Optical Implementations of QIP

Kevin ReschIQC, Department of Physics

University of Waterloo

Quantum optics and Quantum Info. group

Entanglement sources

Tomography

Quantum computing

Tests of nonlocality

Interferometry

Optical Imaging

Goal of the talk

• To understand from basic principles how a quantum information protocol works in theory and in practice using optics

• I chose quantum teleportation where we can understand the discrete (polarization) and continuous variable versions of this protocol

Quantum teleportation

• A process for transmitting quantum information using a classical channel and shared entanglement

Figure credit: Bouwmeester et al. Nature 390, 575 (1997).

Outline

• Introduction to quantum optics– photons, encodings, entanglement

• QIP with polarization– waveplates, CNOT, teleportation

• QIP with continuous variables– Wigner function, measurements, teleportation

Photons

Field quantization

• A procedure for finding a quantum description of light

• Starting point is Maxwell’s Equations in vacuum

• Introducing the potentials and gauge

Field quantization

• From these derive wave equation for the vector potential

• Spatial mode expansion (exact form depends on boundary conditions)

Plane wave solutionsPeriodic BC, cubic volume

Field quantization

• Also from classical physics, the energy stored in an EM field

• Energy for a single mode

• Rewriting complex A in terms of real quant

• Gets us onto familiar territoryClassical Harmonic Oscillator(mass = 1)

Field quantization

Field quantization

• Promote the classical parameters to operators

• Which defines field operators

Field quantization

• And find the energy for each mode

• Which simplifies to

Field quantization

• Harmonic oscillator

The excitations of the EM modes are “photons” – particles of light

“number” operator q

Experimental evidence for photons

• Particles can only be detected in one place

Grangier, Roger, Aspect Europhysics Lett 1, 173 (1986)

Ca Atomic Cascade

Properties of photons

• A single photon has just three properties:

– Colour/energy,

– Polarization,

– Direction/momentum,

• Its quantum state can be described as a superposition of these properties

Single photon QI encodings

|H> |V>

+ =i

• Polarization

• Time-bin

• Spatial modes

• Freq. encoding

QIP with optics

• Pros: – Low decoherence*– High speed– Flexible encodings

• Cons: – Negligible photon-photon interactions– Loss– Hard to keep in one place– Some encodings unsuitable for some

situations, ex., polarization/modes in fibre

Ideal for quantum communication

*can be susceptible to coupling internal DOF

But an optical mode is more complicated…

• Photons are bosons, so we can have many per mode

• Important multi-photon states of a single mode:– Fock or number state– Coherent state– Squeezed state– Thermal state

• (Things can get very complicated with a large number of modes and all the DOF)

“ Mode” observables: Quadratures

• We can write quadrature operators analogous to x and p (but do not correspond to pos/mom of the photon!)

• Since , there must be an uncertainty relation

Useful operator identities

• Baker-Campbell-Hausdorff lemma

• Glauber’s identity

valid when [A,[A,B]]= [A,[A,B]]=0.

Phase shift operators• Phase shift operator (exp free-field)

• Using BCH

• Or

• Free-field evolution converts one quadrature into the other in the form of a rotation

Quadratures

• These observables correspond to components of the electric field

• There is an uncertainty relation between the E field ‘now’ and the E field a quarter cycle ‘later’

Coherent states

• Defined as eigenstates of lowering operator

a is not Hermitian so α can be complex• Uncertainties in mode variables:

• Min uncertainty, equal between q and p

Displacement operator

• Coherent states can be generated using the displacement operator:

• This can be seen by rewriting the operator using Glauber’s identity and comparing

Displacement operator

• Useful identities and properties:

Coherent states in quantum optics

• Coherent states play an important role as a basis in quantum optics

• But coherent states with different amplitudes are orthogonal

• And the basis is “overcomplete” (projectors do not sum to identity)

Entanglement

The characteristic trait of QM

Math. Proc. Camb. Philos. Soc. 31, 555 (1935).

