shift. According to our model, the increase of damping rates with energy for high momentum excitations is a result of stronger coupling between the excitation and the empty bath due to increasing spatial overlaps, along with a small increase in the number of available modes. For low momentum excitations there are also more available modes as energy is increased, but this is compensated by quantum interference which tends to suppress the damping rate. These phenomena cancel each other, leaving a damping rate with very weak energy dependence.

·       A theoretical offshoot of our measurement of the Beliaev interaction between excitations is the study of bosonic wave mixing, which arises when several Bogoliubov modes become significantly occupied. We present a novel dressed state basis for analyzing the problem and explain both the coherent evolution and decay of these coupled Bogoliubov states. Interestingly, the Schwinger boson mapping (of a two-dimensional harmonic oscillator to an angular momentum basis) is the key to understanding this system, predicting, for example a quantum uncertainty decay of Rabi oscillations and a transition to instability for intermediate mixing strengths .

·       We calculated the roton-like correlations in the ground-state of a realistic ⁸⁷Rb condensate. We derived a simple analytical expression for low density and tested its limits with a quantum Monte-Carlo simulation, taking the true scattering wave-function into account (from the solution of inter-atomic potential). We found a peak in the structure factor that corresponds to a basic correlation of the ground-state. This peak occurs at the inverse scattering length. In liquid Helium, all the relevant length scales are similar (scattering length, inter-particle spacing and healing length) and therefore it is not obvious as to where the peak in the structure factor would appear in BEC, where these length scales are well separated.

·       When the condensate was placed suddenly in a strong traveling optical lattice a new regime of interest is revealed. We observed Rabi oscillations between momentum modes due to beating between the Bloch states of the lattice onto which the condensate wave packet is projected. Remarkably, we measured an order of magnitude increase in the coherence time of these oscillations, as compared to the measured coherence of perturbative excitations. This effect is explained by use of Gross-Pitaevskii energy functional and the analysis of the resulting nonlinear two-level system.

·       We found that the excitation spectrum from such an oscillating state is split and that the decay products strongly deviate from the usual s wave halo of perturbative states.  Using the stochastically seeded GPE method and a simple Bloch state collision model, we are able to both simulate and explain our results. We found that by imposing the strong optical lattice potential, the BEC state is coupled to a nontrivial quasi-continuum of decay products. Both the coupling and decay product states are controlled by the lattice. Such control is useful for observing the quantum Zeno effect (or anti-Zeno effect).

Figure 8: Top: absorption images after 38 msec time of flight following a Bragg pulse. (a) Strong pulse with Rabi frequency of 8.6 kHz (b) Weak pulse with Rabi frequency   <2 kHz. Dotted circles represent the predicted s-wave shell. The collisional manifold for the strong pulse is clearly shifted inwards as  compared to that of the weak pulse, which agrees with the expected s-wave shell. Bottom: corresponding numerical simulations of the stochastic GPE, confirming the main experimental observations.

·       The ability to observe many-body Zeno effects was explored theoretically. Both the Zeno and anti-Zeno effects appear to be measurable, depending on the pulse used (detuning and intensity). These effects provide a unique probe into the many-body correlation time of ground-state of the condensate. It is interesting to note that in atomic physics the time scale for observing these correlations typically less than femtosecond, rendering them observable only in artificially engineered states where the coherence is much extended. The weakly interacting BEC was shown to be a natural setting for experiments in this direction.

·       We studied, both theoretically and experimentally, super radiant excitation of a BEC in the strong coupling regime. We used a pair of collinear pump beams with a controllable frequency difference between them to perform for the first time spectroscopic measurements of the super radiance process, through the population of the backward scattering atomic modes. These backward modes are off-resonant from the single-frequency super radiance by 4 times the recoil frequency. We observed a shift of the backward scattering two-frequency resonance to lower frequency in the averaged-phase spectrum, due to the exponential nature of the super radiance process, which is captured by the calculation. In addition, we performed coherent time dynamics, revealing oscillations of the mode populations and a peaked structure of the fixed-phase spectrum. This is in qualitative agreement with our theoretical model, which is semi-classical, and indicates the coherence of the measured dynamics. These novel results open up further possibilities for studying quantum fluctuations and buildup of super radiance in ultra cold atomic gases.

Figure 9: (a) Experimental setup: super radiant emission builds up along the BEC’s long axis, creating forward modes of scattered atoms. The backward atomic modes result from stimulated absorption of super radiant photons and emission into the pump laser beam. (b) Absorption image after 38 msec time-of-flight of forward super radiance modes obtained with a single-frequency pump and (c) an X pattern (forward and backward super radiance modes) obtained for a two-frequency pump with f = 15kHz, both for a pulse duration of 200 μsec.

