Tuesday, June 30, 2015

Six Optical Miracles

Paul Peter Ewald (1888-1985) and Carl Wilhelm Oseen (1879-1944)
Exactly 100 years ago a remarkable paper appeared in Annalen der Physik, a prestigious German physics journal in which both Max Planck and Albert Einstein published their findings. This paper, by German physicist Paul Peter Ewald explained how light behaves when it strikes a sheet of glass. Ewald, in 1915, explained, in effect, how a window works. A few year later, Swedish physicist Carl Wilhelm Oseen extended these findings to explain how light behaves when it strikes a crystal. Together the work of these two men is known as the Ewald-Oseen Extinction Theorem.

What's so mysterious about how a window works. Isn't a window merely a sheet of glass?

A light wave traveling through a sheet of glass
When light strikes a window some of it bounces off (about 10%) and the remainder is refracted (bent) into the glass at an angle that depends on a number "n" called the refractive index which is different for different materials. If n is greater than 1, the light bends deeper into the material; if n is less than 1, the light bends towards the surface. 

In a vacuum the speed of light is equal to a constant c. But in material media, the velocity v of light is equal to: v = c/n. For visible light in glass, the refractive index is about 1.5, so light travels at about 70% of its vacuum speed: inside glass, light travels SLOW. On the other hand, for X-rays in many materials, the refractive index is less than 1, so X-ray light travels FAST -- faster than light in a vacuum.

But those "in the know" realize that not only did Einstein prohibit anything from traveling faster than light, he also showed that light always travels at the same speed no matter who's looking at it. So both the notion of fast X-ray light and slow light in glass would seem to be outlawed by relativity. However this notion of fast and slow light in material media is being taught to high school students. It's called Snell's Law (and seems to have been discovered first by Arab physicist ibn Sahl during the Islamic Golden Age.

Snell's Law (Ibn Sahl's Law) does not violate relativity, because Einstein and Snell (ibn Sahl) are talking about two different ways of measuring velocity. Einstein's prohibition refers to group velocity, how fast a packet of light can move, while Snell's Law refers to phase velocity, how fast the peaks of a wavelet can move. The difference between group and phase velocity is often illustrated by the difference between how fast a caterpillar moves (group velocity) and how fast one of his humps moves (phase velocity). It is clear from this analogy that the phase velocity can be faster or slower than whatever speed limit the caterpillar must obey. Here's a nice animation showing the difference between group and phase velocity for deep ocean waves. (In this ocean wave case, phase velocity is faster than group velocity.)

To fully appreciate the miraculousness of the Ewald-Oseen Extinction Theorem, we must first realize that glass consists of a random arrangement of electrically-excitable molecules. And that light is a traveling electromagnetic wave that will excite each of the molecules it encounters. Each excited molecule will emit a omni-directional light wave (of the same frequency as the incident light) in somewhat the same manner as an ocean buoy will emit a scattered wave when struck by an incident wave train.

Incident ocean wave striking a buoy produces scattered wave.
But light approaching a sheet of glass is not just going to strike one buoy, but approximately 10^24 buoys or about 1 million billion billion buoys (electrical excitable molecules). And these buoys (in glass) are arranged in a random fashion. What is the first thing you would imagine happening if a wave of light impinged on a medium composed of zillions of randomly arranged excitable molecules? Total chaos. That's what I would guess. The window will turn into some sort of deeply frosted glass that will randomly scatter light in every direction.

If windows are made of zillions of randomly arranged excitable molecules, then windows can not be transparent. That's what I would predict.

The first optical miracle is that windows are transparent. But the second, third and fourth miracle are even better.

Ewald proves for random media like glass (and Oseen does the same for ordered media such as crystals), that the net result of all of those zillions of little excited light waves add up to zero in every direction but three.

Only three big waves survive the fierce destructive interference between zillions of little waves. The first surviving wave travels in the same direction as the incident wave but vibrates exactly 180 degrees out of phase with the incident wave. Thus the incident wave is extinguished as it passes into the glass over a length of a few tenths of a micron (the extinction length). This exceedingly unlikely mechanism for extincting the incident wave gives the extinction theorem its odd name.

Two more waves survive the Great Destruction -- the reflected wave that seems to bounce off the glass but in reality is created by zillions of tiny molecules all radiating with the proper timing to direct a beam only in the reflected direction (and no other) much in the manner of phased-array radar antennas which do not physically move but are aimed by changing the phase relation between separate fixed transmitters. The glass molecules (like the phased-radars) are radiating in all directions but in both radar and window pane destructive interference removes light from every direction except one.

The third wave that survives the Great Cancellation is the internally transmitted wave that travels into the glass at "Snell's angle" with a "slow" phase velocity of c/n. Everywhere inside the glass the little wavelets travel at velocity c, but the net product of all their activity is an effective slow-phase wave, produced, like the reflected wave, by a kind of internal phased-antenna array consisting of zillions of glass molecules.

Here then are four optical miracles:

1. That glass is transparent despite its consisting of zillions of randomly situated electrically-excitable molecules;

2. That the excitable medium, all by itself, completely extinguishes the incident wave;

3. That the excitable medium, all by itself, synthesizes a "reflected wave" traveling in exactly the right direction and no other.

4. That the excitable medium, all by itself, synthesizes an "internal wave" traveling in exactly the right direction (Snell's Law) and no other.

These four miracles apply not just to windows but to anything made of glass, or to anything transparent for that matter, such as the lenses in telescopes, the lenses in your cameras, your eyeglasses, your contact lenses, the living lens and the clear fluid that fills your eye. The inner operation of each of these optical devices is explained by the four-fold miraculous Ewald-Oseen Extinction Theorem. (Happy Hundreth Anniversary!)

But wait. There's more. The EOET was derived in 1915 before quantum mechanics was devised. And quantum mechanics has one final miracle to add to the wonder of classical window panes and lenses.

Classical physics considered all of its waves to be "real", that is, made of something actually vibrating at every location where the wave exists. Quantum physics, on the other hand, considers its waves to be "unreal", that is, made only of sheer possibility. A light wave, for instance, represents only the possibility for a "photon" (or quantum of light) to be observed.

Quantum waves only become real in the act of observation, in a still mysterious process called by some "the collapse of the wave function".

