Czesław RadzewiczWarsaw University

Poland

Konrad BanaszekNicolaus Copernicus University Toruń, Poland

Alex LvovskyUniversity of Calgary

Alberta, Canada

Squeezing eigenmodesin parametric down-conversion

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

Wojciech Wasilewski

Agenda

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

Fiber optical parametric amplifier

c(2)tp

• Pump remains undepleted• Pump does not fluctuate

Linear propagation

Highorder effects

Group velocity

dispersion

Group velocity

Phase velocity

Three wave mixing

kp, p

p =+ ’k,

k’, ’

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 relations

Quantization: etc.

Decomposition

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 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 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 regime

kp, p

p = + ’

L

k,

k’, ’Amplitude S sin(k L/2)/k

k = kp-k-k’

Single pair generation regime

’

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

Single pair generation

’

p

S(,’)=ei… ,’|out

=Σ j fj()gj(’)

Gaussian approximation of S

2

1

k=0

1+2=p

“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 regime

• 1 mm waveguide in BBO• 24 fs pump @ 400nm

Squeezing parameters

RMS quadrature squeezing: e-2

Spectral intensity of eigenmodes

Input and ouput modes

| =| | |0 02 2

arg 0

arg 0

First mode vs. pump intensity

| |02

arg 0

L =100mmNL

L =1/ 15mmNL

Homodyne detection

LO

Noise budget

Detected squeezing vs. LO duration

1/LNL=1

2

3

4

ts

Contribution of various modes

Mn

n

15fstLO

30fs50fs

Optimal LOs

345

Optimizing homodyne detection

–

SHG

PDC

Conclusions• 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