Correlations in noisy Landau-Zener transitions

We analyze the influence of colored classical Gaussian noise on Landau-Zener transitions during a two-level crossing in a time-dependent regular external field. Transition probabilities and coherence factors become random due to the noise. We calculate their two-time correlation functions, which describe the response of this two-level system to a weak external pulse signal. The spectrum and intensity of the magnetic response are derived. Although the noise enters the equation of motion for the Bloch vector in a multiplicative way, nonperturbative analytic results are obtained by a resummation of diagrams in the limit of a short-noise correlation time. Our results also cover regimes where fluctuations are of the same order of magnitude as averages.

Figure 1

Diagrammatic representation of $w$ and contributions to $G_{\alpha ,\alpha }(t,t_0)$. The time axis is chosen to point to the left-hand side. An encircled $+$ or $-$ represents a noise vertex ${\tilde{\eta }}_{\pm }$. (a) A pair of vertices connected by a gray bond represents $-w(t_1,t_2)$. A pair of bold circles contains the sum over both possible polarities. (b) To leading order in small ${\tau }_{\mathrm{n}}$, only diagrams with noncrossing vertex contractions contribute. $G_{z,z}(t,t_0)$ consists only of noise contraction (wiggly lines) within vertex pairs. (c) $G_{+,+}(t,t_0)$ consists of a negative excess vertex as the first vertex (closest to $t_0$) and a positive excess vertex as the last vertex. Contractions can be performed only from pair to pair.

Figure 2

Diagrams contributing to $G_{zz,zz}(t,t_0)$. First, there is the “ladder” without rungs. The diagrams with rungs can be represented as propagators $G_{z,z}(t,t_1)$ on both legs of the ladder, following the ring contraction between the last rungs as times $t_1$ and $t_2$. Prior to $t_2$ additional leg propagators and rings may occur. In the Bethe-Salpeter equation, these additional contractions sum up to $G_{zz,zz}(t_2,t_0)$.

Figure 3

The propagator $G_{zz,+-}$ consists of diagrams in which the first vertices are excess vertices. They can be contracted with a certain number of pairs until the first rung occurs at time $t_1$. The contraction of all later pairs yields $G_{zz,zz}(t,t_1)$.

Figure 4

The diagrams $G_{+-,zz,}$ are analogous to the diagrams of $G_{zz,+-}$ with the only difference being that now the excess vetices are the last vertices.

Figure 5

$G_{+-,+-}$ consists of diagrams with excess vertices as first and last vertices on both time legs. There are disconnected diagrams (where the “ladder” has no rungs) and connected diagrams (where the ladder has rungs). In the second case, the excess vertices are are connected by rungs as $t_1$ and $t_2$, and between $t_1$ and $t_2$ additional pairs can be contracted as in $G_{zz,zz}(t_1,t_2)$.

Figure 6

The diagrams of $G_{+-,-+}$ as similar to these of $G_{+-,+-}$ with the difference being that the polarity of the first excess vertices is inverted. Therefore, no disconnected diagrams can be formed.