Download Atomic quantum phase studies with a longitudinal Stern

Atomic quantum phase studies with a longitudinal
Stern-Gerlach interferometer
J. Robert, Ch. Miniatura, O. Gorceix, S. Le Boiteux, V. Lorent, J. Reinhardt,
J. Baudon
To cite this version:
J. Robert, Ch. Miniatura, O. Gorceix, S. Le Boiteux, V. Lorent, et al.. Atomic quantum
phase studies with a longitudinal Stern-Gerlach interferometer. Journal de Physique II, EDP
Sciences, 1992, 2 (4), pp.601-614. <10.1051/jp2:1992155>. <jpa-00247659>
HAL Id: jpa-00247659
https://hal.archives-ouvertes.fr/jpa-00247659
Submitted on 1 Jan 1992
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J.
Phys.
II
France
(1992)
2
601-614
1992,
APRIL
601
PAGE
Classification
Physics
Abstracts
03.65W
07.60L
32.60V
phase
quantum
Atomic
with
studies
longitudinal
a
Stern-Gerlach
interferometer
Robert,
J.
and
Laboratoire
J-B-
Av.
de Physique des Lasers(*), Institut
C16ment, 93430 Villetaneuse, France
(Received
31
R4sumd
January1992,
On
donne
diffusion.
de
rateur
sentd
O. Gorceix, S. Le
Miniatura,
Ch.
Boiteux, V. Lorent, J.
par
5
montre
l'action
que
polar1s6
d'onde
se
ramkne
atomiques
longitudinal sur
interf4romktres
des
interf6rom+tre
d'un
£
A
Abstract
given.
The
polarized
general description ofatomic
interferometers
of a longitudinal
interferometer
on
action
wavepacket is shown
atome
un
On
examine
champ pr4cessant
m6tastables
d'hydrog+ne est
pr6c6dentes sont
discut6s.
On
des champs de quelques 10 G,
et
celui
d'un
scattering
by an
leading to a
of the
terms
described
atom
operator
incident
"beaded"
transformation,
magnetic field and that of a spatially
examined.
precessing field are
metastable
hydrogen atoms is presented
and the results
obtained in both
situations
mentioned
above are
discussed.
Properties of beaded
produced by relatively strong fields (od 10 G) are investigated by means of the intensity
atoms
diagram of their electrically induced
radiative decay.
The
atom.
special
of
case
to
pure
a
be
equivalent
in
an
d'opdreprd-
termes
en
r4fdrentiel.
de
transformation
une
cas
is
Paris-Nord,
Universitd
February 1992)
description g6n6rale
une
On
paquet
un
accepted
Gali16e,
particulier d'un champ magn4tique I gradient longitudinal
spatialement. Une
r4alisation
exp4dmentale
utilisant
des
atomes
prdsent6e et les r6sultats
ensuite
obtenus
dans les deux
situations
chalelet"
dtudie
enfin les firopr16t6s des
"en
produits par
atomes
£ partir du
induit
champ 61ectrique.
rayonnement
par
un
le
Reinhardt
Baudon
J.
to
a
frame
longitudinal gradient
An experiment «sing
of
Introduction.
1
In
the
states
representation,
arises
generally
definite
(*)
the
system
assoc16
based
problem of
problem of bra
the
au
of kets
CNRS,
is
the
upon
the phase
and
used
URA
282.
as
superposition principle, of
associated
conjugation.
representation
to
ket
a
of
quantum
any
given state (cf.
a
This question is of
states
insofar
as
it
Dirac
system by
[ii)
and
importance
leads
a
more
when
priori
to
a
a
JOURNAL
602
phase-dependent representation of
cancels
perform
measurements
choice of phases is adopted.
observables.
the
N°4
II
by chance, the usual
However
difficulties,
these
all
to
PHYSIQUE
DE
least
at
if
a
way
and
permanent
we
use
coherent
necessity of a renewal of phase studies (for a
that,
with systems containing several degrees of
answer
see
relative phases. The idea
freedom, it is possible in a real or "gedanken" experiment to exhibit
(by
filter) whereas
the
degrees
of
freedom
of
is to lock with respect to each other
some
some
of
interferometric
the
evolution
observes,
by
of
the
other degrees
measurement,
means
an
one
that has been made on the
of freedom, in an elTective sub-space depending on the preparation
locked degrees of freedom.
It may be noticed
that this type of methods is commonly used in
scattering experiments on atoms or molecules, as well as in light-matter interaction
studies.
atomic
interferometry experiments is not the evidence for
What is really new in the
recent
the de Broglie wavelength but rather the variety of methods to "freeze" or "lock"
degrees
some
made feasible by the use of atomic
control techniques. AH these experiments
of freedom
beam
that each particle (atom) interferes
with itself. So far
of the one-particle type in the
sense
are
coherent elTect has been observed or even, to our knowledge, proposed.
collective
no
Atomic
systems are particularly very well fitted for fine studies of quantum phases »ince
external
degrees of freedom can be manipulated rather easily by means of external
inner and
of a rich internal
in atoms is an advantage, compared to photons
fields. The
structure
occurence
be
exploited
which
the
external degrees of freedom
via the
internal
to act on
neutrons,
can
or
provides
and
This
flexibility
the
methods
control
vice
in
specify
to
great
a
versa.
