# Squeezing Creation and Interaction with Environment

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## Non Linear Cavity

We consider a non linear crystal with an index n in a cavity like in Figure 4.5. We suppose some light at frequency 2! in the cavity, which causes some non linear gain (see chapter on parametric amplification subsection 2.3.3) to generate some light at the frequency ! in the crystal. We suppose the light at frequency ! at point A to be given by: E0e−iwt−ikz with E0 real. (The imaginary part would be de-amplified until it reaches zero after a few round trips in the cavity, so we can consider the field to be real there). We simplify the calculation considering the crystal to be touching mirror 2. Using the equation Eq. 2.20 twice, and multiplying by the reflectivity of mirror 2, we get the field in point C. Supposing that there is no light at frequency ! coming from outside, then a simple propagation of the field to the point A gives the equation: E0 = E0e2 lr1r2e2k(d+nl).

The operators ˆa and ˆa† are not Hermitian operators, so they do not correspond to any measurable quantities. It is some times useful to define two new operators X1 = ˆa + ˆa† and X2 = ˆa−ˆa† i called quadratures, which are twice the real part (amplitude quadrature) and imaginary part (phase quadrature) of the operator ˆa. These operators are Hermitian and can be measured with a homodyne detector. In function of the quadratures, the electric field operator, expressed in Heisenberg representation, becomes: ˆE (r, t) = X l El0 (X1(t) cos(kl · r) + X2(t) sin(kl · r)) l.

#### Different Light States and Wigner Function

In this thesis we use three different states of light: the Fock states; the coherent states; and the squeezed states. One way to characterize them is by using the Wigner Function. In classical mechanics, it is possible to define a probability for the state to be in any point of a phase diagram. But in quantum mechanics, the Heisenberg uncertainty makes the notion of defined points in the phase diagram not anymore correct . It is not possible to define at the same time the two quadratures X1 and X2. However, it is still possible to define a quasi-density of probability, which is the Wigner function. It is defined by: W(X1,X2) = 1 2 Z 1  1 dqhX1 − q||X1 + qieiX2q.

Single pass squeezing generation in a non-linear crystal

One way to create squeezing state of light is to use non linear crystals. Classically the energy of interaction between the field and matter is given by D.E. In a non linear crystal, it can be seen in Appendix A that the energy of interaction due to the non linear polarization is given by: Up2 = « 0 2 E(2!) 2(2)(!, !)E(!)2 + cc with E(2!) and E(!) the fields at frequency 2! and !, and 2(2)(!, !) the effective non linear susceptibility of the material at frequency !. The Hamiltonian corresponding to this classical energy is given by: ˆH = gˆa† 2!ˆa†2 ! + gˆa2!ˆa2.

Squeezing Interaction in a Lossy Channel

The squeezed state as defined in the previous section subsection 5.3.3 minimizes the Heisenberg inequality X1X2 = 1. But when losses are introduced into the system, this is no longer the case and we obtain a mixed state X1X2 > 1. As long as min(X) < 1, we still call this mixed state a squeezed state. It is possible to model losses in the system by supposing a pure squeezed state like that described in subsection 5.3.3 going into a beam splitter, mixing the state with a vacuum state and thus changing the amount of squeezing S = min(X) and anti squeezing A = max(X) that we measure (Figure 5.6). We define ˆXs and ˆXa the quadratures of the pure state in the direction of squeezing and anti-squeezing, and ˆX 0s and ˆX 0a the quadratures after the beam splitter of reflectivity : ˆX 0s = p ˆXs + p1 − ˆ.

