Abstract
This chapter presents the essential theoretical background necessary to explain some of the most important concepts discussed throughout this thesis. The aim is to provide the reader with the basic tools to understand the many fundamental equations and approximations used in the contexts of cavity quantum electrodynamics (CQED) and quantum chemistry. We start by addressing the question of what is the quantum Hamiltonian for the light–matter interaction and illustrating what approximations play an important role in its definition. We then focus on the matter part of the light–matter Hamiltonian in order to provide the best possible description of a complex molecule. In this section we address the Born–Oppenheimer approximation, widely used in molecular and solid-state physics and in quantum chemistry. Additionally, we present the description of different characteristic phenomena of organic molecules such as chemical structure and reactions, and their response to the electromagnetic field. Then, we focus on this last part, discussing the important features of CQED and the different theoretical descriptions that study them. Finally we present the fundamentals of the two different regimes of light–matter interaction: weak and strong coupling.
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Notes
- 1.
This expression is valid both in the presence and absence of sources. It is worth mentioning that in the case of the free field it is possible to write each term as \(A_{\mathbf {k},\lambda }(\mathbf {r},t) = A_{\mathbf {k},\lambda }(\mathbf {r})e^{-i\omega (\mathbf {k})t}\), where the \(A_{\mathbf {k},\lambda }(\mathbf {r})\) satisfy the homogeneous Helmholtz equation \((\varvec{\nabla }^2 + k^2)A_{\mathbf {k},\lambda }(\mathbf {r})=0\). In this case the explicit time dependence of \(\mathcal {H}_\mathrm {EM}\) disappears, as the energy is conserved.
- 2.
We see that in the vacuum state of the system the energy is \(E_\mathrm {0} = \frac{1}{2}\sum _{\mathbf {k},\lambda } \hbar \omega (\mathbf {k})\). The frequencies \(\omega (\mathbf {k})\) have no upper bound, so \(E_\mathrm {0}\) diverges. However, this is not a problem since expectation values only depend on energy differences and not absolute energies, so the divergence of the vacuum state does not appear in any physical observable.
- 3.
The more fundamental definition for the polarization field is \(\hat{\mathbf {P}}(\mathbf {r}) = \sum _{i,j} q_i^{(j)}(\mathbf {r}_i-\mathbf {r}_j) \int _0^1 ds \delta ^3 \left[ \mathbf {r} - \mathbf {r}_j -s(\mathbf {r}_i-\mathbf {r}_j) \right] \). This is more cumbersome and less intuitive, so we instead present \(\hat{\mathbf {P}}(\mathbf {r})\) directly as a multipole expansion. The connection between each expression can be found in [2].
- 4.
This approximation is completely equivalent to the long-wavelength approximation, in which the charges conforming each dipole are very close compared to the EM wavelength and thus experience the same fields, i.e. for \(\varvec{\mu } = q_i\mathbf {r}_i + q_j\mathbf {r}_j\) the fields satisfy \(\mathbf {A}(\mathbf {r}_i) \approx \mathbf {A}(\mathbf {r}_j)\). This is analogous to neglect the effects of higher electric multipoles, as they are more significant as the spatial structure of the collection of charges becomes important.
- 5.
Note that, depending on the context, we use the indices i and j to represent either individual charged particles or molecules.
- 6.
This effect emerges because the speed of light is finite. When the information of a particular charge configuration reaches an emitter situated far away, these charges have already rearranged, so that the emitter response is no longer in phase. This arises for distances much larger than the wavelength corresponding to the characteristic absorption frequency of the emitters [20]. In the systems that we are concerned with in this thesis, these wavelengths are of the order of hundreds of nanometers, so we can completely disregard retardation effects when dealing with dipole–dipole interactions.
- 7.
Note that while in one dimension the transition state is a local maximum, in general multi-dimensional scenarios this is a saddle point.
- 8.
In Sect. 2.3 we analyze the relevance of this term when not all EM modes are explicitly considered.
- 9.
In this thesis we are mainly interested in molecules, but this occurs for any set of charges such as atoms, quantum dots, nanoparticles, etc.
- 10.
The Franck–Condon principle states that the intensity of a vibronic transition (i.e., a change of electronic and vibrational states) is directly proportional to the overlap between the corresponding nuclear wavefunctions. This is based on the assumption of a vertical transition, i.e., that there is some adiabatic separation between nuclear and electronic timescales, similarly as in the Born–Oppenheimer approximation.
- 11.
We note that theoretical efforts in this regard have been made for small plasmonic nanoparticles using time-dependent density functional theory [38].
- 12.
This also includes the emitter–emitter interactions in the multiple-emitter case.
- 13.
The exact diagonalization of Eq. (2.63) can be done without invoking the RWA. However, for the sake of brevity we present the diagonalization within the single-excitation subspace.
- 14.
Note that this also depends on the choice of quantum emitter, as very large Q factors are not useful if the emitter has a broader linewidth than the cavity.
- 15.
This is equivalent to treating the many-particle TC Hamiltonian with collective operators, thus representing the coupling between the bright state and the cavity mode.
- 16.
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Galego Pascual, J. (2020). Theoretical Background. In: Polaritonic Chemistry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-48698-3_2
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