Quantum-mechanical systems having two discrete energy levels are ubiquitous in nature. For crossing energy levels, depending on how fast they approach each other, there is a possibility of a transition between them. This phenomenon is known as Landau-Zener tunneling and it forms the physical basis of the Zener diode, for example.

The traditional treatment of the Landau-Zener tunneling, however, ignores quantum-mechanical interference. Here we report an observation of phase-sensitive interference between consecutive Landau-Zener tunneling attempts in an artificial two-level system formed by a Cooper-pair-box qubit.
We interpret the experiment in terms of a multi-pass analog to the well-known optical Mach-Zehnder interferometer. In our case, the beam splitting occurs by Landau-Zener tunneling at the charge degeneracy, while the arms of the Mach-Zehnder interferometer in energy space are represented by the ground and excited state. Our Landau-Zener interferometer can be used as a high-resolution detector for phase and charge owing to interferometric sensitivity enhancement.
The findings also demonstrate new methods for qubit manipulations.

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 φ_{L,R} are accumulated between O and the turning points. The uppermost dashed line represents the odd parity state E^{odd}_{0}. (b) Interpretation of one cycle of LZ interference as four spin rotations on the Bloch sphere, with one possible set of φ_{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 I2ne> are aligned along the z axis.

Schematics of our experiment. (a) The resonant frequency f_{0} ~ 800 MHz of the lumped-element LC circuit is tuned by the Josephson capacitance C_{eff} of the CPB shown in the scanning electron micrograph. The maximum CPB Josephson energy 2E_{J<>/sub> = 12:5 GHz could be tuned down to 2.7 GHz by magnetic flux Φ. The total junction capacitance amounts to CJ = C1 + C2 ~ 0:44 fF, yielding a Coulomb energy of e2/2(CJ + Cg) = 1.1 K. (b) Ceff calculated for the two lowest levels of our CPB with EJ=EC = 0.27 and asymmetry d = 0.22, at Φ = 0.}

Interference patterns, measured via the microwave phase shift. (a) f_{rf} = 4 GHz and phase Φ = 0 (i.e., level repulsion 2Δ = 2E_{J} = 12:5 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: φ_{L} = 2Φ_{S} (solid lines) and φ_{R} = 2Φ_{S} (dashed line) are multiples of 2π [see Eq. (7)], with the v-dependent Stokes phase Φ_{S}. The highest (red) population of the upper state is expected when both conditions are satisfied. The equicapacitance contour C_{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_{rf} = 7 GHz. (c) The average gate spacing between the central interference peaks [see (a)], for the phase bias 0 (square) and π (circle). The expected linear behavior yields a fit E_{C} = 1.1, about 25% higher than we obtained by rf spectroscopy (M. A. Sillanpää et al., Phys. Rev. Lett. 95, 206806 (2005).).

Calculated C_{eff} , using Bloch equations and linear-response theory, with α = 0.04, δn_{ac} = 0.06e 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_{eff} = 0 fF.

Related publications

Continuous-time monitoring of Landau-Zener interference in a cooper-pair box

M.A. Sillanpää, T. Lehtinen, A. Paila, Yu. Makhlin, and P.J. Hakonen