Vibronic Spectroscopy of an Artificial Molecule

We have performed microwave reflection experiments on a charge-phase qubit coupled to an $LC$ oscillator. We find that the system behaves like an artificial molecule showing vibronic sideband transitions. The reflected signal is determined by a combination of the Franck-Condon principle and resolved-sideband cooling or heating of the oscillator.

Figure 1

The potential energy and energy eigenstates in the artificial molecule (1). (a) The total potentials $U_{\pm }$ (solid curves) are formed as a sum of qubit energy (dashed line) and a harmonic vibrational potential. The six lowest energy eigenstates are shown for both electronic states. The bars represent the energy differences $\Delta E=h\nu$ corresponding to the drives of low- and high-frequency, ${\nu }_{\mathrm{LF}}$ and ${\nu }_{\mathrm{HF}}$, respectively. The parameters are for the bias point $({\Phi }_b,n_g)=(0.60{\Phi }_0,1.16)$. (b) A close-up at the lowest eigenstates together with wave functions (shaded areas). The strongest transitions according to the Franck-Condon principle are shown by arrows and are labeled by the change of vibrational quanta $k$.

Figure 2

Scanning electron microscopy micrographs and circuit diagram of the artificial molecule. (a) General view showing top electrodes of the oscillator capacitors (2 large pads) with capacitances $2C$ each. (b) Enlargement of the white square area in (a) showing the $300\mu \mathrm{m}$-long microstrip loop that forms the inductance $L$ of the oscillator. (c) Magnification of the white diamond in (b) showing the SCPT formed by two Josephson junctions with coupling energies $E_{J1}$ and $E_{J2}$ and the gate electrode. Also visible is a reference SCPT (lower left) which enables one to estimate the resistances of the junctions in the sample. (d) The circuit diagram shows the SCPT and the $LC$ oscillator connected in parallel. The resistance $R=2.8\mathrm{k}\Omega$ represents the ac dissipation of the capacitors as well as of other parts of the circuit. The artificial molecule is biased by a gate voltage $V_g$ and an external magnetic flux ${\Phi }_b$. The low-frequency drive (LF) is applied through a $Z_0=50\Omega$ transmission line via a lumped-element coupling capacitor $C_c=7\mathrm{pF}$, which is wire bonded to the top plate of one of the $2C$ capacitors. The high-frequency drive (HF) is applied through a separate $Z_0=50\Omega$ transmission line and is wire bonded to the back gate of the sample. The back gate is in turn wire bonded to the ground. The inductances of the bond wires give voltage division and thus a finite $V_{\mathrm{HF}}$ over the SCPT.

Figure 3

Measured spectra of the artificial molecule. (a) The absolute value of the reflection coefficient $\Gamma =V_{\mathrm{out}}/V_{\mathrm{in}}$ as a function of the bias point $({\Phi }_b,n_g)$. The color bar gives the scale for $|\Gamma |$. The dashed arch indicates the pure electronic transition at $\Delta E/h={\nu }_{\mathrm{HF}}=22\mathrm{GHz}$. The cross denotes the bias point of Fig. 1 where $k=-2$. The applied low-frequency power $P_{\mathrm{LF}}=-129\mathrm{dBm}$, and the high-frequency flux amplitude over the SCPT is approximately $8\times {10}^{-4}{\Phi }_0$. (b) The same as (a) but at higher $P_{\mathrm{LF}}=-123\mathrm{dBm}$. (c) A cut along $n_g=1.27$ as indicated by the horizontal lines in (a) and (b). (d) A cut along ${\Phi }_b=0.56{\Phi }_0$.

Figure 4

Simulated properties of the artificial molecule. (a) The reflection amplitude $|\Gamma |$ calculated as a function of the bias point in the flux-charge plane. The parameters correspond to the measurement plotted in Fig. 3a. (b) The energy flow from the qubit to the resonator showing damping for lower sideband transitions (blue) and amplification for the upper sideband transitions (red). The power unit is $P_{\mathrm{LF}}=-129\mathrm{dBm}$. The simulation uses cone approximation mentioned in the text as the difference to full Mathieu bands is small. The parameter values are given in the text including the asymmetry $d$, $T_1=40\mathrm{ns}$, and $T_2=2\mathrm{ns}$.