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'''Quantum-optical spectroscopy'''<ref name="KiraKoch2006">Kira, M.; Koch, S. (2006).
"Quantum-optical spectroscopy of semiconductors". ''Physical Review A'' '''73''' (1).
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevA.73.013813 10.1103/PhysRevA.73.013813]. [[ISSN]]
1050-2947.</ref><ref name="KochKira2006">Koch, S. W.; Kira, M.; Khitrova, G.; Gibbs, H. M. (2006).
"Semiconductor excitons in new light". ''Nature Materials'' '''5''' (7): 523–531.
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1038/nmat1658 10.1038/nmat1658]. ISSN 1476-1122.</ref> is a
[[Quantum_optics|quantum-optical]] generalization of [[laser]] [[spectroscopy]] where matter is
excited and probed with a sequence of [[laser pulse]]s. Classically, such pulses are defined by
their spectral and temporal shape as well as phase and amplitude of the [[electromagnetic field]].
Besides these properties of light, the phase-amplitude aspects have [[intrinsic]] quantum
fluctuations that are of central interest in [[quantum optics]]. In ordinary laser spectroscopy,<ref
name="Stenholm2005">Stenholm, S. (2005). ''Foundations of laser spectroscopy''. Dover Pubn. Inc.
[[ISBN]] 978-0486444987.</ref><ref name="Demtroder2008_1">Demtröder, W. (2008). ''Laser
Spectroscopy: Vol. 1: Basic Principles''. Springer. [[ISBN]] 978-3540734154.</ref><ref
name="Demtroder2008_2">Demtröder, W. (2008). ''Laser Spectroscopy: Vol. 2: Experimental
Techniques''. Springer. [[ISBN]] 978-3540749523.</ref> one utilizes only the classical aspects of
laser pulses propagating through matter such as [[atom]]s or [[semiconductor]]s. In quantum-optical
spectroscopy, one additionally utilizes the [[Quantum fluctuation|quantum-optical fluctuations]] of
light to enhance the spectroscopic capabilities by directly shaping and/or detecting the [[quantum fluctuations]] of light. Quantum-optical spectroscopy has applications in controlling and
characterizing quantum dynamics of many-body states because one can directly access a large set of
[[many-body]] states,<ref name="KiraKoch2011">Kira, M.; Koch, S. W.; Smith, R. P.; Hunter, A. E.;
Cundiff, S. T. (2011). "Quantum spectroscopy with Schrödinger-cat states". ''Nature Physics''
'''7''' (10): 799–804. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1038/nphys2091 10.1038/nphys2091]. [[ISSN]]
1745-2473.</ref><ref name="SQOBook">Kira, M.; Koch, S. W. (2011). ''Semiconductor Quantum Optics''.
Cambridge University Press. [[ISBN]] 978-0521875097.</ref> which is not possible in classical
spectroscopy.

==Quantum-optical state injection==

A generic [[electromagnetic field]] can always be expressed in terms of a [[Eigenmode expansion|mode expansion]] where individual components form a [[Complete set of commuting observables|complete set]] of modes. Such modes can be constructed with different methods and they can, e.g., be energy
eigenstate, generic spatial modes, or temporal modes. Once these light [[Normal mode|mode]] are
chosen, their effect on the quantized electromagnetic field can be described by [[Boson]] [[creation and annihilation operators]] <math>\hat{B}^\dagger</math> and <math>\hat{B}</math> for
[[photons]], respectively.<ref name="Walls2008">Walls, D. F.; Milburn, G. J. (2008). ''Quantum
Optics''. Springer. [[ISBN]] 978-3540285731.</ref> The quantum fluctuations of the light field can
be uniquely defined<ref name="KiraKoch2008">Kira, M.; Koch, S. (2008). "Cluster-expansion
representation in quantum optics". ''Physical Review'' A '''78''' (2).
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevA.78.022102 10.1103/PhysRevA.78.022102]. [[ISSN]]
1050-2947.</ref> by the photon [[correlation]]s <math>\Delta\langle\left [ B^ {\dagger}\right]^J\,
B^K\rangle</math> that contain the pure <math>(J+K)</math>-particle correlations as defined with the
[[cluster-expansion approach]]. Using the same [[Second quantization|second-quantization formalism]]
for the matter being studied, typical electronic excitations in matter can be described by
[[Fermion]] operators for electronic excitations and holes, i.e.~electronic vacancies left behind to
the many-body [[ground state]].<ref name="Ashcroft1976">Ashcroft, N. W.; Mermin, N. D. (1976).
''Solid state physics''. Saunders College. [[ISBN]] 978-0030839931.</ref> The corresponding
electron–hole excitations can be described by operators <math>\hat{X}^\dagger</math> and
<math>\hat{X}</math> that create and annihilate an electron–hole pair, respectively.

