PHYSICAL
REVIEW
A
VOLUME
44,
NUMBER
9
1
NOVEMBER
1991
Atom-
and field-state
evolution
in
the
Jaynes-Cummings model for
large
initial
fields
Julio Gea-Banacloche
Department
of
Physics,
Uniuersity
of
Arkansas,
Fayetteuille,
Arkansas,
72701
and Instituto
de
Optica, Consej
o
Superior
de Inuestigaciones
Cientificas,
Serrano
121,
28006
Madrid,
Spain
(Received 19 March
1991)
An
asymptotic
result
is derived for
the
Jaynes-Cummings model of
a
two-level
atom
interacting with
a
quantized
single-mode
field,
which
is valid
when the field is
initially
in
a coherent
state
with
a
large
aver-
age
photon
number.
It is
shown that for certain
initial atomic
states the
joint
atom-field
wave
function
factors into an
atomic
and
a
field
part
throughout
the
interaction, so that each
system
remains
separately
in
a
pure
state.
The atomic
part
of the
wave function
displays
a
crossing
of
trajectories
in the
atom
Hil-
bert
space
that
leads
to a
unique
state
for the
atom, independent
of
its initial
state,
at
a
specific
time
to
(equal
to
half
the revival
time).
The field
part
of the wave
function
resembles
a
crescent
squeezed
state.
The
well-known
collapses
and revivals
are
investigated
from this
perspective.
The
collapse
appears
to
be
associated with
a
"measurement"
of
the initial
state
of
the atom with
the field
as
the
measuring ap-
paratus.
The
measurement is not
complete
for finite
average
photon
number: the
system
is instead left
in
a
coherent
superposition
of
macroscopically
distinct
states. At the
half-revival
time
to
this
superposition
of
states
is
entirely
in
the field
part
of
the
state
vector,
so that the
(pure)
state of the field
at that time
is of
the
form sometimes referred
to as a
"Schrodinger
cat.
"
The
revivals of the
population
inversion
are seen
to be
entirely
due to the fact that
the linear
superposition
of the
two
macroscopically
distinct field
states
is coherent
(i.
e.
,
a
pure
state),
as
opposed
to
an incoherent
mixture.
PACS
number(s): 42.50.
—
p,
03.
65.
Bz,
42.52.+x
I. INTRODUCTION
AND OVERVIEW
The
Jaynes-Cummings model
(JCM)
[1]
is
one of the
simplest
of
quantum-electrodynamical
systems:
a
two-
level
atom
interacting
with
a
single
mode of the
quan-
tized
radiation
field,
in
the
so-called
rotating-wave
ap-
proximation
(RWA).
In addition
to its
being
exactly
solvable,
it
has become
increasingly
well
approximated
by
recent
experiments
involving
the
passage
of
single
atoms
through
superconducting
microwave
cavities
[2].
In
spite
of its
apparent
simplicity,
it
has
provided
theorists
with
many
nontrivial
and
unexpected
results
through
the
years, beginning
with
the
well-known
phenomena
of
col-
lapses
and revivals of
the atomic
population
inversion
[3,
4].
Much of
the
recent
research
has
focused
on
the
squeezing
of the field
predicted
by
the model
[5,
6).
One of the
most
interesting
features of
the JCM
is that
it
involves
two
systems
in
interaction, with
every
feature
of the
interaction
and of each
system
described
quantum
mechanically:
there
are
no
external,
c-number-type
forces
or
potentials.
(In fact,
one reason
why
the
model
was
originally
introduced
[1]
was
to find out in which
way
the
quantization
of the
field affected
the
predictions
for the evolution
of the
atom,
that
is,
to
compare
with the
semiclassical
theory,
where the field is
not treated
as a
quantized
variable.
)
It is
therefore
possible
to use
the
JCM to
investigate
many
issues of interest
involving
in-
teracting
quantum
systems,
quantum
correlations and
en-
tanglement,
and
perhaps
even
state
preparation
and
mea-
surernent.
Especially
noteworthy
is the fact that one of
the
two
systems,
namely,
the
field,
has a
well-defined
clas-
sical
limit,
which is
usually
taken
to be a
Glauber
coherent
state
[7]
with
a
very
large
number
of
photons.
Not
many
previous
studies have
emphasized
this
as-
pect
of
the JCM
[8,
9].
