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