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.[1]
Typically the photon contained 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 the two possible states. The two possible states are if an electron is found in its ground state or the excited state. In many experiments an atom of lithium is used because it can be closely modeled to a two-level atom as the excited states of the singular electron are separated by large enough energy gaps to significantly reduce the possibility of the electron jumping to a higher excited state. Thus it allows for easier frequency tuning of the applied laser as frequencies further off 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.[2]
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,
\hat{Ri
\hat{Rj
\hat{Rk
\hat{H}=
| 1 | |
| 2 |
\int(\epsilon0\hat{\vec{E}}2(\vec{r},t)+
| 1 | |
| \mu0 |
\hat{\vec{B}}2(\vec{r},t))d3x+\hbar\omega0\hat{Rk
After quantizing the electromagnetic field, the Heisenberg Equation as well as Maxwell's equations can then be used to find the resulting equations of motion for
\hat{Rk
\hat{b}(t)
| \hat{R |
| \hat{b |
where
\beta
\gamma
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 is when an excited electron arbitrarily decays to the ground state emitting a photon. 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
|e\rangle ⊗ |\{0\}\rangle ⇒ |g\rangle ⊗ |\{1\}\rangle
\langle\hat{R}k(t)\rangle+
| 1 | |
| 2 |
=[\langle\hat{R}k(0)\rangle+
| 1 | |
| 2 |
]e-2\beta
\langle\hat{b}s(t)\rangle=\langle\hat{b}s(0)\ranglee(-\beta
So the atom decays exponentially and the atomic dipole moment shall oscillate. 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
|\{v\}\rangle
\langle\hat{\vec{E}}(-)(\vec{r},t) ⋅ \hat{\vec{E}}(+)(\vec{r},t)\rangle=\left(
| |||||||
| 4\pi\epsilon0c2 |
\right)2\left(
| \mu2 | |
| r2 |
-
| (\vec{\mu | |
| ⋅ |
\vec{r})2
where
\psi
\hat{\mu}
\hat{r}
There are two general types of excitations produced by fields. The first is one that dies out as
V=0,t ⇒ infty
\hat{V}(t)=\hat{\epsilon}\alpha
| i(\omega0-\omega1)t+i\phi | |
| e |
Here
\alpha
\phi
\hat{\epsilon}
Thus as
t ⇒ infty
\langle\hat{R}k(t)\rangle+1/2 ⇒
| |||||
|
As
\Omega
There are several limits that can be analyzed to make the study of resonance fluorescence easier. The first of these is the approximations associated with the Weak Field Limit, where the square modulus of the Rabi frequency of the field that is coupled to two-level atom is much smaller than the rate of spontaneous emission of the atom. 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.[3] 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
\rhoab(t)=
| -i(\OmegaR/2)e-i\nu | |
| i(\omega-\nu)+\Gamma/2 |
[\rhoaa(0)-\rhobb(0)]
\OmegaR
\Gamma
\langle\vec{E}(+)(\vec{r},t)\rangle=
| \omega2\mu\sin\psi | |
| 4\pi\epsilon0c2|\vec{r |
|}\hat{x}\langle\sigma-(t-
| |\vec{r | |
| |}{c})\rangle |
The weak field approximation is also used in approaching two-time correlation functions. In the weak-field limit, the correlation function
\langle
| \dagger | |
| \hat{b} | |
| s |
(t)\hat{b}s(t+\tau)\rangle
\langle
| \dagger | |
| \hat{b} | |
| s |
(t)\hat{b}s(t+\tau)\rangle=
| 1 | |
| 4 |
| |||||||||||
| \beta2(1+\theta2) |
\left(1-
| \Omega2 | ||||
|
\right)+
| |||||||||||
| 8\beta4\theta(1+\theta2)2 |
x [\sin(\beta\theta|\tau|)+\theta\cos(\beta\theta\tau)]
t ⇒ infty
From the above equation we can see that as
t ⇒ infty
\tau
t ⇒ infty
\tau ⇒ infty
t ⇒ infty,\tau ⇒ infty
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
|e\rangle ⊗ |\{n\}\rangle
|g\rangle ⊗ |\{n+1\}\rangle
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
\langle\sigma-(t)\rangleei\omega
\langle\sigma-(t)\rangleei\omega=
| 1 | |
| 4 |
\{[2\rho++
| |||||
| (0)-1]e |
-[\rho+-
| ||||||||||
| (0)e |
-c.c]\}
| \Gamma | |
| 2 |
| 3\Gamma | |
| 4 |
\Gamma
\rho++(0) ⇒
| 1 | |
| 2 |
\rho+-(0) ⇒ 0
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
\langle\sigma+(0)\sigma-(\tau)\rangle=
| 1 | |
| 4 |
\left(
| |||||
| e |
+
| 1 | |
| 2 |
| |||||
| e |
| -i\OmegaR\tau | ||
| e | + |
| 1 | |