Among
them, as
especially
relevant
to
the
present
paper,
one
must
note the research
of
Phoenix and
Knight
[10],
who used the
entropy
to
study
the correlations between
the field
and atom
as
well
as the
"purity"
of the
state
of
each.
They
showed
that
the
field
in
the JCM
was
essentially a
two-state
quantity
[11].
Their numerical
calculations
with
states with
not-
too-large numbers of
photons
also
show that the field
(and
hence the
atom) most
closely
returns
to
a
pure
state
somewhere within the
so-called
"collapse
region.
"
It was
recently
shown
by
the
present
author
[12]
that
for
a
large
average
number of
photons,
if the initial
state
of the field is
a coherent
state,
the
states
of
the atom
and
field become in
fact
arbitrarily
pure
at a
specific
time
to
equal
to
one-half
the
conventionally defined
"revival
time"
t~;
that
is,
the
atom
and field
spontaneously
be-
come
disentangled
at this time.
Moreover,
it
was found
that the
state
of the
atom at the time
to
is
completely
in-
dependent
of the initial
atomic
state. One of the
purposes
of
the
present
paper
is
to
investigate
the
consequences
of
this
result,
as
well
as
to
present
a
more
detailed and
care-
ful
proof
than
could be
given
in
Ref.
[12];
this
is done in
Sec. II
and
Appendix
A,
respectively.
It turns
out
that for
large
numbers
of
photons
the
most
convenient basis of
atomic
states is not
provided
by
the
energy
eigenstates
but
by
special
states
~+
)
and
~
—
),
defined in
Sec.
II,
which are
eigenstates
of the
semiclassi-
cal
interaction Hamiltonian
(assuming,
as.
will
be
done
throughout
this
paper,
exact
resonance
between
the
atom
and the
field). These
states, as
shown
in this
paper,
have
the
remarkable
property
that if
the
atomic
system
is
ini-
tially
prepared
in
one of
them,
the state vector for the
44 5913
1991
The
American
Physical Society
5914
JULIO 6
EA-BANACLOCHE
atom-field
system
at
later times factors
out,
to
a
good
ap-
proximation,
into
an
atom
part
and a
field
part;
that
is,
the two
systems
do
not become
entangled
but remain
each in an
approximately
pure
state.
In
spite
of this
property,
however,
the
evolution
of
each
(atomic or
field)
part
of the wave function under
these
circumstances is
decidedly
nontrivial,
and in
particular
nonunitary,
in
the
sense that initially
orthogonal
atomic states
do
not
remain
orthogonal
in
the course
of
their
evolution.
(The
total wave function
for
the
joint
atom-field
system
does
evolve
unitarily,
of
course,
since
this
is
taken
to
be
a
closed
system.
)
The
result is
like
a
"crossing
of trajec-
tories"
in
the atom Hilbert
space
at t
=tp.
This is
dis-
cussed and
explored
in
Sec. II.
The
field
part
of
the
wave function that evolves
when
the initial atomic state is
one of
the
states
~+
)
or
~
—
)
also has
interesting
properties,
which
are
studied
in some
detail in
Appendix
B;
it
corresponds to a field
state
which
(always
in
the limit
of
very large
number
of
photons)
has
a
large
amplitude
and
becomes
squeezed,
as time
passes,
in
a
way
analogous
to
the
"crescent
states"
discussed
by
Yamamoto and co-workers
[13,
14].
The field state
corre-
sponding
to an initial
atomic
state
~
+
)
(~
—
))
has
a
ma-
croscopically
well-defined
phase
which
grows
(decreases)
with
time;
in the
field's
phase
space
the
two
states are
represented
by
counterrotating
phasors.
This
phenomenon underlies
the
recent
observations
of
Eiselt
and Risken
[15,
16]
(see also
Refs.
[17]
and
[18])
regarding
the
quasiprobability
distribution for
the
field
in
the JCM
when the atom
is
prepared
in
a
state
(such
as
the
energy
eigenstates) which is
a
linear
superposition
of
~+
)
and
—
).
When
this fact
is taken
together
with
the
observa-
tion that
all initial
atomic
states
lead
to the same
atomic
state as
tp,
it follows
that
the state
of the Geld
at this
time,
for an initial
atomic
energy
eigenstate,
is
in fact
a
coherent
linear
superposition
of
macroscopically distinct
states
(states of
opposite
phase);
such
a
superposition
has
been
dubbed a "Schrodinger
cat"
[19].
