3.1.3 Rotating wave approximation and Lindblad form

We assume that the interaction has the form

$$H\_{\rm int} = \sum\_{a} A\_{a} \otimes \mathcal{B}\_{a}. \tag{40}$$

already been used when in the Markov approximation ρ<sup>S</sup> t

eigenstates <sup>∣</sup>ϕn<sup>i</sup> of <sup>H</sup>S. We introduce the eigen-energies Es

<sup>n</sup>∣ϕni, and with <sup>Ð</sup> <sup>∞</sup>

ð<sup>∞</sup> �∞

¼ 2πℏ ∑ nm

�Að Þ ω : Now, we find for the influence term

1 ℏ2

ð Þ int

We can perform the integral over t

ð<sup>∞</sup> 0 dτe

> ð<sup>∞</sup> �∞

<sup>S</sup> <sup>ð</sup>t; <sup>t</sup> � <sup>t</sup>0ÞA†

functions of bath variables are related to Γð Þ ω .

ð Þ int

ð Þ int

dω 2π ð<sup>∞</sup> �∞

(the response function of the bath)

Γð Þ¼ ω

bath observables) we have <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>Ae�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

ðt

�∞ dt 0 ð<sup>∞</sup> �∞

Að Þ¼ ω

changes of the observables.

Nonequilibrium Statistical Operator

DOI: http://dx.doi.org/10.5772/intechopen.84707

<sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ¼� <sup>t</sup> � <sup>t</sup><sup>0</sup>

� B t h i ð Þ <sup>0</sup> � t B <sup>B</sup>½Að Þ ω ; ρ

exp �iH<sup>B</sup> t ½ � ð Þ <sup>0</sup> � t =ℏ .

short notation

<sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ¼� <sup>t</sup> � <sup>t</sup><sup>0</sup>

� <sup>Γ</sup>2ð Þ <sup>ω</sup>″ <sup>A</sup>ð Þ <sup>ω</sup> ; <sup>ρ</sup>

The expression ρ

15

to HS∣ϕni ¼ Es

This corresponds to the Heisenberg picture where the state of the system does not change with time. The time dependence of averages is attributed to the temporal

�<sup>∞</sup> exp ½ � ikx dx <sup>¼</sup> <sup>2</sup>πδð Þ<sup>k</sup> ,

<sup>∣</sup>ϕn〉 <sup>ϕ</sup><sup>n</sup> h i <sup>j</sup>Ajϕ<sup>m</sup> 〈ϕm∣<sup>δ</sup> Es

dω 2π ð<sup>∞</sup> �∞

with the time-dependent bath operators B tð Þ¼ <sup>0</sup> � t exp iH<sup>B</sup> t ½ � ð Þ <sup>0</sup> � t =ℏ B

<sup>i</sup>ð Þ <sup>ω</sup>þi<sup>ϵ</sup> <sup>τ</sup>=<sup>ℏ</sup>TrB ρBB†

dω″ <sup>2</sup><sup>π</sup> <sup>e</sup>

n o h i

ð Þ <sup>ω</sup>″ h i <sup>þ</sup> <sup>Γ</sup>1ð Þ <sup>ω</sup>″ <sup>A</sup>†ð Þ <sup>ω</sup>″ <sup>ρ</sup>

enters via equilibrium auto-correlation functions of the time-dependent bath operators Bαð Þτ . We introduce the Laplace transform of the bath correlation function

that is a matrix Γαβð Þ ω if the observable B has several components. We find in

<sup>i</sup>ð Þ <sup>ω</sup>″�<sup>ω</sup> ð Þ <sup>t</sup>�t<sup>0</sup>

after the transformation <sup>ω</sup><sup>0</sup> ! �ω″ and using Eq. (45). Note that this expression for the influence term is real because the second contribution is the Hermitean conjugated of the first contribution. Using symmetry properties, all correlation

time t because in the Heisenberg picture (we consider the lowest order of interaction) the state of the system does not depend on time. Oscillations with e<sup>i</sup>ð Þ <sup>ω</sup>�ω″ ð Þ <sup>t</sup>�t<sup>0</sup> occur that vanish for <sup>ω</sup>″ <sup>¼</sup> <sup>ω</sup>. The rotating wave approximation (RWA) takes into

<sup>S</sup> ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þ<sup>t</sup> <sup>e</sup>�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> is not depending on

ð Þ<sup>τ</sup> <sup>B</sup> � � <sup>¼</sup> <sup>1</sup>

2

ð Þ int

(the index α in (40) is dropped). In interaction picture (A commutes with the

dω<sup>0</sup> <sup>2</sup><sup>π</sup> <sup>e</sup><sup>ε</sup> <sup>t</sup>ð Þ <sup>0</sup>

<sup>S</sup> ðt; t � t0ÞA ω<sup>0</sup> ð Þ� þ BB t h i ð Þ <sup>0</sup> � t <sup>B</sup>½A ω<sup>0</sup> ð Þρ

n o

<sup>d</sup>teiωð Þ <sup>t</sup>�t<sup>0</sup> <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup>ℏAe�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> <sup>¼</sup> <sup>A</sup>†

<sup>n</sup> � Es <sup>m</sup> <sup>þ</sup> <sup>ℏ</sup><sup>ω</sup> � �

�<sup>t</sup> e�iω<sup>0</sup> <sup>t</sup>ð Þ <sup>0</sup>

To include the interaction, we characterize the dynamics of the system observable A introducing the spectral decomposition with respect to the (discrete)

<sup>0</sup> ð Þ is replaced by ρSð Þt .

<sup>n</sup> of the system S according

(45)

ð Þ �ω

�<sup>∞</sup> <sup>d</sup>ω=ð Þ <sup>2</sup><sup>π</sup> exp ½ � �iωð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup>

�<sup>t</sup> e�<sup>i</sup> <sup>ω</sup>þω<sup>0</sup> ð Þð Þ <sup>t</sup>�t<sup>0</sup>

ð Þ int

<sup>0</sup> that concerns the bath observables. The bath

γ ωð Þþ i

1

<sup>S</sup> ðt; t � t0Þ; Að Þ ω

<sup>S</sup> ðt; t � t0Þ; Að Þ� ω

<sup>ℏ</sup> <sup>S</sup>ð Þ <sup>ω</sup> (47)

(46)

(48)

