Field Matter Coupling and Two-Level Systems

Abstract

In this chapter, we start with the theoretical description of the coupling of a classical light field realized, e.g., by a laser, to a quantum mechanical system. Different gauges, related by unitary transformations are then introduced. After the study of the Volkov solution of the time-dependent Schrödinger equation for the free particle in a laser field, some analytically solvable driven two-level systems will be discussed in the remainder of this chapter. The phenomenon of Rabi oscillations and the fundamental rotating wave approximation are thereby going to be reviewed.

Appendices

Appendix 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 $e\mathcal{E}_{0}x\sin(\omega t)$, the extended Hamiltonian $\hat{\mathcal{H}}$ in (2.121) is invariant under the generalized parity transformation

$$ \hat{\mathcal{P}}: x\to-x,\quad t\to t+\frac{T}{2}. $$

(3.69)

The Floquet functions thus transform according to

$$ \hat{\mathcal{P}}\psi_{\alpha^{\prime}}(x,t)=\pm \psi_{\alpha ^{\prime}}(x,t), $$

(3.70)

i.e., they have either positive or negative generalized parity. With the help of (2.133) it follows that ψ _α′(x,t), ψ _β′(x,t) have the same or different generalized parity, depending on (α−k)−(β−l) being odd or even.

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 [21]. 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. (2.162), and therefore the same reasoning applies with parity replaced by generalized parity.

For the investigations to be presented in Sect. 5.4.1 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 [21] 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 [22]. Pictorially this is very nicely represented in the example of the driven quantum well, depicted in Fig. 3.3, taken out of [23], where for reasons of better visualization the Husimi transform of the quasi-eigenfunctions as a function of action angle variables (J,Θ) [24] is shown.

Fig. 3.3

Avoided crossing of Floquet energies (here denoted by Ω _α ) as a function of field amplitude (upper panel) and associated change of character of the Floquet functions in the driven quantum well (lower panels (a–f)); from [23]

Appendix B: 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 (3.49), (3.50) 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

$$ \mathrm {i}\dot{d}_2=\nu_{21}(t)\exp[\mathrm {i}\omega_{21}t]. $$

(3.71)

This equation can be integrated immediately to yield

$$ d_2=-\mathrm {i}\int_0^{t}{ \mathrm{d}}t'\nu_{21}\bigl(t'\bigr)\exp\bigl[ \mathrm {i}\omega_{21}t'\bigr], $$

(3.72)

analogous to the first order iteration in (2.21). The field shall consist of a superposition of waves with uniformly distributed, statistically independent phases ϕ _j

$$ \boldsymbol {\mathcal{E}}(t)=\frac{1}{2}\sum_{\omega_j>0} \boldsymbol {\mathcal{E}}_j \exp[\mathrm {i}\phi_j-\mathrm {i}\omega_jt]+\mbox{c.c.} $$

(3.73)

If we insert this into the equation above, we get

$$\begin{aligned} d_2 =&-\frac{\mathrm {i}}{2\hbar}\sum_{j}\boldsymbol { \mathcal{E}}_j\cdot \boldsymbol {\mu}_{21} \exp[\mathrm {i}\phi_j]\int_0^{t}{ \mathrm{d}}t'\exp\bigl[\mathrm {i}(\omega _{21}- \omega_j)t'\bigr] \end{aligned}$$

(3.74)

$$\begin{aligned} =&-\frac{\mathrm {i}}{2\hbar}\sum_{j}\boldsymbol { \mathcal{E}}_j\cdot \boldsymbol {\mu}_{21} \exp[\mathrm {i}\phi_j]S_j, \end{aligned}$$

(3.75)

where the definition

$$ S_j=\bigl[\mathrm {i}(\omega_{21}-\omega_j) \bigr]^{-1} \bigl\{\exp\bigl[\mathrm {i}(\omega_{21}- \omega_j)t\bigr]-1 \bigr\} $$

(3.76)

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

$$ |d_2|^2=(2\hbar)^{-2}\sum _j\sum_{j'}\exp\bigl[ \mathrm {i}(\phi_j-\phi_{j'})\bigr] \boldsymbol {\mathcal{E}}_j \cdot \boldsymbol {\mu}_{21} \boldsymbol {\mathcal{E}}_{j'}\cdot \boldsymbol { \mu}_{21}^{\ast} S_jS_{j'}^{\ast}. $$

(3.77)

Averaging over the phases is now performed and denoted by 〈 〉, yielding

$$ \bigl\langle \exp\bigl[\mathrm {i}(\phi_j-\phi_{k})\bigr]\bigr \rangle =\delta_{jk}. $$

(3.78)

One of the sums in (3.77) therefore collapses and

$$ \bigl\langle \bigl |d_2(t)\bigr |^2\bigr\rangle =\biggl \vert \frac{\boldsymbol {e}\cdot \boldsymbol {\mu}_{21}}{\hbar }\biggr \vert ^2 \sum_j| \mathcal{E}_j|^2(\omega_{21}- \omega_j)^{-2} \sin^{2}\bigl[( \omega_{21}-\omega_j)t/2\bigr] $$

(3.79)

follows for identical polarization of the light waves e.

Now we have to sum over the distribution of frequencies. To this end we consider the time derivative of the expression above⁵

$$ \frac{\mathrm {d}}{\mathrm {d}t}\bigl\langle \bigl |d_2(t)\bigr |^2\bigr\rangle = \biggl \vert \frac{\boldsymbol {e}\cdot \boldsymbol {\mu}_{21}}{\sqrt{2}\hbar}\biggr \vert ^2 \sum _j|\mathcal{E}_j|^2( \omega_{21}-\omega_j)^{-1} \sin\bigl[( \omega_{21}-\omega_j)t\bigr]. $$

(3.80)

With the definition of an energy density per angular frequency interval $\rho(\omega_{j})=\frac{1}{2}\epsilon_{0}|\mathcal{E}_{j}|^{2}/\varDelta \omega_{j}$, assuming that the frequencies are distributed continuously, and replacing ρ(ω _j ) by its resonance value ρ(ω ₂₁), due to

$$ \int_{-\infty}^{\infty} {\mathrm{d}}\omega\sin(\omega t)/ \omega =\pi, $$

(3.81)

we get

$$ \frac{\mathrm {d}}{\mathrm {d}t}\bigl\langle \bigl |d_2(t)\bigr |^2\bigr\rangle = \frac{\pi}{\varepsilon_0} \biggl \vert \frac{\boldsymbol {e}\cdot \boldsymbol {\mu}_{21}}{\hbar}\biggr \vert ^2 \rho(\omega_{21}). $$

(3.82)

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 [25]

$$ B=\frac{2\pi^2}{\varepsilon_0} \biggl \vert \frac{\boldsymbol {e}\cdot \boldsymbol {\mu}_{21}}{h}\biggr \vert ^2 $$

(3.83)

is found for Einstein’s B coefficient.