It is
analyzed
in
Sec.
IV,
where it
is shown
that
the
signature
of
the
mutu-
al
coherence
between
the two
macroscopically distinct
parts
of the
field
wave function
at the
time
t
p
is
none
oth-
er than the well-known
revival of
the
population
inver-
sion
at
the
later time
tz.
Section
III
deals
with
another
interesting
question.
As
said
above,
the
two
orthogonal
initial
atomic states
~+
)
and
~
—
)
lead to
two field states
which
become in
time
macroscopically distinct.
Hence,
information
about
the
initial
state
of
the atom
becomes stored in
the
field, which
is a
potentially
macroscopic
system
with a
classical
limit.
Does this
mean
that a measurement
of the
initial
state
of
the atom
has
been carried
out? The
analysis
in
Sec.
III
points
to
a number of
interesting
analogies between this
process
and
some
aspects
of
the
quantum
theory
of
mea-
surement;
it
is
shown,
for
instance,
that at
the time
of the
well-known
JCM
collapse
the
reduced
density
operator
for
the
atom
becomes
diagonal
in
the
basis of
the states
~+
)
and
~
—
),
and that this
collapse
takes
place
as soon
as
the
two field
states
associated with
~+
)
and
~
—
)
be-
come
macroscopically
distinguishable.
Whether
it is
con-
sistent or
not to
postulate
that
a
"collapse
of the
wave
function,
"
in
the
sense
of
quantum
measurement
theory,
takes
place
at the
traditional
collapse
time
is
a
question
dealt with
briefly
in
Sec.
III
also.
In
any
case,
for
a
finite
number of
photons
in
the
field it is clear
that
one
could
have at most
a sort
of
"incomplete
measurement,
"
that
is,
the total
system
is
left in
a
coherent
superposition
of
ma-
croscopically
distinct
states. The
spontaneous
disentan-
glement
of
atom
and field
that takes
place
at
tp
causes
this
superposition to afFect
only
the field
part
of the
state
vector
at
this
time,
which
results in
the
Schrodinger cat
discussed in
Sec. IV.
In
addition
to the results
that
have
been
summarized
above,
the
present work
brings
together,
in
a
special
lim-
it,
many
observations
made
by
a
large
number of
people
over
the
years,
showing
how in
this limit
all
these
obser-
vations
Inay
be
derived from
a
simple
expression for
the
atom
and
field
evolution.
It
is
quite possible that the
re-
sults
of this
paper
might
lead to
further
insights
in
other
aspects
of
the
JCM
not
covered
here, as well as
in
the
general
study
of the
dynamical and
coherence
properties
of
"open"
quantum
systems,
and
possibly
also in
quan-
tum
measurement
theory.
A brief
outlook
on
some
possi-
ble
directions
in
which
further
research
might
proceed
is
presented
in
Sec. V.
II.
NONUNITARY, QUASI-PURE-STATE
EVOLUTION
IN
THE
JCM
A. General
results
This
section
introduces
the JCM
and
presents
the
main
analytical
result
of
the
paper.
The JCM
evolution
turns
out to
be
very
simple,
for
large average
number
of
pho-
tons,
if
one
looks
at
special
initial
states of the
atom,
cor-
responding
to well-defined
values
of
the
dipole
moment
amplitude.
The
evolution
of
any
other
initial
state
may
be understood
as
a linear
superposition
of
these.
The
JCM
involves
only
two
interacting
quantum
sys-
tems:
a
two-level
atom,
whose
upper
and lower
states
may
be
written,
respectively,
as
~a
)
and
~b
),
and a
single
mode of the
quantized
electromagnetic
field,
whose
an-
nihilation
and
creation
operators are
denoted
by
a and
a .
Assuming
exact
resonance, and
performing
the
rotating-wave
approximation, the interaction
Hamiltoni-
an
adopts
the
simple
form
Ht
=fig(~a )(b~a+a"
b
)(a
~),
where
g=d(co/RVeo)'
is a
coupling
constant
(d
is the
atomic
dipole
matrix
element
for
the
transition, co
is the
transition
frequency,
and V the
mode
volume).
The
exact
solution
for
an initial
atomic
state
~g(0))„,
=n~a
)+P~b)
and
field
state
~i/'(0))„„~
=y.
