Spontaneous emission from a two-level atom tunneling in a double-well potential

Spontaneous emission from a two-level atom tunneling in a double-well potential

Daniel Braun and John Martin

Laboratoire de Physique Théorique–IRSAMC, CNRS, Université Paul Sabatier, Toulouse, France

共Received 27 July 2007; revised manuscript received 15 January 2008; published 10 March 2008兲 We study a two-level atom in a double-well potential coupled to a continuum of electromagnetic modes 共black-body radiation in three dimensions at zero absolute temperature兲. Internal and external degrees of freedom of the atom couple due to recoil during emission of a photon. We provide a full analysis of the problem in the long wavelengths limit up to the border of the Lamb-Dicke regime, including a study of the internal dynamics of the atom 共spontaneous emission兲, the tunneling motion, and the electric field of the emitted photon. The tunneling process itself may or may not decohere depending on the wavelength corre-sponding to the internal transition compared to the distance between the two wells of the external potential, as well as on the spontaneous emission rate compared to the tunneling frequency. Interference fringes appear in the emitted light from a tunneling atom, or an atom in a stationary coherent superposition of its center-of-mass motion, if the wavelength is comparable to the well separation, but only if the external state of the atom is post selected.

DOI:10.1103/PhysRevA.77.032102 PACS number共s兲: 03.65.Xp, 03.65.Yz

I. INTRODUCTION

Young’s double slit experiment, in which interference is observed from light passing through two small slits or holes in a plate placed at a distance comparable to the wavelength of the light, constitutes one of the experiments at the base of quantum mechanics. Theory and experiment have been re-fined over the years to the point that the two holes have been replaced by two trapped atoms or ions which scatter incom-ing laser light关1–6兴. With the advance of the coherent con-trol of the external degrees of freedom of atoms 共see, e.g., Refs.关7–10兴兲, the realization of atom interferometers 关11–13兴 共see Ref. 关14兴 for a recent review兲, and in particular the re-alization of macroscopic quantum superposition 共so-called “Schrödinger cat”兲 states of the center-of-mass coordinate of a single atom or ion关15兴, it is natural to ask if interference could be observed in the light emitted from a single atom superposed coherently in two different positions. A similar question was answered to the negative in a paper by Cohen-Tannoudji et al.关16兴 for the case of scattering of light from a massive object brought into orthogonal position states. The physical reason is clear: interference can only arise if the probe particle can distinguish the two locations of the target. But then the probe particle must get entangled with the tar-get. If the target is massive its two orthogonal position states remain unaltered during scattering and therefore lead to van-ishing overlap of the scattered probe states after tracing out the target. Later the scattering problem was reconsidered for lighter targets, where it was shown that interference can arise. In particular, Rohrlich et al.关17兴 analyzed the general situation of the scattering of two free particles, a probe with mass m and a target with mass M. Interference was predicted for the case of m⯝M, and even perfect visibility of interfer-ence fringes for m = M in one dimension.

Similarly, Schomerus and co-workers 关18兴 analyzed the scattering of particles from a “quantum obstacle,” an ob-stacle brought into a coherent superposition of positions. An important difference from Ref.关17兴 lies in the fact that the target was supposed to be bound in a double-well potential

and to tunnel coherently between the two wells with tunnel-ing frequency⌬. For the case of one-dimensional scattering, they showed that the quantum obstacle leads to almost the same transmission resonances as two fixed obstacles, if the kinetic energy⑀of the incident particle satisfies⑀Ⰶ⌬. In the opposite limit, interference can still be recovered by post-selecting the elastic scattering channel.

Spontaneous emission is not the same as scattering, and it is a priori unclear if these results apply to spontaneous emis-sion alone as well. Furthermore, the properties of the emitted light are only a small part of the interesting physics that can arise, if internal and external degrees of the tunneling atom are coupled. Indeed, one might ask, if spontaneous emission itself共e.g., the decay rates of the excited level兲 changes, if the atom is brought into a coherent superposition of different external states. Also, what happens with the tunneling mo-tion? To what extent does the emitted photon cause decoher-ence of the external degree of freedom? Spontaneous emis-sion from a tunneling two-level atom was considered in Ref. 关19兴 for the case of the transmission of an atom through a rectangular energy barrier in one dimension with the atom coupled to a one-dimensional mode continuum. It was shown that the recoil from photon emission can lift the atom over the tunnel barrier and thus increase significantly the trans-mission.

In the present paper we examine spontaneous emission of a two-level atom trapped in a double-well potential, where the atom interacts with the full three-dimensional continuum of electromagnetic modes共the interaction with a single cav-ity mode was treated in Ref.关20兴兲. We carefully investigate the effective dynamics of all three subsystems involved: the internal degree of freedom of the atom, the tunneling motion, and the electric field created by the emitted photon in a re-gime where the photon wavelength is at most comparable to the well separation. The tunneling motion may suffer deco-herence from the emission of a photon from the atom, but this depends, among other things, on the timing of the emis-sion of the photon. Interference in the light emitted from the atom is very weak, but interference with perfect visibility of

the fringes arises if the external state of the atom is post-selected in the energy basis.

II. MODEL

A. Derivation of the Hamiltonian

We consider a trapped two-level atom共with levels 兩g典, 兩e典 of energy ⫿ប␻0/2, respectively兲 interacting with traveling modes of the electromagnetic field as illustrated in Fig. 1. The atom is assumed to be tightly bound in the x-y plane at the equilibrium position x = y = 0 and to experience a sym-metric double-well potential V共z兲 along the z direction.

The Hamiltonian of this system is given by

H = HA+ HF+ HAF, 共1兲

where HA= HA

ex_{+ H}

A

in_{denotes the Hamiltonian of the trapped} atom, HFis the Hamiltonian of the free field, and HAFis the

interaction Hamiltonian describing the atom-field interaction. Explicitly, HAex= pz 2 2M + V共z兲, 共2兲 HA in =ប␻0 2 ␴z in , 共3兲 HF=ប

兺

k ␻kak†ak, 共4兲

and, in the dipole approximation

HAF= −

兺

k

d · Ek, 共5兲

where d is the atomic dipole operator and

Ek=Ek⑀k共akeik·R+ ak

†

e−ik·R兲 共6兲

is the electric field operator,Ek=

冑

ប␻k

2⑀0V,⑀0is the permittivity of free space, V the electromagnetic mode quantization vol-ume,⑀k the electric field polarization vector共normalized to

length one兲, k stands for wave number k=共kx, ky, kz兲, and

polarization ␭=1,2 of the electromagnetic modes with

fre-quency ␻k= c兩k兩 共where c is the speed of light in vacuum兲

and R =共x,y,z兲 for the center-of-mass position of the atom. Note that in Eqs. 共2兲–共6兲, z is still an operator, with pz

its conjugate momentum for the atomic center-of-mass motion along the z axis; M denotes the atomic mass ␴z

in_{=兩e典具e兩−兩g典具g兩 and a}

k共ak

†_{兲 the annihilation 共creation兲} op-erator of mode k of the radiation field.

