Figure 1

(Color online) (a) Double-well potential for a buckled nanobar. As shown in the left panel, the lowest energy level is split into two levels for $f_{\perp }=0$ due to tunneling from the right potential well to the left one. The lowest (blue or dark gray) and the first excited (red or lighter gray) levels correspond to the symmetric and antisymmetric combinations of the wave functions localized in the left and right potential wells. The energy level splitting between the left and right states could be controlled by the transverse force $f_{\perp }$, as shown in the top right panel. (b) A buckled rod qubit, where the compressed longitudinal force $f$ applied to the rod ends controls the potential shape [$\alpha$ and $\beta$ in Eq. (33)] and, therefore the energy splitting at the degeneracy point. The transverse force $f_{\perp }$ allows one to drive the rod to its degeneracy point. (c) Correlated noise produced by some phonons can be reduced using a decoherence-free subspace (see, e.g., Ref. 35): nearby qubits Q1 and Q2 (in red) can be associated with one logic qubit.Reuse & Permissions

Figure 2

(Color online) (a) Experimental resonance (Ref. 33) of the effective dissipation (which is proportional to the average energy $⟨E⟩$) as a function of the dimensionless reduced detuning frequency $(\omega -{\omega }_0)∕{\omega }_0$, for a suspended carbon nanotube, for different values of the power $P$ of the externally applied ac electromagnetic wave. (b) The resonance in the response of the three-junction SQUID probed by measuring (Ref. 34) the voltage for the resonant tank $V_T$ for different driving amplitudes ($V_d=23.8\mu \mathrm{V}$ for the single peak resonance and $V_d=30\mu \mathrm{V}$ for the split resonance) as a function of $(\omega -{\omega }_0)∕{\omega }_0$. For these different systems, a similar splitting of the resonance peak was observed (Refs. 33, 34) Schematic diagrams for a driven suspended carbon nanotube and three junction SQUID are also shown at the top. The simulated resonance [average energy $⟨E⟩$ versus the dimensionless detuning frequency $(\omega -{\omega }_0)∕{\omega }_0$] for quantum (c) and classical (d) particles moving in the double-well potential shown in the inset of (c). A quantum description (c) agrees with experiments (Refs. 33, 34) [(a) and (b)]: For quantum particles, the standard Lorentz-form resonance peak for weak ac drives (c, dashed red curve, drive amplitude $A=0.005$, i.e., about a factor of 0.02 of the barrier height $U_0$ and half of the level splitting) separates into two subpeaks due to tunneling at stronger drives [dotted blue curve, $A=0.01\approx 0.04U_0$ (i.e., almost equal to the level splitting); solid green curve, $A=0.0135\approx 0.055U_0$ (i.e., about 1.35 times the level splitting)]. A classical description (i.e., when the level splitting is negligible) presented in (d) cannot describe experiments (Refs. 33, 34) [(a) and (b)]: For classical particles, the weak-driving Lorentz-form peak (dashed red curve, $A=0.005\approx 0.02U_0$) becomes asymmetric and exhibits a discontinuous jump $⟨E⟩\left(\omega \right)$ (solid green curve, $A=0.0135\approx 0.055U_0$) due to the nonlinearity of $U\left(y\right)$.Reuse & Permissions

Figure 3

(Color online) (a) The dependence of the normalized average energy $⟨E⟩∕E_J$ versus normalized frequency detuning $(\omega -{\omega }_{12})∕{\tilde{\omega }}_p$, with $E_J∕\hslash {\tilde{\omega }}_0=40$, $\tilde{\alpha }=1.6$, $\tilde{\beta }=0.4$, semiclassical matrix elements $\mid x_1\mid =\mid x_2\mid \approx {10}^{-4}$, ${\Phi }_{\mathrm{ampl}}=0.5$ (red solid line), ${\Phi }_{\mathrm{ampl}}=0.4$ (green dotted line), and ${\Phi }_{\mathrm{ampl}}=0.1$ (blue dashed line). Note that when the driving increases, the barrier height decreases according to Eq. (29) as follows: $U_0\approx 0.75$ for ${\Phi }_{\mathrm{ampl}}=0.1$, $U_0\approx 0.7$ for ${\Phi }_{\mathrm{ampl}}=0.4$, and $U_0=0.67$ for ${\Phi }_{\mathrm{ampl}}=0.5$. (b) The black solid line presents the frequency difference ${\Delta }_{\text{peaks}}$ between subpeaks of the split resonance, versus power, for the same parameters used in panel (a). The dashed part of this curve corresponds to the situation when the two-subpeak structure cannot be resolved. The dotted red line shows the dependence of the split-peak (or subpeak) height, normalized by its value at ${\Phi }_{\mathrm{ampl}}=0.2$. (c) Schematic diagram of the four energy levels participating in the splitting resonance.Reuse & Permissions

Figure 4

(Color online) Schematic diagram of a procedure for controlling the quantum state of a nanobar via coherent oscillations at the degeneracy point (see also Ref. 26). Applying the perpendicular force $f_{\perp }$ initializes the system during the stage (i). Turning off $f_{\perp }$ brings the rod to its degeneracy point, and the nanorod starts to oscillate a time $t_{\Delta }$ during stage (ii). To observe coherent oscillations, measurements (e.g., optically or electrically) must be done for many values of $t_{\Delta }$.Reuse & Permissions