ones,
or
interferometer.
phaseshifts and consequently to build an
interferometers
is presented and
In part 2 of this article, a general description of atomic
discussed.
the problem of dynamical and topological phases h
In part 3, the principle of the
Stern-Gerlach
interferometer
longitudinal
is given and the
theoretical
methods
used to study
presented.
is
devoted
Part
4
the
description
of
the
its operation
experiJnent
and to
to
are
either with a
the results
obtained
magnetic field with a longitudinal gradient (pure
transverse
dynamical phases) or with a conical field configuration (dynamical and topological phases). In
part 5, experiments using a strong field are presented. They are devoted to the study of special
optical properties exhibited by the atoms emerging from the interferometer l'~beaded" atoms).
this
In
context,
2.
may
one
about
wonder
the
[2]). One possible
general review
Description
ofatomic
is
interferometers.
Any atonfic
interferometer
consists of an
arrangement
slits, grating, etc...) which perturbs the atomic
within a finite
domain of space.
evolution
From this view point, it is quite s1nlilar (except
device.
Just like a collision, the operation of the
interferometer
for the size) to a scattering
is
contained
in a "scattering operator" § which
transforms
asymptotic (free) incident
state
any
(~l~~) into an asymptotic outgoing state (~1°~~)
GENERAL
2.I
of
external
CONSIDERATIONS.
fields
(among
which
includes
one
§ j~inj
l~outj
Let
us
variable
define
£
now
'~principal path
the
(discrete
or
continuous),
being
/
=
A
Stieltjes
in
discrete
integral
and
with
continuous
a
measure
cases.
ofthe
states"
as
interferometer, [p(£)), labelled by
eigenstates of this operator:
has
been
quantities
used
s(£)
are
in
a
vectorial
(2)
dP(f)lw(f))e'~~~~(~2(t)1
p(£)
The
the
(~)
order
real
to
phases
get
a
insofar
similar
as
all
expression
scattering
QUANTUM
ATOMIC
N°4
channels
is
case
the
atomic
actually
is
ultra-cold
of
of the
that
tion
This
open.
are
(the
beams
is in
describe
to
further
then
ossible
m
the
of
=
set
Is
with
#~,
flint
[k,
by states
iven
basis
and
dk
/
SELECTED
~
genvalues
=
e(x),
dE
such
E,x)lk,
P(x)a"(k,
tt°(r,t, d
restrict
We further
bution of which is
choice
our
peaked
at
value
some
With
a°(k, E,x)
jr,
kP
=
+ u,
[u[
«
-kfl
=
m
asymptotic
the
jr,
=
=
kP,
i,y j~t
outgoing) states
analysed) wave-packet,
or
(4)
°
these
~l
E
a°(k,x)b
exp[I(k
~
distri-
momentum
corresponds
that
such
by projection
obtained
is
wavepackets, the
the (y[ selection
asumptions and reminding that
and
to
a
asymptotic
these
(x)j
~
Et/h)]
r
~~~~
EP~
=
=
(particle) velocity,
group
+
2m
and
wavepacket now labelled with kfl and
slowly variable envelop:
a
(x~)
~
h2 (kfl)2
Ef
x~
=
can
be
~ m
written
In
such
in
the
conditions,
form
of
a
the
carrier
multiplying
1~"
the
(kfl, x~;
r,
t)
; kfl
=
exp
r
envelop is given by:
£"
The
spectively.
(kfl
h
vfl
=
k, E)
t
E
where
e(x)
or
quasi-monochromatic
to
given eigenstate of flint, say (x~[
wavepackets are free, one gets:
wave
+
~~
(3)
described, for each atom in the beam, by a selected
implies a choice of a phase reference insofar as the wavepacket
of [~l") onto the bra jr, t, y[, where y labels the internal
state:
are
Let
~
asymptotic (incomin g
If
WAVE-PACKETS.
ASYMPTOTIC
#
Ei
this
k
are
#,
E,
~~,
(prepared
with
which
'~mixers"
the
after
m
lt~°)
2
choije
the
interferometer, I.e. to
of the principal path states [~g(£)). In
combinations
linear
include
not
and
before
=
il,
of
eigenstates
2
a
the
does
It
the atomic
interferometer
state, one can expand
on
in, out) on a basis set realizing a complete description of the
interferometer
being of the short-range
The interaction, in the
asymptotic states [~1°) (a
#~.
Hamiltonian
asymptotic
tonian,
atomic
experiments using thermal
carefully).
descripthis
Actually
more
separatrices and the arms), in which
of the
elTect
the
the
internal ariable
(say
beam
atomic
603
of the
most
adiabatic.
cases
charge to prepare or analyse the
specify (~li~, °~~) which are definite
STUDIES
examined
be
interferometer
most
in
order
in
case
should
of the
"core"
evolution
the
atoms
PHASE
special
(r
case
vflt; x~)
of
a
plane
=
exp
wave
[-ie (x~) t/h]
is
readily
/
obtained
(~ t)j
du a°
£°
(u, x~)
by setting
(r
exp
in a°
vflt; x~)
(I
u
(5)
(r
(u,x~)
vflt))
=
a°
(x~) b(u).