Now imagine the situation of a single photon striking a window pane. I remind you that a window pane consists of a million billion billion electrically excitable molecules. And photons like to excite molecules. 

What do you suppose the odds are for the photon to exit the window pane without collapsing its wave function? Because there seem to be a zillion ways for that photon to lose its phase, to shed its dignity, to become entangled with the randomly arrayed electrically-excitable mess that we call "a pane of glass", I would naively guess that a pane of glass must collapse the photon wave function with 100% certainty.

But this conclusion is wrong. For exactly the same reason that the window pane is transparent (namely the Ewald-Oseen Extinction Theorem), the window pane has a lot of fun with the photon but does not actually collapse its wave function.

This is the fifth optical miracle: Window panes do not collapse the photon wave function.

Lenses do not collapse the photon wave function. Your contact lens does not collapse the photon wave function The living lens in your eye does not collapse the photon wave function. Your aqueous and vitreous humours do not collapse the photon wave function.

But inside your retina the photon wave function does somehow finally collapse into an actual event.

Which gives rise to this marvelous visual experience.

The sixth optical miracle.




Thursday, June 25, 2015

The Philosopher's Bone

Nick Herbert aka Dr Jabir 'abd al-Khaliq

THE PHILOSOPHER'S BONE

(for Kelly Evans)

I'd love to own the Philosopher's Bone
As big and hard as Allah's own
With Physics, Joy and Math inside It
And me be man enough to ride It
On my board that buggers the imagination
I'd surf the Curly Crimp of Creation
I'm strong and smart; I'm a quantum knave
But where can I go to catch Her Wave?

One hundred quantum billiard waves

Wednesday, June 17, 2015

Tom Herbert (1938 - 2015)


Tom Herbert was born on June 13, 1938 in Pittsburgh, PA and died on the same day in 2015 in Vienna, VA from complications following a stroke. He was the second-oldest of 5 children: Nick, Tom, Marilyn, Donald ("Duke") and Nancy who all grew up together in the Linden neighborhood of Columbus, OH, which was then populated mainly by immigrants from Italy, Ireland and Eastern Europe. Our Father spoke Ukrainian; Mom spoke Slovak. And both grew up in coal mining towns in Pennsylvania and Ohio, migrating North to Lorain, Ohio where some of the family found work in the steel mills along Lake Erie.

Thomas Anthony Herbert (1938 - 2015)
Tom was younger than Nick but stronger so we two were physically an even match to wrestle, box, sled, skate and explore the woods and creeks within easy reach of our neighborhood. Tom and I both went to grade school at St Augustine Catholic school (where Mom ran the cafeteria) and both completed high school at St Charles Borromeo Preparatory School. Tom was fond of animals and of water -- in our back yard he raised rabbits, chickens and ducks in addition to the obligatory dogs and cats. Not only did Tom work as a lifeguard at our local Morningside pool, but he also organized special events at the pool including water ballets. Tom also bought a kayak which he piloted on the Olentangy and Scioto rivers -- and he imagined that someday he would follow the route Huck Finn took down the Mississippi on his raft. The floor of our boy's bedroom was covered with a big linoleum map of the USA, so we could rehearse Tom's future trip down the big river using bottle caps for boats.

Tom, Nick and Duke Herbert at Betsy's Memorial Celebration
Both Tom and I obtained Bachelor's Degrees from Ohio State University, Tom in Electrical Engineering and Nick in Physics. For the next stage in his life, Nick made a move to California -- our parents gave me the family station wagon as a graduation present. Tom and I decided to use this opportunity to explore the country we had only seen printed on a floor mat. So we packed camping gear in the car and embarked on a long road trip. In those days (1960), the US was relatively uncrowded and on the few occasions we could not find a campground we just set up camp near the road and were never bothered. We explored Colorado, Utah, the Grand Canyon, Las Vegas and large parts of California together. Tom was a congenial companion and good camping buddy -- no matter what hardships we endured he would always see its humorous side. And, yes the two Ohio boys finally got to experience the mighty Mississippi (near St Louis) and to cool our feet in the Father of Waters.

Tom liked to eat and was especially fond of the Slovak food that Mom put on the table, which included stuffed peppers, stuffed cabbages (holubki), cabbage, nut and prune roll pastries. Tom enjoyed eating this kind of food so much that, unlike his brothers, he actually learned to duplicate Mom's recipes for himself. And in addition to cooking holubki, etc, Tom learned how to make Perfect Spaghetti Sauce which Mom probably learned from Leona Vanelli, her best friend. Mom used to say that Tom preferred a meal of her spaghetti and meatballs over steak.

Tom cooking Perfect Spaghetti Sauce
After our road trip, Tom flew back to Ohio State where he completed his degree, met and married his first wife Dawn and began an engineering career with the Air Force at Wright-Patterson in Dayton, Ohio. Tom and Dawn produced five children: Mark, Julie, Terri, Dan and Joe. After their divorce, Tom met a female physicist who had learned how to turn silicon into gold -- Gail Walters was a brilliant computer chip designer. Their combined careers took them to Colorado, to California's Livermore Valley and finally to their present home in Virginia near Washington, DC. Tom and Gail produced one child Meghan who, when asked what she wanted to do, answered: "Anything but what my parents do."

Tom Herbert and Gail Walters
Tom and Gail on a jet ski.
After leaving Wright-Patterson, Tom took on another government job that was so secret  he could not talk about it. Since this job often involved trips to exotic locations near the equator, I assumed it had something to do with the tracking of military spy satellites. Whatever the nature of his clandestine work, Tom always found time to explore the underwater beauties of these tropical locales in his SCUBA gear. Later Tom, Gail and Meghan would vacation in exotic places like the Caymen Islands that Tom had no doubt reconnoitered on his "business trips"

Tom Herbert in SCUBA gear
Becalmed?
Tom Herbert worked hard, played hard (and often underwater) raised two fine families, was possessed of a contagious cheerfulness. I do not know whether he believed in reincarnation, but I'd like to think that in his next life Tom might come back as a smart, sexy, good-hearted dolphin.

Tom embraces his alter ego
Bon voyage, wonderful Tom. I will miss you, little brother.




