coherent control

The initial idea was to control the outcome of chemical reactions. Two approaches were pursued:

• in the time domain, a "pump-dump" scheme where the control is the time delay between pulses{{cite journal | last1=Tannor | first1=David J. | last2=Rice | first2=Stuart A. | title=Control of selectivity of chemical reaction via control of wave packet evolution | journal=The Journal of Chemical Physics| volume=83 | issue=10 | date=1985-11-15 | issn=0021-9606 | doi=10.1063/1.449767 | pages=5013–5018}}{{cite journal | last1=Tannor | first1=David J. | last2=Kosloff | first2=Ronnie | last3=Rice | first3=Stuart A. | title=Coherent pulse sequence induced control of selectivity of reactions: Exact quantum mechanical calculations | journal=The Journal of Chemical Physics| volume=85 | issue=10 | date=1986-11-15 | issn=0021-9606 | doi=10.1063/1.451542 | pages=5805–5820| s2cid=94455480 }}
• in the frequency domain, interfering pathways controlled by one and three photons.{{cite journal | last1=Brumer | first1=Paul | last2=Shapiro | first2=Moshe | title=Control of unimolecular reactions using coherent light | journal=Chemical Physics Letters| volume=126 | issue=6 | year=1986 | issn=0009-2614 | doi=10.1016/s0009-2614(86)80171-3 | pages=541–546}}

The two basic methods eventually merged with the introduction of optimal control theory.{{cite journal | last1=Peirce | first1=Anthony P. | last2=Dahleh | first2=Mohammed A. | last3=Rabitz | first3=Herschel | title=Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications | journal=Physical Review A| volume=37 | issue=12 | date=1988-06-01 | issn=0556-2791 | doi=10.1103/physreva.37.4950 | pmid=9899641 | pages=4950–4964}}{{cite journal | last1=Kosloff | first1=R. | last2=Rice | first2=S.A. | last3=Gaspard | first3=P. | last4=Tersigni | first4=S. | last5=Tannor | first5=D.J. | title=Wavepacket dancing: Achieving chemical selectivity by shaping light pulses | journal=Chemical Physics| volume=139 | issue=1 | year=1989 | issn=0301-0104 | doi=10.1016/0301-0104(89)90012-8 | pages=201–220}}

Experimental realizations soon followed in the time domain{{cite journal | last1=Baumert | first1=T. | last2=Engel | first2=V. | last3=Meier | first3=C. | last4=Gerber | first4=G. | title=High laser field effects in multiphoton ionization of Na₂. Experiment and quantum calculations | journal=Chemical Physics Letters| volume=200 | issue=5 | year=1992 | issn=0009-2614 | doi=10.1016/0009-2614(92)80080-u | pages=488–494}} and in the frequency domain.{{cite journal | last1=Zhu | first1=L. | last2=Kleiman | first2=V. | last3=Li | first3=X. | last4=Lu | first4=S. P. | last5=Trentelman | first5=K. | last6=Gordon | first6=R. J. | title=Coherent Laser Control of the Product Distribution Obtained in the Photoexcitation of HI | journal=Science| volume=270 | issue=5233 | date=1995-10-06 | issn=0036-8075 | doi=10.1126/science.270.5233.77 | pages=77–80| s2cid=98705974 }} Two interlinked developments accelerated the field of coherent control: experimentally, it was the development of pulse shaping by a spatial light modulator{{cite journal | last=Weiner | first=A. M. | title=Femtosecond pulse shaping using spatial light modulators | journal=Review of Scientific Instruments| volume=71 | issue=5 | year=2000 | issn=0034-6748 | doi=10.1063/1.1150614 | pages=1929–1960|access-date=2010-07-06 |archive-url=https://wayback.archive-it.org/all/20070417213918/http://cobweb.ecn.purdue.edu/~fsoptics/articles/Femtosecond_pulse_shaping-Weiner.pdf |archive-date=17 April 2007 |url-status=live|url=http://cobweb.ecn.purdue.edu/~fsoptics/articles/Femtosecond_pulse_shaping-Weiner.pdf }}

Liquid Crystal Optically Addressed Spatial Light Modulator, [http://www-optique.enst-bretagne.fr/18_LCOASLM.htm] {{Webarchive|url=https://web.archive.org/web/20120204234540/http://www-optique.enst-bretagne.fr/18_LCOASLM.htm|date=2012-02-04}}

