Nanomechanical readout of a single spin

The spin of a single electron in a suspended carbon nanotube can be read out by using its coupling to the nanomechanical motion of the nanotube. To show this, we consider a single electron confined within a quantum dot formed by the suspended carbon nanotube. The spin-orbit interaction induces a coupling between the spin and one of the bending modes of the suspended part of the nanotube. We calculate the response of the system to pulsed external driving of the mechanical motion using a Jaynes-Cummings model. To account for resonator damping, we solve a quantum master equation, with parameters comparable to those used in recent experiments, and show how information about the spin state of the system can be acquired by measuring the mechanical motion of the nanotube. The latter can be detected by observing the current through a nearby charge sensor.

Figure 1

Schematic of the proposed setup of the carbon nanotube (CNT) resonator with a quantum dot containing a single electron. The gates on both sides of the CNT can be electrostatically adjusted to control the number of confined electrons. The back gate is used to tune the resonance frequency of the CNT into resonance with the splitting of the Zeeman sublevels of the electron which make up the qubit. The charge sensor in the vicinity of the vibrating nanotube serves as a detection device for changes in the amplitude due to the coupling of the spin to the vibrational motion. It can be a quantum point contact or a quantum dot which needs to be operated in a regime where the current depends nonlinearly on the gate voltage. A current meter is used to measure a nonlinear current $I$ which flows through the device.

Figure 2

(a) The eigenenergies of $H_0$ are shown in red. The black dotted lines correspond to the uncoupled, i.e., $g=0$, case. Note that the ground state is not modified by the coupling $g$. The coupling introduces a splitting $2g\sqrt{n}$, which is nonlinear in the phonon number $n$. (b) The time-averaged squared resonator displacement $\overline{X^2}\left(\tau \right)=\frac{1}{\tau }{\int }_0^{\tau }dt{X}^2\left(t\right)$ as a function of the detuning of the driving frequency $\delta \omega =\omega -{\omega }_p$ for a fixed integration time of $\tau =100\mu \mathrm{s}$. Here $X^2\left(t\right)=\langle x^2\left(t\right)\rangle =\mathrm{Tr}\left[x^2\rho \left(t\right)\right],$ and $\rho$ is the oscillator density matrix. The upper (lower) blue dashed line is the evolution at $T=0\mathrm{mK}$ of the initial state $|\uparrow 0\rangle$ ($|\downarrow 0\rangle$). The strongest difference between the amplitudes of the two aforementioned initial states is found for a detuning of $\delta \omega /2\pi =\pm 0.07\mathrm{MHz}$ (see mark C). The upper and lower red solid lines correspond to the same initial spin states at a temperature of $T=30\mathrm{mK}$. The other parameters are $\lambda /2\pi =0.04\mathrm{MHz}$, ${\omega }_p/2\pi ={\omega }_q/2\pi =1.5\mathrm{GHz}$, $\Gamma =5\times {10}^4$ $\mathrm{Hz}$, and $g/2\pi =0.3\mathrm{MHz}$. At $30\mathrm{mK}$ the thermal energy $k_{B}T$ is 2.4 times smaller than $\hslash {\omega }_q$. The label A in both panels (a) and (b) marks the transition from the ground state to the first excited state with positive detuning. The line marked B corresponds to the next transition between states with positive parity. A transition between the states with negative parity requires a negative detuning. In between A and B only two-phonon transitions can occur. The main peak around C at $\delta \omega \approx 0.07\mathrm{MHz}$ is due to several transitions, with the main contribution from the transition between eigenstates $|{\psi }_{+,4}\rangle \to |{\psi }_{+,5}\rangle$. This is consistent with the magnitude of the main peak around $n\sim \overline{X^2}/l_0^2\sim 4$. The inset shows the squared amplitude of the CNT as a function of the elapsed time $t$ for a detuning of the driving frequency of $\delta \omega /2\pi =0.07\mathrm{MHz}$. Within the first $10\mathrm{ns}$ the fast Rabi oscillations with the frequency of the coupling strength $g$ are damped, and following oscillations are caused by the driving strength $\lambda$. The resonator reaches a steady state after an interaction time $1/\Gamma$.

Figure 3

(a) The squared amplitude $X^2\left(t\right)=\langle x^2\left(t\right)\rangle =\mathrm{Tr}\left[x^2\rho \left(t\right)\right]$ as a function of the elapsed time $t$ is shown for the cases of continuous (dashed blue line), pulsed (dotted green line), and variable (solid red line) driving. The change of magnitude in the current $|I_2{X}^2\left(t\right)|=|I\left(t\right)-I_0|$ of spin up and spin down is shown on the right vertical axis. The same parameters as in Fig. 2 are used. The driving frequency $\omega$ is changed at the time $t=19\mu \mathrm{s}$ right before the amplitude is largest. (b) The difference of the time-averaged squared amplitudes between spin up and spin down $\overline{X^2}{\left(\tau \right)}_{\uparrow }-\overline{X^2}{\left(\tau \right)}_{\downarrow }$, on the left vertical axis. The three curves show the integrated signal for the same driving modes as in (a). On the right vertical axis we indicate the difference of the corresponding time-averaged squared current through the QD, $\overline{I}{\left(\tau \right)}_{\uparrow }-\overline{I}{\left(\tau \right)}_{\downarrow }$. The current is integrated over an interval $[0,\tau ]$ given by the integration time $\tau$. The variable driving scheme results in an increase in current signal by a factor of about 2.

Figure 4

Capacitor model for the charge readout of the motional degree of freedom of a CNT. The CNT (2), ground plane (3), and charge-sensor gate (1) are modeled as cylinders of length $l$. The distance between (1) and (2) [(2) and (3)] is $d+X$ ($d-X$), where $X$ denotes the vibrational displacement of the CNT. The corresponding capacitances are denoted $C\left(X\right)$ and $C(-X)$. The electrostatic potential of the CNT (2) is set to zero (ground). If the potential of (3) is also fixed at a voltage $V_3$ (e.g., $V_3=0$), then the motion of the CNT (2) by an amount $X$ leads to an extra charge on the gate (1), which is manifested as in the voltage $V_1$. The charge-sensor gate voltage $V_1$ can then be used to bias the current $I$ through a QD with voltage bias $V_{\mathrm{sd}}$.