We use the interaction picture that coincides at t<sup>0</sup> with the Schrödinger picture,

$$\mathbf{O}^{(\text{int})}(t - t\_0) = e^{i(H\_\text{S} + H\_\text{B})(t - t\_0)/\hbar} \mathbf{O} e^{-i(H\_\text{S} + H\_\text{B})(t - t\_0)/\hbar} \tag{41}$$

for any operator O. In particular, we denote

$$\begin{aligned} \mathcal{D}^{(\text{int})} [\rho\_{\text{S}}(t)](t - t\_{0}) &= e^{i(H\_{\text{S}} + H\_{\text{B}})(t - t\_{0})/\hbar} \mathcal{D}[\rho\_{\text{S}}(t)] e^{-i(H\_{\text{S}} + H\_{\text{B}})(t - t\_{0})/\hbar}, \\ \rho\_{\text{S}}^{(\text{int})} (t; t - t\_{0}) &= e^{iH\_{\text{S}}(t - t\_{0})/\hbar} \rho\_{\text{S}}(t) e^{-iH\_{\text{S}}(t - t\_{0})/\hbar} \end{aligned} \tag{42}$$

(note that H<sup>B</sup> commutes with ρSð Þt which is defined in the Hilbert space HS). Then, the dynamical evolution of the system is given by

$$\frac{\partial}{\partial t}\rho\_{\rm S}^{(\rm int)}(t;t-t\_{0})=\mathcal{D}^{(\rm int)}[\rho\_{\rm S}(t)](t-t\_{0}).\tag{43}$$

On the left-hand side, we cancel H<sup>B</sup> because it commutes with the system variables. The right-hand side, the influence term, has the form (note that ρ<sup>B</sup> commutes with HB)

$$D^{(\rm int)}[\rho\_{\rm S}(t)](t - t\_0) = -\frac{1}{\hbar^2} \int\_{-\infty}^{t} \mathbf{d}t' e^{-s(t - t')} \text{Tr}\_{\rm B} \left[ H\_{\rm int}^{(\rm int)}(t - t\_0), \left[ H\_{\rm int}^{(\rm int)}(t' - t\_0), \rho\_{\rm S}^{(\rm int)}(t; t - t\_0) \right] \right] \rho\_{\rm B} \tag{44}$$

In zeroth order of interaction, ρ ð Þ int <sup>S</sup> ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þ<sup>t</sup> <sup>e</sup>�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> is not depending on t because the derivative with respect to t vanishes. This fact has

Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

ΔρðÞ¼ t

<sup>H</sup>intð Þ¼ <sup>τ</sup> <sup>e</sup>� <sup>1</sup>

ρsðÞ�t

1 iℏ

higher orders of Hint see [4, 5].

∂ ∂t ðt

Non-Equilibrium Particle Dynamics

dt 0 e �ε t�t <sup>0</sup> ð Þe 1 <sup>i</sup> <sup>ℏ</sup> t�t

<sup>i</sup> <sup>ℏ</sup>τð Þ HsþH<sup>B</sup> Hinte

Hs; ρ<sup>s</sup> ½ �¼� ð Þt

We can use the unperturbed dynamics to replace ρ<sup>s</sup> t

1 ℏ2 ð 0

�∞

dτ e

Hint ¼ ∑ α

1

3.1.3 Rotating wave approximation and Lindblad form

We assume that the interaction has the form

<sup>O</sup>ð Þ int ð Þ¼ <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>e</sup>

for any operator O. In particular, we denote

<sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ¼ <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>e</sup>

<sup>S</sup> ð Þ¼ t; t � t<sup>0</sup> e

∂ ∂t ρ ð Þ int

1 ℏ2 ðt

In zeroth order of interaction, ρ

�∞ dt 0 e �ε t�t

ρ ð Þ int

with HB)

14

<sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ¼� <sup>t</sup> � <sup>t</sup><sup>0</sup>

<sup>0</sup> ð Þð Þ HsþH<sup>B</sup> 1

<sup>i</sup><sup>ℏ</sup> <sup>H</sup>int; <sup>ρ</sup><sup>s</sup> <sup>t</sup> <sup>0</sup> ð Þρ<sup>B</sup> ½ �e

Inserting the solution (38) into the equation of motion of ρsð Þt (34), a closed equation of evolution is obtained eliminating ρð Þt . In the lowest (second) order with respect to the interaction considered here, memory effects are neglected.

This result is described as quantum master equation in Born approximation. For

We use the interaction picture that coincides at t<sup>0</sup> with the Schrödinger picture,

i Hð Þ <sup>S</sup>þH<sup>B</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup><sup>D</sup> <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> <sup>e</sup>

(note that H<sup>B</sup> commutes with ρSð Þt which is defined in the Hilbert space HS).

On the left-hand side, we cancel H<sup>B</sup> because it commutes with the system variables. The right-hand side, the influence term, has the form (note that ρ<sup>B</sup> commutes

<sup>0</sup> ð ÞTrB Hð Þ int

ð Þ int

depending on t because the derivative with respect to t vanishes. This fact has

�iHSð Þ t�t<sup>0</sup> =ℏ

<sup>S</sup> ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup> : (43)

int ð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup> ; <sup>H</sup>ð Þ int

int t

<sup>S</sup> ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þ<sup>t</sup> <sup>e</sup>�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> is not

h i h i

<sup>0</sup> ð Þ � t<sup>0</sup> ; ρ

ð Þ int <sup>S</sup> ð Þ t; t � t<sup>0</sup>

iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þ<sup>t</sup> <sup>e</sup>

Then, the dynamical evolution of the system is given by

� 1 <sup>i</sup> <sup>ℏ</sup> t�t

<sup>0</sup> ð Þ¼ <sup>e</sup>� <sup>1</sup>

<sup>i</sup> <sup>ℏ</sup>τð Þ HsþH<sup>B</sup> so that after a shift of the integration variable

<sup>i</sup> <sup>ℏ</sup> t�t <sup>0</sup> ð ÞHs

ετ TrB <sup>H</sup>int; <sup>H</sup>intð Þ<sup>τ</sup> ; <sup>ρ</sup>sð Þ<sup>t</sup> <sup>ρ</sup><sup>B</sup> <sup>½</sup> ½ �� ¼ <sup>D</sup> <sup>ρ</sup><sup>s</sup> ½ � ð Þ<sup>t</sup> : (39)

A<sup>α</sup> ⊗ Bα: (40)

i Hð Þ <sup>S</sup>þH<sup>B</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>Oe�i Hð Þ <sup>S</sup>þH<sup>B</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> (41)

�i Hð Þ <sup>S</sup>þH<sup>B</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>,

(42)

ρB:

(44)

<sup>0</sup> ð Þð Þ HsþH<sup>B</sup> : (38)

ρsð Þt e 1 <sup>i</sup> <sup>ℏ</sup> t�t <sup>0</sup> ð ÞHs and

�∞

already been used when in the Markov approximation ρ<sup>S</sup> t <sup>0</sup> ð Þ is replaced by ρSð Þt . This corresponds to the Heisenberg picture where the state of the system does not change with time. The time dependence of averages is attributed to the temporal changes of the observables.

