Decoherence

In this post I delve into the current view of what happens to a wave function as it interacts with its environment and tell the story of how I anticipated the idea of this view around 1971 or 1972 some 10 years before a crucial paper was published in 1991. If you have a non-technical background, I hope you can skim through without too much puzzlement. In the next post I will revert to writing which is entirely non mathematical.

Back around 1970, when I first became interested in the “collapse of the wave function”, I noticed at some point while thinking about the situation, that this collapse entailed more than simply the materialization of, say, a particle in accord with its probability distribution. For the wave function is more than a probability distribution. It contains, in addition, information which allows it to be transformed into a new “representation” in which it gives a probability distribution for a different physical quantity. For example, if we have a wave function from which we can find a probability for a particle’s position, we can transform this wave function into a new form from which we can find the distribution for the particle’s energy. With the “collapse”, however, one loses the information that would allow such transformations. One loses the “phases” of the wave function. To understand what are meant by phases I need to point out that a complex number can be viewed as a little arrow, lying in a plane. The length of the arrow can represent a positive real number, i.e. a probability. The arrow, lying in its plane, can point in any 360-degree direction and the angle at which it points is called its “phase”. A wave function consists of many complex numbers, each of which can be looked upon as a little arrow with magnitude and phase. Looking at an entire collection of these little arrows, one can consider their lengths (actually length squared) as a probability distribution for one physical quantity, and the pattern of their phases as additional information about other physical quantities.

Collapse occurs when a quantum system interacts with its environment. With the “collapse”, one of the probabilities becomes realized; and ALL of the phases simply disappear from the record. The information associated with the phases’ pattern goes missing. These days people have realized something I missed back in the 1970’s: the information contained in the phases doesn’t actually go missing, but leaks into the environment where it can show up, giving us information about the quantum system of interest. People no longer talk much about collapse, concentrating on the disappearance of a system’s phase pattern, which may or may not actually be linked to collapse. The modern buzz word for this possible way-station to collapse is “decoherence”. The phase pattern is “coherent” and when it goes away, we have “quantum decoherence”. Back in 1971, long before the word “decoherence” had ever appeared in this context, I wondered if there might be a way of calculating how the phases go away as a quantum system interacts with its environment, and, through blind luck, came to realize that there was indeed the possibility of such a calculation. In reading various papers about “measurement theory” I came across an essay by Eugene Wigner, a Nobel prize winning theorist, who pointed out that a quantum expression called “the density matrix” might possibly throw some light on the whole “measurement-collapse” situation because with the density matrix phases went away. Wigner said, however, that this possibility was of no use, because the density matrix belongs not to a single quantum system, but always to an “ensemble”. An ensemble is a collection of a number of similar systems, while the “collapse” happens with a single system. So, the essay’s conclusion was: forget about the density matrix as being of any help in understanding what was going on. I noted what Wigner had said and thought no more about it until I was browsing in a quantum text by Lev Landau and Evgeny Lifshitz, translated ten or so years earlier from the Russian. There on pages 35 – 38 was a definition and discussion of the density matrix; and the definition was definitely for a single system interacting with its environment. I remembered that Lev Landau had independently defined the density matrix along with von Neumann in 1927. Perhaps Landau’s version had simply been forgotten. In any case, being defined for a single system, to me it showed great promise for calculating how wave function phases could disappear. (See Landau and Lifshitz, Quantum Mechanics: Non-Relativistic Theory, First English Edition, 1958.)

Lev Landau was still another of the geniuses associated with the development of quantum mechanics. Born in June, 1908, in Baku, Azerbaijan, of Russian parents, he was enough younger than the Pauli – Heisenberg generation that he missed out on the first 1925 – 1926 wave of the quantum revolution. By the time he was 19 or so he had caught up enough to independently define a version of the density matrix. Later he spent time in Europe, visiting the Bohr institute on several occasions between 1929 and 1931. A wonderful book about that time period is Faust in Copenhagen: A Struggle for the Soul of Physics by Geno Segrè. Dr. Segrè is a neutrino physicist who is also a talented writer. Warning! If you’re not a physics buff by now, this book might well make you into one. Geno Segrè’s uncle was Emilio Segrè, a famous member of Fermi’s group in Italy and later one of the atomic bomb developers. Talking about Landau, known by his nickname, Dau, Segrè says, “Dau, who became Russia’s greatest theoretical physicist and one of the twentieth century’s major scientific figures was never intimidated by anybody, …”. “As the Dutch physicist, Casimir remembered, ‘Landau’s was perhaps the most brilliant and quickest mind I have ever come across.’ This is high praise from someone who knew well both Heisenberg and Pauli.”

With the Landau Lifshitz definition in hand I tried to see if I could prove that the right sort of environmental interaction could make the phases of the wave function fade away. The density matrix for discrete states is a square with the real probabilities running down the main diagonal from upper left to lower right. The off-diagonal elements are complex and contain the relevant phase information. (The matrix is Hermitian, though that fact is somewhat irrelevant in the context of interest here.)

