Radiation damping optical enhancement in cold atoms

The typically tiny effect of radiation damping on a moving body can be amplified to a favorable extent by exploiting the sharp reflectivity slope at one edge of an optically induced stop-band in atoms loaded into an optical lattice. In this paper, this phenomenon is demonstrated for the periodically trapped and coherently driven cold 87Rb atoms, where radiation damping might be much larger than that anticipated in previous proposals and become comparable with radiation pressure. Such an enhancement could be observed even at speeds of only a few meters per second with less than 1.0% absorption, making radiation damping experimentally accessible. Subjecting an ensemble of cold rubidium atoms to a particular combination of optical beams should enable the study of ‘radiation damping’, a typically minuscule velocity-dependent effect that a light source exerts on a moving body. This is the finding of a theoretical study performed by Jin-Hui Wu at Jilin University, China, in cooperation with researchers from the university of Exeter (UK), the Lens Laboratory and Scuola Normale Superiore (IT). Their proposal is based on exploiting reflections at the boundary between allowed and forbidden frequency bands in the photonic crystal made by periodically distributed cold atoms. This approach should provide a way of amplifying radiation damping to a level suitable for experimental study, making it useful for manipulating cold atoms and potentially other material systems.


INTRODUCTION
A body in motion relative to a light source experiences not only a light pressure but also a damping force that depends on its velocity, akin to a viscous friction. 1 Pressure is a result of the momentum exchange between light and matter due to photons scattering, while damping arises from the Doppler effect and opposes, for instance, the velocity of a moving mirror. 2 This velocity-dependent force damps mechanical oscillations at a rate of the order of P5mc 2 , with P being the light power and m the mirror mass. Radiation damping could be used to control macroscopic moving objects or reversibly to implement precision motion sensing schemes. Gaining control over radiation damping forces is interesting in its own right. [3][4][5][6] However, this capability can also develop into a new approach to manage the dynamics of hybrid optomechanical atom-membrane interfaces, 7 where the motion of cold atoms is strongly coupled to the vibration of a micro-mechanical membrane. 8 Controlling the coherent motion of cold atoms via radiation damping, particularly if this control can be performed all-optically as we will demonstrate below, would permit substantial engineering of the mechanical oscillator, including the quantumlimited read-out of its position [9][10][11][12] and the efficient exchange of quantum states among light, the oscillator and cold atoms. 13 However, for ordinary velocities and a light power that is not too large to destroy the mirror, radiation damping is usually several orders of magnitude smaller than the pressure and, hence, particularly difficult to detect, even for the smallest diffraction-limited mirrors of m<10 212 g. [14][15][16] With such a small figure, radiation damping can hardly be of any practical use at variance, e.g., with radiation pressure that is routinely employed to cool atoms. 17 Methods to enhance such a typically tiny damping are now being sought for. This enhancement is performed by exploiting the sharp optical response near one edge of the photonic stop-band in a dielectric Bragg mirror 15 or in a onedimensional (1D) ordered structure of cold 87 Rb atoms. 18 In the latter case, this enhancement hinges on the formation of stop-bands near an atomic resonance in a periodic atomic structure, a subject that is relatively unexplored. [19][20][21][22][23][24] Only very recent experiments have reported quite sharp reflectivity profiles in this regime. 25,26 Hence, we propose here a new mechanism to achieve a large radiation damping effect in a 1D atomic photonic crystal. The enhancement is found to occur around a narrow stop-band opening up in the cold confined 87 Rb atoms when driven into a L configuration 27 of levels by a pump beam (cf. Figure 1) that allows for an efficient alloptical (external) control. The expected enhancement occurs in the Autler-Townes (AT) splitting regime and arises from the ultrahigh sensitivity of reflectivity (transmissivity) near such an optically induced stop-band. Consequently, the enhancement can be controlled on demand by changing the pump intensity and frequency. In a cold sample of density 6.0310 12 cm 23 , e.g., for atoms sloshing within the confining optical potential at characteristic velocities (about 1.0 m s 21 ), the radiation damping force may become so strong as to compare with the radiation pressure. Controllable damping enhancement of various orders of magnitude, with a concomitant large suppression of absorption, makes this proposal a remarkable improvement over previous ones 15,18 and a viable option for the observation of radiation damping effects.

