Atom Binding and Reflection by Spatially Inhomogeneous ...

VOLUME 80, NUMBER 17

PHYSICAL REVIEW LETTERS

27 APRIL 1998

Atom Binding and Reflection by Spatially Inhomogeneous Spontaneous Emission Y. Japha,1 V. M. Akulin,2,3 and G. Kurizki1 1

Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel 76100 2 Laboratoire Aimé Cotton, CNRS II, Bâtiment 505, 91405 Orsay Cedex, France 3 University of Marne la Vallée, France (Received 30 September 1997)

Remarkable quantum effects are predicted for an initially excited two-level wave packet incident on a broadband vacuum-field reservoir with sharp spatial boundaries: (a) reflection of the excited wave packet from the boundary (skin effect); (b) transient three-dimensional binding of the decayed wave packet to the boundary by photons carrying more energy than the incident wave packet. These effects are realizable for cold atoms in cavities where excitation decay is nearly exponential but strongly enhanced. [S0031-9007(98)05834-7] PACS numbers: 42.50.Ar, 03.65.Bz

Although dissipation of quantum systems is a fundamental, all-pervasive theme of contemporary physics, its research has until recently been limited to two kinds of simple models: (a) structureless particles whose translational motion is dissipated by a thermal reservoir, or (b) excited static systems with few levels (typically two), which relax into or exchange energy with the reservoir [1]. Yet, new, important effects can be revealed by studying the dissipative dynamics of composite quantum systems governed by the interplay between internal and translational degrees of freedom, e.g., atom optics in the presence of spontaneous emission [2] or electron (spin-1y2 particle) propagation in dissipative mesoscopic structures [3]. In particular, dissipation by spontaneous emission can affect the translational motion of atoms along the axis of an optical cavity [4] or result in an exponential enhancement of atomic tunneling probabilities [5]. It is our purpose to address the basic question: how is the free motion of a two-level atom affected by a spatially confined reservoir and by measurements of the reservoir state? Specifically, we study the spontaneous emission from a free-moving cold two-level atom into a broad band of electromagnetic (EM) cavity modes initially in the vacuum state. Two remarkable effects are predicted: (a) partial reflection of excited atomic wave packets from the interface between free space and the cavity, which bears a similarity to the well-known skin effect; (b) transient binding of a decayed atom to the interface by emission of photons carrying more energy than the incident atom. The dynamics of a two-level atom coupled to the EM field is described in the dipole and rotating wave approximations (RWA) by the Hamiltonian X P2 1 vq aqy aq hv ¯ 0 sz 1 1 h¯ H 2 2m q X 2 h¯ fmEq srdaq s1 1 mp Eqp srdaqy s2 g . (1)

operator. The free-field Hamiltonian is written in terms of number operators aqy aq of the photon modes (of the entire Universe) with spatial eigenfunctions Eq srd, which may have a large amplitude inside the confining region. In the last term, m is the matrix element of the atomic dipole, which is assumed to interact only with one polarization of the field and the Pauli matrices s6 stand for the level raising and lowering operators. If initially there are no photons in the field, then the wave function of the total atom plus field system can be written in the following general form, valid in the RWA: X c˜ q sr, td j g, hqjl , (2) jcsr, tdl c˜ e sr, td je, h0jl 1 q

where the ket-vector je, h0jl denotes the atom in the excited state with no photons in the field, whereas j g, hqjl corresponds to the ground state of the atom with a photon emitted at a mode q, and c˜ esqd are the corresponding amplitudes. We obtain from (1) and (2) the following coupled Schrödinger equations for the slowvarying envelopes of these states upon factorizing out their fast time dependence c˜ e,q sr, td e7iv0 ty2 ce,q sr, td: X h= ¯ 2 ≠ (3) ce 2 m Eq srdcq , i ce 2 ≠t 2m q i

≠ h= ¯ 2 cq 2 cq 1 Dq cq 2 mp Eqp srdce , ≠t 2m

where Dq vq 2 v0 is the frequency detuning of photon mode q from the atomic resonance. We start the analysis of these coupled equations by eliminating the amplitude cq of a given j g, ql state and expressing it in terms of the excited state amplitude ce sr, t 0 d at earlier times as Z t 0 dt 0 eiDq st2t d cq sr, td 2mp

q

Here the first term is the atomic internal energy term, v0 being the resonance frequency, and sz the population inversion operator. The second term is the kinetic-energy 0031-9007y98y80(17)y3739(4)$15.00