In several relevant cases, the light–matter interaction can be described using the dipole
interaction<ref name="SQOBook" />

<math>
\hat{H}_{\mathrm{lm}}=-\sum\mathcal{F}\,\hat{B}\hat{X}^{\dagger}+\mathrm{h.c.}\,,
</math>

where the summation is implicitly taken over all possibilities to create an electron–hole pair (the
<math>\hat{X}^\dagger</math> part) via a photon absorption (the <math>\hat{B}</math> part); the
Hamiltonian also contains the [[Hermitian conjugate]] (abbreviated as h.c.) of the terms that are
explicitly written. The [[coupling strength]] between light and matter is defined by
<math>\mathcal{F}</math>.

When the electron–hole pairs are excited resonantly with a single-mode light <math>\hat{B}</math>,
the photon correlations are directly injected into the many-body correlations. More specifically,
the fundamental form of the light–matter interaction inevitably leads to a '''correlation-transfer
relation'''<ref name="KiraKoch2006" /><ref name="SQOBook" />

<math>
\Delta\langle\left[\hat{X}^{\dagger}\right]^J\hat{X}^K\rangle=\eta^{\frac{J+K}{2}}
\Delta\langle\left [ B^ { \dagger
}\right]^JB^K\rangle\,,
</math>

between photons and electron–hole excitations. Strictly speaking, this relation is valid before the
onset of scattering induced by the [[Coulomb]] and [[phonon]] interactions in the solid. Therefore,
it is desirable to use laser pulses that are faster than the dominant scattering processes. This
regime is relatively easy to realize in present-day laser spectroscopy because lasers can already
output [[femtosecond]], or even [[attosecond]], pulses with a high precision in controllability.

==Realization==

Physically, the correlation-transfer relation means that one can {\it directly} inject desired
many-body states simply by adjusting the quantum fluctuations of the light pulse, as long as the
light pulse is short enough. This opens a new possibility for studying properties of distinct
many-body states, once the quantum-optical spectroscopy is realized through controlling the quantum
fluctuations of light sources. For example, a [[coherence (physics)|coherent]]-state laser is described entirely by its
single-particle [[expectation value]] <math>\langle \hat{B} \rangle</math>. Therefore, such
excitation directly injects property <math>\langle \hat{X} \rangle</math> that is polarization
related to electron–hole transitions. To directly excite bound electron–hole pairs, i.e.,
[[excitons]], described by a two-particle correlation <math>\Delta \langle\hat{X}^\dagger \hat{X}
\rangle</math>, or a [[biexciton]] transition <math>\Delta \langle \hat{X}\, \hat{X} \rangle</math>,
one needs to have a source with <math>\Delta\langle \hat{B}^\dagger \hat{B} \rangle</math> or
<math>\Delta \langle \hat{B} \hat{B} \rangle</math> photon correlations, respectively.

To realize quantum-optical spectroscopy, high-intensity light sources with freely adjustable
[[quantum statistics]] are needed which are currently not available. However, one can apply
projective methods<ref name="Sudarshan1963">Sudarshan, E. (1963). "Equivalence of Semiclassical and
Quantum Mechanical Descriptions of Statistical Light Beams". ''Physical Review Letters'' '''10'''
(7): 277–279. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevLett.10.277 10.1103/PhysRevLett.10.277].
[[ISSN]] 0031-9007. </ref><ref name="LobinoKorystov2008">Lobino, M.; Korystov, D.; Kupchak, C.;
Figueroa, E.; Sanders, B. C.; Lvovsky, A. I. (2008). "Complete Characterization of Quantum-Optical
Processes". ''Science'' '''322''' (5901): 563–566.
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1126/science.1162086 10.1126/science.1162086]. [[ISSN]] 0036-8075.</ref>
<ref name="KiraKoch2011" /> to access the quantum–optical response of matter from a set of
classical measurements. Especially, the method presented in Ref.<ref name="KiraKoch2011" /> is
robust in projecting quantum-optical responses of genuine many-body systems. This work has shown
that one can indeed reveal and access many–body properties that remain hidden in classical
spectroscopy. Therefore, the quantum-optical spectroscopy is ideally suited for characterizing and
controlling complicated many-body states in several different systems, ranging from [[molecules]] to
[[semiconductor]]s.