(6)
JOURNAL
604
SPATIAL
2. 3
AND
TEMPORAL
SHIFTS
PHYSIQUE
DE
II
DEGENERATE
IN A
N°4
LONGITUDINAL
INTERFEROMETER.
degenerate longitudinal
interferometer
h a device such that (I) the asymptotic
channels
internal
variables,
an
degenerated
with
the
I.e.
there
exists
single
value
respect
to
e of
are
a
e(x) (we shall take e
0 in the following) (it) any incoming beam of atoms at a given direction
results
propagating in the same direction, regardless to the external
into an outgoing beam
A
=
(all trajectories
motion
such
In
concept is relevant) within the
of the § operator
reduces
if this
matrix
the
case,
a
form
(k, E',x'(S(k, E, #
and
principal path
the
states
stands
y
for
internal
the
E')
simply:
are
z)
1~2(t)ilk, E,
where
k') b (E
(k, E, x'(S[k, E, x) b (k
=
interferometer.
of the
arms
to
phaseshifts
corresponding
The
state.
(7)
by equation (2)
defined
are:
S(t)
An
incoming
The
prepared (polarized) along (x')
state
(l#°~~)
dk
#
outgoing
dp (X') dp(y) (X'
(r,t,x°)
/
dk
=
(r,t,x°
=
of
powers
u
s(k, y)
(Vk s)~,
outgoing one:
where s'
one
=
~1°~~
/
makes
fl
explicitely
appear
k,
x') a~" (k, x') e'~~
((°)
refers
and
to
X'
(8)
~'~~
~~~
~
+
u
(9)
s'(k", £
incident
the
wavepacket within the expression
(r,t, x°)
=
dv(y) (x° y) (y x") tt~" [r
+ Ar
(k', fl
t + At
(k', fl
,
spatial
-j
2m
,
=
The
J~2 ~2
fit°~~)
s° (k",
ci
state:
k' in the vicinity of k"
k
=
outgoing
the
to
then:
is
dp(y) (x° y) e'~(~,Y) (y
By expanding s(k, y) in
leads
y) e'~~~'~~ (y X') a~'~ (k, X')
packet polarized along (x°)
wave
lV°~~
of the
S(k,z)
=
temporal
shifts
fl
Ar
(k~,
At
(k",fl
the s°
are
s°
h
~
(k') ~
~~
=
(k') ~
dependent part
(lo)
given by:
~
=2
x"]
,
(s°
of Ar
s'
2
k"
and
~'
(k') ~
s')
At)
(k' s')
At°
=
=
+ At'
Ar° + Ar'
(11)
QUANTUM
ATOMIC
N°4
noting
packet.
worth
It is
envelop
of the
In the
that
(spatial)
one
Ar°/At°
as
dimension
case,
V", the phase velocity, which
packet.
conclude
To
this
phaseshift
the
part,
well
as
carrier
as
In
wave.
averaging
the
a
Ar
that
these
on
of k'.
one
The
interference
then
does
result
not
h
source
(cf
used
similar
is
the
carrier
=
of the
wave
and
adiabatic
dealing with atoms
adiabatic topological phaseshifts
The
the
Az'/At'
has:
one
the
alTect
not
dynamical parts of
The topological
case.
aiect
the envelop but only the
of dilTerent velocities, I.e. with
the
is
pattern.
light
monochromatic
k" and
involve
not
in
topological
the
about
k~6z,
=
absent
are
At° do
and
hi and k"
shifts
wavepacket, in
the
605
Ar°
shifts
Az
=
remark
a
elTects
experiment
real
of the
extended
an
does
distribution
incoherent
an
Ar°,
as
make
us
their
v',
=
where
indicates
let
part,
regards
hk'/m
=
STUDIES
PHASE
to
Young's
will
that
holes
involved
be
not
optics
experiment used
observed
in
in
when
as
a
spatial correlator),
longitudinal
The
3.
PRINCIPLE
3. I
AND
profile B(z)
opposite directions
restricted
field
on
the
section
t6
a
which
means
The
2.3.
finite
in the in and
zero,
the
to
energy
The
izer,
is
atom
in
device
This
DESCRIPTION.
described
type
of the
eter
polarization
Stern-Gerlach
core
The
range.
interferometer.
is
longitudinal-degenerated
a
of this
interferometer
gradient
of its
modulus
integral
out-sides
that
the
that
net
transfer
in such a way
there is finally no
is
of
interferom-
magnetic
longitudinal, with
consists
of
of the
a
force
exerted
neither
momentum
of
atom.
core
of the
which
selects
interferometer
a
specific
preceeded by
is
Zeeman
preparation
by
a
(x"),
state
followed
section
a
mixer
consisting
of
builds
which
a
polar-
a
linear
superposition of such states.
Experimental details will be given further.
The principle of this
(with
the
direction
of
field
rapidly
mixer is basically to turn
respect to the Larmor
a magnetic
frequency). Similarly the core is followed by a second mixer and an analyzer selecting (x°).
