# Coherent control

Coherent control is a quantum mechanical based method for controlling dynamical processes by light. The basic principle is to control quantum interference phenomena typically by shaping the phase of a laser pulses [1] .[2] The basic ideas have proliferated finding vast application in spectroscopy mass spectra, quantum information processing, laser cooling, ultracold physics and more.

## Brief History

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 [3] [4] and in the frequency domain, interfering pathways controlled by one and three photons.[5] The two basic methods eventually merged with the introduction of optimal control theory [6] .[7]

Experimental realisations soon followed. In the time domain [8] and in the frequency domain.[9] Two interlinked developments accelerated the field of coherent control: Experimentally it was the development of pulse shaping by a spatial light modulator [10][11] and its employment in coherent control.[12] The second development was the idea of automatic feedback control [13] and its experimental realization .[14] [15]

## Controllability

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 generalisation 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.[16][17][18]

Controllability of a closed quantum system has been addressed by Tarn and Clark.[19] 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,[20] 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 factorially 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.

Once constraints are imposed controllability can be degraded. For example, what is the minimum time required to achieve a control objective .[21] this is termed quantum speed limit.

## Constructive approach to coherent control

The constructive approach uses a set of predetermined control fields for which the control outcome can be inferred. The pump dump scheme [3][4] in the time domain and the three vs one photon interference scheme in the frequency domain [5] are prime examples. Another constructive approach is based on adiabtic ideas. The most well studied method is Stimulated raman adiabatic passage STIRAP [22] 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.[23][24]

## Optimal control

Optimal control as applied in coherent control seeks the optimal control field for steering a quantum system to its objective.[6][7] For state-to-state control the objective is defined as the maximum overlap at the final time T with the state ${\displaystyle |\phi _{f}\rangle }$:

${\displaystyle J=|\langle \psi (T)|\phi _{f}\rangle |^{2}}$

where the initial state is ${\displaystyle |\phi _{i}\rangle }$. The time dependent control Hamiltonian has the typical form:

${\displaystyle H(t)=H_{0}+\mu \cdot \epsilon (t)}$

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

${\displaystyle J'=J+\int _{0}^{T}\langle \chi (t)|\left(i{\frac {\partial }{\partial t}}-H(\epsilon (t))\right)|\psi (t)\rangle dt+\lambda \int _{o}^{T}|\epsilon (t)|^{2}dt}$

where ${\displaystyle |\chi \rangle }$ is a wavefunction like Lagrange multiplier and the ${\displaystyle \lambda }$ parameter regulates the integral intensity. Variation of ${\displaystyle J'}$ with respect to ${\displaystyle \delta \epsilon }$ and ${\displaystyle \delta \psi }$ leads to two coupled Schrödinger equations A forward equation for ${\displaystyle |\psi \rangle }$ with initial condition ${\displaystyle |\psi (0)\rangle =|\phi _{i}\rangle }$ and a backward equation for the Lagrange multiplier ${\displaystyle |\chi \rangle }$ with final condition ${\displaystyle |\chi (T)\rangle =|\phi _{f}\rangle }$. Finding a solution requires an iterative approach. Different algorithms have been applied for obtaining the control field such as the Krotov method.[25]

A local in time alternative method has been developed,[26] where at each time step the field is calculated to direct the state to the target. A related method has been called tracking [27]

## Experimental applications

Coherent control has been applied to unimolecular chemical reactions controlling the final outcome [28] .[29]

Coherent control has been applied to bimolecular chemical reactions.[30]

Coherent control has been applied to biological photoisomerization of Retinal.[31][32]

Coherent control has been applied in the field of nuclear magnetic resonance NMR.[33]

Coherent control has been applied in the field of ultracold matter for photoassociation [34] and for laser cooling of internal degrees of freedom .[35][36]

Applications of coherent control to quantum information processing [37] [38] .[39]

An important issue is the spectral selectivity of two photon coherent control .[40] These ideas can be applied to single pulse Raman spectroscopy and microscopy.[41]

Coherent control has been applied in attosecond physics.[42][43]

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. [44]