„",
C„~n)
is
~g(t))=
g
[[aC„cos(g&n+lt)
n
=p
i/3C„+,
sin(g&—
n
+
lt
)]~a )
+[
iaC„,
sin(g&—
n
t)
+PC„cos(gv'n t
)
]
ib
)
]
~
n
)
.
(2)
ATOM-
AND
FIELD-STATE
EVOLUTION
IN THE
JAYNES-.
.
.
5915
In
general
the
state described
by
(2)
is
a
highly
entan-
gled
state
of the
field
and
the atom. For
an initial
coherent
state field
with
large
photon
number,
however,
the result
simplifies
considerably
if
one
looks
at
the
evolu-
tion of the
initial atomic
states
~
+
&
and
~
—
&,
defined
by
~+&=
—
(e
'~la
&+lb
&),
2
(3)
where
P
is the
phase
of
the
initial
field
coherent
state
~
v
&:
(4)
(the
average
number of
photons
is
clearly n
=
~
v
~
).
Formally,
what is
special
about the states ~+
&
is
that
they
are
eigenstates
of
the
"semiclassical
(sc)
Hamiltoni-
an"
which
is
obtained
when
replacing
the
operators
a
(a
)
in
(1)
by
the
c-number
field
amplitude
v
(v
*):
H,
=A'g(~a
&(b~v+v*~6
&(a~)
.
(5)
~+&~v&~,
=,
~
—
(e
'&e
''"
"~a&+~b&)
2
—
n/2
—
n
/2
~
—
I'np
—
Igt+n
I
„~,
&n|
(6a)
and
the
corresponding
equation
for the evolution
of a sys-
tem
prepared
in the initial state
~
—
&
v
&:
Therefore,
in the
classical
limit,
these
states do
not
really
evolve
at
all
(except
for an overall
phase).
With
a quan-
tized
field,
as
in
the
Hamiltonian
(1),
they
are
found
indeed
to
evolve,
but
in
a
way
which,
in
the
limit of
large
n,
is
very
simple,
as
will
be
shown
presently.
Physically,
the
states
~+&
correspond
to
well-defined
values of the atomic
dipole
amplitude
which are either in
phase
or
180'
out
of
phase
with the
applied
field,
which is
the reason
why,
classically,
no
exchange
of
energy
with
the field takes
place.
The
fact that these
states do
not
evolve in the semiclassical
theory
was
pointed
out
by
Puri
and
Agarwal
[20],
who
suggested
that
they
could be used
as
sensitive
probes
of the differences between
semiclassi-
cal or neoclassical and
fully
quantized
theories
[21];
Zaheer and
Zubairy
[22]
referred
to
them
as "trapping
states"
and studied the
emission
spectrum
for an atom
in-
itially
prepared
in
one of these states.
[Other
"trapping"
states
in the
JCM for other
(not coherent)
initial
field
states have been
studied
in
Ref.
[23].
]
This
paper
s
main
analytical
result is the
following
ex-
pression
for the time
evolution of the
atom-field
system,
when the initial state is
~+ &
~
v
&
[with
~
v
&
given
by (4)],
in the limit as n
—
+
~:
The
result holds for
any
finite
time,
and even for
t~
~,
provided,
as
the derivation
in
Appendix
A
shows,
that t
go
to infinity
slowly enough
to
have
t/n~0. This last
provision
is
important,
because,
as one
can
see
from
Eq.
(6),
as n
~oo
the evolution
of the
atomic
part
of the wave
function becomes
"infinitely
slow";
in
particular,
it is
well
known
[3,
4]
that
the
relevant
time
scale
for the JCM
re-
vivals
is
tie
=
2v—
r+n
jg.
Since,
however,
t~
In~0
as
n
~
oo,
the
result
(6)
holds
accurately
over
an arbitrarily
large
number of revivals, provided
n
is
large
enough.
In
practice,
for
a
finite
time,
the results
in
Appendix
A
may
be
used to
estimate
how
large
n has to be
in order
for the
asymptotic
solution to
be a
good
approximation
over that
time. For atomic
operators,
whose
expectation
values are
typically
of the order
of
unity,
the
error in
the
expectation
values calculated
using
the
asymptotic
ap-
proximation
should
not
be greater
than the norm
of the
terms
neglected.
