Download - Atomic entangled states with BEC
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Atomic entangled states with BECAtomic entangled states with BEC
SFB Coherent Control€U TMR
A. Sorensen
L. M. Duan
P. Zoller
J.I.C.
(Nature, February 2001)
KIAS, November 2001.
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Entangled states of atomsEntangled states of atoms
Motivation:• Fundamental.
• Applications: - Secret communication - Computation - Atomic clocks
• NIST: 4 ions entangled.
• ENS: 3 neutral atoms entangled.
Experiments:
j ª i6= j '1 i j '2 i : : : j 'N i
'
E ' 4
E ' 3
E 103This talk: Bose-Einstein condensate.
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OutlineOutline
1. Atomic clocks
2. Ramsey method
3. Spin squeezing
4. Spin squeezing with a BEC
5. Squeezing and atomic beams
6. Conclusions
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1. Atomic clocks1. Atomic clocks
To measure time one needs a stable laser
click
The laser frequency must be the same for all clocks
click
click
Innsbruck
Seoul
The laser frequency must be constant in time
click
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Solution: use atoms to lock a laser
detector
feed back
frequencyfixeduniversal
In practice:Neutral atoms ions
! L = ! 0 + ±!
! 0
! L
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Independent atoms:
Entangled atoms:
• 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:
We want
±! =1
tpn r e p
pN
±! e n t =1
tp n r e p f (N )
±! e n t = ±!
»2 ¿ 1
»2 =(n r e p )e n t
n r e p=
T e n t
T=
pN
f (N )
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2. Ramsey method2. Ramsey method
# of atoms in |1>
single atom
single atom
single atom
j0i !1p2(j0i + j1i )
!1p2(j0i + e¡ i (! 0¡ ! L )t j1i )
P1 = cos2·12(! 0 ¡ ! L )t
¸
sin·12(! 0 ¡ ! L )t
¸j0i+ co s
·12(! 0 ¡ ! L ) t
¸j1i
• Fast pulse:
• Wait for a time T:
• Fast pulse:
• Measurement:
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Independent atomsIndependent 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:
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Another way of looking at itAnother way of looking at it
J x
J y
J z
J x
J y
J z
Initial state: all atoms in |0> First Ramsey pulse:
J x
J y
J z
J x
J y
J z
Free evolution:
J x
J y
J z
Measurement:
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In generalIn general
where the J‘s are angular momentum operators
Remarks:
• We want
• Optimal:
• If then the atoms are entangled.
That is,
measures the entanglement between the atoms
»2 =N (¢ J z )2
hJ x i 2 + hJ y i 2
J ® =NX
k=1
j (k)®
»2 ¿ 1
»2 ¸ 1=N
»2 < 1
½ 6=X
n
pn½1 ½2 : : :½N
»2
J x
J y
J z
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3. Spin squeezing3. Spin squeezing
No gain!
·1p2(j0i + j1i )
̧ N
hJ x i = N=2 ¢ J x = 0
¢ J y = ¢ J z =pN =2
»2 =N (¢ J z )2
hJ x i 2 + hJ y i 2= 1
J x
J y
J z
• Product states:
hJ y i = hJ z i = 0
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• Spin squeezed states:(Wineland et al,1991)
These states give better precission in atomic clocks
hJ x i ' N =2
¢ J z <pN =2
hJ y i = hJ z i = 0
»2 =N (¢ J z )2
hJ x i 2 + hJ y i 2< 1
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How to generate spin squeezed states?How to generate spin squeezed states?