In the following we will resort to the two-level approxi-mation of the motion in the external potential which amounts to taking into account only the two lowest energy levels of the Hamiltonian HAex. We denote by ⌬ the tunnel splitting,

i.e., the energy spacing between the two lowest energy states 共the symmetric 兩−典 and antisymmetric 兩+典 states兲 of the double-well potential. Within this approximation, Hamil-tonian共2兲 becomes

HAex=

ប⌬ 2 ␴z

ex _共7兲

with␴_zex=兩+典具+兩−兩−典具−兩. We can form states that are mainly concentrated in the left and right wells by superposing the symmetric兩−典 and antisymmetric 兩+典 states

兩L典 =兩 + 典 − 兩− 典

_冑

2 , 兩R典 =

兩 + 典 + 兩− 典

冑

2 . 共8兲

The position operator z reads z = b␴xex/2 in the two-level

ap-proximation, where ␴_xex=兩−典具+兩+兩+典具−兩=兩R典具R兩−兩L典具L兩 and

b/2=具+兩z兩−典=具R兩z兩R典 is the average z position of the atom

localized in the right well共see Fig.1兲.

The two-level approximation is justified if the higher vi-brational energy levels are not populated during the sponta-neous emission process. This is well satisfied, if the recoil energy共ប␻0/c兲2/2M is much smaller than the difference in energy to the next highest vibrational levelប␻23共see Fig.1兲 in the external potential共Lamb-Dicke regime with respect to vibrational excitations兲. This implies that the wavelength ␭ of the emitted photon is much larger than the extension of the ground state wave function. However, the level splitting ប⌬ between the two lowest states in the double-well poten-tial retained in the two-level approximation is typically much smaller than the energy differenceប␻₂₃ to the next highest vibrational level, and hence one should distinguish two dif-ferent Lamb-Dicke regimes: one with respect to vibrational excitations and the other with respect to tunneling. While remaining in 共and up to the border of兲 the Lamb-Dicke re-gime with respect to vibrational excitations, our analysis is not restricted to the Lamb-Dicke regime with respect to the ground-state doublet. As we want to reach the regime where ␭⬃b in order to observe interference in the emitted light, we have to go up to the border of the Lamb-Dicke regime with respect to vibrational excitations. The localization of the states兩R典 and 兩L典 compared to ␭ is measured by the param-eter␤=␻0b/c=2␲b/␭. Numerical simulations show that for ␤⬃3 共i.e., ␭⬃2b兲, the restriction of the dynamics to the two lowest states is still a reasonable approximation 共see Sec. II E for further details兲.

−b 2 b 2 z ∆ ω23 ω0 |e
|g
z x y k ℘ η θ φ

FIG. 1. Two-level atom in a double-well potential interacting with a continuum of electromagnetic waves. Right panel: coordi-nate system used, with the atom tunneling in the z direction, and the atomic dipole in the x-z plane.

By expressing the dipole operator as d =具e兩d兩g典␴_xin ⬅㜷⑀d␴x

in_{, the interaction Hamiltonian共5兲 can be cast in the} form

HAF=ប

兺

k

gk␴xin共akeik·R+ ak†e−ik·R兲 共9兲

with atom-field coupling strength gk= −㜷⑀d·⑀kEk/ប, dipole

matrix element㜷, and the unit vector⑀d in the direction of

the vector of dipole matrix elements, which we take without restriction of generality in the x-z plane with components ⑀d=共sin␩, 0 , cos␩兲 共see Fig.1兲. Furthermore,␴x

in

=␴₊in+␴₋in, ␴−in=兩g典具e兩, ␴+in=兩e典具g兩, and exp共ik·R兲=cos␬+ i sin␬ ␴xex

with␬= kzb/2. We will consider in the following the

situa-tion where ␻₀Ⰷ⌬, and introduce the small parameter ␦ ⬅⌬/␻0Ⰶ1. Experimentally, this is the most accessible situ-ation共see Sec. II E兲. A rotating wave approximation is then in order, which leads to the interaction Hamiltonian

HAF=ប

兺

k gk关cos␬共ak␴+ in + ak †_␴ − in_兲 + i sin␬ ␴xex共ak␴in+− ak†␴−in兲兴. 共10兲 Note that in the case of⌬⬃␻0 two additional terms would have to be kept, i sin␬共ak␴+

ex_␴ − in_{− a} k †_␴ − ex_␴ + in_{兲. Equation 共10兲} makes clear that different electromagnetic waves will act in quite different ways: waves with sin␬= 0 will only interact with the internal degree of freedom, but leave the atom po-sition untouched. Indeed, these waves do not distinguish be-tween the left and the right well. In particular, in the long wavelength limit ␭Ⰷb 共␬Ⰶ1兲 the atom stays in its initial motional state. Waves with sin␬⫽0, however, will couple to both internal and external degrees of freedom at the same time and can thus modify the tunneling behavior of the atom. It is evident from Eq.共10兲 that the tunneling motion can in principle reduce spontaneous emission: The coupling con-stant of the second term in Eq.共10兲 changes its sign with the position of the atom in the double-well关“+” in the right 共z ⬎0兲 well, “−” in the left well兴, as 兩R典 and 兩L典 are eigenstates of␴xexwith⫾1 as eigenvalues. In the case of rapid tunneling

motion, the sign of that part of the Hamiltonian is therefore reverted, so is the corresponding time evolution, and sponta-neous emission is thus reduced. However, reverting the time evolution has to happen on the time scale of the dominating bath modes共i.e., 1/␻0兲 in order to give a significant effect. Based on Eq.共10兲, one may expect at most a reduction by a factor 2 in the rate of spontaneous emission for⌬⬃␻0as the first term in Eq.共10兲 is independent of the external degree of freedom of the atom. In the limit␦Ⰶ1 which we consider in this paper, the change of⌫ will turn out to be very small.

We will describe all dynamics in the interaction picture with the free Hamiltonian H0= HAin+ HAex+ HF. The

corre-sponding time dependent field and atomic operators read

ak共t兲=exp共iH0t/ប兲akexp共−iH0t/ប兲=akexp共−i␻kt兲, ␴+ ex_共t兲 =␴₊exexp共i⌬t兲, and ␴₊in共t兲=␴₊inexp共i␻0t兲. We thus arrive at the final form of the Hamiltonian

HAF共t兲 = ប

兺

k gk关cos␬共ei共␻0−␻k兲tak␴+ in + e−i共␻0−␻k兲t_a k †_␴ − in_兲 + i sin␬共ei共␻0+⌬−␻k兲t_a k␴+ex␴+in+ e i_共␻0−⌬−␻k兲t_a k␴−ex␴+in − e−i共␻0+⌬−␻k兲t_a k †_␴ − ex_␴ − in − e−i共␻0−⌬−␻k兲t_a k †_␴ + ex_␴ − in_兲兴. 共11兲

B. Internal dynamics—spontaneous emission

Let us first examine the process of spontaneous emission for the tunneling atom. We write a general pure state of the entire system共atom+field兲 as

兩␺共t兲典 =

兺

n,␴=⫾,␮=g,e

cn␴␮兩n␴␮典, 共12兲

where兩n␴␮典⬅兩n典兩␴典ex兩␮典in,兩n典=兿k兩nk典 is a product state of

all the field modes, nk= 0 , 1 , 2 , . . ., denotes the occupation

number of mode k, and the sum over n is over all sets 兵n0, n1, . . .其, 兩␴典exdenotes the atomic external state, and 兩␮典in the atomic internal state. We start with a general initial state without any photon, but with the atom excited internally, and externally in an arbitrary pure state