In
fact,
numerical calculations
(to
be
presented
elsewhere)
show
this
to be a
rather
conserva-
tive
estimate,
and that the
asymptotic
approximation
is
remarkably
good
over
one revival
time even for
photon
numbers as
small
as,
say,
n
=25
(compare
also the
results
in
Ref.
[12]
for n
=49).
This
is
fortunate from the
point
of view
of the
possibility
of eventually observing
some
of
these
results
experimentally,
which is
being
studied
presently
and
will
be
discussed at
length
in
a
later
publi-
cation. One
may
note in
particular
that
for such
low
photon
numbers the
Rabi
frequency
g+n
is
typically
much
smaller than
the atomic
transition
frequency
~
(e.
g.
,
for the
recent
micromaser experiments
of
the
Mun-
ich
group
[2],
g/2m.
=44
kHz
and
c/02m.
=21.
5GHz),
which means
that the
RWA used to
write the
Hamiltoni-
an
(1)
would be
quite appropriate
to
describe
the
atom-
field interaction (for
a treatment
of
the
atom-field
interac-
tion
without
the
RWA,
see
Ref.
[24]).
The
main restriction
to
the usefulness of
Eq.
(6)
stems
from
the fact
that some
of the terms
neglected
corre-
spond
to
vectors
whose
norm
squared
may
go
to
zero
only
as
fast as 1/n
If
the s.tate
vector
(6)
is
used to
calcu-
late expectation
values of
field
operators,
the result
may
not be
quite
reliable if for
some reason the
leading
order
in
n vanishes;
thus
(
a
&
and
(
a
&,
for instance,
will
be
correct to
leading
order
in
n,
but
the difference
(a
&
—
(a
&
may
not
be
given
correctly
by
(6)
if the
lead-
ing
order
in n
cancels,
as
is the case
when
calculating
squeezing.
In
spite
of
this,
the
vectors
~
n/2
~@+(t)&=e
n/2
y
—
e
inge+igtVn~n
&
,
&n)
~
—
&~v
&~
~
(e
'&e'~'
"~a
&
—
~b
&)
1
2
—
n/2
Xe
""
y
"
'"&e''
"~n&.
,
&n!
(6b)
The
proof
of
Eqs. (6)
is found in
Appendix
A. The
result
is
rigorous
in
the
following
sense: the
difference
between
the exact solution
(2)
and the
right-hand
side of
(6a)
[or
(6b)]
is a
vector
whose norm vanishes in
the limit
n
~
oo.
appearing
in
Eq.
(6)
are useful
to
predict,
at
least
qualita-
tively,
many
properties
of
the field in
the
JCM in
the
large
IT limit. This
is further
discussed
in
Appendix
B.
Since the states
~+
&
and
~
—
&
form
a basis set
for
the
atom,
the evolution of
any
other
initial state can
be
ex-
pressed
as
a
simple
linear
combination of
(6a)
and
(6b).
For
instance,
if
the atom is
initially
in the excited
state
~a
&
=e'~(~+
&+
~
—
&)/&2
the
total
state
vector evolves
into
5916
JULIO
GEA-BANACLOCHE
—
(e
's'
"
a
)+e'~Ib
) )le+(r))
2
(8)
are
rather
special
themselves,
i.
e.
,
far
from generic.
It
will be
shown
in Sec.
III that
this
may
not
necessarily
be
the
case.
B.
Atomic-state
preparation
at
t
=
f
p
As
will be
shown later,
the field
states
I@+(t)
),
given
by
Eq.
(7),
correspond
to
a
macroscopic
field
rotating,
in the
phasor
plane,
clockwise and
counterclockwise,
respec-
tively.
Thus
Eq.
(8)
exhibits the
splitting
of
the field
quasiprobability
distribution
found in
Refs.
[15
—
18].
This
point
will
be
discussed
at
length
in the
following
sections.
Returning
to
the
evolution
equations
(6),
their most
re-
markable
property
is that the states
appearing
in them
are
product
states
[unlike,
e.
g.
,
the
general
form
(1)
or
the
special
case
(8)].
This means
that,
in the
limit
of
large
n
in which
(6)
is
valid,
both
the
field and
the atom
remain
in
pure
states throughout
the
interaction
if the atom
is
ini-
tially prepared
in
one of the
states
I+
).
Clearly,
no
other
atomic
states
have
this
property.
