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arXiv:cond-mat/0410161 · Physical Review B

Fast partial decoherence of a superconducting flux qubit in a spin bath

Jacek Dziarmaga

Published Feb 25, 2005• 1 citation
arXivHTMLPDF

Abstract

The superconducting flux qubit has two quantum states with opposite currents which generate different magnetic fields. Environment of nuclear spins can find out the magnetic field after a decoherence time ${\ensuremath{\tau}}_{0}$ inversely proportional to the magnetic field and the square root of the number of spins. When the Hamiltonian of the qubit drives fast coherent Rabi oscillations between the states with opposite current, then the magnetic field is flipped at a constant rate $\ensuremath{\omega}$ and the decoherence time $\ensuremath{\tau}=\ensuremath{\omega}{\ensuremath{\tau}}_{0}^{2}$ is much longer than ${\ensuremath{\tau}}_{0}$. However, on closer inspection decoherence actually takes place on two time scales. The long time $\ensuremath{\tau}$ is a time of full decoherence but a part of quantum coherence is lost already after the much shorter time $\ensuremath{\pi}∕\ensuremath{\omega}$. This fast partial decoherence biases coherent flux oscillations towards the initial flux direction and it can affect performance of the superconducting devices as qubits.

DOI
10.1103/physrevb.71.054516

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The superconducting flux qubit has two quantum states with opposite currents which generate different magnetic fields. Environment of nuclear spins can find out the magnetic field after a decoherence time \ensuremathτ0{\ensuremath{\tau}}_{0}\ensuremathτ0​ inversely proportional to the magnetic field and the square root of the number of spins. When the Hamiltonian of the qubit drives fast coherent Rabi oscillations between the states with opposite current, then the magnetic field is flipped at a constant rate \ensuremathω\ensuremath{\omega}\ensuremathω and the decoherence time \ensuremathτ=\ensuremathω\ensuremathτ02\ensuremath{\tau}=\ensuremath{\omega}{\ensuremath{\tau}}_{0}^{2}\ensuremathτ=\ensuremathω\ensuremathτ02​ is much longer than \ensuremathτ0{\ensuremath{\tau}}_{0}\ensuremathτ0​. However, on closer inspection decoherence actually takes place on two time scales. The long time \ensuremathτ\ensuremath{\tau}\ensuremathτ is a time of full decoherence but a part of quantum coherence is lost already after the much shorter time \ensuremathπ∕\ensuremathω\ensuremath{\pi}∕\ensuremath{\omega}\ensuremathπ∕\ensuremathω. This fast partial decoherence biases coherent flux oscillations towards the initial flux direction and it can affect performance of the superconducting devices as qubits.
A persistent current qubit has two quantum states with opposite currents flowing in a superconducting loop. Their magnetic field couple to nuclear spins. The qubit state is not only perturbed by the spins but it also gets entangled with the spins' state on a very short timescale. However, when the same spins are exposed to a strong but less than critical external magnetic field, then the qubit field is just a small perturbation on top of the external field and the entanglement with each spin is negligible. For a qubit which is more microscopic than certain threshold this partial entanglement results in negligible partial decoherence.
An exactly solvable model for the decoherence of one and two-qubit states interacting with a spin bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi oscillations between the spin states of a single qubit is shown to depend on the size of bath, the distribution of qubit-bath couplings, and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the qubit shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-qubit states can be controlled by tuning the frequency of the rotating field. The decay rates of entanglement and purity for all the Bell states are the same when the qubits are noninteracting, and different when they are interacting.
The interaction of solid-state qubits with environmental degrees of freedom strongly affects the qubit dynamics, and leads to decoherence. In quantum information processing with solid-state qubits, decoherence significantly limits the performances of such devices. Therefore, it is necessary to fully understand the mechanisms that lead to decoherence. In this review, we discuss how decoherence affects two of the most successful realizations of solid-state qubits, namely, spin qubits and superconducting qubits. In the former, the qubit is encoded in the spin 1/2 of the electron, and it is implemented by confining the electron spin in a semiconductor quantum dot. Superconducting devices show quantum behaviour at low temperatures, and the qubit is encoded in the two lowest energy levels of a superconducting circuit. The electron spin in a quantum dot has two main decoherence channels, a (Markovian) phonon-assisted relaxation channel, due to the presence of a spin–orbit interaction, and a (non-Markovian) spin bath constituted by the spins of the nuclei in the quantum dot that interact with the electron spin via the hyperfine interaction. In a superconducting qubit, decoherence takes place as a result of fluctuations in the control parameters, such as bias currents, applied flux and bias voltages, and via losses in the dissipative circuit elements.
The research work discussed in this thesis deals with the study of superconducting Josephson qubits. Superconducting qubits are solid-state artificial atoms which are based on lithographically defined Josephson tunnel junctions properties. When sufficiently cooled, these superconducting devices exhibit quantized states of charge, flux or junction phase depending on their design parameters. This allows to observe coherent evolutions of their states. The results presented can be divided into two parts. In a first part we investigate operations of superconducting qubits based on the quantum coherence in superconducting quantum interference devices (SQUID). We explain experimental data which has been observed in a SQUID subjected to fast, large-amplitude modifications of its effective potential shape. The motivations for this work come from the fact that in the past few years there have been attempts to interpret the supposed quantum behavior of physical systems, such as Josephson devices, within a classical framework. Moreover, we analyze the possibility of generating GHZ states, namely maximally entangled states, in a quantum system made out of three Josephson qubits. In particular, we investigate the possible limitations of the GHZ state generation due to coupling to bosonic baths. In the second part of the thesis we address a particular cause of decoherence of flux qubits which has been disregarded until now: thermal gradients, which can arise due to accidental non equilibrium quasiparticle distributions. The reason for these detrimental effects is that heat currents flowing through Josephson tunnel junctions in response to a temperature gradient are periodic functions of the phase difference between the electrodes. The phase dependence of the heat current comes from Andreev reflection, namely an interplay between the quasiparticles which carry heat and the superconducting condensate which is sensitive to the superconducting phase difference. Generally speaking, the flux qubit states are characterized by different values of the phase difference through their Josephson junctions. Consequently, the phase-dependent thermal current through a device subject a temperature gradient is related to the phase-dependent qubit states. We study how the thermal currents change according to the state of the qubits hence yielding a measurement of the qubit state. This in turn leads to an impact of temperature gradient on the dynamics of the system. We show that flux qubits in the Delft qubit design can have limitations of the decoherence time to the order of microseconds as a result of this newly discovered source of decoherence. In contrast, the fluxonium qubit is found to be well protected due to its superinductance.
Superconducting, flux-based qubits are promising candidates for the construction of a large scale quantum computer. We present an explicit quantum mechanical calculation of the coherent behavior of a flux based quantum bit in a noisy experimental environment such as an integrated circuit containing bias and control electronics. We show that non-thermal noise sources, such as bias current fluctuations and magnetic coupling to nearby active control circuits, will cause decoherence of a flux-based qubit on a timescale comparable to recent experimental coherence time measurements.
We have observed coherent time evolution between two quantum states of a superconducting flux qubit comprising three Josephson junctions in a loop. The superposition of the two states carrying opposite macroscopic persistent currents is manipulated by resonant microwave pulses. Readout by means of switching-event measurement with an attached superconducting quantum interference device revealed quantum-state oscillations with high fidelity. Under strong microwave driving, it was possible to induce hundreds of coherent oscillations. Pulsed operations on this first sample yielded a relaxation time of 900 nanoseconds and a free-induction dephasing time of 20 nanoseconds. These results are promising for future solid-state quantum computing.