Transcript Lecture 1
Atomic entangled states with BEC
A. Sorensen
L. M. Duan
P. Zoller
J.I.C.
(Nature, February 2001)
KIAS, November 2001.
SFB Coherent Control
€U TMR
Entangled states of atoms
j ª i6= j ' 1 i
j '2 i
:::j ' N i
Motivation:
• Fundamental.
• Applications:
- Secret communication
- Computation
- Atomic clocks
Experiments:
• NIST: 4 ions entangled.
• ENS: 3 neutral atoms entangled.
E
E
'
'
4
3
This talk: Bose-Einstein condensate.
E
'
103
Outline
1.
Atomic clocks
2.
Ramsey method
3.
Spin squeezing
4.
Spin squeezing with a BEC
5.
Squeezing and atomic beams
6.
Conclusions
1. Atomic clocks
To measure time one needs a stable laser
click
The laser frequency must be the same for all clocks
Innsbruck
click
Seoul
click
The laser frequency must be constant in time
click
Solution: use atoms to lock a laser
!L
!0
detector
feed back
In practice:
Neutral atoms
ions
!
=
!
+
±
!
L
0
frequency
fixed
universal
Independent atoms:
Entangled atoms:
1
±
!=p p
tn
r
e
pN
1
±
!
=
p
e
n
t
tn
f
(
N
)
r
e
p
• N is limited by the density (collisions).
• t is limited by the experiment/decoherence.
• We would like to decrease the number of repetitions (total
time of the experiment).
Figure of merit:
• To achieve the same uncertainity:
±
!
±
!
e
n
t=
p
»2 =
(nrep )ent
Tent
N
=
=
nr ep
T
f (N)
2
We want »
¿1
2. Ramsey method
• Fast pulse:
single atom
1
j
0
i! p
(
j
0
i+
j
1
i
)
2
• Wait for a time T:
single atom
1 ¡
i
(
!
¡
!
)
t
0
L
!p
(
j
0
i
+
e
j
1
i
)
2
• Fast pulse:
single atom
· ¸· ¸
1
1
(
!
¡
!
)
t
(
!
¡
!
)
t
s
i
n
j
0
i
+
c
o
s
j
1
i
0
L
0
L
2
2
• Measurement:
# of atoms in |1>
·
¸
1
2
P
=
c
o
s
(
!
¡
!
)
t
1
0
L
2
Independent atoms
Number of atoms in state |1> according to the binomial distribution:
where
If we obtain n, we can then estimate
The error will be
If we repeat the procedure we will have:
Another way of looking at it
Initial state: all atoms in |0>
First Ramsey pulse:
Jz
Jz
Jy
Jy
Jx
Jx
Free evolution:
Measurement:
Jz
Jz
Jy
Jx
Jx
Jy
In general
2
N
(
¢
J
)
z
»
= 2
2
h
J
i+
h
J
i
x
y
Jz
2
Jy
where the J‘s are angular momentum operators
Jx
N
X
(k
)
J
j®
®=
Remarks:
k
=
1
2
¿
1
• We want »
2
• Optimal: »
¸1
=
N
2
• If »
<1then the atoms are entangled.
That is,
X
½
=p
6
½
½
:
:
:
½
n
1
2
N
n
»2 measures the entanglement between the atoms
3. Spin squeezing
• Product states:
h
J
i
=
N
=
2
x
·
-̧
N
1
p
(
j
0
i+
j
1
i)
2
¢
J
=
0
x
2
N
(
¢
J
)
z
= 2
=
1
p »
2
h
J
i
+
h
J
i
x
y
¢
J
=
¢
J
=
N
=
2
h
J
i
=
h
J
i
=
0
y
z
y
z
2
No gain!
Jz
Jy
Jx
• Spin squeezed states:
(Wineland et al,1991)
h
J
i
'N
=
2
x
h
J
i
=
h
J
i
=
0
y
z
p
¢
J
<
N
=
2
z
2
N
(
¢
J
)
z
»
= 2
<
1
2
h
J
i
+
h
J
i
x
y
2
These states give better precission in atomic clocks
How to generate spin squeezed states?