Equation
(6)
implies
that
one can at
all
times
assign
a
well-defined
state
to
the atom
prepared
in
one
of the
states
I+
);
yet
the
evolution of
such a state
is
nonunitary.
This
is seen
in its
most
extreme instance
at the
special
time
to=
t~
/2=m+—
n
Ig,
where
the atomic
wave
func-
tions
in
(6a)
and
(6b)
become,
in
fact,
identical
(except
for
an overall
minus
sign)
and
equal
to
lou&=
-(
—
ie
'~la
&+lb&),
Thus atomic
systems
prepared
initially
in the
orthogonal
states
I
+
)
and
I
—
)
find themselves in the
same
atomic
state
at
the time
to,
while
remaining
throughout
their
evolution,
to a
good
approximation,
in a
pure
state. This
is
in
contrast with normal
unitary
evolution,
which
preserves
orthogonality
between states. Here neither
the
field nor the
atom
separately
evolve
unitarily,
even
though
a state
vector for each of them is well defined
at
all
times.
[The
total
wave function,
naturally,
does
evolve
unitarily,
since
the
atom-field
system
is assumed to
be
closed.
This
means,
in
particular,
that the
orthogonality
lost
by
the
atomic state must be taken
up
by
the field
state
at
the time
to,
i.e.
,
that
the
states
l@+(to))
and
(to)
)
must
be
orthogonal,
which,
in
the
limit
n
~
oo,
they
are.
]
Clearly,
the
nonunitarity
arises from the fact
that
nei-
ther the
atom nor the
field alone is
a
closed
system;
the
remarkable
fact is
that,
in
spite
of
this,
a
well-defined
state vector
for
each of them
exists
at
all times.
One is
tempted
to consider an
analogy
with
classical mechanics:
there,
the
trajectories
of
a
conservative
system
cannot
cross,
whereas for
a dissipative
system,
on the other
hand,
it is
quite
common
for
trajectories
to,
e.
g.
,
con-
verge
to a
point
attractor
(or
to more
complicated
attrac-
tors).
What
the
present
example
illustrates
is
an
instance
of
"crossing
of quantum-mechanical
trajectories"
for an
open
quantum system.
The
analogy
is
intriguing,
and it
might
be worth
investigating
whether
this kind
of
phenomenon
might
be
common in
open
quantum
systems
in
general.
One
might
think that the atomic
"trajectories"
that
have the
special
states
I+)
and
I
—
)
as
starting points
Since atomic
systems
prepared
initially
in either of the
two basis states
evolve towards the same state
at
t
=
to,
this
means
that in
every
case,
regardless
of the
initial
atomic
state,
at
the time t
=to the atom wi11
be
in
a
pure
state,
and
that
state,
independent
of the
initial
conditions,
is
I/0)
given
by
(9).
This
was recently pointed
out
by
the
present
author
[12].
(As
mentioned in
the
Introduction,
Phoenix and
Knight
[10]
had
previously
noticed,
in
cal-
culations
involving
not-too-large
values
of
n,
that
if
the
initial states of the atom and
field were
both
pure,
they
were
both
closer to
pure
again
around the time
to.)
This means
that
the JCM could be used
to
prepare
a
specific
atomic
state,
at
a
given
time,
in a
way
which is
completely
independent
of
what
the initial state of the
atom
might
be.
It
is
perhaps
worth
emphasizing
the
no-
velty
of this result.
Semiclassically, a
state
such
as
lgo)
could be
prepared
by
a
~/2
pulse,
beginning
with an
atom in the
ground
state;
but
if
the atom
is
initially
in
the
upper
state
instead,
the
same
m.
/2 pulse
would
yield
not
the state
lgo)
but one
orthogonal to
it
(by
unitarity).
For
the quantized-field
system
considered
here,
however,
the
ground
and excited
states,
as well as
any
linear
combina-
tion
of
them,
all
evolve
towards
I
i)'jo)
at
t
=
to.
In
fact,
the state
of the
atom
at
t
=to is
the
pure
state
I/0)
even
if
initially
the
atom
was not
prepared
in a
pure
state at
all.
Suppose
in fact that the
initial atomic state
is
described
by
a
density
operator
p„(0),
which,
without
any
loss
of
generality, may
be
taken to be
diagonal
in
some
basis
[lg,
),
lg2)].
