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 P=mc2, 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 quantum-limited read-out of its position9,10,11,12 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≈10−12 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 mirror15 or in a one-dimensional (1D) ordered structure of cold 87Rb 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 87Rb atoms when driven into a Λ configuration27 of levels by a pump beam (cf. Figure 1) that allows for an efficient all-optical (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 6.0×1012 cm−3, e.g., for atoms sloshing within the confining optical potential at characteristic velocities (about 1.0 m s−1), 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 ones15,18 and a viable option for the observation of radiation damping effects.

Figure 1
figure 1

(a) Atomic Bragg mirror obtained by loading cold 87Rb atoms into an optical lattice formed by retro-reflecting a laser beam of wavelength λo that is far-detuned from relevant atomic resonances (dipole trap). The confined atoms arrange themselves into a 1D chain of pancake-shaped layers with an average thickness d and are located at the standing-wave antinodes of period a=λo/2. Such an ordered atomic structure can be set to move coherently inside the lattice.28,29 When a weak probe pulse impinges from the left, the moving Bragg mirror experiences a (viscous) friction force that depends on its velocity υ and can be all-optically controlled through the frequency and intensity of an external pump. (b) The slightly detuned probe (Ωp) and pump (Ωc) fields drive all the confined cold atoms into a three-level Λ configuration. The three states correspond to the hyperfine split levels of the D2 line in 87Rb.

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Materials and methods

Our ensemble of cold 87Rb atoms is loaded into an optical lattice of period a formed by retro-reflecting a light beam of wavelength λo=2a (Figure 1). In each period of such a dipole trap, the confined atoms occupy a small region d<<a with a homogeneous volume density N0. 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 ωp 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(ωp) 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 κ=κ′+iκ″. The refractive index associated with the collection of cold trapped atoms, that enters M(ωp) is obtained from:

where γ12 (γ13) is the dephasing rate of the spin (optical) coherence ρ12 (ρ13), Δp=ωp−ω31 (Δc=ωc−ω32) is the probe (pump) detuning relative to transition (), and μ13 denotes the relevant electric-dipole moment. Here, we assume that the cold 87Rb atoms are driven into a three-level Λ-type configuration27 on the D2 line by a probe of Rabi frequency Ωp traveling in the x-direction and a pump of Rabi frequency Ωc traveling in the y- (or z-) direction (Figure 1). Note that the probe field may have a small angle of incidence (e.g., θ≈20) 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 (Ωp<<Ωc), within the rotating-wave and electric-dipole approximations. In particular, Equation (1) recovers the case of a single absorption line when Ωc→0, but implies a narrow electromagnetically induced transparency(EIT) window near the probe resonance when due to the quantum destructive interference between two dressed-state transitions. However, for Ωc>>γ13, the dressed susceptibility evolves into that for two well-separated AT split levels with the quantum destructive interference becoming unimportant.

Solving Equation (1) for κ′ and κ″ as a function of ωp(Δp) yields photonic band-gaps that appear at the boundary of the first Brillouin zone (κ′=π/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 χ(ωp) in the case without the pump (Ωc=0), two wide stop-bands open up in negative and positive detuning regions near a single absorption line. However, when the pump is on (Ωc≠0), as illustrated in Figure 2a and 2c, a third narrow stop-band arises24,26 and fits between the two wide ones separated by the AT split absorption lines.

Figure 2
figure 2

Real (black-solid) and imaginary (red-dashed) parts of the Bloch wave vector κ as a function of probe detuning Δp for an infinite periodic structure of cold 87Rb atoms as determined from Equations (1)–(2) with (a) and without (b) the pump. The relevant parameters are γ13=6.0 MHz, Δc=0, N0=6.0×1012 cm−3, μ13=1.5×10−29 C m, λp=780.792 nm, a=780.787/2 nm and d=a/20. (c) and (d) are enlargements of (a) and (b), respectively, in the resonance region. (e) and (f) present the reflectivity (black-solid) and transmissivity (red-dashed) profiles corresponding to (c) and (d), respectively, for a L=2.0 cm long periodic structure of cold 87Rb atoms as determined from Equation (3).