(4)

3

Z

2`

d 3 r 0 Gsr 2 r 0 , t 2 t 0 dEqp sr 0 dce sr 0 , t 0 d , (5)

© 1998 The American Physical Society

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VOLUME 80, NUMBER 17 m

PHYSICAL REVIEW LETTERS 2imr 2

3y2 expf 2 ht where Gsr, td s 2pi ht ¯ d ¯ g is the free-space (spherical-wave) Feynman propagator. By substituting Eq. (5) in Eq. (3) we obtain a closed equation of motion for ce : ≠ h= ¯ 2 i ce 2 ce ≠t 2m Z Z ` 1 dt 0 d 3 r 0 Gsr, r 0 , t 0 dce sr 0 , t 2 t 0 d , (6) 0

with Gsr, r 0 , td Gsr 2 r 0 , td jmj2

X

Eq srdEqp sr 0 de iDq t (7)

q

as a nonlocal complex response function, determined by the mode amplitudes Eq srd and the detunings Dq . Equation (6) is completely general and allows for the entanglement of the quantized motion with an arbitrary field-dipole coupling. Whenever the kinetic-energy term in (6) is negligible, it yields the equation for the evolution of a static atom coupled to an arbitrary field-mode reservoir [6]. In the limit of a single discrete mode (“good cavity”), where Gsr, r 0 , td is highly nonlocal, Eq. (6) describes oscillatory photon emission and reabsorbtion, along with excited-state reflection or transmission (the mazer) in an empty cavity [7,8]. Here we shall assume that the fieldmode continuum that is appreciably coupled to the atomic transition is a Lorentzian of linewidth hc around a central frequency vc . This description fits Fabry-Perot and confocal cavities. From here on we will concentrate on a “bad” cavity having moderately high finesse, such that we have irreversible (nearly exponential) excitation decay at a rate which is strongly enhanced compared to the natural atomic linewidth, i.e., the free-space decay rate. We shall consider p the limit where the atomic propagation ¯ distance ,c , hymh c during the memory (correlation) time hc21 of the emitted photon is much shorter than the deBroglie wavelength lDB and the excitation decay disp ¯ tance ,d , hy2mg c , which is traversed during its spontaneous lifetime in the confined reservoir gc21 . This limit is equivalent to having the reservoir linewidth hc much larger than the atomic kinetic energy and its spontaneous emission rate in the reservoir gc . In this limit ce can be factored out of the integral in (6), Rwhich then becomes a local complex potential Gsr, td d 3 r 0 Gsr, r 0 , td, indicating that the atomic motion and decay in the reservoir are Markovian, i.e., have negligible correlation (memory) time and distance. Then, by Fourier transforming Eq. (6) with respect to t, one obtains the following timeindependent equation for an atomic excited wave packet that is incident with kinetic energy E (E ø hh ¯ c ) on a field-confined region where spontaneous emission is strongly enhanced: 2m ¯ (8) =2 ce 1 2 fE 2 hGsrdgc e 0. h¯ In order to simplify the expression for Gsrd we make the additional assumptions that hc is much larger than the 3740

