Physics:Quantum beats: Difference between revisions

From HandWiki Test
← Older editNewer edit →
Maintenance script (talk | contribs)
5,209 edits
Normalize Book I Quantum page structure
Clean Book I red links, intro, and image slots
Line 1: Line 1:
{{Short description|Quantum Collection topic on Quantum beats}}
{{Short description|Quantum Collection topic on Quantum beats}}


{{Quantum book backlink|Atomic and spectroscopy}}
{{Quantum book backlink|Atomic and spectroscopy}}
Line 10: Line 10:


<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
In [[HandWiki:Physics|physics]], '''quantum beats''' are simple examples of phenomena that cannot be described by semiclassical theory, but can be described by fully quantized calculation, especially [[Physics:Quantum electrodynamics|quantum electrodynamics]]. In semiclassical theory (SCT), there is an interference or [[Physics:Beat (acoustics)|beat note]] term for both V-type and <math>\Lambda</math>-type atoms.{{Clarify|date=March 2009}} However, in the quantum electrodynamic (QED) calculation, V-type atoms have a beat term but <math>\Lambda</math>-types do not. This is strong evidence in support of [[Physics:Quantum electrodynamics|quantum electrodynamics]].
'''beats''' is a Book I topic in the Quantum Collection. 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. 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.
</div>
</div>


<div style="width:300px;">
<div style="width:300px;">
[[File:not found]]
[[File:Quantum_book1_beats_yellow.png]]
</div>
</div>


Line 20: Line 20:


== Historical overview ==
== Historical overview ==
The observation of quantum beats was first reported by A.T. Forrester, R.A. Gudmundsen and P.O. Johnson in 1955,<ref>{{cite journal | url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.99.1691 | doi=10.1103/PhysRev.99.1691 | title=Photoelectric Mixing of Incoherent Light | date=1955 | last1=Forrester | first1=A. Theodore | last2=Gudmundsen | first2=Richard A. | last3=Johnson | first3=Philip O. | journal=Physical Review | volume=99 | issue=6 | pages=1691–1700 | bibcode=1955PhRv...99.1691F | url-access=subscription }}</ref> in an experiment that was performed on the basis of an earlier proposal by A.T. Forrester, W.E. Parkins and E. Gerjuoy.<ref>A.T. Forrester, W.E. Parkins, E. Gerjuoy: ''On the possibility of observing beat frequencies between lines in the visible spectrum'', Physical Review, vol.&nbsp;72, pp.&nbsp;241–243, 1947</ref> 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|spectral line]] into several components in the presence of a [[Magnetic field|magnetic field]] due to the [[Physics:Zeeman effect|Zeeman effect]]. These light components were mixed at a [[Physics:Photoelectric effect|photoelectric]] surface, and the electrons emitted from that surface then excited a [[Physics:Microwave cavity|microwave cavity]], which allowed the output signal to be measured in dependence on the magnetic field.<ref>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.&nbsp;83–102, [https://books.google.com/books?id=3U2HSMHsouMC&pg=PA97 p. 97]</ref><ref>Paul Hartman: ''A Memoir on The Physical Review: A History of the First Hundred Years'', Springer, 2008, {{ISBN|978-1-56396-282-0}}, [https://books.google.com/books?id=tPxu74KtAeQC&pg=PA193 p. 193]</ref>
The observation of quantum beats was first reported by A.T. Forrester, R.A. Gudmundsen and P.O. Johnson in 1955,<ref>{{cite journal | url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.99.1691 | doi=10.1103/PhysRev.99.1691 | title=Photoelectric Mixing of Incoherent Light | date=1955 | last1=Forrester | first1=A. Theodore | last2=Gudmundsen | first2=Richard A. | last3=Johnson | first3=Philip O. | journal=Physical Review | volume=99 | issue=6 | pages=1691–1700 | bibcode=1955PhRv...99.1691F | url-access=subscription }}</ref> in an experiment that was performed on the basis of an earlier proposal by A.T. Forrester, W.E. Parkins and E. Gerjuoy.<ref>A.T. Forrester, W.E. Parkins, E. Gerjuoy: ''On the possibility of observing beat frequencies between lines in the visible spectrum'', Physical Review, vol.&nbsp;72, pp.&nbsp;241–243, 1947</ref> 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.<ref>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.&nbsp;83–102, [https://books.google.com/books?id=3U2HSMHsouMC&pg=PA97 p. 97]</ref><ref>Paul Hartman: ''A Memoir on The Physical Review: A History of the First Hundred Years'', Springer, 2008, {{ISBN|978-1-56396-282-0}}, [https://books.google.com/books?id=tPxu74KtAeQC&pg=PA193 p. 193]</ref>


