My previous blog post discussed open quantum systems that can lose phase information to their quantum or quantum and classical environments in what is called “decoherence.” Decoherence means that the capacity of the open quantum system to exhibit constructive and destructive interference to yield the famous light dark bands of the two slit interference pattern is gradually lost as decoherence proceeds.
As we saw, decoherence is real. During decoherence, the Schrodinger equation does not apply as it does in closed quantum systems where it is time reversible. In the presence of decoherence, a dissipative process, we discussed new physics, such as the Quantum Anti-Zeno effect where quantum systems change states faster than the normal exponential of radioactive half life fame.
The “weird” quantum superpositions, or “mixed states”, where Schrodinger’s cat is simultaneously “dead” AND “alive” until measured, decay by decoherence rapidly to leave still quantum pure states, either dead, or alive, but not both, and which of these is then determined by measurement. In short fundamental new physics arises in the presence of decoherence. I also discussed a theorem by Shor and quantum coherence in chlorophyll wrapped by its antenna protein suggesting that RECOHERENCE from classical (for all practical purposes, FAPP) to more or fully quantum coherent behavior is possible. This opens new realms of science.
In the blog post after this one, I will make use of the astonishing fact that superpositions of cats, simultaneously dead and alive, decay rapidly in the presence of decohrence, to suggest a new approach to what is called the “Measurement Problem” in Quantum Mechanics as axiomatized by John von Neumann. That approach builds upon the current blog post.
I wish now to define what I and my colleagues Samuli Niiranen and Gabor Vattay of Tampere University of Technology Finland, and Eotvos University, Budapest, respectively, call “The Poised Realm, (2).
Consider a two-dimensional coordinate system. The vertical axis runs from fully quantum at the origin to classical FAPP as one ascends the Y axis. Decoherence, as modeled mathematically by the Lindblad operator, takes an open quantum system up the Y axis from fully quantum towards classicity FAPP. Recoherence, for example by physical realization of the Shor theorem or perhaps the antenna protein, takes the open quantum system down the Y axis towards more or fully quantum coherent behavior.
The X axis is new. It runs starting at the origin from Order to Criticality to Chaos. At present two and perhaps a third means are known to control position on the X axis. The first is the character of the Hamiltonian describing the behavior of the system.
A classical pendulum is ordered. It is a quantum harmonic oscillator with discrete energy levels and is ordered. In the classical case, the system is conservative and nested orbits arise if the frictionless pendulum is released at different displacements from vertical in a gravitational field. Thus, the neighboring orbits neither converge nor diverge in their trajectories in state space, so the Lyapunov exponent that measures divergence, if positive, or convergence if negative, is zero. Mathematical results show that as the Hamiltonian is modified gradually, the Lyapunov exponent remains 0 in the ordered regime on the classical X axis, but undergoes a second order phase transition at a critical value, after which chaos appears. (A second order phase transition means that the derivative of the line showing the Lyapunov exponent has a kink, or discontinuity, at criticality.) Such chaotic Hamiltonians are one way to study quantum chaos.
A second established way to control position on the X axis is through use of the kicked quantum rotor, hit by delta distributed, i.e. instantaneous, momentum kicks via the impact of light. By tuning the intensity, K, of light kicks to the quantum rotor, or the frequency of kicks to the quantum rotor, the rotor can be driven from the ordered regime on the X axis to critical then to chaotic behavior. This system is commonly used to study quantum chaos.
A third potential means to control position on the X axis concerns the graph structure of models that may generalize to models of molecules. Random graphs were studied by Erdos and Renyi. Here N nodes are randomly chosen in pairs and connected by M edges. A component in such a graph is a connected set of nodes. Erdos and Renyi showed that when the ratio of edges to nodes, M/N reaches 0.5, so the ends of edges equal the number of nodes, a first order phase transition occurs and a giant connected component with most of the nodes in the graph arises. This is called the critical giant component.
Vattay has studied what are called the “eigenvalues” of the adjacency matrix of critical and supracritical (M/N > 0.5 ) graphs and derived the absorption emission energy spectrum. (An adjacency matrix is an N by N matrix showing which of the N nodes are connected to one another. For critical ER graphs, this spectrum is a negative exponential, where the size of the quantum emitted is plotted on the X axis and the probability on the Y axis. Thus, the peak occurs for quanta of 0 energy. In sharp contrast, for more richly connected graphs, the peak of the absorption emission spectrum occurs at a finite emitted or absorbed quantum of energy. This begins to suggest that molecules with the same number of atoms, but different connectivities (counting single, double and triple bonds as just “connected”), may have different positions positions on the X axis. This may turn out to have biological, medical and other practical implications.
The Poised Realm is real. It may be the only “passage” pathway from the quantum world via open quantum systems to the classical world.
I now describe some very early simulations of the kicked quantum rotor carried out by Vattay. These results must be repeated and extended. Vattay tuned the frequency of kicking the kicked quantum rotor from once per period to ten times per period. This simulation has been carried out in the presence of the Lindblad operator which can be tuned from weak to strong decohrence. Our aim is to explore the Poised Realm, so far, only with decohrence, not recoherence.
Vattay has reported to Niiranen and me that when the rotor is kicked once per period, a large number of quantum amplitudes are propagating. Given Lloyd’s claim that in the presence of decohrence, superpositions decohere rapidly, what are propagating correspond to pure states, (The cat is either dead or alive, not simultaneously both as in a mixed state. But the cat remains quantum until measured.) Vattay then increased the frequency of kicking the quantum rotor for fixed values of the Lindblad operator and the number of propagating amplitudes DECREASED TO A FEW. At some point, as position on the X axis increases from the origin, and a critical intensity of kicking is reached the system SWITCHES TO CLASSICAL BEHAVIOR WITH DIFFUSIVE BEHAVIOR IN MOMENTUM SPACE. This switch is a roughly or truly sudden transition to classical behavior.
Thus, it begins to appear that classical behavior can be obtained by motion up the Y axis to classicity for all practical purposes, FAPP, or out the X axis to classicity which may be classicity “simpliciter” ie just classical. In the next blog post I will use the results above, which we are investigating at this moment, to suggest what might turn out to be a solution to most or all of the quantum Measurement Problem. Our results are not advanced enough at present to know, but we are encouraged that we may have a useful approach via the Poised Realm to the measurement problem.