After viewing the images of individual atoms as provided by electron micro- ... tunneling-probability, the radiation pattern will in general contain structure that ... interference, e.g., in molecules with an appropriate structure2, by selecting suitable ... excited hydrogen-like (alkali) atom bound in a double well potential that is a. 2 ...
:
I -3 - -- r..-
SLAC-PUB-6158 April 1993 T
Atomic Self-Coherence*
RICHARD BLANKENBECLER Stanford Linear Accelerator Stanford
. ._. .-_ -
University, Stanford,
Center
California
94309
ABSTRACT
There is nothing new in this paper.
It contains a pedagogical discussion of
a very interesting interference effect involving the radiation pattern from a single atom which is confined by a multi-well
potential.
The interference phenomenum
is a quantum effect with a pattern strength that is proportional probability.
to the tunneling
Extensions of the phenomenon are briefly discussed.
Submitted to Journal of Physics A.
* Work supported by the Department of Energy, contract DE-AC03-76SF00515.
r
.-‘-.-. ---
:
1. Introduction
After viewing the images of individual
atoms as provided by electron micro-
scopes, especially when they have been arranged to spell out the initials of a rather well-known company ‘, rt’ is easy to slip into the error of thinking of atoms as classical billiard balls with quantum phenomena becoming important tance scale-except
at a smaller dis-
for exceptional circumstances such as superconductivity.
purpose of this note is pedagogical; we will describe the familiar radiation
of a single photon by an atom.
The
process of the
The unusual feature is that the radia-
tion interferes with itself by virtue of the fact that the atom is in a non-localized .-state with a net symmetry. tunneling-probability,
If the atom is in a multi-well
the radiation
potential,
with a finite
pattern will in general contain structure that
reflects the geometric arrangement of the wells. The magnitude of this variation is proportional
to the quantum tunneling probability.
New laboratory
techniques in the manipulation
of individual atoms have opened
up new possibilities in both experiment and theory. This paper discusses the interference pattern expected from a single atom bound in a multi-well
potential
is
discussed. Suitable binding potentials should be available to realize this type of interference, e.g., in molecules with an appropriate structure2, by selecting suitable defects on the surface of a solid3, or by an appropriate electromagnetic
fields in a Stern-Gerlach
configuration
of external
interferometer4.
To simplify our discussion and to permit an emphasis on the physics of the interference
phenomena,
excited hydrogen-like
we will consider the problem of radiative
(alkali)
decay of an
atom bound in a double well potential 2
that is a
-
.1-- r. .-
function of the center-of-mass coordinate only. The nonrelativistic
Hy;
7
be + eA(re))2 + &
P
(&I - eA(rp))2 + V(q
where r’ = Fe - Fp and R’ = (Mer’, + Mprfp)/M~ set of definitions).
The potential
V(r)
Hamiltonian
+ ~(2)
(see Appendix
(14
,
B for a complete
is the atomic binding potential;
W(R)
is
the center-of-mass potential that is assumed to have two well-separated minima at 2 E &I?/2 in the example treated below. The standard electromagnetic
interaction
is then given by the linear current
interaction: J -A = + .-,...
e
p’c . &re)
- +
gp . ;b(pp) , P
(l-2)
.
. which can be rewritten
in relative coordinates (see Appendix B) as
The last term can be safely neglected in the limit of large Mp and small k (the dipole approximation
limit).
Let us now turn to a discussion of photon emission
from a free and then a bound atom.
I
-
.P --
r, .-
2. Single Atom-Dipole
Emission
As a warmup, and for Iater use, we will first consider the radiation photon of momentum
k from an atom in the dipole approximation
of a single
for two different
situations:
a free atom and an atom bound in a center-of-mass potential.
