Tru Physics

Rabi Oscillations

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Introduction

Rabi oscillations, named after Isidor Isaac Rabi, are coherent oscillations of an atomic system in a two-level quantum mechanical state due to the application of an external oscillating field. The phenomenon is of fundamental importance in quantum mechanics and is a key principle behind technologies like nuclear magnetic resonance (NMR) and quantum computing.

Basic Concept

Consider a two-level atomic system with energies E_1 and E_2, where E_1 < E_2, and the energy difference is given by \Delta E = E_2 - E_1. The system can exist in either the lower energy state, typically referred to as the “ground state” or the “spin-down” state, or the higher energy state, often called the “excited state” or the “spin-up” state.

If an oscillating external field with frequency \omega close to the resonance frequency \omega_0 = \Delta E / \hbar is applied, the system can absorb energy from the field and transition between the two states. The probability of finding the system in the excited state oscillates with time, giving rise to Rabi oscillations.

Rabi Frequency

The frequency of the Rabi oscillations, called the Rabi frequency \Omega_R, is proportional to the strength of the external field and the matrix element of the dipole moment between the two states. If the external field is represented by a sinusoidal function F(t) = F_0 \cos(\omega t), then the Rabi frequency can be expressed as:

\Omega_R = \dfrac{\langle 2 | \hat{D} | 1 \rangle F_0}{\hbar}

Here, \langle 2 | \hat{D} | 1 \rangle is the matrix element of the dipole moment operator \hat{D} between the two states, and \hbar is the reduced Planck’s constant.

Rabi Oscillations in Quantum Computing

In the context of quantum computing, Rabi oscillations are critical for performing quantum gate operations on qubits. By carefully controlling the duration and strength of the external field (often a microwave pulse), a quantum system can be manipulated between its two states, effectively implementing quantum gates.

For instance, a \pi pulse (a pulse with a duration that causes the system to undergo a half-period of a Rabi oscillation) can flip a qubit from the ground state to the excited state or vice versa, acting as a NOT gate.

Conclusion

Rabi oscillations provide a fundamental mechanism for controlling and manipulating two-level quantum systems. This principle is at the heart of numerous technologies, from magnetic resonance imaging (MRI) to cutting-edge quantum computers. Understanding Rabi oscillations is thus crucial for both fundamental physics and technological applications.

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