Acknowledgments
Abstract
I. Theoretical Background
1. Introduction
1.1. The Maxwell Equations and the Wave Equations
1.2. Energy Considerations
1.3. Fourier Transforms
1.4. Dielectric Medium and the Wave Equation
1.4.1. Properties of Susceptibilities
2. Light Propagation
2.1. Linear Homogeneous Isotropic (LHI) Behavior
2.1.1. Gaussian Propagation Solution
2.1.2. High Order Propagation Mode Solutions
2.1.3. Astigmatism
2.1.4. ABCD Matrix
2.2. Linear Anisotropic Medium
2.2.1. Ordinary Beam
2.2.2. Extraordinary Beam
2.2.3. Dispersion Angle
2.3. Non Linear Medium
2.3.1. Propagation Equation
2.3.2. Second Harmonic Generation
2.3.3. Degenerate Parametric Amplification
3. Interface Conditions 25
3.1. Field Interface Conditions
3.1.1. Normal Fields
3.1.2. Tangential Fields
3.1.3. Poynting Vector
3.2. Fresnel Equations
3.2.1. TE Polarization or S-Polarization
3.2.2. TM Polarization or P-Polarization
4. Cavity
4.1. Beam Splitter Conventions
4.2. Cavity Transmission
4.3. Stability
4.4. Non Linear Cavity
5. Quantum Optics
5.1. Quantization of the Field
5.2.2. Optics Components
5.2.3. Homodyne
5.3. Different Light States and Wigner Function
5.3.1. Fock States
5.3.2. Coherent States
5.3.3. Squeezed States
5.4. Squeezing Creation and Interaction with Environment
5.4.1. Single pass squeezing generation in a non-linear crystal
5.4.2. Squeezing Interaction in a Lossy Channel
5.4.3. Squeezing generation in a non linear crystal in a cavity
II. Fibered Mini OPO
6. Introduction
6.1. Introduction
6.2. Squeezing Generation
6.3. Toward an all-fibered squeezer
7. Experimental Method
7.1. The OPO Cavity, Description of the Experiment
7.1.1. Coupling Mirror
7.1.2. Crystal and Crystal Mount
7.1.3. Fiber and Fiber Mount
7.1.4. Base
7.2. Other experimental consideration
7.2.1. Laser Source
7.2.2. Fibered Elements
7.2.3. Crystal Temperature Control
7.2.4. High Voltages Amplifiers:
7.2.5. Homodyne Detectors
7.2.6. Optical Suspension Table
7.3. Alignment of the Cavity
7.3.1. Schematic of the Set Up
7.3.2. Crystal Alignment with White Light Interferometry
7.3.3. Temperature Tuning
7.3.4. Homodyne Alignment:
7.3.5. Curved Mirror:
7.3.6. Alignment of the Green and Red:
7.4. System Limitations
7.4.1. Curvature Matching
7.4.2. Asphericity of the phase surfaces
7.4.3. Grey Tracking and Damaging:
7.5. Locking the System
7.5.1. PDH locking
7.5.2. Self Locking
7.5.3. Locking with a Micro-Controller
8. Results
8.1. Second Harmonic Generation and Amplification De-Amplification .
8.2. Squeezing
9. Conclusion
III. Square Monolithic Resonator
10.Introduction:
11.Resonator Coupling
11.1. Evanescent Prism Coupling
11.1.1. S-Polarization
11.1.2. P-Polarization
11.2. Resonator
11.3. Phase Control
11.4. TEM Modes
11.5. ABCD Matrix Considerations and Stability of the resonator
12.Experimental Methods
12.1. Creation of Resonators
12.1.1. Polishing
12.2. Prisms
12.3. Mechanical System
12.4. High Voltage
12.5. Alignment
12.5.1. How to Align Prisms and Resonators
12.5.2. How to Align the Beam with Contra-Propagating Beams
12.5.3. Temperature Tuning of the Crystal
12.5.4. Homodyne Alignment
12.5.5. Prism Switching and alignment for Squeezing
12.6. Lock and Self Locking
13.Results
13.1. Non Linearity
13.1.1. SHG and De-amplification
13.1.2. Squeezing
13.1.3. Conclusion
A. Fourier Transform of the Polarisation Field versus the Susceptibility
B. Index calculation for the green calcite coupler in p-polarization

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