Coherent control is a
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
-based method for controlling dynamic processes by
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
. The basic principle is to control quantum interference phenomena, typically by shaping the phase of
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
pulses. The basic ideas have proliferated, finding vast application in
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
mass spectra,
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
processing,
laser cooling
Laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) a ...
, ultracold physics and more.
Brief History
The initial idea was to control the outcome of
chemical reactions
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
. Two approaches were pursued:
* in the time domain, a "pump-dump" scheme where the control is the time delay between pulses
* in the frequency domain, interfering pathways controlled by one and three photons.
The two basic methods eventually merged with the introduction of
optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering an ...
theory.
Experimental realizations soon followed in the time domain and in the frequency domain. Two interlinked developments accelerated the field of coherent control: experimentally, it was the development of
pulse shaping In electronics and telecommunications, pulse shaping is the process of changing the waveform of transmitted pulses to optimize the signal for its intended purpose or the communication channel. This is often done by limiting the bandwidth of the tran ...
by a
spatial light modulator
A spatial light modulator (SLM) is an object that imposes some form of spatially varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the term SLM is used, it means that the transparency ca ...
and its employment in coherent control. The second development was the idea of automatic feedback control and its experimental realization.
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 generalization is steering simultaneously an arbitrary set of initial pure states to an arbitrary set of final states i.e. controlling a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
. Such an application sets the foundation for a quantum gate operation.
Controllability of a closed quantum system has been addressed by Tarn and Clark. Their theorem based in
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
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 if the control operators and the unperturbed Hamiltonian generate the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of all
Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
s. 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
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...
. The algorithmic task of finding the field that generates a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
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.
Once constraints are imposed controllability can be degraded. For example, what is the minimum time required to achieve a control objective? This is termed the "quantum speed limit". The speed limit can be calculated by quantizing Ulam's control conjecture.
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
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 which employs an auxiliary state to achieve complete state-to-state population transfer.
One of the most prolific generic pulse shapes is a
chirp
A chirp is a signal in which the frequency increases (''up-chirp'') or decreases (''down-chirp'') with time. In some sources, the term ''chirp'' is used interchangeably with sweep signal. It is commonly applied to sonar, radar, and laser system ...
ed pulse a pulse with a varying frequency in time.
Optimal control
Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering an ...
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
:
:
where the initial state is
. The time dependent control Hamiltonian has the typical form:
:
where
is the control field.
Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering an ...
solves for the optimal field
using the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
introducing
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
s. A new objective functional is defined
:
where
is a wave function like
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
and the
parameter regulates the integral intensity.
Variation of
with respect to
and
leads to two coupled
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
s. A forward equation for
with initial condition
and a backward equation for the
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
with final condition
. Finding a solution requires an iterative approach. Different algorithms have been applied for obtaining the control field such as the Krotov method.
A local in time alternative method has been developed, where at each time step, the field is calculated to direct the state to the target. A related method has been called tracking
Experimental applications
Some applications of coherent control are
* Unimolecular and bimolecular
chemical reactions
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
.
* The biological photoisomerization of
Retinal
Retinal (also known as retinaldehyde) is a polyene chromophore. Retinal, bound to proteins called opsins, is the chemical basis of visual phototransduction, the light-detection stage of visual perception (vision).
Some microorganisms use retina ...
.
* The field of
nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
.
* The field of ultracold matter for photoassociation.
* Quantum information processing.
*
Attosecond physics
Attosecond physics, also known as attophysics, or more generally attosecond science, is a branch of physics that deals with light-matter interaction phenomena wherein attosecond (10−18 s) photon pulses are used to unravel dynamical processes in ...
.
Another important issue is the spectral selectivity of two photon coherent control. These concepts can be applied to single pulse
Raman spectroscopy
Raman spectroscopy () (named after Indian physicist C. V. Raman) is a spectroscopic technique typically used to determine vibrational modes of molecules, although rotational and other low-frequency modes of systems may also be observed. Raman s ...
and microscopy.
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.
References
Further reading
*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).
Chemical reactions
Quantum mechanics
Control theory