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

V-type and Λ-type atoms

There is a figure in Quantum Optics that describes V-type and Λ-type atoms clearly.

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

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

The derivation below follows the reference Quantum Optics

Calculation based on semiclassical theory

In the semiclassical picture, the state vector of electrons is

$|\psi(t)\rangle=c_aexp(-i\omega_at)|a\rangle+c_bexp(-i\omega_bt)|b\rangle+c_cexp(-i\omega_ct)|c\rangle$.

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 Λ-type atoms,

then each atom has two microscopic oscillating dipoles

$P(t)=\mathcal{P}_{ac}(c_a^*c_c)exp(i\nu_1t)+\mathcal{P}_{bc}(c_b^*c_c)exp(i\nu_2t)+c.c.$ for V-type, when ν₁ = ω_a − ω_c,ν₂ = ω_b − ω_c,

$P(t)=\mathcal{P}_{ab}(c_a^*c_b)exp(i\nu_1t)+\mathcal{P}_{ac}(c_a^*c_c)exp(i\nu_2t)+c.c.$ for Λ-type, when ν₁ = ω_a − ω_b,ν₂ = ω_a − ω_c.

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

$E^{(+)}=\mathcal{E}_1exp(-i\nu_1t)+\mathcal{E}_2exp(-i\nu_2t)$,

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

$|E^{(+)}|^2=|\mathcal{E}_1|^2+|\mathcal{E}_2|^2+\lbrace\mathcal{E}_1^*\mathcal{E}_2exp\lbrack i(\nu_1-\nu_2)t\rbrack+c.c.\rbrace$.

Calculation based on quantum electrodynamics

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

Let

$E_n^{(+)}=a_nexp(-i\nu_nt)$ is a creation operator and

$E_n^{(-)}=a_n^\dagger exp(i\nu_nt)$ is an annihilation 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 Λ-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 Λ-type.

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

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

Conclusion

As a result of calculation, V-type atoms have quantum beats but Λ-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 ν₁ and ν₂. Since both transitions decayed to the same state, one cannot determine along which path each decayed, similar to Young's double-slit experiment. However, Λ-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.

See also

• Quantum electrodynamics
• Double-slit experiment

Further reading

• Marlan Orvil Scully & Muhammad Suhail Zubairy (1997). Quantum optics. Cambridge UK: Cambridge University Press. p. 541. ISBN 0521435951. http://books.google.com/books?id=20ISsQCKKmQC&pg=PA430&sig=d5TzC9UTl7CGU3PIJiCV0c0M6HU#PPA541,M1.