To include the interaction, we characterize the dynamics of the system observable A introducing the spectral decomposition with respect to the (discrete) eigenstates <sup>∣</sup>ϕn<sup>i</sup> of <sup>H</sup>S. We introduce the eigen-energies Es <sup>n</sup> of the system S according to HS∣ϕni ¼ Es <sup>n</sup>∣ϕni, and with <sup>Ð</sup> <sup>∞</sup> �<sup>∞</sup> exp ½ � ikx dx <sup>¼</sup> <sup>2</sup>πδð Þ<sup>k</sup> ,

$$\begin{split} A(\boldsymbol{\omega}) &= \int\_{-\infty}^{\infty} \mathbf{d} \mathbf{t} e^{i\boldsymbol{\alpha}(t-t\_0)} \mathbf{e}^{iH\_\mathbb{S}(t-t\_0)/\hbar} A \mathbf{e}^{-iH\_\mathbb{S}(t-t\_0)/\hbar} = A^\dagger(-\boldsymbol{\alpha}) \\ &= 2\pi\hbar \sum\_{nm} |\phi\_n\rangle \langle \phi\_n| A |\phi\_m\rangle \langle \phi\_m| \delta\left(E\_n^\circ - E\_m^\circ + \hbar \boldsymbol{\alpha}\right) \end{split} \tag{45}$$

(the index α in (40) is dropped). In interaction picture (A commutes with the bath observables) we have <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>Ae�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup> �<sup>∞</sup> <sup>d</sup>ω=ð Þ <sup>2</sup><sup>π</sup> exp ½ � �iωð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup> �Að Þ ω : Now, we find for the influence term

$$\mathcal{D}^{(\text{int})}[\rho\_{\text{S}}(t)](t - t\_{0}) = -\frac{1}{\hbar^{2}} \int\_{-\infty}^{t} \text{d}t' \int\_{-\infty}^{\infty} \frac{\text{d}\omega}{2\pi} \int\_{-\infty}^{\infty} \frac{\text{d}\omega'}{2\pi} \,\text{e}^{c(t' - t)} \text{e}^{-i o^{\prime}(t' - t)} \text{e}^{-i(a + o^{\prime})(t - t\_{0})}$$

$$\times \left\{ \langle \mathbf{B}(t' - t) \mathbf{B} \rangle\_{\text{B}}[A(\alpha), \rho\_{\text{S}}^{(\text{int})}(t; t - t\_{0}) A(\alpha^{\prime})] + \langle \mathbf{B}(t' - t) \rangle\_{\text{B}}[A(\alpha^{\prime}) \rho\_{\text{S}}^{(\text{int})}(t; t - t\_{0}), A(\alpha)] \right\} \tag{46}$$

with the time-dependent bath operators B tð Þ¼ <sup>0</sup> � t exp iH<sup>B</sup> t ½ � ð Þ <sup>0</sup> � t =ℏ B exp �iH<sup>B</sup> t ½ � ð Þ <sup>0</sup> � t =ℏ .

We can perform the integral over t <sup>0</sup> that concerns the bath observables. The bath enters via equilibrium auto-correlation functions of the time-dependent bath operators Bαð Þτ . We introduce the Laplace transform of the bath correlation function (the response function of the bath)

$$\Gamma(\boldsymbol{\alpha}) = \int\_0^\infty d\tau e^{i(\boldsymbol{\alpha} + \boldsymbol{i}\boldsymbol{\epsilon})\tau/\hbar} \text{Tr}\_\mathbf{B} \left\{ \rho\_\mathbf{B} \mathbf{B}^\dagger(\boldsymbol{\tau}) \mathbf{B} \right\} = \frac{1}{2} \boldsymbol{\gamma}(\boldsymbol{\alpha}) + \mathrm{i}\frac{\mathbf{1}}{\hbar} \mathcal{S}(\boldsymbol{\alpha}) \tag{47}$$

that is a matrix Γαβð Þ ω if the observable B has several components. We find in short notation

$$\begin{split} \mathcal{D}^{(\text{int})} [\rho\_{\text{S}}(\text{t})](\mathbf{t} - \mathbf{t}\_{0}) &= -\int\_{-\infty}^{\infty} \frac{\mathbf{d}\rho}{2\pi} \Bigg[ \frac{\mathbf{d}\rho^{\*}}{2\pi} \frac{\mathbf{d}\rho^{\*}}{2\pi} e^{i(\alpha^{\*} - \omega)(\mathbf{t} - \mathbf{t}\_{0})} \\ &\times \left\{ \Gamma\_{2}(\boldsymbol{\alpha}^{\*}) \Big[ A(\boldsymbol{\alpha}), \rho\_{\text{S}}^{(\text{int})}{\mathbf{s}}(\mathbf{t}; \mathbf{t} - \mathbf{t}\_{0}) A^{\dagger}(\boldsymbol{\alpha}^{\*}) \Big] + \Gamma\_{1}(\boldsymbol{\alpha}^{\*}) \Big[ A^{\dagger}(\boldsymbol{\alpha}^{\*}) \rho\_{\text{S}}^{(\text{int})}{\mathbf{s}}(\mathbf{t}; \mathbf{t} - \mathbf{t}\_{0}), A(\boldsymbol{\alpha}) \Big] \right\} \end{split} \tag{48}$$

after the transformation <sup>ω</sup><sup>0</sup> ! �ω″ and using Eq. (45). Note that this expression for the influence term is real because the second contribution is the Hermitean conjugated of the first contribution. Using symmetry properties, all correlation functions of bath variables are related to Γð Þ ω .

The expression ρ ð Þ int <sup>S</sup> ð Þ¼ <sup>t</sup>; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>e</sup>iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þ<sup>t</sup> <sup>e</sup>�iHSð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup> is not depending on time t because in the Heisenberg picture (we consider the lowest order of interaction) the state of the system does not depend on time. Oscillations with e<sup>i</sup>ð Þ <sup>ω</sup>�ω″ ð Þ <sup>t</sup>�t<sup>0</sup> occur that vanish for <sup>ω</sup>″ <sup>¼</sup> <sup>ω</sup>. The rotating wave approximation (RWA) takes into

account only contributions with <sup>ω</sup>″ <sup>¼</sup> <sup>ω</sup> that are not depending on <sup>t</sup>0. Oscillations with ei <sup>ω</sup>�ω<sup>0</sup> ð Þð Þ <sup>t</sup>�t<sup>0</sup> , <sup>ω</sup><sup>0</sup> � <sup>ω</sup> 6¼ 0 exhibit a phase, depending on <sup>t</sup>0. Any process of dephasing will damp down these oscillations.

