In physics, quantum beats are simple examples of phenomena that cannot be described by semiclassical theory, but can be described by fully quantized calculation, especially quantum electrodynamics. In semiclassical theory (SCT), there is an interference or beat note term for both V-type and
Λ
Λ
The observation of quantum beats was first reported by A.T. Forrester, R.A. Gudmundsen and P.O. Johnson in 1955,[1] in an experiment that was performed on the basis of an earlier proposal by A.T. Forrester, W.E. Parkins and E. Gerjuoy.[2] This experiment involved the mixing of the Zeeman components of ordinary incoherent light, that is, the mixing of different components resulting from a split of the spectral line into several components in the presence of a magnetic field due to the Zeeman effect. These light components were mixed at a photoelectric surface, and the electrons emitted from that surface then excited a microwave cavity, which allowed the output signal to be measured in dependence on the magnetic field.[3] [4]
Since the invention of the laser, quantum beats can be demonstrated by using light originating from two different laser sources. In 2017 quantum beats in single photon emission from the atomic collective excitation have been observed.[5] Observed collective beats were not due to superposition of excitation between two different energy levels of the atoms, as in usual single-atom quantum beats in
V
Λ
V
Λ
Simply, V-type atoms have 3 states:
|a\rangle
|b\rangle
|c\rangle
|a\rangle
|b\rangle
|c\rangle
|a\rangle
|b\rangle
|c\rangle
In
Λ
|a\rangle
|b\rangle
|c\rangle
|a\rangle
|b\rangle
|c\rangle
|a\rangle
|b\rangle
|c\rangle
The derivation below follows the reference Quantum Optics.
In the semiclassical picture, the state vector of electrons is
|\psi(t)\rangle=caexp(-i\omegaat)|a\rangle+cbexp(-i\omegabt)|b\rangle+ccexp(-i\omegact)|c\rangle
l{P}ac=e\langlea|r|c\rangle,l{P}bc=e\langleb|r|c\rangle
l{P}ab=e\langlea|r|b\rangle,l{P}ac=e\langlea|r|c\rangle
Λ
P(t)=l{P}ac
*c | |
(c | |
c)exp(i\nu |
1t)+l{P}bc
*c | |
(c | |
c)exp(i\nu |
2t)+c.c.
\nu1=\omegaa-\omegac,\nu2=\omegab-\omegac
P(t)=l{P}ab
*c | |
(c | |
b)exp(i\nu |
1t)+l{P}ac
*c | |
(c | |
c)exp(i\nu |
2t)+c.c.
Λ
\nu1=\omegaa-\omegab,\nu2=\omegaa-\omegac
E(+)=l{E}1exp(-i\nu1t)+l{E}2exp(-i\nu2t)
|E(+)
*l{E} | |
| | |
2exp\lbrack |
i(\nu1-\nu2)t\rbrack+c.c.\rbrace
For quantum electrodynamical calculation, we should introduce the creation and annihilation operators from second quantization of quantum mechanics.
Let
(+) | |
E | |
n |
=anexp(-i\nunt)
(-) | |
E | |
n |
\dagger | |
=a | |
n |
exp(i\nunt)
\langle\psiV(t)|E
(-) | |
1 |
(+) | |
(t)E | |
2 |
(t)|\psiV(t)\rangle
\langle\psiΛ(t)|E
(-) | |
1 |
(+) | |
(t)E | |
2 |
(t)|\psiΛ(t)\rangle
Λ
|\psiV(t)\rangle=\sumi=a,b,ci|i,0\rangle+c1|c,1
\nu1 |
\rangle+c2|c,1
\nu2 |
\rangle
|\psiΛ(t)\rangle=\sumi=a,b,ci'|i,0\rangle+c1'|b,1
\nu1 |
\rangle+c2'|c,1
\nu2 |
\rangle
The beat note term becomes
\langle\psiV(t)|E
(-) | |
1 |
(+) | |
(t)E | |
2 |
(t)|\psiV(t)\rangle=\kappa\langle
1 | |
\nu1 |
0 | |
\nu2 |
\dagger | |
|a | |
1 |
a2|0
\nu1 |
1 | |
\nu2 |
\rangleexp\lbracki(\nu1-\nu2)t\rbrack\langlec|c\rangle=\kappaexp\lbracki(\nu1-\nu2)t\rbrack\langlec|c\rangle
\langle\psiΛ(t)|E
(-) | |
1 |
(+) | |
(t)E | |
2 |
(t)|\psiΛ(t)\rangle=\kappa'\langle
1 | |
\nu1 |
0 | |
\nu2 |
\dagger | |
|a | |
1 |
a2|0
\nu1 |
1 | |
\nu2 |
\rangleexp\lbracki(\nu1-\nu2)t\rbrack\langleb|c\rangle=\kappa'exp\lbracki(\nu1-\nu2)t\rbrack\langleb|c\rangle
Λ
\langlec|c\rangle=1
\langleb|c\rangle=0
Therefore, there is a beat note term for V-type atoms, but not for
Λ
As a result of calculation, V-type atoms have quantum beats but
Λ
|c\rangle
\nu1
\nu2
Λ
The calculation by QED is correct in accordance with the most fundamental principle of quantum mechanics, the uncertainty principle. Quantum beats phenomena are good examples of such that can be described by QED but not by SCT.