Actually, (x'>°) are eigenstates of the component of the atomic spin J along a fixed axis u".
fl~ reduces to the kinetic energy operator. The principal
Hamiltonian
Here the asymptotic
determined
following the lines
path states [~g(y)), belonging to the [k E J) manifold, can be
developed by Majorana [3]. Within both mixers and core domains, the
Hamiltonian
also
can
be
written
in
the
form:
fi
~
fl(r)
where
B(r)
3.2
is
g
=
the
Land6
B(r)fiB(r),
THE
with
SCHWINGER'S
factor
and
pB
LimjrjcoB
=
TREATMENT.
gvBJ
=
the
0,
so
Bohr
that
In
B(r)
The
magneton.
fl
his
reduces
paper
to
fl~
field
in the
"Non-adiabatic
has
limit
general
[r[oo.
the
processes
in
form:
non-
homogeneous fields" [4], Schwinger has developed all the theoretical background needed in
time-dependent description of the spin evolution in the spin-state subspace.
However
to
a
properly the
interferometer
describe
simply transpose this time-dependent theory
cannot
we
the
external
motion is given by the
classical
expression z
One has
in assuming that
vi.
solve
the
complete
atomic
motion,
by
adding
the
kinetic
the
internal
operator
to
to
energy
The field configuration will be assumed
Hamiltonian.
In a first part the field
to be as follows:
has a longitudinal gradient but a fixed direction, its magnitude rhing from
zero
up to
some
value B. In a second part of length L it keeps
direction
in magnitude but its
constant
preBfiB(z), with a (spatial) angular velocity Q
2~/L and
uniformly around z : B(z)
cesses
making a constant angle
with
z-axis.
In a third part, the field goes down to zero keeping
=
=
=
JOURNAL
606
direction.
constant
a
referred
the
to
concentrate
be
and
attention
our
z-dependent
a
fl is
fltM
with:
£
(6)
As the
be
restritted
£
fit
of the
coupled
FM(z)fltM (6(z)).
=
M
We
shall
adiabatic.
Let
fi(z)
fl
+ sin
a(z)iiy
sin
eigenstates fltM
The
+
phi
cos
u(z)
of J
parametrized
are
(hi
Mi
~0)FM
=
FM (z) describing the
Functions
by 6,
external
definite
of
energy
will
problem
The
the IVM
over
motion
(6)
solutions
are
equations:
dilTerential
FM(z)
E
Qz.
states
difficulty.
is time-independent it is allowed to look for a state
being independent of z, y, these variables are separated.
only z coordinate.
The complete state
be expanded
can
the
to
basis:
a(z)6~
=
any
not
spin
the
among
occurs
Hamiltonian
total
interaction
The
transition
no
form:
l8~, ~,(0, -fl, -a(z))lV
=
Mi
E.
of the
cos
a(z)
and
constant
a
fl
sin
N°4
without
parts can be treated
central part in which the
evolution is
vector
=
parts
II
these
the
unitary
third
and
then
to
6
where
first
the
In
ads
B
PHYSIQUE
DE
£ (-gpB)
+
~2 (M (0z( M')
(M(J B(M')
m
0zFM/-
m
-~ (M (0)( M') MJj
~,
(12)
m
where
matrix
all
algebra [5]
allows
(M
Using
JWKB
the
0z FM is of the
value
totic
energy.
terms
K
M')
(0z
in
-iQ
=
(M
2~/A
the
because
hand
(M (0)( if')
are
order
the
the
compare
local
of
dilTerent
very
As L »
to
those in
of Jz
momentum
J)
and
:
(M (J) M')
-Q~
=
are
I/L.
negligible with respect
those
of
terms
equation (12):
close
to
the
asymp-
small
compared
to
the
kinetic
A,
wavenumber
energies
Zeeman
Q is of
to
K is the
angular
standard
The
0) by
and
(M (0)( M')
M')
where
basis.
of 0z
elements
is able
one
K(z)FM,
of
(3~
(6)
the fltM
in
matrix
approximation
order
=
taken
are
replace
to
one
another
On
the
elements
(M
very
Q~
one
has
QK,
then
(0z
M'). Obviously
these
«
interaction
compared to E but not compared to the "static"
terms
(M [-gpBJ B[ M') and they must be kept in the
form of the ampliThe eikonal
treatment.
and SM a rapidly
tudes is FM = AM(z)exp [iSM(z)], where AM is a slowly varying function
S + aM with aM < S. The approximate
JWKB
form of
varying real phase such that SM
the coupled equations is now:
latter
terms
are
small
=
E AM
(z)
=
£ (M gpBJ. B
)S'~AM
~~'
~,
The
due
parts
oIT-diagonal coupling
to
the
cancel
z
terms
contain
dependence of the basis
eachother by a
convenient
~~
B +
gfiBm
hQ6z
=
a
set.
static
In
choice
Be6(z)
the
of
=
Q6zj
+
gPBm
M')
part (in J B) and a dynamical part (in
present case it is possible to make these
6(z)
:
if 6 is
such
that:
Be(elTective magnetic field)
QS')
two
QUANTUM
ATOMIC
N°4
the
then
equations decouple
~2
solution
takes
607
in:
the
(M (gpBJ Be M')
-S'~
E
Finally the
STUDIES
PHASE
=
form:
+CO
FM
Reafisation
of
4. I
GENERAL
DESCRIPTION
already
sented
here.
continuous,
(TOF).