==Relation to semiconductor quantum optics==

Quantum-optical spectroscopy is an important approach in general semiconductor [[quantum optics]].
The capability to discriminate and control many-body states is certainly interesting in extended
semiconductors such as [[quantum well]]s because a typical classical excitation indiscriminately
detects contributions from multiple many-body configurations; With quantum-optical spectroscopy one
can access and control a desired many-body state within an extended semiconductor.<ref
name="SQOBook" /> At the same time, the ideas of quantum-optical spectroscopy can also be useful
when studying simpler systems such as [[quantum dot]]s.

Quantum dots are a semiconductor equivalent to simple atomic systems where most of the first
quantum-optical demonstrations have been measured.<ref name="Walls2008" /> Since quantum dots are
man-made, one can possibly customize them to produce new quantum-optical components for
[[information technology]]. For example in [[Quantum information science|quantum-information science]], one is often interested to have light sources that can output photons on demand or
[[Quantum entanglement|entangled]] photon pairs at specific frequencies. Such sources have already
been demonstrated with quantum dots by controlling their photon emission with various schemes.<ref
name="Michler2000">Michler, P. (2000). "A Quantum Dot Single-Photon Turnstile Device". ''Science''
'''290''' (5500): 2282–2285.
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1126/science.290.5500.2282 10.1126/science.290.5500.2282]. [[ISSN]]
00368075.</ref><ref name="BensonSantori2000">Benson, Oliver; Santori, Charles; Pelton, Matthew;
Yamamoto, Yoshihisa (2000). "Regulated and Entangled Photons from a Single Quantum Dot". ''Physical
Review Letters'' '''84''' (11): 2513–2516.
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevLett.84.2513 10.1103/PhysRevLett.84.2513]. [[ISSN]]
0031-9007.</ref><ref name="StevensonYoung2006">Stevenson, R. M.; Young, R. J.; Atkinson, P.; Cooper,
K.; Ritchie, D. A.; Shields, A. J. (2006). "A semiconductor source of triggered entangled photon
pairs". ''Nature'' '''439''' (7073): 179–182. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1038/nature04446 10.1038/nature04446]. [[ISSN]] 0028-0836.</ref> In the same way, quantum-dot [[lasers]] may exhibit
unusual changes in the conditional probability<ref name="UlrichGies2007">Ulrich, S. M.; Gies, C.;
Ates, S.; Wiersig, J.; Reitzenstein, S.; Hofmann, C.; Löffler, A.; Forchel, A.; Jahnke, F.; Michler,
P. (2007). "Photon Statistics of Semiconductor Microcavity Lasers". ''Physical Review Letters''
'''98''' (4). [[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevLett.98.043906 10.1103/PhysRevLett.98.043906]. [[ISSN]] 0031-9007.</ref> to emit a photon when already one photon
is emitted; this effect can be measured in the so- alled [[correlation function|''g''<sup>2</sup>
correlation]]. One interesting possibility for quantum- ptical spectroscopy is to pump quantum dots
with quantum light to contr l their light emission more precisely.<ref
name="AßmannBayer2011">Aßmann, Marc; Bayer, Manfred (2011). "Nonlinearity sensing via
photon-statistics excitation spectroscopy". ''Physical Review A'' '''84''' (5).
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevA.84.053806 10.1103/PhysRevA.84.053806]. [[ISSN]]
1050-2947.</ref>

Quantum-dot [[Distributed Bragg reflector|microcavity]] investigations have progressed rapidly ever
since the experimental demonstration<ref name="ReithmaierSęk2004">Reithmaier, J. P.; Sęk, G.;
Löffler, A.; Hofmann, C.; Kuhn, S.; Reitzenstein, S.; Keldysh, L. V.; Kulakovskii, V. D.; Reinecke,
T. L.; Forchel, A. (2004). "Strong coupling in a single quantum dot–semiconductor microcavity
system". ''Nature'' '''432''' (7014): 197–200. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1038/nature02969 10.1038/nature02969]. [[ISSN]] 0028-0836.</ref><ref name="YoshieScherer2004">Yoshie, T.; Scherer,
A.; Hendrickson, J.; Khitrova, G.; Gibbs, H. M.; Rupper, G.; Ell, C.; Shchekin, O. B. et al. (2004).
"Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity". ''Nature''
'''432''' (7014): 200–203. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1038/nature03119 10.1038/nature03119].
[[ISSN]] 0028-0836.</ref> of vacuum [[Rabi problem|Rabi splitting]] between a single dot and a
cavity resonance. This regime can be understood on the basis of the [[Jaynes–Cummings model]] while
the semiconductor aspects provide many new physical effects<ref name="FörstnerWeber2003">Förstner,
J.; Weber, C.; Danckwerts, J.; Knorr, A. (2003). "Phonon-Assisted Damping of Rabi Oscillations in
Semiconductor Quantum Dots". ''Physical Review Letters'' '''91''' (12).
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevLett.91.127401 10.1103/PhysRevLett.91.127401]. [[ISSN]]
0031-9007.</ref><ref name="CarmeleRichter2010">Carmele, Alexander; Richter, Marten; Chow, Weng W.;
Knorr, Andreas (2010). "Antibunching of Thermal Radiation by a Room-Temperature Phonon Bath: A
Numerically Solvable Model for a Strongly Interacting Light-Matter-Reservoir System". ''Physical
Review Letters'' '''104''' (15). [[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevLett.104.156801 PhysRevLett.104.156801]. [[ISSN]] 0031-9007.</ref> due to the electronic coupling with the [[lattice vibration]]s.