·       Recently, we studied the decay of excitations over a BEC in a 1D optical lattice. By changing the depth of the lattice we modify the dispersion relation of the excitations from the Bogoliubov spectrum (with no lattice) to the Bloch-Bogoliubov spectrum. For deep lattices such spectrum becomes convex, thus "protecting" the excitation from decay, which violates by energy and momentum conservation (similar to low k excitations in super fluid helium). However, in our experiments we are sensitive to finite time effects where the excitations can decay via off-shell processes, smilingly violating energy conservation. We observe experimentally substantial decay for deep lattices, confirming this surprising prediction, as well as characterizing the coupling strength between the excitations and the decay product which is found to dramatically increase with the depth of the lattice.    

·       Recently, we experimentally studied the scattering of phonons over a BEC. Due to the linear spectrum of the phonons, the resonance condition for an excitation of a quasi-particle q is the same as that for scattering a quasi-particle q to 2q. When the q excitation is perturbative, one might expect the scattering of a q excitation to 2q to be negligible. We showed both experimentally and theoretically that this is not the case. While there is a quantum destructive interference in the creation of the q excitation, we showed there is a constructive interference of the amplitude for scattering the q to 2q. As the suppression in the structure factor Sq, also this enhancement is a result of momentum correlations due to the many-body nature of the system. This effect made the scattering significant in Bragg excitation experiments at small momenta, despite the lack of Bosonic amplification by the condensate wave.

3. Electromagnetic Induced Transparency (EIT)

When on-resonance light impinge upon atoms in an alkali vapor cell, it is strongly scattered, and the amount of light intensity is exponentially decreasing. If however, a second, stronger "pump" beam, is added, which resonantly couples a second unoccupied state and the same excited state, the original beam will manage to pass through – a phenomenon known as EIT. Moreover, the atoms in this scheme are pumped into a dark state which is very stable to decay. This stability stems from destructive quantum interference between two indistinguishable decay pathways. The transparency window can be extremely narrow, thus marking EIT as a promising candidate for light-weight and compact atomic clocks. EIT has drawn much attention in recent years due to the possibility of light slowing and stopping. To stop a "probe" light pulse, one has to close the "pump" beam while the probe is contained in the medium, thus imprinting the amplitude and phase information of the probe pulse onto the atomic coherences. While the temporal aspects of EIT were extensively studied in recent years, the spatial aspects received relatively little attention. In collaboration with the theory group of A. Ron from the Technion, our joint students M. Shuker and O. Firtstenberg, and my student R. Pugatch, we concentrated on these spatial aspects in several theoretical and experimental studies.

Dicke Narrowing - Theory and Experiment

Dicke narrowing is a phenomenon that dramatically reduces the Doppler width of spectral lines, due to frequent velocity-changing collisions. We showed theoretically that such a narrowing mechanism indeed exists for EIT resonances. The narrowing factor is the ratio between the atom’s mean free path and the "Raman" wavelength associated with the wave-vector difference of the two radiation fields and facilitates ultra narrow spectral features in room-temperature vapor. Next, we measured this EIT Dicke narrowing by studying the dependence of the EIT line shapes on the angle between the pump and probe beams. For small angles, our model predicts that both the residual Doppler width and the EIT-Dicke parameter are proportional to the angle. Therefore, the resultant broadening is proportional to the square of the angle. The measurements we performed were in excellent agreement with our analytical calculation with no fit parameters. The results demonstrate that Dicke narrowing can increase substantially the tolerance of hot-vapor EIT to angular deviations.

Topological stability of a stored optical vortex

We stored for the first time an optical vortex beam in our rubidium vapor cell. We showed that due to its 2π phase twist, this mode, also known as the Laguerre-Gauss mode, is topologically stable and cannot unwind even under conditions of strong diffusion. For comparison, we stored a Gaussian beam with a dark center and a uniform phase. Contrary to the optical vortex, which stays stable for over 100 sec, the dark center in the retrieved flat-phased image was filled with light after a storage time as short as 10 sec (see Fig. 10). This experiment proved that higher electromagnetic modes can be converted into atomic coherences and that modes with phase singularities are robust to decoherence effects such as diffusion. This opens the possibility to more elaborate schemes for classical and quantum information storage in atomic vapors.

We also developed a full theoretical model for the diffusion/diffraction propagation of stored and slow light and showed that in fact all transverse electromagnetic modes (e.g. Laguerre-Gauss, and Hermite-Gauss modes of diffraction) are also modes of the diffusion/diffraction propagation. We performed quantitative measurements with several Laguerre-Gauss modes which verified this rather surprising prediction. 

Figure 10: The effect of diffusion on a helical beam and a flat phase beam. The left column shows the initial shape of a helical (top) and a flat phase (bottom) beam before any diffusion. The right column shows both beams after storage of 30 sec. The center remains dark in the helical beam while completely filling with light in the flat phase beam.