27 APRIL 1998

atomic recoil energy Erec ; h¯ 2 v02 y2mc2 and the Doppler shift 6k ? qym, k being the mean atomic momentum. Then the local complex potential Gsrd in Eq. (8) is nearly independent of the atomic and photon energies (momenta) P jmj2 q jEq srdj2 Gsrd ø 2 , (9) Dc 1 ihc where Dc vc 2 v0 is the detuning of the reservoir central frequency vc from the atomic resonance v0 . In this limit RehGsrdj ; dc srd is the local Lamb shift and ImhGsrdj ; gc srd is the local rate of spontaneous emission. When the atomic motion becomes classical and Gfrstdg depends parametrically on rstd, we recover from Eq. (3) the previous results concerning fast-moving atoms in spatially varying reservoirs [9]. We can illustrate the quantitative applications of Eqs. (8) and (9) by the simple model Gsrd 2igc Qsxd, where Qsxd is the Heaviside step function. We assume that the excited atomic wave packet is normally incident from the free-space region x , 0 on the interface at x 0 and that the Lorentzian reservoir is resonant with the atom, Dc 0, so that the potential 2igc is purely imaginary. In this case, Eq. (8) becomes one dimensional and has the solution ce sx . 0d seikc x , ce sx , 0d eikx 1 re 2ikx , p p where k 2mEyh, ¯ kc 2msE 1 i hg ¯ c dyh, ¯ and the coefficients of reflection (r) and transmission (s) are 2k k2k s k1kc and r k1kcc . The wave vector kc in the dissipative region x . 0 assumes a complex value with positive imaginary part. Hence, the probability to detect an excited atom at x . 0 decreases exponentially with x and only the fraction jrj2 of excited atoms remains at large negative x (to the left of the interface). This reflection ¯ c . E is analogous to the increases with gc , and, for hg skin effect of light reflection from metals (Fig. 1). The novelty here is that the excited atoms are reflected by an imaginary potential, which is induced by a broadband vacuum-field reservoir, due to its sharp confinement. When does the imaginary part of Gsrd significantly contribute to the reflection of excited atoms? For this to happen it is required thatp hg ¯ c . E, or equivalently, that the decay length ,d , hy2mg ¯ c be shorter than lDB . The stronger requirement is that the imaginary potential should rise from its free-space value jGsrdj gfree to a much larger value, such that jGsrdj . E, on a length scale shorter than both ,d and lDB . Since the mode amplitudes Eq srd contributing to Gsrd vary on the scale of their wavelengths lopt , this requirement can be satisfied only for lDB . lopt , i.e., for subrecoil energies E , Erec . The atomic skin effect is realizable for cold atoms weakly coupled to a single-mode confocal cavity having mirrors with high reflectivity R. The requirements are that (i) the decay rate gc be strongly enhanced [6,10], gc ygfree ø 3fy2s1 2 Rd ¿ 1, where f is the solid angle



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(a) Cavity

(b)

Atom

Filter

∆E

1/τD

0

E

2E ∆q Forbidden

FIG. 1. Numerical simulation of an initially excited atomic wave packet (dashed) approaching a sharp interface between a cavity with enhanced spontaneous emission and free space: (a) reflected excited envelope (solid) and total ground-state envelope (shaded); (b) the incident and reflected wave packets at the moment of incidence on the interface (solid) and a bound component (shaded) associated with “forbidden” photon emission ( hv ¯ q . E 1 hv ¯ 0 ). Inset: Spectra of incident atomic wave packet and detector “filter” for “forbidden” photons.

fraction subtended by the cavity mirrors; (ii) gc be larger than Eyh, ¯ but much smaller than the cavity spectral linewidth hc s1 2 RdcyL, L being the mirror separation. These conditions hold for f * 1023 , 1 2 R * 1024 , L & 1 cm, and gfree ø 106 s21 . We note that ce srd and cq srd in a single-mode cavity are diffracted perpendicular to the incidence axis x, along the cavity axis z, due to the periodicity of Eq srd [2]. A simplified one-dimensional character of the above effects obtains in a high-Q confocal cavity whose mirror diameter D is comparable to the distance L between them. Such a cavity has very large mode degeneracy that smears out the oscillations of the different jEq srdj2 contributing to Gsrd, thereby canceling the diffraction in directions perpendicular to the axis of incidence x and rendering the Gsrd only x dependent (Fig. 2, inset). If we take D 0.04 mm and L 0.05 mm ( f ø 0.4), then R 0.999 yields gc ygfree 400, hc 6 3 1011 sec21 ¿ Erec yh¯ , 10 kHz. We assume that hg ¯ free , 0.1E, whence (Fig. 2) a purely imaginary potential Gsrd 2igc (Dc 0) gives a reflection of 2% if lDB 0.05L 2.5 mm). When Dc , hc the potential has a real part (Lamb shift) RehGj dc , gc Dc yhc as well, for which the reflectivity can be much stronger (Fig. 2). Another intriguing effect is revealed by considering the ground-state atomic distribution. Once the spatial dependence of ce srd is evaluated from (8) for a given Gsrd, we can infer the ground-state amplitude cq for an excited atom incident with energy E that has emitted a photon with wave vector q, using the Fourier transform of Eq. (5)

FIG. 2. Reflection probability (see text for parameters) of an atom from a high-Q highly degenerate confocal cavity (left inset) as a function of lDB yL for different Dc . Right inset: 2GsrdyGs0d as a function of x (normalized to the waist), for y z 0 (a sum over 676 degenerate modes).

