Model for cavity quantum electrodynamics (cQED) with imperfect mirrors 

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Model for cavity quantum electrodynamics (cQED) with imperfect mirrors

The quantization procedure for the electromagnetic modes of a perfect cavity is, nowadays, well known. The quantization leads to a set of discrete normal modes with annihilation and creation operators assigned to each of them. The wave functions associated with those modes are the solutions of the one-dimensional Helmholtz equation, with zero boundary conditions at the positions of the mirrors: the result is a set of sine functions with different discrete frequencies !n = nc ` , where ` is the length of the one-dimensional cavity.
The conventional way to describe a cavity field must, however, take losses via imperfect (yet not absorbing) mirrors into account. Deriving models that are closer to real experiments, having imperfect mirrors has its own interest, since it allows to transfer photons from the cavity to propagating modes. In this work we will consider that one mirror is perfect while the other is not in order to identify a unique channel for leaking photons. The procedure usually adopted describes the damping of the radiation field in the cavity with a phenomenological system-reservoir approach [60,61], where the cavity system is described by a set of quantized harmonic oscillators associated to the discrete modes the cavity would have in the absence of damping. The reservoir is another set of quantized harmonic oscillators associated with the continuum of external free-space modes. It is often assumed that the coupling strength between cavity and outside modes is independent of the frequency, leading to a Markovian damping. The input-output relations, which relate in principle the outside field to the intracavity field, are also based on this property. However, it has been shown by Dutra and Nienhuis that this property is valid up to the first order in the transmission coefficient jtj, based on the derivation of the phenomenological Hamiltonian from first principles [62, 63].  It allows one to obtain an explicit expression for the coupling strength. They show that the phenomenological Hamiltonian is valid up to the first order transmission of the leaking mirror, i.e. it is well justified for high-Q cavities. We present this approach in this section, the approximations it involves and derive its consequences on the input-output formulation. We also show that it allows one to characterize the leaking photon by including the Poynting vector to this formulation.
In section 2.2, we will focus on another way of deriving effective models for cQED and show that it is consistent with the derivation of Dutra and Nienhuis, since it leads to the effective Hamiltonian up to the first order in the transmission jtj. This description will be used to link cQED and plasmonic QED, studied in the next part of the thesis. Indeed, the input-output description is valid for high-Q cavities, but in the case of plasmons the quality factor is low.
This requires stepping back to the global field description and building the effective model from this starting point. As we show the equivalence of the input-output and the global field approaches, this paves the way for cQED-like description of quantum plasmonics. The chapter is organized as follows:
In the first section we present the general field quantization, and its application to a one-dimensional cavity and spontaneous emission in a homogeneous medium. The global field operators are introduced, and we provide a description of the input-output equivalence, i.e. the decomposition of the global field into a perfect cavity field coupled to a set of continuous reservoir operators. We base the derivation of the photon flux and the master equation on the input-output approach.
The second section is an alternative derivation of the atom-cavity field model, based on the coupling between the atom and the global field operators. We show the equivalence of this derivation with the input-output model, which is useful as the plasmonic system will be studied under the structure of this alternative derivation.
The last section presents results on the control of photon wavepackets leaking from a cavity, with the use of single and multiple atoms coupled to a cavity field.

Quantization of the electromagnetic field

In this section we derive the quantization of the electromagnetic field in a linear, passive medium. The standard quantization procedure is slightly tricky because unlike atoms, the electromagnetic field is a continuous distribution of harmonic oscillators. A single harmonic oscillator is quantized by writing the canonical variables q; p and substituting them by operators bq; bp acting on a well-defined Hilbert space L2(R; dq). An infinite collection of harmonic oscillators would then bring a total Hilbert space as an infinite tensor product of L2(R; dq), and since d1q is not a well-defined Lebesgue measure, the total space would not be defined either [64]. The difficulty with the definition of the Hilbert space is avoided by reformulating the theory in a well-defined Fock space, which is isomorphic to L2(RN; dNq) for models with finite degrees of freedom N. Another difficulty arises due to the electromagnetic field transversality constraint r A = 0 (in a homogeneous medium), leading to the canonical variables A; not being independent. The redundancy is removed by making a canonical change of variables such that in the new coordinate system, the constraint is automatically satisfied.

Three-level atoms in a cavity

The production of quantum light can be achieved with the use of attenuated laser sources. With low intensity lasers, one can design non-classical light beams, and nowadays such systems are being commercialized for quantum key distribution (QKD), using the horizontal and vertical polarization of anti-bunched photons as flying qubits [8]. With such non-classical sources, portable devices have been developed. The production of controlled single or Nphoton states allows to generate entanglement and interference between them, hence quantum information and high security protocols for QKD.
Experiments with single quantum emitters, such as atoms, placed in an optical cavity showed anti-bunched and indistinguishable single photon signals [69]. Hence, we require models for understanding the interaction between single or few atoms with a quantized cavity field. Moreover, three-level atoms trapped in a cavity can be controlled by laser pulses (in the transverse direction with respect to the cavity axis), enabling the control of the single photon time envelope [70–73].

