Field-Matter Coupling and Two-Level Systems

Abstract

With this chapter, we start the applications part of this book by considering the interaction between lasers and matter. Lasers have already been discussed in Chap. 1.

Appendices

3.A Generalized Parity Transformation

In the case of a symmetric static potential $V(x)=V(-x)$ and in length gauge, with a sinusoidal laser potential of the form $e\mathcal{E}_0x\sin (\omega t)$, the extended Hamiltonian $\hat{\mathcal{H}}$ in (2.138) is invariant under the generalized parity transformation

$$\begin{aligned} \hat{\mathcal{P}}:\qquad x\rightarrow -x,\quad t\rightarrow t+\frac{T}{2}\;. \end{aligned}$$

(3.80)

The Floquet functions thus transform according to

$$\begin{aligned} \hat{\mathcal{P}}\psi _{\alpha ^{\prime }}(x,t)=\pm \psi _{\alpha ^{\prime }}(x,t)\;, \end{aligned}$$

(3.81)

i.e., they have either positive or negative generalized parity. With the help of (2.150) it follows that $\psi _{\alpha ^{\prime }}(x,t),\psi _{\beta ^{\prime }}(x,t)$ have the same or different generalized parity, depending on $(\alpha -k)-(\beta -l)$ being even or odd.

As we will see in Chap. 5, exact crossings of the quasienergies as a function of external parameters are of utmost importance for the quantum dynamics of periodically driven systems. For stationary systems, the possibility of exact crossings has been studied in the heyday of quantum theory by von Neumann and Wigner [32]. These authors found that eigenvalues of eigenfunctions with different parity may approach each other arbitrarily closely and may thus cross exactly. This is in contrast to eigenvalues of the same parity, which always have to be at a finite distance, a fact which is sometimes referred to as the non-crossing rule. The corresponding behavior in the spectrum as a function of external parameters is called allowed, respectively avoided crossing. In the Floquet case, the Hamiltonian can also be represented by a Hermitian matrix, see e.g. (2.183), and therefore the same reasoning applies, with parity replaced by generalized parity.

For the investigations to be presented in Sect. 5.5.1 it is decisive if these exact crossings are singular events in parameter space or if they can occur by variation of just a single parameter. In [32] it has been shown that for Hermitian matrices (of finite dimension) with complex (real) elements, the variation of three (two) free parameters is necessary in order for two eigenvalues to cross. Using similar arguments, it can be shown that for a real Hermitian matrix with alternatingly empty off-diagonals (as it is e.g., the case for the Floquet matrix of the periodically driven, quartic, symmetric, bistable potential) the variation of a single parameter is enough to make two quasienergies cross.

In the case of avoided crossings an interesting behavior of the corresponding eigenfunctions can be observed. There is a continuous change in the structure in position space if one goes through the avoided crossing [33]. Pictorially this is very nicely represented in the example of the driven quantum well, depicted in Fig. 3.4, taken out of [34], where for reasons of better visualization the Husimi transform of the quasi-eigenfunctions as a function of action angle variables $(J,\varTheta )$ [35] is shown.

Fig. 3.4

Avoided crossing of Floquet energies (here denoted by $\varOmega _\alpha$) as a function of field amplitude (upper panel) and associated change of character of the Floquet functions, corresponding to the two levels labelled by A and B in the driven quantum well (lower panels (a–f)); from [34]

3.B Pauli Spin Matrices and the Two-Level Density Matrix

The Pauli spin matrices

$$\begin{aligned} \varvec{\sigma }_x = \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix} ; \ \varvec{\sigma }_y = \begin{pmatrix} 0 &{} -\mathrm{i}\\ \mathrm{i}&{} 0 \end{pmatrix} ; \ \varvec{\sigma }_z = \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix}, \end{aligned}$$

(3.82)

together with the 2 $\times$ 2 unit matrix, form a complete basis in the space of complex hermitian 2 $\times$ 2 matrices. In their terms our Hamiltonian (3.45) reads

$$\begin{aligned} \mathbf{H}=\hbar \nu \varvec{\sigma }_x-\hbar \epsilon \varvec{\sigma }_z. \end{aligned}$$

(3.83)

Furthermore, a general density operator can be written as

$$\begin{aligned} \hat{\rho }=\frac{1}{2}\left( \hat{1}+\varvec{r}\cdot \varvec{\hat{\sigma }}\right) , \end{aligned}$$

(3.84)

with a vector $\varvec{r}$ that is of unit length for all times in the case of pure state dynamics, and a vector-operator $\varvec{\hat{\sigma }}$, composed of the Pauli operators. This then allows for a geometrical interpretation of two-level dynamics by going to the Feynman-Vernon-Hellwarth (or Bloch sphere) representation, discussed in the book by Tannor [36].

The pure state density matrix, in the case of a two-level system with energies $E_1,E_2$, in the basis of the corresponding eigenstates is given by

$$\begin{aligned} \varvec{\rho }= \begin{pmatrix} |d_1|^2 &{} d_1d_2^*\exp \{-\mathrm{i}(E_1-E_2)t/\hbar \} \\ d_1^*d_2\exp \{\mathrm{i}(E_1-E_2)t/\hbar \} &{} |d_2|^2 \end{pmatrix}, \end{aligned}$$

(3.85)

with the populations of the different energy levels on the diagonal and where the off-diagonal elements are sometimes called coherences.

