Atomic%20entangled%20states%20with%20BEC

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Atomic entangled states with BEC
A. Sorensen L. M. Duan P. Zoller J.I.C.
(Nature, February 2001)
KIAS, November 2001.
SFB Coherent ControlU TMR
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Entangled states of atoms
Motivation

• Fundamental.
• Applications -
  Secret communication -
  Computation -
  Atomic clocks

Experiments

• NIST 4 ions entangled.
• ENS 3 neutral atoms entangled.

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E
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This talk Bose-Einstein condensate.
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Outline

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 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
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Solution use atoms to lock a laser
frequency fixed universal
detector
feed back
In practice
Neutral atoms
ions
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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
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1
1
1
1
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2. Ramsey method

• Fast pulse
• Wait for a time T
• Fast pulse
• Measurement

single atom
single atom
single atom
of atoms in 1gt
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Independent atoms
Number of atoms in state 1gt 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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1
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Another way of looking at it
Initial state all atoms in 0gt
First Ramsey pulse
Free evolution
Measurement
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In general
where the Js are angular momentum operators

• Remarks
• We want
• Optimal
• If then the atoms are entangled.
  

That is,
measures the entanglement between the atoms
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3. Spin squeezing

• Product states

No gain!
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• Spin squeezed states

(Wineland et al,1991)
These states give better precission in atomic
clocks
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How to generate spin squeezed states?
(Kitagawa and Ueda, 1993)
1) Hamiltonian
It is like a torsion
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2) Hamiltonian
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Explanation
are like position and momentum operators
Hamiltonian 1
Hamiltonian 2
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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
• Atomic configuration

laser
trap
laser interactions
Lit JILA, ENS, MIT ...

• optical trap

AC Stark shift via laser no collisions
FORT as focused laser beam
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A toy model two modes

• we freeze the spatial wave function
• Hamiltonian
• Angular momentum representation

• Schwinger representation

spatial mode function
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A more quantitative model ... including the motion

• Beyond mean field (Castin and Sinatra '00)wave
  function for a two-component condensatewith
• Variational equations of motion
• the variances now involve integrals over the
  spatial wave functions decoherence
• Particle loss

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Time evolution of spin squeezing

• Idealized vs. realistic model

• Effects of particle loss

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Can one reach the Heisenberg limit?
We have the Hamiltonian
We would like to have
Idea Use short laser pulses.
short evolution
short evolution
short pulse
short pulse
Conditions
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Stopping the evolution
Once this point is reached, we would like to
supress the interaction
The Hamiltonian is
Using short laser pulses, we have an effective
Hamiltonian
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In practice
wait
short pulse
short pulses
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5. Squeezing and entangled beams
L.M Duan, A. Sorensen, I. Cirac and PZ, PRL '00

• Atom laser
• Squeezed atomic beam
• Limiting cases
• squeezing
• sequential pairs

• atomic configurationcollisional
  Hamiltonian

atoms
pairs of atoms
condensate as classical driving field
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Equations ...

• Hamiltonian 1D model
• Heisenberg equations of motion linear
• Remark analogous to Bogoliubov
• Initial condition all atoms in condensate

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Case 1 squeezed beams

• Configuration
• Bogoliubov transformation
• Squeezing parameter r
• Exact solution in the steady state limit

input vaccum
condensate
output
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(No Transcript)
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Case 2 sequential pairs

• Situation analogous to parametric downconversion
• Setup
• State vector in perturbation theorywith wave
  function consisting of four pieces
• After postselection "one atom left" and "one atom
  right"

collisions
symmetric potential
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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.

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Quantum repeaters with atomic ensembles
L. M. Duan M. Lukin P. Zoller J.I.C.
(Nature, November 2001)
SFB Coherent ControlU TMR U EQUIP (IST)
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Quantum communication
Classical communication
Quantum communication
Bob
Bob
Alice
Alice
Quantum Mechanics provides a secure way of secret
communication
Classical communication
Quantum communication
Bob
Bob
Alice
Alice
Eve
Eve
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In practice photons.
laser
vertical polarization
horizontal polarization
optical fiber
photons
Problem decoherence.
1. Photons are absorbed
2. States are distorted
_
L

L
P

e
Probability a photon arrives
0
ª
j
i
½
Alice
Quantum communication is limited to short
distances (lt 50 Km).
Bob
We cannot know whether this is due to
decoherence or to an eavesdropper.
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Solution Quantum repeaters.
(Briegel et al, 1998).
laser
repeater
ª
ª
½
j
i
j
i
Questions
1. Number of repetitions
2. High fidelity
3. Secure against eavesdropping.
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Outline

1. Quantum repeaters
2. Implementations
3. With trapped ions.
4. With atomic ensembles.
5. Conclusions

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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)
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Key ideas
1. Entanglement creation
Establish pairs over a short distance
Small number of trials
2. Connection
Connect repeaters
Long distance
3. Pufication
Correct imperfections
High fidelity
4. Quantum communication
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2. Implementation with trapped ions
Entanglement creation
(Cabrillo et al, 1998)
Internal states
ion A
ion A
ion B
laser
ion B
laser
- 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.
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Description
Initial state
ion A
ion B
After laser pulse
Evolution
Detection
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Repeater
Entanglement creation
Gate operations
Connection
Purification
Entanglement creation
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3 Implementation with atomic ensembles
Atomic cell
Internal states
Atomic cell
- 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.
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Description
Initial state
After laser pulse
Evolution
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
Detection
do not spoil the entanglement
1 photon towards the detectors and others in
several directions
2 photon towards the detectors and others in
several directions
negligible
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Atomic collective operators
and similarly for b
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
Sample B
Measurement
Apply operator
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(A) Ideal scenareo
A.1 Entanglement generation
Sample A
After click
(1)
Sample R
After click
(2)
Sample B
Thus, we have the state
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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
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A.3 Secret Communication
- Check that we have an entangled state

• Enconding a phase
• Measurement in A
• Measurement in B

The probability of different outcomes /- depends
on
One can use this method to send information.
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(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
Fix the final fidelity F
Number of repetitions
Example
Detector efficiency 50 Length L100 L0
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Time T10 T0
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(to be compared with T10 T0 for direct
communication)
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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.
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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
• No trapping/cooling is required.
• No (high-Q) cavity is required.
• Atomic collective effects make it more efficient.
• No high efficiency detectors are required.

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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 EU networksCoherent
Matter Waves, Quantum Information EU
(IST) EQUIP Austrian Industry Institute for
Quantum Information Ges.m.b.H.