• Slinger, C.; Cameron, C.; Stanley, M.; [http://www.macs.hw.ac.uk/modules/F24VS2/Resources/Holography.pdf "Computer-Generated Holography as a Generic Display Technology"] {{Webarchive|url=https://web.archive.org/web/20110927101623/http://www.macs.hw.ac.uk/modules/F24VS2/Resources/Holography.pdf|date=2011-09-27}}, IEEE Computer, Volume 38, Issue 8, Aug. 2005, pp 46–53

and its employment in coherent control.{{cite journal | last1=Kawashima | first1=Hitoshi | last2=Wefers | first2=Marc M. | last3=Nelson | first3=Keith A. | title=Femtosecond Pulse Shaping, Multiple-Pulse Spectroscopy, and Optical Control | journal=Annual Review of Physical Chemistry| volume=46 | issue=1 | year=1995 | issn=0066-426X | doi=10.1146/annurev.pc.46.100195.003211 | pmid=24341370 | pages=627–656}} The second development was the idea of automatic feedback control{{cite journal | last1=Judson | first1=Richard S. | last2=Rabitz | first2=Herschel | title=Teaching lasers to control molecules | journal=Physical Review Letters| volume=68 | issue=10 | date=1992-03-09 | issn=0031-9007 | doi=10.1103/physrevlett.68.1500 | pmid=10045147 | pages=1500–1503}} and its experimental realization.{{cite journal | last=Assion | first=A. | title=Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses | journal=Science| volume=282 | issue=5390 | date=1998-10-30 | doi=10.1126/science.282.5390.919 | pmid=9794756 | pages=919–922}}{{cite journal | last1=Brif | first1=Constantin | last2=Chakrabarti | first2=Raj | last3=Rabitz | first3=Herschel | title=Control of quantum phenomena: past, present and future | journal=New Journal of Physics| volume=12 | issue=7 | date=2010-07-08 | issn=1367-2630 | doi=10.1088/1367-2630/12/7/075008 | page=075008|doi-access=free| arxiv=0912.5121 }}

Coherent control aims to steer a quantum system from an initial state to a target state via an external field. For given initial and final (target) states, the coherent control is termed state-to-state control. A generalization is steering simultaneously an arbitrary set of initial pure states to an arbitrary set of final states i.e. controlling a unitary transformation. Such an application sets the foundation for a quantum gate operation.{{cite journal | last1=Tesch | first1=Carmen M. | last2=Kurtz | first2=Lukas | last3=de Vivie-Riedle | first3=Regina | title=Applying optimal control theory for elements of quantum computation in molecular systems | journal=Chemical Physics Letters| volume=343 | issue=5–6 | year=2001 | issn=0009-2614 | doi=10.1016/s0009-2614(01)00748-5 | pages=633–641}}{{cite journal | last1=Palao | first1=José P. | last2=Kosloff | first2=Ronnie | title=Quantum Computing by an Optimal Control Algorithm for Unitary Transformations | journal=Physical Review Letters| volume=89 | issue=18 | date=2002-10-14 | issn=0031-9007 | doi=10.1103/physrevlett.89.188301 | pmid=12398642 | arxiv=quant-ph/0204101 | page=188301| s2cid=9237548 }}{{cite journal | last1=Rabitz | first1=Herschel | last2=Hsieh | first2=Michael | last3=Rosenthal | first3=Carey | title=Landscape for optimal control of quantum-mechanical unitary transformations | journal=Physical Review A| volume=72 | issue=5 | date=2005-11-30 | issn=1050-2947 | doi=10.1103/physreva.72.052337 | page=052337}}

