Interplay of charge and spin coherence in Landau-Zener-Stückelberg-Majorana interferometry

We study Landau-Zener dynamics in a double quantum dot filled with two electrons, where the spin states can become correlated with charge states and the level velocity can be tuned in a time-dependent fashion. We show that a correct interpretation of experimental data is only possible when finite-time effects are taken into account. In addition, our formalism allows the study of partial adiabatic dynamics in the presence of phonon-mediated hyperfine relaxation and charge-noise-induced dephasing. Our findings demonstrate that charge noise severely impacts the visibility of Landau-Zener-Stückelberg-Majorana interference fringes. This indicates that charge coherence must be treated on an equal footing with spin coherence.

Figure 1

(a) The Landau-Zener-Stückelberg-Majorana (LZSM) problem. The LZSM model addresses the problem of a two-level system that is swept through an anticrossing. It assumes an infinitely long ramp (i.e., propagation from $t_{\mathrm{i}}\to -\infty$ to $t_{\mathrm{f}}\to \infty$), a constant coupling constant $\lambda$ leading to a splitting $2\left|\lambda \right|$ at $t=0$, and an energy difference between the levels that varies linearly with time $\Delta E\left(t\right)=\alpha t$. Its main result gives the nonadiabatic transition probability $P_{\mathrm{LZSM}}$. An extension of the model to finite times resolves the problem of infinite energies and undefined phases. (b) Singlet and triplet ${\text{T}}_{+}$ energies in a DQD as a function of detuning $\varepsilon$, where ${\varepsilon }_{\mathrm{i}}=\varepsilon \left(t_{\mathrm{i}}\right)$ and ${\varepsilon }_{\mathrm{s}}=\varepsilon \left(t_{\mathrm{f}}\right)$. The two-electron DQD is a physical realization of the LZSM model. Here, the hyperfine coupling of the two spin states leads to a splitting ${\Delta }_{\mathrm{HF}}$. (c) The LZSM model assumes a constant level velocity $\alpha$. In order to increase adiabaticity, one can lower $\alpha$, but in order to keep short pulses it is preferable to use multi rise-time pulses. (d) Comparison of possible pulses that can be used to manipulate the $\text{S}$-${\text{T}}_{+}$ qubit. In terms of total propagation time, “double-hat” pulses are a good compromise between conventional trapezoid and convolved pulses.

Figure 2

Comparison of the adiabatic transition probability $P_{\mathrm{a}}$ for a double hat (orange) and a linear pulse (red) with $t_{\mathrm{w}}=0$ as a function of $\Delta E_{\mathrm{f}}$. (Inset) Pulse profiles used to obtain $P_{\mathrm{a}}$. The maximal offset for the single rise-time pulse is chosen such that the system is never driven through the anticrossing ($\Delta E_{\mathrm{f}}^{\mathrm{single}}<0$). The maximal offset for double-hat pulses is determined by the condition that the system cannot be driven with the slow component of the pulse through the anticrossing ($\Delta E_{\mathrm{sr}}^{\mathrm{dh}}<0$). For double-hat pulses, the values of $P_{\mathrm{a}}$ between $0<\Delta E_{\mathrm{f}}<0.5\mu \mathrm{eV}$ originate from the slow portion of the pulse, which brings the system close to the anticrossing. In this range, the magnitude of $P_{\mathrm{a}}$ is not due to the second fast rise-time portion of the pulse, which drives the system through the anticrossing.

Figure 3

Comparison of the adiabatic transition probability $P_{\mathrm{a}}$ for a double hat (orange) and a truncated double hat (green), which is missing the second fast rise-time component, as a function of $\Delta E_{\mathrm{f}}$ at the end of the slow detuning pulse. (Inset) Pulse profiles used to obtain $P_{\mathrm{a}}$. The results obtained with a double hat are shifted by an amount $\Delta E_{\mathrm{fast}2}$ on the $x$ axis to allow for comparison. We conclude from these results that the state of the system is hardly changed during the second fast detuning pulse.

Figure 4

Energy diagram for the relevant states in the DQD as a function of $\varepsilon$. The spin states for the implementation of the qubit are the hybridized singlet $\text{S}$ and the triplet ${\text{T}}_{+}$.

