Posted on 12/03/2005 10:24:55 PM PST by sourcery
Scientists at the Commerce Department’s National Institute of Standards and Technology (NIST) have coaxed six atoms into spinning together in two opposite directions at the same time, a so-called Schrödinger “cat” state that obeys the unusual laws of quantum physics. The ambitious choreography could be useful in applications such as quantum computing and cryptography, as well as ultra-sensitive measurement techniques, all of which rely on exquisite control of nature’s smallest particles.
The experiment, which was unusually challenging even for scientists accustomed to crossing the boundary between the macroscopic and quantum worlds, is described in the Dec. 1 issue of Nature.* NIST scientists entangled six beryllium ions (charged atoms) so that their nuclei were collectively spinning clockwise and counterclockwise at the same time. Entanglement, which Albert Einstein called “spooky action at a distance,” occurs when the quantum properties of two or more particles are correlated. The NIST work, along with a paper by Austrian scientists published in the same issue of Nature, breaks new ground for entanglement of multiple particles in the laboratory. The previous record was five entangled photons, the smallest particles of light.
“It is very difficult to control six ions precisely for a long enough time to do an experiment like this,” says physicist Dietrich Leibfried, lead author of the NIST paper.
The ability to exist in two states at once is another peculiar property of quantum physics known as “superposition.” The NIST ions were placed in the most extreme superposition of spin states possible with six ions. All six nuclei are spinning in one direction and the opposite direction simultaneously or what physicists call Schrödinger cat states. The name was coined in a famous 1935 essay in which German physicist Erwin Schrödinger described an extreme theoretical case of being in two states simultaneously, namely a cat that is both dead and alive at the same time.
Schrödinger’s point was that cats are never observed in such states in the macroscopic “real world,” so there seems to be a boundary where the strange properties of quantum mechanics—the rule book for Nature’s smallest particles—give way to everyday experience. The NIST work, while a long way from full entanglement of a real cat’s roughly 1026 atoms, extends the domain where Schrödinger cat states can exist to at least six atoms. The Austrian team used a different approach to entangle more ions (eight) but in a less sensitive state.
In the NIST experiment, the ions are held a few micrometers apart in an electromagnetic trap. Ultraviolet lasers are used to cool the ions to near absolute zero and manipulate them in three steps. To create and maintain the cat states, the researchers fine-tuned trap conditions to reduce unwanted heating of the ions, improved cooling methods, and automated some of the calibrations and other formerly manual processes. One run of the experiment takes about 1 millisecond; the cat states last about 50 microseconds (about 1/20 as long). The team ran the experiment successfully tens of thousands of times, including numerous runs that entangled four, five, or six ions.
Entanglement and superpositions are being exploited in laboratories around the world in the development of new technologies such as quantum computers. If they can be built, quantum computers could solve certain problems in an exponentially shorter time than conventional computers of a similar size. For example, current supercomputers would require years to break today’s best encryption codes, (which are used to keep bank transactions and other important information secret) while quantum computers could quickly decipher the codes. Quantum computers also may be useful for optimizing complex systems such as airline schedules and database searching, developing "fraud-proof" digital signatures, or simulating complex biological systems for use in drug design.
Cat states, because they are superpositions of opposite overall properties that are relatively easy to verify, could be useful in a NIST-proposed design for fault-tolerant quantum computers. In addition, cat states are more sensitive to disturbance than other types of superpositions, a potentially useful feature in certain forms of quantum encryption, a new method for protecting information by making virtually all eavesdropping detectable.
The entangled cat states created by the NIST researchers also might be used to improve precision instruments, such as atomic clocks or interferometers that measure microscopic distances. Six ions entangled in a cat state are about 2½ times more sensitive to external magnetic fields than six unentangled ions, offering the possibility of better magnetic field sensors, or (for fixed external magnetic fields) better frequency sensors, which are components of atomic clocks. In addition, correlations between entangled ions could improve measurement precision, because a measurement of the spin of one of the entangled ions makes it possible to predict the spin of all remaining ions with certainty.
The research was funded by the Advanced Research and Development Activity/ National Security Agency, the Department of Defense Multidisciplinary University Research Initiative Program administered by the Office of Naval Research, and NIST.
More information about NIST research on quantum computing and cryptography, and spin-off applications in measurement science, is available at http://qubit.nist.gov.
As a non-regulatory agency of the Commerce Department’s Technology Administration, NIST promotes U.S. innovation and industrial competitiveness by advancing measurement science, standards and technology in ways that enhance economic security and improve our quality of life.
1 = 1
1/-1 = -1/1
(cross multiply)
-1 * -1 = 1 * 1
(-1) ** 2 = (1)**2
(take square roots)
-1 = 1
Tee hee!
Which points up that an operation performed on both sides of an equation had better return a single, unique value.
You are absolutely correct - this is a common misconception about entanglement that I've even seen a few physicists fall into.
Quote from a 1997 physics news article Signal Travels Farther and Faster Than Light
"Whatever the nature of the connection between entangled particles may be, nearly all physicists agree that it cannot be used to transmit messages faster than the speed of light. All it can do is assure that a random choice by one entangled particle is instantly echoed by its distant partner. This is not the same thing as transmitting information, the experts say, and therefore it does not violate relativity theory."
Hmm. Is that supposed to refer to wife beating in the Bible? How?
for making my brain hurt after a long work day:
*SMACK!!!*
Now my HEAD is spinning in both directions :-)
But I deserved it...
HA! Let's see the muslimes do this!
It is a classic formatting problem, which I'm surprised to see is mirrored in the original source article.
Cat Burglars beware.
