Basis choice, perturbation theory and the FGR

Dissipation in a quantum system will be the result of `driving' a parameter $x$ on which the Hamiltonian depends. The parameter is externally controllable with any (real) function of time $x(t)$; it does not have any dynamics of its own (it is not a degree of freedom). We start by linearizing about $x=0$,

$${\mathcal{H}}(x) \; \approx \; {\mathcal{H}}(0) + \frac{\partial \mathcal{H}}{\partial x}x .$$ (2.34)

The hermitian operator $-\partial {\mathcal{H}} / \partial x\equiv {\mathcal{F}}$ is the `generalized force' conjugate to the parameter $x$: it is an actual force if $x$ is a displacement, it is a dipole moment if $x$ is an applied electric field magnetic field, it is a current around a ring (magnetization) if $x$ is magnetic flux enclosed by the ring, and so forth. I shall now use time-dependent perturbation theory, choosing to perform this in a fixed basis of the eigenstates of ${\mathcal{H}}(0)$, which we call the `unperturbed' Hamiltonian. This choice of basis is conventional in textbooks [128,99,118] so is ideal for introductory purposes. However this choice does not allow an understanding of the limitations of conventional LRT; to understand breakdown and possible extensions to LRT it is necessary to consider transitions in the so-called `adiabatic' basis of the local eigenstates of ${\mathcal{H}}(x)$ ([200,46,48]. The adiabatic basis is much more natural for unifying with the classical picture [46]. It also removes certain difficulties, for instance that of the infinite matrix elements $\frac{\partial \mathcal{H}}{\partial x}$ which would otherwise occur for any deformation $x \neq 0$ of a hard-walled billiard system. As another example, the case of constant velocity driving $x = Vt$ cannot be shown to give diffusive $E$ spreading in a fixed basis--the adiabatic basis is required (Appendix B of [46]).

The Schrodinger-picture wavefunction is represented as

$$\left\vert\psi(t)\right\rangle \; = \; \sum_n a_n(t) \left\vert n... ...\right\rangle \; = \; e^{\frac{1}{i\hbar} E_n t} \left\vert n(0)\right\rangle ,$$ (2.35)

where the time-independent states $\left\vert n\right\rangle \equiv \left\vert n(0)\right\rangle$ are eigenstates of the unperturbed Hamiltonian: ${\mathcal{H}}(0) \left\vert n\right\rangle = E_n \left\vert n\right\rangle$. The basis vectors are fixed but include phase evolution under ${\mathcal{H}}(0)$ (following [128,160]), thus the coefficients $a_n(t)$ are equivalent to an `interaction picture' representation. The full time-evolution

$$i\hbar \frac{d}{dt} \left\vert\psi(t)\right\rangle \; = \; {\mathcal{H}}(x(t)) \left\vert\psi(t)\right\rangle$$ (2.36)

gives a first-order differential equation for the coefficients

$$\frac{da_n}{dt} \; = \; \frac{1}{i\hbar} x(t) \sum_m {\mathcal{F}}_{nm}(t) a_m$$ (2.37)

where ${\mathcal{F}}_{nm}(t) \equiv (\partial {\mathcal{H}} / \partial x)_{nm} e^{-i \omega_{nm} t}$ and $\hbar \omega_{nm} \equiv E_n - E_m$. The formal solution (using notation ${\mathbf a} \equiv \{ a_n \}$, and the matrix $\mathcal{F}$) is a time-ordered exponential (Dyson series [174])

$${\mathbf a}(t) \; = \; \exp \left( \frac{1}{i\hbar} \int_0^t dt' x(t') {\mathcal{F}}(t') \right) {\mathbf a}(0) .$$ (2.38)