Then,
the
total
initial
state
for
the
system
will
be
p.
..
(0)=p„lq,
&lu
&&ul&q,
l+p„ly,
&lv
&&ul&y,
l
.
(10)
But,
since the states
I@i)
lu )
and
If@)
lv )
must evolve,
respectively,
toward states
I
@0)
I/i
)
and
I
@0)
I
Pz)
at time
to
(where
I
P,
)
and
I /2
)
are
some
orthogonal
field
states),
the total state for the
system
at
to
will
be
p
.
(
to
)
=
I
pi
& &
&OI
(p
i i
I
pi
& &
p i
I
+
p22I
&~
& &
&2I
),
which
again
shows
the atom
to be
in
the
pure
state
I
go),
and
the
field,
incidentally,
in
a mixed
state.
In
terms
of
entropy,
one
could
say
that
all the
entropy
of the
atom
at
the time
t
=0
is transferred
to the field
at
t =to.
One can
think of
the
following
optical
analogy.
The
polarization
states of a
photon
may
be described
in
a
two-dimensional
state
space
entirely analogous
to
that of
the
two-level
atom. In terms
of
photons,
the
present
de-
vice
might
be
likened
to
an
ideal
polarizer
that would
transmit
every photon
with unit
probability
and,
regard-
less of
their initial
polarization,
place
them all
in
the
same
polarization state
upon
transmission. No
such
de-
vice exists for
photons,
yet
the
present study
shows how
for
a
two-level
atom one could
conceivably
be
built.
Note also that this
hypothetical
ideal
polarizer
would
work
even if
the
initial
photon
did not have a
definite po-
ATOM-
AND
FIELD-STATE EVOLUTION
IN THE
JAYNES-.
. . 5917
larization
at
all;
for
instance,
if it were a
member
of
a
pair
in
an
entangled
polarization
state,
such as
those used
in
the experimental
tests of
the Bell
inequalities
[25].
For
the atom
in the
JCM,
this
is also
easy
to
verify explicitly
from
the
results
presented
so
far: Even
if
at
t
=0
the
atom is
an
entangled
state with some
other
system,
at the
time
to
it is
in
a
pure,
nonentangled
state;
all
the
entan-
glement
is
transferred to the field.
Consider,
for
instance,
the
evolution of the initial state
le(0)
&=
—
(l~
&I
&
&+
Ib &I&
&)e
IU
&,
2
(12)
where
~
A
&
and
~8 &
are states of
some
hypothetical
third
system,
which
may
even be
infinitely
distant.
At t
=0
no
pure
state
exists for the
atom,
yet
at
to,
the state
of the
total
system
(including
the
field) is
naturally
le(ro)
&=lgo&
—
(IP.
&l~
&+IN
&I&
&),
2
(13)
where
~P,
&
and
~Pi,
&
are
the
(pure)
field
states
which are
obtained at
to
when the atom is
initially
prepared
in
~a &
and
~b
&,
respectively. Equation
(13)
indeed shows the
atom in a
pure
state and the field
entangled
with the
dis-
tant
system.
From
these
examples
one can see that the
JCM in the
limit
n
~
ao
becomes an ideal device
to
perform
a state
preparation
on a
quantum
system
(the
atom),
regardless
of
this
system
s
initial
state or
previous entanglement
his-
tory.
This is
a
unique
example,
to the
best
of this
author's
knowledge.
One
must mention in this context
the work of
Meystre
and
co-workers
[9],
who showed
in
a
system
also
evolving according
to the JCM
dynamics
how
a
pure
state
might
be
prepared
under certain
condi-
tions. Their
system
is,
however,
the
quantized
elec-
tromagnetic field,
whose
space
of states is much
larger
than
the
two-level
atom's,
and thus
a
specific
state
can
be
prepared
only
for initial
states
which
satisfy
some ap-
propriate
conditions. For
many
initial
states,
in
fact,
convergence
to a
pure
state
does not occur
in the
Meystre-Slosser-Braunstein
system.
To conclude this
section,
one
may
remark
again
that
even
though
all the initial atomic states evolve towards
the
pure
state
~
$0
&
at
t
=
to,
only
when the
initial
state is
one of the
special
states
~+
&
and
~
—
&
does the atomic
system
remain in
a
pure
state
throughout
its
evolution.