(Kitagawa and Ueda, 1993)
2
1) Hamiltonian: H
=
Â
J
z
¡
i
(
Â
t
J
)
J
z
z
U
=
e
It is like a torsion
»2
1
t=
0
»2m in » N ¡
2=3
Â
t'
Ât min» 1=2N 2=3
Ât
1
2
=
3
2
N
2
»
=1
2
»
'
1
2
=
3
N
22
=
Â
(
J
¡
J
)
2) Hamiltonian: H
z
y
t=
0
Ât'
1
N
Â
t' 1
2
»
=1
»2'
1
N
1
j
ª
i
'p
(
j
0
;
:
:
:
;
0
i
+
j
1
;
:
:
:
;
1
i
)
2
»2
1
»2m in » N ¡ 1
Ât min» 1=2N
Ât
Explanation
J
'N
=
2
x
"
#
J
J
Jx
p y ;p z
=i
' i
N=2
N=2
N=2
J
y
X´ p
N
=
2
J
z
P´ p
N
=
2
2
¡
x
ª
(
x
;
0
)
/e
are like position and momentum operators
Hamiltonian 1:
t=
0
t>
0
N
2Â
2
H
=
Â
J
=
P
z
2
Hamiltonian 2:
¢
ÂN ¡ 2
2
2
2
H = Â(Jz ¡ Jy ) =
P ¡ X
2
t=
0
t>
0
4. Spin squeezinig with a BEC
A. Sorensen, L.M Duan, J.I. Cirac and P. Zoller, Nature 409, 63 (2001)
•
Weakly interacting two component
BEC
2 2 V T
H d3 r
r
r jr
j
2m
ja,b
laser
1
2
ja,b
U ab d3 r
r
rar
b
r
a
b
b
trap
U jj d 3rj rj rj rj r
a
+ laser interactions
Lit: JILA, ENS, MIT ...
•
Atomic configuration
•
optical trap
AC Stark shift via laser:
no collisions
FORT as focused laser beam
|0
| 1
| 1
!
aaa a bb a ab
F 1
A toy model: two modes
•
we freeze the spatial wave function
a
x a
x
a
b
x b
x
b
a
x
spatial mode function
•
Hamiltonian
•
Angular momentum representation
b
x
H 1 U aa a 2 a 2 U ab a ab b 1 U bb b 2 b 2
2
2
a b ab
H1
Uaa Ubb 2Uab
J2z Jx
2
2
=
Â
J
¡
J
x
z
•
Schwinger representation
Jx 1
a b ab
2
Jy i
a b ab
2
Jz 1
a a b b
2
A more quantitative model ... including the motion
• Beyond mean field: (Castin and Sinatra '00)
wave function for a two-component condensate
N
|
c Na Nb |Na : a
N a :t
; Nb : b
Nb:t
b
Na 0
N bNN a
a
with
xaNa :
x, t
d 3 x
a
Na !
Na
xb
Nb :
x, t
d 3x
b
N b!
Nb
|vac
• Variational equations of motion
• the variances now involve integrals over the spatial wave functions: decoherence
• Particle loss
Time evolution of spin squeezing
•
•
Idealized vs. realistic model
Effects of particle loss
21
1
10-1
10
2
loss
-1
including motion
-2
-2
10
10
-3
ideal
-3
10
10
0
-4
10
-4
4
idealized model
8
12
t
16
20
10
0
4
8
t
20 % loss
12
16
20
X
-4
10
Can one reach the Heisenberg limit?
We have the Hamiltonian:
2
H
=
Â
J
¡
J
x
z
2
2
2
2
J
+
J
+
J
=
J
=
c
o
n
s
t
a
n
t
x
y
z
|
{
z
}
H2 = Â(Jx2 ¡ Jz2) = Â(2Jx2 + Jy2 ¡ J 2)
We would like to have:
Idea: Use short laser pulses.
short evolution
short evolution
short pulse
short pulse
2
¼
2
¼
2
2
¡ i 2 Jx ¡ i ±tJz i 2 Jx ¡ i Â2±tJx
e
e {z e }e
' 1¡ i±t(2Jx2 + Jy2) ' e¡ i±t(Jx¡ Jz)
|
2
¡iÂ
±
t
J
y
e
Conditions:
-t =
¼
2
t¿
±
t
Stopping the evolution
»2
1
»2m in » N ¡ 1
Ât min» 1=2N
Ât
Once this point is reached, we would
like to supress the interaction
The Hamiltonian is:
2
H
=
Â
J
z
Using short laser pulses, we have an effective Hamiltonian:
2 2
2
2
2
H
=
Â
J
J
+
J
+
J
=
J
=
c
o
n
s
t
a
n
t
z
x
y
z
In practice:
wait
short pulse
short pulses
5. Squeezing and entangled beams
L.M Duan, A. Sorensen, I. Cirac and PZ, PRL '00
•
•
Atom laser
atomic configuration
Stark shift by laser:
switch collisions on
and off
atoms
•
collisions
Squeezed atomic beam
|0
| 1
| 1
F 1
condensate
pairs of atoms
•
Limiting cases
squeezing
sequential pairs
collisional Hamiltonian
2
x
x
x
1
1
0
x
x
20
x
e i2t
1
1
condensate as classical driving field
Equations ...