interaction
Hamiltonian
is approximated
as
G
H’ = J./l=
P;.x(,)
,-ik.R
T
(2.1) L
HI
=
M
e
where e is the photon polarization
4
r
-
The
P, - 6 e
-ik.R
Ze
-ik.r
-ik.R
Eve
-z
7
(2.2)
vector, and standard manipulations
lead to
First we will discuss the emission from a ‘spinless’ free atom and thereby choose the wave functions as: Initial State
Ip) = 2p state
1;) = eipsR lp)
(24 Final State
If)
Is) = 1s state
= eiP”R Is)
with energies EP and E,, respectively. The transition - M = (s 1v 1p) /
e-i(P’+k’P)‘Rd3R
matrix element then becomes
= (s I v I p) (ZTL)~S(P” + $ - $) ,
(2.5)
_
and the differential
decay rate is written
- Rate
succinctly as
- c PI2 = c lb I ?J IP)
12P
=
Rfree
PO1
where EP - E, = ko = I ICI, and recoil energy has been neglected. 4
(2.6)
-
-
.ir --
F. .*
If th e a to m is in a simple c o n fin i n g p o te n tial W s(R), in a state o f e n e r g y E s with well-separated e n e r g y levels, th e w a v e fu n c tio n s for th e transition in w h i c h only th e state o f th e relative w a v e fu n c tio n c h a n g e s c a n b e written Initial S ta te
I4 = d(R) IP ) (2.7)
Final S ta te
If> = 4(R) I4 7
with e n e r g i e s g i v e n b y E s + E P a n d E s + E ,, respectively. T h e transition m a trix e l e m e n t for r e m a i n i n g in th e state E s while th e internal state c h a n g e s is th e n
.--
w ith-Ff.1 :E_,
M=
(slvlp)/d3Re
-ik’R $ 2 ( R )
E (s 1v lp) F(k)
,
( 2 .8 )
= k,, = lkl a s b e fore. T h e d e c a y rate is written a s
R a te = R ~ ,,IF(k)12
,
( 2 .9 )
w h e r e F(k) is th e fo r m factor for th e c e n ter-of-mass (C-M) b o u n d state, m e a s u r i n g its ability to a b s o r b a m o m e n tu m transfer o f k.
3. One Atom-Double
Well
Following the notation and level of approximation
of the preceding discussion,
let us now consider the case in which the atom is in a C-M double well potential. The initial state is assumed to be the lowest symmetric eigenstate in the wellseparated double well and will be approximated as
where the normalized functions d(R) peak at R = 0. The energy of the state is E+ +.-EP where E+ = ER - TET/~
(T
is the exponentially behaved overlap
factor defined below), and ET is a positive reasonable number of order ER which is easily ,computed given the form of the potential.
For clarity, we write the above
as
Qi = I+) = N+ Id@+) + w->I The corresponding anti-symmetric
I-)
with energy E_
The normalization 1 Nl
(3.2)
(spatial) state is
= N- [4@+)
+ Ep where E-
IP) *
- @-)I
(3.3)
1~) >
= ER + TET/~.
factors are:
= 2[1 f T] ,
where
T =
J
d3R$(Rt)$(R-)
.
(34
For a gaussian trial function, the C-M wave function, tunneling factor T and form 6
-
.-
--
F.
.-
factor F(k) $(R)
are = N e-R2/2a2
,
T = e-D2/4a2
and
F(k)
= emazk214.
The decay process will lead to both symmetric and antisymmetric
(3.5)
final states:
I+) = N+ MW + 4uw I4 (3.6) I-> = iv- W W - W -)I I4 with energies Eh + E,, respectively. The transition
matrix element to the sym-
metric final state is (tl
._-
H’I \E‘i) = Nt = Ni
(S I v
I p)
J
d3R emikaR[gi(R-t-) + 4(R-)12
(s 1v lp) { ei(1/2)k’DF(k)
. ._, ... _ . wherek=E++Ep-E+-E, Jti
+ e-i(1/2)k*DF(k)
J
swww-)
oc
J
&R
e -(1/a2)[R2+(1/4)D2]
,-ik.R
(3.8)
Thus Ikl = Ep - E,, as before, and
(+IH’I Qi) = (SI v IP) F(k)
(-IH’I
,
,
for a gaussian wave function.
The transition
+- 2J}
=EP-E,,and
d3R ,-ik.R
J = F(k)T
P-7)
(cosz;T)
matrix element to the antisymmetric
Qi) =
(S
Iv
Ip)
N+N-
{
ei(112)D’i-F(k-)
.