So should another name be used for the states of the entangled particles. Dog states, anyone?
But, for the record, the positive square root of the square of a quantity is equal to the absolute value of the quantity; it is not equal to the quantity with its original sign. That is,
√[ x2 ] = |x| .
Hence, since
√[ (-1)2 ] = |-1| = 1
and
√[ (1)2 ] = |1| = 1 ,
there's no contradiction.
Shame on you, g_w, for giving KP a headache!
Tee hee.
It was a jest.
(I saw it on in a T-shirt catalog once, next to the "Reunite Gondwanaland" and "If you're not part of the solution, you're part of the precipitate".)
Cheers!
...and BTW, great job on the radical signs in HTML in your post.
It was a jest.
Mine was, too. I guess I'm getting heavy-handed (not to mention heavy-headed) in my dotage!
Wow, that must be one small cat!
I still have a problem with the generalization that an obervation cannot be made without affecting the outcome. I would accept that in some cases, even most cases, that an observation would affect the outcome, but absolutes seem to be dangerous territory.
Ah! The time-dependent one-dimensional Schrödinger equation! Be still my beating heart! The gentle curves of your partial derivatives are so much more romantic than the harsh Bras and Kets of the Dirac formulation! ;-)
Bump. Neat.
For example, we wish to know "a" and "b" from the following equation:
x^3 +1 = (x+1)(x-a)(x-b)=0
The values of "a" and "b" will be the other two cube roots of "-1".
I'm not aware of a "procedure" for factoring polynomials which have complex roots.
But there is a way to find these roots using the known geometric properties of roots in the complex number plane.
Every complex number can be represented in the form "a+bi", where a is the real component and b is the imaginary component.
Mapping such a number in the complex number plane results in a single point.
There is an alternative method for designating complex numbers which use the magnitude and direction of the vector which is represented by that same point.
The direction along the positive real axis is at angle zero. The direction along the positive imaginary axis is at angle 90 degrees. Any complex number can be represented by the angle and magnitude of the vector to that point from the origin.
The real number "1" is represented as (1,0), which means we have the vector which is one unit long pointed along the zero degrees line to the right of the origin.
The real number "-1" is represented as (1,180), which means we have the vector one unit long which is pointed leftward along the negative real axis.
The imaginary number "i" is represented as (1,90) which means we have the vector one unit long which is pointed upward along the postive imaginary axis.
The imaginary number "-i" is represented as (1,270) which means we have the vector one unit long which is pointed downward along the negative imaginary axis.
Note that there is NOT a unique representation for each complex number. For any complex number (r,theta) there are an infinite number of representations in the form, (r,theta + n*360) where n can take on any integer value.
The neat part is that finding "roots" of numbers using this "magnitude-angle" representation is relatively simple. One simply finds the real root of the magnitude and divides the angle by the root being sought.
To find one of the "second" roots of a number ( the square root) simply find the square root of the magnitude and divide the angle by 2.
The remaining roots have the same magnitude but are rotated through angles which cause all of the roots to be equally spaced throughout 360 degrees.
A first example: find the two square roots of "1". The number "1" has a magnitude-angle representation which is (1,0). The positive square root of the magnitude is "1", so this is the magnitude of both square roots we are looking for. To find the angle for the first root, we divide the angle of the original number by 2, giving "0". So the first square root of (1,0) is (1,0).
To find the other square root, we recognize that two equally spaced roots will be 180 degrees apart, so that the angle for the second square root is 180. The magnitude of the second square root is the same magnitude as the other root, so the second square root is (1,180).
The first square root of "1" is therefore the vector which is "1" unit long and has angle "0", so it is identical to the original number.
The second square root of "1" is the vector which is "1" unit long and has angle "180", so it is the number which is one unit to the left of the origin of the complex number plane, or "-1".
Now lets find the cube roots of "-1".
We start by representing "-1" in magnitude-angle form, or (1,180). Then we find the first of the three cube roots by taking the cube root of the real-number magnitude, which is "1". We find the angle of the first root by dividing the original angle by "3", which gives us "60". So the first cube root of "1" is (1,60). That is, it has magnitude "1" and is in the direction sixty degrees above the positive real number axis".
The other two cube roots have the same magnitude as the first cube root, but are equally spaced around 360 degrees. This means we add 120 degrees to the angle of the first root to get the angle of the second cube root. We add 240 degrees to the angle of the first root to get the angle of the third cube root. The two roots have magnitude-angle representations of (1,180) and (1,300).
The final step is to convert the cube roots to "a+bi" form to get the complex roots being sought.
The first cube root was (1,60). That is, magnitude "1" at angle "60". This means that the real-coodinate is "1/2" and the imaginary coordinate is "sqrt(3)/2". The complex number is then "1/2 +i*sqrt(3)/2".
The second cube root was (1,180). This is simply the real number "-1", as expected.
The third cube root was (1,300). That is, magnitude "1" at angle "300". This means that the real-coodinate is "1/2" and the imaginary coordinate is "-sqrt(3)/2". The complex number is then "1/2 - i*sqrt(3)/2".
Now we can plug these values back into the original equation as follows:
x^3 +1 = 0 = (x+1)*(x - 1/2 - i*sqrt(3)/2 )*(x - 1/2 + i*sqrt(3)/2 )
I'll let you multiply out the expression on the right to demonstrate that its value is "0" (Let's hope I made no errors.)
Note that the method above can be used to find arbitrary roots of any complex number. Multiplications, divisions, and other arithmetic operations can be similarly performed.
"Doesn't work that way. You can't send information faster than light."
I believe the jury is still out on that one.
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