| 2 |
| |||||
| e |
| i\OmegaR\tau | |
| e |
\right)e-i\omega\tau
Since this correlation function includes the steady state limits of the density matrix, where
| s.s | |
| \rho | |
| ++ |
⇒
| 1 | |
| 2 |
| s.s | |
| \rho | |
| +- |
⇒ 0
The study of correlation functions is critical to the study of quantum optics as 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
\tau
\langle
| \dagger | |
| \hat{b} | |
| s |
(t)\hat{b}s(t+\tau)\rangle
| i(\omega1-\omega0)\tau | |
| e |
\equivg(t,\tau)
\langle
| \dagger | |
| \hat{b} | |
| s |
(t)\hat{b}s(t+\tau)\rangle
| i(\omega1-\omega0)(2t+\tau | |
| e |
e2i\phi\equivf(t,\tau)
\langle
| \dagger | |
| \hat{b} | |
| s |
(t)\hat{R}k(t+\tau)\rangle
| i(\omega1-\omega0)t | |
| e |
ei\phi\equivg(t,\tau)
where
g(t,\tau)=[\langle\hat{R}k(t)\rangle+
| 1 | |
| 2 |
]e-\beta(1-i\theta)\tau+\Omega
| \tau | |
| \int\limits | |
| 0 |
dt'h(t,t')e\beta(1-i\theta)(t'-\tau)
f(t,\tau)=\Omega
| \tau | |
| \int\limits | |
| 0 |
dt'h(t,t')e\beta(1+i\theta)(t'-\tau)
h(t,\tau)=-
| 1 | |
| 2 |
| \dagger | |
| \langle\hat{b} | |
| s |
(t)\rangle
| i(\omega0-\omega1)t | |
| e |
ei\phi-
| 1 | |
| 2 |
\Omega
| \tau | |
| \int\limits | |
| 0 |
dt'[f(t,t')+g(t,t')]e2\beta(t'-\tau)
Two-time correlation functions are generally shown to be independent of
t
\tau
t ⇒ infty
S(t,\omega)
S(t,\omega)=K
| infty | |
| \int\limits | |
| 0 |
d\tau
| i(\omega-\omega1)\tau | |
| g(t-\tau,\tau)e |
+c.c
where K is a constant. The spectral density can be viewed as the rate of photon emission of photons of frequency
\omega
t
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
S(\vec{r},\omega0)=
| 1 | |
| \pi |
Re
| infty | |
| \int\limits | |
| 0 |
d\tau\langleE(-)(\vec{r},t)E(+)(\vec{r},t+\tau)\rangle
| i\omega0\tau | |
| e |
This is related to the intensity by
\langleE(-)(\vec{r},t)E(+)(\vec{r},t+\tau)\rangle=I0(\vec{r})\langle\sigma+(t)\sigma-(t+\tau)\rangle
In the weak field limit when
\OmegaR\ll
| \Gamma | |
| 4 |
S(\vec{r},\omega0)=I0(\vec{r})\left(
| \OmegaR | |
| \Gamma |
\right)2\delta(\omega-\omega0)
In the strong field limit, the power spectrum is slight more complicated and found to be
S(\vec{r},\omega0)=
| I0(\vec{r | |||
|
+
| \Gamma | |
| (\omega-\omega0)2+(\Gamma/2)2 |
+
| 3\Gamma/4 | |
| (\omega+\OmegaR-\omega0)2+(3\Gamma/4)2 |
\right]
From these two functions it is easy to see that in the weak field limit a single peak appears at
\omega0
\omega=\omega0\pm\OmegaR
| \Gamma | |
| 2 |
| 3\Gamma | |
| 4 |
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
p(\tau)
p(\tau) ⇒ 0
\tau ⇒ 0
g(2)(\tau)=1-\left(cos\mu\tau+
| 3\Gamma | |
| 4\mu |
sin\mu\tau\right)e-3\Gamma\tau/4
g(2)(0)=0
g(2)(\tau)>0
g(2)(\tau)>g(2)(0)
\tau=0
g(2)(\tau)
\tau
g(2)(\tau)
\tau ⇒ infty
Double Resonance[4] is the phenomena when an additional magnetic field is applied to a two-level atom in addition to the typical electromagnetic field used to drive resonance fluorescence. This lifts the spin degeneracy of the Zeeman energy levels splitting them along the energies associated with the respective available spin levels, allowing for not only resonance to be achieved around the typical excited state, but if a second driving electromagnetic associated with the Larmor frequency is applied, a second resonance can be achieved around the energy state associated with
mB=0
mb=\pm1
Any two state system can be modeled as a two-level atom. This leads to many systems being described as an "Artificial Atom". For instance a superconducting loop which can create a magnetic flux passing through it can act as an artificial atom as the current can induce a magnetic flux in either direction through the loop depending on whether the current is clockwise or counterclockwise.[5] The hamiltonian for this system is described as
\hat{H}=\hbar
| 2 | |
| \sqrt{\omega | |
| 0 |
+\epsilon2
\hbar\epsilon=2Ip\delta\Phi
A quantum dot is a semiconductor nano-particle that is often used in quantum optical systems. This includes their ability to be placed in optical microcavities where they can act as two-level systems. In this process, quantum dots are placed in cavities which allow for the discretization of the possible energy states of the quantum dot coupled with the vacuum field. The vacuum field is then replaced by an excitation field and resonance fluorescence is observed. Current technology only allows for population of the dot in an excited state (not necessarily always the same), and relaxation of the quantum dot back to its ground state. Direct excitation followed by ground state collection was not achieved until recently. This is mainly due to the fact that as a result of the size of quantum dots, defects and contaminants create fluorescence of their own apart from the quantum dot. This desired manipulation has been achieved by quantum dots by themselves through a number of techniques including four-wave mixing and differential reflectivity, however no techniques had shown it to occur in cavities until 2007. Resonance fluorescence has been seen in a single self-assembled quantum dot as presented by Muller among others in 2007.[7] In the experiment they used quantum dots that were grown between two mirrors in the cavity. Thus the quantum dot was not placed in the cavity, but instead created in it. They then coupled a strong in-plane polarized tunable continuous-wave laser to the quantum dot and were able to observe resonance fluorescence from the quantum dot. In addition to the excitation of the quantum dot that was achieved, they were also able to collect the photon that was emitted with a micro-PL setup. This allows for resonant coherent control of the ground state of the quantum dot while also collecting the photons emitted from the fluorescence.
In 2007, G. Wrigge, I. Gerhardt, J. Hwang, G. Zumofen, and V. Sandoghdar developed an efficient method to observe resonance fluorescence for an entire molecule as opposed to its typical observation in a single atom.[8] Instead of coupling the electric field to a single atom, they were able to replicate two-level systems in dye molecules embedded in solids.They used a tunable dye laser to excite the dye molecules in their sample. Due to the fact that they could only have one source at a time, the proportion of shot noise to actual data was much higher than normal. The sample which they excited was a Shpol'skii matrix which they had doped with the dyes they wished to use, dibenzanthanthrene. To improve the accuracy of the results, single-molecule fluorescence-excitation spectroscopy was used. The actual process for measuring the resonance was measuring the interference between the laser beam and the photons that were scattered from the molecule. Thus the laser was passed over the sample, resulting in several photons were scattered back, allowing for the measurement of the interference in the electromagnetic field that resulted. The improvement to this technique was they used solid-immersion lens technology. This is a lens that has a much higher numerical aperture than normal lenses as it is filled with a material that has a large refractive index. The technique used to measure the resonance fluorescence in this system was originally designed to locate individual molecules within substances.
The largest implication that arises from resonance fluorescence is that for future technologies. Resonance fluorescence is used primarily in the coherent control of atoms. By coupling a two-level atom, such as a quantum dot, to an electric field in the form of a laser, you are able to effectively create a qubit. The qubit states correspond to the excited and the ground state of the two-level atoms. Manipulation of the electromagnetic field allows for effective control of the dynamics of the atom. These can then be used to create quantum computers. The largest barriers that still stand in the way of this being achievable are failures in truly controlling the atom. For instance true control of spontaneous decay and decoherence of the field pose large problems that must be overcome before two-level atoms can truly be used as qubits.