Quantum beats

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 $\Lambda$ -type atoms. However, in the quantum electrodynamic (QED) calculation, V-type atoms have a beat term but $\Lambda$ -types do not. This is strong evidence in support of quantum electrodynamics.

Historical overview

The observation of quantum beats was first reported by A.T. Forrester, R.A. Gudmunsen 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.

V-type and $\Lambda$ -type atoms

There is a figure in Quantum Optics^[5] that describes V-type and $\Lambda$ -type atoms clearly.

The derivation below follows the reference Quantum Optics^[6]

Calculation based on semiclassical theory

In the semiclassical picture, the state vector of electrons is

|\psi (t)\rangle =c_{a}exp(-i\omega _{a}t)|a\rangle +c_{b}exp(-i\omega _{b}t)|b\rangle +c_{c}exp(-i\omega _{c}t)|c\rangle

If the nonvanishing dipole matrix elements are described by

{\mathcal {P}}_{{ac}}=e\langle a|r|c\rangle ,{\mathcal {P}}_{{bc}}=e\langle b|r|c\rangle

for V-type atoms,

{\mathcal {P}}_{{ab}}=e\langle a|r|b\rangle ,{\mathcal {P}}_{{ac}}=e\langle a|r|c\rangle

for

\Lambda

-type atoms,

then each atom has two microscopic oscillating dipoles

P(t)={\mathcal {P}}_{{ac}}(c_{a}^{*}c_{c})exp(i\nu _{1}t)+{\mathcal {P}}_{{bc}}(c_{b}^{*}c_{c})exp(i\nu _{2}t)+c.c.

for V-type, when

\nu _{1}=\omega _{a}-\omega _{c},\nu _{2}=\omega _{b}-\omega _{c}

P(t)={\mathcal {P}}_{{ab}}(c_{a}^{*}c_{b})exp(i\nu _{1}t)+{\mathcal {P}}_{{ac}}(c_{a}^{*}c_{c})exp(i\nu _{2}t)+c.c.

for

\Lambda

-type, when

\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

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

so it is clear that there is an interference or beat note term in a square law detector

|E^{{(+)}}|^{2}=|{\mathcal {E}}_{1}|^{2}+|{\mathcal {E}}_{2}|^{2}+\lbrace {\mathcal {E}}_{1}^{*}{\mathcal {E}}_{2}exp\lbrack i(\nu _{1}-\nu _{2})t\rbrack +c.c.\rbrace

Calculation based on quantum electrodynamics

For quantum electrodynamical calculation, we should introduce the creation and annihilation operators from second quantization of quantum mechanics.

Let

E_{n}^{{(+)}}=a_{n}exp(-i\nu _{n}t)

is an annihilation operator and

E_{n}^{{(-)}}=a_{n}^{\dagger }exp(i\nu _{n}t)

is a creation operator.

Then the beat note becomes

\langle \psi _{V}(t)|E_{1}^{{(-)}}(t)E_{2}^{{(+)}}(t)|\psi _{V}(t)\rangle

for V-type and

\langle \psi _{\Lambda }(t)|E_{1}^{{(-)}}(t)E_{2}^{{(+)}}(t)|\psi _{\Lambda }(t)\rangle

for

\Lambda

-type,

when the state vector for each type is

|\psi _{V}(t)\rangle =\sum _{{i=a,b,c}}c_{i}|i,0\rangle +c_{1}|c,1_{{\nu _{1}}}\rangle +c_{2}|c,1_{{\nu _{2}}}\rangle

and

|\psi _{\Lambda }(t)\rangle =\sum _{{i=a,b,c}}c_{i}'|i,0\rangle +c_{1}'|b,1_{{\nu _{1}}}\rangle +c_{2}'|c,1_{{\nu _{2}}}\rangle

The beat note term becomes

\langle \psi _{V}(t)|E_{1}^{{(-)}}(t)E_{2}^{{(+)}}(t)|\psi _{V}(t)\rangle =\kappa \langle 1_{{\nu _{1}}}0_{{\nu _{2}}}|a_{1}^{\dagger }a_{2}|0_{{\nu _{1}}}1_{{\nu _{2}}}\rangle exp\lbrack i(\nu _{1}-\nu _{2})t\rbrack \langle c|c\rangle =\kappa exp\lbrack i(\nu _{1}-\nu _{2})t\rbrack \langle c|c\rangle

for V-type and

\langle \psi _{\Lambda }(t)|E_{1}^{{(-)}}(t)E_{2}^{{(+)}}(t)|\psi _{\Lambda }(t)\rangle =\kappa '\langle 1_{{\nu _{1}}}0_{{\nu _{2}}}|a_{1}^{\dagger }a_{2}|0_{{\nu _{1}}}1_{{\nu _{2}}}\rangle exp\lbrack i(\nu _{1}-\nu _{2})t\rbrack \langle b|c\rangle =\kappa 'exp\lbrack i(\nu _{1}-\nu _{2})t\rbrack \langle b|c\rangle

for

\Lambda

-type.

By orthogonality of eigenstates, however $\langle c|c\rangle =1$ and $\langle b|c\rangle =0$ .

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

Conclusion

As a result of calculation, V-type atoms have quantum beats but $\Lambda$ -type atoms do not. This difference is caused by quantum mechanical uncertainty. A V-type atom decays to state $|c\rangle$ via the emission with $\nu_1$ and $\nu_2$ . Since both transitions decayed to the same state, one cannot determine along which path each decayed, similar to Young's double-slit experiment. However, $\Lambda$ -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 of quantum mechanics, the uncertainty principle. Quantum beats phenomena are good examples of such that can be described by QED but not by SCT.

References

↑ A.T. Forrester, R.A. Gudmunsen, P.O. Johnson, Physical Review, vol. 99, pp. 1691–1700, 1955 (abstract)
↑ A.T. Forrester, W.E. Parkins, E. Gerjuoy: On the possibility of observing beat frequencies between lines in the visible spectrum, Physical Review, vol. 72, pp. 241–243, 1947
↑ Edward Gerjuoy: Atomic physics, In: H. Henry Stroke (ed.): The Physical Review—the First Hundred Years: A Selection of Seminal Papers and Commentaries, Springer, 1995, ISBN 978-1-56396-188-5, pp. 83–102, p. 97
↑ Paul Hartman: A Memoir on The Physical Review: A History of the First Hundred Years, Springer, 2008, ISBN 978-1-56396-282-0, p. 193
↑ Marlan Orvil Scully & Muhammad Suhail Zubairy (1997). Quantum optics. Cambridge UK: Cambridge University Press. p. 18. ISBN 0-521-43595-1.
↑ Marlan Orvil Scully & Muhammad Suhail Zubairy (1997). Quantum optics. Cambridge UK: Cambridge University Press. pp. 16–19. ISBN 0-521-43595-1.

Quantum beats

Historical overview

V-type and $\Lambda$ -type atoms

Calculation based on semiclassical theory

Calculation based on quantum electrodynamics

Conclusion

See also

References

Further reading

Quantum beats

Historical overview

V-type and -type atoms

Calculation based on semiclassical theory

Calculation based on quantum electrodynamics

Conclusion

See also

References

Further reading

V-type and $\Lambda$ -type atoms