## References

1. ^ Gordon, Robert J., and Stuart A. Rice. "Active control of the dynamics of atoms and molecules." Annual Review of Physical Chemistry 48, no. 1 (1997): 601-641.
2. ^ Shapiro, Moshe, and Paul Brumer. "Coherent control of atomic, molecular, and electronic processes." Advances in atomic, molecular, and optical physics 42 (2000): 287-345.
3. ^ a b Tannor, David J., and Stuart A. Rice. "Control of selectivity of chemical reaction via control of wave packet evolution." The Journal of chemical physics 83, no. 10 (1985): 5013-5018.
4. ^ a b Tannor, David J., Ronnie Kosloff, and Stuart A. Rice. "Coherent pulse sequence induced control of selectivity of reactions: Exact quantum mechanical calculations." The Journal of chemical physics 85, no. 10 (1986): 5805-5820.
5. ^ a b Brumer, Paul, and Moshe Shapiro. "Control of unimolecular reactions using coherent light." Chemical physics letters 126, no. 6 (1986): 541-546.
6. ^ a b Peirce, Anthony P., Mohammed A. Dahleh, and Herschel Rabitz. "Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications." Physical Review A 37, no. 12 (1988): 4950.
7. ^ a b Kosloff, Ronnie, Stuart A. Rice, Pier Gaspard, Sam Tersigni, and D. J. Tannor. "Wavepacket dancing: Achieving chemical selectivity by shaping light pulses." Chemical Physics 139, no. 1 (1989): 201-220.
8. ^ Baumert, T., V. Engel, Ch Meier, and G. Gerber. "High laser field effects in multiphoton ionization of ${\displaystyle Na_{2}}$. Experiment and quantum calculations." Chemical physics letters 200, no. 5 (1992): 488-494.
9. ^ Zhu, Langchi, Valeria Kleiman, Xiaonong Li, Shao Ping Lu, Karen Trentelman, and Robert J. Gordon. "Coherent laser control of the product distribution obtained in the photoexcitation of HI." SCIENCE-NEW YORK THEN WASHINGTON- (1995): 77-77.
10. ^ A.M. Weiner. "Femtosecond pulse shaping using spatial light modulators" (PDF). REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 71, NUMBER 5 MAY 2000. Retrieved 2010-07-06.
11. ^ Liquid Crystal Optically Addressed Spatial Light Modulator, [1]
12. ^ Kawashima, Hitoshi, Marc M. Wefers, and Keith A. Nelson. "Femtosecond pulse shaping, multiple-pulse spectroscopy, and optical control." Annual Review of Physical Chemistry 46, no. 1 (1995): 627-656.
13. ^ Judson, Richard S., and Herschel Rabitz. "Teaching lasers to control molecules." Physical Review Letters 68, no. 10 (1992): 1500.
14. ^ Assion, Andreas, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber. "Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses." Science 282, no. 5390 (1998): 919-922.
15. ^ Brif, Constantin, Raj Chakrabarti, and Herschel Rabitz. [2]"Control of quantum phenomena: past, present and future." New Journal of Physics 12, no. 7 (2010): 075008.
16. ^ Tesch, Carmen M., Lukas Kurtz, and Regina de Vivie-Riedle. "Applying optimal control theory for elements of quantum computation in molecular systems." Chemical Physics Letters 343, no. 5 (2001): 633-641.
17. ^ Palao, José P., and Ronnie Kosloff. "Quantum computing by an optimal control algorithm for unitary transformations." Physical review letters 89, no. 18 (2002): 188301.
18. ^ Rabitz, Herschel, Michael Hsieh, and Carey Rosenthal. "Landscape for optimal control of quantum-mechanical unitary transformations." Physical Review A 72, no. 5 (2005): 052337.
19. ^ Huang, Garng M., Tzyh J. Tarn, and John W. Clark. "On the controllability of quantum‐mechanical systems." Journal of Mathematical Physics 24, no. 11 (1983): 2608-2618.
20. ^ Ramakrishna, Viswanath, Murti V. Salapaka, Mohammed Dahleh, Herschel Rabitz, and Anthony Peirce. "Controllability of molecular systems." Physical Review A 51, no. 2 (1995): 960.
21. ^ Caneva, Tommaso, Michael Murphy, Tommaso Calarco, Rosario Fazio, Simone Montangero, Vittorio Giovannetti, and Giuseppe E. Santoro. "Optimal control at the quantum speed limit." Physical review letters 103, no. 24 (2009): 240501.
22. ^ Unanyan, R., M. Fleischhauer, B. W. Shore, and K. Bergmann. "Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states." Optics communications 155, no. 1 (1998): 144-154.
