Continuous-Time Monitoring of Landau-Zener Interference in a Cooper-Pair Box

Landau-Zener (LZ) tunneling can occur with a certain probability when crossing energy levels of a quantum two-level system are swept across the minimum energy separation. Here we present experimental evidence of quantum interference effects in solid-state LZ tunneling. We used a Cooper-pair box qubit where the LZ tunneling occurs at the charge degeneracy. By employing a weak nondemolition monitoring, we observe interference between consecutive LZ-tunneling events; we find that the average level occupancies depend on the dynamical phase. The system’s unusually strong linear response is explained by interband relaxation. Our interferometer can be used as a high-resolution Mach-Zehnder–type detector for phase and charge.

Figure 1

Landau-Zener interference in a CPB. (a) The energy diagram: As $n_g$ is modulated, the CPB evolves from the initial state A through the avoided crossing O ($n_g=1$) towards B (no LZ tunneling) or C (with LZ tunneling). On the return journey, the final state D is reached by remaining on the excited band (from C) or by LZ tunneling (from B). The dynamical phases ${\phi }_{L,R}$ are accumulated between O and the turning points. The uppermost dashed line represents the odd parity state $E_0^{\mathrm{odd}}$. (b) Interpretation of one cycle of LZ interference as four spin rotations on the Bloch sphere, with one possible set of ${\phi }_{L,R}$ yielding constructive interference (see text). The black arrows indicate the final position of the Bloch vector after each step. Number states of the island charge $|2ne⟩$ are aligned along the $z$ axis.

Figure 2

Schematics of our experiment. (a) The resonant frequency $f_0\sim 800\mathrm{MHz}$ of the lumped-element $LC$ circuit is tuned by the Josephson capacitance $C_{\mathrm{eff}}$ of the CPB shown in the scanning electron micrograph. The maximum CPB Josephson energy $2E_J=12.5\mathrm{GHz}$ could be tuned down to 2.7 GHz by magnetic flux $\Phi$. The total junction capacitance amounts to $C_J=C_1+C_2\sim 0.44\mathrm{fF}$, yielding a Coulomb energy of $e^2/2(C_J+C_g)=1.1\mathrm{K}$. (b) $C_{\mathrm{eff}}$ calculated for the two lowest levels of our CPB with $E_J/E_C=0.27$ and asymmetry $d=0.22$, at $\varphi =0$.

Figure 3

Interference patterns, measured via the microwave phase shift. (a) $f_{\mathrm{rf}}=4\mathrm{GHz}$ and phase $\varphi =0$ (i.e., level repulsion $2\Delta =2E_J=12.5\mathrm{GHz}$). The color codes indicate the equivalent capacitance obtained using standard circuit formulas. Around $n_{g0}=-1$, the conditions of constructive Landau-Zener interference are illustrated: ${\phi }_L-2{\tilde{\varphi }}_S$ (solid lines) and ${\phi }_R-2{\tilde{\varphi }}_S$ (dashed line) are multiples of $2\pi$ [see Eq. (7)], with the $v$-dependent Stokes phase ${\varphi }_S$. The highest (red) population of the upper state is expected when both conditions are satisfied. The equicapacitance contour $C_{\mathrm{eff}}=0$ around $n_g=1$, obtained from the simulation of the Bloch equations (Fig. 4), agrees well with the predicted resonance grid and with the data. (b) The corresponding measurement with $f_{\mathrm{rf}}=7\mathrm{GHz}$. (c) The average gate spacing between the central interference peaks [see (a)], for the phase bias 0 (square) and $\pi$ (circle). The expected linear behavior yields a fit $E_C=1.1$, about 25% higher than we obtained by rf spectroscopy [12].

Figure 4

Calculated $C_{\mathrm{eff}}$, using Bloch equations and linear-response theory, with $\alpha =0.04$, $\delta n_{\mathrm{ac}}=0.06e\mathrm{pp}$. The inclined white lines indicate the threshold of the LZ tunneling, where the driving signal $n_g(t)$ touches, but does not cross, a degeneracy point. The comparison with data in Fig. 3 is performed by the equicapacitance contours for $C_{\mathrm{eff}}=0\mathrm{fF}$.