Proposal for a Nanomechanical Qubit

Mechanical oscillators have been demonstrated with very high quality factors over a wide range of frequencies. They also couple to a wide variety of fields and forces, making them ideal as sensors. The realization of a mechanically based quantum bit could therefore provide an important new platform for quantum computation and sensing. Here, we show that by coupling one of the flexural modes of a suspended carbon nanotube to the charge states of a double quantum dot defined in the nanotube, it is possible to induce sufficient anharmonicity in the mechanical oscillator so that the coupled system can be used as a mechanical quantum bit. However, these results can only be achieved when the device enters the ultrastrong coupling regime. We discuss the conditions for the anharmonicity to appear, and we show that the Hamiltonian can be mapped onto an anharmonic oscillator, allowing us to work out the energy level structure and find how decoherence from the quantum dot and the mechanical oscillator is inherited by the qubit. Remarkably, the dephasing due to the quantum dot is expected to be reduced by several orders of magnitude in the coupled system. We outline qubit control, readout protocols, the realization of a CNOT gate by coupling two qubits to a microwave cavity, and finally how the qubit can be used as a static-force quantum sensor.

Popular Summary

Quantum information processing, including computation, communication, and sensing, may deliver the next technological revolution through the promise of unprecedented computational capabilities, security, and detection sensitivities. At the heart of a quantum information system is the quantum bit, or qubit, for which only a handful of platforms have demonstrated all the requirements, such as high-fidelity controlled gates, easy qubit-qubit coupling, and good isolation from the environment. Here, we investigate the theoretical potential for quantum information processing in an electromechanical system with mechanical qubits.

In our proposed qubit, the mechanical vibrations of a nanotube resonator couple to a double quantum dot formed in the suspended nanotube. The large electromechanical coupling possible in this system results in a strongly anharmonic coupled system operating in the quantum regime, in which the decoherence times are predicted to be remarkably long. In addition to working out the expected performance of such a qubit, we show how to implement two-qubit entangling gates and how to carry out quantum sensing of static forces.

Mechanical qubits based on nanotubes and other systems potentially offer long coherence times and could enable straightforward coupling to other quantum systems such as photons, spins, and superconducting qubits, a potential not easily exploited in other qubit systems.

Figure 1

Schematic of the proposed setup. A suspended carbon nanotube hosting a double quantum dot, whose one-electron charged state is coupled to the second flexural mode. (a) Sketch of the electronic confinement potential and of the two main parameters, the hopping amplitude $t$ and the energy difference $\epsilon$ between the two single-charge states. (b) Physical realization. One of the gate electrodes is connected to a microwave cavity for dispersive qubit readout.

Figure 2

Effective potentials ${ϵ}_{+}(x)$ (red) and ${ϵ}_{-}(x)$ (blue) from Eq. (2) for $t/\hslash {\omega }_{\mathrm{m}}=20$ and the values of $(4g/{\omega }_{\mathrm{m}}{)}^2=0$, 10, 20, 30, 40, 50, with the first and last lines explicitly indicated in the figure. The potential for $g=g_c^{\mathrm{sc}}={\omega }_{\mathrm{m}}\sqrt{5}$ is shown with a thicker line.

Figure 3

Lowest-lying energy eigenvalues $E_n$ of the Hamiltonian (1) for $\epsilon =0$ and $t=20\hslash {\omega }_{\mathrm{m}}$ as a function of $g/{\omega }_{\mathrm{m}}$. The Born-Oppenheimer potential given by Eq. (2) and the energy levels are shown in the insets for $g=0$ and $g=3.2{\omega }_{\mathrm{m}}$. The dashed line indicates the lowest noninteracting electronic level $-t/2$. The semiclassical critical value for the bistability is $g_c^{\mathrm{sc}}/{\omega }_{\mathrm{m}}=\sqrt{5}\approx 2.23$. The value of $g=g_{5\%}\approx 1.8{\omega }_{\mathrm{m}}$ for which the anharmonicity is 5% is also shown.