The
OF
dz
EXPERIMENT
details
elsewhere
but
a
it
TOF
thermal
molecular
beam.
In
results.
(see Fig. I).
[6] and only its
general principle developed in parts
Hydrogen atom H* (2si/2) beam.
metastable
of
wM(z)
experimental
THE
in
~co
The
of a
bombardment
case
the
described
been
ikz +
interferometer,
4.
has
Amexp
=
most
2.3
and
3.I
beam
This
of the
The
main
is
experiments
H~m~
'~
~~
~j)fj~j~i~~j~flj)~
~
@~~~
C
P
C'B~
[--,--)
setup
be
pre-
been
bombardment
this
easily pulsed in view of an analysis or selection
can
dhtribution
is well fitted by the following function:
Bp
will
applied to the
produced by electronic
has
be
KA
experimental
features
of the
time
of
h
flight
la)
D
j
--_z
P
(b)
(a) Scheme of the apparatus. K, A: electron gun, Bp,A Poladzing and analysing fields,
quasi-zero field chambers, p: magnetic shieldings, iC
(b)
axisparallel intensities, D detector.
Principle of the experiment. P,A: polarizer, analyzer; M, M' : mixers; R: region where the longitudinal
(a J
magnetic field gradient induces phaseshifts among Zeeman
1 value being assumed).
states
Fig.
1.
CC'
=
f(z)
where
vo
=10
z
t/to,to being
km/s.
=
The preparation
lowing steps:
of the
the
most
beam
at
=
C~~z~~exp
probable
the
entrance
time
(-z~)
of
of the
flight, corresponding
interferometer
consists
to
the
of the
velocity
two
fol-
JOURNAL
608
PHYSIQUE
DE
N°4
II
Lamb-Retherford
method [7],
(I) a partial polarization of the beam h achieved by use of the
by passage through a transverse field Bp parallel to z-axis, of about 600 G in magnitude:
only the two hyperfine Zeeman states F
I, MF
+1, 0 remain within the beam, with equal
between
them.
populations and no coherence
(ii) Starting from one or the other of these filtered levels, a linear superposition of hyperfine
sublevels F
I, MF
+1, 0, -1, referred to the y-axis is built in a "mixer" (AI in Fig. I).
reduce
the fringe field of the polarizer (parallel to z-axis) to
Its principle is simply to first
distance
y-direction,
of 5 mm.
The
the field to the
few tens of mG and then
turn
over
a
I-e-
=
=
#
=
evolution
non-adiabatic
spin makes
of the
the
desired
result
without
appreciable
any
elTect
on
According to a general treatment given by Schwinger [4] applied to a spin
show
J
I (which is fully justified owing to the low magnitude of the magnetic field), one
can
that the operation of the mixer is representable by a Wigner matrix Dl') in which the Euler
characteristics
of the field in the mixer.
angles depend on the
The mixer is followed by a region 7l where a magnetic field B(Z), with a longitudinal gradient
dilTerent
interferometer
of B, induces
phaseshifts on the dilTerent eigenstates ("arms") of the
external
motion of the atom.
and acts at the
time on the
Two dilTerent field configurations
same
field parallel to y-axis, (it) a conical magnetic field.
These
have been used: (I) a
transverse
the
external
motion.
=
two
ending
consists
of
discussed
be
will
cases
The
side
(M')
parallel or anti-parallel
emerging metastable
either
further.
detail
more
(analysis)
apparatus
mixer
second
a
in
of the
followed
by
a
symmetric to entrance
polarisation analyzer, the
is
(preparation).
side
field BA of
which
It
is
Bp.
to
finally counted in a detector specific of H*(2s) atoms.
atoms
are
other species (detectable by a standard
requirement
since
electron
many
an
multiplier) are produced by the electronic
bombardment
of H2 (metastable molecules, ions).
detector a static
electric field of 40 V/cm induces
the 2s-2p
transition.
The Lyman a
In this
photons resulting from the decay of the 2p level are detected by a channel electron multiplier
through a MgF~, window.
The
This
important
is
4.2
TRANSVERSE
described
conical
in
to
to
states
part 2.3, this is
of
wave
the
are
shown
and
t.
In
the
AZM
recalled
a
definite
here
GRADIENT.
in
order
to
This
experiment, already
emfihasize the contrast
with the
following paragraph.
