User:PhysicsyPaul/Cavity Quantum Electrodynamics

Cavity Quantum Electrodynamics (Cavity QED) is the study of the interaction of light trapped inside a reflective cavity with atoms or particles. The cavity is constructed such that the photons of light will exhibit quantum effects rather than only classical.

By introducing new boundary conditions on the field, the cavity acts to restrict the allowed modes of light inside the cavity.

Although there are now many different typed of experiments in Cavity QED. They all rely on the cavity restricting the allowed modes of the light field. They also tend to fall into one of two regimes. The weak coupling regime, where the primary cavity modes are off resonance with the atom’s transmission frequency. The second is the strong coupling regime, where the primary cavity modes are near (or on) resonance with the atom transmission frequency.

One of the most important features of Cavity QED is the ability to make a quantum non-demolition measurement (QND measurement) of photons. Previously to measure a photon it was necessary to use something like a photodetector. Which in the process of measuring the photon will always irreversibly destroy that photon. Whereas a QND measurement does not destroy the photon meaning that further interaction and measurements can be done with the same photon(s).

In Cavity QED the QND measurements are possible because instead of measuring the photon itself, the photon is entangled to an atom/particle. This way instead of ‘asking’ the photon about itself we can instead get information about the photon by measuring the atom.

Modification of Spontaneous Emission
Because the cavity acts to restrict the allowed modes of light inside it, this means that the spontaneous emission of a photon from an exited atom can be restricted if the cavity is tuned far enough from the atoms transmission frequency. In a perfect cavity an atom caould be held in an excited state indefinitely. However, in real world situations it extremely difficult to create a perfect cavity for a number of reasons.

One of the primary difficulties in creating a perfect cavity is that this would require that the cavity completely surround the atom. The reason for this need is that modes are only restricted for the solid angel in which the atom ‘sees’ the cavity. For those other angles the transmission modes will not be restricted. An atom that was placed inside such a cavity would be difficult to interact with.

Another difficulty in creating a perfect cavity is that no mirrors can fully restrict specific modes of the field. In practice there is not a perfect node at the mirror walls the waves can penetrate the cavity walls to an extent, however the further off resonance the less likely this is. This effect is highly dependent on the quality of the mirror. Purcell showed that for a cavity that is tuned to resonance the effect on spontaneous emission by the quality of the Cavity is given by

$$\frac{\gamma_c}{\gamma_f}=\frac{2\pi Q}{\omega_0^2 V_c}=\frac{Q\lambda_0^2}{4\pi^2 V_c}$$ where $$\gamma_c, \gamma_f$$ are the density of modes in the cavity and field. $$V_c$$ is the volume of the cavity. $$\lambda_0$$ is the transition wavelenth of the atom. This means, while a detuned cavity can decrease the spontaneous emission rate a cavity tuned to resonance with a volume $$\lambda_0^3$$ will increase the emission rate by Q.

QND Measurement and Quantum Feedback
Because Cavity QED allows for a QND measurement of a photon, it allows for further measurements or interaction with the same photons. This is particularly true for high Q cavities where the photon lifetime inside the cavity is long enough that it can see multiple atoms.

For such a cavity that is tuned so that it is near resonance with the atomic transmission frequency, and is prepared so that the field present is in a Fock state. An atom that is sent through the cavity will undergo Rabi oscillations, in accordance with the Jaynes-Cummings model. such that the atom and cavity will oscillate between the states $$|e,n-1\rangle\leftrightarrow |g,n\rangle$$. This will even be true if the cavity is prepared such that there are no photons inside the cavity. In this case an excited atom sent through this cavity will undergo vacuum Rabi Oscillations.

A particularly interesting consequence of this is that, in such a system it is possible to implement quantum feedback controls on the cavity light field. Classically a coherent field is one that does not have a definite number state. Instead a coherent field is given by: $$|\alpha > = \sum_{n=0}^{\infty} c_n |n\rangle$$ this means that a classical field is a superposition of the stationary Fock number states. However due to this Rabi oscillation effect in Cavity QED it is possible to prepare non classical number states using quantum feedback. The simplest theory for how this can be understood by considering the two level system with the cavity initially prepared in the vacuum. If a beam of excited atoms is sent through the cavity such that no more that one atom is present inside of the cavity at a time, and the cavity quality is high enough that the photon(s) will last long enough that the see many atoms before they escape. Then a specific number state can be prepared by counting the number of atoms exiting the cavity in the ground state. Based on that number either another exited atom can be sent through to increase the number state inside the cavity, or a ground state atom can be sent through to absorb one of the photons decreasing the number state in the cavity.

The major difficulty of this kind of quantum control is that those number states tend to be very fragile. Any thermal photons inside the cavity will decohere the system. For creating higher number states a number of difficulties can arise using this method. Higher number states require higher and higher quality cavities to prepare in this way, because the photons must last for the time that it takes many atoms to cross the cavity. A second difficulty is that of interaction time. This model follows the Jaynes-Cummings model which has different energy splitting for different n's. This will cause the Rabi Frequency to be different for each number state. Because of as the n's get larger the required interaction time to guarantee a photon has been emitted by the atom will also grow. Because of this once the n state gets high enough there is only a small probability that each excited atom will add a photon to the cavity.

This effect as well as photon's escaping the cavity means that typically the number of photons inside the cavity is not exactly equal to the number of atoms that where counted in the ground state. Because of this it is difficult to create pure number states higher than n=1. Typically this type of method will create a distribution of states around a specific $$\langle n \rangle$$.

Microwave Cavity QED
How strongly the atom couples to the cavity is dependant on the stength of it's dipole moment inside of the cavity. This means that in practice is is difficult to couple an atom to optical light. Because of this many of the early experiments in Cavity QED where done in microwave frequencies with Rydberg Atoms. Due to their size Rydberg atoms have large dipole moments inside of the cavity. Which allows for strong coupling with the field. Because of this much of the early work with Cavity QED and quantum feedback was done in the microwave range of frequencies with Rydberg Atoms.

Serge Haroche, who won the 2012 Nobel Prize for Physics along with David Wineland, was one of the major early contributors to this field. For his experiments Haroche used mirrors made out of copper that where coated with a layer of superconductic niobium. The cavity used by Haroche was of such high quality that Photons trapped inside of it would travel between the mirrors for about 40,000 kilometers. This meant that the photons would last long enough to have thousands of atoms cross the cavity before it escapes. Haroche used this cavity to create Schrodinger cat states with the atom and field. As well to show that it is possible to create semi-stable photon number states

An interesting application of the type of cavity created by Haroche is that of the micromaser. Where a beam of exited atoms is pumped into the cavity which is tuned to resonance. When the number state(s) is low enough there is a high probability that the atom will emit a photon. Ideally this is done with an approximate $$\pi$$-pulse. The beam of atoms entering the cavity will act to pump the field inside of it. Steady state of this type of system is achieved due to cavity losses.

Other Types of Cavity QED
Although in early Cavity QED experiments it was only possible to reach the strong coupling regime in the Microwave range, now it is possible to show these effect in other situations.

In the early 1990s QED experiments where able to reach the strong coupling regime in the optical frequency range. These type of experiments are typically done with extremely cold atoms inside of Fabry-Perot type cavities.

Circuit QED is another field which has developed and can show strong coupling Cavity QED effects. In this case instead of using atoms small superconducting circuits with Josephson junctions are used. These circuits act as a two level system that obeys Jaynes-Cummings model. These types of experiments are particularly interesting for there much more rapid dynamics.

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