Rabi Oscillations and the Rabi Frequency in Two-Level Systems

]Key Takeaways

• Rabi oscillations describe the exact state of matter interacting with a driving field. The Rabi frequency is the frequency of Rabi oscillations existing in two-level systems when interacting with near-resonant driving fields. 

• The Rabi frequency is 2 times the number of cycles in Rabi oscillations per sec. It is the angular frequency of Rabi oscillations measured in radians. 

• The generalized Rabi frequency rule states “the oscillation frequency grows while the amplitude drops with increasing detuning of the driving field from the natural resonance of the two-level system.”

Figure 1. MRI scans utilize magnetic resonance.

Modern medical science has advanced to new heights with the growth of technology, especially in the field of imaging technology. Among non-invasive imaging technology methods, Magnetic Resonance Imaging (MRI) is a sophisticated disease detection method that provides 3-D images of human anatomy. In MRI, the rotation of protons in the human body are aligned to a strong external magnetic field and made to spin out of equilibrium by passing a radio frequency wave through the body. In order to better understand MRI, it is important to understand the fundamental concepts of Nuclear Magnetic Resonance (NMR) and the Rabi frequency. 

Rabi Oscillations and the Rabi Frequency

The Rabi frequency is a semi-classical concept based on quantum mechanics and electrodynamics. The interactions between light and matter cause Rabi oscillations, which is named after the Nobel prize-winning American physicist Isidor Isaac Rabi. Rabi oscillations describe the exact state of a matter interacting with a driving field. The Rabi frequency is the frequency of Rabi oscillations existing in two-level systems (systems having two transition states of lower and higher energy) when interacting with near-resonant driving fields. 

The atom is the smallest unit of matter, and it is a classic example of a two-level system with two levels or states: ground state and excited state. When light interacts with atoms, energy is exchanged between them. This energy exchange is periodic in nature and forms oscillations called Rabi oscillations. The Rabi frequency is 2 times the number of cycles in Rabi oscillations per sec. It is the angular frequency of Rabi oscillations measured in radian. The Rabi frequency is proportional to the oscillatory driving field applied and to the strength of the light-atom coupling. 

Atoms exhibit a cyclic behavior when in resonance with the oscillatory driving field, and this phenomenon is called the Rabi cycle. When the energy associated with the two states of the atom are different, then the atoms will undergo the Rabi cycle. The atoms subjected to a coherent beam of incident light absorbs photons cyclically. The energized photon interacts with electrons in the excited atoms and transitions it from the excited state to the ground state. The energy from the transitioned atom is emitted back to the driving field in the form of a new photon. The new photon moves from the atom to the incident light, causing the light-atom interaction. The new photon possesses the same parameters as the incident light—frequency, phase, polarization, and orientation. This phenomenon is called the Rabi cycle, and the reciprocal of the Rabi cycle duration can be called the Rabi frequency. 

The Generalized Rabi Frequency Rule

The Rabi frequency is often applied in optics, radio, and microwave frequency systems. The response of the two-level system is driven by near-resonant fields and can be described using the generalized Rabi frequency rule, which states that “the oscillation frequency grows while the amplitude drops with increasing detuning of the driving field from the natural resonance of the two-level system.” However, the rule fails in inhomogeneous systems where oscillation frequency rigidity results from driving field frequency change—a contrast to the generalized rule. The frequency rigidity in inhomogeneous systems leads to spectral broadening.

Rabi Oscillations in Graphene

Graphene

Graphene is the honeycomb lattice structure of carbon atoms. It is the basic element in graphite, carbon nanotubes, and fullerene, and it is used for various applications in electronics. Graphene is used in high-frequency electronics manufacturing, printable circuit components, and thermal management products like thermal pads and thermal paste. 

Graphene has several peculiar properties, including pseudospin degrees of freedom. Graphene is an example of a two-level system and is vulnerable to the Rabi frequency. Apart from conventional Rabi oscillations, there are anomalous Rabi oscillations far from resonance in graphene. Anomalous Rabi oscillations are responsible for the pseudospin degree of freedom in graphene. A similar phenomenon occurs in the single-photon limit, as well as in vacuum. 

When graphene is subjected to a quantized electromagnetic field, it gives anomalous Rabi oscillations. The anomalous Rabi oscillations consist of a sum of oscillating terms. Each of these terms corresponds to a photon number ‘n’ and oscillations at a particular Rabi frequency. The Rabi frequency is dependent and proportional to the square root of (n+1). 

In graphene, the distribution of Rabi frequency far from resonance produces collapse and revival Rabi oscillations or anomalous Rabi oscillations. The collapse and revival oscillations are due to the destructive and constructive interference between the oscillating terms mentioned above. The initial collapse in the Rabi oscillations is called initial dephasing. It is due to the destructive interference where the neighboring oscillating terms are 180° out of phase. If the neighboring oscillating terms are in-phase, revival Rabi oscillations are created due to constructive interference. The revival leads to the resurrection of the decaying oscillations from the previous destructive interference. 

The collapse and revival oscillations in graphene depend on the coherence state of the photons and the distribution of photon numbers. If the photon distribution is continuous, it would produce only collapse oscillations as in the classical field. The collapse oscillations are the trademark of two-level systems coupled to the single-mode photon. The collapse and revival simultaneously occur when the two-level graphene-like systems are subjected to a quantized electromagnetic field. 

Rabi oscillations have great technological significance, as two-level systems are present in fields such as optics, semiconductors, microwave, and radio-frequency systems. Any two-level system subjected to the light would interact and get coupled with driving fields and produce Rabi oscillations at the Rabi frequency. Whenever you work with ferromagnets, battery film materials, or any material constituting of two-level building blocks in the presence of an electromagnetic field, remind yourself about the non-equilibrium atomic spins due to Rabi oscillations.