D<sup>0</sup> ρ<sup>S</sup> ½ �¼ ð Þt ∑

<sup>H</sup><sup>S</sup> <sup>¼</sup> <sup>1</sup>

x þ i=ð Þ 2ℏmω<sup>0</sup>

with the creation <sup>a</sup>† <sup>¼</sup> ð Þ <sup>m</sup>ω0=2<sup>ℏ</sup> <sup>1</sup>=<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffi

<sup>a</sup> <sup>¼</sup> ð Þ <sup>m</sup>ω0=2<sup>ℏ</sup> <sup>1</sup>=<sup>2</sup>

<sup>ϕ</sup>njajϕ<sup>n</sup> h i0 <sup>¼</sup> ffiffiffi

ð Þ¼ ω 2π ∑

ρ0

<sup>n</sup> <sup>p</sup> <sup>δ</sup><sup>n</sup><sup>0</sup>

n

<sup>B</sup>,mm<sup>0</sup> ¼ m<sup>0</sup>

a and a†. In interaction picture we have

Hð Þ int

the bath Γð Þ ω (47) we find

Es

reads

a†

17

<sup>2</sup><sup>m</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup>

1=2

are the well-known harmonic oscillator states, with eigen-energies

construct the statistical operator for the canonical distribution as

1

We introduce a weak coupling between the system and the bath

e

jρ<sup>B</sup> h i jm ¼ δmm<sup>0</sup>

int ð Þ¼ <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>λ</sup> <sup>a</sup>†

3.1.4 Example: harmonic oscillator in a bath

DOI: http://dx.doi.org/10.5772/intechopen.84707

Nonequilibrium Statistical Operator

k

<sup>γ</sup><sup>k</sup> AkρSð Þ<sup>t</sup> <sup>A</sup>†

with the atomic system is (dipole approximation, dipole moment D ¼ er)

sider a one-dimensional harmonic oscillator with the eigen-frequency ω0,

<sup>n</sup> ¼ ℏω0ð Þ n þ 1=2 . The matrix elements of the construction operators are

<sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>∣</sup>ϕ<sup>n</sup>þ<sup>1</sup>i〈ϕnjδ ωð Þ <sup>þ</sup> <sup>ω</sup><sup>0</sup> , að Þ¼ <sup>ω</sup> <sup>2</sup><sup>π</sup> <sup>∑</sup>

motion are da†ð Þ<sup>t</sup> <sup>=</sup>d<sup>t</sup> <sup>¼</sup> <sup>i</sup>ω0a†ð Þ<sup>t</sup> , da tð Þ=d<sup>t</sup> ¼ �iω0a tð Þ. The spectral representation

At this moment, we do not specify the bath any more in detail. Suppose we have the solutions ∣ni of the energy eigenvalue problem HB∣mi ¼ EB,m∣mi, then we can

where the operator B acts only on the variables of the bath and commutes with

The influence term is calculated as given above. With the response function of

<sup>Z</sup> <sup>e</sup>�EB,m=kBT, Z <sup>¼</sup> <sup>∑</sup>

<sup>i</sup>ω0ð Þ <sup>t</sup>�t<sup>0</sup> <sup>þ</sup> ae�iω0ð Þ <sup>t</sup>�t<sup>0</sup>

mω<sup>2</sup> 0 2

A typical example is the absorption or emission of light. An isolated atom (e.g., hydrogen) is usually treated with the Schrödinger equation which gives the wellknown energy eigenvalues and the corresponding eigenstates. However, this is not correct, and the finite (natural) linewidth indicate that the energetically sharp eigenstates have not an infinite life-time. The coupling to the environment, the electromagnetic field (even in the vacuum at T ¼ 0) leads to transitions and a finite life-time. The electromagnetic field which is considered as bath can be represented as a system of harmonic oscillators (for each mode of the field), and the interaction

We discuss this phenomenon of radiation in a simplified version [5]. We con-

<sup>x</sup><sup>2</sup> <sup>¼</sup> <sup>ℏ</sup>ω<sup>0</sup> <sup>a</sup>†

x � i=ð Þ 2ℏmω<sup>0</sup>

<sup>k</sup> � <sup>1</sup> 2 A†

<sup>k</sup>Ak; <sup>ρ</sup>Sð Þ<sup>t</sup> � � � �: (54)

Hint ¼ �er � E ¼ �D � E: (55)

a þ 1 2

<sup>p</sup> ( <sup>a</sup>; <sup>a</sup>† ½ �¼ 1). The discrete eigenstates <sup>∣</sup>ϕn<sup>i</sup> of <sup>H</sup><sup>S</sup>

n

m

� �Bð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup> : (60)

<sup>H</sup>int ¼ �exE <sup>¼</sup> <sup>λ</sup> <sup>a</sup>† <sup>þ</sup> <sup>a</sup> � �B, (59)

ffiffiffi

1=2

�1,n and its adjoint complex. In interaction picture, the equations of

� �, (56)

p and destruction operator

<sup>n</sup> <sup>p</sup> <sup>j</sup>ϕ<sup>n</sup>�<sup>1</sup><sup>i</sup> <sup>ϕ</sup><sup>n</sup> <sup>h</sup> <sup>∣</sup>δ ωð Þ � <sup>ω</sup><sup>0</sup> :

e�EB,m=kB<sup>T</sup>: (58)

(57)

In the case of a discrete spectrum, the spectral function (45) can be used, and the integrals over <sup>ω</sup>,ω″ are replaced by sums over the eigenstates <sup>∣</sup>ϕn<sup>i</sup> of the system S:

$$\begin{split} \mathcal{D}^{(\text{int})} [\rho\_{\text{S}}(t)](t - t\_{0}) &= -\frac{1}{\hbar^{2}} \sum\_{nn',mm'} \mathsf{c} (\underline{\boldsymbol{E}}\_{n}^{\prime} - \underline{\boldsymbol{E}}\_{n'}^{\prime} - \underline{\boldsymbol{E}}\_{n}^{\prime} + \underline{\boldsymbol{E}}\_{n'}^{\prime})(t - t\_{0})/\hbar} \Gamma \Big( (\underline{\boldsymbol{E}}\_{n}^{\prime} - \underline{\boldsymbol{E}}\_{m}^{\prime})/\hbar \Big) \\ & \times \Big[ |\phi\_{n}\rangle\langle\phi\_{n}|A|\phi\_{m}\rangle\langle\phi\_{m}|\text{e}\_{\left(\underline{\boldsymbol{E}}\_{n}^{\prime} - \underline{\boldsymbol{E}}\_{n'}^{\prime}\right)(t - t\_{0})/\hbar} \rho\_{\text{S}}(t) |\phi\_{m'}\rangle\langle\phi\_{m'}|A|\phi\_{n'}\rangle\langle\phi\_{n'}| \\\\ & \qquad - |\phi\_{m'}\rangle\langle\phi\_{m'}|A|\phi\_{n'}\rangle\langle\phi\_{n'}|\phi\_{n}\rangle\langle\phi\_{n}|A|\phi\_{m}\rangle\langle\phi\_{m}|\text{e}\_{\left(\underline{\boldsymbol{E}}\_{n}^{\prime} - \underline{\boldsymbol{E}}\_{n'}^{\prime}\right)(t - t\_{0})/\hbar} \rho\_{\text{S}}(t) \Big] + \text{h.c.} \end{split} \tag{49}$$

