School of something FACULTY OF OTHER School of Physics and Astronomy FACULTY 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 Entanglements WILDPEDIGREE Bunnies CatsBats 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 information NOON 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 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)(link) 2. Evaporative Cooling (link)(link)

What is a BEC? For our purposes, a BEC is a ‘macroscopic’ quantum entity - thickness of a human hair All the atoms (~ ) are in the same quantum state

Phase of a BEC Coherent state:    “Most classical” quantum state

BEC Localisation  NN Conservation of atom number: ?

BEC Localisation  NN Conservation of atom number: ? Experiment

BEC Localisation First detection:  NN a b We don’t know which BEC the atom came from x Position-dependent phase

BEC Localisation First detection:  NN a b We don’t know which BEC the atom came from x Position-dependent phase

BEC Localisation  NN a b x Probability density of second detection: :

BEC Localisation  NN a 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 NN a c N b

NN a c N b

NN a c N b

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 position Flat distribution

Position Localisation Can do the same for position and momentum Initial state of the mirrors: Relative position Flat distribution Photon with momentum k, state before N:

Position Localisation

Detection at D 1 : Detection at D 2 :

Position Localisation  1. Rau, Dunningham, Burnett, SCIENCE 301, 1081 (2003) 2. Dunningham, Rau, Burnett, SCIENCE 307, 872 (2005)

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|  |  c n,m |n, m, E-n-m> x 23 x 12 ?

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 Entanglements WILDPEDIGREE Bunnies CatsBats Quantum Information Everyday World

Experimental progress 4 Be + ions (2000) C 60 molecules (1999) ~ 10 9 Cooper pairs (2000)

Experimental progress 4 Be + ions (2000) C 60 molecules (1999) ~ 10 9 Cooper pairs (2000) Future Micro-mirrors Biological systems? (E. Coli)

a b c Coupling between wellsInteractions between atoms Ref: Boyer et al, PRA 73, (2006) Superfluid cats

a b c Coupling between wellsInteractions between atoms Ref: Boyer et al, PRA 73, (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?

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? +

+ 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