Two wires
The
interference
figure
2a.
case
3s[ /K
pattern
It is well
of the
of VB
eTect
a
magnetic
field
Bc(z)
-4l
on
the
-1
interferometer.
of the
create
in region lZ is
y-axis. In this particular
mono-kinetic, each of
If the beam
was
resp.). According to the theory given in
+1, 0,
=
given phaseshift (+4l, 0,
evolution
the
referred
to
motion:
external
an
incident
atomic
diTerentially shifted outgoing plane waves.
momentum
almost
monokinetic
incident
wavepacket is used, the 3 outgoing
an
incident
providing
convenient
changes in thi origins of z
to the
one
gives
K
present
"
"arms"
a
MF
states
manifestation
before, when
wavepackets are identical
As
LONGITUDINAL
A
Zeeman
accumulate
would
plane
for
adiabatic
these
case
them
WITH
[6], is briefly
configuration which will be considered in the
(+ ic)
z-axis
and transporting opposite
intensities
the
y-axis- As the direction of
field is not varied,
field
parallel
parallel
strictly
FIELD
reference
(M
and
to 3
+1, 0, -1)
=
AtM
obtained
interpreted by
1=
rise
/~
o
#
4ms[ / (hk~)
by scanning ic,
use
of the
dz
f(z)~
s[
where
without
any
=
~°~~~~~
8
~ ~
~'~~~~
j~~~~~~
velocity selection, is shown
formula:
~ ~~
M
~~~
in
QUANTUM
ATOMIC
N°4
PHASE
STUDIES
609
a
b
N
-300
300
o
mA
mA
DA
,04
ic,A
Fig. 2.
-Interference
experimental data
TOF
(8t/t
selection
where
with
=
The
data.
with
magnetic
transverse
a
fives
the
field
region
in
(a)
prediction.
theoretical
R.
give
Points
selection;
TOF
no
(b)
0.3)
parameter
the
the
obtained
pattern
the
brokenline
while
characteristic
broadness
mixers
of the
has
adjusted
been
(bt/t
distribution
TOF
order
interferences
of the
I)
ci
to
causes
a
get the
best
fit
damping
fast
of
visibility of the fringes when the
increases.
By chopping the beam and
selecting a slice in the TOF spectrum (St It ci 0.3), a significant improvement of the visibility is
achieved, as expected (Fig. 2b). Whilst a wavepacket description would be able to reproduce
the loss of visibility due to the
distribution
width, it is not possible in the present
TOF
experiment to
discriminate
in between
wavepackets (which should be
the roles of the "true"
source)
determined
by the operation of the beam
and that of the
statistical
distribution
of the
these packets. An other experiment including a velocity selector
with a
group velocity among
the
well-defined
transmission
CONICAL
4.3
experiment
Of the
factor
FIELD
has
CONFIGURATION.
to
by
Its
creates
spatial frequency
The
order
to
get
field BH
a
is Q
=
information
some
typical time
compare
a
the magnetic field. It
of 0.I
TOPOLOGICAL
G,
spin
one
gets T
evolution
ci
and
of
flight
turns
7L,
the
2~/L.
On
almost
diTerent
another
constant
B is
(BH/Bj)
T of
an
that
Therefore
correlative
the
a
constant
Dubbers
around
the
and
on
period (L
one
hand
over
Artan
and
turn
one
where
about
out
making
(see Fig. 3).
This type of
conditions
regarding the value
PHASES
somewhat
spin dud the adiabaticity of the evolution) by Bitters
parallel wires are now replaced by
field B II which can be considered
as
B then takes the form: B = 86(z),
of axis z and apex angle f =
a
cone
In
[8].
necessary
already performed (in
been
[9]. In region 7l, the two
helix (intensity iH) which
[0, L].
be
would
the
solenoid
a
interval
6
a
z
=
cm)
of
over
the
axis
creates
vector
a
double
interval
longitudinal
a
resulting field
[0, L]. The
unit
beam
neutron
a
6
which
describes
validity of the adiabatic approximation one has
Larmor period 7L imposed
atom through [0, L] to the
at a velocity of10 km/s and a typical magnetic field
adiabatic
approximation is not valid here in general.
the
eTect
on
the
external
motion
have
been
examined
in
JOURNAL
610
PHYSIQUE
DE
II
N°4
a
',
Z
'
'
'
'
iH
b
~
i
e
~e
E
i
i
e
~
~
~
~
l
Z
---+
e
II
Fig.
Conical
3.
field
configuration:
field
Be
=
configuration (a) experimental setup: H double helix; S
B + Afiz where A is proportional to spatial frequency
part 3.3. As explained
spin
the
Be
lated
B +
=
hQ/g~Bi1z.
along path
the
of the
axis
For
L is: lbM
principal paths
the
before
along
component
each
of
states,
these
interferometer
magnetic field:
labelled by M,
the
the 3
are
phase
total
eigenstates
shift
of
accumu-
Mlb, with:
=
16
Actually Be
of the
effective
(b) magnetic
solenoid
Q.
l)
(~(~~~~
h QK
2~
=
ill and K. Because of the K depenaveraging
the TOF
distribution.
an
over
The topological part fl of the phase is proportional to the solid angle of the cone
described by
with its geometric
Be. Therefore it depends explicitely on K, which
to be contradictory
seems
non-adiabatic
character.
In fact however this is not paradoxial since, in the
case, the geometric
dence
the
character
followed
does
not
refer
but
rather
to
to
encloses
+0A A
+6 mA
(+
(+
worthnoting
of
the
experiment
same
cone.