W)
state can be written - e-i(1~2)k-*DF(k-)}
(3.10)
where kk-
=E+$E,-E--E,
T, and the difference between k and
=k-TE
has been retained in the phase factor only. 7
Assuming that the detector sums the counts from photons of energy k and k-, the total decay rate from the symmetric initial state is then
Rate = heelF
(3.11)
I2 A+ 3
where the angular modulation factor A+ contains the interference terms of interest; it is given by
A+ = {(f$g2t&}
(3.12)
)
where for convenience we have introduced .--
t9 =
and
;Dek
8-
= ;D.k-
E ep-T$)
(3.13)
. ._, .-_ _
and defined c = case (3.14) = sine-
S-
E
S-CT
Thus to first order in T we find
A+
c(l-c)-csTe
=
ET
1 .
(3.15)
In the limit where ET 12 (+lH.‘$i) l = ” * d3RcikaR > {J = ...
d3R ,-ik*R
{J
has the expected peaks at f D/2 anti-symmetric .--
(4.1)
[4 2 (R+)+cj2(R-)+2)(R+)4(R-)I}
7
and a small (overlap) peak at R = 0. The
final state produces no peak at R = 0,
(--1~‘1-2cli) = , . *
= .*.
{J {J
d3Re-ik’R
d3R
,-ik.R
MR+) - W-)1 [SW+>+ @-)I > [4
2
CR+)
-
81olj
(44
.
If there were no peak at R = 0 and if ET
IR) = & 2
(A4
w h e r e IL) h a s th e a to m in th e left well w a v e fu n c tio n a n d IR) h a s th e a to m m o s tly .-in th e right well w a v e (but orthogonality is retained). T h e n starting from a n initial state with ._ ,.-. a_ g i v e n s y m m e try, th e relevant m a trix e l e m e n ts are:
IQ i) =
WW
l&&T
(LIH’I@ i)= fiim (RIH’I
S i) = 5
d&
d&
-
f W-)1
1~)
(31 ~IP )F(k) [e iet T] ( S I v lp) F(k)
(A4 ( A .3)
[ztT + e D i e F T e i o - T ’] (A4
R a te = R e e e IF(k)j2 A &
A A= {(~ )2t& T } ~ w h i c h is exactly-the s a m e result a s b e fo r e in this lim it o f E T = 0 .
12
(A4
(A-6)
APPENDIX
B
Standard Coordinate Transformation In the text we have used the standard coordinate transformations
to relative
and center-of-mass variables:
NOW
use--the fact $hat & e ?e + & e r’P = &OT * 2 i- p’, o 7 to write: . _, ,_. _ I%oT
= gp t 2% (B-2)
j% =
and, of course,
&OT
7
P.3)
where the reduced mass is given by -E1 Mr
1
$+-* e
MP
ACKNOWLEDGEMENTS The author wishes to thank Claude 1tz;vkson for introducing for interesting gratefully
and helpful discussions.
this problem and
Discussions with Hans Evertz are also
acknowledged. 13
REFERENCES 1. D. M. Eigler and E. K. Schweizer, Nature 344, 524 (1990). 2. Molecules with adjacent indentical binding sites, for example. 3. A surface with adjacent empty sites, or other interesting voids. 4. J. Robert,
Ch. Miniatura,
0. Gorceix, S. Le Boiteux, V. Lorent, J. Rein-
hardt, and J. Baudon, ‘Atomic Quantum Phase Studies with a Longitudinal Stern-Gerlach
Interferometer,’
Galilee, Universite Paris-Nord,
Laboratoire
de Physique des Lasers, Institute
submitted to J. Phys. Paris II (1992).
5. We are interested in the electromagnetic coupling and its dependence on .-the center of mass as well as the relative coordinate. For a treatment of . relativistic ._, .-_ -
effects and of spin, see S. J. Brodsky and J. R. Primack, Phys.
Rev. 174, 2071 (1968). 6. The factor c(1 - c) is the well known ‘kilroy’ pattern if plotted against the angle 0 , with a minimum (l/4)
of (-2)
at 6’= 7r and two symmetric maxima of
at 0 = 7r/3 and 57r/3.
14
