The Hamiltonian of a charge qubit, which consists of two Josephson junctions is found within well known quantum mechanical procedure. The inductance of the qubit is included from the very beginning. It allows a selfconsistent derivation of the current operator in a two state basis. It is shown that the current operator has nonzero nondiagonal matrix elements both in the charge and the eigenstate basis. It is also shown that the interaction of the qubit with its own LC resonator has a noticeable influence on the qubit energies. The influence of the junctions asymmetry and the gate capacitance on the matrix elements of the current operator and on the qubit energies are calculated. The results obtained in the paper are important for the circuits where two or more charge qubits are coupled with the aid of inductive coil.
Qubits based on the magnetic flux degree of freedom require a flux bias, whose stability and precision strongly affect the qubit performance, up to a point of forbidding the qubit operation. Moreover, in the perspective of multiqubit systems, it must be possible to flux-bias each qubit independently, hence avoiding the traditional use of externally generated magnetic fields in favour of on-chip techniques that minimize cross-couplings. The solution discussed in this paper exploits a persistent current, trapped in a superconducting circuit integrated on chip that can be inductively coupled with an individual qubit. The circuit does not make use of resistive elements that can be detrimental for the qubit coherence. The trapping procedure allows to control and change stepwise the amount of stored current; after that, the circuit can be completely disconnected from the external sources. We show in a practical case how this works and how to drive the bias circuit at the required value.
An effective Hamiltonian is derived for two coupled three-Josephson-junction (3JJ) qubits. This is not quite trivial, for the customary ``free'' 3JJ Hamiltonian is written in the limit of zero inductance
L. Neglecting the self-flux is already dubious for one qubit when it comes to readout, and becomes untenable when discussing inductive coupling. First, inductance effects are analyzed for a single qubit. For small
L, the self-flux is a ``fast variable,'' which can be eliminated adiabatically. However, the commonly used junction phases are not appropriate ``slow variables,'' and instead one introduces degrees of freedom that are decoupled from the loop current to leading order. In the quantum case, the zero-point fluctuations (
LC oscillations) in the loop current diverge as $L\ensuremath{\rightarrow}0$. While their effect thus formally dominates over the classical self-flux, it merely renormalizes the Josephson couplings of the effective (two-phase) theory. In the coupled case, the strong zero-point fluctuations render the full (six-phase) wave function significantly entangled in leading order. However, in going to the four-phase theory, this uncontrollable entanglement is integrated out completely, leaving a computationally usable mutual-inductance term of the expected form as the effective interaction.
This paper considers the behavior of a model persistent current qubit in the presence of a time-dependent electromagnetic field. A semi-classical approximation for the electromagnetic field is used to solve the time- dependent Schrodinger equation (TDSE) for the qubit, which is treated as a macroscopic quantum object. The qubit is describe3d by a Hamiltonian involving the enclosed magnetic flux (Phi) and the electric displacement flux Q, which obey the quantum mechanical commutation relation. The paper includes a brief summary of recent work on quantum mechanical coherence in persistent current circuits, and the solution of the TDSE in superconducting rings. Of particular interest is the emergence of strongly non-perturbative behavior that corresponds to transitions between the energy levels of the qubit. These transitions are due to the strong coupling between the electromagnetic fields and the superconducting condensate and can appear at frequencies predicted by conventional methods based on perturbations around the energy eigenstate of the time-independent system. The relevance of these non-perturbative processes to the operation of quantum logic gates based on superconducting circuits and the effect of the resultant non linearities on the environmental degrees of freedom coupled to the qubit are considered.
An optimal effective current operator for flux qubit has been investigated by taking account of the inductive effects of the circuit loop. The whole system is treated as two interacting subsystems: one is the inductance-free flux qubit consisting of three Josephson junctions and the other a high frequency
LC oscillator. As the composite system hardly affords one excessively high-energy
LC photon, an effective theory for the inductive flux qubit providing its physical variable operators has been achieved, which can take account of the inductive effects but does not include the additional degree of freedom for the
LC oscillator. Considering the tradeoff between simplicity and accuracy, it has been revealed that the optimal effective current operator resulting in an error only on the order of
L3/2 provides an approximation of high accuracy, which is also verified numerically.