Bj,16
has
=
that
a
to
sign
16 is
single
out
rotation
an
ilz).
even
minimum
ibm;n
iH is
way:
z), for fixed
ill, Bj is
iH
For
BH
=
one
"
2~
has the same sign as Bj. This
means
only when Bjj < 0. Consequently the
"
the
present
values
about
scanned
of
((gpBBjjm/(hKQ)
the
zero-order
dose
15
value 16
=
+
I(
0 is
bright fringe
in
the
range
curve
when
the
the
cyclic
-0A
A up
ip ranging from -6 mA
times larger than BH. It
of iH> not of I II. As a function
minimum
0. The
value:
that
the
case,
states
around
function
at
(in
[10]: fl remains the same
provided that Be (which defines
of the
following
in the
iH>
also
parameters
space
another
into
the
Bj
of the
Hilbert
carried
of
here
include
space
the
to
changed
is
refers
the
to
given values of
for
will
is
hamiltonian
states)
defined
are
pattern
by B)
Zeeman
The
then 16
and
interference
of
BH for
a
fixed
up
is
value
I]
obtainable
still
exists
at
at
some
value of (BH(
of
negative values
QUANTUM
ATOMIC
N°4
STUDIES
PHASE
611
it disappears at positive values of Bjj.
peak
present in all patterns is only due
seen
complete calculation
qualitative:
This
solely
mechanism
is
mixing
not
[6]).
agreement
to the
a
4b
in
figure
which
in good agreement
the
shown
interference
fives
of the
patterns
are
curves
with the experiment, except at large values of BH because of a failure of the model adopted to
Bjj (but
it is
shifted
behaviour
This
describe
the
is
at
value
some
clearly
in
(BH( # 0),
figure
whereas
(the
4a
central
mixers.
b,mA
;,,,~A
a
N
o
2
2
4
OA
-O.4
iH,A
Fig. 4.- (a)
intensity iH in
predictions.
5.
Interference
the
Experiment
5. I
helix is
in
a
patterns
scanned
strong
As
EXPERIMENT.
longitudinal gradient
of
i~,A
using
for
field.
mentioned
magnetic field
conical
the
fixed
values
properties
Optical
before
can
be
configuration depicted in figure 3. The
intensity ill in the solenoid. (b) Theoretical
field
of the
the
spatial
increased
of
shifts
beaded
the
induced
1
on
atoms.
an
by use
up to few 10~
out of the complete
by a
wave
"strong~ fields
atomic
of
of Gauss. In such
conditions
interferometer
will
atom going
an
following properties: (I) its internal state is we[I defined (by the analyzer), for instance
I); (it) its external motion consists of 3 wavepackets the envelops and the
I, M
(2sij2, F
largely shifted in space and time as explained in 2:3 (beaded atom).
carrier
of which
waves
are
Because of the large values of the corresponding phaseshifts (few 10~ rad) the visibility of the
distribution.
interference fringes is completely cancelled by the averaging over the TOF
atomic
preserved
multiple
wavepackets
generated
Nevertheless
coherence is
&om a given
among the
a
if
it
through
interference
of
the
shortness
incident one, even
be seen
the atomic
because
cannot
associated wavelength.
of the
coherence by
The present experiment has been carried out in view of investigating this
electrostatic field, the
transition 2s-2p is
beaded
another
atom is passed through an
way. If the
induced
without
external
motion.
It is followed by the emission
any appreciable change in the
conformation
of the
of a Lyman a photon. The question which arises
is: does the special
now
external
motion
affect in some
emission of the atom ?
way the optical
of few
have
tens
the
=
=
JOURNAL
612
PHYSIQUE
DE
II
N°4
diagram of such atoms, the range of the
electrostatic
field
along
z-axis
and
the
photons
Lyman
detected
at
a
mm
are
angle
this
axis,
(Az,
At)
given
with
through
finite
solid
angle
As
shifts
respect
to
a
a
a
w.
K-dependent, a TOF technique can be used to scan these quantities experimentally. The
are
experiment has been carried out as follows: In a first step for given a and w, a TOF
distribution
field
in
shift).
is obtained
without
magnetic
regibn
without
7l
(I.e.
This
distribution
any
any
I)
(maximum
normalized
equal
is
practically
independent
of
and
In
second
to
once
a
a
w.
step a longitudinal magnetic field Bp produced by a solenoid (2000 tums~ 10 cm in length) is
applied in 7l. The difference A between the intensities
measured
without
and with this field is
then plotted as a function of th~e time of flight, in the
scale as that used for the
normalized
same
order
In
has
study
to
(standard)
of B II,
ues
A is
been
Table
(£
in
6il of the
Characteristics
figures 5a,b
modulation
both
been
in
carried
The
with
out
figures 5a,b,c,d.
long acquisition
rather
bars.
error
have
shown
are
and
is
main
in
various
The
val-
difference
(few 10~ s)
times
features
the
cases
these
distribution,
which
results
have
are
the
magnetic
field;
the
to d.
a
/BjjdZ(G.cm) AZ(I)
1200
45
1000
1200
45
0.2
c
2000
2400
45
0.5
d
1000
1200
90
0.2
at
with
a
are
45°
=
with
2
better
contrast
normalized
with
a
that
means
are
elongating
strong
a
figures 5,
in
1000
A(t)
curves
with
shown
a
A
on
(d)
to
b
obtained
seen
experiments
the
of
experiments (a)
of the
In
results
flux)
total
get sufficiently small
to
Expt.