Nevertheless, the [[quantum Rabi splitting]]—stemming directly from the quantized light
levels—remained elusive because many experiments were monitoring only the intensity of
[[photoluminescence]]. Following the ideology of quantum-optical spectroscopy, Ref.<ref
name="SchneebeliKira2008">Schneebeli, L.; Kira, M.; Koch, S. (2008). "Characterization of Strong
Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics
Spectroscopy". ''Physical Review Letters'' '''101''' (9).
[[Digital object identifier|doi]]:[http://dx.doi.org/10.1103/PhysRevLett.101.097401 10.1103/PhysRevLett.101.097401]. [[ISSN]]
0031-9007.</ref> predicted that quantum-Rabi splitting could be resolved in photon-correlation
measurement even when it becomes smeared out in photoluminescence spectrum. This was experimentally
demonstrated<ref name="ReinhardVolz2011">Reinhard, Andreas; Volz, Thomas; Winger, Martin; Badolato,
Antonio; Hennessy, Kevin J.; Hu, Evelyn L.; Imamoğlu, Ataç (2011). "Strongly correlated photons on a
chip". ''Nature Photonics'' '''6''' (2): 93–96. [[Digital object identifier|doi]]:[http://dx.doi.org/10.1038/nphoton.2011.321 10.1038/nphoton.2011.321]. [[ISSN]] 1749-4885.</ref> by measuring the so-called ''g''<sup>2</sup>
correlations that quantify how regularly the photons are emitted by the quantum dot inside a
microcavity.

==References==
{{Reflist}}

==Related articles==

* [[Semiconductor Bloch equations]]
* [[Semiconductor luminescence equations]]
* [[Quantum optics]]
* [[Cluster-expansion approach]]
* [[Resonance fluorescence]]
* [[Laser spectroscopy]]
* [[Ultrafast laser spectroscopy]]
* [[Photon antibunching]]

==Further reading==

* {{cite book|last1=Jahnke|first1=F.|title=Quantum Optics with Semiconductor
Nanostructures|year=2012|publisher=Woodhead Publishing Ltd.|isbn=978-0857092328}}.
* {{cite book|last1=Kira|first1=M.|last2=Koch|first2=S. W.|title=Semiconductor Quantum
Optics|year=2011|publisher=Cambridge University Press|isbn=978-0521875097}}.
* {{cite book|last1=Walls|first1=D. F.|last2=Milburn|first2=G. J.|title=Quantum
Optics|year=2008|publisher=Springer|isbn=978-3540285731}}.
* {{cite book|last1=Vogel|first1=W.|last2=Welsch|first2=D.-G.|title=Quantum Optics: An
Introduction|year=2006|publisher=Wiley-VCH Verlag GmbH & Co. KGaA|isbn=978-3527405077}}.
* {{cite book|last1=Gerry|first1=C. C.|last2=Knight|first2=P. L.|title=Introductory Quantum
Optics|year=2010|publisher=Cambridge University Press|isbn=978-0521527354}}.
* {{cite book|last1=Scully|first1=M. O.|last2=Zubairy|first2=M. S.|title=Quantum
Optics|year=1997|publisher=Cambridge University Press|isbn=978-0521435956}}.
* {{cite book|last1=Schleich|first1=W. P.|title=Quantum Optics in Phase
Space|year=2001|publisher=Wiley-VCH Verlag GmbH & Co. KGaA|isbn=978-3527294350}}.

[[Category:Spectroscopy]]

Multivariate Behrens–Fisher problem - Revision history

en>ChrisGualtieri: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB