在論文主要在研究電磁引發透明條件下光和物質交互作用的量子性質。此文中使有限數量的原子與兩個電磁波交互作用，第一個是很強的古典場，第二個是可用量子力學描述很弱的探測電磁場。一種臨界的耦合強度會在量子相變發生的時候被揭露。我們提供了它在某些特定狀態的應用，做為介觀數量原子的量子記憶體。我們用數值計算來證明透過古典場的絕熱控制，把量子態轉移到原子系統是可行的，且這個原子系統達到拉曼共振的條件時，會將已儲存的光子以高保真度取回量子場。此外，可證明不必使用無限數量的原子來把光子態儲存到原子系統，只要光子的希爾伯特空間的維度等於或小於原子的數量。為了證明記憶體可用在非平凡態，我們用二相式態做為例子：用在有限大小的光子態類相干的性質。

In this thesis we study the quantum properties of light-matter interaction under the electrically induced transparency configuration. Here, a finite number of atoms interact with two electromagnetic fields one assumed to be a strong field treated classically and the second one, a weak probe, which description is done by means of a quantum mechanics. A critical coupling strength is revealed at which quantum phase transition occurs. The application of this scheme, with a mesoscopic number of atoms, as a quantum memory is also provided for particular states. Numerical calculations are used to proof that by adiabatic control of the classical field it is possible to transfer the quantum state of the field into the atomic system, with the retrieval of this stored photons back to the field with a high fidelity if Raman resonance condition is achieved. Additionally, It is shown that an infinite number of atoms are not required to store a photonic state in the atomic system as long as the Hilbert space of the photons is of equal or smaller dimensions than the number of atoms. In order to justify the use of this memory on non trivial states binomial states are introduced as an example of finite-sized photonic states whit coherent-like properties.

Abstract vii
List of Figures xi
List of Tables xiii
1 Introduction 1
1.1 Quantum interaction of light and matter . . . . . . . . . . . . . . . . . 1
2 Binomial States 5
2.1 Introduction to Binomial States . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Properties of the binomial state . . . . . . . . . . . . . . . . . . 8
2.2 Propagation of two coupled fields . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Twomode
Binomial state . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Coherent states in twomode
propagation . . . . . . . . . . . . 13
2.2.3 Additional properties . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Quantum Phase Transitions on EIT media 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 Reduced Hilbert subspaces. . . . . . . . . . . . . . . . . . . . 25
3.2.3 First Quantum Phase Transition . . . . . . . . . . . . . . . . . 26
3.2.4 Quantum Phase transition in limiting parameters . . . . . . . . 29
3.3 Alternative configurations . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Dark State Polaritons in finite atomic ensembles 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Dark state polariton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Dark State Polariton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3.1 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 41
4.3.2 Time evolution of the ground state . . . . . . . . . . . . . . . . 43
4.4 AtomField
entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
ix
5 Projective Quantum State Tomography 47
5.1 Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.1 Quadrature Measurement . . . . . . . . . . . . . . . . . . . . . 49
5.1.2 Estimation of quadratures probabilities . . . . . . . . . . . . . 53
5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.1 results from experiments . . . . . . . . . . . . . . . . . . . . . 54
5.2.2 Applications of PQST beyond quantum mechanics . . . . . . . 55
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Conclusions 59
References 61
x

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