Since the invention of the [[Physics:Laser|laser]], quantum beats can be demonstrated by using light originating from two different laser sources. In 2017 quantum beats in single [[Physics:Photon|photon]] emission from the atomic collective excitation have been observed.<ref>{{Cite journal|last1=Whiting|first1=D. J.|last2=Šibalić|first2=N.|last3=Keaveney|first3=J.|last4=Adams|first4=C. S.|last5=Hughes|first5=I. G.|date=2017-06-22|title=Single-Photon Interference due to Motion in an Atomic Collective Excitation|journal=Physical Review Letters|volume=118|issue=25|article-number=253601|doi=10.1103/PhysRevLett.118.253601|pmid=28696754|arxiv=1612.05467|bibcode=2017PhRvL.118y3601W |s2cid=5126428}}</ref> Observed collective beats were not due to [[Superposition principle|superposition]] of excitation between two different [[Physics:Energy level|energy level]]s of the atoms, as in usual single-atom quantum beats in <math>V</math>-type atoms.<ref name=":0">{{Citation|last=Haroche|first=S.|chapter=Quantum beats and time-resolved fluorescence spectroscopy|date=1976|pages=253–313|publisher=Springer Berlin Heidelberg|isbn=978-3-540-07719-0|doi=10.1007/3540077197_23|title=High-Resolution Laser Spectroscopy|volume=13|series=Topics in Applied Physics}}</ref> 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,<ref name=":0" /> that acquire relative phase due to [[Doppler effect]].
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.<ref>{{Cite journal|last1=Whiting|first1=D. J.|last2=Šibalić|first2=N.|last3=Keaveney|first3=J.|last4=Adams|first4=C. S.|last5=Hughes|first5=I. G.|date=2017-06-22|title=Single-Photon Interference due to Motion in an Atomic Collective Excitation|journal=Physical Review Letters|volume=118|issue=25|article-number=253601|doi=10.1103/PhysRevLett.118.253601|pmid=28696754|arxiv=1612.05467|bibcode=2017PhRvL.118y3601W |s2cid=5126428}}</ref> 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 <math>V</math>-type atoms.<ref name=":0">{{Citation|last=Haroche|first=S.|chapter=Quantum beats and time-resolved fluorescence spectroscopy|date=1976|pages=253–313|publisher=Springer Berlin Heidelberg|isbn=978-3-540-07719-0|doi=10.1007/3540077197_23|title=High-Resolution Laser Spectroscopy|volume=13|series=Topics in Applied Physics}}</ref> 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,<ref name=":0" /> that acquire relative phase due to Doppler effect.


== V-type and <math>\Lambda</math>-type atoms ==
== V-type and <math>\Lambda</math>-type atoms ==
Line 36: Line 36:
In the semiclassical picture, the state vector of electrons is
In the semiclassical picture, the state vector of electrons is
:<math>|\psi(t)\rangle=c_aexp(-i\omega_at)|a\rangle+c_bexp(-i\omega_bt)|b\rangle+c_cexp(-i\omega_ct)|c\rangle</math>.
:<math>|\psi(t)\rangle=c_aexp(-i\omega_at)|a\rangle+c_bexp(-i\omega_bt)|b\rangle+c_cexp(-i\omega_ct)|c\rangle</math>.
If the nonvanishing [[Physics:Dipole|dipole]] matrix elements are described by
If the nonvanishing dipole matrix elements are described by
:<math>\mathcal{P}_{ac}=e\langle a|r|c\rangle, \mathcal{P}_{bc}=e\langle b|r|c\rangle</math> for V-type atoms,
:<math>\mathcal{P}_{ac}=e\langle a|r|c\rangle, \mathcal{P}_{bc}=e\langle b|r|c\rangle</math> for V-type atoms,
:<math>\mathcal{P}_{ab}=e\langle a|r|b\rangle, \mathcal{P}_{ac}=e\langle a|r|c\rangle</math> for <math>\Lambda</math>-type atoms,
:<math>\mathcal{P}_{ab}=e\langle a|r|b\rangle, \mathcal{P}_{ac}=e\langle a|r|c\rangle</math> for <math>\Lambda</math>-type atoms,
Line 44: Line 44:
In the semiclassical picture, the field radiated will be a sum of these two terms
In the semiclassical picture, the field radiated will be a sum of these two terms
:<math>E^{(+)}=\mathcal{E}_1exp(-i\nu_1t)+\mathcal{E}_2exp(-i\nu_2t)</math>,
:<math>E^{(+)}=\mathcal{E}_1exp(-i\nu_1t)+\mathcal{E}_2exp(-i\nu_2t)</math>,
so it is clear that there is an interference or ''beat note'' term in a [[Square-law detector|square-law detector]]
so it is clear that there is an interference or ''beat note'' term in a square-law detector
:<math>|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</math>.
:<math>|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</math>.