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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 γ12 of cold atoms confined in the optical lattice26 do not play a crucial role in determining the optical response as long as Ωc>>γ13>>γ12, and our results do not significantly change for the range of values 1.0 kHz<γ12<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 A=1−R−T from the transfer matrix MN of the entire stack

where I is the unity matrix and N is the number of primitive cells (L=Na). From Equation (3), we compute the probe reflectivity and transmissivity for a sample of density N0=6.0×1012 cm−3 and length L=2.0 cm; relevant results are plotted in 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 κ″, a measure of the extinction rate of evanescent waves, has very different values within these stop-bands. 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(ωp) 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 87Rb atoms oscillating in a 1D optical lattice is obtained by first computing the transfer rate of the four-momentum dPμ/dt in the atoms’ rest frame with the help of a Maxwell stress tensor. Upon Lorentz-transforming to the lab (primed) frame, where the atoms move with a velocity υ along x, the relevant component of the average four-force is:18

where and result from the Lorentz transformation and is the force in units of the incident light mean power (P′) over the speed of light (c). The momentum transfer rate and hence depend on the atoms’ velocity υ 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 η≈1−υ/c such that it is viable to expand and , which appear in . To the first order in υ/c, one arrives at the simplified expression on the right-hand side of Equation (4), where and are, respectively, the velocity-independent and velocity-dependent force components. This expansion is valid provided that the Doppler shift is smaller than the frequency range over which sharply rises to unity at the stop-band edge. Such a frequency range is approximately 1.5γ13 in Figure 2e, but becomes four times as much in Figure 4a.

It follows from Equation (4) that, for a lossless (R=1−T) and non-dispersive () medium, the velocity-independent term reduces to the pressure force , while the velocity-dependent term reduces to the damping force . The latter is much smaller than the former, which makes the damping force difficult to observe for practical lattice velocities υ and light powers P. To have and attain a large enough radiation damping, we should try to find a highly dispersive medium with accompanied by negligible absorption. This objective can be achieved with cold 87Rb atoms loaded into an optical lattice and driven into the Λ configuration as discussed above.

Another case may be relevant when the probe field is a Gaussian light pulse (containing 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 ΔPx imparted in the lab frame by a light pulse can be obtained by integrating the scaled force over the pulse (normalized) frequency distribution .

The main panels of Figure 3a and 3c indicate a (normalized) damping force and absorption A≈0.3% near Δp=0.5γ13 (denoted by a red square) in the resonance region when a pump with Ωc=100 MHz is on. Conversely, in the insets of Figure 3a and 3c where the pump is off (Ωc=0), we have and A≈7.6% near Δp=97.0γ13 (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 (L=0.5 cm), the probe reflectivity and transmissivity are not as steep as before; however, we still obtain rather favorable results with and A≈0.6% near Δp=1.5γ13 at one edge of the optically induced stop-band as demonstrated separately in Figure 4. It is important to note that the damping force may be enhanced to be comparable with the pressure force for υ as small as several tens of cm s−1, 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 is computed by the exact expression of Equation (4) rather than the approximate expansion following it.

Figure 3
figure 3

Normalized damping force (a) and absorption (c) as a function of probe detuning Δp for a 2.0-cm long periodic structure of cold 87Rb atoms in the presence of the pump. The insets show the corresponding force and absorption when the pump is off. The relevant parameters are the same as those in Figure 2. Normalized momentum transfer from a Gaussian light pulse to the same atomic structure of cold 87Rb as a function of central-frequency detuning Δp0 with pulse width δp0=5.0 MHz (b); δp0=20.0 MHz (d). The black-solid, red-dashed and blue-dotted curves refer to velocities υ=0.0, +6.0 and −6.0 m s−1, respectively.

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Figure 4
figure 4

Reflectivity (a), normalized damping force (b), transmissivity (c) and absorption (d) as a function of probe detuning Δp for a shorter periodic structure of cold 87Rb atoms in the presence of the pump. The relevant parameters are the same as in Figure 2 except L=0.5 cm.

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Figure 3b and 3d shows the momentum transfer ΔPx, in units of the incident light total momentum , for a pulse with a Gaussian frequency distribution centered at ωp0. This quantity depends critically on the lattice velocity υ when the pulse width δp0 is small enough that it doesn't extend beyond the steep edge of the stop-band. In numerical calculations, we have defined Δp0=ωp0−ω31 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 υ=−6.0 m s−1 (υ=+6.0 m s−1), the normalized momentum transfer is increased from 0.38 to 1.62 (decreased from 1.70 to 0.66) at Δp0≈0.0γ13 (Δp0≈1.33γ13) compared with that attained for υ=0.0 m s−1. 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 87Rb atoms driven into the Λ 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−1. Although our treatment concerns a semiclassical atomic velocity υ, 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.