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"All we do is draw little arrows..." Richard P. Feynman.
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Sunday, January 4, 2009
State of polarization of a triphoton
The indeterminacy in the state of polarization of a combined state of three photons (a triphoton) may be pictured on a sphere. Krister Shalm, Rob Adamson and Aephraim Steinberg of University of Toronto's Department of Physics and Centre for Quantum Information and Quantum Control, published their work in Nature and have some very interesting pictures and animation on the squeezing of the state of polarization of a triphoton. I wonder how we could picture this with spinning arrows. I should have a closer look at it.
Saturday, January 3, 2009
Quantum probabilities with ordinary objects
It is commonly asserted that quantum probability distributions may not be obtained with ordinary everyday objects. For example, in his sixth Stanford lecture on Quantum Mechanics, Professor Leonard Susskind says that "it is quite hard to think of a classical setup that would produce the same kind of probability distributions and how they depend on the orientation of the polarizers. I don't think anybody has ever successfully designed such a thing, at least not in any great generality that would do the right thing to polarized photons". There have already been some tries to design such experiments, for example with Aerts' Quantum Machine. I have always wondered why there has not been more research on such models, especially using Bohmian pilot-waves. The following video gives a proposal for such a setup with ordinary objects.
Here is the videoscript:
Hello, I’m Arjen the Common Sense Quantum Physicist. This sequence is a video comment on a point mentioned by professor Leonard Susskind in his sixth lecture on QM. I highly recommend this lecture for those who want to be introduced to QM and to the way how quantum state vectors are used to compute measurement probabilities on the polarization of photons. At one point in this lecture, at about the 12th minute, professor Susskind asserts that “and it is quite hard to think of a classical setup that would produce the same kind of probability distributions and how they depend on the orientation of the polarizers. I don't think anybody has ever successfully designed such a thing, at least not in any great generality that would do the right thing to polarized photons”. Here it is commonly accepted that no classical model could reproduce the full outcome of polarization measurements on photons. There are no means to obtain two distinct outcomes with the quantum probabilities cos² theta and sin² theta for instance with flying bullets that are flagged with a direction theta. However, and here is the point I want to put forward, if one makes use of so called pilot-waves, it is possible to reproduce the quantum probabilities with a model that uses ordinary everyday objects. Pilot-waves were introduced by Louis de Broglie in the 1920's in order to make some intuitive sense of Quantum Mechanics and were later rediscovered by David Bohm. Pilot-waves have the ability to steer the motion of particles. So let’s see how we could design such an experiment with ordinary objects and that reproduces the quantum probabilities of polarized photons.
We start with an object that we may describe with the help of a quantum state vector, for instance a spinning needle of unit length that we shoot in the z-direction. For reasons of simplicity, we restrict the example to the situation where the needle spins in a plane parallel to the z-direction. If the needle spins parallely to the x-z-plane, we denote its state vector by |x> or (1 0). If the needle spins parallely to the y-z-plane, we denote its state vector by |y> or (0 1) and if the needle spins in an arbitrary plane of angle theta with the x-z-plane, still parallel to the z-direction, we denote its state vector |theta> by cos(th) |x> + sin(th) |y>. So if th=0, we check that |theta=0> = 1.|x> + 0.|y> = |x>. The same for theta=90°, we check that |theta=90°> = 0.|x> + 1.|y> = |y>. So we’ve assigned a quantum state vector to the spinning state of an ordinary object. So that's the easy part!
The next step is to describe a two-valued measurement R_M where one value is obtained with probability cos²th and the alternative value with probability sin²th if the needle is in state |theta>. Let us use a wire-grid whose wires are spaced by the length of the needle and put it on the path of the needles, perpendicularly to the propagation direction. Let us then define the result RM = +1, if the needle passes through the grid without touching any wire: then it wins a point. If the needle touches a wire it looses one point: the result of the measurement will be R_M = -1.
We are considering the ideal case where the needle and the wires of the grid are infinitely thin. We may do this because this is a thought experiment. In real life, needles and wires always have a finite thickness and that would change a bit the results of the experiment. But for the sake of simplicity let us ignore that. Let us now fix the direction of the wire grid such that the wires are vertically aligned along the x-direction.
So, if the needle is in the |x> state, that is if it is spinning parallely to the x-z-plane, there is zero probability for that needle to touch a wire of the grid because we are considering the ideal case of infinitely thin needles and wires. The result for a needle with th = 0 is always +1. So there is a probability cos² th = cos²0 = 1 that it will pass the grid unaffected. So we have a probability 1 for that possibility.
If the needle is in the |y> state, that is if it is spinning parallely to the theta = 90° y-z-plane, we want the needle to have cos² theta = cos²90° = zero probability to pass the grid unaffected. So we want it to always touch a wire of the grid. Well, remembering that the spacing between the wires is equal to the unit-length of the needle, this may be achieved if the needle always has an angle of 90° with the z-axis when it arrives at the plane of the wiregrid. For theta = 90°, we need a kind of pilot-wave that steers the spinning motion of the needle in such a way that the phase of the needle with respect to the z-direction is always 90° when it arrives at the plane of the wire-grid.