• Hamiltonian: 1D model
2
xx
H
x
Vx
xdx
i
i
2m
i1
x, t
,
x , tij
x x
i
j
g
x, t
x
x
e i2t h. c.
dx,
1
1
• Heisenberg equations of motion: linear
2
e i2 t
i
x, t xx V
x
x, tg x, t
x, t
t
1
1
1
2m
i
x, t
t
1
2
i2 t
xx V
x
x, tg x, t
x, t
e
1
1
2m
• Remark: analogous to Bogoliubov
• Initial condition: all atoms in condensate
Case 1: squeezed beams
• Configuration
g
(x
,t)
Â1
input: vaccum
Â1
B
output
1
B
1
condensate
a
0
x
• Bogoliubov transformation
B
1
Â1
1Â
1
1
B
1
Â
1Â1
1
1
• Squeezing parameter r
|
|
|
|
tanh
r |1| | 1|
1
• Exact solution in the steady state limit
1
Squeezing parameter r versus dimensionless detuning /g 0 and
interaction coefficient g 0 t
broadband two-mode squeezed state with the squeezing bandwidth g 0 .
numbers: g 0 20kHz, a 3 m, v
output flux of approx. 680 atoms/ms
squeezing r 0 2 (large)
2 /m 9cm/s
Case 2: sequential pairs
• Situation analogous to parametric downconversion
• Setup:
collisions
|0
| 1
| 1
F 1
symmetric potential
• State vector in perturbation theory
|
t
f
x,y, t
x
ydx dy |vac
1
1
with wave function consisting of four pieces
f
x,y f LR
x, y f RL
x, y f LL
x,y f RR
x,y
• After postselection "one atom left" and "one atom right"
| eff f LR
x,y
x
y
x
y
dxdy|vac
1
1
1
1
|
1,1 LR |1,
1 LR
6. Conclusions
• Entangled states may be useful in precission measurements.
• Spin squeezed states can be generated with current technology.
- Collisions between atoms build up the entanglement.
- One can achieve strongly spin squeezed states.
• The generation can be accelerated by using short pulses.
• The entanglement is very robust.
• Atoms can be outcoupled: squeezed atomic beams.
Quantum repeaters with atomic ensembles
L. M. Duan
M. Lukin
P. Zoller
J.I.C.
(Nature, November 2001)
SFB Coherent Control
€U TMR
€U EQUIP (IST)
Quantum communication:
Classical communication:
1
Alice
101
Quantum communication:
j
Á
i
0 0
1
Bob
Alice
j
Á
i j
Á
i
Á
i
j
Á
i j
Bob
Quantum Mechanics provides a secure way of secret communication
Classical communication:
1
1 0 1
Alice
0
01
Quantum communication:
jÁ
i
Bob
Alice
Eve
jÁ
i jÁ
i
½½
Bob
Eve
In practice: photons.
laser
y
j0
i =a
jv
a
c
i vertical polarization
0
jÁi
photons
y
j1
i =a
jv
a
c
i horizontal polarization
1
optical fiber
Problem: decoherence.
1. Photons are absorbed:
2. States are distorted:
_
Probability a photon arrives: P = e L =L 0
Quantum communication is limited
to short distances (< 50 Km).
ª i
j
Alice
½
We cannot know whether this is due to
decoherence or to an eavesdropper.
Bob
Solution: Quantum repeaters.
(Briegel et al, 1998).
laser
jª i
repeater
½
jª i
Questions:
L
=
L
0
1. Number of repetitions <
e
2. High fidelity: F
=
h
ª
j
½
j
ª
i
'1
3. Secure against eavesdropping.
Outline
1.
Quantum repeaters:
2.
Implementations:
3.
1.
With trapped ions.
2.
With atomic ensembles.
Conclusions
1. Quantum repeaters
The goal is to establish entangled pairs:
(i) Over long distances.
(ii) With high fidelity.
(iii) With a small number of trials.
Once one has entangled states, one can use the Ekert protocol for secret communication.
(Ekert, 1991)
Key ideas:
1. Entanglement creation:
Establish pairs over a short distance
2. Connection:
Connect repeaters
Long distance
3. Pufication:
Correct imperfections
4. Quantum communication:
High fidelity
Small number of trials
2. Implementation with trapped ions
Entanglement creation:
(Cabrillo et al, 1998)
ion A
laser
Internal states
ion A
ion B
jxi
jxi
ion B
laser
j0i
j1i
- Weak (short) laser pulse, so that the excitation probability is small.
- If no detection, pump back and start again.
