Inphysics,quantum beats are simple examples ofphenomena that cannot be described by semiclassical theory, but can be described by fully quantized calculation, especiallyquantum electrodynamics. In semiclassical theory (SCT), there is an interference orbeat note term for both V-type and${\displaystyle \mathrm{\Lambda }}$-type atoms.^{[clarification needed]} However, in the quantum electrodynamic (QED) calculation, V-type atoms have a beat term but${\displaystyle \mathrm{\Lambda }}$-types do not. This is strong evidence in support ofquantum electrodynamics.

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 thespectral line into several components in the presence of amagnetic field due to theZeeman effect. These light components were mixed at aphotoelectric surface, and the electrons emitted from that surface then excited amicrowave cavity, which allowed the output signal to be measured in dependence on the magnetic field.^[3][4]

Since the invention of thelaser, quantum beats can be demonstrated by using light originating from two different laser sources. In 2017 quantum beats in singlephoton emission from the atomic collective excitation have been observed.^[5] Observed collective beats were not due tosuperposition of excitation between two differentenergy levels of the atoms, as in usual single-atom quantum beats in${\displaystyle V}$-type atoms.^[6] Instead, single photon was stored as excitation of the same atomic energy level, but this time two groups of atoms with different velocities have been coherently excited. These collective beats originate from motion between entangled pairs of atoms,^[6] that acquire relative phase due toDoppler effect.

There is a figure inQuantum Optics^[7] that describes${\displaystyle V}$-type and${\displaystyle \mathrm{\Lambda }}$-type atoms clearly.

Simply, V-type atoms have 3 states:${\displaystyle |a⟩}$,${\displaystyle |b⟩}$, and${\displaystyle |c⟩}$. The energy levels of${\displaystyle |a⟩}$ and${\displaystyle |b⟩}$ are higher than that of${\displaystyle |c⟩}$. When electrons in states${\displaystyle |a⟩}$ and :${\displaystyle |b⟩}$ subsequently decay to state${\displaystyle |c⟩}$, two kinds of emission are radiated.

In${\displaystyle \mathrm{\Lambda }}$-type atoms, there are also 3 states:${\displaystyle |a⟩}$,${\displaystyle |b⟩}$, and :${\displaystyle |c⟩}$. However, in this type,${\displaystyle |a⟩}$ is at the highest energy level, while${\displaystyle |b⟩}$ and :${\displaystyle |c⟩}$ are at lower levels. When two electrons in state${\displaystyle |a⟩}$ decay to states${\displaystyle |b⟩}$ and :${\displaystyle |c⟩}$, respectively, two kinds of emission are also radiated.

The derivation below follows the referenceQuantum Optics.^[7]

In the semiclassical picture, the state vector ofelectrons is

${\displaystyle |\psi (t)⟩=c_{a}exp(-i{\omega }_{a}t)|a⟩+c_{b}exp(-i{\omega }_{b}t)|b⟩+c_{c}exp(-i{\omega }_{c}t)|c⟩}$.

If the nonvanishingdipole matrix elements are described by

${\displaystyle {\mathcal{P}}_{ac}=e⟨a|r|c⟩,{\mathcal{P}}_{bc}=e⟨b|r|c⟩}$ for V-type atoms,

${\displaystyle {\mathcal{P}}_{ab}=e⟨a|r|b⟩,{\mathcal{P}}_{ac}=e⟨a|r|c⟩}$ for${\displaystyle \mathrm{\Lambda }}$-type atoms,

then each atom has two microscopic oscillatingdipoles

${\displaystyle P(t)={\mathcal{P}}_{ac}(c_a^{\ast }{c}_c)exp(i{\nu }_{1}t)+{\mathcal{P}}_{bc}(c_b^{\ast }{c}_c)exp(i{\nu }_{2}t)+c.c.}$ for V-type, when${\displaystyle {\nu }_1={\omega }_a-{\omega }_c,{\nu }_2={\omega }_b-{\omega }_c}$,

${\displaystyle P(t)={\mathcal{P}}_{ab}(c_a^{\ast }{c}_b)exp(i{\nu }_{1}t)+{\mathcal{P}}_{ac}(c_a^{\ast }{c}_c)exp(i{\nu }_{2}t)+c.c.}$ for${\displaystyle \mathrm{\Lambda }}$-type, when${\displaystyle {\nu }_1={\omega }_a-{\omega }_b,{\nu }_2={\omega }_a-{\omega }_c}$.