23. ^ Ruhman, S., and R. Kosloff. "Application of chirped ultrashort pulses for generating large-amplitude ground-state vibrational coherence: a computer simulation." JOSA B 7, no. 8 (1990): 1748-1752.
24. ^ Cerullo, G., C. J. Bardeen, Q. Wang, and C. V. Shank. "High-power femtosecond chirped pulse excitation of molecules in solution." Chemical Physics Letters 262, no. 3 (1996): 362-368.
25. ^ Somlói, József, Vladimir A. Kazakov, and David J. Tannor. "Controlled dissociation of ${\displaystyle I_{2}}$ via optical transitions between the X and B electronic states." Chemical physics 172, no. 1 (1993): 85-98.
26. ^ Kosloff, Ronnie, Audrey Dell Hammerich, and David Tannor. "Excitation without demolition: Radiative excitation of ground-surface vibration by impulsive stimulated Raman scattering with damage control." Physical review letters 69, no. 15 (1992): 2172.
27. ^ Chen, Yu, Peter Gross, Viswanath Ramakrishna, Herschel Rabitz, and Kenneth Mease. "Competitive tracking of molecular objectives described by quantum mechanics." The Journal of chemical physics 102, no. 20 (1995): 8001-8010.
28. ^ Levis, R. J., and H. A. Rabitz. "Closing the loop on bond selective chemistry using tailored strong field laser pulses." The Journal of Physical Chemistry A 106, no. 27 (2002): 6427-6444.
29. ^ Dantus, Marcos, and Vadim V. Lozovoy. "Experimental coherent laser control of physicochemical processes." Chemical reviews 104, no. 4 (2004): 1813-1860.
30. ^ Coherent Control of Bond Making Liat Levin, Wojciech Skomorowski, Leonid Rybak, Ronnie Kosloff, Christiane P. Koch, Zohar Amitay arXiv:1411.1542 (2015)
31. ^ Prokhorenko, Valentyn I., Andrea M. Nagy, Stephen A. Waschuk, Leonid S. Brown, Robert R. Birge, and RJ Dwayne Miller. "Coherent control of retinal isomerization in bacteriorhodopsin." Science 313, no. 5791 (2006): 1257-1261.
32. ^ Wohlleben, Wendel, Tiago Buckup, Jennifer L. Herek, and Marcus Motzkus. "Coherent control for spectroscopy and manipulation of biological dynamics." ChemPhysChem 6, no. 5 (2005): 850-857.
33. ^ Khaneja, Navin, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J. Glaser. "Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms." Journal of Magnetic Resonance 172, no. 2 (2005): 296-305.
34. ^ Wright, Matthew J., S. D. Gensemer, J. Vala, R. Kosloff, and P. L. Gould. "Control of ultracold collisions with frequency-chirped light." Physical review letters 95, no. 6 (2005): 063001.
35. ^ Viteau, Matthieu, Amodsen Chotia, Maria Allegrini, Nadia Bouloufa, Olivier Dulieu, Daniel Comparat, and Pierre Pillet. "Optical pumping and vibrational cooling of molecules." Science 321, no. 5886 (2008): 232-234.
36. ^ Lien, Chien-Yu, Christopher M. Seck, Yen-Wei Lin, Jason HV Nguyen, David A. Tabor, and Brian C. Odom. "Broadband optical cooling of molecular rotors from room temperature to the ground state." Nature communications 5 (2014).
37. ^ Garcia-Ripoll, Juan José, Peter Zoller, and J. Ignacio Cirac. "Speed optimized two-qubit gates with laser coherent control techniques for ion trap quantum computing." Physical review letters 91, no. 15 (2003): 157901.
38. ^ 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).
39. ^ Scharfenberger, Burkhard, William J. Munro, and Kae Nemoto. "Coherent control of an NV− center with one adjacent 13C." New Journal of Physics 16, no. 9 (2014): 093043.
40. ^ Meshulach, Doron, and Yaron Silberberg. "Coherent quantum control of two-photon transitions by a femtosecond laser pulse." Nature 396, no. 6708 (1998): 239-242.
41. ^ Silberberg, Yaron. "Quantum coherent control for nonlinear spectroscopy and microscopy." Annual Review of Physical Chemistry 60 (2009): 277-292.
42. ^ Corkum, P. B., and Ferenc Krausz. "Attosecond science." Nature Physics 3, no. 6 (2007): 381-387.
43. ^ Boutu, W., S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski et al. "Coherent control of attosecond emission from aligned molecules." Nature Physics 4, no. 7 (2008): 545-549.
44. ^ Glaser, Steffen J., Ugo Boscain, Tommaso Calarco, Christiane P. Koch, Walter Köckenberger, Ronnie Kosloff, Ilya Kuprov et al. "Training Schrödinger’s cat: Quantum optimal control." The European Physical Journal D 69, no. 12 (2015): 1-24.