MATERIALS AND METHODS
Our ensemble of cold 87 Rb atoms is loaded into an optical lattice of period a formed by retro-reflecting a light beam of wavelength l o 52a ( Figure 1). In each period of such a dipole trap, the confined atoms occupy a small region d,,a with a homogeneous volume density N 0 . This periodic 1D index modulation will cause pronounced photonic stop-bands as expected. 25 We use the transfer matrix formalism to describe the propagation of a probe field of frequency v p through such an atomic stack, referring to Ref. 22 for a detailed description of this formalism in a similar context. From the frequency-dependent transfer matrix M(v p ) for a single period and with the use of the Bloch theorem, we obtain the following dispersion relation: whose solutions yield the complex Bloch wave vector k5k91ik0. The refractive index associated with the collection of cold trapped atoms, where c 12 (c 13 ) is the dephasing rate of the spin (optical) coherence r 12 (r 13 ), D p 5v p 2v 31 (D c 5v c 2v 32 ) is the probe (pump) detuning relative to transition 1 j i< 2 j i ( 1 j i< 3 j i), and m 13 denotes the relevant electric-dipole moment. Here, we assume that the cold 87 Rb atoms are driven into a three-level L-type configuration 27 on the D 2 line by a probe of Rabi frequency V p traveling in the x-direction and a pump of Rabi frequency V c traveling in the y-(or z-) direction ( Figure 1). Note that the probe field may have a small angle of incidence (e.g., h<2 0 ) in the x-direction of an oscillating atomic lattice. 22,25 The dressed susceptibility in Equation (1) holds in the limit of a weak probe (V p ,,V c ), within the rotating-wave and electric-dipole approximations. In particular, Equation (1) recovers the case of a single absorption line when V c R0, but implies a narrow electromagnetically induced transparency (EIT) window near the probe resonance when V c fc 13 due to the quantum destructive interference between two dressed-state transitions. However, for V c ..c 13 , the dressed susceptibility evolves into that for two well-separated AT split levels with the quantum destructive interference becoming unimportant.
Solving Equation (1) for k9 and k0 as a function of v p (D p ) yields photonic band-gaps that appear at the boundary of the first Brillouin zone (k95p/a); 22 these band gaps are shown in the left and right panels of Figure 2, respectively, in the presence or absence of the pump. We recall that in a non-resonant multilayer stack, as for a standard distributed Bragg reflector, 15 a single stop-band opens up near the Bragg frequency; conversely, in the presence of a resonant absorption line near the Bragg frequency, 18 two stop-bands will arise from the interplay between the polaritonic gap due to the resonance and the Bragg gap due to the periodic index modulation. The latter case is featured in Figure 2b and 2d where, for a susceptibility x(v p ) in the case without the pump (V c 50), two wide stop-bands open up in negative and positive detuning regions near a single absorption line. However, when the pump is on (V c ?0), as illustrated in Figure 2a and 2c, a third narrow stop-band arises 24,26 and fits between the two wide ones separated by the AT split absorption lines.
In the following, we focus on this additional stop-band by applying a strong pump to attain AT splitting beyond the EIT regime. 27 The width and position of this narrow stop-band can be manipulated through the modulation of pump frequency and intensity, which then allows us to efficiently control the optomechanical properties of our atomic multilayer. The most relevant damping effects we address here occur in the steepest region of the reflectivity and transmissivity profiles near one edge of this third stop-band, where absorption is significantly suppressed. This result is particularly relevant in that the excited absorption and spontaneous emission cycles may randomize the atomic motion and hamper the experimental observation of radiation damping. [30][31][32][33][34] However, the actual values of the spin dephasing rate c 12 of cold atoms confined in the optical lattice 26 do not play a crucial role in determining the optical response as long as V c ..c 13 ..c 12 , and our results do not significantly change for the range of values 1.0 kHz,c 12 ,100 kHz.
In a typical experimental situation, it is no longer adequate to only obtain the band dispersion via Equation (1). For a finite sample, we would need to compute the reflectivity R, transmissivity T and absorption A512R2T from the transfer matrix M N of the entire stack where I is the unity matrix and N is the number of primitive cells (L5Na). From Equation (3) Figure 2 in the three stop-bands region near resonance. The probe reflectivity behaves quite differently within each stop-band, which may be attributed to the fact that k0, a measure of the extinction rate of evanescent waves, has very different values within these stopbands. The central narrow stop-band, at variance with the other two, exhibits a much steeper optical response at the edge near atomic resonance, where R(v p ) sharply drops to zero from a homogeneous 100% reflectivity plateau as illustrated in Figure 2e. We further compare in Figure 2f the reflectivity (transmissivity) when the pump is off.
In the absence of the central stop-band, we observe a much milder frequency-sensitivity around resonance yet accompanied by a remarkable absorption.

