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Advances in Applied Physics

Simulation of laser manipulation of Bloch vector in adiabatic regime

Ryuzi Yano, Hiroyuki Shinojima

Published Jan 1, 2014• 0 citations

Abstract

Copyright © 2014 Ryuzi Yano and Hiroyuki Shinojima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Simulations of the Bloch vector manipulation of two operations (one that utilizes the geometric phase (geometric operation) and the other that does not utilize the geometric phase (simple operation) in the adiabatic regime enabled us to study the tolerance of the geometric phase utilized operation against the laser noise. We simulated the Bloch vector manipulation by the laser field with the amplitude, the frequency, and the constant phase are proportionally, integrally, and derivatively controlled. The manipulation of the Bloch vector is a /2 rotation around 2-axis. There are few advantage by using the operation of the geometric phase over the simple operation. It has an advantage over only when the geometric phase operation is utilized such that the magnitude of the frequency of the noise is large, and that the noise correlation time is small.

DOI: 10.12988/aap.2014.41013

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Copyright © 2014 Ryuzi Yano and Hiroyuki Shinojima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Simulations of the Bloch vector manipulation of two operations (one that utilizes the geometric phase (geometric operation) and the other that does not utilize the geometric phase (simple operation) in the adiabatic regime enabled us to study the tolerance of the geometric phase utilized operation against the laser noise. We simulated the Bloch vector manipulation by the laser field with the amplitude, the frequency, and the constant phase are proportionally, integrally, and derivatively controlled. The manipulation of the Bloch vector is a /2 rotation around 2-axis. There are few advantage by using the operation of the geometric phase over the simple operation. It has an advantage over only when the geometric phase operation is utilized such that the magnitude of the frequency of the noise is large, and that the noise correlation time is small.

Quantum manipulation protocols for quantum sensors and quantum computation often require many single-qubit rotations. However, the impact of phase noise in the field that performs the qubit rotations is often neglected or treated only for special cases. We present a general framework for calculating the impact of phase noise on the state of a qubit, as described by its equivalent Bloch vector. The analysis applies to any Bloch vector orientation and any rotation axis azimuthal angle for both a single pulse and pulse sequences. Experimental examples are presented for several special cases. We apply the analysis to commonly used composite $\ensuremath{\pi}$ pulse sequences for suppression of static amplitude or detuning errors and also to spin-echo sequences. We expect the formalism presented will help guide the development and evaluation of future quantum manipulation protocols.

The quantum states of cold ensemble of two-level atoms were rotated around axis 2 on the Bloch sphere geometrically using a laser-controlled resonant pulse sequence with pulse areas of $\ensuremath{\pi}∕2$, $\ensuremath{\pi}$, and $\ensuremath{\pi}∕2$, whose central pulse has a relative phase difference of $\ensuremath{\pi}+\ensuremath{\delta}∕2$. It was ascertained that the population probability of the excited state varies according to a sinusoidal function of $\ensuremath{\delta}$ within an uncertainty of 1%. Hybrid operations of two-level atoms were realized combining the geometrical operation with the Rabi excitation around axis 2 or the time evolution around axis 3.

The geometric manipulation of the quantum states of two-level atoms with two laser-controlled resonant $\ensuremath{\pi}$ pulses with a relative phase difference was evaluated using a time-domain atom interferometer comprising a cold sodium atom ensemble and two copropagating two-photon Raman pulses. It was verified that the relative phase shift between two states is double the control phase difference and that the rotation around the axis 3 of the Bloch sphere can be controlled by geometric manipulation.

Dynamical decoupling protocols provide a versatile toolbox for robust control of quantum systems, suppressing noise that would otherwise limit their performance. This tutorial aims to provide an intuitive interpretation of these protocols. Written from an experimentalist's perspective, we use the Bloch sphere picture to visualise quantum dynamics for a range of sequences. Here each dynamical decoupling operation can be viewed as an "effective field vector", whilst the phase and population of the two-level system is encoded into a "Bloch vector". The two-level dynamics are calculated by the cross product between these two vectors, such that the quantum state represented by the Bloch vector rotates around the effective field vector. Several prominent control schemes are simulated using this picture, starting with a Rabi oscillation, followed by the Ramsey sequence and a selection of pulsed and continuous dynamical decoupling techniques. These sequences are engineered to preserve control of the quantum system for as long as possible, with their efficacy often quantified by a coherence time. We clarify the definition of the longitudinal

T_1

, transverse

T_2

and inhomogeneous

T_2^*

coherence times, which are used across the literature. Each simulation has an accompanying animation to illustrate the protocol's two-level dynamics on the Bloch sphere. The custom simulation and Bloch sphere plotting scripts are also provided in an open access repository to allow the reader to reproduce, modify, and explore these results. Dynamical decoupling sequences have established themselves as an invaluable tool across the entire quantum technologies remit; by providing an open framework alongside this tutorial, we aim to make these protocols more accessible to new researchers discovering this thriving field.

We consider the effect of classical noise applied to the geometric quantum phase of a spin

1∕2

in a revolving magnetic field. The Berry phase shows some sensitivity to the noise because the Bloch vector cannot return to its original direction, and the variance caused by noise is proportional to the evolution time.

With carefully prepared resonant laser-controlled pulses, the two quantum states of a cold ensemble of sodium atoms were manipulated by pure geometric rotations of the Bloch vector around axes 2 and 3 on the Bloch sphere. The fidelity for two geometric rotations was improved to be 90%. A universal single-qubit operation on the ground-state wave function was demonstrated by a pure geometric operation and the phase shift due to the rotation around axis 3 was measured using a geometric atom interferometer.