Controllability of a closed quantum system has been addressed by Tarn and Clark.{{cite journal | last1=Huang | first1=Garng M. | last2=Tarn | first2=T. J. | last3=Clark | first3=John W. | title=On the controllability of quantum-mechanical systems | journal=Journal of Mathematical Physics| volume=24 | issue=11 | year=1983 | issn=0022-2488 | doi=10.1063/1.525634 | pages=2608–2618}} Their theorem based in control theory states that for a finite-dimensional, closed-quantum system, the system is completely controllable, i.e. an arbitrary unitary transformation of the system can be realized by an appropriate application of the controls{{cite journal | last1=Ramakrishna | first1=Viswanath | last2=Salapaka | first2=Murti V. | last3=Dahleh | first3=Mohammed | last4=Rabitz | first4=Herschel | last5=Peirce | first5=Anthony | title=Controllability of molecular systems | journal=Physical Review A| volume=51 | issue=2 | date=1995-02-01 | issn=1050-2947 | doi=10.1103/physreva.51.960 | pmid=9911672 | pages=960–966}} if the control operators and the unperturbed Hamiltonian generate the Lie algebra of all Hermitian operators. Complete controllability implies state-to-state controllability.

The computational task of finding a control field for a particular state-to-state transformation is difficult and becomes more difficult with the increase in the size of the system. This task is in the class of hard inversion problems of high computational complexity. The algorithmic task of finding the field that generates a unitary transformation scales factorial more difficult with the size of the system. This is because a larger number of state-to-state control fields have to be found without interfering with the other control fields. It has been shown that solving general quantum optimal control problems is equivalent to solving Diophantine equations. It therefore follows from the negative answer to Hilbert's tenth problem that quantum optimal controllability is in general undecidable.{{Cite journal|last1=Bondar|first1=Denys I.|last2=Pechen|first2=Alexander N.|date=2020-01-27|title=Uncomputability and complexity of quantum control|journal=Scientific Reports|language=en|volume=10|issue=1|pages=1195|doi=10.1038/s41598-019-56804-1|pmid=31988295 | pmc=6985236 |arxiv=1907.10082|issn=2045-2322}}

Once constraints are imposed controllability can be degraded. For example, what is the minimum time required to achieve a control objective?{{cite journal | last1=Caneva | first1=T. | last2=Murphy | first2=M. | last3=Calarco | first3=T. | last4=Fazio | first4=R. | last5=Montangero | first5=S. | last6=Giovannetti | first6=V. | last7=Santoro | first7=G. E. | title=Optimal Control at the Quantum Speed Limit | journal=Physical Review Letters | volume=103 | issue=24 | date=2009-12-07 | issn=0031-9007 | doi=10.1103/physrevlett.103.240501 | pmid=20366188 | page=240501|arxiv=0902.4193| s2cid=43509791 }} This is termed the "quantum speed limit". The speed limit can be calculated by quantizing Ulam's control conjecture.{{cite journal | last1=Gruebele | first1=M. | last2=Wolynes | first2=P. G. | title=Quantizing Ulam's control conjecture | journal=Physical Review Letters | volume=99 | issue=6 | date=2007-08-06 | issn=0031-9007 | doi=10.1103/PhysRevLett.99.060201 | page=060201| pmid=17930806 }}

The constructive approach uses a set of predetermined control fields for which the control outcome can be inferred.

The pump dump scheme in the time domain and the three vs one photon interference scheme in the frequency domain are prime examples. Another constructive approach is based on adiabatic ideas. The most well studied method is Stimulated raman adiabatic passage STIRAP {{cite journal | last1=Unanyan | first1=R. | last2=Fleischhauer | first2=M. | last3=Shore | first3=B.W. | last4=Bergmann | first4=K. | title=Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states | journal=Optics Communications| volume=155 | issue=1–3 | year=1998 | issn=0030-4018 | doi=10.1016/s0030-4018(98)00358-7 | pages=144–154}} which employs an auxiliary state to achieve complete state-to-state population transfer.

One of the most prolific generic pulse shapes is a chirped pulse a pulse with a varying frequency in time.{{cite journal | last1=Ruhman | first1=S. | last2=Kosloff | first2=R. | title=Application of chirped ultrashort pulses for generating large-amplitude ground-state vibrational coherence: a computer simulation | journal=Journal of the Optical Society of America B| volume=7 | issue=8 | date=1990-08-01 | issn=0740-3224 | doi=10.1364/josab.7.001748 | pages=1748–1752}}{{cite journal | last1=Cerullo | first1=G. | last2=Bardeen | first2=C.J. | last3=Wang | first3=Q. | last4=Shank | first4=C.V. | title=High-power femtosecond chirped pulse excitation of molecules in solution | journal=Chemical Physics Letters| volume=262 | issue=3–4 | year=1996 | issn=0009-2614 | doi=10.1016/0009-2614(96)01092-5 | pages=362–368}}

