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arXiv:quant-ph/0511143 · Molecular Physics

Simulations of ‘decoherence’ with noise pulses

Leo Stodolsky

Published Mar 31, 2006• 0 citations
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Abstract

A simulation of decoherence as random noise in the Hamiltonian is studied. The full Hamiltonian for the rf Squid is used, with the parameters chosen such that there is a double-potential well configuration where the two quasi-degenerate lowest levels are well separated from the rest. The results for these first two levels are in quantitative agreement with expectations from the ‘spin 1/2’ picture for the behaviour of a two-state system.

DOI
10.1080/00268970500525960

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A simulation of decoherence as random noise in the Hamiltonian is studied. The full Hamiltonian for the rf Squid is used, with the parameters chosen such that there is a double-potential well configuration where the two quasi-degenerate lowest levels are well separated from the rest. The results for these first two levels are in quantitative agreement with expectations from the ‘spin 1/2’ picture for the behaviour of a two-state system.
We study the effects of noise and decoherence for a double-potential well system, suitable for the fabrication of qubits and quantum logic elements. A random noise term is added to the hamiltonian, the resulting wavefunction found numerically and the density matrix obtained by averaging over noise signals. Analytic solutions using the two-state model are obtained and found to be generally in agreement. In particular, a simple formula for the decoherence rate in terms of the noise parameters in the two-state model is reviewed and verified for the full simulation with the multi-level system. The formalism is extended to describe multiple sources of noise or different "dephasing" axes at the same time. Furthermore, the old formula for the "Turing-Watched Pot" effect is generalized to the case where the environmental interactions do not conserve the "quality" in question. Various forms for the noise signal are investigated. An interesting result is the importance of the noise power at low frequency. If it vanishes there is, in leading order, no decoherence. This is verified in a numerical simulation where two apparently similar noise signals, but differing in the power at zero frequency, give strikingly different decoherence effects. A short discussion of situations dominated by low frequency noise is given.
Methods and results for numerical simulations of one and two interacting rf-SQUID systems suitable for adiabatic quantum gates are presented. These are based on high accuracy numerical solutions to the static and time dependent Schr\"odinger equation for the full SQUID Hamiltonian in one and two variables. Among the points examined in the static analysis is the range of validity of the effective two-state or ``spin-12\frac{1}{2}21​'' picture. A range of parameters is determined where the picture holds to good accuracy as the energy levels undergo gate manipulations. Some general points are presented concerning the relations between device parameters and ``good'' quantum mechanical state spaces. The time dependent simulations allow the examination of suitable conditions for adiabatic behavior, and permits the introduction of a random noise to simulate the effects of decoherence. A formula is derived and tested relating the random noise to the decoherence rate. Sensitivity to device and operating parameters for the logical gates NOT and CNOT are examined, with particular attention to values of the tunnel parameter $\ensuremath{\beta}$ slightly above one. It appears that with values of $\ensuremath{\beta}$ close to 1, a quantum CNOT gate is possible even with rather short decoherence times. Many of the methods and results will apply to coupled double-potential well systems in general.
A particular superconducting quantum interference device (SQUID) qubit, namely the double SQUID qubit, can be manipulated by rapidly modifying its potential with the application of fast flux pulses.In this system we observe coherent oscillations exhibiting non-exponential decay, indicating an unconventional decoherence mechanism.Moreover, via tuning the qubit in different conditions (different oscillation frequencies) by changing the pulse height, we observe a crossover between two distinct decoherence regimes and the existence of an 'optimal' point where the qubit is only weakly sensitive to intrinsic noise.We find that this behavior is in agreement with a model considering the decoherence caused essentially by low-frequency noise contributions, and we discuss the experimental results and possible issues.
At low temperature and low damping, the magnetic flux of a rf Superconducting QUantum Interference Device (SQUID) acts as a macroscopic quantum variable whose dynamic is described by a double well potential. A rf SQUID consists of a superconducting loop interrupted by a thin insulating barrier, known as a Josephson junction. The modified rf SQUID design discussed in this thesis has independent, in situ, controls that allow varying the barrier height between the potential wells and their relative tilt. This high level of control of the potential together with the capability to implement an experimental setup optimized to reduce coupling to external sources of noise, made this system ideal not only for tests of quantum transition at a macroscopic level, but also as a scalable solid state device for quantum computation. A variety of measurements of quantum phenomena are reported in this thesis, including microwave level spectroscopy, coherent oscillations, and macroscopic resonant tunneling (MRT). These measurements are compared with theoretical models to study the possible sources of decoherence in a rf SQUID qubit. The analysis confirmed that the main source of decoherence is not due to coupling to the external environment, but it can be linked to intrinsic low frequency flux noise. Work from many groups over the last several years has shown that the performance of a superconductive qubit is limited by coupling to spurious defects in the materials used for the device fabrication. In this work we analyze the possibility of using the MRT as a tool for future studies of effects of materials and fabrication process in the realization of improved qubits, and we also explore the feasibility for measurements of quantum phenomena that involve fast pulsing of the potential barrier height.
A double SQUID qubit (Superconducting Quantum Interference Device) can be handled by applying microwave trains, but also by using fast flux pulses. In this second case the manipulation is based on the fast and radical modification of the qubit potential shape that induces non-adiabatic transitions between the computational states (the two lowest energy eigenstates), still avoiding transitions to upper levels. This modality is interesting because it allows faster operations with respect to other techniques, but also because it gives access to interesting nontrivial physical features, concerning in particular decoherence and adiabaticity. About decoherence, we observed experimentally the existence of an “optimal” bias region and the transition between two distinct decoherence regimes. These results can be explained by considering the effect of first and second order slow fluctuations which dominate on high frequency noise contributions. This allows a deep insight in the qubit decoherence mechanisms.