Quantum Control Theory: Applying Classical Concepts to Manipulate Quantum Systems, Papers of Physics

Download Quantum Control Theory: Applying Classical Concepts to Manipulate Quantum Systems and more Papers Physics in PDF only on Docsity! QUANTUM CONTROL (Term paper – PHY 392T) Swaroop Ganguly Department of Electrical and Computer Engineering In this term paper, we review the theory and applications of quantum (or coherent) control. Classical control theory has been developed and used extensively by engineers and applied mathematicians for much of the last century. The program of applying the concepts of control theory to quantum systems is relatively recent. Here, we will give a brief introduction to classical control theory, followed by a non-rigorous exposition of the theory of quantum control. Then, we will look at how these concepts are applied to manipulate the dynamics of a variety of quantum systems, chemical reactions, and, perhaps most pertinently for us, semiconductor nanostructures. 2 I. INTRODUCTION The study of classical control systems has been ubiquitous across engineering disciplines for most of the last century. It has been studied and developed for two major reasons. The first is to develop the abstract mathematical tools, viz. control theory, to describe and understand dynamical systems. The second is to precisely control the evolution of systems in a noisy environment, usually via feedback. Examples where control theory has been successfully applied include airplane pitch and yaw control, chemical reactors, precision electrical circuits, and limiting vibrations in mechanical systems. We shall start by looking at some of the basic concepts of this incredibly successful theory, in its simplest form: linear, time-invariant (LTI). The rapid evolution of electronic and mechanical systems toward the nanometer scale makes it attractive to develop and study a control theory that might be applicable to these systems. This interest started in the 1980’s, and gathered steam after a series of landmark papers [1] describing a scheme to control the dynamics of a laser cavity by continuous feedback via homodyne detection (i.e. detection of a signal using a local oscillator of the same frequency). We shall review the basics of the theory of quantum feedback control [2], pointing out how the peculiarities of quantum mechanical systems and measurements performed thereupon inform this theory. Finally, we shall look at a few representative proposed or realized applications of quantum control theory. These will include coherent control of chemical reactions and carrier dynamics in nanostructures. The latter provides a hint to the deep connection between quantum control and quantum computing/information. In fact, it is now expected that robust quantum computing and/or information processing will require use of the concepts of quantum control. 5 intimately tied to quantum computing and/or information processing). We will look at the physical picture emerging from this theory, in the simplest possible terms. First, it might be useful to point out the similarities and correspondences between the classical and quantum control theory. The linear version of classical theory mentioned here, of course, shares the linearity property with quantum systems. The state space in the classical case corresponds to a (possibly infinite dimensional) Hilbert space in the quantum case. The state vector corresponds to the density matrix in the quantum system. The system matrix A is analogous to the Liouvillian, which propagates the density matrix as follows: . In the special case where the Liouvillian is a Hamiltonian, the equation of motion for the system is the von Neumann equation: . The forcing term corresponds, in the quantum case, to the interaction Hamiltonian , which adds a feedback control term to the system Liouvillian: . Physically this arises from the interaction of the system and the external control field, usually from a laser. Lastly, the observation process in the classical system corresponds to conditional evolution in the quantum case, where it is well known that observing the system will affect the evolution in the form of backaction noise. The way to carry out a weak (non-projective) measurement on a quantum system is via what is called an indirect measurement [6]. This involves entangling the system with a disposable meter (e.g. laser pulse) first, switching off the interaction between the meter and system and then carrying out a projective measurement on the meter. Though the quantum system is never actually perturbed by the measurement, it still encounters (Heisenberg limited) backaction noise because of the entanglement. 6 Following [2], let us suppose the system belongs to a Hilbert space , and the meters to a Hilbert space for the reservoir (collection of meters). The scheme is illustrated below: The initial density operator for the composite system (system and reservoir) can be written as a tensor product of the system density operator and the density operator for the reservoir mode respectively, since the system and reservoir do not interact at this time: At time , the system interacts with the i-th meter according to the interaction Hamiltonian . Since the meter is usually a laser mode it travels at the speed of light and therefore, the 7 interaction time is very small. At time , a measurement is performed on the mode by way of projecting the combined system-reservoir state onto the measurement operator . A partial trace over the reservoir yields the conditional system density operator . The subscript denotes that the density operator is conditioned on the random outcome of the measurement. The evolution path obtained by conditioning according to the measurement outcome is called a quantum trajectory. Now we assume that (i) the system-reservoir interaction is time-independent, that is, , where is the measurement (coupling) strength (this is justified by the assumption that the