兩␺共0兲典 = c0+e共0兲兩0 + e典 + c0−e共0兲兩0 − e典. 共13兲

Normalization imposes 兩c_0+e共t兲兩2_+兩c

0−e共t兲兩2= 1. From the

Schrödinger equation in the interaction picture iបd dt兩␺共t兲典

= HAF共t兲兩␺共t兲典 we obtain the equations of motion for the

rel-evant coefficients,

ic˙0−e=

兺

k

gkei共␻0−␻k兲t共cos␬c1_k−g+ i sin␬e−i⌬tc1_k+g兲, 共14兲 ic˙0+e=

兺

k gkei共␻0−␻k兲t共cos␬c1_k+g+ i sin␬ei⌬tc1_k−g兲, 共15兲 ic˙₁ k−g=

兺

k

gke−i共␻0−␻k兲t共cos␬c0−e− i sin␬e−i⌬tc0+e兲,

共16兲

ic˙1_k+g=

兺

k

gke−i共␻0−␻k兲t共cos␬c0+e− i sin␬ei⌬tc0−e兲,

共17兲 where the overdot means derivative with respect to the time

t. We can formally integrate Eqs. 共16兲 and 共17兲 and insert

them into Eqs.共14兲 and 共15兲. This leads to a closed system of equations for the coefficients c0⫾e. In order to avoid

unnec-essarily heavy notations, we focus momentarily on the equa-tion for c0−e, which can be compactly summarized as

c˙0−e=G共␻0,c0−e兲 + G共␻0−⌬,c0−e兲

+Gc共␻0,c0−e兲 − Gc共␻0−⌬,c0−e兲, 共18兲 G共␻0,c0−e兲 ⬅ − 1 2

兺

k gk 2

冕

0 t dt

⬘

ei共␻0−␻k兲共t−t⬘兲_c 0−e共t

⬘

兲, 共19兲

Gc共␻0,c0−e兲 ⬅ − 1 2

兺

k gk2cos 2␬

冕

0 t dt

⬘

ei共␻0−␻k兲共t−t⬘兲_c 0−e共t

⬘

兲. 共20兲 The equation for c0+e can be found by substituting⌬→−⌬,

and c_0−e→c_0+e in Eq. 共18兲. In principle, Eq. 共18兲 contains two more terms, one given by

i 2

兺

k gk 2 sin 2␬

冕

0 t dt

⬘

ei共␻0−␻k兲共t−t⬘兲_e−i⌬t⬘_c 0+e共t

⬘

兲 共21兲

and the other by an almost identical term with opposite sign and the phase factor e−i⌬t⬘ replaced by e−i⌬t. However, we will find that the overwhelming part of the time integrals comes from t

⬘

⯝t⫾b/c, such that the two phases differ only by ⌬b/c. This quantity represents a tunneling speed com-pared to the speed of light, and has to be necessarily much smaller than one, as otherwise the tunneling would have to be described in relativistic terms. Indeed, even for very large tunneling splittings 共⬃MHz兲 and well separation 共⬃␮m, say兲, this ratio is of order 10−8 _{and thus entirely negligible.} Therefore, the two additional terms cancel to very good ap-proximation.

We replace the sum over k by an integration in the limit of infinite quantization volume V, use polar coordinates for k, and find

Gc共␻0,c0−e兲 = Gc共+兲共␻0,c0−e兲 + Gc共−兲共␻0,c0−e兲, 共22兲

Gc共⫾兲共␻0,c0−e兲 = − 㜷2 64␲2⑀0បc3

冕

₋₁ 1 d␮h共␩,␮兲 共23兲 ⫻

冕

0 t dt

⬘

冕

0 ⬁ d␻ ␻3ei关⫾␮b/c−共t−t⬘兲兴␻ei␻0共t−t⬘兲_c 0−e共t

⬘

兲, 共24兲

h共␩,␮兲 = 关sin2␩+ 2 cos2␩+␮2共sin2␩− 2 cos2␩兲兴, 共25兲 which is still exact, but makes Eq.共18兲 a complicated diff-erointegral equation. In order to proceed, we resort to the Wigner-Weisskopf approximation 关21,22兴. This amounts to realizing that the main contribution to the integral over ␻ will arise from a narrow interval around␻=␻0, with a width of the order of the rate of spontaneous emission⌫. The stan-dard Wigner-Weisskopf vacuum spontaneous emission rate reads

⌫共␻0兲 = ␻03㜷2

3␲⑀0បc3, 共26兲

which means that ⌫共␻0兲/␻0=共4/3兲␣共␻0d/c兲2Ⰶ1, where ␣ ⯝1/137 is the fine-structure constant and d=㜷/e0is the di-pole length 共dipole matrix element divided by electron charge兲. The ratio␻0d/c must be much smaller than one for the dipole approximation to hold. In our problem, the rate of spontaneous emission will be hardly modified. Thus, for all values of␻₀, there is indeed a sharp peak of width⬃⌫ in the integrand of the integral over␻ 共if one was to perform the

integration over t

⬘

first兲. The factor␻3_{varies only slowly on} that scale, and can therefore be pulled out of the integral. Moreover, the lower bound of the␻integral can be extended to −⬁, such that the␻integral leads to a Dirac-delta function 2␲␦(⫾␮b/c−共t−t

⬘

兲). The slight retardation b/c

correspond-ing to the time of travel of a light signal between the two wells is important here as it determines the␮-interval that contributes, but can be neglected in c_0−e after performing the integration over t

⬘

, as c0−e will evolve on a time scale

1/⌫Ⰷb/c. We thus arrive at Gc共␻0,c0−e兲 = − d共␩,␤兲 ⌫共␻0兲 4 c0−e共t兲, d共␩,␤兲 =3 4

冉

共sin 2_{␩+ 2 cos}2_␩兲i ␤共1 − e i␤_兲 +共sin2␩− 2 cos2␩兲− 2i + e i␤_{共2i + 2␤} − i␤2兲 ␤3

冊

, 共27兲 with␤=␻0b/c. The functional G共␻0, c0−e兲 is obtained from

Eq.共27兲 by taking the limit␤→0 and thus gives

G共␻0,c0−e兲 = −

⌫共␻0兲

4 c0−e共t兲, 共28兲

as lim_␤→0d共␩,␤兲=1. Note that the dependence on␻0is both in⌫共␻0兲 and in␤. Altogether we have

c˙0_⫾e= − ⌫_⫾ 2 c0⫾e, 共29兲 ⌫⫾=1 2

冋

⌫共␻0兲 + ⌫共␻0⫾ ⌬兲 + ⌫共␻0兲d

冉

␩, ␻0b c

冊

−⌫共␻0⫾ ⌬兲d

冉

␩, 共␻0⫾ ⌬兲b c

冊

册

, 共30兲

with the obvious solution

c0⫾e共t兲 = c0⫾e共0兲e−⌫⫾t/2. 共31兲

It is convenient to express ⌫_⫾ in terms of the standard Wigner-Weisskopf rate⌫共␻0兲 for a localized atom, and use the dimensionless parameter ␦=⌬/␻0Ⰶ1 introduced previ-ously. We then have

