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| Oktay Pashaev & Aygul Koçak, Izmir Institute of Technology |
Most mornings I begin my day by looking at two of my favorite web sites -- NASA's
Astronomy Picture of the Day (APOD) where you are sure to find some stunning view of our Universe to lift you out of your daily grind and the
Cornell/Los Alamos ArXiv which publishes preprints of fresh new science papers in dozens of different specialties, putting anyone with an iPad in daily touch with some of the most brilliant minds on the planet. All this while sipping a cup of exotic coffee from
my friends at Boardwalk Beans in New Jersey.
A few days ago, I discovered a paper on the quantum physics arXiv by two mathematical physicists from Izmar, Turkey (formerly known as "Smyrna") entitled
"Kaleidoscope of Quantum Coherent States". These two researchers, Oktay Pashaev and Ayguy Koçak, had devised an infinite set of brand new breeds of Schrödinger Cats.
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| Schrödinger's Cat in bra-ket notation |
In quantum mechanics it is commonplace for a system to be in a SUPERPOSITION of states. An (unmeasured) electron's spin, for instance, can simultaneously exist in a spin-up state |UP> and a spin-down state |DOWN>. When measured, however, the electron is always observed to be in one definite spin state. Austrian physicist Erwin Schrödinger, shortly after he invented his famous quantum wave equation, argued that if unmeasured electrons could exist in two states at once, so could cats, and he devised a famous thought experiment in which an unobserved cat could, according to the laws of quantum physics, exist simultaneous as a live cat |ALIVE> and as a dead cat |DEAD>. Schrödinger's famous alive/dead cat conjecture has generated thousands of physics papers on the possible application of quantum superposition to macroscopic objects and numerous jokes, cartoons and T-shirts ("Schrödinger's Cat is a zombie" reads a T-shirt my neighbor Debi gave me for my birthday.).
Schrödinger's cat walks into a bar. And doesn't.
A brief note on notation. When physicists write down their quantum equations, they commonly use the compact and powerful bra-ket notation devised by British physicist Paul Dirac. In Dirac notation, a quantum initial state A is symbolized by a ket symbol |A> and a quantum final state B by a bra symbol <B|. When multiplied together <B|A> represents the
probability amplitude that a quantum system A will be measured to have property B. The
probability (different from
probability amplitude) that A will be measured to have property B is given by the absolute square of the quantity: <B|A>
As a rough example of this kind of physics talk, let the ket |p,p> represent the initial quantum state of two protons. Let transformation T represent the act of accelerating each of these protons to an energy of 6 Gev in CERN's Large Hadron Collider and nudging them into a head-on collision. And let the bra <H,a| represent the final state that contains a Higgs boson and anything else.
Then, in Dirac's concise notation:
<H,a|T|p,p>
represents a number that expresses the probability amplitude of observing a Higgs boson. Square this quantity to get the probability of observing a Higgs boson.
Dirac's simple notation tells you basically what's going on by concealing a ton of detailed math that you really don't want to know about.
So, using Dirac's bra-ket notation, the quantum state of Schrödinger's cat can be simply represented as:
|ALIVE> + |DEAD>
Or, in a more picturesque description, as:
This is the picture one usual gets about Schrödinger's famous cat -- he's both dead PLUS alive.
Quantum mechanics, however, is more complicated than that, and allows for many more existential possibilities for this hapless quantum cat. Quantum mechanical superposition uses COMPLEX NUMBERS (which possess a direction: North, South, East, West, for instance) as well as a magnitude. (Numbers that possess only magnitude but not direction -- the kind of numbers we use every day -- are called REAL NUMBERS).
Using the extra degrees of freedom provided by complex numbers, the |ALIVE> and |DEAD> states can be "added together" in an infinite number of ways. If we let the direction "East" represent "+", then the direction "West" will represent "-". Using "West addition" to combine the two cat states we obtain what might be called a MINUS CAT KET.