Heisenberg treatment of spontaneous emission

Having derived the effective Heisenberg operator for the transmission of the field through the semi-transparent mirror, we may now derive, at z = 0, the integrated field operator corresponding to the transverse three-dimensional reservoir: bb F(t) = X Z d3k g k; bb k;(t): (2.1.72).
This operator models the dynamics of the spontaneous emission of the transition jei $ jfi, corresponding to the emission of photons outside of the cavity mode. We formally integrate equation (2.1.52b) to get: bb k;(t) = e􀀀i!k(t􀀀t0)bbk;(t0) + gk;pN Z t t0 dt0b(t0)e􀀀i!k(t􀀀t0􀀀t0).

An alternative derivation of the cQED effective model

Starting from the general quantization procedure developed in section 2.1.1, we derive an alternative form of the effective model corresponding to Hamiltonian (2.1.33). In the preceding sections, we quantized the global field and derived effective perfect cavity field operators bcn; bc y n coupled to a flat reservoir whose excitations are toggled bybb(!);bby(!), and lastly we described the interaction of the inside cavity field with atoms. The field in and out of the cavity must match at the boundaries, and it does in the high-Q limit when we consider first order series expansion in jtj2. The alternative derivation we present here is different in the order of the procedure steps: we start with the global field operators ba(!); bay(!) corresponding to the classical mode (x; !) given by equation (2.1.29), and describe its interaction with a single atom when it is placed inside of the cavity.

Table of contents :

I Introduction 
II Quantum control with trapped ions and optical cavities 
1 Trapped ions controlled by lasers and quantum information
1.1 Quantum Information in the adiabatic limit
1.1.1 Basic notions of quantum computing
1.1.2 Adiabatic theorem
1.1.3 Stimulated Raman adiabatic passage (STIRAP)
1.2 Ion trap as a model for quantum information
1.2.1 Linear Paul trap – trapping of a single ion
1.2.2 Quantization of the vibrational modes
1.2.3 Manipulating ions by laser – Lamb-Dicke regime
1.3 Building arbitrary gates by adiabatic passage
1.3.1 Householder reflections by adiabatic passage
1.3.2 Quantum Fourier transform on a quartit and energy study
1.A Derivation of the Householder reflection for a qubit
2 Quantum optics with atoms in cavities
2.1 Model for cavity quantum electrodynamics (cQED) with imperfect mirrors
2.1.1 Quantization of the electromagnetic field
2.1.2 The one-dimensional cavity field
2.1.3 Three-level atoms in a cavity
2.1.4 Cavity input-output relation and photon flux
2.1.5 Heisenberg treatment of spontaneous emission
2.1.6 Master equation
2.2 An alternative derivation of the cQED effective model
2.2.1 Atom-field interaction
2.2.2 Mode-selective quantum dynamics and effective Hamiltonian
2.3 Production of photon states with atoms in a cavity
2.3.1 Single photons with one atom in a cavity
2.3.2 Single and two-photon states with two atoms in a cavity
2.3.3 Characterization of the outgoing two-photon state
2.A Canonical quantization of the electromagnetic field in a dielectric medium
2.B Lorentzian structure of the cavity spectral response function
2.C Complex plane integration of the atom-field coupling
2.D Numerical solution of _~X (t) = M(t)~X (t) + ~Y (t)
III Quantum control of emitters coupled to plasmons 91
3 Mode-selective quantization procedure in a spherically layered medium
3.1 Light-emitter interactions and quantum plasmonics
3.1.1 Nano-optics and plasmonics
3.1.2 Nano-emitters near plasmonic structures
3.1.3 Localized plasmons and nanoparticles
3.2 Mode expansion in a spherically layered medium
3.2.1 Spherical vector harmonics and orthogonality relations
3.2.2 Green’s tensor expansion
3.3 Mode-selective quantization
3.3.1 Field quantization
3.3.2 Addressing harmonic excitations
3.3.3 Spherical mode-structured field and quantum emitters
3.A Green’s tensor in a spherically layered medium
4 Effective models for quantum plasmonics
4.1 Continuous model with multiple emitters
4.1.1 Single emitter – dark and bright operators
4.1.2 Multiple two-level emitters
4.2 Discrete model
4.2.1 Single emitter
4.2.2 Multiple emitters
4.3 Application: two emitters and a single metallic nanoparticle
4.3.1 Continuous model
4.3.2 Discrete effective model
4.A Single Lorentzian model – continuous and discrete Hamiltonian
4.A.1 Discretization
4.A.2 Dynamics
5 Quantum plasmonics with metallic nanoparticles 132
5.1 Quantum emitter coupled to a metallic nanoparticle
5.1.1 Local density of states
5.1.2 Dynamics and strong coupling regime
5.2 Adiabatic passage mediated by plasmons
5.2.1 Population transfer: STIRAP
5.2.2 Entanglement: fractional STIRAP
5.2.3 Simplified model and discussion
5.2.4 General model and perspectives
IV Conclusion

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