Consistency of Ground State and Spectroscopic Measurements on Flux Qubits

We compare the results of ground state and spectroscopic measurements carried out on superconducting flux qubits which are effective two-level quantum systems. For a single qubit and for two coupled qubits we show excellent agreement between the parameters of the pseudospin Hamiltonian found using both methods. We argue that by making use of the ground state measurements the Hamiltonian of $N$ coupled flux qubits can be reconstructed as well at temperatures smaller than the energy level separation. Such a reconstruction of a many-qubit Hamiltonian can be useful for future quantum information processing devices.

Figure 1

Scanning electron micrograph showing two coupled aluminum flux qubits, placed inside a niobium tank coil and dc and microwave lines. The qubits are fabricated on an oxidized silicon substrate by making use of electron-beam lithography and shadow evaporation. The loops share a large Josephson junction visible in the center. This junction provides a coupling between the qubits with a coupling energy $J/h=1.9\mathrm{GHz}$. The qubit loops are interrupted by three junctions each. Two of them, located in the inner sides, have nominal areas of $200\times 700\mathrm{nm}$ while the outer junctions are 70% smaller. The qubit system is placed inside a 30 turn superconducting niobium pancake coil which forms a resonant tank circuit, with a resonance frequency of 20.8 MHz, with a capacitor mounted on the sample holder. The resonator typically has a quality $Q=300$. Individual control of the dc-flux bias of the individual qubits (${\Phi }_a$, ${\Phi }_b$), is provided by the niobium bias lines visible on top of the coil windings. The antenna used for supplying the MW excitation is visible at the top of the picture.

Figure 2

Comparison of the ground state and spectroscopic measurements for qubit $b$. (a) Ground state measurements. Presented is the dependence of the phase shift between the tank circuit voltage and bias current on the flux bias. The solid lines are experimental data fitted by the theoretical curves (dashed curves) for qubit parameters $I_p=225\mathrm{nA}$ and $\Delta /h=1.75\mathrm{GHz}$. The curves correspond to various values of the rf-bias current on the tank circuit resulting in rf voltage amplitudes, from top to bottom, $V_T=4.3$, 2.9, 0.5, and $0.3\mu \mathrm{V}$. (b) Amplitude of the tank voltage as a function of the normalized magnetic flux in the qubit at the driving frequencies, from top to bottom, $\omega /2\pi =18$, 5, and 3.5 GHz. The curves have been shifted for clarity. The resonant excitation in the flux qubit results in the peak-and-dip at the positions defined by the condition $\Delta E({\Phi }_{\mathrm{dc}})=\hslash \omega$. (c) Energy gap $\Delta E$ between the qubit energy levels determined from the positions of the mid points of the peak-and-dip structures (solid squares). The solid line is the theoretical curve calculated from Eq. (2) using the parameters $I_p=225\mathrm{nA}$ and $\Delta /h=1.75\mathrm{GHz}$ obtained from the ground state measurements. The effective temperature $T\approx 70\mathrm{mK}\approx 1.4\mathrm{GHz}h/k_B$ is smaller than the minimal energy level separation $2\Delta$.

Figure 3

Landau-Zener interferometry for qubit $b$. Dependence of the tank voltage phase shift $\Theta$ on the dc-flux bias ${\Phi }_{\mathrm{dc}}$ and the ac flux amplitude ${\Phi }_{\mathrm{ac}}$ (the microwave amplitude). The spots along the ${\Phi }_{\mathrm{dc}}$ axis correspond to the multiphoton resonances at the positions defined by the relation $\Delta E({\Phi }_{\mathrm{dc}})\approx n\hslash \omega$; the numbers from 1 to 7 show the position of the $n$-photon resonances. The changes along the ${\Phi }_{\mathrm{ac}}$ axis are due to Stückelberg oscillations in the qubit. The calibration of the driving power of the ac flux can be done either with the distance between these oscillations (black arrow) or from the slope of the interference fringes (white line).

Figure 4

Spectroscopy of the system of two coupled flux qubits. The dependence of the tank voltage phase shift $\Theta$ on the flux biases in qubits $a$ and $b$ is presented for measurements without microwave excitation in (a) and for microwave excitation with $\omega /2\pi =14.125$, 17.625 and 20.75 GHz in (b) till (d), respectively. Inset shows the transition to the third excited level. The blue, magenta and white-dotted lines in the pictures with microwave excitation show the expected positions of the resonant excitations of the qubits to the first, second and third excited levels, respectively, calculated from the energy eigenvalues of Hamiltonian (10) with parameters: ${\Delta }_a/h=7.9\mathrm{GHz}$, ${\Delta }_b/h=1.75\mathrm{GHz}$, $I_p^{(a)}=120\mathrm{nA}$, $I_p^{(b)}=225\mathrm{nA}$, and $J/h=1.9\mathrm{GHz}$. The trough around ${\Phi }_b=0$ is due to the ground state curvature of qubit $a$ and corresponds to the ground state measurements of Fig. 2. The shallow trough around ${\Phi }_a=0$ visible in figure (a), is due to qubit $a$.