⌫_⫾ ⌫共␻0兲= 1 2兵1 + d共␩,␤兲 + 共1 ⫾␦兲 3_{关1 − d„␩,␤共1 ⫾␦兲…兴其.} 共32兲 For␦= 0 and␤ⱗ3 共i.e., for finite b a double-well potential with infinite barrier兲, we are immediately led back to the standard Wigner-Weisskopf rate for both initial states兩0⫾e典, ⌫_⫾=⌫共␻0兲. Thus, as long as there is no tunneling, an arbi-trary coherent superposition of the atom in the right and in the left well does not change the spontaneous emission at all. Similarly we get back ⌫_⫾=⌫共␻0兲 for ␤= 0. For ␤ⲏ1,

d共␩,␤兲 vanishes as 1/␤, and we have approximately ⌫_⫾/⌫共␻0兲=1

interpre-tation of arising from two independent decay channels, from 兩⫾e典 to 兩⫾g典 or 兩⫿g典. Transitions which do not change the external state see the same level spacing␻0as an atom with-out tunneling degree of freedom, whereas a flip of the exter-nal state changes the level spacing by⫾⌬. Both transitions come with the corresponding Wigner-Weisskopf rate, ad-justed for the correct overall level spacing, and the total rate is the average rate from the two decay channels.

Contrary to the Wigner-Weisskopf case, the rates⌫_⫾ are in general complex. For the decay of the probabilities兩c0_⫾e兩2,

only the real part of d共␩,␤兲,

Re d共␩,␤兲 =3 2

冋

共sin

2_{␩+ 2 cos}2_␩兲sin␤ ␤

+共sin2␩− 2 cos2␩兲2␤cos␤+共␤

2_{− 2兲sin␤}

␤3

册

共33兲 is relevant. This function equals one for␤= 0, depends only slightly on␩, and decays with some slight oscillations as a function of␤. However, for␤ⲏ1,␦Ⰶ1 共exponentially small overlap of wave functions兲, and for␦⬃1,␤Ⰶ1 共small␻0兲, i.e., in the regime attainable within the two-level approxima-tion made in this paper, we have that⌫_⫾deviates from⌫共␻0兲 only very slightly and we will set⌫_⫾=⌫共␻0兲⬅⌫ in the rest of the paper.

C. External dynamics—decoherence of the atomic tunneling

The quantum mechanical expectation value for the aver-age position 具z共t兲典 and thus the dynamics of the external degree of freedom follow from

具z共t兲典 =b 2关␳+−

ex_共t兲e−i⌬t_{+ c.c.兴, 共34兲} where the matrix element␳₊₋ex共t兲 in the interaction picture is obtained from tracing out the internal and field degrees of freedom, ␳+−ex共t兲 = c0+e共t兲c0−eⴱ 共t兲 + K共t兲, 共35兲 K共t兲 ⬅

兺

k c₁ k−g ⴱ _共t兲c1 k+g共t兲. 共36兲

The first term in Eq.共35兲 can be obtained immediately from Eq.共31兲. In order to get K共t兲, we insert the solutions 共31兲 for

c0_⫾e共t兲 into Eqs. 共16兲 and 共17兲. Direct integration of the latter

equations gives c1k⫾g共t兲 = gk

冋

c0_⫾e共0兲cos␬ 1 − e关−i共␻0−␻k兲−⌫/2兴t 共␻k−␻0兲 + i⌫/2 − ic0_⫿e共0兲sin␬ 1 − e关−i共␻0⫿⌬−␻k兲−⌫/2兴t 共␻k−␻0⫾ ⌬兲 + i⌫/2

册

, 共37兲 where we have set ⌫_⫾=⌫ in accordance with the previous section. It is possible to calculate the tunneling motion in-cluding terms of order␦, but the calculations are tedious and the additional information gained compared to order zero in ␦ not illuminating. We therefore present here a simplified calculation which leads to a result valid up to corrections

O共␦兲. The shifts ⫾⌬ in the denominator and exponent in Eq.

共37兲 have to be kept. Without the ⫾⌬ in the exponent, there will be obviously no tunneling at all. Formally, the need to keep ⌬ in the denominator and exponent of Eq. 共37兲 arises from the fact that there⌬ has to be compared not to␻₀but to ␻0−␻k, which can vanish, as␻kvaries from 0 to⬁.

We restrict ourselves to real initial amplitudes c0⫾e共0兲 and

proceed in a similar fashion as for the spontaneous emission to evaluate the sum over all modes k in Eq.共35兲, after insert-ing Eq.共37兲. This leads to

K共t兲 =㜷 2_c 0−e共0兲c0+e共0兲 16␲2c3⑀0ប

冕

₀ ⬁ d␻ ␻3

冕

−1 1 d␮h共␩,␮兲 ⫻

冋

cos2

冉

␮␤ ˜ 2

冊

e−⌫t− 2 cos关共␻−␻0兲t兴e−⌫t/2+ 1 共␻−␻0兲2+⌫2/4 + sin 2

冉

␮␤˜ 2

冊

e2i⌬te−⌫t− 2ei⌬tcos关共␻−␻0兲t兴e−⌫t/2+ 1

共

␻−␻0+⌬ + i⌫2

兲共

␻−␻0−⌬ − i⌫2

兲

册

⯝㜷2␻03c0−e共0兲c0+e共0兲 16␲2c3⑀0ប

冕

−⬁ ⬁ d␻

冋

a共␩,␤˜ 兲e −⌫t_{− 2 cos关共␻−␻0兲t兴e}−⌫t/2_{+ 1} 共␻−␻0兲2+⌫2/4 +

冉

8 3− a共␩,␤˜ 兲

冊

e2i⌬te−⌫t− 2ei⌬tcos关共␻−␻0兲t兴e−⌫t/2+ 1

共

␻−␻0+⌬ + i⌫2

兲共

␻−␻0−⌬ − i⌫2

兲

册

, 共38兲

with␤˜ =␻b/c and

a共␩,␤兲 = 共sin2_{␩+ 2 cos}2_␩兲

冉

_{1 +}sin␤

␤

冊

+共sin2␩− 2 cos2␩兲

冉

1 3+

2␤cos␤+共␤2− 2兲sin␤

␤3

冊

. 共39兲

We have made the same approximations as for the calculation of the rates of spontaneous emission, i.e., pulled out a factor␻₀3 from the integral over␻, and extended the lower bound of the integral to −⬁. In principle the␻integral is uv divergent and would need a cutoff 共see Ref. 关23兴 for a discussion of experimentally relevant cutoffs兲. However, in order to conserve

probability, the same approximations as for the rates of spontaneous emission need to be made here. Also, while there are two resonances now␻0⫾⌬, they differ only at order␦, so that at lowest order in␦ it is indeed enough to pull out␻0

3

from the integral. The remaining␻integral is performed by contour integration, and we find the final result

具z共t兲典 = bc0−e共0兲c0+e共0兲 ⫻

再

e−⌫tcos共⌬t兲 +3 8

冋

a共␩,␤兲共1 − e −⌫t_{兲cos共⌬t兲 +}

冉

8 3− a共␩,␤兲

冊

␥/2 1 +␥2/4

冉

共1 + e −⌫t_{兲sin共⌬t兲 +}␥ 2共1 − e −⌫t_{兲cos共⌬t兲}

冊

册

冎

⫻ 关1 + O共␦兲兴, 共40兲

where we have introduced␥⬅⌫/⌬.