E. Schrödinger

Definition of entanglement

• Any state that can be written,

is said to be separable, otherwise it is entangled

• Pure states:

are separable, otherwise entangled

ÃA B = ÃA ÃB

Ã(x1;x2) = Ã(x1)Ã(x2)

Superposition and entanglement

Superposition Entanglement

The characteristic trait of QM

Entanglementhttp://www.eng.yale.edu/rslab/

Quantum Computing

Quantum Communication

Foundations of QM

Phase transitions

Quantum relativisticeffects

Figure credit: Rupert UrsinJennewein et al. PRL 84, 4729 (2000)

Enhanced Sensors

S. Hawking Illustrated Brief History of Time

http://www.ligo.caltech.edu

http://www.quantum-munich.de

Nonlinear optics

• Direct photon-photon interactions too weak

• Instead atoms can mediate interactions between photons – Nonlinear Optics

• Ex. Second-order nonlinearity

Creates pairs ofphotons

Destroys pairs ofphotons

Nonlinear coefficient

Nonlinear optics

• Instead of oscillating only at the frequency of the driving field, the charge can oscillate at new frequencies

ω 2ω

Χ(2) material(such as BBO or KTP)

Example: Second Harmonic Generation

Second-harmonic generation

Entangled photons

“ blue” photon two “red” photons

(2)

Phase matching:pump = s + i

kpump = ks + ki

Parametric Down-conversion

• Reverse of SHG

‘Conservation laws’ constrain the pair withoutconstraining the individual entanglement

Also: QD, at. casc

Down-conversion movie

www.quantum.at

KTP – nonlinear crystal

PPKTP source

PPKTP source

Multiphoton sources: Pulsed SPDC

• Down-conversion can sometimes emit two pairs. • If a short pulse is used for an entangled photon source,

the pair are properly described by a 4-photon state

H = gayH 1ay

V 2 + gayV 1ay

H 2 + h:c:

H 2j0i ! (ayH 1ay

V 2 + ayV 1ay

H 2)2j0i

= j2H1;2V2i + jH1;V1;H2;V2i + j2V1;2H2i

GHZ Correlations

Bouwmeester PRL 82, 1345 (1999)Lavoie NJP 11, 073501 (2009)

• Measured 4-photon coincidences to post-select GHZ state

• Needs at least 1H and 1V

in mode 1

j2H1;2V2i + jH1;V1;H2;V2i + j2V1;2H2i V

H

H

V

VH

H

VHaHbVc

H

V

V

VaVbHc

Three-photon GHZ states

• ~4 four-fold coincidence counts per minute (3-fold coincidence + trigger)

• Fidelity with target GHZ 84% from tomography

2nd method: Cascaded down-conversion

Bulk crystal (BBO)

PPKTP

Waveguide PPLN

~1 in a billion years

~1 in a hundred thousand years

~2 per month

~1 per day

~3 per hour

~1 per second

10-11

10-9

10-6

*assuming 106 s-1 primary photons, no loss, perfect detectors

Experimental cascaded down-conversion

4.7 ± 0.6 counts/hr

See also Shalm Nature Physics 9, 19 (2013);Hamel arxiv: 1404.7131

Two-mode squeezed vacuum

• Two mode squeezing operator

• Creates or destroys photons in pairs

• Properties

• Warning: can’t use Glauber’s theorem

Two-mode squeezed vacuum

• The interesting properties show up in the correlations between quadrature obs.

Two-mode squeezed vacuum

• The commutator,

• And so we have the same uncertainty relation between these joint observables as the quadratures themselves:

Two-mode squeezed vacuum

• We can calculate the uncertainty in these observables for the TMSV

• Recall

• To calculate this requires several applications of the squeeze operator identities, ex.,

Two-mode squeezed vacuum

• After some algebra

• Choosing

• We can “squeeze” the uncertainty in one observable at the expense of the other

Einstein Podolsky Rosen correlations

• If we consider a different pair of joint quadrature observables, ex.

• These operators commute (thus the uncertainty relation is trivial) and for the TMSV

Einstein Podolsky Rosen Correlations

• For infinite squeezing, the state is an eigenstate of both

• Highly entangled state central to:

Two-mode squeezed vacuum

• This state is the most entangled state for a given amount of energy (its subsystems are thermal states, which have the highest entropy for a fixed energy)

• As such it plays the role of the Bell states in CV protocols