(Kitagawa and Ueda, 1993)
1) Hamiltonian:
It is like a torsion
Ât '1
2N 2=3
t=0 »2=1
»2 '1
N 2=3
H = ÂJ 2z
U = e¡ i (ÂtJ z )J z
»2
»2m in » N ¡ 2=3
ÂtÂtmin» 1=2N 2=3
1
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2) Hamiltonian:
t=0 »2=1
Ât'1N
»2 '1N
Ât ' 1 jª i '1p2(j0; : : : ; 0i + j1; : : : ; 1i )
H = Â(J 2z ¡ J 2
y )
»2
Ât
1
»2m i n » N ¡ 1
Âtmin» 1=2N
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ExplanationExplanation
Hamiltonian 1:
Hamiltonian 2:
J x ' N =2"
J ypN=2
;J zpN=2
#
= iJ xN =2
' i
X ´J ypN=2
H = ÂJ 2z =
ÂN2
P 2
H = Â(J 2z ¡ J 2
y ) =ÂN2
¡P 2 ¡ X 2
¢
P ´J zpN=2
t = 0
t = 0
t > 0
t > 0
ª (x; 0) / e¡ x2
are like position and momentum operators
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4. Spin squeezinig with a BEC4. Spin squeezinig with a BEC
• Weakly interacting two component BEC
• Atomic configuration• optical trap
A. Sorensen, L.M Duan, J.I. Cirac and P. Zoller, Nature 409, 63 (2001)
laser
trap
F 1| 1
|0| 1
aaa! abb aab
AC Stark shift via laser:no collisions
H j a,b
d3r jr 2
2m 2 VTr jr
12 j a,b
U jj d3r jr jr jr jr
Uab d3r a r b r ar br
+ laser interactions
FORT as focused laser beam
Lit: JILA, ENS, MIT ...
a
b
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A toy model: two modesA toy model: two modes
• we freeze the spatial wave function
• Hamiltonian
• Angular momentum representation • Schwinger representation
ax b x
ax axa bx bxb
spatial mode function
H 12Uaaa2a2 Uaba abb 1
2Ubbb2b2
ab ab
Jx 12 a b ab
Jy i2
a b ab
Jz 12
a a bb
H 12
Uaa Ubb 2UabJz2 Jx
= ÂJ 2z ¡ J x
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A more quantitative model ... including the motionA more quantitative model ... including the motion
• Beyond mean field: (Castin and Sinatra '00)wave function for a two-component condensate
with
• Variational equations of motion• the variances now involve integrals over the spatial wave functions: decoherence• Particle loss
| Na 0 Nb NNa
N
cNaNb |Na: aNa:t;Nb : bNb:t
d3x a x aNa : x, t
Na
Na !
d3x b x bNb : x, t
Nb
Nb!|vac
a
b
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Time evolution of spin squeezingTime evolution of spin squeezing
• Idealized vs. realistic model • Effects of particle loss
1
10-1
10-2
10-3
10-4
0
4 8 12 16 20
t
2
idealized model
including motion
1
10-1
10-2
10-3
10-40 4 8 12 16 20
2
t 10X-4
loss
20 % loss
ideal
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Can one reach the Heisenberg limit?Can one reach the Heisenberg limit?
H = ÂJ 2z ¡ J x
H 2 = Â(J 2x ¡ J 2
z ) = Â(2J 2x + J 2
y ¡ J 2)
J 2x + J 2
y + J 2z = J 2 = constant
e¡ i ¼2 J xe¡ i±tJ 2z ei
¼2 J x| {z }e
¡ iÂ2±tJ 2x ' 1 ¡ i±t(2J 2
x + J 2y ) ' e¡ i±t(J 2
x ¡ J 2z )
e¡ i ±tJ 2y
t =¼2
t ¿ ±t
We have the Hamiltonian:
We would like to have:
| {z }
short pulseshort pulseshort evolution short evolution
Conditions:
Idea: Use short laser pulses.
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H = ÂJ 2z
Stopping the evolutionStopping the evolution
»2
Ât
1
»2m i n » N ¡ 1
Âtmin» 1=2N
Once this point is reached, we wouldlike to supress the interaction
H = ÂJ 2z
The Hamiltonian is:
Using short laser pulses, we have an effective Hamiltonian:
J 2x + J 2
y + J 2z = J 2 = constant
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In practice:In practice:
wait
short pulses
short pulse
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5. Squeezing and entangled beams5. Squeezing and entangled beams
• Atom laser
• Squeezed atomic beam
• Limiting cases squeezing sequential pairs
• atomic configuration
collisional Hamiltonian
L.M Duan, A. Sorensen, I. Cirac and PZ, PRL '00
atoms
condensate as classical driving field
collisions
F 1| 1
|0| 1
condensate
Stark shift by laser:switch collisions onand off
pairs of atoms
1 x 1
x 02xe i2 t
1 x 1
x 02x
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Equations ...Equations ...