cq srd 2mp

Z

d 3 r 0 Eqp sr 0 dGsr 2 r 0 , E 2 hD ¯ q dce sr 0 d , (10)

where Gsr, Ed smyhd ¯ expsikrdyr [the Fourier transform of the propagator in (5)] hasp the shape of a spheri¯ The atomic cal wave with wavevector k 2mEy h. ground qstate consists of partial waves with momenta ¯ q d. The atomic kinetic energy Eq hk ¯ q 2msE 2 hD E 2 hD ¯ q associated with each partial wave is then equal to the total energy of the incident excited atom E 1 hv ¯ 0 minus the energy hv ¯ q that is carried away by the emitted photon. Remarkably, the ground state contains bound atomic states, with Eq , 0 and imaginary kq , created by the emission of photons with hv ¯ q . E 1 hv ¯ 0 . These bound atomic states are superpositions of evanescent (exponentially diminishing) spherical waves confined at maxima of jEq srdj near the interface, where the decay rate is peaked. Typically, they extend over radius ,lopt from the interface. But is it possible to observe these bound atomic states? These states can be projected out of the superposition with other atomic momentum states by measuring a “forbidden” photon centered at energy hv ¯ q . E 1 hv ¯ 0 , which is correlated with the bound atomic wave function. However, nonzero atomic bandwidth DE is essential for forbidden photon detection, since any measurement which localizes a particle to a finite region in space requires nonzero initial momentum uncertainty. The bound atomic wave function and the correlated forbidden photon have a finite existence time, which is essentially the dwell time 3741



tD , hyDE ¯ of the excited wave packet ce sr, td near the interface [Eq. (5)]. For cold atoms with DEyh¯ , Erec , 10 kHz this implies tD , 1024 s, which corresponds to the huge photon localization distance of ctD , 30 km. The time resolution of the detector must coincide with tD [Fig. 1(b), inset]. This detection projects the atomic ground-state wave packet onto a freely propagating state with average energy jEq j, as would any measurement attempting to localize a tunneling particle with the barrier region. Hence, energy is not conserved in this measurement and after it the exponentially localized atomic wave function starts spreading according to free-space evolution. Binding is possible also for atoms incident with E ¿ ¯ c , so that photons with hD ¯ q. Erec , provided E , hg E fall within the spontaneous-emission linewidth. A quantitative estimation of the binding effect can be gained by considering the quasi-one-dimensional atomic decay occurring in a highly degenerate cavity. For an atom incident on such a cavity with E ¿ Erec , the emission probability of photons causing binding (with hD ¯ q . E) is found to be Ç Ç2 dPsvq d 2k 1 geff 2 2 dvq k" 1 kc 8p # Dq 1 geff DE jkc j2 3 11 x2 , (11) jkq j hjk ¯ qx jy0 where y0 is the group q velocity of the incident atomic x wave packet, kq kq2 2 qy2 2 qz2 is the imaginary x component of kq , and geff is the spontaneous emission rate at distance ,lopt (an optical wavelength) from the interface. Under these conditions, the total probability to detect a transient photon causing atomic binding (localization) for atomic spectral widths DE , 0.1hg ¯ eff and energies E & 0.3hg ¯ eff , reaches more than 10%. This regime is realizable for the 33 P1 -31 S1 transition in 40 Ca atoms (lopt 657.46 nm, gfree 2.5 kHz, Erec 11.5 kHz) cooled to the recoil temperature kB T Erec and accelerated to an average velocity of 0.5 cm s21 . The gravity potential across the cavity interface is comparable to the binding energy and to Gsrd. Hence, its effect must be reckoned with. Alternatively, the effect of gravity can be avoided by accelerating the atom (exiting from a magneto-optical trap) in a horizontal laser waveguide to the required velocity.

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To conclude, we have analyzed a hitherto unexplored dynamical regime of wave packets with internal-state dissipation: spatially local (Markovian) propagation of excited cold atoms in a complex potential induced by a broadband vacuum-field reservoir with a sharp interface. The combined reflection of atomic excited and ground states from the interface in this regime can be ascribed to a dipole force induced by the vacuum-field reservoir. However, only the fully quantum scattering equations (8) and (10) predict the detailed evolution of the atom and field and reveal the intriguing effect of atomic binding. This work has been supported by the EU TMR grant and by GIF.

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