For all other
states,
a
total or
partial
"collapse
of the
state
purity"
takes
place
at
the conventional JCM
"col-
lapse
time"
t,
—
1/g
of
the
Rabi
oscillations,
as
was
illus-
trated
in Ref.
[12]
(Fig.
1;
see
also
Ref.
[10]).
One
is
tempted
to
inquire,
in
view of the
apparent
loss
of
memory
exhibited
by
the
atom,
at
t
=
to,
regarding
its
ini-
tial
state,
whether
a
collapse
of the wave function in the
sense
of
the
quantum theory
of measurement takes
place
around t
=
t„with
the
information
about the
initial state
of the
atom
being
stored
in the field. In
fact,
one
may
wonder whether in some sense a
measurement of the
ini-
tial state of the atom can be
said
to be
carried out
by
the
field
at the time of
collapse
t„'
the
field,
from
this per-
spective,
would be
the
"apparatus"
which does
the
measuring
and stores the
information. This
intriguing
hypothesis
is
thoroughly
explored
in
the
following
sec-
tion,
where the
analogies
with
quantum
measurement
theory
will indeed
be
seen
to
be
many
and remarkable.
III. THE JCM
COLLAPSE
AS
A
QUANTUM
MKASURKMKNT
A.
The measurement
analogy
gti/n
=gr
v'
tT +
—
t
—
—
r+.
. . .
g
n
n—
g
(n
n)—
2
Q„—
8
(14)
For all the
number
states
having
an
appreciable weight
in
the sum
(7),
one
expects
(n n)
to
—
be
of the order
of,
or
smaller
than,
n;
hence,
as
long
as t
«
tz
=2m'1/n
/g.
,
the
third term
on
the
right-hand
side of
(14)
and
all the
higher-order
terms
may
be
ignored
to
yield
oo
—
n
/2
~Ct+(t)
&
=e
"
g
e
'"~e+'g'
"
~n
&
,
i/n!
—
n/2
=e
+igt
t/
n
/2
—
n/2
~
n
—
in(p+gt/2"(/
n
)
„~,
&n!
e
TigtV n
/2!
Ue
Tigt/2V n
g
(15)
where, as
in
Eq. (4),
U
="t/n
e
'~,
and the notation
in
the
last line means a
coherent
state with the
complex
ampli-
tude shown.
It
follows, then,
from
Eq.
(6)
that,
for the
time
considered,
the evolution of
a
system
prepared
ini-
tially
in
one
of the states
~+
&
~U &
may
be
written
as
~+&~ &
Tigt
n/2
~
(
e
T
igt
/2+
n
(
t2
&
+.
~
b
& )
~
Ue
V
igt
/2
+
n
&
Tigt
t/n
/2(+
&
(
+igt/2i/n
&
(16)
The
phase
factor
neglected
in
going
from
the first
line
to
the
second
line of
Eq.
(16)
is
very
close
to
unity
for
times
t
«
tz.
Around
the
collapse
time
t„
the
phase
in
ques-
tion is of
the order
of
gt,
/2n
'/
—
I/n
'/,
and
thus,
for
large
average
photon
number,
negligible.
The same
phase
factor e
+' ' "
cannot, however,
be neglected
in
the field
coherent
state
in
(16),
due
to the
large
(propor-
tional
to
+n
)
amplitude
of that
field,
as
will
be explained
It
has
long
been known
[3,
4]
that for an
atom
prepared
initially
in
one
of the
energy
eigenstates
the JCM
predicts
Rabi oscillations of
decreasing
amplitude,
which all but
vanish around
the
so-called
collapse
time
t,
—
1/g
[more
precisely,
the
oscillations
have
a
Gaussian
envelope
2t2
which
decays
as e
g
';
see
below,
Eq.
(20)].
Partial
re-
vivals of these oscillations
take
place
around the revival
time
tlt
=2m')/
n
/g
introduced in
the
preceding
section.
Note
that for
large
n
the
collapse
and
revival involve
completely
di8'erent
time
scales,
with
t,
«
tz.
The
short-time
evolution
of the
JCM,
where short
means t
«
t~,
but,
possibly,
t
)&t„can
be
easily
ob-
tained from
the
expressions
(6a)
and
(6b).
The
field
states
~@+(t)
&
[see Eq.
(7)]
are
in this
limit, to
a
good
approxi-
mation,
very
similar
to
coherent states: one has