Storing of Images

Equipped with insight from the optical vortex storing experiment, we turned to the storage and retrieval of short light pulses carrying arbitrary transverse two-dimensional images by reversibly converting them to atomic coherences. We showed that both the temporal and transverse properties are preserved. We first stored images of digits for several micro-seconds (see Fig. 11). By introducing a phase-shift technique analogue to phase shift-lithography, we demonstrated significant improvement in the immunity of certain images against atomic diffusion. This experiment also served as a proof that the phase pattern of the light pulse is also stored, and opens the possibility of storing an array of qubits in a single vapor cell.

Figure 11: Images of digits slowed and stored for different durations in warm atomic vapor. The left column shows the original images sent to the storage medium; they were taken by tuning the laser away from any atomic resonances. The middle column shows the image of the digit '2' that was slowed using the EIT system (the slowing delay was about 6 micro second). The right column shows the images after they have been stored and retrieved.

4. Mode selection and coherent combining of lasers

In collaboration with the group of A. Friesem from Weizmann we investigated two approaches to increase the power and improve the beam quality of lasers. The first approach is based on selecting pure high order modes using novel intra-cavity phase elements that we developed, and the second involves passive phase locking and coherent combining of several lasers using novel interferometric optical elements inserted into a common cavity of all lasers.

Mode selection

We developed a novel method in which single high order mode laser operation is obtained, by inserting into the laser resonator novel phase elements with either discontinuous or continuous phase. These elements both select and shape specific higher order modes which yield a combination of high output power (up to 5 fold improvement), improved beam quality and even stability. Moreover, special laser modes, such as helical beams or beams with either radial or azimuthal polarization were directly and efficiently generated within the laser resonator. Radial and azimuthal polarization was also achieved by controlling the birefringence-induced thermal lensing in high power YAG lasers in collaboration with the group of S. Jackel in Soreq. More complicated resonators that contain a combination of two modes with orthogonal polarization can produce even higher power . Finally, the beam quality was characterized and further improved, up to the quality of the fundamental Gaussian mode. These methods were successfully demonstrated in a large variety of laser systems including gas lasers, solid state lasers, micro lasers and fiber lasers both in CW, pulsed and Q-switched operation (see Fig. 12).

Figure 12: Measured intensity distributions of TEM_0,4, - TEM_4,4 modes (left to right) selected in a Q-switched Nd:YAG laser using intra-cavity phase elements . 

Coherent combining

When the field distributions of several laser output beams are incoherently combined, the resulting beam-quality factor (M²) is relatively poor with low optical brightness. But when the field distributions are coherently added, with the proper phase relations, the combined beam quality factor can be as good as that of a single low-power laser, while the combined power is greater by a factor equal to the number of the combined lasers. We developed a unique approach for coherently combining various transverse field distributions of separate laser channels within a laser cavity. This includes the coherent combining of Gaussian channels, of single high-order mode channels, and even of spatially incoherent multimode channels. With this approach, self phase locking and coherent combining is achieved by use of intra-cavity interferometric combiners, so as to obtain a compact, stable and practical configuration. The combined laser will tend to operate so that the losses are minimum, whereby the phases of the individual beams will be automatically matched such that coherent addition takes place. This can be achieved only for those longitudinal modes (frequencies) that are common in the two laser channels. Thus, care must be taken to imbalance the optical length of the two resonator channels in such a manner so as to obtain one or more common longitudinal modes.

To verify our approach, we performed experiments with a pulsed Nd:YAG laser setup, in free running and Q-switched operation. The channels were formed within the laser cavity with multiple apertures, designed to enable single mode operation within each channel. The separate laser channels were combined using specially designed interferometric combiners, made of fused silica etalons with several dielectric coatings. With this setup we experimentally demonstrated coherent phase locking and coherent combining of 2, 4 , 9 16 and 25 intra-cavity Gaussian channels with ~90% combining efficiency and excellent beam quality. The schematic arrangement for coherent addition of 16 Gaussian channels along with representative experimental results are shown in Fig. 13.

Recently we applied our coherent combining techniques to fiber lasers. Since the high power of the combined output light propagates in free space rather than inside, fiber optical damage and deleterious nonlinear effects are substantially reduced. Two different configurations were investigated. One involves conventional intra cavity coupling between the lasers. The other is a novel configuration where the coupling is done out of the combined cavities. The latter configuration requires much less coupling for obtaining the same output power, so the damage to the fiber is further reduced.

Figure 13: Intra-cavity coherent combining of 16 channels. (a) Laser resonator; (b) intensity distribution at rear laser mirror; (c) intensity distribution of the combined output beam.