Let us consider a few special cases of this general result. First of all, Eq.共40兲 shows that for c0−e共0兲=0 or c0+e共0兲=0,

具z共t兲典=0 for all t. This corresponds to putting the atom ex-ternally into one of the two energy eigenstates 兩⫾典 of the uncoupled system, which are symmetric with respect to z = 0. The two decay channels do not introduce a position bias either, and therefore the atom stays on average always at z = 0. Tunneling with full amplitude needs an initial prepara-tion in the right or left well, i.e., c0−e共0兲c0+e共0兲= ⫾1/2, and

we will therefore assume from now on c0−e共0兲c0+e共0兲=1/2.

The limit␤→0 leads with lim_␤→0a共␩,␤兲=8/3 immedi-ately to具z共t兲典=₂bcos共⌬t兲, i.e., undisturbed tunneling motion, as if the atom had no internal structure at all. The physical reason for this is of course that the emitted photon has in this case a wavelength much larger than the distance between the two wells such that it does not carry any information about the position of the atom, and therefore no decoherence of the tunneling motion arises.

The case of ␤ⲏ1 is more subtle. At ␤⯝3 we have

a共␩,␤兲⯝4/3 and hence 具z共t兲典 ⯝b 2

冋

冉

1 2共1 + e −⌫t_{兲 +}␥2/8共1 − e−⌫t兲 1 +␥2/4

冊

cos共⌬t兲 + ␥/4 1 +␥2/4共1 + e −⌫t_{兲sin共⌬t兲}

册

_{, 共41兲} which for tⰇ⌫−1_{, i.e., when a photon has certainly been} emitted, settles down to

具z共t兲典 ⯝b 2

冋

冉

1 2+ ␥2_/8 1 +␥2/4

冊

cos共⌬t兲 + ␥/4 1 +␥2/4sin共⌬t兲

册

, 共42兲 ⯝b 2A cos共⌬t +␸兲, 共43兲 with A =

冑

␥ 2_{+ 1} ␥2_{+ 4}. 共44兲

As a consequence, after a period of initial damping, tunnel-ing with a finite amplitude Ab/2, which is in general reduced compared to the full possible value b/2, and phase shift ␸ persists.

This is very much in contrast to standard decoherence scenarios of a particle tunneling through a potential barrier 关24兴, where the continued coupling to a heat bath normally destroys all coherence共even though exceptions are possible in other contexts, in particular for heat baths with small cut-off frequency, which can lead to incomplete decoherence as well 关25兴兲. Here, the decoherence is switched off once the photon is emitted, as the center-of-mass coordinate of the atom does not couple directly to the electromagnetic modes. Furthermore, the time at which the photon is emitted, plays a crucial role. If we take␥=⌫/⌬→⬁ in Eq. 共44兲, we find A = 1, i.e., in spite of strong dissipation and short wavelength of the photon, there is no decoherence of the tunneling mo-tion at all. The reason lies in the fact that the photon is emitted immediately after preparation of the atom in the right well, i.e., at a time, when it is not in a coherent superposition of eigenstates of its center-of-mass position. Thus, no coher-ence can get destroyed, and since after the emission of the photon decoherence is switched off, tunneling proceeds in the ground state with full amplitude. One may also see the emission of the photon as a measurement process which should, for␭ⱗb, project the atom either into the right or left well. However, since the atom was prepared in the right well just before, one projects the state back into the right well, therefore keeping the coherence of the initial external state.

On the other hand, if we take␥→0 in Eq. 共44兲, we find that the amplitude of the tunneling motion in the long time limit reduces to A = 1/2. In the many runs of the experiment necessary to verify Eq. 共40兲, the time when the photon is emitted is averaged over many tunneling periods, such that roughly speaking in half the runs the atom is in a coherent superposition of eigenstates of its center-of-mass position z, half of the time it is in an eigenstate of z. Therefore, it is natural that on the average tunneling with half the full am-plitude persists for tⰇ⌫−1_.

This physical picture is confirmed by the result for the tunneling amplitude for the case of an initial state 共兩+典 + i兩−典兲/

冑

2, which is neither an eigenstate of position nor of energy. Without dissipation 共␥→0兲 this state would be reached from the initial state after a quarter period of the tunneling motion. For tⰇ⌫−1 and␤ⲏ3, the formula corre-sponding to Eq.共44兲 reads

A =

_冑

1

␥2_{+ 4}. 共45兲

Thus, indeed for ␥→0, one obtains, as in the case of the initially localized external state, a reduction of the tunneling

amplitude by a factor 2. For very large␥, on the other hand, the coherence of the superposition is destroyed immediately, and the resulting unbiased mixture of the states兩R典 and 兩L典 leads to vanishing tunneling amplitude.

An alternative interpretation can be found by considering the frequencies of the emitted photon. If the atom is initially in a position eigenstate, photons with frequencies ␻0, ␻0 +⌬, and␻0−⌬, can be emitted corresponding to the transi-tions 兩⫾典→兩⫾典, 兩+典→兩−典, and 兩−典→兩+典, respectively. These frequencies are smeared out over a width ⌫/2. For ⌫Ⰷ⌬, the four transitions cannot be distinguished and the coherence between the states兩⫾典 remains intact, leading to

A = 1. On the other hand, for ⌫Ⰶ⌬, the observation of the

emitted photon reveals the path how the atom arrived in a given external state 兩+典 or 兩−典 in half of the cases 共␻ =␻0⫾⌬兲 and therefore destroys half of the coherence be-tween兩+典 and 兩−典, giving A=1/2.

The limits␥→0 and t→⬁ do not commute, which is a consequence of the factors exp共−⌫t兲 in 具z共t兲典. If we take the limit ␥→0 already in Eq. 共40兲, i.e., without considering t

→⬁ first, we get 具z共t兲典=b

2cos共⌬t兲. The atom tunnels with full amplitude, i.e., A = 1, and shows no decoherence, as it should be, of course. Therefore, in a finite time interval starting with the preparation of the initial state, decoherence is most effec-tive in an intermediate regime ⌫⬃2⌬, when a photon is likely to be emitted at the time when tunneling has estab-lished a coherent superposition of兩R典 and 兩L典. For general ␩,␥ and large times, tⰇ⌫−1, i.e., after the damping has settled down,具z共t兲典/共b/2兲 oscillates with an amplitude

A =1

4

冑

9a共␩,␤兲2_{+ 16␥}2

4 +␥2 , 共46兲

a function which we show in Fig.2. This makes clear that also at very small but finite ␥, A reduces to 1/2 for suffi-ciently large␤共A=1/2 is reached asymptotically for␤→⬁, which is, however, beyond the validity of the theory兲. When plotting具z共t兲典 in a fixed time interval as in Fig.3, the reduc-tion of A is not visible yet at small values of␥.