Sunday, May 31, 2015

Muldoon the solid man


In the latter half of the 19th Century, Irish politicians dominated the political life of New York and Boston. The Irish musical producers Harrigan and Hart (the Rogers and Hammerstein of their day) caricatured these immigrant upstarts in the figure of Michael Muldoon, the solid man. The song has long outlived its Victorian origins and recently appeared on a Chieftains album. This version that Nick performs in Santa Cruz's Floral Park was inspired by Dublin singer Frank Harte.

 For Californians, the "Tombs" and the "Island" in the third verse, where Muldoon's constituents go "to enjoy their summer recreation and the refreshing East River air",  refer to Manhattan's infamous Houses of Detention and Blackwell's Island. Then, as now, the line between criminals and lawmakers was not easy to discern.

Sunday, May 24, 2015

Thirteen Unnatural Acts

Doctor Jabir 'abd al-Khaliq

THIRTEEN UNNATURAL ACTS

Our fore-fathers and fore-mothers twisted
This natural human need to breed
Into pornography, into romance,
Into the practice of recreational sex.

Our distant ancestors perverted
This natural human need to breathe
Into speech, into singing
Into making music with wind instruments.

Our ingenious predecessors debased
This natural human need to eat
Into gluttony, into usury, into greed
Into grabbing more than we need.

And in these times we ourselves abuse
This natural human need to excrete
To poison the bees, to shit in the seas

To bequeath to our kids burning turds

Whose quantum atomic fires
Will long outlast our human deeds.

Friday, May 22, 2015

More Subatomic Particles

First 13 GeV collisions at CERN: May 20, 2015
Physicists at the Large Hadron Collider at CERN have for the fiirst time begun to collide protons at the record-breaking energy of 13 GeV. They are currently adjusting the collimators that protect parts of the accelerator from leakage of its own Gigavolt radiation. Especially vulnerable to fast-particle impact are the 1232 superconducting dipole magnets which guide both beams into a roughly circular path with a circumference of 27 kilometers. Early next month, serious experiments will begin at the highest particle energy every achieved in a physics lab. Congratulations, guys and gals!

CERN's new high-water mark reminds me of the first science fiction story I ever published -- in a collection called Semiotext(e) SF, edited by Rudy Rucker and Peter Lamborn Wilson (aka Hakim Bey). This theme of the book was the exploration of dangerous edges and includes such gems as William Gibson's "Hippy Hat Brain Parasite", "The New Boy" by William S. Burroughs, Hakim Bey's "Antarctic Autonomous Zone", "We See Things Differently" by Bruce Sterling, "Six Kinds of Darkness" by John Shirley and Rudy Rucker's "Rapture in Space".

Compared to these literary heavyweights, my contribution to Semiotext(e) may seem paltry. But I'm going to take the liberty of sharing it with you anyway:

MORE SUBATOMIC PARTICLES

Only a few weeks before the Esalen Seminar on the Nature of Reality, where top physicists and philosophers gather each year to decide what portion of the Great Secret to release to the public, private researchers at the Strongly Non-linear Accelerator Center (SNAC), Bonny Doon, California, announced the discovery of a handful of new particles that sent theorists back to their blackboards and experimentalists into their tunnels for ritual confirmation. The SNAC facility, the largest of its kind, produces the strongest weird-particle beam in the Solar System.

The first of the new particles -- christened the "Sigma" in honor of San Francisco-based theorist Saul-Paul Sirag, who predicted its non-existence -- appears to be another heavy lepton, resembling in all properties (save mass) its lighter brothers Mu and Tau. Chief SNAC scientists Drs. Kenneth Goffman and Alison Kennedy are modest about their discovery. "We see weirder things on our coffee breaks," they said, insisting that the Nobel Prize be shared with their reclusive colleague Dr. Nic Harvard, who provided the drugs necessary for the fundamental break-through.

After months of secrecy, the SNAC group's unorthodox techniques were finally revealed in Kennedy's controversial article "Quantum Tantra" published in last month's "Physical Review", the major scholarly journal of the American physics community. Kennedy's paper was immediately repudiated by established physicists for its blatant violation of normal standards of professional conduct, scientific rigor, and common decency.

Unlike the Sigma, which can be observed by any physicist with a research budget the size of America's National Debt, SNAC's other new particles are invisible to everyone not in the proper state of consciousness, the first of many examples of "state-specific physics" emerging from research projects initiated in the 1960's and only now beginning to bear fruit. In addition to their unanticipated physical properties, the new particles seem to possess mental attributes as well. Making contact is easy, but scientific study of these psychic quanta -- or "Persons", as they are called at SNAC -- is possible only by researchers with strong egos, since the experience is more akin to demonic possession than data collection.


"Expect to be changed a lot, when you mess around with Persons," warn researchers Goffman and Kennedy. "Each one is unique -- an utterly alien island of consciousness. The're around us all the time now, and the only thing they want to do is get inside your mind."


"We can't keep them in the lab. They fly right through our thickest shields. Before we could shut SNAC down, trillions of merge-hungry Persons had already escaped through the walls. They're slithering into restaurants now, into laundromats, stock exchanges and shopping malls, even into your bathtubs and bedrooms, attracted by concentrations of human emotion,"


"We're terribly sorry, folks, but everyone alive is involved in this experiment now. The era of state-specific physics has arrived."


"Tantra Door" by Iona Miller

Tuesday, May 19, 2015

Kiss My Bare Art

Nick Herbert: Das Gym, May 2015
KISS MY BARE ART

We drink our Muse; we smoke our Muse
We duct Her thru our gaping pores
Invent new sins to fan Her whims
We're lovely Muse's lowly whores.

We follow Muse beyond the stars
To bomb labs, muscle gyms and porno bars
Where Life beckons, there we go
Seek deeper meat than Jacques Cousteau.

We crave that rush, that punch, that flood
We love that dark orgasmic drain
Then pick ourselves up off the rug
And open up another vein.

For just one glance we drop our pants
Her prostitutes and renegades
Yet every kiss burns like the first
We're virginal as new-born babes.

Friday, May 15, 2015

Nick Meets Werner von Braun

President J. F. Kennedy and Werner von Braun: both shared a dream of landing a man on the Moon.
In 1957, I was an undergraduate at Ohio State University, a year away from a degree in Engineering Physics. I answered an ad for a summer job at Redstone Arsenal in Huntsville, Alabama. I look upon my trip to late '50s Alabama as my first visit to a foreign country.