The rotating wave approximation means that n ¼ n<sup>0</sup> , m ¼ m<sup>0</sup> so that

$$\begin{split} \mathcal{D}^{(\text{int})} [\rho\_{\text{S}}(t)](t - t\_{0}) &= -\frac{1}{\hbar^{2}} \sum\_{n,m} \Gamma \left( \left( E\_{n}^{\prime} - E\_{m}^{\prime} \right) / \hbar \right) \\ &\times \left[ |\phi\_{n}\rangle \langle \phi\_{n}| A | \phi\_{m}\rangle \langle \phi\_{m}| \rho\_{\text{S}}(t) | \phi\_{m}\rangle \langle \phi\_{m}| A | \phi\_{n}\rangle \langle \phi\_{n}| - |\phi\_{m}\rangle \langle \phi\_{m}| A | \phi\_{n}\rangle \langle \phi\_{n}| A | \phi\_{m}\rangle \langle \phi\_{m}| \rho\_{\text{S}}(t) \right] + \text{h.c.} \tag{50} \end{split} \tag{51}$$

The generalization to a more complex coupling to a bath (40) is straightforward, leading to matrices. More difficult is the discussion if the spectral function Að Þ ω is continuous, see [5]. Going back to the Schrödinger picture we have

$$\mathcal{D}[\rho\_{\mathcal{S}}(t)] = \int d\boldsymbol{\alpha} \,\sum\_{a\boldsymbol{\beta}} \Gamma\_{a\boldsymbol{\beta}}(\boldsymbol{\alpha}) \left[ A\_{\boldsymbol{\beta}}(\boldsymbol{\alpha}) \rho\_{\mathcal{S}}(t) A\_{a}^{\dagger}(\boldsymbol{\alpha}) - A\_{a}^{\dagger}(\boldsymbol{\alpha}) A\_{\boldsymbol{\beta}}(\boldsymbol{\alpha}) \rho\_{\mathcal{S}}(t) \right] + \text{h.c.} \tag{51}$$

The influence term D ρ<sup>S</sup> ½ � ð Þt cannot be given in the form of a commutator of an effective Hamiltonian with ρSð Þt that characterizes the Hamiltonian dynamics. Only a part can be separated that contributes to the reversible Hamiltonian dynamics, whereas the remaining part describes irreversible evolution in time and is denoted as dissipator D<sup>0</sup> ρ<sup>S</sup> ½ � ð Þt .

With Γαβð Þ¼ ω γαβð Þ ω =2 þ iSαβð Þ ω , we introduce the Hermitian operator <sup>H</sup>infl <sup>¼</sup> <sup>Ð</sup> <sup>d</sup>ω∑αβSαβð Þ <sup>ω</sup> <sup>A</sup>† <sup>α</sup>ð Þ ω Aβð Þ ω and obtain the quantum master equation

$$\frac{\partial}{\partial t}\rho\_{\mathcal{S}}(t) - \frac{1}{\mathbf{i}\,\hbar}[H\_{\mathcal{S}},\rho\_{\mathcal{S}}(t)] - \frac{1}{\mathbf{i}\,\hbar}[H\_{\text{inf}},\rho\_{\mathcal{S}}(t)] = D^{'}[\rho\_{\mathcal{S}}(t)].\tag{52}$$

The dissipator has the form

$$\mathcal{D}'[\rho\_{\mathcal{S}}(t)] = \left[ d\boldsymbol{\alpha} \sum\_{a\boldsymbol{\beta}} \boldsymbol{\gamma}\_{a\boldsymbol{\beta}}(\boldsymbol{\alpha}) \left[ \mathbf{A}\_{\boldsymbol{\beta}}(\boldsymbol{\alpha}) \rho\_{\mathcal{S}}(t) \mathbf{A}\_{a}^{\dagger}(\boldsymbol{\alpha}) - \frac{1}{2} \left\{ \mathbf{A}\_{a}^{\dagger}(\boldsymbol{\alpha}) \mathbf{A}\_{\boldsymbol{\beta}}(\boldsymbol{\alpha}), \rho\_{\mathcal{S}}(t) \right\} \right] \tag{53}$$

where f g A; B ¼ AB þ BA denotes the anticommutator. The influence Hamiltonian Hinfl commutes with the system Hamiltonian, HS; Hinfl ½ �¼ 0, because the operator A† <sup>α</sup>ð Þ ω Aβð Þ ω commutes with HS. It is often called the Lamb shift Hamiltonian since it leads to a shift of the unperturbed energy levels influenced by the coupling of the system to the reservoir, similar to the Lamb shift in QED. The Lindblad form follows by diagonalization of the matrices γαβð Þ ω ,

Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

account only contributions with <sup>ω</sup>″ <sup>¼</sup> <sup>ω</sup> that are not depending on <sup>t</sup>0. Oscillations with ei <sup>ω</sup>�ω<sup>0</sup> ð Þð Þ <sup>t</sup>�t<sup>0</sup> , <sup>ω</sup><sup>0</sup> � <sup>ω</sup> 6¼ 0 exhibit a phase, depending on <sup>t</sup>0. Any process of

In the case of a discrete spectrum, the spectral function (45) can be used, and the integrals over <sup>ω</sup>,ω″ are replaced by sums over the eigenstates <sup>∣</sup>ϕn<sup>i</sup> of the

�jϕ<sup>m</sup>0〉 <sup>ϕ</sup><sup>m</sup><sup>0</sup> <sup>j</sup>Ajϕ<sup>n</sup> h i0 〈ϕ<sup>n</sup><sup>0</sup> <sup>j</sup>ϕn〉〈ϕnjAjϕm〉〈ϕmj<sup>e</sup> <sup>E</sup><sup>s</sup>

� jϕn〉 ϕ<sup>n</sup> h i jAjϕ<sup>m</sup> 〈ϕmjρSð Þj t ϕm〉〈ϕmjAjϕn〉〈ϕnj�jϕm〉〈ϕmjAjϕn〉〈ϕnjAjϕm〉〈ϕmjρ<sup>S</sup> ½ ð Þt � þ h:c:

The generalization to a more complex coupling to a bath (40) is straightforward, leading to matrices. More difficult is the discussion if the spectral function Að Þ ω is

