Interference in BEC

1
Interference in BEC

• Interference of 2 BECs - experiments
• Do Bose-Einstein condensates have a macroscopic
  phase?
• How can it be measured?
• Castin Dalibard solution
• summary

2
Separation between the 2 condensates d
Relative velocity in the x direction d/t
3
Interference fringes
4
giant matter wave interference andrews et. el.
Science 31 / 1/ 1997
closing one slit
5
GPE calculation
Degenerate ground state fi2ppx/h f0
z
r
6
A simple model (Castin 2003)
Initial state
Ka
Kb
There is no relative phase between the two states
gt no interference
Mean density
Conclusion (?) no interference by beating two
condensates with a definite number of particles
(Fock state).
But in one realization we do observe fringes !
7
What happened?
Hint the magnetization of an ideal ferromagnet
(broken ergodicity gt time average ? ensamble
average). in a single realization we will observe
fringes BUT the position of the fringes will be
random from one realization to the next. How to
derive this from quantum theory ? All the
information about the outcome of the experiments
is stored in the N-body density matrix In one
single realization we can only have one outcome
(particle at x1, another at x2 etc..) and the
probability density of that outcome is given by
where O is the projection operator for the
quantity that is measured. In the above example P
is the probability density for finding N
particles at positions X1,..,XN so the projection
operator is
8
Calculation of the 2-body distribution function
Calculation the N body distribution function is
hard. Calculation of the 2 body distribution
function already reveals that there are
correlations. Define the 2nd quantized field
operator
The 2 body distribution function is
and
9

An interference term
When Ngtgt1
10
What is being Measured in the experiment?
In the experiment, we observe the position of the
atoms by sending photons against the expanding
cloud/s. After pinning down the position of the
first absorbing atoms, the next position will be
correlated to theirs. As more and more detections
occur, the correlation is enhanced and we get
one realization of the N-body wave
function. Quantum mechanics only allows us to
calculate the probability to observe a particular
image. The one body density function is the
average over many realizations gt interference is
washed out since the position of the fringes
on each realization (determined by the detection
of the first few atoms) is random from one
realization to the next.

11
Phase states
Definition
A phase state has a well defined phase ? between
the two modes a and b.
Let Na and Nb be Poissonian with the same avrage
N/2
12
Two views on the density matrix
When N is large
The corresponding density matrix is
The same density matrix can be written by using
the phase states
13
Coherent states
14
One more representation for the density matrix
15
Conclusions so far
So does 2 condensates that had never seen each
other have a definite phase between them?

comment any which way information will spoil
the interference !
16
Another example
17
(No Transcript)
18
Operational definition of phase in this
experiment
since, tan2(F)
19
On each realization of the experiment, we have a
random phase, hence phase difference between the
2 condensates. What is the average probability
(over different realizations) ltP(k)gt for k (out
of k) detections in the left detector? In a
single realization with phase difference
it is
So
ltP(k)gt
A non classical behavior since classically this
should go to zero exponentially (like 2-k).
20
We would like to show that when we start with a
definite number of particles (random on each
realization) in each condensate a definite phase
(as defined) will form as we detect more and more
particles. As explained earlier, the distribution
of the number of particles that hit the left (and
right) detectors in ALL the realizations, will be
the same in both cases.
21
second experiment each condensate has a
definite number of particles (assumed equal for
convenience)
The experiment reveals the relative phase
2
22
if the left detector clicks for the first time,
we know that there are 2 amplitudes for that
The opposite detector is an orthogonal state so
it is equal to
23
What is the probability amplitude for a second
click on left and right detectors?


Photon bunching given a first click at the
left detector, the probability for a second
click in the same detector is 3 times larger
than the probability of a (2nd) right click
(when Ngtgt1).
Remark Feynmans intuition.
24
Lets simulate what happens


25
When repeating this experiment several times we
observe that on any realization there will be a
different phase, but over all (the number
realizations) the phase will be random. Also,
the time averaged phase does not equate with the
ensemble average (broken ergodicity).


26
Continuous measurement theory a brief review
So the probability for at least k detections at
arbitrary times is
27
Why N times G ? Because the probability to of a
given particle to be emitted is Gdt
(dtltlt1). Monte-carlo wave function simulation

Move ahead in time using UeiHdt (H
non-hermitian).
check what is the probability of emission 1-
Begin with ?0
Choose randomly if there is a jump and where
(left/right).
No jump
Normalize the state vector
jump
Operate with the left/right jump operator to get
to the new state
28
Back to
Jump operators (2)
The probability density for at least k detections
at times
And at either of the detectors

The probability density for at least k detections
at times
29
So the probability of getting at least kkk-
detections after a long time, when the first k
measurements have k and k- counts is
30
Lets calculate the probability of (k,k-)
detection
For coherent states after each measurement the
condensates maintain their relative phase since a
coherent state is an eigenstate of the
annihilation operators. Each count occurs with
probability . Given
k detections we have
For an initial state that has well defined total
number of particles
To analyze the evolution of this state due to the
measurements we use the (over complete) phase
states defined as
31
The phase state is almost orthogonal for large
N
32
Using the formula () we get
Expand
where
Now we calculate the state

33
Using the almost orthogonallity for large N we
get
where
34
Castin Dalibard 1997




35
Vortex Interference
36
Vortex Interference