Optimal control as applied in coherent control seeks the optimal control field for steering a quantum system to its objective. For state-to-state control the objective is defined as the maximum overlap at the final time T with the state $|\backslash phi\_f\; \backslash rangle$:

:$J=\; |\backslash langle\backslash psi\; (T)|\; \backslash phi\_f\backslash rangle|^2$

where the initial state is $|\; \backslash phi\_i\backslash rangle$. The time dependent control Hamiltonian has the typical form:

:$H(t)\; =\; H\_0\; +\; \backslash mu\; \backslash cdot\; \backslash epsilon(t)$

where $\backslash epsilon\; (t)$ is the control field. Optimal control solves for the optimal field $\backslash epsilon(t)$using the calculus of variations introducing Lagrange multipliers. A new objective functional is defined

:$$

J' = J + \int_0^{T} \langle \chi (t)|\left( i \frac{\partial}{\partial t}-H(\epsilon(t))\right)|\psi(t) \rangle dt +\lambda \int_o^T |\epsilon(t)|^2 dt

where $|\backslash chi\backslash rangle$ is a wave function like Lagrange multiplier and the $\backslash lambda$ parameter regulates the integral intensity.

Variation of $J\text{'}$ with respect to $\backslash delta\; \backslash epsilon$ and $\backslash delta\; \backslash psi$ leads to two coupled Schrödinger equations. A forward equation for $|\backslash psi\backslash rangle$ with initial condition $|\backslash psi(0)\backslash rangle=|\backslash phi\_i\backslash rangle$and a backward equation for the Lagrange multiplier $|\backslash chi\backslash rangle$ with final condition $|\backslash chi(T)\backslash rangle=|\backslash phi\_f\backslash rangle$. Finding a solution requires an iterative approach. Different algorithms have been applied for obtaining the control field such as the Krotov method.{{cite journal | last1=Somlói | first1=József | last2=Kazakov | first2=Vladimir A. | last3=Tannor | first3=David J. | title=Controlled dissociation of I₂ via optical transitions between the X and B electronic states | journal=Chemical Physics| volume=172 | issue=1 | year=1993 | issn=0301-0104 | doi=10.1016/0301-0104(93)80108-l | pages=85–98| doi-access=free }}

A local in time alternative method has been developed,{{cite journal | last1=Kosloff | first1=Ronnie | last2=Hammerich | first2=Audrey Dell | last3=Tannor | first3=David | title=Excitation without demolition: Radiative excitation of ground-surface vibration by impulsive stimulated Raman scattering with damage control | journal=Physical Review Letters| volume=69 | issue=15 | date=1992-10-12 | issn=0031-9007 | doi=10.1103/physrevlett.69.2172 | pmid=10046417 | pages=2172–2175}} where at each time step, the field is calculated to direct the state to the target. A related method has been called tracking {{cite journal | last1=Chen | first1=Yu | last2=Gross | first2=Peter | last3=Ramakrishna | first3=Viswanath | last4=Rabitz | first4=Herschel | last5=Mease | first5=Kenneth | title=Competitive tracking of molecular objectives described by quantum mechanics | journal=The Journal of Chemical Physics| volume=102 | issue=20 | date=1995-05-22 | issn=0021-9606 | doi=10.1063/1.468998 | pages=8001–8010}}

