Landau-Zener Transition in a Continuously Measured Single-Molecule Spin Transistor

We monitor the Landau-Zener dynamics of a single-ion magnet inserted into a spin-transistor geometry. For increasing field-sweep rates, the spin reversal probability shows increasing deviations from that of a closed system. In the low-conductance limit, such deviations are shown to result from a dephasing process. In particular, the observed behaviors are successfully simulated by means of an adiabatic master equation, with time averaged dephasing (Lindblad) operators. The time average is tentatively interpreted in terms of the finite time resolution of the continuous measurement.

Figure 1

(a) Artistic view of the molecular spin transistor with the ${\mathrm{TbPc}}_2$ molecule embedded between the gold electrodes. (b) Schematics of the molecular system: one of the phthalocyanine ligands acts as a read-out quantum dot, where the spins of the localized electrons are exchange coupled to the total angular momentum ($J=6$) of the ${\mathrm{Tb}}^{3+}$ ion, whose $M_J=\pm 6$ states define an effective two-level system. (c) The system is prepared in the initial ground state $|\uparrow ⟩$, evolves under the effect of a magnetic field $B_z$ that depends linearly on time, and ends up in the final ground state $|\downarrow ⟩$ with probability $P_{\text{gs}}$. (d) If the system is prepared in the initial excited state, the spin reversal leads to the final excited state with probability $P_{\text{es}}$.

Figure 2

(a) Measured values of the spin-reversal probabilities $P_{\text{gs}}$ (black squares) and $P_{\text{es}}$ (red), obtained after preparing the spin in the initial ground and exited states, respectively (the solid lines are drawn as a guide for the eye). This set of probabilities has been obtained with a conductance of $g=0.245\mu \mathrm{S}$. (b) Difference between $P_{\text{gs}}$ and $P_{\text{es}}$ as a function of the conductance, for different values of the field-sweep rate $(dB/dt)$. The solid lines correspond to linear fits of the experimental results (symbols).

Figure 3

(a) Computed spin-reversal probability $P_c$ as a function of the inverse field-sweep rate, for different values of the averaging time ${\tau }_{\text{av}}$, normalized to ${\tau }_{\mathrm{\Delta }}$. For a given $\mathrm{\Delta }$, the quantity reported in the horizontal axis can also be identified with the time that the spin takes to go through the anticrossing, being ${\tau }_{\text{ac}}/{\tau }_{\mathrm{\Delta }}=({\mathrm{\Delta }}^2/\hslash g_J{\mu }_B)(dB/dt{)}^{-1}$. For all the solid curves, the dephasing time is ${\tau }_d={\tau }_{\mathrm{\Delta }}$, while the dotted curve corresponds to the coherent case (${\tau }_d=\infty$). (b) Dependence of $P_c$ on the inverse sweeping rate for a fixed averaging time, ${\tau }_{\text{av}}=20{\tau }_{\mathrm{\Delta }}$, and for different values of the dephasing time ${\tau }_d$.

Figure 4

Simulated values of the spin-reversal probability as a function of the inverse sweeping rate, for different values of the dephasing time ${\tau }_d$ and of the averaging time ${\tau }_{\text{av}}=20{\tau }_d$. The reported values of the inverse sweeping rate and of the dephasing time correspond to a zero-field gap $\mathrm{\Delta }\simeq 3.4\mu \mathrm{K}$ or, equivalently, to a time scale ${\tau }_{\mathrm{\Delta }}\simeq 2.25\mu \mathrm{s}$.