• Hamiltonian: 1D model
• Heisenberg equations of motion: linear
• Remark: analogous to Bogoliubov
• Initial condition: all atoms in condensate
H i 1
i
x xx22m
Vx ixdx
gx, t 1
x 1 xe i2 t h.c. dx,
ix, t, jx , t ij x x
i t 1x, t xx22m
Vx 1x, t gx, t 1 x, te i2 t
i t 1 x, t xx2
2m Vx 1
x, t gx, t 1x, te i2 t
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Case 1: squeezed beamsCase 1: squeezed beams
• Configuration
• Bogoliubov transformation
• Squeezing parameter r
• Exact solution in the steady state limit
B 1 1 Â 1 1 Â 1
B 1 1 Â 1
1 Â 1
tanhr | 1 || 1 |
| 1 |
| 1 |
g (x ,t)
0 a x
condensate
 1  1 B 1 B 1
input: vaccum
output
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S q u eez ing p a ram e ter r v e rsu s d im ens io n le ss d e tu n in g /g 0 an d
in te rac tio n co effic ien t g 0 t
b ro ad b an d tw o -m o d e sq ueezed s ta te w ith th e sq ueez in g b an dw id th g 0 .
n u m b ers : g 0 20k H z, a 3 m , v 2 /m 9cm /s
o u tp u t f lu x o f ap p ro x . 680 a to m s /m s
sq ueez in g r 0 2 (la rg e)
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Case 2: sequential pairsCase 2: sequential pairs
• Situation analogous to parametric downconversion
• Setup:
• State vector in perturbation theory
with wave function consisting of four pieces
• After postselection "one atom left" and "one atom right"
| eff fLRx,y 1 x 1
y 1 x 1
ydxdy|vac
| 1, 1LR | 1, 1LR
F 1| 1
|0| 1
symmetric potential
collisions
| t fx,y, t 1 x 1
ydx dy |vac
fx,y fLRx, y fRLx, y fLLx,y fRRx,y
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6. Conclusions6. 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.
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Quantum repeaters with atomic ensemblesQuantum repeaters with atomic ensembles
SFB Coherent Control€U TMR
€U EQUIP (IST)
L. M. Duan
M. Lukin
P. Zoller J.I.C.
(Nature, November 2001)
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Quantum communication:Quantum communication:
Classical communication: Quantum communication:
Quantum Mechanics provides a secure way of secret communication
AliceBob Alice
Bob
Classical communication:
AliceBob
Quantum communication:
AliceBob
Eve
0
1010 1
1
jÁi jÁi
jÁi
jÁi jÁi
0
1010 1
1
½jÁi jÁi
jÁi½
Eve
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Problem: decoherence.
We cannot know whether this is due to decoherence or to an eavesdropper.
Probability a photon arrives:
2. States are distorted:
Alice Bob
1. Photons are absorbed:
Quantum communication is limitedto short distances (< 50 Km).
j ª i ½
P =e_ L=L 0
In practice: photons.
laser
optical fiberphotons
vertical polarization
horizontal polarization
j0i = ay0jvaci
j1i = ay1jvacijÁi
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laser repeater
Questions:
1. Number of repetitions
2. High fidelity:
3. Secure against eavesdropping.
j ª i j ª i½
< eL =L 0
F = hª j½jª i ' 1
Solution: Quantum repeaters.(Briegel et al, 1998).
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OutlineOutline
1. Quantum repeaters:
2. Implementations:
1. With trapped ions.
2. With atomic ensembles.
3. Conclusions
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1. Quantum repeaters1. 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)
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Establish pairs over a short distance Small number of trials
Connect repeaters
Correct imperfections
Long distance
High fidelity
Key ideas:Key ideas:
1. Entanglement creation:
2. Connection:
3. Pufication:
4. Quantum communication:
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2. Implementation with trapped ions2. Implementation with trapped ions
ion A ion Blaser
laser
ion A
ion B
Internal states
- 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.