And if the needle is in the |theta> state = cos(th) |x> + sin(th) |y>, we want the probability to pass the grid unaffected to be cos²(th). Equivalently this means that the probability to touch a wire of the grid must be 1-cos²(th) = sin²(th), which physically means that the length of the projection of the needle on the y-axis must be sin²(th) when it arrives at the plane of the wire-grid. Well, we can verify that this is arranged if the pilot-wave steers the needle in such a way that the phase of the needle with respect to the z-direction is always theta (or 180°-theta) when it arrives at the plane of the wire-grid.
So we've managed to retrieve quantum probabilities with ordinary spinning needles, provided that the orientation of the needles is steered by a pilot-wave. Building such an experiment is not trivial, but with some creativity it is in principle possible. At least someone could simulate it with some nice computer animation. This would allow easier visualization of quantum processes.
Here is the videoscript:
Hello, I’m Arjen the Common Sense Quantum Physicist. This sequence is a video comment on a point mentioned by professor Leonard Susskind in his sixth lecture on QM. I highly recommend this lecture for those who want to be introduced to QM and to the way how quantum state vectors are used to compute measurement probabilities on the polarization of photons. At one point in this lecture, at about the 12th minute, professor Susskind asserts that “and it is quite hard to think of a classical setup that would produce the same kind of probability distributions and how they depend on the orientation of the polarizers. I don't think anybody has ever successfully designed such a thing, at least not in any great generality that would do the right thing to polarized photons”. Here it is commonly accepted that no classical model could reproduce the full outcome of polarization measurements on photons. There are no means to obtain two distinct outcomes with the quantum probabilities cos² theta and sin² theta for instance with flying bullets that are flagged with a direction theta. However, and here is the point I want to put forward, if one makes use of so called pilot-waves, it is possible to reproduce the quantum probabilities with a model that uses ordinary everyday objects. Pilot-waves were introduced by Louis de Broglie in the 1920's in order to make some intuitive sense of Quantum Mechanics and were later rediscovered by David Bohm. Pilot-waves have the ability to steer the motion of particles. So let’s see how we could design such an experiment with ordinary objects and that reproduces the quantum probabilities of polarized photons.
We start with an object that we may describe with the help of a quantum state vector, for instance a spinning needle of unit length that we shoot in the z-direction. For reasons of simplicity, we restrict the example to the situation where the needle spins in a plane parallel to the z-direction. If the needle spins parallely to the x-z-plane, we denote its state vector by |x> or (1 0). If the needle spins parallely to the y-z-plane, we denote its state vector by |y> or (0 1) and if the needle spins in an arbitrary plane of angle theta with the x-z-plane, still parallel to the z-direction, we denote its state vector |theta> by cos(th) |x> + sin(th) |y>. So if th=0, we check that |theta=0> = 1.|x> + 0.|y> = |x>. The same for theta=90°, we check that |theta=90°> = 0.|x> + 1.|y> = |y>. So we’ve assigned a quantum state vector to the spinning state of an ordinary object. So that's the easy part!
The next step is to describe a two-valued measurement R_M where one value is obtained with probability cos²th and the alternative value with probability sin²th if the needle is in state |theta>. Let us use a wire-grid whose wires are spaced by the length of the needle and put it on the path of the needles, perpendicularly to the propagation direction. Let us then define the result RM = +1, if the needle passes through the grid without touching any wire: then it wins a point. If the needle touches a wire it looses one point: the result of the measurement will be R_M = -1.
We are considering the ideal case where the needle and the wires of the grid are infinitely thin. We may do this because this is a thought experiment. In real life, needles and wires always have a finite thickness and that would change a bit the results of the experiment. But for the sake of simplicity let us ignore that. Let us now fix the direction of the wire grid such that the wires are vertically aligned along the x-direction.
So, if the needle is in the |x> state, that is if it is spinning parallely to the x-z-plane, there is zero probability for that needle to touch a wire of the grid because we are considering the ideal case of infinitely thin needles and wires. The result for a needle with th = 0 is always +1. So there is a probability cos² th = cos²0 = 1 that it will pass the grid unaffected. So we have a probability 1 for that possibility.
If the needle is in the |y> state, that is if it is spinning parallely to the theta = 90° y-z-plane, we want the needle to have cos² theta = cos²90° = zero probability to pass the grid unaffected. So we want it to always touch a wire of the grid. Well, remembering that the spacing between the wires is equal to the unit-length of the needle, this may be achieved if the needle always has an angle of 90° with the z-axis when it arrives at the plane of the wiregrid. For theta = 90°, we need a kind of pilot-wave that steers the spinning motion of the needle in such a way that the phase of the needle with respect to the z-direction is always 90° when it arrives at the plane of the wire-grid.
And if the needle is in the |theta> state = cos(th) |x> + sin(th) |y>, we want the probability to pass the grid unaffected to be cos²(th). Equivalently this means that the probability to touch a wire of the grid must be 1-cos²(th) = sin²(th), which physically means that the length of the projection of the needle on the y-axis must be sin²(th) when it arrives at the plane of the wire-grid. Well, we can verify that this is arranged if the pilot-wave steers the needle in such a way that the phase of the needle with respect to the z-direction is always theta (or 180°-theta) when it arrives at the plane of the wire-grid.
So we've managed to retrieve quantum probabilities with ordinary spinning needles, provided that the orientation of the needles is steered by a pilot-wave. Building such an experiment is not trivial, but with some creativity it is in principle possible. At least someone could simulate it with some nice computer animation. This would allow easier visualization of quantum processes.