D. The electromagnetic field

We now calculate the electromagnetic field in the limit of large times tⰇ⌫_⫾−1, such that the atom has emitted a photon with certainty. We will approximate again ⌫+⯝⌫−⯝⌫共␻0兲

⬅⌫ 共we are restricted to the regime ␻0Ⰷ⌬,⌫兲. The wave function of the entire system has then the form

兩␺共⬁兲典 =

兺

k 关c1_k−g共⬁兲兩1k− g典 + c1_k+g共⬁兲兩1k+ g典兴 共47兲 with c1_k⫾g共⬁兲 = gk

冉

c0⫾e共0兲cos␬ ␻k−␻0+ i⌫/2 − i c0⫿e共0兲sin␬ ␻k−␻0⫾ ⌬ + i⌫/2

冊

共48兲 关see Eq. 共37兲兴. Equations 共47兲 and 共48兲 show that after the emission the system is in a superposition of infinitely many states of two different categories: One with the atom staying in the same external state with an emitted photon in a fre-quency band centered at␻0, and the other with the external state of the atom flipped and an emitted photon in a fre-quency band centered at␻0⫿⌬. The width of the frequency bands is⌫/2 in both cases. For sin␬= 0, there is no contri-bution from states of the second category. This is approxi-mately the case in the regime of long wavelengths 共␬Ⰶ1兲 where the atom remains in the same motional state. The first order correlation function of the electric field G共1兲共r,r;t,t兲 关22兴 is given for large times by

G共1兲共r,r;t,t兲 = 具␺共⬁兲兩E共−兲共r,t兲 · E共+兲共r,t兲兩␺共⬁兲典 = 兩I+兩2+兩I−兩2, 共49兲

I⫾=

兺

k

c1k⫾g具0兩E共+兲共r,t兲兩1k典, 共50兲

with the positive frequency electric field operator E共+兲共r,t兲 =兺kEk⑀kakei共k·r−␻kt兲. The difference between I+ and I− rests

upon c_1k⫾g. We focus for the moment on I+_{. The} correspond-ing results for I− _{are obtained in a completely analogous} fashion. In fact, according to Eqs. 共48兲 and 共50兲 one only needs to exchange c0+e↔c0−e and replace ⌬ by −⌬ in the

final result to obtain I−from I+. We choose the vector r to lie in the x-z plane, convert the sum over k into an integral as before, and are thus led to

A ββ ln γ ln γ 0.2 0.4 0.6 0.8 0 0 0 33 2 2 2 2 2 2 4 4 4 4 1 1 1

FIG. 2. 共Color online兲 Amplitude of the tunneling motion for

tⰇ⌫−1for␩=␲/2 as a function of␤ and ln ␥.

z(t) ττ ln γ ln γ 0.5 0.5 0 0 0 0 0 10 10 20 20 15 15 5 5 5 5 5 5 1 1

FIG. 3.共Color online兲 Tunneling motion 具z共t兲典 in units of b/2 as a function of␶⬅⌬t and ln ␥ for ␤=3,␩= 0 with the initial value 具z共0兲典=b/2.

I+= − 1 16␲3⑀0c3

冕

0 ␲ sin␪d␪

冕

0 2␲ d␾ ⫻

冕

0 +⬁ d␻␻3

冉

㜷 − kk ·㜷 k2

冊

e i_{共k·r−␻t兲} ⫻

冉

c0+e共0兲cos␬ ␻−␻₀+ i⌫/2− i c_0−e共0兲sin␬ ␻−␻₀+⌬ + i⌫/2

冊

, 共51兲 =Ic + + Is + , 共52兲

where k · r =共␻/c兲共z cos␪+ x sin␪cos␾兲. The integral splits into two parts Ic+, Is+with denominators ␻−␻0+ i⌫/2 and␻ −␻₀+⌬+i⌫/2. As in the Wigner-Weisskopf theory of spon-taneous emission, we assume that␻3 varies little around␻ =␻₀ 共respectively, ␻=␻₀−⌬兲 so that we can replace ␻3 _by ␻0

3 _{关respectively, 共␻}

0−⌬兲3兴 and extend the lower limit of integration to −⬁. Only the x and z components of Ic+and Is+,

denoted as I_c,+_␰, I_s,+_␰,␰僆兵x,z其, give a contribution 共the y com-ponent is identically zero兲

I_c,+_␰= −␻0 3_㜷c 0+e共0兲 16␲3_c3_⑀ 0

冕

0 ␲ sin␪d␪

冕

0 2␲ d␾f_␰共␩,␪,␾兲 ⫻

冕

−⬁ +⬁ d␻e

i␻共z/c cos␪+x/c sin ␪ cos ␾−t兲

cos

共

␻b_2ccos␪

兲

␻−␻0+ i⌫/2

, 共53兲

fx共␩,␪,␾兲 = 关sin␩− sin␪cos␾共sin␪cos␾sin␩

+ cos␪cos␩兲兴, 共54兲

fz共␩,␪,␾兲 = 关cos␩− cos␪共sin␪cos␾sin␩

+ cos␪cos␩兲兴. 共55兲

Writing cos

共

␻b_2ccos␪

兲

as a sum of exponentials, the␻integral can be easily evaluated using the contour method and leads to a Heaviside ⌰ function ⌰(t−关共z⫾b/2兲cos␪ + x sin␪cos␾兴/c) for the positive 共negative兲 frequency part of the cos

共

␻b_2ccos␪

兲

function, which illustrates the fact that the electric field cannot spread faster than the speed of light. In order to avoid the complications which arise from the angle dependence of the ⌰ function, we will restrict our-selves to times t⬎1 c共r+b/2兲 with r=

冑

x2+ z2. We find I_c,+_␰= i␻0 3_㜷c 0+e共0兲 8␲2c3⑀0

冕

0 ␲ sin␪d␪

冕

0 2␲ d␾f_␰共␩,␪,␾兲 ⫻ei共␻₀−i⌫/2兲共z/c cos␪+x/c sin ␪ cos ␾−t兲

⫻cos

冉

b

2c共␻0− i⌫/2兲cos␪

冊

. 共56兲 Integration over the angle␾yields for the x component

I_c,x+ = i␻0 3_㜷c

0+e共0兲

4␲c3⑀0

冕

0 ␲

sin␪d␪ei共␻₀−i⌫/2兲共z/c cos␪−t兲_cos

冋

b

2c共␻0− i⌫/2兲cos␪

册

⫻

再

sin␩J0

冉

x

c共␻0− i⌫/2兲sin␪

冊

− i sin␪cos␪cos␩J1

冉

x c共␻0− i⌫/2兲sin␪

冊

− sin2␪sin␩

冋

J1

„

x c共␻0− i⌫/2兲sin␪

…

x c共␻0− i⌫/2兲sin␪ − J2

冉

x c共␻0− i⌫/2兲sin␪

冊

册

冎

, 共57兲

where J0, J1, and J2are Bessel functions. A closed form of the type of the remaining␪ integral has been found very recently by Neves et al.关26兴. Using their formula, we can evaluate the␪ integration analytically and find in the far-field region