RESULTS AND DISCUSSION
The overall light force experienced by a cold sample of 87 Rb atoms oscillating in a 1D optical lattice is obtained by first computing the and v 0 p~v p g result from the Lorentz transformation and 2 x (gv 0 p ) is the force in units of the incident light mean power (P9) over the speed of light (c). The momentum transfer rate and hence 2 x (gv 0 p ) depend on the atoms' velocity u through the reflectivity and transmissivity in a complicated manner. However, for velocity ranges typical of experiments involving cold atoms in moving optical lattices, one has g<12u/c such that it is viable to expand R gv 0 p and T gv 0 p , which appear in 2 x (gv 0 p ). To the first order in u/c, one arrives at the simplified expression on the right- are, respectively, the velocity-independent and velocity-dependent force components. This expansion is valid provided that the Doppler shift u=c ð Þv 0 p is smaller than the frequency range over which R v 0 p sharply rises to unity at the stop-band edge. Such a frequency range is approximately 1.5c 13 in Figure 2e, but becomes four times as much in Figure 4a. It follows from Equation (4) that, for a lossless (R512T) and nondispersive (LR . Lv 0 p~L T . Lv 0 p~0 ) medium, the velocity-independent term reduces to the pressure force z2R v 0 c, while the velocity-dependent term reduces to the damping force {4R v 0 p P 0 u . c 2 . The latter is much smaller than the former, which makes the damping force difficult to observe for practical lattice velocities u and light powers P. To have F x (0) &(u=c)F x (1) and attain a large enough radiation damping, we should try to find a highly dispersive medium with Lv 0 p &c= uv 0 p accompanied by negligible absorption. This objective can be achieved with cold 87 Rb atoms loaded into an optical lattice and driven into the L configuration as discussed above.
Another case may be relevant when the probe field is a Gaussian light pulse (containing Nww1 photons) instead of a monochromatic plane light wave. In this case, we can verify how the momentum transfer from a light pulse to a periodic atomic structure depends on its oscillating velocity (i.e., the Doppler shift). Relevant results will be used as direct evidence to determine whether the damping force is comparable to or even larger than the light pressure. The net momentum DP x imparted in the lab frame by a light pulse can be obtained by integrating the scaled force 2 gv 0 p over the pulse (normalized) frequency distribution f v 0 p . The main panels of Figure 3a and 3c indicate a (normalized) damping force F x (1) &z9:7|10 8 and absorption A<0.3% near D p 50.5c 13 (denoted by a red square) in the resonance region when a pump with V c 5100 MHz is on. Conversely, in the insets of Figure 3a and 3c where the pump is off (V c 50), we have F x (1) &z1:3|10 7 and A<7.6% near D p 597.0c 13 (denoted by a red square) away from resonance; otherwise, the absorption would be too large. 18 This result clearly indicates that the damping force can be further amplified 70 times and the probe absorption can be further suppressed 25 times when working with the central stop-band rather than the other two. When shorter samples are used (L50.5 cm), the probe reflectivity and transmissivity are not as steep as before; however, we still obtain rather favorable results with F x (1) &z2:3|10 8 and A<0.6% near D p 51.5c 13 at one edge of the optically induced stop-band as demonstrated separately in Figure 4. It is important to note that the damping force [! u=c ð ÞF x (1) ] may be enhanced to be comparable with the pressure force [!F 0 (1) ] for u as small as several tens of cm s 21 , making the tiny optomechanical effect of radiation damping accessible to potential experiments. Note that larger velocities are considered in the more relevant case of a light pulse, such as in Figure 3b and 3d where the four-force component dP x =dt h i ' is computed by the exact expression of Equation (4) rather than the approximate expansion following it. Figure 3b and 3d shows the momentum transfer DP x , in units of the incident light total momentum N h À k, for a pulse with a Gaussian ! centered at v p0 . This quantity depends critically on the lattice velocity u when the pulse width d p0 is small enough that it doesn't extend beyond the steep edge of the stop-band. In numerical calculations, we have defined D p0 5v p0 2v 31 as the central-frequency detuning of the light pulse relative to the probe resonance. From Figure 3b, we can see that for a negative (positive) velocity u526.0 m s 21 (u516.0 m s 21 ), the normalized momentum transfer is increased from 0.38 to 1.62 (decreased from 1.70 to 0.66) at D p0 <0.0c 13 (D p0 <1.33c 13 ) compared with that attained for u50.0 m s 21 . This result indicates that the radiation damping will significantly enhance (weaken) the radiation pressure when a medium moves toward (away from) the light pulse and, thus, always tends to prevent the motion of this medium. Figure 3d further demonstrates that an increase of the pulse width will lead to a decrease of the sensitivity of momentum transfer on the medium velocity.

CONCLUSIONS
In summary, we have investigated the steady optical response of a periodic structure of cold 87 Rb atoms driven into the L configuration and discussed its potential application for the enhancement of typically tiny radiation damping forces. This periodic atomic structure has three photonic stop-bands of high reflectivity near the probe resonance, among which the central narrower one is optically induced by a pump field. At one edge of this dynamically controlled stop-band, the reflectivity is very sensitive to the probe frequency yet accompanied by little absorption. This peculiar property has been exploited to obtain a greatly amplified effect of radiation damping, accessible to experiments when an optical lattice oscillates at a velocity of approximately 1.0 m s 21 . Although our treatment concerns a semiclassical atomic velocity u, the predicted radiation damping would also affect the atomic motion even in a fully quantum regime. 13,35 Finally, our scheme does not rely on any concept unique to atoms. Well-developed stop-bands can also be optically induced, e.g., in a diamond crystal doped with N-V color centers, 36 where radiation damping enhancement may then be anticipated to occur.