== Calculation based on quantum electrodynamics ==
== Calculation based on quantum electrodynamics ==
For quantum electrodynamical calculation, we should introduce the creation and annihilation operators from [[Physics:Second quantization|second quantization]] of [[Physics:Quantum mechanics|quantum mechanics]].
For quantum electrodynamical calculation, we should introduce the creation and annihilation operators from second quantization of [[Physics:Quantum mechanics|quantum mechanics]].


Let
Let
Line 63: Line 63:
:<math>\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</math> for V-type and
:<math>\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</math> for V-type and
:<math>\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</math> for <math>\Lambda</math>-type.
:<math>\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</math> for <math>\Lambda</math>-type.
By [[Orthogonality|orthogonality]] of eigenstates, however <math>\langle c|c\rangle=1</math> and <math>\langle b|c\rangle=0</math>.
By orthogonality of eigenstates, however <math>\langle c|c\rangle=1</math> and <math>\langle b|c\rangle=0</math>.


Therefore, there is a beat note term for V-type atoms, but not for <math>\Lambda</math>-type atoms.
Therefore, there is a beat note term for V-type atoms, but not for <math>\Lambda</math>-type atoms.


== Conclusion ==
== Conclusion ==
As a result of calculation, V-type atoms have quantum beats but <math>\Lambda</math>-type atoms do not. This difference is caused by quantum mechanical [[Physics:Uncertainty|uncertainty]]. A V-type atom decays to state <math>|c\rangle</math> via the emission with <math>\nu_1</math> and <math>\nu_2</math>. Since both transitions decayed to the same state, one cannot determine along ''which path'' each decayed, similar to Young's [[Physics:Double-slit experiment|double-slit experiment]]. However,  <math>\Lambda</math>-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.
As a result of calculation, V-type atoms have quantum beats but <math>\Lambda</math>-type atoms do not. This difference is caused by quantum mechanical uncertainty. A V-type atom decays to state <math>|c\rangle</math> via the emission with <math>\nu_1</math> and <math>\nu_2</math>. Since both transitions decayed to the same state, one cannot determine along ''which path'' each decayed, similar to Young's double-slit experiment. However,  <math>\Lambda</math>-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 [[Physics:Quantum mechanics|quantum mechanics]], the [[Uncertainty principle|uncertainty principle]]. Quantum beats phenomena are good examples of such that can be described by QED but not by SCT.
The calculation by QED is correct in accordance with the most fundamental principle of [[Physics:Quantum mechanics|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 ==
== See also ==
*[[Physics:Quantum electrodynamics|Quantum electrodynamics]]
*[[Physics:Quantum electrodynamics|Quantum electrodynamics]]
*[[Physics:Double-slit experiment|Double-slit experiment]]
*Double-slit experiment
*[[Physics:Coherent effects in semiconductor optics#Quantum beats involving excitons and exciton complexes|Quantum beats in semiconductor optics]]
*[[Physics:Coherent effects in semiconductor optics#Quantum beats involving excitons and exciton complexes|Quantum beats in semiconductor optics]]


Revision as of 08:13, 20 May 2026



← Back to Atomic and spectroscopy Book I

beats is a Book I topic in the Quantum Collection. 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. 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.

Historical overview

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-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 to Doppler effect.

V-type and Λ-type atoms

There is a figure in Quantum Optics[7] that describes V-type and Λ-type atoms clearly.

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

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

The derivation below follows the reference Quantum Optics.[7]

Calculation based on semiclassical theory

In the semiclassical picture, the state vector of electrons is

|ψ(t)⟩=caexp(−iωat)|a⟩+cbexp(−iωbt)|b⟩+ccexp(−iωct)|c⟩.