I_c,x+ = i␻0 3_㜷c 0+e共0兲 4␲c3⑀0 e−i共␻0−i⌫/2兲t ⫻

再

sin␩

冋

sin R+ R+ +sin R− R−

册

− sin␩

冋

sin 2_␣ +sin R+ R+ +sin 2_␣ −sin R− R−

册

−1 2cos␩

冋

sin共2␣+兲sin R+ R+ +sin共2␣−兲sin R− R−

册

冎

+ O共1/R_⫾2_{兲, 共58兲} with R_⫾=共␻0− i⌫/2兲 c r⫾, r⫾=

冑

x 2_{+共z ⫾ b/2兲}2_{, tan␣} ⫾=_{z⫾ b/2}x . 共59兲

The expression for I_c,z+ is obtained from I_c,x+ by a rotation of the coordinate system 共␩→␩+␲/2, ␣_⫾→␣_⫾+␲/2兲, which amounts in Eq.共58兲 to exchanging sin␩↔cos␩ and replacing sin2␣_⫾→cos␣_⫾2. In the far field, sin R_⫾⯝eiR⫾/2i, and the expressions can be further simplified,

I_c,x+ =␻0 3_㜷c 0+e共0兲 8␲c3⑀0 e−i共␻0−i⌫/2兲t

再

e iR₊ R+ sin共␩−␣+兲cos␣++ eiR− R−

sin共␩−␣−兲cos␣−

冎

+ O共1/R⫾2兲. 共60兲 Similarly, we get for the second part Is+of the integral

I_s,x+ = −共␻0−⌬兲 3_㜷c 0−e共0兲 8␲c3⑀0 e−i共␻0−⌬−i⌫/2兲t

再

e iR˜₊ R ˜ + sin共␩−␣₊兲cos␣₊−e iR˜₋ R ˜ −

sin共␩−␣₋兲cos␣₋

冎

+ O共1/R˜_⫾2兲 共61兲 with R ˜ ⫾=共␻0 −⌬ − i⌫/2兲 c r⫾. 共62兲

The formulas for I_c,z+ and I_s,z+ are obtained by changing the global sign and replacing cos␣_⫾→sin␣_⫾ where it appears explicitly in Eqs.共60兲 and 共61兲, respectively. We have

I_c,x+ + I_s,x+ ⯝ ␻0 2_㜷 8␲c2⑀0 e−⌫/2共t−r/c兲 r e −i␻0t

⫻兵c0+e共0兲关ei␻0r+/csin共␩−␣+兲cos␣++ ei␻0r−/csin共␩−␣−兲cos␣−兴 − c_0−e共0兲ei⌬t_关ei共␻₀−⌬兲r₊/c_{sin共␩−␣+兲cos␣}

+− ei共␻0−⌬兲r−/csin共␩−␣−兲cos␣−兴其 + O共⌬/␻0,⌫/␻0,1/r2兲 共63兲 and again the corresponding expression for I_c,z+ + I_s,z+ can be found from Eq. 共63兲 by just changing the global sign and the explicitly printed factors cos␣_⫾into sin␣_⫾. We can summarize the results for both I+and I− as

兩I⫾_兩2_=兩I c,x ⫾ _{+ I} s,x ⫾_兩2_+兩I c,z ⫾ _{+ I} s,z ⫾_兩2 ⯝ ␻0 4_㜷2 64␲2c4⑀02 e−⌫共t−r/c兲 r2 兵兩c0⫾e共0兲关e i␻₀␦r_/csin共␩

−␣+兲cos␣++ sin共␩−␣−兲cos␣−兴 − c0⫿e共0兲e⫾i⌬共t−r−/c兲关ei共␻0⫿⌬兲␦r/csin共␩−␣+兲cos␣+− sin共␩−␣−兲cos␣−兴兩2 +兩c0⫾e共0兲关ei␻0␦r/csin共␩−␣+兲sin␣++ sin共␩−␣−兲sin␣−兴

− c0_⫿e共0兲e⫾i⌬共t−r−/c兲关ei共␻0⫿⌬兲␦r/csin共␩−␣+兲sin␣+− sin共␩−␣−兲sin␣−兴兩2+ O共⌬/␻0,⌫/␻0,1/r2兲其 共64兲 with␦r = r₊− r₋. In the limit b→0, we have␦r = 0,␣_⫾=␣with tan␣= x/z, and we recover the well-known result 关27兴 共valid for t⬎r/c兲 G共1兲共r,r;t,t兲 = ␻0 4_㜷2 16␲2c4⑀₀2 e−⌫共t−r/c兲 r2 sin 2_{共␩−␣兲关1 + O共⌬/␻} 0,⌫/␻0,1/r2兲兴, 共65兲

where␩−␣is the angle between the dipole moment and the observer.

Using ␦r⯝b cos␣, cos␣_⫾⯝cos␣−␦␣_⫾sin␣, and sin␣_⫾⯝sin␣+␦␣_⫾cos␣ with ␦␣_⫾=␣_⫾−␣⯝ ⫿_2rbsin␣ in the far field, we get for an initial delocalized state, c0_⫾e共0兲=1 and c0_⫿e共0兲=0,

G共1兲共r,r;t,t兲 ⯝ ␻0 4_㜷2 16␲2_c4_⑀ 0 2 e−⌫共t−r/c兲 r2 sin 2_{共␩−␣兲}

再

_{1⫾ sin}

冉

⌬b 2ccos␣

冊

sin

冉

冋

␤⫾ ⌬b 2c

册

cos␣

冊

冎

关1 + O共⌬/␻0,⌫/␻0,1/r 2_{兲兴, 共66兲} where the upper and lower sign now refer to the initial condition. If the atom is initially located in the right well, c_0+e共0兲 = c0−e共0兲=1/

冑

2 and we find

G共1兲共r,r;t,t兲 ⯝ ␻0 4_㜷2 16␲2c4⑀₀2 e−⌫共t−r/c兲 r2 sin 2_{共␩−␣兲} ⫻

再

1 + sin

冉

⌬b

2ccos␣

冊

冋

cos共␤cos␣兲sin

冉

⌬b

2ccos␣

冊

册

− sin

冉

⌬

冋

t −

r

c

册

冊

关1 + cos共␤cos␣兲兴

冎

⫻关1 + O共⌬/␻0,⌫/␻0,1/r2兲兴. 共67兲

In both cases the interference term is proportional to sin

共

⌬b_2ccos␣

兲

which scales in the nonrelativistic limit as⌬b/c and is thus extremely small.

The radiation from a classical oscillating dipole was ex-amined recently by Bolotovskii and Serov关28兴. They predict interference effects in the regime where ␤⯝1, but do not provide any analytical results for the visibility of the inter-ference fringes. It appears, however, that a classically mov-ing dipole has to move a distance comparable to the wave-length during the time 1/␻0 if waves from different origins are to combine in a remote location—otherwise the radiation pattern only follows its source adiabatically. Indeed, it is easy to show that at order zero in⌬b/c a classical oscillating dipole does not lead to interference共where ⌬ means now the classical oscillation frequency of the center-of-mass motion of the dipole兲. Therefore, the absence of interference in Eq. 共67兲 at lowest order in ⌬b/c agrees with the classical result. The initially delocalized states兩⫾典 do not have a classical analog.

