Czesław RadzewiczWarsaw University

Poland

Konrad BanaszekNicolaus Copernicus University Toruń, Poland

Alex LvovskyUniversity of Calgary

Alberta, Canada

Squeezing eigenmodesin parametric down-conversion

Squeezing eigenmodesin parametric down-conversion

National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland

Wojciech Wasilewski

AgendaAgenda

• Classical description• Input-output relations• Bloch-Messiah reduction• Single-pair generation limit• High-gain regime• Optimizing homodyne detection

Fiber optical parametric amplifier

Fiber optical parametric amplifier

c(2)tp

• Pump remains undepleted

• Pump does not fluctuate

Linear propagationLinear propagation

Highorder effects

Group velocity

dispersion

Group velocity

Phase velocity

Three wave mixingThree wave mixing

kp, p

p =+ ’k,

k’, ’

Classical optical parametric amplifier

Classical optical parametric amplifier

[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band,Opt. Comm. 221, 337 (2003)]

c(2)

Linear propagation

3WMInteraction

strength

Input-output relationsInput-output relations

Quantization: etc.

DecompositionDecomposition

As the commutation relations for the output field operators must be preserved, the two integral kernels can be decomposed using the Bloch-Messiah theorem:

S. L. Braunstein,Phys. Rev. A 71, 055801 (2005).

The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields:

Squeezing modesSqueezing modes

The characteristic eigenmodes evolve according to:

• describe modes that are described by pure squeezed states

• tell us what modes need to be seeded to retain purity

a(0) a(z)

.... ....

G1G2G3G4U V

bin bout

.... ....

a(0) a(z)

Squeezing modesSqueezing modes

a(0) a(z)

.... ....

G1G2G3G4U V

bin bout

.... ....

a(0) a(z)

The operation of an OPA is completely characterized by:• the mode functions nand n• the squeezing parameters n

Single pair generation regimeSingle pair generation regime

kp, p

p = + ’

L

k,

k’, ’Amplitude S sin(k L/2)/k

k = kp-k-k’

Single pair generation regimeSingle pair generation regime

’

pAmplitude S Pump x sin(k L/2)/k

Single pair generationSingle pair generation

’

p

S(,’)=ei… ,’|out

=Σ j fj()gj(’)

Gaussian approximation of SGaussian approximation of S

2

1

k=0

1+2=p

“Classic” approach“Classic” approach

Schmidt decomposition for a symmetric two-photon wave function:

C. K. Law, I. A. Walmsley, and J. H. Eberly,Phys. Rev. Lett. 84, 5304 (2000)

We can now define eigenmodes which yields:

The spectral amplitudes characterize pure squeezing modes

The wave function up to the two-photon term:

W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997);T. E. Keller and M. H. Rubin, Phys. Rev A 56, 1534 (1997)

Intense generation regimeIntense generation regime

• 1 mm waveguide in BBO

• 24 fs pump @ 400nm

Squeezing parametersSqueezing parameters

RMS quadrature squeezing: e-2

Spectral intensity of eigenmodes

Spectral intensity of eigenmodes

Input and ouput modesInput and ouput modes

| =| | |0 02 2

arg 0

arg 0

First mode vs. pump intensityFirst mode vs. pump intensity

| |02

arg 0

L =100mmNL

L =1/ 15mmNL

Homodyne detectionHomodyne detection

LO

Noise budgetNoise budget

Detected squeezing vs. LO duration

Detected squeezing vs. LO duration

1/LNL=1

2

3

4

ts

Contribution of various modesContribution of various modes

Mn

n

15fs

tLO

30fs

50fs

Optimal LOsOptimal LOs

345

Optimizing homodyne detectionOptimizing homodyne detection

–

SHG

PDC

ConclusionsConclusions

• The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields

• For low pump powers, usually a large number of modes becomes squeezed with similar squeezing parameters

• Any superposition of these modes (with right phases!) will exhibit squeezing

• The shape of the modes changes with the increasing pump intensity!

• In the strong squeezing regime, carefully tailored local oscillator pulses are needed.

• Experiments with multiple beams (e.g. generation of twin beams): fields must match mode-wise.

• Similar treatment applies also to Raman scattering in atomic vapor WW, A. I. Lvovsky, K. Banaszek, C. Radzewicz, quant-ph/0512215

A. I. Lvovsky, WW, K. Banaszek, quant-ph/0601170WW, M.G. Raymer, quant-ph/0512157