In the semiclassical picture, the field radiated will be a sum of these two terms

${\displaystyle E^{(+)}={\mathcal{E}}_{1}exp(-i{\nu }_{1}t)+{\mathcal{E}}_{2}exp(-i{\nu }_{2}t)}$,

so it is clear that there is aninterference orbeat note term in asquare-law detector

${\displaystyle |E^{(+)}{|}^2=|{\mathcal{E}}_1{|}^2+|{\mathcal{E}}_2{|}^2+\{{\mathcal{E}}_1^{\ast }{\mathcal{E}}_{2}exp[i({\nu }_1-{\nu }_2)t]+c.c.\}}$.

For quantum electrodynamical calculation, we should introduce the creation and annihilation operators fromsecond quantization ofquantum mechanics.

Let

${\displaystyle E_n^{(+)}=a_{n}exp(-i{\nu }_{n}t)}$ is anannihilation operator and

${\displaystyle E_n^{(-)}=a_n^{†}exp(i{\nu }_{n}t)}$ is acreation operator.

Then the beat note becomes

${\displaystyle ⟨{\psi }_V(t)|E_1^{(-)}(t)E_2^{(+)}(t)|{\psi }_V(t)⟩}$ for V-type and

${\displaystyle ⟨{\psi }_{\mathrm{\Lambda }}(t)|E_1^{(-)}(t)E_2^{(+)}(t)|{\psi }_{\mathrm{\Lambda }}(t)⟩}$ for${\displaystyle \mathrm{\Lambda }}$-type,

when the state vector for each type is

${\displaystyle |{\psi }_V(t)⟩=\sum _{i=a,b,c}{c}_i|i,0⟩+c_1|c,1_{{\nu }_1}⟩+c_2|c,1_{{\nu }_2}⟩}$ and

${\displaystyle |{\psi }_{\mathrm{\Lambda }}(t)⟩=\sum _{i=a,b,c}{c}_i^{\prime }|i,0⟩+c_1^{\prime }|b,1_{{\nu }_1}⟩+c_2^{\prime }|c,1_{{\nu }_2}⟩}$.

The beat note term becomes

${\displaystyle ⟨{\psi }_V(t)|E_1^{(-)}(t)E_2^{(+)}(t)|{\psi }_V(t)⟩=\kappa ⟨1_{{\nu }_1}{0}_{{\nu }_2}|a_1^{†}{a}_2|0_{{\nu }_1}{1}_{{\nu }_2}⟩exp[i({\nu }_1-{\nu }_2)t]⟨c|c⟩=\kappa exp[i({\nu }_1-{\nu }_2)t]⟨c|c⟩}$ for V-type and

${\displaystyle ⟨{\psi }_{\mathrm{\Lambda }}(t)|E_1^{(-)}(t)E_2^{(+)}(t)|{\psi }_{\mathrm{\Lambda }}(t)⟩={\kappa }^{\prime }⟨1_{{\nu }_1}{0}_{{\nu }_2}|a_1^{†}{a}_2|0_{{\nu }_1}{1}_{{\nu }_2}⟩exp[i({\nu }_1-{\nu }_2)t]⟨b|c⟩={\kappa }^{\prime }exp[i({\nu }_1-{\nu }_2)t]⟨b|c⟩}$ for${\displaystyle \mathrm{\Lambda }}$-type.

Byorthogonality ofeigenstates, however${\displaystyle ⟨c|c⟩=1}$ and${\displaystyle ⟨b|c⟩=0}$.

Therefore, there is a beat note term for V-type atoms, but not for${\displaystyle \mathrm{\Lambda }}$-type atoms.

As a result of calculation, V-type atoms have quantum beats but${\displaystyle \mathrm{\Lambda }}$-type atoms do not. This difference is caused by quantum mechanicaluncertainty. A V-type atom decays to state${\displaystyle |c⟩}$ via the emission with${\displaystyle {\nu }_1}$ and${\displaystyle {\nu }_2}$. Since both transitions decayed to the same state, one cannot determine alongwhich path each decayed, similar to Young'sdouble-slit experiment. However,${\displaystyle \mathrm{\Lambda }}$-type atoms decay to two different states. Therefore, in this case we can recognize the path, even if it decays via two emissions as does V-type. Simply, we already know the path of the emission and decay.

The calculation by QED is correct in accordance with the most fundamental principle ofquantum mechanics, theuncertainty principle. Quantum beats phenomena are good examples of such that can be described by QED but not by SCT.

• Quantum electrodynamics
• Double-slit experiment
• Quantum beats in semiconductor optics

• F.G. Major (2007).The Quantum Beat: Principles and Applications of Atomic Clocks. Springer.ISBN 978-0-387-69533-4.
• Marlan Orvil Scully & Muhammad Suhail Zubairy (1997).Quantum optics. Cambridge UK: Cambridge University Press. p. 541.ISBN 978-0-521-43595-6.

• Quantum electrodynamics

• Wikipedia articles needing clarification from March 2009