School of somethingFACULTY OF OTHER
School of Physics and AstronomyFACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES
“Classical entanglement” and cat states
Jacob Dunningham
Paraty, August 2007
Overview
The consequences of entanglement:
• The emergence of classicality from the quantum world
Number and phase of BEC
Position and momentum of micro-mirrors
Energy and time?
• Schrodinger cat states
How can we make them
How can we see them
What can we do with them
Multi-particle
EntanglementsWILD PEDIGREE
Bunnies Cats Bats
Quantum Information
Everyday World
Annihilation and creation operators (bosons)
annihilation
creation
Annihilation and creation operators (bosons)
annihilation
creation
Annihilation and creation operators (bosons)
Eigenvalue equation
is the number operator
annihilation
creation
Annihilation and creation operators (bosons)
annihilation
creation
In the Fock (number state) basis, these can be written as the matrices:
Annihilation and creation operators (bosons)
annihilation
creation
In the Fock (number state) basis, these can be written as the matrices:
An exercise in matrix multiplication confirms the bosonic commutation relation:
Emergence of classicality
One of the most perplexing aspects of quantum theory is that microscopic objects can be in superpositions but macroscopic objects cannot
Schrödinger’s cat
To ‘see’ a coherent superposition, we need interference
How do we see them?
Detect interference of probe state corresponding to phase
Macroscopic variables
Detect interference of probe state corresponding to phase
No interference if the macroscopic states are orthogonal
Macroscopic variables
Need coupling between them - “Lazarus operator”
The key is to wash out the which-way informationNOON state
The key is to wash out the which-way information
There is the problem of the environment
Tracing over the environment gives:
Described in detail by A. Ekert
yesterday
Classical entanglement
Can also understand the emergence of classicality in terms of entanglement
Classical entanglement
Can also understand the emergence of classicality in terms of entanglement
First it is helpful to consider BECs
• Macroscopic quantum entity
• Can probe quantum / classical divide
• Cold enough to enable quantum phase transitions
What is a BEC?
Predicted 1924... ...Created 1995
S. Bose A. Einstein
What is a BEC?
Bose-Einstein distribution:
What is a BEC?
Bose-Einstein distribution:
Take
For consistency:
What is a BEC?
Bose-Einstein distribution:
Take
For consistency:
Onset of BEC:
Cold and dilute
How do we make them?
Trap them with magnetic and/or optical fields
Cool them using two main techniques:
1. Laser Cooling (link)
2. Evaporative Cooling (link)
What is a BEC?
For our purposes, a BEC is a ‘macroscopic’ quantum entity - thickness of a human hair
All the atoms (~103 - 109) are in the same quantum state
Phase of a BEC
Coherent state:
“Most classical” quantum state
BEC Localisation
N NConservation of atom number:
?
BEC Localisation
N NConservation of atom number:
?
Experiment
BEC Localisation
First detection:
N Na b
We don’t know which BEC the atom came from
x
Position-dependent phase
BEC Localisation
First detection:
N Na b
We don’t know which BEC the atom came from
x
Position-dependent phase
BEC Localisation
N Na b
x
Probability density of second detection::
BEC Localisation
N Na b
x
Probability density of second detection::
Feedback gives fringes with visibility ~ 0.5
After ~ N measurements:
Robust relative phase state - classical
The phase of each condensate is still undefined:
Fluffy bunny state
Phase standard
N Na c
Nb
Phase standard
N Na c
Nb
Phase standard
N Na c
Nb
Phase standard
Properties
Absolute versus relative variables
a b c
• Robustness: subsequent measurements do not change the result – classical-like
• Transitivity: ingrained in our classical perception of the world
Entanglement is all around us – not just a “quantum phenomenon”!
Position Localisation
Can do the same for position and momentum
Initial state of the mirrors:
Relative positionFlat
distribution
Position Localisation
Can do the same for position and momentum
Initial state of the mirrors:
Relative positionFlat
distribution
Photon with momentum k, state before N:
Position Localisation
Position Localisation
Detection at D1:
Detection at D2:
Position Localisation
1. Rau, Dunningham, Burnett, SCIENCE 301, 1081 (2003)
2. Dunningham, Rau, Burnett, SCIENCE 307, 872 (2005)
Time
‘time’ ‘time’
No need to go through ‘middle-man’ of time
Angle of hour hand
Position of sun
Barbour view:
Position of sun
Angle of hour hand
Entanglement of three particles
H|
|cn,m |n, m, E-n-m>
x23
x12
?
Don’t need measurements
For every sequence of scattering events, a well-defined relative position (or phase) builds up
If we don’t measure the scattered particles the relative position is uncertain (classically)
Tracing over the scattered particles gives:
Don’t need measurements
Just by shining light on particles they acquire a classical relative position - yet each particle remains highly quantum!
For every sequence of scattering events, a well-defined relative position (or phase) builds up
If we don’t measure the scattered particles the relative position is uncertain (classically)
Tracing over the scattered particles gives:
Well-localised state
Classical mixture
Multi-particle
EntanglementsWILD PEDIGREE
Bunnies Cats Bats
Quantum Information
Everyday World
Experimental progress
4 Be+ ions (2000)C60 molecules
(1999)
~ 109 Cooper pairs (2000)
Experimental progress
4 Be+ ions (2000)C60 molecules
(1999)
~ 109 Cooper pairs (2000)
Future
Micro-mirrors
Biological systems?
(E. Coli)
a
bc
Coupling between wells Interactions between atoms
Ref: Boyer et al, PRA 73, 031402 (2006)
Superfluid cats
a
bc
Coupling between wells Interactions between atoms
Ref: Boyer et al, PRA 73, 031402 (2006)
No rotation
Clockwise
Anticlockwise
No rotation
Clockwise
Anticlockwise
Flow is quantized in units of 2 around the loop -- vortices
How do we make them?
How do we make them?
How do we make them?
How do we make them?
Cat
Entanglement witness
Separable states
How do we see them?
Metastable states
Spectroscopically scan the energy gap -- see it directly
How do we see them?
What can we do with them?
+
What can we do with them?
+For superfluid flows:
• Bell state experiments with macroscopic objects
• Precision measurements - quantum-limited gyroscopes
Precision measurements of angular momentum
Gyroscopes
Precision measurements of angular momentum
Can measure to within 1/N
Gyroscopes
Summary
Next lecture: An even better way of using entanglement to make measurements
• The emergence of classicality from the quantum world
Number and phase of BEC
Position and momentum of micro-mirrors
• Schrodinger cat states
How can we make them
How can we see them
What can we do with them