Figure 4

Contour plot of the anharmonicity $a$ in the $(t,g)$ plane. The contour line for $a=0.05$ is thicker, and it defines the function $g_{5\%}(t)$. The kink at $t\approx 1.54\hslash {\omega }_{\mathrm{m}}$ of this function, better seen in the inset, is due to the avoided crossing between the charge and oscillator eigenstates that occurs at that value of $t$. It indicates the region where the eigenstate begins to have a predominantly charge nature.

Figure 5

Wave functions of the two qubit states $|0⟩$ (upper panels) and $|1⟩$ (lower panels) for $t/\hslash {\omega }_{\mathrm{m}}=20$, $g/{\omega }_{\mathrm{m}}=0.1$, 1.8, and 3.0. We plot $-⟨{\sigma }_x⟩(x)$ (yellow), $⟨{\sigma }_z⟩(x)$ (green), and ${\psi }_{n+}(x{)}^2+{\psi }_{n-}(x{)}^2$ (blue). Note that for small coupling, the yellow and blue lines perfectly overlap. The probability of occupation of the first single-harmonic oscillator states is indicated in the insets.

Figure 6

Comparison between numerical (solid line) and analytical (dashed line) dependence of the anharmonicity parameter $a$ for three values of the coupling $g/{\omega }_{\mathrm{m}}=0.5$, 1, and 1.5.

Figure 7

Coefficients ${\beta }_i$ of the operator projections in the qubit space, obtained by numerical diagonalization (solid lines) and from the analytical approximation to fourth order in $g$ (dashed lines). The value of $t$ is fixed here to $20\hslash {\omega }_{\mathrm{m}}$.

Figure 8

Quantity $|\chi |{\omega }_{\mathrm{m}}/g_{\mathrm{ec}}^2$ for $t/\hslash {\omega }_{\mathrm{m}}=10$, $g/{\omega }_{\mathrm{m}}=1.2164$ (for which $a=0.05$) as a function of ${\omega }_c/{\omega }_{\mathrm{m}}$ in two different regions of the spectrum: close to ${\omega }_{10}<{\omega }_{12}$ and close to ${\delta }_{00}<{\delta }_{11}$, left and right panels, respectively. The thick blue line gives the numerical calculation, the thin red line the expressions (22) and (23), shown in the left and right panels, respectively.

Figure 9

Ratio of the decay rate ${\mathrm{\Gamma }}_{1\to 0}^{\mathrm{qb}}(g_{5\%})/{\mathrm{\Gamma }}_{1\to 0}^{\mathrm{qb}}(g=0)$ and pure decoherence rate ${\mathrm{\Gamma }}_{10}^{\mathrm{qb},\varphi }(g_{5\%})/{\mathrm{\Gamma }}_{10}^{\mathrm{qb},\varphi }(g=0)$ as a function of $t/\hslash {\omega }_{\mathrm{m}}$. We assume $R^D=R^{\varphi }=1$, and we neglect oscillator damping ($\gamma =0$). The vertical dashed line indicates the beginning of the region where the qubit becomes dominated by the two charge states, i.e., where $t<1.54\hslash {\omega }_{\mathrm{m}}$ (cf. also inset of Fig. 4).

Figure 10

Coefficient $K$ of the contributions to $J_{zx}$ that diverge like ${\omega }_{\mathrm{m}}^{(1)}/({\omega }_c-{\omega }_{10}{)}^{(2)})$ [divided by the two coupling constants and the driving intensity $g_{\mathrm{ec}}^{(1)}{g}_{\mathrm{ec}}^{(2)}A$] as a function of ${\omega }_{\mathrm{m}}^{(2)}$. The solid line is the numerical result, and the dashed line is the analytical one, Eq. (43). The other parameters are $g^{(1)}=g^{(2)}=1.264\hslash {\omega }_{\mathrm{m}}^{(1)}$, $t^{(1)}=10\hslash {\omega }_{\mathrm{m}}^{(1)}$, and $t^{(2)}=10.5\hslash {\omega }_{\mathrm{m}}^{(1)}$.

Figure 11

Network of capacitances representing the (a) single- and (b) double-dot circuits. The capacitances $C_B$ are used to model the voltage sources.