My summer job at Redstone Arsenal was exceedingly boring, having to do mainly with documenting the mechanical parts that go into the construction of a rocket weapon. Perhaps because I possessed TOP SECRET security clearance and was almost a physicist, my bosses took mercy on me and awarded a few assignments that took me out of the blueprint trenches and into the upper atmosphere. Every week or so, I would visit an office labeled "nuclear warheads" to check if the production of these crucial components was still on schedule in line with our corporate flow chart.

Then our documentation group was assigned to publish an internal Glossary of Missile Terms for the enlightenment of the troops.  I was chosen to interview the boss of Redstone concerning, among other things, the definition of "ballistic missile". Our boss was Werner von Braun.

I elevated up to von Braun's office, was ushered in and shook his hand. He offered me a seat and I interviewed him for almost an hour about the basic physics of ballistic missiles. In 1957, von Braun was 45 years old. He appeared big and physical like a retired football player, restless and uneasy behind a desk, aching for some sort of action.

Von Braun taught me a lot about ballistic missiles. After the initial burn, a ballistic missile travels in "free fall" like a rock tossed into the air. Ideally such a rock or rocket would travel along a gravity-defined trajectory called a "parabola" (a curve that quirky author Thomas Pynchon dubbed Gravity's Rainbow). But other forces are at work on the rocket besides gravity-- primarily air resistance and fictitious forces due to the rotation of the Earth.

Inside the missile, engineers have installed an "inertial platform" constructed of gyroscopes and accelerometers. If gravity were the only player in the game, this inertial platform would feel no forces and the trajectory of the missile would perfectly track gravity's rainbow. However in the atmosphere of a rotating Earth, the missile will experience additional forces as it travels from launch site to target. A new trajectory is calculated on the ground for a model atmosphere and rotational corrections, this "more realistic" trajectory will produce an expected pattern of forces on the inertial platform.

If the inertial platform feels only these expected forces, it does nothing, but if new forces are encountered different from those predicted for the path stored in its computer memory, the missile fires small attitude-control rockets to bring the rocket back into its calculated course.

The flight of a ballistic missile then is not truly ballistic. Its theoretical trajectory is calculated with the best corrections available -- producing a non-parabolic arc between launch and target site. En route to the target, the inertial platform measures the deviation between this theoretical path and the path the missile is actually following and applies appropriate rocket thrust to bring the course in line with what theory says it ought to be.

After a few more questions about other topics, I thanked the boss for giving me so much of his time, shook his hand a second time and went back to my pals in the trenches. I returned to school in Ohio before the missile glossary was completed and a few months later, the Russians sent up Sputnik, whose beep-beep-beep from space forever changed the world.

Alabama in 1957 was indeed for me a foreign country, with its incomprehensible deep South dialect, its peculiar patterns of racial segregation, its colorful Bible-belt fundamental preachers on the stairs of the Huntsville Court House, born-again Baptist tent revivals in the countryside and the ever-present feeling in the air (and in the public statues of Confederate heroes) that for many residents of Alabama the War Between the States was still current news. I was often referred to as a "Yankee"

And only much later, back home in Columbus, did I realize that in that casual interview with Werner von Braun, Nick Herbert's skin had come in contact with real psychic power --  Nick's hand one mere handshake away from the hand of officially designated Grand Satan Unsurpassed.

Postage stamp honoring Werner von Braun



Saturday, May 9, 2015

Some Notes on Quantum Entanglement

Fig. 1: Kim et al experimental realization of path-entangled photons

SOME NOTES ON QUANTUM ENTANGLEMENT

As pop philosopher Robert Anton Wilson was fond of saying: "Quantum mechanics is as queer as a three-legged duck." The crux of the problem is that even physicists don't know how to speak correctly about quantum reality.

We physicists possess a sophisticated and exact quantum theory -- it's never been wrong. We possess delicate and sophisticated instruments, that Bohr and Heisenberg never dreamt of, for measuring happenings in the quantum world. Quantum theory and quantum measurement are in perfect shape. What more could one ask for?

One thing we might reasonably require is a quantum reality -- we could wish for the ability to tell a story about "what's really happening in the world" -- a story that does justice to the radical queerness of the quantum phenomena. And a quantum reality story is precisely what we physicists simply ain't got. Or rather, we have lots of quantum stories, each differing wildly from one another, and none of them quite satisfactory. Wanna stump your neighborhood physicist? Ask him or her "what really happens during a quantum measurement?"

To describe a complicated quantum experiment like the one pictured in Fig. 1 is almost impossible to do without employing some sort of tentative narrative about "what is really going on". So in addition to putting forth theory and experiment, both of which rest on solid ground but which for most of us seem colorless and boring, these notes will necessarily trespass into the dubious quantum reality zone and slip in some talk about "what seems to be really happening". So let the reader beware!

Everything in this world is ultimately made up of quantum systems. And physicists "represent" each quantum system the same way -- by means of a mathematical object they call the "wave function" (or, more generally, the "wave vector"). I say "represent" instead of "describe" because the relationship of the wave function to observation, let alone to reality, is more indirect than a "description".

To every object, a physicist associates a wave function, which he can point to but never write down. For instance, a physicist can say "Wave Function of the Hydrogen atom" but there is no mathematical image that corresponds to this name. Every representation of a quantum system must specify how you intend to measure it. If I intend to measure the momentum of the Hydrogen atom, I can write down the math for its "momentum representation" -- symbolized |H(p)>. If my intent is to measure position, I can write down its "position representation" -- symbolized |H(x)>. But physics provides no mathematical picture of "the thing in itself" -- no mathematical picture of the Hydrogen atom "as it really is" independent of your measurement intent.

[ CORRECTION: a friend reminds me that most simple quantum systems actually do possess "intrinsic qualities" called "eigenstates". For such systems, there exist measurements of particular observables that always give the same answer -- no probabilities here. For the Hydrogen Atom, such eigen-observables include its energy E and its angular momentum J. Independent of your intention to measure it, a Hydrogen atom can be said to actually possess a particular value of E and J. However, in the H-atom's position/momentum space, Nature has neglected to select a favored eigen-representation. In this space, writing down a wave function needs you to choose a measurement intent.] 