Some applications of coherent control are

• Unimolecular and bimolecular chemical reactions.{{cite journal | last1=Levis | first1=R. J. | last2=Rabitz | first2=H. A. | title=Closing the Loop on Bond Selective Chemistry Using Tailored Strong Field Laser Pulses | journal=The Journal of Physical Chemistry A| volume=106 | issue=27 | year=2002 | issn=1089-5639 | doi=10.1021/jp0134906 | pages=6427–6444}}{{cite journal | last1=Dantus | first1=Marcos | last2=Lozovoy | first2=Vadim V. | title=Experimental Coherent Laser Control of Physicochemical Processes | journal=Chemical Reviews| volume=104 | issue=4 | year=2004 | issn=0009-2665 | doi=10.1021/cr020668r | pmid=15080713 | pages=1813–1860}}{{cite journal | last1=Levin | first1=Liat | last2=Skomorowski | first2=Wojciech | last3=Rybak | first3=Leonid | last4=Kosloff | first4=Ronnie | last5=Koch | first5=Christiane P. |author-link=5| last6=Amitay | first6=Zohar | title=Coherent Control of Bond Making | journal=Physical Review Letters | volume=114 | issue=23 | date=2015-06-10 | issn=0031-9007 | doi=10.1103/physrevlett.114.233003 | pmid=26196798 | page=233003|arxiv=1411.1542| s2cid=32145743 }}
• The biological photoisomerization of Retinal.{{cite journal | last=Prokhorenko | first=V. I. | title=Coherent Control of Retinal Isomerization in Bacteriorhodopsin | journal=Science| volume=313 | issue=5791 | date=2006-09-01 | issn=0036-8075 | doi=10.1126/science.1130747 | pmid=16946063 | pages=1257–1261| s2cid=8804783 }}{{cite journal | last1=Wohlleben | first1=Wendel | last2=Buckup | first2=Tiago | last3=Herek | first3=Jennifer L. | last4=Motzkus | first4=Marcus | title=Coherent Control for Spectroscopy and Manipulation of Biological Dynamics | journal=ChemPhysChem| volume=6 | issue=5 | date=2005-05-13 | issn=1439-4235 | doi=10.1002/cphc.200400414 | pmid=15884067 | pages=850–857}}
• The field of nuclear magnetic resonance.{{cite journal | last1=Khaneja | first1=Navin | last2=Reiss | first2=Timo | last3=Kehlet | first3=Cindie | last4=Schulte-Herbrüggen | first4=Thomas | last5=Glaser | first5=Steffen J. | title=Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms | journal=Journal of Magnetic Resonance| volume=172 | issue=2 | year=2005 | issn=1090-7807 | doi=10.1016/j.jmr.2004.11.004 | pmid=15649756 | pages=296–305}}
• The field of ultracold matter for photoassociation.{{cite journal | last1=Wright | first1=M. J. | last2=Gensemer | first2=S. D. | last3=Vala | first3=J. | last4=Kosloff | first4=R. | last5=Gould | first5=P. L. | title=Control of Ultracold Collisions with Frequency-Chirped Light | journal=Physical Review Letters| volume=95 | issue=6 | date=2005-08-01 | issn=0031-9007 | doi=10.1103/physrevlett.95.063001 | pmid=16090943 | page=063001| url=http://eprints.maynoothuniversity.ie/4493/1/JV_Ultracold_Collisions.pdf }}
• Quantum information processing.{{cite journal | last1=García-Ripoll | first1=J. J. | last2=Zoller | first2=P. | last3=Cirac | first3=J. I. | title=Speed Optimized Two-Qubit Gates with Laser Coherent Control Techniques for Ion Trap Quantum Computing | journal=Physical Review Letters | volume=91 | issue=15 | date=2003-10-07 | issn=0031-9007 | doi=10.1103/physrevlett.91.157901 | pmid=14611499 | page=157901|arxiv=quant-ph/0306006| s2cid=119414046 }}Larsen, T. W., K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, and C. M. Marcus. "Coherent control of a transmon qubit with a nanowire-based Josephson junction." Bulletin of the American Physical Society 60 (2015).{{cite journal | last1=Scharfenberger | first1=Burkhard | last2=Munro | first2=William J | last3=Nemoto | first3=Kae |author3-link= Kae Nemoto | title=Coherent control of an NV⁻ center with one adjacent ¹³C | journal=New Journal of Physics | volume=16 | issue=9 | date=2014-09-25 | issn=1367-2630 | doi=10.1088/1367-2630/16/9/093043 | page=093043|doi-access=free|arxiv=1404.0475}}{{cite journal | last1=Weidinger | first1=Daniel | last2=Gruebele | first2=Martin | title=Quantum computation with vibrationally excited polyatomic molecules: Effects of rotation, level structure, and field gradients | journal=Molecular Physics | volume=105 | issue=13–14 | date=2007-07-01 | doi=10.1080/00268970701504335 | page=1999-20087| s2cid=122494939 | url=https://zenodo.org/record/895615 }}
• Attosecond physics.{{cite journal | last1=Corkum | first1=P. B. | last2=Krausz | first2=Ferenc | title=Attosecond science | journal=Nature Physics | publisher=Springer Science and Business Media LLC | volume=3 | issue=6 | year=2007 | issn=1745-2473 | doi=10.1038/nphys620 | pages=381–387| bibcode=2007NatPh...3..381C }}{{cite journal | last1=Boutu | first1=W. | last2=Haessler | first2=S. | last3=Merdji | first3=H. | last4=Breger | first4=P. | last5=Waters | first5=G. | last6=Stankiewicz | first6=M. | last7=Frasinski | first7=L. J. | last8=Taieb | first8=R. | last9=Caillat | first9=J. | last10=Maquet | first10=A. | last11=Monchicourt | first11=P. | last12=Carre | first12=B. | last13=Salieres | first13=P. |display-authors=5| title=Coherent control of attosecond emission from aligned molecules | journal=Nature Physics | publisher=Springer Science and Business Media LLC | volume=4 | issue=7 | date=2008-05-04 | issn=1745-2473 | doi=10.1038/nphys964 | hdl=10044/1/12527 | pages=545–549| hdl-access=free }}

