2. History of Cavity QED
Bohr-Einstein debate photon-box is
considered one of the first ideas of a
resonance cavity.
1930
Edward Purcell noticed increased
spontaneous photon emission for atoms
near metallic structures.
1946
Hendrik Casimir prediction of the
quantum mechanical interaction
between conducting plates known as
Casimir effect.
1948
First described in 1963 (Jaynes &
Cummings).
• Comparison of Quantum and
Semiclassical Radiation Theories with
Application to the Beam Maser*
1963
Second major paper in 1965
(Cummings).
• A single mode radiation field coupled
to a two-level atom
1965
Experimentally confirmed in the 1980s.
• Vacuum Rabi splitting as a feature of
linear-dispersion theory: Analysis and
experimental observations
1980s
3. Types of CQED
a) Microwave
Determine state of qubit and cavity decay
Gather information about photons inside the cavity
b) Optical
Drives cavity with laser
Monitor changes in transmission due to emitters
4. Jaynes-Cummings Model
Illustration of the Jaynes-Cummings model.
Defining characteristics of the model.
• Atom in a lossless cavity.
• Single-mode electric field.
• Two accessible atomic levels.
• Atom oscillates between energy levels.
5. Excited state
probability of JCM
• Plot of the probability to find the
system in the excited state as a
function of the unit-less parameter
for a system with mean photon
number <n>.
• At later time (gt) there is a revival
of the probability caused by discrete
quanta frequencies whereas classical
continuous frequency spectrum
wouldn’t show the revival.
• Assumes no detuning so
6. Karatsuba, Anatolii A., and Ekatherina A. Karatsuba. "A resummation formula for collapse and
revival in the Jaynes–Cummings model." Journal of Physics A: Mathematical and
Theoretical 42.19 (2009): 195304.
7. Jaynes–Cummings–Hubbard (JCH) model
• Tunnelling of photons between coupled cavities Κ is the tunnelling
rate of photons.
• Κ in is also referred to the photon cavity decay.
• Photon tunneling can be modulated by driving strong photon-atom
coupling and emerging single polariton to jump between cavities.
• Tunneling can also be inhibited by create multiple polariton
repulsion between cavities. (anti-JCH).
Ohira, Ryutaro, et al. "Polariton blockade in the Jaynes–Cummings–Hubbard
model with trapped ions." Quantum Science and Technology 6.2 (2021): 024015.
8. Numerical Simulation of single polariton hopping.
Ohira, Ryutaro, et al. "Polariton blockade in the Jaynes–Cummings–Hubbard
model with trapped ions." Quantum Science and Technology 6.2 (2021): 024015.
9. Examples of CQED
• Quantum system can be an atom
or ion.
• Dispersive regime where readout is
ain/aout transmission.
• E|1> - E|0> - hfc >> ħgeff
• Resonant regime forms
superposition of light and matter.
• E|1> - E|0> ≈ hfc
Burkard, Guido, et al. "Superconductor–semiconductor hybrid-circuit quantum
electrodynamics." Nature Reviews Physics 2.3 (2020): 129-140.
10. Cavity-coupled
double dot
• A quantum dot (QD) is a nanoscale object that confines an electron
in all three spatial dimensions.
• Dipole interactions couple the electron to the cavity photon
• Dipole moment is significantly larger than a single atom due to dot
spacing d.
Burkard, Guido, et al.
"Superconductor–
semiconductor
hybrid-circuit
quantum
electrodynamics." Na
ture Reviews
Physics 2.3 (2020):
129-140.
11. Circuit QED
• Atom is superconducting
artificial atom.
• Microwave cavity replaced by
resonator circuit.
Image from:
https://www.nict.go.jp/en/frontier/mqp/projects.html
Editor's Notes
I originally started my research on Optical Cavity Electrodynamics, but what I found is that most research went into the cavity quantum electrodynamics, so I figured I’d investigate both.
I enjoyed reading about this field mostly because it is closely connected to quantum computing qubit research.
If I was continuing with Physics this would be a specialization, I would’ve been interested in.
We’ll start by looking into the history of Cavity QED. One of the earliest ideas of a cavity of photons was the photon-box debate between bohr-Einstein. A lot of papers I read referenced this thought experiment.
1946 Edward Purcell was researching atom spontaneous photon emission and noticed that moving the atoms closer to a metallic surface changed the photon emission.
1948 the Casimir effect was predicted where quantum vacuum fluctuations caused a force between two plates to occur.
1963 the Jaynes Cummings paper was published that described differences between semiclassical and quantum cavity radiation.
1965 was another paper by Cummings that described a simpler two-level atom system which we’ll look at it in a bit.
Lastly, in the 1980s the results predicted in the 1960s which we’ll go into was shown experimentally using microwave cavities.
There are two types of cavity systems. The first a is the microwave type where you have a cavity with some beam of atoms prepared in a specific state are sent through a microwave cavity.
After going through the cavity the atoms are then investigated using what is called selective state field ionization. My understanding is that you have a tier of possible interactions then depending on which the atom interacts with gives information about the system.
Some of the values shown are omega c which is the photon frequency in the cavity omega q is the oscillation between the ground and excited state. Kappa is photon cavity decay and omega might be an ion trap florescence decay.
B is the optical cavity where you have some capital N number of emitters typically would be some group of atoms that are passed through the cavity.
Then the atoms themselves are investigated to gather information about the cavity. g is the coupling strength between the atoms and photons where it scales by the square of the total emitters.
Here is the two state atom cavity called the Jaynes cummings model. Typically it is a cavity of two mirrors which are concave with some nearly perfect reflectivity.
The cavity has some specific width to cause an electromagnetic standing wave.
The photon frequency is omega c or omega the atom frequency if omega a.
The oscillation difference between omega a and omega c is showing as delta on the right. On the left you have the 2 quantum states of the system with n-1 photons or n photons also determining the excited state and ground state.
gN is the coupling strength mentioned before.
On the right g is that coupling strength and frequency.
D is the dipole moment of the atom.
Omega frequency of photon and V is the model taken up by the modes minimizing this is important to focus the cavity photons on the atoms.
Typically why you might see an hourglass shape.
Omegan is the rabi oscillation frequency, rabi oscillations are just the description of an atom emitting and absorbing photons in a cavity like system.
Pe is the probability of the excited state in rabi oscillations.
One interesting thing to note is the situation where there is no photon n=0 and no detuning frequency delta.
That omega0 becomes double the coupling strength this gives a nonzero probability of the atom being in the excited state.
Here is a probability graph of the JCM model the Pe equation is valid for short times where omegat/2 is much less then the square of the average photon emission.
You can see in that shorter time on the horizontal axis that the probability decays off quickly where it settles to 0.5.
The interesting thing to look at is that the probability after some interval revives and there is a spike of probability amplitude.
For a short time then goes back to 0.5.
This is caused by discrete integer quanta of frequencies whereas if it was only semi-classical the system would decay quickly then immediately flat line.
One other important thing is the see the relation between the effective frequency of the system is proportional to the square of the average number of photons times omega.
Here is a 3d graph. Where time is on the right average number of photons on the left and the atom inversion intensity on the left.
You can see the revivals after the initial decay of the system at some specific interval.
It is interesting to see the relation to the number of photons as well in comparison to the inversion intensity.
Here is an interesting model of the Jaynes cummings system call the hubbard model.
I like to think of it as an array of some n number of cavities.
Where there is some kappa tunneling rate of the photon decay.
Kappa is also mentioned as the photon decay rate out of the cavity.
A research paper I was reading was talking about a situation where you drive the coupling strength to the point of creating a photon-atom polariton quasi-particle.
If you were to have a two ion cavity system if you were to have one polarition it would tunnel across the cavity rapidly whilst there is a way to inhibit tunnels by driving both cavities to contain a polariton.
Here are a few graphs showing a numerical simulation of the polariton coupling where if there is 1 polariton you can see the tunnel rate is high.
Whilst if there are two polaritons in the two ion system, you can see that the tunnel is inhibited and there are few time where there is a two-0 polaritons in the ion cavities.
This is just a few examples of cavity qed systems. Where you have an atom, a color center where this might be a diamond where the cavity drives a photon coupling using an atom.
The double dot and superconducting qubit we will look at in a minute.
There are two regimes for these cavity systems one is the dispersive regime where the coupling strength is low and the readout occurs of the system.
The readout is done using some a-in and a-out where this could be some transmission wave with a very low coupling strength then see how it was affected in a out.
The resonant regime is where there is a maximal coupling between photon quantum systems and the superposition of light and matter occurs.
One system is the double quantum dot.
Where there is some nanoscale set of dots with some spacing that hold electrons.
In the cavity system the coupling strength between the photon and electron is driven in order to induce tunneling across the quantum dots.
Where there is some energy state of left and right and the tunneling induces a significant dipole moment.
Lastly here is an analog of the circuit and cavity QED.
Circuit QED from my reading seems to be a dominant field of qubit research.
The atom itself is referenced as a faux atom or simulated atom in a superconductor.
Where the qubit is excited then moved into the resonator circuit which would be an LC type circuit.