(A shorter word for "representation" is "basis". The Hydrogen wave function can be expressed in the momentum basis or the position basis. (The plural of "basis" = "bases"). A beautiful collection of visualizations of the Hydrogen atom's various position representations is Dean Dauger's Atom in a Box.  

 [CORRECTION: In keeping with the above correction, the different Hydrogen atoms in Dauger's collection are each labeled according to their eigen-values of E and J.]

It goes without saying that I can recommend no better guide to the quantum reality question than my own best-selling book on the topic -- recently made available by Doubleday as an e-book.

The numerical value of the wave function represents the "possibility" that a particular intent will be fulfilled. The square of this "possibility" represents the "probability of fulfillment" of some aspect of that intent -- the probability, say, that in the position representation, the photon will be found at position location x.

QUANTUM REALITY

Mother Nature is ready to give us
Anything we're smart enough to ask for
Where "asking' means choosing
Some unique receptive device.

Each kind of asking brings forth
Its own eigen-sack of possibilities
From which -- unpredictably --
She gives out just one, if we're nice.

Intentions, possibilities, probabilities, actualities? Cut to the chase: what's really going on in the quantum world? Today's physicists can't really answer this question.

Or as Albert Einstein once put it: "Who could have guessed that we would know so much and understand so little?"

But on to "quantum entanglement", which founding father Erwin Schrödinger characterized as not one but the feature of quantum theory that most distinguishes it from classical expectations.

The simplest example of quantum entanglement is a pair of path-entangled photons A and B whose wave function |ψ (A, B)>  can be written:

|ψ (A, B)> = S{ |A1>|B1> + |A2>|B2> }             (1)

This wave function represents a pair of photons A and B, each of which can travel along two paths, represented by numbers 1 and 2. The motions of the photons are correlated such that when photon
A takes path A1, then photon B always takes path B1. And when photon A takes path A2, then photon B always takes path B2. (S is just the number 1/(sqrt(2)).

In addition to being correlated, the two particles are entangled. The wave function |ψ (A, B)> represents a SUPERPOSITION of both possibilities. Thus (here comes a "forbidden story") each correlated photon "takes both paths at once" just like Schrödinger's Cat somehow is "both alive and dead". These human-baffling pictures of photons and cats are (probably misleading) attempts to explain a humanly incomprehensible quantum reality that lies beneath the visible world. In this note you will run across several such dubious stories, but this will be your last warning. Trust in mathematics (such as EQ (1)) and in measurement and you will never go wrong, but take all stories (even my own) about "what's really going on" with a grain of salt.

By the way, the wave function EQ (1) is written in the "path representation" or "path basis".

For me there is no more elegant realization of the classic double-slit experiment than that of Yoon-Ho Kim and his buddies at University of Maryland (henceforth Kim et al). The basic message of the double-slit is that if a quantum particle goes through both slits at once, it will produce interference fringes when the beams from the slits are combined at a screen. But if you measure which slit the photon went through, even if that measurement is "non-disturbing", then the interference at the screen will vanish. So sayeth the high priests of quantum theory.

The beauty of the Kim et al experiment is that Kim is able to effectively send one photon (photon A) through both slits, but also is able to measure which slit photon A went through without disturbing it -- simply by measuring its 100% path-correlated partner photon B.

The Kim et al experiment (Fig. 1) exploits a phenomenon exhibited by certain crystals called "photon down conversion" in which one photon of energy E enters the crystal and two photons with energy E/2 come out the other side. Since these two photons (photons A and B) are created in the same place, their paths are correlated. Now shine the initial source of photons thru a double slit onto the down-conversion crystal and two correlated photon pairs are simultaneously possible. These kinds of superposed possibilities are the meat and potatoes of quantum theory. For this situation quantum theory predicts that from these two slits will emerge from time to time, two photons A and B that enjoy the peculiar quantum situation described by EQ (1), that is, these two photons emerge simultaneously from both slits (and so in principle can produce two-slit interference fringes when combined at a screen.  But also, each photon, say A, is accompanied by a second photon, photon B, which in principle is able to measure (without disturbance) thru which slit A passed -- hence destroying any possibility of interference.

A very subtle and complex state of affairs. But at its core a very simple and computable application of elementary quantum theory. Congratulations, Kim et al for devising (and actually carrying out) this lovely experiment.

So what actually happens in this experiment? Does photon A produce interference fringes or not?

The short answer is: No Fringes. But the longer answer is, under certain conditions: Yes, Fringes.

Let's examine the question of what basis to best describe this quantum experiment so as to better understand the quantum facts. EQ (1) is expressed in the path basis and shows that path A is correlated and entangled with path B.

The experiment illustrated in Fig. 1, shows the B photons being measured in the path basis, but the A photons are combined by a converging lens into a state best described in what I will call the "screen basis". I will also introduce a third representation called the "eraser basis". If you expected that this note was going to be a bit of poetic fluff, you will be seriously disappointed. Warning! Raw, uncensored quantum physics ahead!

About the path basis (photon B for instance in Fig. 1): Photon B takes either path B1 or path B2. And this fact can be verified using photon detectors B1 and B2. If B1 clicks, the photon took path B1; If B2 click, the photon took path B2. It's that simple. Remember, Mother Nature on the quantum level is responsive to the questions that you ask. If you go looking for photon paths, Nature will give you photon paths.

Fig. 2: Screen pattern of A photons

What about photon A, that no longer travels in two beams but has been merged by a converging lens onto a photon sensitive screen? Well, the rules of quantum theory state that if you know which path a photon took, it cannot form an interference pattern. We can determine which path A took by looking at the B counters. If B1 clicks then photon A took path A1, giving rise to pattern # 1 of Fig. 2. If B2 clicks then photon A took path A2, giving rise to pattern #2 of Fig. 2. Both these patterns are featureless blurs. Adding them together gives a featureless blur with twice the intensity. No matter how hard you stare at these pixels, you will never see an interference pattern.

One way of understanding this lack of interference is that because of the perfect A-B path correlation,  observation of the B photon path tells us precisely which path the A photon took. If we possess path info about photon A, no interference is possible -- if photon A took one path, it simply could not have gone thru both slits.

This argument depends on the fact that the paths of the B photons carry info about the paths of the A photons. But suppose we alter our experiment by destroying B's path information with a "quantum eraser"? If the eraser leaves us unable to tell which path A took, will we then be able to observe the A photon interfering at the screen? Let's take a look at how a quantum eraser works.

