In atomic, molecular, and optical physics, resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom. When discussing nuclear physics and gamma rays, it is known as nuclear resonance fluorescence.
General theory Typically the electromagnetic field is applied to the two-level atom through the use of a monochromatic laser. A two-level atom is a specific type of two-state system in which the atom can be found in two possible states: where an electron is either in its ground state or in its excited state. In many experiments a lithium atom is used because it can be modeled as a two-level atom, since the excited states of its singular valence electron are far enough apart that the possibility of the electron jumping to a higher excited state can be ignored. Thus it allows for easier frequency tuning of the applied laser as frequencies further from resonance can be used while still driving the electron to jump to only the first excited state. Once the atom is excited, it will release a photon with the same energy as the energy difference between the excited and ground state. The mechanism for this release is the spontaneous decay of the atom. The emitted photon is released in an arbitrary direction. While the transition between two specific energy levels is the dominant mechanism in resonance fluorescence, experimentally other transitions will play a very small role and thus must be taken into account when analyzing results. The other transitions will lead to emission of a photon of a different atomic transition with much lower energy which will lead to "dark" periods of resonance fluorescence.
The dynamics of the electromagnetic field of the monochromatic laser can be derived by first treating the two-level atom as a spin-1/2 system with two energy eigenstates which have energy separation of ħω. The dynamics of the atom can then be described by the three rotation operators, ,,, acting upon the Bloch sphere. Thus the energy of the system is described entirely through an electric dipole interaction between the atom and field with the resulting hamiltonian being described by
.
After quantizing the electromagnetic field, the Heisenberg equation and Maxwell's equations can be used to find the resulting equations of motion for as well as for , the annihilation operator of the field,
,
where and are frequency parameters used to simplify equations, and denotes the Hermitian conjugate of the previous expression.
Now that the dynamics of the field with respect to the states of the atom has been described, the mechanism through which photons are released from the atom as the electron falls from the excited state to the ground state, spontaneous emission, can be examined. Spontaneous emission, in contrast to stimulated emission, is when an excited electron decays to a lower state by emitting a photon, without any external stimulus. As the electromagnetic field is coupled to the state of the atom, and the atom can only absorb a single photon before having to decay, the most basic case then is if the field only contains a single photon. Thus spontaneous decay occurs when the excited state of the atom emits a photon back into the vacuum Fock state of the field . During this decay process, the expectation values of the above operators obey the following relations
,
.
So both decay exponentially, but the atomic dipole has an oscillating factor. The dipole moment oscillates due to the Lamb shift, which is a shift in the energy levels of the atom due to fluctuations of the field.
It is imperative, however, to look at fluorescence in the presence of a field with many photons, as this is a much more general case. This is the case in which the atom goes through many excitation cycles. In this case the exciting field emitted from the laser is in the form of coherent states . This allows for the operators which comprise the field to act on the coherent state and thus be replaced with eigenvalues. Thus we can simplify the equations by allowing operators to be turned into constants. The field can then be described much more classically than a quantized field normally would be able to. As a result, we are able to find the expectation value of the electric field for the retarded time.
,
where is the angle between and .
There are two general types of excitations produced by fields. The first is one that dies out as , while the other one reaches a state in which it eventually reaches a constant amplitude, thus .
Here is a real normalization constant, is a real phase factor, and is a unit vector which indicates the direction of the excitation.
Thus as , then
.
As is the Rabi frequency, we can see that this is analogous to the rotation of a spin state around the Bloch sphere from an interferometer. Thus the dynamics of a two-level atom can be accurately modeled by a photon in an interferometer. It is also possible to model as an atom and a field, and it will, in fact, retain more properties of the system such as Lamb shift, but the basic dynamics of resonance fluorescence can be modeled as a spin-1/2 particle.
Weak field limit There are several limits that can be analyzed to make the study of resonance fluorescence easier. The first of these is the approximation associated with the weak field limit, where the square modulus of the Rabi frequency of the field coupled to two-level atom is much smaller than the atom's rate of spontaneous emission. This means that the difference in the population between the excited state of the atom and the ground state of the atom is approximately independent of time. If we also take the limit in which the time period is much larger than the time for spontaneous decay, the coherences of the light can be modeled as , where is the Rabi frequency of the driving field and is the spontaneous decay rate of the atom. Thus it is clear that when an electric field is applied to the atom, the dipole of the atom oscillates according to driving frequency and not the natural frequency of the atom. If we also look at the positive frequency component of the electric field,
we can see that the emitted field is the same as the absorbed field other than the difference in direction, resulting in the spectrum of the emitted field being the same as that of the absorbed field. The result is that the two-level atom behaves exactly as a driven oscillator and continues scattering photons so long as the driving field remains coupled to the atom.