A frequently considered mixed state is the thermal density matrix at temperature T with only diagonal elements

$$\begin{aligned} \rho _{mn}=\frac{\mathrm {e}^{-\beta E_n}}{Q}\delta _{mn}, \end{aligned}$$

(3.86)

where $\beta =1/(k_\mathrm{B}T)$ with Boltzmann constant $k_\mathrm{B}$ and where $Q=\sum _{n=1}^2\mathrm {e}^{-\beta E_n}$ is the partition function. An initial pure state evolves into a thermal mixed state by relaxation (due to coupling to an environment) which is governed by the time scale for population decay $T_1$ and the dephasing or coherence decay time scale $T_2$, which are related via

$$\begin{aligned} \frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_2^*}, \end{aligned}$$

(3.87)

with the pure dephasing time $T_2^*$ [36].

3.C Two-Level System in an Incoherent Field

As the starting point of the perturbative treatment of a two-level system in an incoherent external field, we use the Schrödinger equation in the interaction representation (3.60) and (3.61) with the initial conditions $d_1(0)=1$ and $d_2(0)=0$. For very small perturbations, the coefficient $d_1$ is assumed to remain at its initial value, leading to

$$\begin{aligned} \mathrm{i}\dot{d}_2= & {} \nu _{21}(t)\exp [\mathrm{i}\omega _{21}t]\;. \end{aligned}$$

(3.88)

This equation can be integrated immediately to yield

$$\begin{aligned} d_2(t)=-\mathrm{i}\int _0^{t}\mathrm{d}t'\nu _{21}(t')\exp [\mathrm{i}\omega _{21}t']\;, \end{aligned}$$

(3.89)

analogous to the first order iteration in (2.28). The field shall consist of a superposition of waves with uniformly distributed, statistically independent phases $\phi _j$

$$\begin{aligned} \varvec{\mathcal{E}}(t)=\frac{1}{2}\sum _{\omega _j>0}\varvec{\mathcal{E}}_j \exp [\mathrm{i}\phi _j-\mathrm{i}\omega _jt]+\mathrm{c.c.}\;. \end{aligned}$$

(3.90)

If we insert this into the equation above, we get

$$\begin{aligned} d_2(t)= & {} -\frac{\mathrm{i}}{2\hbar }\sum _{j}\varvec{\mathcal{E}}_j\cdot \varvec{\mu }_{21} \exp [\mathrm{i}\phi _j]\int _0^{t}\mathrm{d}t'\exp [\mathrm{i}(\omega _{21}-\omega _j)t'] \nonumber \\= & {} -\frac{\mathrm{i}}{2\hbar }\sum _{j}\varvec{\mathcal{E}}_j\cdot \varvec{\mu }_{21} \exp [\mathrm{i}\phi _j]S_j\;, \end{aligned}$$

(3.91)

where the definition

$$\begin{aligned} S_j=[\mathrm{i}(\omega _{21}-\omega _j)]^{-1} \left\{ \exp [\mathrm{i}(\omega _{21}-\omega _j)t]-1\right\} \end{aligned}$$

(3.92)

has been introduced. The occupation probability of the second level is then given by the double sum

$$\begin{aligned} |d_2(t)|^2=(2\hbar )^{-2}\sum _j\sum _{j'}\exp [\mathrm{i}(\phi _j-\phi _{j'})] \varvec{\mathcal{E}}_j\cdot \varvec{\mu }_{21} \varvec{\mathcal{E}}_{j'}\cdot \varvec{\mu }_{21}^{*} S_jS_{j'}^{*}\;. \end{aligned}$$

(3.93)

Averaging over the phases is now performed and denoted by $<>$, yielding

$$\begin{aligned} <\exp [\mathrm{i}(\phi _j-\phi _{k})]> =\delta _{jk}\;. \end{aligned}$$

(3.94)

One of the sums in (3.93) therefore collapses and

$$\begin{aligned} <|d_2(t)|^2>=\left| \frac{\varvec{e}\cdot \varvec{\mu }_{21}}{\hbar }\right| ^2 \sum _j|\mathcal{E}_j|^2(\omega _{21}-\omega _j)^{-2} \sin ^{2}[(\omega _{21}-\omega _j)t/2] \end{aligned}$$

(3.95)

follows for identical polarization, $\varvec{e},$ of the light waves.

Now we have to sum over the distribution of frequencies. To this end we consider the time derivative of the expression above⁸

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}<|d_2(t)|^2>= \left| \frac{\varvec{e}\cdot \varvec{\mu }_{21}}{\sqrt{2}\hbar }\right| ^2 \sum _j|\mathcal{E}_j|^2(\omega _{21}-\omega _j)^{-1} \sin [(\omega _{21}-\omega _j)t]\;. \end{aligned}$$

(3.96)

With the definition of an energy density per angular frequency interval $\rho (\omega _j)=\frac{1}{2}\varepsilon _0|\mathcal{E}_j|^2/\varDelta \omega _j$, assuming that the frequencies are distributed continuously, and replacing $\rho (\omega _j)$ by its resonance value $\rho (\omega _{21})$, due to

$$\begin{aligned} \int _{-\infty }^{\infty } \mathrm{d}\omega \sin (\omega t)/\omega =\pi \;, \end{aligned}$$

(3.97)

we get

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}<|d_2(t)|^2>= \frac{\pi }{\varepsilon _0} \left| \frac{\varvec{e}\cdot \varvec{\mu }_{21}}{\hbar }\right| ^2 \rho (\omega _{21})\;. \end{aligned}$$

(3.98)

The right hand side of this expression is a constant and therefore consistent with the assumptions made in the derivation of Planck’s radiation law in Chap. 1.

Comparing the equation above with (1.2) for $N_1$=1 and after switching from the angular to the linear frequency case [37]

$$\begin{aligned} B=\frac{2\pi ^2}{\varepsilon _0} \left| \frac{\varvec{e}\cdot \varvec{\mu }_{21}}{h}\right| ^2 \end{aligned}$$

(3.99)

is found for Einstein’s B coefficient.