Fig 3: Beam splitter operating as Quantum Eraser


The quantum eraser consists of a half-silvered mirror. Beam B1 comes in, is half reflected and half transmitted and sent into outputs B3 and B4. Likewise beam B2 comes in, is half reflected and half transmitted and is also sent into outputs B3 and B4. The eraser adds together these two input beams in such a manner as to entirely destroy information about which path the input photon took before entering the beam splitter.

The equations for this path info erasure operation are:

|B3> = S {|B1> + |B2>}                EQ (2)
|B4> = S {|B1> - |B2>}

The minus sign in the second equation looks a little out of place, but is absolutely necessary. If that minus sign were not there, more energy would come out of the eraser than went in -- humans could
create endless energy out of mirrors. However the law of conservation of energy is a very strong prejudice in the physics community and seems to be obeyed to the letter by Nature as well. This minus sign is a special case of the Stokes relations for light traveling across interfaces. Sir George Stokes was an Irish physicist from County Sligo, who also made important discoveries in fluid mechanics.

A little algebra gives us the inverse equation to EQ (2)

|B1> = S {|B3> + |B4>}                               EQ (3)
|B2> = S {|B3> - |B4>}

Substituting EQ (3) in EQ (1) we obtain:

|ψ (A, B)> = S{ S(|A1> + |A2>)|B3> + S(|A1> - |A2>)|B4> }                EQ (4)

Defining  |M(A)> = S(|A1> + |A2>)
and |W(A)> = S(|A1> - |A2>)                        EQ (5)

we get for the system wave function:

|ψ (A, B)> = S{ |M(A)>|B3> + |W(A)>|B4> }                  EQ (6)

EQ (6) expresses a perfect entanglement between photon A (whose wave function is represented in the "screen basis") and photon B (whose wave function is expressed in the "eraser basis". The names for the bases are my own inventions but the physics equations are entirely conventional.

THE SCREEN BASIS consists of two wave functions |M(A)> and |W(A)>. In the case of |M(A)>, both of A's photon paths are ADDED TOGETHER before being combined at the screen. In the case of |W(A)>, both of A's photon paths are SUBTRACTED FROM EACH OTHER before being combined at the screen. I chose the symbols M and W to label these two wave functions because the letters M and W are flipped versions of one another, just as the two wave functions are in the interference sense, precisely one another's opposites, as we shall see.

THE ERASER BASIS consists of the two wave functions |B3> and |B4> which emerge from the beam splitter, aka "quantum eraser". These two wave functions by themselves contain no information about "which path" the B photon took from the source, but taken together the two wave functions could in principle still be combined (in an "anti-eraser") which would resurrect the "which path" info of photon B. However if photon B is actually detected after the eraser either by "Bobski" at detector B3 or by "Boris" at detector B4, then B's path information is definitively erased and cannot ever be recovered. A signal from either Boris or Bobski means: "Da! Dat photon's path? Vorry you not. He is erased."

Fig. 4: Quantum Eraser produces 2 sets of fringes.

Does erasing B photon's which-path information (and via their perfect entanglement also erasing A photon's path info as well) now produce interference when both of A's paths converge on the same screen. The answer is Yes. Erasing B's path info produces interference fringes on A's screen.

But B's eraser and A's screen could be light years apart. If B's action can instantly change A's physical situation, then we can call "B" Bob and A "Alice". In the physics literature, Bob and Alice are iconic figures constantly obsessed with exploiting quantum entanglement to exchange superluminal messages between space-like separated locations. And on the surface, it looks as though path-entangled photons can do the job, because erasing Bob's path will produce Alice's fringes. And not erasing Bob's path will make Alice's fringes disappear.

Of course this entangled eraser scheme can't possibly work. Such a scheme would violate Einstein's laws of relativity. But one can't simply invoke relativity to explain away the eraser scheme. Any such alleged FTL signaling proposal must be refuted on its own terms, by using only the laws of quantum mechanics.

The first thing we notice about Alice's fringes is that there is not just ONE SET OF FRINGES produced on the screen but TWO SETS OF FRINGES. When Bobski announces that he has erased photon B's path, all of photon A's screen pixels that correlate with Bobski's message form a fringe pattern Z. When Boris announces that he has erased photon B's path, all of photon A's screen pixels that correlate with Boris's message form a second fringe pattern "Anti-Z". As shown in Fig. 4, the Z fringes and the anti-Z fringes exactly cancel -- the peaks of one set of fringes fitting exactly into the valleys of the other set of fringes. So Alice's fringes appear when triggered by messages either from Bobski or Boris, but no fringes appear when there is no way to tell whether the dot on Alice's screen was correlated with a click at counter B3 or a click at counter B4. Thus, absent a trigger signal (which must be sent at light speed or slower), what happens at Bob's location remains at Bob's location. Conclusion: FTL signaling using this entangled eraser scheme is impossible.

Fig. 5: Computing the observed pattern of photons on Alice's screen.


To see how this fringe and anti-fringe business works, we calculate the intensity of the patterns at a single location on Alice's screen. Fig. 5 illustrates where the two beams M(A) and W(A) impinge in the screen. These two beams have been slightly shifted for clarity, In reality they would be exactly superposed.

The wave function |M> represents the possibility for a photon to hit the screen. Squaring the wave function we obtain the probability for a photon to hit the screen. This probability is proportional to the INTENSITY of the light produced by the |M> beam at a particular location x on the screen, illustrated by the vertical line in Fig. 5.

If we represent possibilities A1 and A2 as amplitudes a1 and a2 times phases exp i(θ1) and exp i(θ2), we obtain for the probability (intensity) at location x for the wave function |M>

                               Intensity (M)  = C + D cos θ                 EQ (9)

where C = (a1 x a1+ a2 x a2), D = (a1 x a2) and θ is the phase difference at location x between |A1> and |A2>


Carrying out this same calculation for the wave function |W> at location x, we obtain:

                                 Intensity (W) = C - D cos θ                 EQ (10)

C is the intensity at x and D represents the magnitude of the interference fringes at x. Here you can explicitly see that the interferences terms at location x cancel when both terms are summed. Since x is an arbitrary position on the screen, if the interference term cancels at x, then the interference terms cancel everywhere. 