The weak field approximation is also used in approaching two-time correlation functions. In the weak-field limit, the correlation function can be calculated much more easily as only the first three terms must be kept. Thus the correlation function becomes as .
From the above equation we can see that as the correlation function will no longer depend on time, but rather that it will depend on . The system will eventually reach a quasi-stationary state as It is also clear that there are terms in the equation that go to zero as . These are the result of the Markovian processes of the quantum fluctuations of the system. We see that in the weak field approximation as well as , the coupled system will reach a quasi-steady state where the quantum fluctuations become negligible.
Strong field limit
The strong field limit is the exact opposite limit to the weak field where the square modulus of the Rabi frequency of the electromagnetic field is much larger than the rate of spontaneous emission of the two-level atom. When a strong field is applied to the atom, a single peak is no longer observed in fluorescent light's radiation spectrum. Instead, other peaks begin appearing on either side of the original peak. These are known as side bands. The sidebands are a result of the Rabi oscillations of the field causing a modulation in the dipole moment of the atom. This causes a splitting in the degeneracy of certain eigenstates of the hamiltonian, specifically and are split into doublets. This is known as dynamic Stark splitting and is the cause for the Mollow triplet, which is a characteristic energy spectrum found in resonance fluorescence.
An interesting phenomena arises in the Mollow triplet where both of the sideband peaks have a width different than that of the central peak. If the Rabi frequency is allowed to become much larger than the rate of spontaneous decay of the atom, we can see that in the strong field limit will become . This equation reveals the cause of the differences in the Mollow triplet's peaks' widths: the central peak has a width of , and the sideband peaks have a width of , where is the rate of spontaneous emission for the atom. Unfortunately this cannot be used to calculate a steady state solution as and in a steady state solution. Thus the spectrum would vanish in a steady state solution, which is not the actual case.
The solution that does allow for a steady state solution must take the form of a two-time correlation function as opposed to the above one-time correlation function. This solution appears as
.
Since this correlation function includes the steady state limits of the density matrix, where and , and the spectrum is nonzero, it is clear to see that the Mollow triplet remains the spectrum for the fluoresced light even in a steady state solution.
General two-time correlation functions and spectral density The study of correlation functions is critical to the study of quantum optics, because the Fourier transform of the correlation function is the energy spectral density. Thus the two-time correlation function is a useful tool in the calculation of the energy spectrum for a given system. We take the parameter to be the difference between the two times in which the function is calculated. While correlation functions can more easily be described using limits of the strength of the field and limits placed on the time of the system, they can be found more generally as well. For resonance fluorescence, the most important correlation functions are
,
,
,
where
,
,
.
Two-time correlation functions are generally shown to be independent of , and instead rely on as . These functions can be used to find the spectral density by computing the transform
,
where K is a constant. The spectral density can be viewed as the rate of photon emission of photons of frequency at the given time , which is useful in determining the power output of a system at a given time.
The correlation function associated with the spectral density of resonance fluorescence is reliant on the electric field. Thus once the constant K has been determined, the result is equivalent to
This is related to the intensity by
In the weak field limit when the power spectrum can be determined to be
.
In the strong field limit, the power spectrum is slight more complicated and found to be
.
From these two functions it is easy to see that in the weak field limit a single peak appears at in the spectral density due to the delta function, while in the strong field limit a Mollow triplet forms with sideband peaks at , and appropriate peak width of for the central peak and for the sideband peaks.
Photon anti-bunching Photon anti-bunching is the process in resonance fluorescence through which rate at which photons are emitted by a two-level atom is limited. A two-level atom is only capable of absorbing a photon from the driving electromagnetic field after a certain period of time has passed. This time period is modeled as a probability distribution where as . As the atom cannot absorb a photon, it is unable to emit one and thus there is a restriction on the spectral density. This is illustrated by the second order correlation function . From the above equation it is clear that and thus resulting in the relation that describes photon antibunching .