Entanglement creation:Entanglement creation:
j0i j0ij1i j1i
(Cabrillo et al, 1998)
jxi jxi
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Initial state:
After laser pulse:
Evolution:
Detection:
Description:Description:
j0; 0i jvaci
j0; 0i jvaci + ²(bk j0; 1i j1k i + ak j1; 0i j1k i ) + o(²2)
bk j0; 1i § ak j1; 0i ' j0; 1i § j1; 0i
ion A ion B
j0i j0ij1i j1i
jxi jxi
(j0i + ²jxi )A (j0i + ²jxi )B jvaci£j0; 0i + ²j0; xi + ²jx; 0i + o(x2)
¤jvaci
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Repeater:Repeater:
Entanglementcreation
Entanglementcreation
Gate operations:ConnectionPurification
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3 Implementation with atomic ensembles3 Implementation with atomic ensembles
Internal states
- 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.
j0ij1i
Atomic cell
Atomic cell
jxi
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Initial state:
After laser pulse:
Evolution:
Detection:
j0i n j0i n jvaci
j0i n j0i n jvaci
+ photons in several directions (but not towards the detectors)
+ 2 photon towards the detectors and others in several directions
+ 1 photon towards the detectors and others in several directions
1 photon towards the detectors and others in several directions
+ 2 photon towards the detectors and others in several directions
Description:Description:
negligible
do not spoil the entanglement
(j0i + ²jxi ) n (j0i + ²jxi ) n jvaci
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ayj =1pn
nX
k=1
ei 2¼kj =n j1i A n h0j
ay0 =1pn
nX
k=1
j1iA nh0j
Atomic „collective“ operators:Atomic „collective“ operators:
and similarly for b
Entanglement creation:
Measurement:
Sample A
Sample B
Apply operator
Apply operator:
(ay § by)
a
Photons emitted in the forward direction are the ones that excite this atomic „mode“.Photons emitted in other directions excite other (independent) atomic „modes“.
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(A) Ideal scenareo(A) Ideal scenareo
After click:
(1)
(2)
After click:
Thus, we have the state:
Sample A
Sample R
Sample B
A.1 Entanglement generation:
(ay+ r y)j0; 0i
(by+ ~r y)j0; 0i
(by+ ~r y)(ay + r y) j0; 0i
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A.2 Connection:
If we detect a click, we must apply the operator:
Otherwise, we discard it.
We obtain the state:
(r + ~r )
(by + ay)j0; 0i
(by+ ~r y)(ay + r y) j0; 0i
jr ij~r i
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A.3 Secret Communication:
- Check that we have an entangled state:
One can use this method to send information.
• Enconding a phase:
• Measurement in A
• Measurement in B:
(~by + ~ay)(by + ay)j0; 0i
(~by + ei±~ay)(by+ ay)j0; 0i
(a + ~a)
(b+ ~b)
The probability of different outcomes +/- depends on ±
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(B) Imperfections:(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
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(C) Efficiency:(C) Efficiency:
Fix the final fidelity: F
Number of repetitions: rN log2 N
Example:
Detector efficiency: 50%
Length L=100 L0
Time T=10 T06
(to be compared with T=10 T0 for direct communication)43
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Advantages of atomic ensembles: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.
4. Purification is built in.
3. Connection is built in. No need for gates.
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4. Conclusions4. 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. No trapping/cooling is required.
2. No (high-Q) cavity is required.
3. Atomic collective effects make it more efficient.
4. No high efficiency detectors are required.
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Institute for Theoretical PhysicsInstitute for Theoretical Physics
€
FWF SFB F015:„Control and Measurement of Coherent Quantum Systems“
EU networks:„Coherent Matter Waves“, „Quantum Information“
EU (IST):„EQUIP“
Austrian Industry:Institute for Quantum Information Ges.m.b.H.
P. ZollerJ. I. Cirac
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