The influence term D ρ<sup>S</sup> ½ � ð Þt cannot be given in the form of a commutator of an effective Hamiltonian with ρSð Þt that characterizes the Hamiltonian dynamics. Only a part can be separated that contributes to the reversible Hamiltonian dynamics, whereas the remaining part describes irreversible evolution in time and is denoted

With Γαβð Þ¼ ω γαβð Þ ω =2 þ iSαβð Þ ω , we introduce the Hermitian operator

1 iℏ

HS; ρ<sup>S</sup> ½ �� ð Þt

γαβð Þ <sup>ω</sup> <sup>A</sup>βð Þ <sup>ω</sup> <sup>ρ</sup>Sð Þ<sup>t</sup> <sup>A</sup>†

Lindblad form follows by diagonalization of the matrices γαβð Þ ω ,

where f g A; B ¼ AB þ BA denotes the anticommutator. The influence Hamiltonian Hinfl commutes with the system Hamiltonian, HS; Hinfl ½ �¼ 0, because

<sup>α</sup>ð Þ� <sup>ω</sup> <sup>A</sup>†

<sup>α</sup>ð Þ ω Aβð Þ ω and obtain the quantum master equation

<sup>α</sup>ð Þ� <sup>ω</sup> <sup>1</sup> 2 A† <sup>α</sup>ð Þ <sup>ω</sup> <sup>A</sup>βð Þ <sup>ω</sup> ; <sup>ρ</sup>Sð Þ<sup>t</sup> � � � �

<sup>α</sup>ð Þ ω Aβð Þ ω commutes with HS. It is often called the Lamb shift

Hamiltonian since it leads to a shift of the unperturbed energy levels influenced by the coupling of the system to the reservoir, similar to the Lamb shift in QED. The

<sup>H</sup>infl; <sup>ρ</sup><sup>S</sup> ½ �¼ ð Þ<sup>t</sup> <sup>D</sup><sup>0</sup>

<sup>α</sup>ð Þ <sup>ω</sup> <sup>A</sup>βð Þ <sup>ω</sup> <sup>ρ</sup>Sð Þ<sup>t</sup> � � <sup>þ</sup> <sup>h</sup>:c: (51)

<sup>m</sup>þEs ð Þ <sup>m</sup><sup>0</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>Γ Es

<sup>n</sup> � <sup>E</sup><sup>s</sup> m � �=ℏ � �

<sup>m</sup>�E<sup>s</sup> ð Þ <sup>m</sup><sup>0</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þj <sup>t</sup> <sup>ϕ</sup><sup>m</sup>0〉〈ϕ<sup>m</sup><sup>0</sup> <sup>j</sup>Ajϕ<sup>n</sup>0〉〈ϕ<sup>n</sup><sup>0</sup> <sup>j</sup>

<sup>m</sup>�Es ð Þ <sup>m</sup><sup>0</sup> ð Þ <sup>t</sup>�t<sup>0</sup> <sup>=</sup><sup>ℏ</sup>ρSð Þ<sup>t</sup>

, m ¼ m<sup>0</sup> so that

i þ h:c: (49)

(50)

ρ<sup>S</sup> ½ � ð Þt : (52)

(53)

dephasing will damp down these oscillations.

Non-Equilibrium Particle Dynamics

1 <sup>ℏ</sup><sup>2</sup> <sup>∑</sup> nn0 , mm<sup>0</sup> e <sup>E</sup><sup>s</sup> n�E<sup>s</sup> n0�E<sup>s</sup>

h

1 <sup>ℏ</sup><sup>2</sup> <sup>∑</sup> n, <sup>m</sup>

� jϕn〉 <sup>ϕ</sup><sup>n</sup> h i <sup>j</sup>Ajϕ<sup>m</sup> 〈ϕmj<sup>e</sup> <sup>E</sup><sup>s</sup>

The rotating wave approximation means that n ¼ n<sup>0</sup>

Γ E<sup>s</sup>

<sup>n</sup> � <sup>E</sup><sup>s</sup> m � �=ℏ � �

continuous, see [5]. Going back to the Schrödinger picture we have

<sup>Γ</sup>αβð Þ <sup>ω</sup> <sup>A</sup>βð Þ <sup>ω</sup> <sup>ρ</sup>Sð Þ<sup>t</sup> <sup>A</sup>†

system S:

<sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ¼� <sup>t</sup> � <sup>t</sup><sup>0</sup>

<sup>D</sup>ð Þ int <sup>ρ</sup><sup>S</sup> ½ � ð Þ<sup>t</sup> ð Þ¼� <sup>t</sup> � <sup>t</sup><sup>0</sup>

D ρ<sup>S</sup> ½ �¼ ð Þt

as dissipator D<sup>0</sup> ρ<sup>S</sup> ½ � ð Þt .

D<sup>0</sup> ρ<sup>S</sup> ½ �¼ ð Þt

the operator A†

16

<sup>H</sup>infl <sup>¼</sup> <sup>Ð</sup>

ð dω ∑ αβ

<sup>d</sup>ω∑αβSαβð Þ <sup>ω</sup> <sup>A</sup>†

∂ ∂t

The dissipator has the form

ð dω ∑ αβ

ρSðÞ�t

1 iℏ

$$\mathcal{D}'[\rho\_{\mathcal{S}}(t)] = \sum\_{k} \gamma\_{k} \left[ A\_{k} \rho\_{\mathcal{S}}(t) A\_{k}^{\dagger} - \frac{1}{2} \left\{ A\_{k}^{\dagger} A\_{k}, \rho\_{\mathcal{S}}(t) \right\} \right]. \tag{54}$$

## 3.1.4 Example: harmonic oscillator in a bath

A typical example is the absorption or emission of light. An isolated atom (e.g., hydrogen) is usually treated with the Schrödinger equation which gives the wellknown energy eigenvalues and the corresponding eigenstates. However, this is not correct, and the finite (natural) linewidth indicate that the energetically sharp eigenstates have not an infinite life-time. The coupling to the environment, the electromagnetic field (even in the vacuum at T ¼ 0) leads to transitions and a finite life-time. The electromagnetic field which is considered as bath can be represented as a system of harmonic oscillators (for each mode of the field), and the interaction with the atomic system is (dipole approximation, dipole moment D ¼ er)

$$H\_{\rm int} = -e\mathbf{r} \cdot \mathbf{E} = -\mathbf{D} \cdot \mathbf{E}.\tag{55}$$