I take the initial condition as

$$a_n(0) \; = \; \delta_{nm} \hspace{.5in} \mbox{microcanonical distribution} ,$$ (2.39)

that is, a single initial pure eigenstate $\left\vert m\right\rangle$, well above the ground state. The generalization to any incoherent ensemble of initial eigenstates (e.g. canonical ensemble) can be achieved by an occupation-weighted average of the results over $m$. Formulations of LRT ([122,84,99,118,200], and Ingold in [65]) using the density matrix are equivalent to this, although the notation may superficially look different. Using (2.39) and keeping only the first-order term in $x$ in (2.38) gives

$$a_n(t) \; = \; \frac{1}{i\hbar} \left( \frac{\partial \math... ...t' \, x(t') e^{-i \omega_{nm} t'} , \hspace{0.5in} n \neq m,$$ (2.40)

which is called first-order perturbation theory (FOPT). Clearly this is a valid approximation only when $a_m(t)$ remains close to unity and $a_n(t) \ll 1$ for all $n \neq m$. If this is true, the response of any expectation value is linear in $x$: this is what is meant by the word `linear' in LRT (rather that the linearization (2.34)).

I will now specialize to the case of periodic driving at amplitude $A$,

$$x(t) \; = \; A \sin(\omega t) \; = \; \mbox{\small$\frac{1}{2}$}A (e^{i\omega t} - e^{-i\omega t}) .$$ (2.41)

Substituting into FOPT and squaring the absolute value gives

$$\vert a_n(t)\vert^2 \; = \; \frac{1}{\hbar^2} \frac{A^2}{4} \left\vert \left( \frac{\partial ... ...ega)t + \mbox{sinc}^2 \mbox{\small$\frac{1}{2}$}(\omega_{nm} + \omega)t \right]$$

$$\hspace{2in} + \mbox{interference terms} ,$$ (2.42)

where sinc$(x)$ as usual means $\sin(x)/x$. After many periods ( $t \gg 1/\omega$) the sinc$^2$ functions become localized delta-like functions of width $\varepsilon \sim 1/t$,

$$\lim_{t\rightarrow\infty} t \, \mbox{sinc}^2 (\alpha t) \; = \; \pi \delta_{\varepsilon}(\alpha),$$ (2.43)

and the interference terms become irrelevant as these functions become separated. In order that these delta-like functions do not become narrow enough to resolve discrete energy levels, we are limited by the Heisenberg time, $t \ll t_{{\mbox{\tiny H}}}$. Substituting the delta functions (from which I now drop the width subscript $\varepsilon$), the probabilities in states $n \neq m$ can be seen to grow linearly, with transition rates

$$\Gamma_{nm}(\omega) \; =\; \frac{2\pi}{\hbar^2} \frac{A^2}{4} \left\vert \left( \frac{\parti... ...} \right\vert ^2 [ \delta(\omega_{nm} - \omega) + \delta(\omega_{nm} - \omega)]$$ (2.44)

$$= \frac{2\pi}{\hbar} \frac{A^2}{4} \left\vert \left( \frac{\partial... ...ht\vert ^2 [ \delta(E_n - E_m - \hbar\omega) + \delta(E_n - E_m + \hbar\omega)]$$ (2.45)

recognizable as the usual Fermi Golden Rule (FGR) in the presence of both positive and negative frequency driving.

In textbook treatments of the FGR [128], the squared matrix element $\vert(\partial {\mathcal{H}} / \partial x)_{nm}\vert^2$ is assumed to be a smooth function of $E_n$, enabling it to be taken as constant over the width of the delta-function-- this is often true in integrable systems. In contrast, in a chaotic system it takes essentially random, uncorrelated values at each $n$ and $m$ (this assumption underlies any Random Matrix Theory[35,146] description of chaos). The average transition rate is given by the local mean value near the location $nm$ in the matrix (see Fig. 2.3). This will next be formalised using the `band profile' of the matrix.

Figure: Relation of the band profile $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ to the squared matrix elements $\vert(\partial {\mathcal{H}} / \partial x)_{nm}\vert^2$ around $E_n \approx E_m \approx E$. The diagram of the matrix has energy units in both axes. The local average of the transition rate $\Gamma$ is taken over an ellipse whose horizontal width is an energy range $M\Delta$, and whose width in the off-diagonal direction is and energy range $\hbar \varepsilon$. Both widths are $\gg \Delta$, the mean level spacing. The width $M\Delta \ll E$ is classically small.