However, it turns out that in the quantum case interfer-ence with perfect visibility arises if the external state of the atom is post-selected in the energy basis. For example, if only runs of the experiments are taken into account where the atom is measured in external state兩+典 before the photon is recorded, only 兩I+_兩2 _{共but not 兩I}−_兩2_{兲 contributes to}

G共1兲共r,r;t,t兲. We find in this case G₊共1兲共r,r;t,t兲

⬀cos2

共

␤

2cos␣

兲

sin2共␩−␣兲 共subscript + for post-selection in 兩+典兲. The interference fringes for post-selection in 兩−典 are phase shifted, i.e., G₋共1兲共r,r;t,t兲⬀sin2

共

␤

2cos␣

兲

sin2共␩−␣兲, such that in the sum G共1兲共r,r;t,t兲=G₊共1兲共r,r;t,t兲

+ G₋共1兲共r,r;t,t兲 the interference terms compensate and only the dipole characteristics G共1兲共r,r;t,t兲⬀sin2_共␩_{−␣兲 is left} 共see Fig. 4 for ␩= 0兲. Note that for post-selection the inter-ference pattern becomes independent of⌬ and should there-fore be observable even for small⌬, as long as the two states can be reliably distinguished. Spectral filtering of the photon with a resolution better than⌬ has the same effect as mea-suring the external energy of the atom. For c0_⫾e共0兲=1,

mea-suring a photon of frequency ␻=␻0 projects on 兩⫾典, whereas the observation of a photon of frequency ␻ =␻0⫾⌬ projects on 兩⫿典.

E. Experimental perspectives

Several requirements have to be met to observe the effects predicted in this paper. First of all, one needs a double-well potential with tunable well-to-well separation and barrier height in order to vary b and⌬. This has been demonstrated with optical dipole traps, e.g., in Refs. 关7,8兴, and on atom chips, e.g., in Refs. 关9,29兴. Second, we considered in our model the same external potential for both internal states兩g典 and兩e典, and these states should be coupled by a dipole tran-sition. It is by now well known that this requirement can be met for Cs, Yb, Sr, and possibly Mg and Ca atoms at certain “magic wavelengths” in optical traps 关30–33兴. Third, we made the two-level approximation for the external degree of freedom. It turns out that in a typical double-well potential at the limit of the Lamb-Dicke regime with respect to vibra-tional excitations ␤⬃3 共i.e., ␭⬃2b兲 still leads to a reason-able restriction of the dynamics to the two lowest states. To show this, let us consider the usual quartic double-well po-tential V共z兲=V0共z2_{− a}2_兲2_/a4_{, where 2a is the well-to-well} separation. Similar results are obtained for other shapes of the double-well potential. The transition probability during emission of a photon between energy eigenstates兩n典 and 兩m典 of this potential is given by兩具n兩eikz_兩m典兩2 _{关34兴. For Cs atoms} and for the parameters V0/ប=0.23 MHz and a=␭/4, where ␭=852.4 nm is the 6

S_1/2→6S_3/2 transition wavelength, this gives less than 10% transition probability out of the ground state doublet with a tunneling splitting ⌬⯝150 Hz and ␤ ⯝2.93. For lighter atoms, such as Mg, tunnel splittings of the order of kHz can be easily achieved for the same␤. The theory thus works well for␤ⱗ3 but not anymore for larger values. Equation共26兲 implies that for a ⌬ in the kHz range, ⌫⬃⌬ for a transition in the near-infrared 共␻0⬃1013 Hz兲, where a well-to-well separation of ⬃␮m leads to ␤⬃3, which still keeps transition probabilities to higher vibrational states at less than about 10%.

In any case, for optical traps the trap frequency and thus the tunnel splitting are determined by the laser power and the focusing 共or the wavelength for optical lattices兲, and can therefore be controlled independently of ⌫ such that both regimes⌬Ⰷ⌫ and ⌬Ⰶ⌫ should be achievable. The sponta-neous emission rate⌫ can be varied over a large interval by using a small static magnetic field to enable electrical dipole transitions between hyperfine levels that would otherwise be forbidden, due to admixture of small amplitudes of other hyperfine levels with weakly allowed dipole transition关35兴. In关33兴, ⌫⬃20 Hz 共including additional external broadening mechanisms兲 was demonstrated for a clock transition at 578.42 nm in Yb trapped with light at the magic wavelength ␭=759.35 nm with this technique. Even smaller values of ⌫ should be achievable according to theoretical predictions 关36兴.

Cooling close to the ground state in a single well trap has been demonstrated and should work down to temperatures

kBT⬍ប⌬ if ⌬ is comparable to the single well vibrational

frequency 关30兴. Finally, one needs to detect the tunneling motion. That should be possible by optical imaging, i.e., dif-fusion of laser light from another transition in the optical regime with smaller wavelength than the well separation. Another possibility might be using the atomic spin as a po-sition meter关37兴.















xxx x zzz z 000 0....5555 000 0....5555 000 0....2222 000 0....2222 0000....4444 000 0....4444 111 1 111 1

FIG. 4. 共Color online兲 Polar plot of the angular dependence of G_⫾共1兲共r,r;t,t兲 共red dashed and blue dotted curves兲 and

G共1兲共r,r;t,t兲=G₊共1兲共r,r;t,t兲+G₋共1兲共r,r;t,t兲 共green solid curve兲 for

III. CONCLUSIONS

A two-level atom which can tunnel between the two wells of a double-well potential allows for a host of interesting phenomena, which we have studied systematically in this paper in the regime ␦=⌬/␻₀Ⰶ1 共tunnel frequency much smaller than the atomic transition frequency兲 and␤=␻0b/c ⱗ3. Whereas the spontaneous emission rate ⌫ of the atom is only slightly modified by putting the atom into a coherent symmetric superposition 共ground state of the external double-well potential兲 of the two states in the right and left well, the tunneling of the atom and the properties of the emitted light are altered more profoundly. The emission of a single photon can cause decoherence of the tunneling mo-tion, but only if共1兲 the photon wavelength is not much larger than the distance between the two potential wells and 共2兲 spontaneous emission is not too fast, i.e.,⌫ⱗ⌬. For ⌫Ⰷ⌬, the photon is emitted even before the atom starts its tunnel-ing motion, i.e., before a coherent superposition of eigen-states of the center-of-mass coordinate of the atom is estab-lished. Hence, despite strong coupling to the environment, the tunneling motion does not suffer from decoherence at all in this regime. After the emission of the photon no more decoherence takes place, and tunneling will then continue with constant amplitude. For very slow spontaneous

emis-sion,⌫Ⰶ⌬, the average amplitude of the tunneling motion reduces by a factor 2. The electric field of the emitted photon shows interference fringes, but their amplitude is very small unless the external state of the atom is post-selected in the energy basis or, alternatively, the photons are spectrally fil-tered with a resolution better than⌬, in which case interfer-ence fringes with perfect visibility arise. The absinterfer-ence of in-terference without post-selection共or spectral filtering兲 is in agreement with the results by Rohrlich et al. in the case of scattering关17兴, whereas the appearance of interference after post-selection or spectral filtering of the photon agrees with the results in Ref.关18兴, where it was predicted that interfer-ence fringes from the scattering of a particle by a quantum scatterer disappear in the limit where the kinetic energy ⑀Ⰶ⌬, but can be recovered by post-selecting the elastic scattering channel. The effects predicted here should in prin-ciple be observable with modern cold-atom technology.

ACKNOWLEDGMENTS

We would like to thank CALMIP共Toulouse兲 for the use of their computers. This work was supported by the Agence National de la Recherche 共ANR兲 project INFOSYSQQ Project No. ANR-05-JCJC-0072 and the EC IST-FET project EUROSQIP.

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