Hyperfine and Spin-Orbit Coupling Effects on Decay of Spin-Valley States in a Carbon Nanotube

The decay of spin-valley states is studied in a suspended carbon nanotube double quantum dot via the leakage current in Pauli blockade and via dephasing and decoherence of a qubit. From the magnetic field dependence of the leakage current, hyperfine and spin-orbit contributions to relaxation from blocked to unblocked states are identified and explained quantitatively by means of a simple model. The observed qubit dephasing rate is consistent with the hyperfine coupling strength extracted from this model and inconsistent with dephasing from charge noise. However, the qubit coherence time, although longer than previously achieved, is probably still limited by charge noise in the device.

Figure 1

(a) Schematic and (b) scanning electron microscopy image of a device lithographically identical to the one measured. The nanotube is suspended between contact electrodes (130 nm $\mathrm{Cr}/\mathrm{Au}$, marked $S$ and $D$) and over gate electrodes (20 nm $\mathrm{Cr}/\mathrm{Au}$, marked $G1-G5$) patterned on a $\mathrm{Si}/{\mathrm{SiO}}_2$ substrate. Field axes are indicated. For imaging, 2 nm of Pt was evaporated over this chip. (c) Current as a function of gate voltages ${\mathit{V}}_{G1}$ and ${\mathit{V}}_{G4}$, mapping out a double quantum dot stability diagram.

Figure 2

(a) Current at a Pauli-blocked transition, with ${\mathit{V}}_{\mathrm{SD}}=8\mathrm{mV}$, ${\mathit{V}}_{G2}=V_{G3}=-210\mathrm{mV}$, $\mathbf{B}=0$. Dashed (dotted) line marks ground-state degeneracy between left and right dots (between right dot and lead). Arrow marks detuning axis. (b) Current as a function of ${\mathit{V}}_{G1}$ along the detuning axis and of the magnetic field parallel to the nanotube. Arrow marks a region of the Pauli blockade leakage current near zero field. (c) As (b) for perpendicular field. (d) As a function of the field angle for $|\mathbf{B}|=0.8\mathrm{T}$ . Color scales in (a),(d) match (c). Dashed and dotted curves highlight the same transitions as in (a).

Figure 3

(a) Field-dependent transition measured with different tunnel barrier gate settings. Upper plots: current as a function of $ϵ$ and $B_Z$ in the Pauli blockade. Lower plots: cuts along dashed lines of constant detuning. Data (dots) are fitted by a model (curves) described in the text. (b) Schematic energy levels in the absence of hyperfine interaction. The zero-field current peak in (a) is associated with the level degeneracy at $B_Z=0$ (highlighted by ellipse), where hyperfine interaction mixes blocked and unblocked states. Side peaks are associated with spin-dependent tunneling. (c) Tunnel coupling $t$ and $t_{\text{spin}}$ extracted from fits as in (a). Error bars represent 95% confidence intervals.

Figure 4

(a) Pulse scheme for qubit manipulation. Upper panel: gate voltage cycle applied to $G1$. Lower panel: double-dot energy levels during each step. (b) Resonance signal (marked by dashed line) as a function of $f$ and $B_Z$, with $\theta =0$, ${\tau }_E=40\mathrm{ns}$, and microwave level $\sim -20\mathrm{dBm}$ at the device. (c) Resonant signal as a function of ${\tau }_E$ and of power at the device (d) Ramsey dephasing measurement. Points: fringe amplitude as a function of ${\tau }_S$. Curve: Gaussian fit. Left inset: Ramsey pulse cycle. Right inset: signal as a function of phase difference, with ${\tau }_S=5\mathrm{ns}$. (e) Hahn echo measurement. Points: fringe amplitude. Line: fit (see text). Right inset: fringes for orthogonal phases of the echo pulse (along $x$ and $y$ axes in the qubit’s rotating frame), with ${\tau }_S=10\mathrm{ns}$. In panels (c)–(e), $\theta =15°$, $B=83\mathrm{mT}$, and $f=2.82\mathrm{GHz}$.

Figure 5

(a) Qubit frequency as a function of pulse amplitude (points) with linear fit. (b) Measured $T_2^{*}$ (circles) and $T_{\text{echo}}$ (squares) as a function of temperature. Above 750 mK, incoherent current leakage prevents EDSR measurement.