We discuss this phenomenon of radiation in a simplified version [5]. We consider a one-dimensional harmonic oscillator with the eigen-frequency ω0,

$$H\_{\rm S} = \frac{1}{2m}p^2 + \frac{m\alpha\_0^2}{2}\pi^2 = \hbar\alpha\_0\left(a^\dagger a + \frac{1}{2}\right),\tag{56}$$

with the creation <sup>a</sup>† <sup>¼</sup> ð Þ <sup>m</sup>ω0=2<sup>ℏ</sup> <sup>1</sup>=<sup>2</sup> x � i=ð Þ 2ℏmω<sup>0</sup> 1=2 p and destruction operator <sup>a</sup> <sup>¼</sup> ð Þ <sup>m</sup>ω0=2<sup>ℏ</sup> <sup>1</sup>=<sup>2</sup> x þ i=ð Þ 2ℏmω<sup>0</sup> 1=2 <sup>p</sup> ( <sup>a</sup>; <sup>a</sup>† ½ �¼ 1). The discrete eigenstates <sup>∣</sup>ϕn<sup>i</sup> of <sup>H</sup><sup>S</sup> are the well-known harmonic oscillator states, with eigen-energies Es <sup>n</sup> ¼ ℏω0ð Þ n þ 1=2 . The matrix elements of the construction operators are <sup>ϕ</sup>njajϕ<sup>n</sup> h i0 <sup>¼</sup> ffiffiffi <sup>n</sup> <sup>p</sup> <sup>δ</sup><sup>n</sup><sup>0</sup> �1,n and its adjoint complex. In interaction picture, the equations of motion are da†ð Þ<sup>t</sup> <sup>=</sup>d<sup>t</sup> <sup>¼</sup> <sup>i</sup>ω0a†ð Þ<sup>t</sup> , da tð Þ=d<sup>t</sup> ¼ �iω0a tð Þ. The spectral representation reads

$$a^\dagger(\omega) = 2\pi \sum\_n \sqrt{n+1} |\phi\_{n+1}\rangle\langle\phi\_n| \delta(\omega + \omega\_0), \qquad a(\omega) = 2\pi \sum\_n \sqrt{n} |\phi\_{n-1}\rangle\langle\phi\_n| \delta(\omega - \omega\_0). \tag{57}$$

At this moment, we do not specify the bath any more in detail. Suppose we have the solutions ∣ni of the energy eigenvalue problem HB∣mi ¼ EB,m∣mi, then we can construct the statistical operator for the canonical distribution as

$$\rho\_{\mathbf{B},mm'}^{0} = \langle m'|\rho\_{\mathbf{B}}|m\rangle = \delta\_{mm'} \frac{\mathbf{1}}{Z} \mathbf{e}^{-E\_{\mathbf{B},m}/k\_{\mathbf{B}}T}, \qquad Z = \sum\_{m} \mathbf{e}^{-E\_{\mathbf{B},m}/k\_{\mathbf{B}}T}. \tag{58}$$

We introduce a weak coupling between the system and the bath

$$H\_{\rm int} = -e\varkappa E = \lambda (a^\dagger + a)\mathbf{B},\tag{59}$$

where the operator B acts only on the variables of the bath and commutes with a and a†. In interaction picture we have

$$H\_{\rm int}^{(\rm int)}(t - t\_0) = \lambda \Big(a^\dagger e^{i\nu\_0(t - t\_0)} + a e^{-i\nu\_0(t - t\_0)}\Big) \mathbf{B}(t - t\_0). \tag{60}$$

The influence term is calculated as given above. With the response function of the bath Γð Þ ω (47) we find

Non-Equilibrium Particle Dynamics

$$\begin{split} &\frac{\partial}{\partial t}\rho\_{\mathcal{S}}(t) - \frac{1}{\mathbf{i}\hbar}[H\_{\mathcal{S}},\rho\_{\mathcal{S}}(t)] - \frac{1}{\mathbf{i}\hbar}[(\mathcal{S}(a\rho\_{0})a^{\dagger}a + \mathcal{S}(-a\rho\_{0})aa^{\dagger},\rho\_{\mathcal{S}}(t)] \\ &= \chi(a\rho\_{0})\Big(a\rho\_{\mathcal{S}}(t)a^{\dagger} - \frac{1}{2}\{a^{\dagger}a,\rho\_{\mathcal{S}}(t)\}\Big) + \chi(-a\rho\_{0})\Big(a^{\dagger}\rho\_{\mathcal{S}}(t)a - \frac{1}{2}\{aa^{\dagger},\rho\_{\mathcal{S}}(t)\}\Big). \end{split} \tag{61}$$

The curly brackets in the dissipator denote the anticommutator. There are eight additional terms containing aa or a†a†. In interaction picture, they are proportional to e�2iω0ð Þ <sup>t</sup>�t<sup>0</sup> and are dropped within the rotating wave approximation. For a bath in thermal equilibrium, using eigenstates the detailed balance relation is easily proven,

$$\chi(-\alpha\_0) = \chi(\alpha\_0) e^{-\hbar \alpha\_0/k\_B T}. \tag{62}$$

γ ωð Þ¼ <sup>4</sup>ω<sup>3</sup>

Nonequilibrium Statistical Operator

DOI: http://dx.doi.org/10.5772/intechopen.84707

<sup>H</sup>infl <sup>¼</sup> <sup>Ð</sup>

D<sup>0</sup> ρ<sup>S</sup> ½ �¼ ð Þt

4ω<sup>3</sup>

4ω<sup>3</sup>

19

∂ ∂t

<sup>d</sup>ωℏSð Þ <sup>ω</sup> <sup>D</sup>†

quantum master equation reads

<sup>d</sup><sup>ω</sup> <sup>4</sup>ω<sup>3</sup>

<sup>d</sup><sup>ω</sup> <sup>4</sup>ω<sup>3</sup>

<sup>3</sup>ℏc<sup>3</sup> nBð Þ ω describing the absorption of photons.

corresponds to a state vector ∣ai in the Hilbert space.

described by a master equation or balance equation

dt <sup>p</sup>1ð Þ¼ <sup>a</sup>; <sup>t</sup> <sup>∑</sup>

a0 6¼a

d

of motion (8) with the Hamiltonian

ð<sup>∞</sup> 0

þ ð<sup>∞</sup> 0

photons. The term D†

3.1.6 The Pauli equation

ρSðÞ�t

<sup>3</sup>ℏc<sup>3</sup> ½ � <sup>1</sup> <sup>þ</sup> <sup>n</sup>Bð Þ <sup>ω</sup> , Sð Þ¼ <sup>ω</sup> <sup>2</sup>

γ ωð Þ¼ <sup>4</sup>ω3½ � <sup>1</sup> <sup>þ</sup> <sup>n</sup>Bð Þ <sup>ω</sup> <sup>=</sup> <sup>3</sup>ℏc<sup>3</sup> ð Þ for <sup>ω</sup>>0 and γ ωð Þ¼ <sup>4</sup>j j <sup>ω</sup> <sup>3</sup>