I)
I. The
table
ones:
I.
results
w
small
necessary
following
given
experiments
Four
DISCUSSION.
AND
a
I
distribution.
and
generally
emission
about
to
TOF
RESULTS
5.2
the
restricted
been
decrease
the
values
of
in
figure
in
seen
(0.5
and
5b.
It
resp.)
srd
0.2
should
be
the
(b)
is not
the
noted
corresponding
to
respect
of A
w
0.5
same
that
standard
due to the
smaller
in
TOF
value
of
w.
ii)
seen
is
In
figure
that
the
iii)
Data
now
90°.
shown
order
In
and
to
the
the
Tab.
A in
Bp
I).
get
In
that
quenching
case
case
and
could
electric
=
be
field is
90°
in the
(again
but
faster
Bjj
has
than
in
conditions
same
normalized
with
multiplied by
been
2.
It is
(a).
as
in
respect
(b)
the angle a
corresponding
but
the
to
wavelength
the
that
would
characteristic
be
obtained
modulation
of the
(with
the
values
same
one
has
of a,
emitting coherently Lyman a photons and located at
of the multiple wavepackets.
Then the
difference 6(t) between
with Bp is calculated (see broken
in Figs. sa~b,c~d and
curves
unlikely
smallness of
obviously
6
is
that the
0.
It
gets
one
ci
attributed
not
(a)
twice
small.
about
light
centers
without
(d) (a
obtained
in
as
about
patterns I(t)
with
the
as
I(t)
of
same
of A
is very
insight
some
interference
points
same
figure 5d are
magnitude
distribution)
the
are
oscillates
of A
the
before)
as
values
in
Clearly
the
calculated
w
modulation
TOF
standard
conditions
the
5c
well
sources
to
a
light polarization
defined
all
over
the
effect
atomic
insofar
beam
as
the
section,
direction
and
also
of the
because
N°4
ATOMIC
-QUANTUM
PHASE
STUDIES
613
a
b
o
o
~
o
o
o
o
o
j
o
~
£
o
q
~
,
,
£
~
o
~
Z
~
Z
'
o
t
o
,
,
i
,
a
t
t
'
'
,
,
,
o
t
',
'
'
',
o
'
t
,
'
a
t
°
,
,
,
'
,
i
'
'
,
1
,
,
,,
_.
Vt~
2
t/tm
2
~o
d
o
C
o
o
o
oo
°
o
o
o
a
o
a
o
o
o
~
~
o
o
o
o
o
o
o
Z
o
q
o
o
£
,
~
Z
o
o
o
~,
~
o
o
o
I
a
i
o
°
i
o
I
i
a
i
o
i
I
a
'
'
,
o
~
~ ~
~
°°o
°
',
°
,
',
%m
Fig.
Time
5.
TOF
spectra
difference A
refer
a
to
small
small
in
the
of
with
flight
Bjj
between
=
signal
these
is
not
without
conditions
observed
conditions.
of
spectra in
presence
normalized
such
and
with
defined
when
Bjj
strong
the
that
0
spectra
experimental
tj~
Bjj.
maximum
Broken
curve:
value
is I.
fields
Full
Young's
holes
difllerence
A
Bjj. Open
circles
and
calculation.
circles:
full
line:
(a,b~c,d)
table 1.
in
=
longitudinal magnetic
0
:
it is
only the
which
is
particularly
JOURNAL
614
conclusion
In
it
this
Lyman a
experiment is
also
because
does
one
priori
a
N°4
phenomenon
characterized
by
optical emission.
The
theoretical
difficult, not only because of the emission
in
occur
rather
wavelength close
interpretation of
a
the
of the
combined
field
II
interference
an
complexity
of the
profile of electric
that
seems
the
to
PHYSIQUE
DE
induced
present
2p
the
with
emission
process
(2s
process
2p
itself
transition
but
in
a
transition).
ls
Conclusion.
6.
opportunities to study more deeply the wave
easily controlable tools one is allowed
systems.
are
(the
the
incident
projectile)
insofar as the
properties
interferometer
to
concentrate
atom
on
configuration (the target) is well defined at a macroscopic scale.
The ability of producing
beaded
with splittings much larger than the atomic wavelength lead to new
atoms
information
about the
correlation
of atomic wavepackets which can be now
considered
experimentally
as
an
testable
quantity rather than as a mere
theoretical
concept.
Atomic
interferometers
properties
of atomic
provide
with
us
new
interferometers
As these
Acknowledgements.
The
Rubin, City College, New-York, for many
wavepackets in an interferometer,
interesting suggestions for future experiments.
authors
cussions
very
wish
about
to
the
thank
Pr
behaviour
enlightening
K.
of
atomic
as
well
as
for
dhhis
References
iii
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