Phase locking, which is achieved by transferring some energy from one oscillator to the others, strongly depends on the coupling strength between the oscillators. Typically, the coupling strength must be above a certain threshold in order to achieve phase locking. Using these two fiber laser configurations we showed analytically, numerically and experimentally, how this threshold can be significantly reduced when phase dependent losses are introduced into the oscillators. Specifically, the coupling strength can be reduced by at least an order of magnitude, thereby substantially decreasing the needed transfer of energy between oscillators (see Fig. 14) . Finally, we used the fiber laser configuration to obtain experimental, analytic and numerical results that clearly reveal that phase locking and synchronization strongly depends on quantum noise when the coupled lasers oscillates very close to threshold. The last two results indicate that even in our applied-oriented research, new and surprising effects were revealed on coupled laser oscillator, which may not only influence the laser research area, but also affects many other areas that involve coupled ensembles.

Figure 14: Experimental and calculated fringe visibility as a function of coupling strength. Stars [dots] denote experimental results for the configuration with [without] phase-dependent loss. Solid curves denote corresponding numerical results, and dashed curves corresponding analytic results.

5. Imaging, light concentration and microscopy

In a long-standing collaboration with Dr. N. Bokor from Budapest we continued to develop new concepts and methods in linear and applied optics. In the past we proposed, investigated and demonstrated novel techniques to shape diffuse light beams using unconventional optical elements. The main motivation was to strongly compress diffuse light in an anamorphic way so as to overcome the one-dimensional thermodynamic limit. This was used for one-dimensional concentration of solar energy, and for improving the resolution of spectrometers. We also introduced and experimentally demonstrated the concepts of faceted diffuse reflectors that are capable of shaping diffuse beams with a single reflection, and adiabatic beam shaping of diffuse light.

We next investigated optical systems that operate at extremely high numerical for both imaging and light concentration. We concentrated on diffractive optical elements (DOEs) that are formed on a curved substrate. Both the substrate shape and the phase function are free  parameters which can be controlled independently to optimize the curved DOE, resulting in much better performance than is possible with DOEs fabricated on flat substrates. In a simplified ray optics picture, the phase determines the directions into which the incoming rays are diffracted by the grating, whereas the shape determines the spatial density of rays propagating in the directions imposed by the phase, thereby serving as an effective local apodization factor. Improved fabrication method along with the appearance of new application areas and design principles, account for the recent increase of interest in curved DOEs. We first studied aplanatic DOES formed on spherical surfaces. We showed how they can provide aberration-free imaging and  Fourier transformation, as well as concentration of  quasi-monochromatic diffuse light onto a flat target at the theoretical limit imposed by brightness conservation (or equivalently, the second law of thermodynamics). By simply reversing the direction of the light rays, we also showed that these spherical DOEs can also uniformly collimate the light emitted from a flat Lambertian source. We next extended the sine condition, and used it to design coma-free imaging and ideal diffuse light concentration and collimation, to arbitrary target and source shapes, such as solar light concentration onto cylindrical water pipes, or efficient and uniform concentration of pumping light on cylindrical laser rods. This generalization leads to curved DOE shapes other than cylindrical operating at the thermodynamic limit.

More recently we concentrated on investigations of light focusing and imaging by optical systems with very high numerical aperture, for applications such as lithography, data storage, microscopy, optical tweezers and optical traps. In such systems the scalar diffraction theory is not applicable and vectorial effects of the focused light field play an important role. We first investigated high numerical aperture focusing of raidally polarized beams. We showed that a parabolic mirror and a flat diffractive lens, due to their large apodization factor can achieve a significant reduction in spot area compared to the conventional aplanatic systems. We then extended our studied for so called "4" focusing and imaging systems (where two high numerical aperture elements are places on both side of the sample) and showed that a nearly spherical central spot in the focal region, with reduced spot volume and uniformly low side-lobe intensities. By using a radially polarized first order Laguerre Gaussian beam (having a helical phase) in the "4" focusing system we could obtain an extremely small dark volume at the focal point, completely surrounded by light in a nearly spherically symmetric manner. We showed how such a dark focal intensity distribution can be used effectively as the erase beam in fluorescence depletion microscopy to achieve a dramatic improvement in resolution beyond the diffraction limited resolution. This significant resolution improvement is achieved simultaneously in all three dimensions, contrary to the axial or transverse resolution improvements obtained so far. In collaboration with the group of M. Fujii from Tokyo we showed how the polarization distribution in the focal region strongly affects the resolution in fluorescence depletion microscopy, and developed methods for its characterization and optimization]. Finally, we developed simpler and more practical alternatives for high numerical aperture bright or dark focusing with uniform and tight intensity and polarization distributions.

Send mail to nir.bar-gill@weizmann.ac.il with questions or comments about this web site.
Last modified: 02/14/08