Creating arbitrary quantum vibrational states in a carbon nanotube

We theoretically study the creation of single- and multiphonon Fock states and arbitrary superpositions of quantum phonon states in a nanomechanical carbon nanotube (CNT) resonator. In our model, a doubly clamped CNT resonator is initialized in the ground state, and a single electron is trapped in a quantum dot which is formed by an electric gate potential and brought into the magnetic field of a micromagnet. The preparation of arbitrary quantum phonon states is based on the coupling between the mechanical motion of the CNT and the electron spin which acts as a nonlinearity. We assume that electrical driving pulses with different frequencies are applied on the system. The quantum information is transferred from the spin qubit to the mechanical motion by the spin-phonon coupling, and the electron spin qubit can be reset by the single-electron spin resonance. We describe Wigner tomography which can be applied at the end to obtain the phase information of the prepared phonon states.

Figure 1

Schematic view of a single electron being trapped in a quantum dot (QD) formed by gate voltages in a suspended carbon nanotube (CNT). The resonance frequency of the CNT can be adjusted by the voltages on the back gates. An external ac electric field is applied on the CNT by the antenna. The micromagnet is deposited in the vicinity of the CNT. The single electron wave function can be electrostatically shifted by applying voltage on the back gates. Therefore the spin splitting of the electron in the slanting magnetic field of the micromagnet can be manipulated.

Figure 2

The energy-level diagram of the spin-phonon states. (a) The electron spin resonance between $|\downarrow 0\rangle$ and $|\uparrow 0\rangle$ (blue dashed). The qubit is detuned from the phonon. The parameter $\alpha$ is the effective strength of the spin operator ${\sigma }_x$ in the effective Hamiltonian (2). (b) The qubit is brought into resonance with the phonon in the slanting magnetic field of the micromagnet by electrostatically moving the QD. The spin-phonon interaction strength is $g$ (red solid). Phase operation ${\sigma }_z$ with the strength $\beta$ is applied to adjust the relative phase of the state (green dotted).

Figure 3

(a) Sequence of operations for obtaining arbitrary quantum phonon states. An external ac electrical field with the frequency $\omega$ (blue dashed) is applied for obtaining the qubit flip in the time intervals $r_1$ and $r_2$, and ${\omega }_p$ is the frequency of the phonon mode (red dotted) in the CNT. The large detuning $\mathrm{\Delta }$ is required for the qubit flip operation $R$. For the qubit-phonon swap $U\left(\tau \right)$, the qubit with the frequency ${\omega }_q$ (black solid) is brought into resonance with the phonon that ${\omega }_p={\omega }_q$ by moving the QD in the slanting magnetic field of the nearby micromagnet. The swap operation's time intervals are ${\tau }_1$ and ${\tau }_2$. The phase operations $P\left(l_{1U}\right), P\left(l_{1R}\right)$, and $P\left(l_{2R}\right)$ adjust the relative phase of the state. (b) Diagram for calculating the operation sequence in a backwards direction for obtaining the superposition of Fock states $|\downarrow \rangle (|0\rangle +i|2\rangle )$ from the ground state $|\downarrow 0\rangle$. The operation $U\left({\tau }_2\right)$ is applied to fully transfer the state $|\downarrow 2\rangle$ to the state $|\uparrow 1\rangle$, and the flip operation $R\left(r_2\right)$ is applied to fully transfer the state $|\uparrow 1\rangle$ to $|\downarrow 1\rangle$ in step number 2. In step number 1, phase operations are applied to adjust the relative phases to cancel some states (red crosses), e.g., the states $|\downarrow 1\rangle$ and $|\uparrow 0\rangle$ in the following qubit flip or qubit-phonon swap.

Figure 4

The Wigner tomography of the quantum phonon states $|0\rangle +|2\rangle , |0\rangle -i|2\rangle$ and $|0\rangle +i|2\rangle$. The change of the relative phase of a two-state superposition of Fock states rotates the Wigner function $W\left(\alpha \right)$.

Figure 5

The fidelity $F$ of obtaining $|\psi \rangle =|0\rangle +i|2\rangle$ at temperature $T=10\mathrm{mK}$ as a function of the damping rate $\mathrm{\Gamma }$. The fidelity $F=\sqrt{\langle \psi |\rho |\psi \rangle }$ shows how close the obtained state $\rho$ is to the target state $|\psi \rangle$. The Wigner tomography of the obtained states at $T=10\mathrm{mK}$ with (a) damping $\mathrm{\Gamma }=0$, (b) $\mathrm{\Gamma }={10}^4{\mathrm{s}}^{-1}$, and (c) $\mathrm{\Gamma }=3\times {10}^4{\mathrm{s}}^{-1}$. The fidelity for the state obtained at (a) is $F=0.999$, for (b) is $F=0.945$, and for (c) is $F=0.859$. The other parameters are $\lambda /2\pi =0.8\mathrm{MHz}, \mathrm{\Delta }=100\mathrm{MHz}, {\omega }_p/2\pi =1.5\mathrm{GHz}$, and $g/2\pi =0.56\mathrm{MHz}$.