We see here precisely in what sense the two "screen basis" wave functions |M> and |W> can be regarded as "opposites". Each of these wave functions produces interference fringes on Alice's screen. But these two sets of interference fringes exactly cancel one another. As far as interference fringes go, |M> and |W> by themselves produce precisely opposite effects.

Path-entangled systems possess a variety of "magical" qualities which I will only mention in passing. There is, for instance, the so-called "delayed-choice" quantum eraser which involves invoking Bob's quantum eraser long after Alice's photon pattern has been indelibly recorded. Hence apparently after-the-fact producing matched sets of mutually canceling Alice fringes that correlate with Bobski's and Boris's detector events.

But wait, there's more. The path eraser pictured in Fig. 3 represents only one out of an infinite array of possible ways of erasing Bob's which-path information. By changing the position of Bob's movable mirror, Bob can add a phase angle φ tο photon path B2. Now when we trigger on Bobski's and Boris's detector outputs we get two complementary patterns of Alice fringes as before, but these fringes have moved to a new location on Alice's screen -- a new location that depends on Bob's choice of the phase angle φ.

Thus altho there are no fringes visible in Alice's total pattern of photons on her screen, Bob can, by changing the phase angle φ, seemingly shift the location of Alice's two mutually canceling fringe patterns. And Bob can carry out this invisible Alice fringe pattern shifting in the "delayed-choice" mode, that is, long after Alice has already recorded every pixel of her pattern "in stone". Given the peculiar behavior of this simplest of all quantum entanglements, it is hard not to imagine that "something" must be being transmitted faster-than-light (or even backwards-in-time) from Bob to Alice. Something must be really being sent (and really fast too) in these experiments. But physicists can't really say what that "something" might be.

We have introduced here three different photon representations, the path basis P, the screen basis S and the eraser basis E. Applying these three bases separately to photon A or photon B, we could obtain nine different experiments on photon entanglement. Since P(A)P(B) is the same as P(B)P(A), the number of unique entanglement experiments reduces to six, namely (PP), (PE), (PA), (EE), (ES) and (SS). We have already considered most of these six experiments, but not the two (SS) and (EE) where both Bob and Alice combine their photons either on screens (SS) or using erasers (EE).

Both (SS) and (EE) have such similar behaviors that I will just consider the (SS) case for which the wave function (in the screen - screen basis) looks like this :

|ψ (A, B)> = S{ |M(A)>|M(B)> + |W(A)>|W(B)> }    EQ (11)

We see from EQ (11) that Alice's interference pattern Z is perfectly correlated with Bob's interference pattern Z. Likewise Alice's anti-interference pattern anti-Z is perfectly correlated with Bob's anti-Z pattern. Thus on both Alice and Bob's screens appear two complementary fringe patterns exactly as illustrated in Fig. 4. The two interference patterns exactly cancel. And again no FTL signaling is possible between Alice and Bob in the (SS) basis.

[ ADDED 5/14: we have seen, that depending on your measurement intention, a simple path-entangled photon pair can be written as a Path-Screen, an Eraser- Screen or a Screen-Screen entanglement -- each a different but equally valid way of envisioning the same quantum wave function. In fact, by adding a phase φ to one side of either the Eraser basis or to the Screen basis, one can generate a continuous infinity of different bases, each of which takes a different experimental perspective on the very simple path-entangled photon pair represented by EQ (1).]

If a photon takes two paths to a single location it can produce interference fringes, If which-path information is available these fringes will vanish. Because we can entangle two photons, it seems that we can distantly decide whether path information for photon A exists or not, depending on what we do with its entangled partner photon B. Distant which-path erasure seems on the face of it to be a viable road to achieving FTL signaling using entangled photons. As these examples show, erasing Bob's which-path information does indeed lead (instantly?) to the appearance of fringes on Alice's screen.

But these Alice fringes always appear in pairs -- as a set Z of fringes and a set anti-Z of complementary fringes which exactly cancel out: with the total result that no fringes appear on Alice's site when Bob chooses to erase his which-path information.

Is this fringe/anti-fringe behavior a general feature of quantum erasure schemes? Or is it merely an accidental feature of this particular method of path erasure?

Recently Demetrios Kalamidas came up with an ingenious FTL signaling scheme that uses a radically new method of which-path erasure. Instead of merely erasing photon paths, the Kalamidas scheme makes the paths "ambiguous" by mixing each path with a weak coherent state. A coherent state has the property that its photon number is uncertain. The result of this mixing is to create a situation in which when you measure a photon in a particular path, you can never be sure whether that path was "full" and you are measuring the "real" entangled photon. Or whether that path was "empty" and you are measuring a "fake photon" originating from the coherent state.

Claiming that if Bob's uses his clever new means of path erasure, uncompensated fringes will be produced on Alice's screen, Kalamidas published his FTL scheme in a well-regarded optics journal. His scheme was immediately refuted in general terms by a number of different physicists in a number of different ways, but it took several months of work before Martin Suda and Nick Herbert were able to finally demonstrate exactly where Kalamidas went off the rails. It turns out that even in the Kalamidas case, Bob's path erasure produces complementary sets of fringes which totally cancel out all interference at Alice's screen.

Even though his clever FTL signaling scheme was eventually refuted, the Kalamidas scheme led to intense discussions of the subtle details of few-photon states and coherent states. And contributed certainly to my knowledge of such topics and perhaps added to the physics community's store of few-photon lore. Thank you, Demetrios.

I would also like to thank Jack Sarfatti for prodding me to reconsider the physics of eraser-based FTL communication schemes.

Despite its enormous successes, quantum theory has left physicists with two big mysteries, both of which involve quantum reality -- the ability to tell a convincing story about "what's really happening in the world" that does justice to quantum theory and the quantum facts.

Quantum Mystery # 1: What does the quantum wave function really represent? What is really happening in the world before any measurements are made?

Quantum Mystery # 2: What really happens during a quantum measurement? How does a quantum possibility decide to turn into an actuality?

To these fundamental questions, the fact of quantum entanglement adds a third:

Quantum Mystery # 3: What (if anything) is actually exchanged between two distant entangled quantum systems? When will physicists get smart enough to be able to tell their kids a believable story about what's really going on between Alice and Bob?

