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## 物理代写|热力学代写thermodynamics代考|Generic Quantum Baths

In the strict quantum-mechanical approach, expounded in Chapters 1 and 2 , the system and its environment taken together form an isolated entity that is governed by a Hamiltonian and may evolve unitarily, depending on its initial state. The division of this entity into a system with a limited number of degrees of freedom (DOF) and its environment with a much larger number of DOF (infinite number, in the thermodynamic limit) is a matter of expediency: we choose a system that we are able to observe and control, as opposed to the unobserved and uncontrolled environment, either for lack of interest or due to its complexity or inaccessibility (or both).

It is convenient and plausible to view the environment as a collection of “reservoirs” (“baths”) that hardly change as a result of their interaction with the system, because such change is distributed over many, possibly infinitely many, DOF of the baths. The standard bath is a thermostat (i.e., a thermal ensemble with a constant temperature), as assumed in Chapter 2, but contemporary theory and experiment allow for a much more detailed characterization of bath quantum states and bath spectra, as detailed below. These properties are key to the dynamical control of quantum systems in contact with a bath.
In general, the quantum state of the bath is mixed because of the lack of knowledge concerning this state, although a bath may also be found in a pure state, particularly its ground state. All the baths discussed in this book are composed of identical quantum objects (particles or quasiparticles) that obey either bosonic or fermionic statistics and may collectively undergo elementary excitations with nontrivial spectra.

## 物理代写|热力学代写thermodynamics代考|Bosonic Bath Models

The Hamiltonian of a bosonic bath is a quadratic functional of a bosonic bath (field) operator $\boldsymbol{B}(\boldsymbol{x})$, that is, it ignores higher-order nonlinearities in $\boldsymbol{B}(\boldsymbol{x})$. Such higher-order nonlinearities may lead to diverse effects, which are not discussed here (e.g., phonon anharmonicity, corrections to magnon dispersion, or nonlinearoptics effects for photons).

We here assume that each normal mode in a bosonic bath is associated with a quantized harmonic oscillator. The resulting decomposition of a bosonic bath operator $\boldsymbol{B}(\boldsymbol{x})$ has the form
$$\boldsymbol{B}(\boldsymbol{x})=\sum_{\Lambda}\left[a_{\Lambda} \phi_{\Lambda}(\boldsymbol{x})+a_{\Lambda}^{\dagger} \boldsymbol{\phi}{\Lambda}^{*}(\boldsymbol{x})\right]$$ where $a{\Lambda}\left(a_{\Lambda}^{\dagger}\right)$ denotes the annihilation (creation) operator of elementary excitations in a mode $\Lambda$ described by wave function $\phi_{\Lambda}(\boldsymbol{x})$. The operators $a_{\Lambda}$ and $a_{\Lambda}^{\dagger}$ obey the boson commutation relations,
$$\left[a_{\Lambda}, a_{\Lambda^{\prime}}^{\dagger}\right]=\delta_{\Lambda \Lambda^{\prime}}, \quad\left[a_{\Lambda}, a_{\Lambda^{\prime}}\right]=\left[a_{\Lambda}^{\dagger}, a_{\Lambda^{\prime}}^{\dagger}\right]=0,$$
where $\delta_{\Lambda \Lambda^{\prime}}$ is the Kronecker symbol.
The choice of $\boldsymbol{B}(\boldsymbol{x})$ for a given bath is not unique. Thus, for an electromagnetic (photonic) bath, one may choose to work with the electric and magnetic field operators. However, it is often appropriate to adopt $\boldsymbol{B}(\boldsymbol{x})$ obtainable by quantizing the vector potential (Sec. 3.1.1) for reasons explained in Chapter 4 .

The same form of $\boldsymbol{B}(\boldsymbol{x})$ and its normal-mode decomposition apply to various elementary excitations: photons, phonons, polaritons, and magnons. The geometry and composition of the bath medium may dictate different choices of normal modes, as opposed to plane waves $\phi_{k}(x) \sim e^{i k \cdot x}$ that are appropriate for free space, where $k$ is the mode wave vector.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Bosonic Bath Models

$$\boldsymbol{B}(\boldsymbol{x})=\sum_{\Lambda}\left[a_{\Lambda} \phi_{\Lambda}(\boldsymbol{x})+a_{\Lambda}^{\dagger} \boldsymbol{\phi} \Lambda^{*}(\boldsymbol{x})\right]$$

$$\left[a_{\Lambda}, a_{\Lambda^{\prime}}^{\dagger}\right]=\delta_{\Lambda \Lambda^{\prime}}, \quad\left[a_{\Lambda}, a_{\Lambda^{\prime}}\right]=\left[a_{\Lambda}^{\dagger}, a_{\Lambda^{\prime}}^{\dagger}\right]=0,$$

## 有限元方法代写

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## MATLAB代写

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