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 ng is modulated, the CPB evolves from the initial state A through the avoided crossing O (ng = 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 φL,R are accumulated between O and the turning points. The uppermost dashed line represents the odd parity state Eodd0. (b) Interpretation of one cycle of LZ interference as four spin rotations on the Bloch sphere, with one possible set of φL,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 f0 ~ 800 MHz of the lumped-element LC circuit is tuned by the Josephson capacitance Ceff of the CPB shown in the scanning electron micrograph. The maximum CPB Josephson energy 2EJ<>/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) frf = 4 GHz and phase Φ = 0 (i.e., level repulsion 2Δ = 2EJ = 12:5 GHz). The color codes indicate the equivalent capacitance obtained using standard circuit formulas. Around ng0 = -1, the conditions of constructive Landau-Zener interference are illustrated: φL = 2ΦS (solid lines) and φR = 2ΦS (dashed line) are multiples of 2π [see Eq. (7)], with the v-dependent Stokes phase ΦS. The highest (red) population of the upper state is expected when both conditions are satisfied. The equicapacitance contour Ceff = 0 around ng = 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 frf = 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 EC = 1.1, about 25% higher than we obtained by rf spectroscopy (M. A. Sillanpää et al., Phys. Rev. Lett. 95, 206806 (2005).).
Calculated Ceff , using Bloch equations and linear-response theory, with α = 0.04, δnac = 0.06e pp. The inclined white lines indicate the threshold of the LZ tunneling, where the driving signal ng(t) touches, but does not cross, a degeneracy point. The comparison with data in Fig. 3 is performed by the equicapacitance contours for Ceff = 0 fF.
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