1 iℏ

has the Lindblad form. The influence Hamiltonian

<sup>3</sup>ℏc<sup>3</sup> <sup>n</sup>Bð Þ <sup>ω</sup> <sup>D</sup>†

<sup>3</sup>πℏc<sup>3</sup> <sup>P</sup>

Note that the Planck distribution satisfies nBð Þ¼� �ω ½ � 1 þ nBð Þ ω such that

pling of atoms to the radiation field Hint ¼ �D � E in dipole approximation,

HS; ρ<sup>S</sup> ½ �� ð Þt

<sup>3</sup>ℏc<sup>3</sup> ½ � <sup>1</sup> <sup>þ</sup> <sup>n</sup>Bð Þ <sup>ω</sup> <sup>D</sup>ð Þ <sup>ω</sup> <sup>ρ</sup>Sð Þ<sup>t</sup> <sup>D</sup>†

The resulting quantum optical master equation which, e.g., describes the cou-

1 iℏ

nian H<sup>S</sup> that is induced by the vacuum fluctuations of the radiation field (Lamb shift) and by the thermally induced processes (Stark shift). The dissipator of the

ð Þ <sup>ω</sup> <sup>ρ</sup>Sð Þ<sup>t</sup> <sup>D</sup>ð Þ� <sup>ω</sup> <sup>1</sup>

where the integral over the negative frequencies has been transformed into positive frequencies. This result can be interpreted in a simple way. The application of the destruction operator Dð Þ ω on a state of the system lowers its energy by the

<sup>3</sup>ℏc<sup>3</sup> ½ � 1 þ nBð Þ ω contains the spontaneous emission as well as the thermal emission of

We consider a system whose state is described by the observable A, and which takes the value a. This can be a set of numbers in the classical case that describe the degrees of freedom we use as relevant variables. In the quantum case, this is a set of relevant observables that describe the state of the system. The eigenvalue a

At time t, we expect a probability distribution p1ð Þ a; t to find the system in state a, if the property A is measured. The change of the probability p1ð Þ a; t with time is

waa0p<sup>1</sup> a<sup>0</sup> ð Þ� ; t wa<sup>0</sup>

In the context of the time evolution of a physical system, this master equation is also denoted as Pauli equation. We derive it from a microscopical approach using perturbation theory. The statistical operator ρð Þt follows the von Neumann equation

ap1ð Þ <sup>a</sup>; <sup>t</sup> � �: (72)

amount ℏω and describes the emission of a photon. The transition rate

ð<sup>∞</sup> 0

dωkω<sup>3</sup> k

ð Þ� ω Dð Þ ω leads to a renormalization of the system Hamilto-

ð Þ� <sup>ω</sup> <sup>1</sup> 2 ( D†

> 2 (

ð Þ ω gives the creation of excitations with transition rate

" # )

" # )

<sup>D</sup>ð Þ <sup>ω</sup> <sup>D</sup>†

1 þ nBð Þ ω<sup>k</sup> ω � ω<sup>k</sup>

Hinfl; ρ<sup>S</sup> ½ �¼ ð Þt D<sup>0</sup> ρ<sup>S</sup> ½ � ð Þt , (70)

ð Þ ω Dð Þ ω ; ρSð Þt

ð Þ ω ; ρSð Þt

,

(71)

þ

� �:

<sup>n</sup>Bð Þ <sup>j</sup>ω<sup>j</sup> <sup>=</sup> <sup>3</sup>ℏc<sup>3</sup> ð Þ for <sup>ω</sup>< 0.

nBð Þ ω<sup>k</sup> ω þ ω<sup>k</sup>

(69)

The evolution equations for the averages <sup>a</sup>† h i<sup>t</sup> <sup>¼</sup> TrS <sup>ρ</sup>Sa† f g, a† h i <sup>a</sup> <sup>t</sup> <sup>¼</sup> TrS <sup>ρ</sup>Sa† f g<sup>a</sup> are immediately calculated as

$$\frac{d}{dt}\langle a^\dagger \rangle^t = \text{Tr}\_{\mathbb{S}}\left\{\frac{\partial}{\partial t}\rho\_{\mathbb{S}}(t)a^\dagger\right\} = \left(\text{i}\boldsymbol{o}\_0^\prime - \frac{1}{2}[\boldsymbol{\gamma}(\boldsymbol{o}\_0) - \boldsymbol{\gamma}(-\boldsymbol{o}\_0)]\right)\left\langle a^\dagger\right\rangle^t \tag{63}$$

with the renormalized frequency ω<sup>0</sup> <sup>0</sup> ¼ ω<sup>0</sup> þ ½ � Sð Þþ ω<sup>0</sup> Sð Þ �ω<sup>0</sup> =ℏ. The solution is

$$
\langle \mathbf{a}^\dagger \rangle^t = \langle \mathbf{a}^\dagger \rangle^{t\_0} e^{[i\alpha\_0 - \gamma(a\nu\_0)/2 + \gamma(-a\nu\_0)/2](t - t\_0)}.\tag{64}
$$

Similar expressions are obtained for h i <sup>a</sup> <sup>t</sup> . We find for the occupation number h i <sup>n</sup> <sup>t</sup> <sup>¼</sup> <sup>a</sup>† h i <sup>a</sup> <sup>t</sup> <sup>¼</sup> pnð Þ<sup>t</sup>

$$\frac{d}{dt}\langle a^\dagger a \rangle^t = \chi(-a\nu\_0) - [\chi(a\nu\_0) - \chi(-a\nu\_0)]\langle a^\dagger a \rangle^t \tag{65}$$

with the solution

$$\left\langle a^{\dagger}a \right\rangle^{t} = \left\langle a^{\dagger}a \right\rangle^{t\_{0}} e^{-[\gamma(a\bullet)-\gamma(-a\bullet)](t-t\_{0})} + \frac{\gamma(-a\bullet\_{0})}{\gamma(a\bullet\_{0})-\gamma(-a\bullet\_{0})} \left[\mathbf{1} - e^{-[\gamma(a\bullet)-\gamma(-a\bullet)](t-t\_{0})}\right]. \tag{66}$$

The asymptotic behavior t � t<sup>0</sup> ! ∞ is determined by the properties of the bath,

$$\frac{\chi(-a\mathbf{o}\_0)}{\chi(a\mathbf{o}\_0) - \chi(a\mathbf{o}\_0)} = \frac{\mathbf{1}}{\mathbf{e}^{-\hbar a\mathbf{o}\_0/k\_B T} - \mathbf{1}} = n\_\mathbf{B}(a\mathbf{o}\_0),\tag{67}$$

the system relaxes to the thermal equilibrium distribution that is independent on the initial distribution <sup>a</sup>† h i <sup>a</sup> <sup>t</sup><sup>0</sup> .