Schrödinger's MINUS CAT KET, in pictures, might look like this:
In addition to the PLUS CAT state and the MINUS CAT state, the arithmetical freedom provided by complex numbers allows us to imagine NORTH CAT, SOUTH CAT and NNW CAT states. And, in fact, LIVE and DEAD cats may be added together along any conceivable compass direction.
Whether actual cats can be subjected to quantum superposition is still a matter of some controversy, but there does exist a class of macroscopic states of light that can be placed in a variety of quantum superpositions.
Today's physicists probably know more about light than about any other natural phenomenon. Starting with all the natural forms of electromagnetic radiation, we have created both in theory and in practice a large variety of "unnatural" forms of light, some of which were recently invented in this new paper by Pashaev and Koçak.
Pashaev and Koçak begin their work with a familiar quantum state of light |α> called the "Glauber State" after optical physicist Roy Glauber.
The Glauber state is a quantum state (also called "coherent state", hence the title of P&K's paper) that most closely approximates a classical state of light, possessing Heisenberg uncertainty and photons (light quanta) which, however, the corresponding classical state of light does not. The quantity "α" which labels the Glauber state is a complex number. The square of α represents the average number of photons in the Glauber state. And the direction of α (North, South, East or West) represents the location of the Glauber state in a flat space physicists call the "optical phase plane".
The larger the number α, the more photons in the Glauber state |α>. The special case of α = 0 represents no photons whatsoever, or the vacuum state. Many books could be written about the properties of |0>, the quantum vacuum state. "I've got plenty of nothing. And nothing's plenty for me." might well be the theme song of this particular Glauber state, a state that is completely empty of photons.
Prior to the P&K paper, the Optical Schrödinger Cat (OSC) was well known. It consisted of two states from which all other OSCs could be constructed: the PLUS OPTICAL CAT STATE
|plus optical cat state> = |α> + |-α>
and the MINUS OPTICAL CAT STATE :
|minus optical cat state> = |α> - |-α>
The heart of the Schrödinger Cat controversy concerns the question of how big a system can get before it becomes impossible to place it in a quantum superposition. Optical Schrödinger Cats are in a particularly fortunate position to investigate this question because the larger the number of photons in an optical S-Cat state, the "bigger" the state -- and the more it resembles a classical "cat". In the other direction, when α is small (close to 1 photon), the resulting optical states are sometimes referred to as "Schrödinger Kittens".
To construct their "Optical Cat Kaleidoscopes", the two Turks take advantage of the fact that both α and the coefficients multiplying the optical quantum states |α> are complex numbers -- that is, they possess direction as well as magnitude.
The well-known plus and minus optical cats may be considered "cats of order two (C2)." The first new cat in P&K's infinite series of kaleidoscopic cats may be labeled "cats of order three (C3)." Cats of order three are constructed by adding particular cats with different kinds of dead/aliveness along directions that are separated by 120 degrees (similar to the Mercedes emblem). A caricature of the P&K "three cat" might look like this:
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| Quantum optical trinity cats in Dirac ket notation. |
Or, in keeping with the kaleidoscopic metaphor, C3 could look like this:
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| Three-fold kaleidoscopic optical Schrödinger's Cat (artist's conception) |
Pashaev and Koçak go on to show how kaleidoscopic optical Schrödinger Cats of any order can be constructed, depending on which angle you tilt your mathematical mirrors. On its own terms, theirs is a simple but beautiful achievement of pure mathematics. But the authors go further and show how their kaleidoscopic optical cats may someday find a practical use in quantum computing -- each order of cat representing a different number of quantum bits. Thus, if I am not mistaken, the eighth-order cat (octopussy?) can encode eight quantum bits, the same byte size as
the ancient Altair computer and many of its successors.
As a poetic reprieve from so much gratuitous quantum math, this may be a good place to quote British mystic William Blake from a letter to his friend Thomas Butts:
Now I a fourfold vision see
And a fourfold vision is given to me
Tis fourfold in supreme delight
And three-fold in soft Beulah's night
And twofold Always. May God us keep
From Single vision & Newton's sleep.