About the time I started working on the matrix there was a talented graduate student, Yashwant Shitoot from India, at Auburn who needed a thesis topic so I suggested that he work on the problem for his Master’s thesis which he did. Shitoot and I came up with somewhat different approaches to the problem. Yashwant observed that in practice the environment potentials could not be exactly specified and thus the off-diagonal elements of the matrix could be considered to be a probability distribution arising from the many unknown environmental potentials. Citing the “central limit theorem” he argued that these distributions were normal distributions and would vanish over time. (See Yashwant Anant Shitoot Theory of Measurement, M.S. Thesis, Auburn University, March, 1973.) The probabilities in Shitoot’s approach are classical probabilities arising from our ignorance; not quantum probabilities arising from the “mind of God”. In my approach I visualized the wave function in a Stern-Gerlach experiment. The classic Stern-Gerlach experiment passes a beam of silver atoms in vacuum between unsymmetrical poles of a magnet. Such poles generate a non-uniform magnetic field which exerts a force on a silver atom which has a magnetic moment due to the spin of its outer electron. A sliver atom wave function splits into a superposition of two spatially separated parts representing the two spin possibilities spin-up or spin down. (This splitting is similar to what occurs with Schrödinger’s unhappy cat.) After passing through the magnet poles the silver beam can either impinge on a barrier where it forms two spots of silver or, instead come to a barrier with a slit positioned where, say, the upper the silver dots would be. In the latter case some of the silver atoms form a dot below and others pass through the slit. The atoms that pass through the slit all have their spin up when passed through a second pole piece oriented like the first; or confirm the way that spin ½ works if the second pole piece is tilted. My interest, however, was not with the spin of the silver atoms, but instead, with a calculation of how the superposition changes as one part of it impinges on the atoms of the barrier. To attack the calculation, I considered a silver atom as the “system” and the atoms of the barrier as the “environment”. In quantum mechanics there are not only representations, but “pictures”. In the Schrödinger picture, the time dependence is carried by the wave function (state vector) while in the Heisenberg picture the time dependence is carried by the quantum mechanical operators. Furthermore, there is a third picture called the interaction picture where one ends up with the time dependence in the interaction part when a system and its environment interact. Using the interaction picture and a model potential consisting of a series of step functions to simulate the atoms of the barrier, I could easily show that the off-diagonal elements of the density matrix “gradually” went to zero. Of course, I’m being facetious in using the word “gradually” because the time involved here is of the order of 10⁻¹⁴ seconds. However, in one’s imagination one can split this time into thousands or millions of increments. Then the change is indeed gradual. Or one can imagine a different physical situation where a quantum particle traveling through an imperfect vacuum encounters the field from a stray atom from time to time. The essential point is that the quantum decoherence in not instantaneous and one can imagine situations where the time interval is experimentally significant. (See below.)

There are two problems with my approach. First, I failed to find a proof that used a realistic interaction potential. Nevertheless, what I did was highly suggestive and over the years gave me the satisfaction of feeling that I understood what was happening whenever I encountered quantum puzzles involving collapse. In particular, the model calculation showed how an interaction of one piece of a superposition would affect another piece where there was no interaction. The second problem I had at the time was how to interpret the physical situation when the off-diagonal elements of the density matrix had gone only part way to zero. In particular, what was the physical meaning of the situation when a particle passed by a weak interaction potential into an area free from interaction so that any decoherence was only partial. I kept thinking about this second difficulty over the years and at some point, an answer dawned on me. (See below.)

In spite of these difficulties, around 1973 I wrote up a paper and sent it to the Physical Review where it was summarily rejected because I had pointed out no ramifications of the calculation which could be experimentally tested. I didn’t follow up for a number of reasons: I had no answer to the second difficulty mentioned above, I was and am somewhat lazy, and my life was falling apart at the time. I left Auburn in 1974 and my only copy of the paper has disappeared.

Currently, quantum decoherence is of interest because it is highly relevant to quantum computing. In a quantum computer a collection of “qubits” which act like spin ½ particles are put into a quantum state where they carry out a calculation provided that they do not “decohere” during the time necessary for the calculation to take place. This means that the qubit collection must be as isolated as possible from any stray potentials. However, it is likely to be impossible to completely isolate the collection. What happens during a partial decoherence? Here is my answer. During an encounter with a stray potential the off-diagonal terms of the density matrix of the system are slightly smaller. One can get a handle on this situation by splitting the density matrix into a linear superposition of two density matrices, one with zero off-diagonal elements and a second with diagonal elements somewhat reduced. Let the two coefficients of the superposition be c₁ and c₂. Then c₁*c₁ is the probability that decoherence has occurred and c₂*c₂ is the probability that the calculation is OK. I have applied a probability interpretation to the situation, a satisfying idea where quantum physics is concerned. In many cases a quantum calculation seeks an answer which takes too long to find with a conventional computer, but which is easily tested if found. With a quantum computer subject to decoherence one simply repeats the calculation until the answer shows up. Provided the isolation of the system is good, this should not require many repeats. Whether or not my ideas about partial decoherence are valid, it is clear that the entire situation about quantum measurement and decoherence will become clear as quantum computers are developed.

To close this post, I want to consider my conscious motivations in talking about quantum decoherence and my engagement with it. One motivation is that this is an interesting story which goes a long way towards answering the puzzles of quantum measurement, decoherence and collapse. I believe that this history makes clear that the long-standing difficulties in this area which have led to much controversy, are puzzles in the Kuhnian sense and require no radical revolution involving quantum mechanics. A second motivation is personal. Although I certainly deserve no credit whatsoever in the story of how quantum decoherence came into being, I did have an understanding of the situation before the march of science explicated it and it gives me satisfaction to make my involvement public. A final motivation involves my hopes for this blog. I hope the story of my involvement with physics makes clear that I was a hard headed, skeptical practitioner of a basic science and that in promoting Western Zen I’m dedicated to a superstition-free insight that provides a unifying sub-structure for all of Western, and indeed, non-Western World thought.

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