Another important issue is the spectral selectivity of two photon resonant excitation coherent control.{{cite journal | last1=Meshulach | first1=Doron | last2=Silberberg | first2=Yaron | title=Coherent quantum control of two-photon transitions by a femtosecond laser pulse | journal=Nature | publisher=Springer Science and Business Media LLC | volume=396 | issue=6708 | year=1998 | issn=0028-0836 | doi=10.1038/24329 | pages=239–242| s2cid=41953962 }} A similar but non-resonant two photon excitation from the 1s1s to the 1s3s state of the He atom was investigated with ab-initio quantum mechanics es well.{{Cite journal |last=Barna |first=Imre Ferenc |date=2005 |title=Coherent control calculations for helium atom in short and intensive XUV laser pulses |url=https://link.springer.com/article/10.1140/epjd/e2005-00049-1 |journal=The European Physical Journal D |volume=33 |pages=307 |doi=10.1140/epjd/e2005-00049-1}} These concepts can be applied to single pulse Raman spectroscopy and microscopy.{{cite journal | last=Silberberg | first=Yaron | title=Quantum Coherent Control for Nonlinear Spectroscopy and Microscopy | journal=Annual Review of Physical Chemistry| volume=60 | issue=1 | year=2009 | issn=0066-426X | doi=10.1146/annurev.physchem.040808.090427 | pmid=18999997 | pages=277–292}}

As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing, and quantum simulation.{{cite journal | last1=Glaser | first1=Steffen J. | last2=Boscain | first2=Ugo | last3=Calarco | first3=Tommaso | last4=Koch | first4=Christiane P. | last5=Köckenberger | first5=Walter | last6=Kosloff | first6=Ronnie | last7=Kuprov | first7=Ilya | last8=Luy | first8=Burkhard | last9=Schirmer | first9=Sophie | last10=Schulte-Herbrüggen | first10=Thomas | last11=Sugny | first11=Dominique | last12=Wilhelm | first12=Frank K. |display-authors=5| title=Training Schrödinger's cat: quantum optimal control | journal=The European Physical Journal D | volume=69 | issue=12 | year=2015 | issn=1434-6060 | doi=10.1140/epjd/e2015-60464-1 | pages=1–24|doi-access=free|arxiv=1508.00442}}

{{Reflist}}

• Principles of the Quantum Control of Molecular Processes, by Moshe Shapiro, Paul Brumer, pp. 250. {{ISBN|0-471-24184-9}}. Wiley-VCH, (2003).
• "Quantum control of Molecular Processes", Moshe Shapiro and Paul Brumer, Wiley-VCH (2012).
• Rice, Stuart Alan, and Meishan Zhao. Optical control of molecular dynamics. New York: John Wiley, 2000.
• d'Alessandro, Domenico. Introduction to quantum control and dynamics. CRC press, 2007.
• David J. Tannor, "Introduction to Quantum Mechanics: A Time-dependent Perspective", (University Science Books, Sausalito, 2007).

Category:Chemical reactions

Category:Quantum mechanics

Category:Control theory