Figure 5

The effective hyperfine-mediated coupling between $|\mathrm{S}\rangle$ and $|\mathrm{T}\rangle$ is a function of detuning. This is a consequence of the state $|\mathrm{S}\rangle$ being a superposition of different charge states.

Figure 6

(a) Relaxation rate ${\Gamma }_1\left(\varepsilon \right)$ obtained for $B=100\mathrm{mT}$. Here, we consider hyperfine-mediated flip-flops of the electron spin accompanied by emission or absorption of phonons. The relaxation rate is maximal for $\varepsilon ={\varepsilon }_{\mathrm{c}}$ where $|E_{\mathrm{S}}^d\left(\varepsilon \right)-E_{\mathrm{T}}^d|\ll k_{\mathrm{B}}T$. This results in a strong mixing of the spin states due to thermal fluctuations. (b) Charge-noise-induced dephasing rate ${\Gamma }_2\left(\varepsilon \right)$. The dephasing rate is assumed to have a functional dependence proportional to $1-c{\left(\varepsilon \right)}^2$ to ensure that dephasing related to charge noise is weaker when the system is in a $(1,1)$ charge configuration.

Figure 7

(a) Measurements of $P_{\mathrm{S}}$ as a function of $t_{\mathrm{w}}$ and ${\varepsilon }_{\mathrm{s}}$ for $A_{1\mathrm{f}}=-0.26\mathrm{meV}$, $t_{\mathrm{slow}}=4\mathrm{ns}$, and $B=55\mathrm{mT}$. Double-hat pulses allow the observation of nonadiabatic transitions when the system is driven slowly to close proximity of the anticrossing. (b) Theoretical predictions obtained using a pulse profile obtained at the output of the waveform generator and parameters from (a). (c) $P_{\mathrm{S}}$ plotted as a function of $B$ and ${\varepsilon }_{\mathrm{s}}$ for $t_{\mathrm{w}}=20\mathrm{ns}$ reveals the spin funnel. Parameters are $A_{1\mathrm{f}}=-0.26\mathrm{meV}$ and $t_{\mathrm{slow}}=4\mathrm{ns}$. (d) Theoretical calculations for the same parameters as in (c).

Figure 8

Experimentally [(a) and (c)] and theoretically [(b) and (d)] obtained LZSM interference patterns and spin funnels. Here, $A_{1\mathrm{f}}=-0.29\mathrm{meV}$, $t_{\mathrm{slow}}=4\mathrm{ns}$, and $B=55\mathrm{mT}$. The waiting time for the spin funnel is $t_{\mathrm{w}}=20\mathrm{ns}$. The interference patterns differ from results presented in Fig. 7, indicating that the evolution of the system is not only sensitive to the level velocity, but also the time at which the change in level velocity happens.

Figure 9

Singlet return probability $P_{\mathrm{S}}$ as a function of the waiting time $t_{\mathrm{w}}$ and final position ${\varepsilon }_{\mathrm{s}}$ obtained for a double-hat detuning pulse. Pulse details are given in the text. Here, we have set $B=100\mathrm{mT}$, $u=4\mathrm{meV}$, and $\tau =5\mu \mathrm{eV}$ for all plots. The parameters ${\gamma }_0$ and ${\gamma }_2$ are (a) ${\gamma }_0={10}^{-2}$, ${\gamma }_2={10}^8{\mathrm{s}}^{-1}$, (b) ${\gamma }_0={10}^{-2}$, ${\gamma }_2={10}^9{\mathrm{s}}^{-1}$, (c) ${\gamma }_0={10}^{-1}$, ${\gamma }_2={10}^8{\mathrm{s}}^{-1}$, and (d) ${\gamma }_0={10}^{-1}{\mathrm{s}}^{-1}$, ${\gamma }_2={10}^9{\mathrm{s}}^{-1}$. Figures (e) and (f) show cuts for case (a) in blue, (b) in dark green, (c) in purple, and (d) in dark red. The cuts are respectively taken before the anticrossing at ${\varepsilon }_{\mathrm{s}}=3.987\mathrm{meV}$ and after at ${\varepsilon }_{\mathrm{s}}=3.908\mathrm{meV}$.