- If detection, an entangled state is created.
j0i
j1i
Description:
Initial state:
j
0
;
0
i
j
v
a
c
i
ion A
ion B
jxi
jxi
After laser pulse:
(
j
0
i
+
²
j
x
i
)
(
j
0
i
+
²
j
x
i
)
j
v
a
c
i
A
B
£
¤
j0;0i + ²j0;xi + ²jx;0i + o(x2) jvaci
Evolution:
j0; 0i jvaci + ²(bkj0; 1i j1ki + akj1; 0i j1ki ) + o(²2)
Detection:
bkj0; 1i § akj1; 0i ' j0; 1i § j1; 0i
j0i
j1i
j0i
j1i
Repeater:
Entanglement
creation
Gate operations:
Connection
Purification
Entanglement
creation
3 Implementation with atomic ensembles
Atomic cell
Internal states
jxi
Atomic cell
j1i
j0i
- Weak (short) laser pulse, so that few atoms are excited.
- If no detection, pump back and start again.
- If detection, an entangled state is created.
Description:
n
n
0
i
j
0
i
j
v
a
c
i
Initial state: j
n
n
After laser pulse: (
j
0
i
+
²
j
x
i
)
(
j
0
i
+
²
j
x
i
)
j
v
a
c
i
Evolution:
n
n
j
0
i
j
0
i
j
v
a
c
i
+ photons in several directions (but not towards the detectors)
+ 1 photon towards the detectors and others in several directions
+ 2 photon towards the detectors and others in several directions
do not spoil the entanglement
Detection:
1 photon towards the detectors and others in several directions
+ 2 photon towards the detectors and others in several directions
negligible
n
Atomic
1 X i 2¼kj =n
p
e
j1i An h0j
„collective“
operators:
n
ay
j =
k= 1
n
X
y 1
p j
a
=
1
i
h
0
j and similarly for b
A
n
0
n
k
=
1
Photons emitted in the forward direction are the ones that excite this atomic „mode“.
Photons emitted in other directions excite other (independent) atomic „modes“.
Entanglement creation:
Sample A
Apply operator
y y
(
a
§
b
)
Sample B
Measurement:
Apply operator:
a
(A) Ideal scenareo
A.1 Entanglement generation:
Sample A
After click:
yy
(
a
+
r
)
j
0
;
0
i
(1)
Sample R
After click:
(2)
yy
(
b
+
r
~
)
j
0
;
0
i
Sample B
y
y
y
y
b
+
r
~
)
(
a
+
r
)
j
0
;
0
i
Thus, we have the state: (
A.2 Connection:
y
y
y
y
(
b
+
r
~
)
(
a
+
r
)
j
0
;
0
i
j~
ri
jri
If we detect a click, we must apply the operator:
(
r+
r
~
)
Otherwise, we discard it.
yy
We obtain the state: (
b
+
a
)
j
0
;
0
i
A.3 Secret Communication:
- Check that we have an entangled state:
y
y
y
y
~
(
b
+
a
~
)
(
b
+
a
)
j
0
;
0
i
• Enconding a phase:
y
i
±
y
y
y
~
(
b
+
e
a
~
)
(
b
+
a
)
j
0
;
0
i
• Measurement in A
(
a
+
a
~
)
• Measurement in B:
(b
+~
b
)
The probability of different outcomes +/- depends on ±
One can use this method to send information.
(B) Imperfections:
- Spontaneous emission in other modes:
No effect, since they are not measured.
- Detector efficiency, photon absorption in the fiber, etc:
More repetitions.
- Dark counts:
More repetitions
- Systematic phaseshifts, etc:
Are directly purified
(C) Efficiency:
Fix the final fidelity: F
l
o
g
N
2
Number of repetitions: r
N
Example:
Detector efficiency: 50%
Length L=100 L0
6
Time T=10 T0
43
(to be compared with T=10 T0 for direct communication)
Advantages of atomic ensembles:
1. No need for trapping, cooling, high-Q cavities, etc.
2. More efficient than with single ions: the photons that change the collective mode
go in the forward direction (this requires a high optical thickness).
Photons connected to the collective mode.
Photons connected to other modes.
3. Connection is built in. No need for gates.
4. Purification is built in.
4. Conclusions
•
Quantum repeaters allow to extend quantum communication over long
distances.
•
They can be implemented with trapped ions or atomic ensembles.
•
The method proposed here is efficient and not too demanding:
1.
2.
No trapping/cooling is required.
3.
4.
Atomic collective effects make it more efficient.
No (high-Q) cavity is required.
No high efficiency detectors are required.
Institute for Theoretical Physics
Postdocs:
- L.M. Duan (*)
- P. Fedichev
- D. Jaksch
- C. Menotti (*)
- B. Paredes
- G. Vidal
- T. Calarco
Ph D:
- W. Dur (*)
- G. Giedke (*)
- B. Kraus
- K. Schulze
P. Zoller
J. I. Cirac
FWF SFB F015:
„Control and Measurement of Coherent Quantum Systems“
€
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„EQUIP“
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