Lossless multimode OPO

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Table of contents

Introduction
I Context and Tools
1 Not an Introduction to Multimode Quantum Optics
1.1 Elements of quantum optics
1.1.1 The quantized electric eld
1.1.2 Quantum states of interests
1.1.3 The density matrix
1.1.4 The Wigner function
1.2 Multimode generalization
1.2.1 Optical modes
1.2.2 Multimode electric eld
1.2.3 Multimode quantum states
1.2.4 Multimode density matrix
1.2.5 Multimode Wigner function
1.3 Multimode Gaussian states
1.3.1 Single-mode Gaussian state
1.3.2 The covariance matrix
1.3.3 Multimode squeezed vacuum
2 A Source of Spectrally Multimode Quantum States
2.1 A crash course of ultrafast optics
2.1.1 Generation of ultrafast pulses
2.1.2 Pulse modeling
2.2 Our light source
2.3 Single-mode OPO in a ring cavity
2.3.1 Input/output relations
2.3.2 OPO threshold
2.3.3 Predicting squeezing
2.4 Measuring single-mode squeezing
2.4.1 A technical description of our OPO
2.4.2 Pump spectrum
2.4.3 Single-mode squeezing measurement
2.5 Lossless multimode OPO
2.5.1 Degenerate parametric down-conversion in a cavity
2.5.2 Parametric down-conversion in BiBO
2.5.3 Hamiltonian of the conversion
2.5.4 Eigenmodes of the conversion
2.5.5 Toward a full model of SPOPO
3 Tunable Projective Measurements
3.1 Homodyne Detection
3.1.1 Single-mode homodyne detection
3.1.2 Homodyne detection as a projective measurement
3.2 Ultrafast Pulse Shaping
3.2.1 Diraction-based amplitude shaping
3.2.2 Pulse-shaper design
3.2.3 Potential problems
3.2.4 Dual beam pulse-shaping
3.3 Measuring multimode squeezed vacuum
3.3.1 Principle
3.3.2 Measurement
3.3.3 The covariance matrices
3.3.4 Eigenvalues and eigenmodes
3.3.5 Going further in multipartite entanglement
II Single-Photon Subtraction
4 A Theoretical Framework for Multimode Single-Photon Subtraction
4.1 Modeling single-photon subtraction
4.1.1 Detecting a single photon
4.1.2 Single-mode case
4.1.3 General multimode case
4.1.4 Application to multimode squeezed vacuum
4.2 Photon subtraction from spectrally/temporally multimode light
4.2.1 Linear and non-linear photon subtraction
4.2.2 Time-resolved detection of a photon
4.2.3 Single-photon subtraction kernel
5 Single-Photon Subtraction via Parametric Up-Conversion
5.1 Theory of sum-frequency
5.1.1 Modes of the process
5.2 Collinear SFG in BiBO
5.2.1 Phase-matching
5.2.2 Choosing the crystal length
5.2.3 Playing with the gate
5.3 Non-collinear SFG in BiBO
5.3.1 Eect of focusing
5.3.2 Eect of birefringence
5.3.3 The problem of gate SHG
5.3.4 Eigenmodes of non-collinear subtraction
III Process Tomography
6 Tomography of the single-photon subtractor
6.1 Quantum process tomography
6.1.1 Tomography of a quantum black box
6.1.2 Checking the quantum process
6.2 Probing the subtraction matrix
6.2.1 The probing basis
6.2.2 Accessing the elements of the subtraction matrix
6.2.3 Experimental subtraction matrices
6.3 Maximum-Likelihood reconstruction
6.3.1 Principle of Maximum-Likelihood reconstruction
6.3.2 Optimizing with evolutionary strategies
6.3.3 Reconstruction in the pixel basis
6.4 Tomography in a natural basis
6.4.1 The measurement basis
6.4.2 Raw measurement data
6.4.3 Maximum-Likelihood reconstruction
Conclusion & outlooks
Appendices
A The SPOPO cavity
B Non-linear optics in BiBO
C The fantasy of type II parametric interaction for QPG
References