If the nonvanishing dipole matrix elements are described by

𝒫ac=e⟨a|r|c⟩,𝒫bc=e⟨b|r|c⟩ for V-type atoms,
𝒫ab=e⟨a|r|b⟩,𝒫ac=e⟨a|r|c⟩ for Λ-type atoms,

then each atom has two microscopic oscillating dipoles

P(t)=𝒫ac(ca*cc)exp(iν1t)+𝒫bc(cb*cc)exp(iν2t)+c.c. for V-type, when ν1=ωa−ωc,ν2=ωb−ωc,
P(t)=𝒫ab(ca*cb)exp(iν1t)+𝒫ac(ca*cc)exp(iν2t)+c.c. for Λ-type, when ν1=ωa−ωb,ν2=ωa−ωc.

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

E(+)=ℰ1exp(−iν1t)+ℰ2exp(−iν2t),

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

|E(+)|2=|ℰ1|2+|ℰ2|2+{ℰ1*ℰ2exp[i(ν1−ν2)t]+c.c.}.

Calculation based on quantum electrodynamics

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

Let

En(+)=anexp(−iνnt) is an annihilation operator and
En(−)=an†exp(iνnt) is a creation operator.

Then the beat note becomes

⟨ψV(t)|E1(−)(t)E2(+)(t)|ψV(t)⟩ for V-type and
⟨ψΛ(t)|E1(−)(t)E2(+)(t)|ψΛ(t)⟩ for Λ-type,

when the state vector for each type is

|ψV(t)⟩=∑i=a,b,cci|i,0⟩+c1|c,1ν1⟩+c2|c,1ν2⟩ and
|ψΛ(t)⟩=∑i=a,b,cci′|i,0⟩+c1′|b,1ν1⟩+c2′|c,1ν2⟩.

The beat note term becomes

⟨ψV(t)|E1(−)(t)E2(+)(t)|ψV(t)⟩=κ⟨1ν10ν2|a1†a2|0ν11ν2⟩exp[i(ν1−ν2)t]⟨c|c⟩=κexp[i(ν1−ν2)t]⟨c|c⟩ for V-type and
⟨ψΛ(t)|E1(−)(t)E2(+)(t)|ψΛ(t)⟩=κ′⟨1ν10ν2|a1†a2|0ν11ν2⟩exp[i(ν1−ν2)t]⟨b|c⟩=κ′exp[i(ν1−ν2)t]⟨b|c⟩ for Λ-type.

By orthogonality of eigenstates, however ⟨c|c⟩=1 and ⟨b|c⟩=0.

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⟩ 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

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ Forrester, A. Theodore; Gudmundsen, Richard A.; Johnson, Philip O. (1955). "Photoelectric Mixing of Incoherent Light". Physical Review 99 (6): 1691–1700. doi:10.1103/PhysRev.99.1691. Bibcode: 1955PhRv...99.1691F. https://journals.aps.org/pr/abstract/10.1103/PhysRev.99.1691. 
  2. ↑ 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
  3. ↑ 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
  4. ↑ 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
  5. ↑ Whiting, D. J.; Šibalić, N.; Keaveney, J.; Adams, C. S.; Hughes, I. G. (2017-06-22). "Single-Photon Interference due to Motion in an Atomic Collective Excitation". Physical Review Letters 118 (25). doi:10.1103/PhysRevLett.118.253601. PMID 28696754. Bibcode: 2017PhRvL.118y3601W. 
  6. ↑ 6.0 6.1 Haroche, S. (1976), "Quantum beats and time-resolved fluorescence spectroscopy", High-Resolution Laser Spectroscopy, Topics in Applied Physics, 13, Springer Berlin Heidelberg, pp. 253–313, doi:10.1007/3540077197_23, ISBN 978-3-540-07719-0 
  7. ↑ 7.0 7.1 Marlan Orvil Scully & Muhammad Suhail Zubairy (1997). Quantum optics. Cambridge UK: Cambridge University Press. p. 18. ISBN 978-0-521-43595-6. https://books.google.com/books?id=20ISsQCKKmQC&pg=PA430. 

Further reading


Author: Harold Foppele


Source attribution: Physics:Quantum beats

Retrieved from "https://handwiki.scholarlywiki.org/index.php?title=Physics:Quantum_beats&oldid=5076"