Wave function (in the position basis) for an excited Hydrogen Atom: from Dean Dauger's Atom in a Box.











Sunday, May 3, 2015

Does Earth Possess a Second Season?

Earth's changing distance from the Sun
DOES EARTH POSSESS A "SECOND SEASON"?

Motivated by the observation that for approximately 18 years, global warming has been essentially constant -- a behavior on Nature's part that not a single climate model predicted -- I decided to look more closely at the scientific basis for Catastrophic Anthropogenic Global Warming (CAGW), recently sanitized (to protect the children?) to the more innocuous "Climate Change". Catastrophic Warmists claim that human production of Carbon Dioxide (CO2) is warming the Earth via the greenhouse effect and that the magnitude of this warming will soon be so immense as to be termed catastrophic. (See below for a "spaghetti graph" of climate model predictions versus Nature's response).

CAGW skeptics (sometimes termed "climate deniers") believe that the Earth is indeed warming and that some of this warming is surely due to human-produced CO2, but skeptics suspect that the magnitude of this CO2-induced warming has yet to be correctly modeled by climate scientists and has probably been exaggerated. While atmospheric CO2 continues to increase -- CO2 concentration recently exceeded 400 ppm compared to a pre-industrial level of 280 ppm (parts per million) -- the average global temperature has remained essentially flat since the beginning of the 21th Century.

Climate scientists quantify the warming strength of a greenhouse gas as a "forcing" measured in Watts per square meter. If you aimed a 200 Watt heat lamp at a 1x1 meter Muslim prayer rug, you would be subjecting that prayer rug to a "forcing" of 200 Watts per square meter. (W/m^2).

The physics of the greenhouse effect predicts the forcing due to a doubling of CO2 to be 3.7 W/m^2 -- about the same effect on that prayer rug as a tiny Xmas tree light instead of a heat lamp. But these tiny Xmas tree lights are spread over the surface of the entire Earth so this seemingly small forcing represents an immense total input of heat into the Earth's atmosphere.

The Earth's temperature rise due to 3.7 W/m^2 of forcing is estimated to be about 1.2 degrees Centigrade.

But CO2 forcing is only a small part of the story. Climate scientists argue that "feedback effects" will amplify this modest temperature rise to truly catastrophic values. Estimates of CO2 - induced warming WITH FEEDBACK range from 2 to 6 Centigrade degrees for a doubling of CO2. Differing estimates of this crucial feedback factor account for the spaghetti-like quality of climate model predictions.

The main source of this conjectured feedback is water vapor which is a more powerful greenhouse gas than CO2. When increased CO2 heats the atmosphere, more water vapor can be stored in the heated air. This extra water vapor causes more global warming than the CO2 alone.

BARE CO2 plus FEEDBACK = Catastrophic Anthropogenic Global Warming.

The crux of the global warming controversy is whether current climate models are reliable enough to guide public policy. The correct estimation of feedback factors is a big part of this controversy.

The global warming debate is much too large a topic for a blog post. Here I want to discuss a discovery that is probably not original but which I have never seen discussed in the global warming literature. Does Earth have a "second season"?

Everyone knows that the Earth goes around the sun and that Earth's orbit is not circular but is shaped like an ellipse (a fact discovered by Johannes Kepler in the 17th Century). Traveling this oval path, the Earth is closest to the Sun in January (producing record high winter tides in Santa Cruz) and furtherest from the Sun in July.

When the Earth is close to the Sun (perihelion) it should be gathering more heat. And at aphelion, less heat will fall upon us. Just for fun I decided to calculate this dependable yearly change in solar radiation and compare this number with the 3.7 Watts/m^2 forcing due to CO2 doubling.

This is a simple physics problem. The change in Earth-Sun distance between perihelion and aphelion is 3.33%. Elementary calculus shows that the inverse-square law turns this 3.33% distance change into a 6.66% intensity change in the amount of solar radiation reaching Earth.

I looked up the intensity of sunlight at the top of the atmosphere. Bare solar radiation is 1350 Watts/m^2. But not all of that sunlight reaches the Earth's surface. The fraction of light reflected from a planet's surface is called its "albedo" (from the Latin term for "whiteness"). Mainly due to its bright white clouds, Earth's albedo is about 30% (compared to the dull Moon's 12% reflectivity). Subtracting reflected sunlight gives a nice round number for solar intensity at the Earth's surface -- 1000 Watts/m^2.

The 6.66% change in solar intensity amounts to a yearly change of 66.6 Watts/m^2 at the Earth's surface. That's a change in radiative forcing that's 18 times greater than the forcing (3.7 Watts/m^2) due to CO2 doubling!!! What a surprise!! Periodic solar forcing is immense!

Not so fast, Nick. Most of this solar radiation strikes the (spherical) Earth at a slant. And none of this radiation strikes the night side. To get a fair estimate of the "effective yearly change in total solar forcing" this number must be divided by 4. (Books on climate change explain why this correction factor for the spherical Earth is just 4 and not some complicated factor of pi.)

Dividing by four gives an effective yearly change in total solar forcing of 16.6 Watts/m^2 -- a factor of 4.5 greater than the calculated forcing due to a doubling of CO2.

Nearly 5 times as intense as the CO2 forcing -- and many times faster: Rate of solar forcing is measured in months compared to decades for CO2!

And that's just the BARE solar forcing before water-vapor FEEDBACK is factored into the equation.

Big BARE ANNUAL SOLAR FORCING FLUCTUATION plus big FEEDBACK = big ANNUAL TEMPERATURE SWING.

This simple calculation suggests that in addition to the familiar seasons due to the tilt of the Earth's axis -- an asymmetric effect that produces Winter in Melbourne while it's Summer in New York, Earth should enjoy a symmetric "second season" during which the whole Earth warms in January and the whole Earth cools in July.

Does Earth's second season actually exist? Is our much vaunted science capable of measuring this periodic pulsation in the Earth's heat budget. Does our planet possess instruments sensitive enough to register this regular thermal heartbeat superimposed on the ordinary seasons that we know so well?

Therefore I am suggesting a new goal for 21th-Century climate scientists comparable to the search for a new planet or a new elementary particle. I propose we mount a well-funded international search for Earth's Second Season.

Spaghetti Graph: comparison of 90 climate models compared with satellite-measured average global temperature..(Click to enlarge.)