interaction takes place over an interval that is small compared to the time scale for the internal dynamics of the system); (ii) the meters are non-interacting, so that which denotes the string of meters (in other words, the reservoir) satisfies since each meter is orthogonal, viz., ; and (iii) the retardation time is insignificant (since the meter for most quantum control application is laser light). With the above assumptions, the propagator for the quantum system is: where is the time-ordering and is a reservoir creation operator prior to the interaction, which occurs at time for an interval . It can be shown that the measurement result (often called the ‘photocurrent’ for historical reasons) is: (2) 10 IV. APPLICATIONS OF QUANTUM CONTROL The first broad class of applications addresses the chemist’s dream: to create novel stable or meta-stable products by selectively making or breaking chemical bonds to drive the chemical system toward the desired (quantum) state [7]. These applications take advantage of the enormous strides in recent times of femtosecond laser pulse-shaping capabilities. The hope is that the laser-driven coherent manipulation of the molecular motion-induced quantum interferences (between different pathways from one state to another) will facilitate products or molecular states, not accessible by conventional chemical or photo-chemical means. For example, Rabitz et al. [7] have illustrated this by considering the analog of a double-slit experiment in the 3s 5s two-photon transition in atomic Na. Consider two (or more, as would be the case for an ultra-short laser pulse) different paths with photon energies (ω1+ ω2) and (ώ1+ ώ2); by choosing the phase difference between the path appropriately (a femtosecond laser is tailored to modulate the phases using a pulse shaper controlled by a learning algorithm), one can achieve constructive (or destructive) interference leading to maximization (minimization) of the two-photon transition probability as shown in the figure below. 11 In a very significant work in the quantum control of chemical systems, Herek et al. [8] have used coherent learning control over the energy flow pathways in the light-harvesting molecular complex in a photosynthetic bacterium. Their experiment on a (soft) condensed phase system not only proves that molecular complexity need not preclude the possibility of coherent control, but also opens up the new application area of the quantum control of biological systems. That brings us to our final topic: the application of coherent control to semiconductors and their nanostructures, specifically, the carrier dynamics therein. It might be noted that, unlike the coherent control of chemical reactions, this application area has not yet witnessed a serious experimental demonstration as far as I am aware. The selected control schemes described are all theoretical proposals. The first example we look at is the control of carrier dynamics in a bulk semiconductor within a two band effective mass model [9]. Dargys has here theorized that the inter-band carrier transfer can be controlled coherently using an optical pulse. The author obtains an equation for coherent transition between the bands. This together with a functional that minimizes the energy of the pulse yields a strategy, in the form of an Euler equation, for optimal control of the system. The second example is a proposal for the laser-actuated coherent control of ferromagnetism in undoped, low Mn mole-fraction II(Mn)-VI dilute doped magnetic semiconductors [10]. Here, Fernandez-Rossier et al. have shown that in these paramagnetic materials, an intense sub-bandgap laser radiation induces coherence between the conduction and valence bands leading to an optical exchange interaction, causing the material to enter a ferromagnetic phase at low temperatures. This, if experimentally validated might be technologically important, since these material systems had never been known to show ferromagnetism under usual conditions. 12 The third example we might wish to consider is the control of spin-spin interaction in doped semiconductors [11]. The authors have developed a theory of bound exciton-mediated interaction between spins localized on two impurity centers. This has been applied to shallow neutral donors in III-V semiconductors. It has been shown that when the control laser energy is tuned to the bonding – anti-bonding gap of the exciton bound by the two impurities, the coupling between the spins increases and may change from ferromagnetic to anti-ferromagnetic. This method of controlling single-spin dynamics suggests application to quantum information processing. The last example we look at is a scheme for the control of exciton dynamics in semiconductor nano-dots for quantum logic operations [12]. In this paper, the authors propose to control the dynamics of two interacting excitons of opposite polarization (control of the spin of excitons has been experimentally demonstrated elsewhere), using circularly polarized optical pulses. The quantum operation uses the lowest four states formed by these two excitons – this is sufficient for universal quantum logic since one and two-qubit operations can be used to realize any logic operation. The femto-second laser pulses are to be shaped to finish the quantum operation within the decoherence time. The authors have then theoretically demonstrated the application of the control to the complete solution of a simple quantum algorithm. I have tried to show that the still-evolving theory of the control of quantum systems might be a highly useful tool for nanoscience. On the one hand, it provides a general purpose mathematical framework to describe the manipulated evolution of quantum systems; this obviously holds enormous promise for experimental nanotechnology. On the other hand, it might stimulate basic questions (and provide answers to) the nature of quantum mechanics, quantum information and computing, and quantum chaos.