Posted in Borges, Fun links, General physics, Perception, Quantum by Mike Stay on 2010 April 27

Lazulinos are quasiparticles in a naturally occurring Bose-Einstein condensate first described in 1977 by the Scottish physicist Alexander Craigie while at the University of Lahore [3]. The quasiparticles are weakly bound by an interaction for which neither the position nor number operator commutes with the Hamiltonian. A measurement of a lazulino’s position will cause the condensate to go into a superposition of number states, and a subsequent measurement of the population will return a random number; also, counting the lazulinos at two different times will likely give different results.

Their name derives from the stone lapis lazuli and means, roughly, “little blue stone”. Lazulinos are so named because even though the crystals in which they arise absorb visible light, and would otherwise be jet black, they lose energy through surface plasmons in the form of near-ultraviolet photons, with visible peaks at 380, 402, and 417nm. Optical interference imparts a “laser speckle” quality to the emitted light; Craigie described the effect in a famously poetic way: “Their colour is the blue that we are permitted to see only in our dreams”. What makes lazulinos particularly interesting is that they are massive and macroscopic. Since the number operator does not commute with the Hamiltonian, lazulinos themselves do not have a well-defined mass; if the population is N, then the mass of any particular lazulino is m/N, where m is the total mass of the condensate.

In a recent follow-up to the “quantum mirage” experiment [2], Don Eigler’s group at IBM used a scanning tunneling microscope to implement “quantum mancala”—picking up the lazulino ‘stones’ in a particular location usually changes the number of stones, so the strategy for winning becomes much more complicated. In order to pick up a fixed number of stones, you must choose a superposition of locations [1].

  1. C.P. Lutz and D.M. Eigler, “Quantum Mancala: Manipulating Lazulino Condensates,” Nature 465, 132 (2010).
  2. H.C. Manoharan, C.P. Lutz and D.M. Eigler, “Quantum Mirages: The Coherent Projection of Electronic Structure,” Nature 403, 512 (2000). Images available at
  3. A. Craigie, “Surface plasmons in cobalt-doped Y3Al5O12,” Phys. Rev. D 15 (1977). Also available at

2 Responses

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  1. Doug said, on 2010 April 27 at 3:36 pm

    I heard these naturally form crystals that ramificate in labyrinthine patterns.

  2. reperiendi said, on 2010 April 28 at 1:51 pm

    For everyone but Doug:

    The link on the third reference goes to a story by Jorge Borges. Borges often referred to books that didn’t exist, to fictional versions of himself and friends, and to problems of self-reference and the infinite. This story, “Blue Tigers”, is a metaphor for infinite series.

    There’s a famous derivation that 3 = 4 (or any other number) using infinite series naively:

    3 = 3 + 0 + 0 + 0 + ...
      = 3 + (1 - 1) + (1 - 1) + (1 - 1) +... 
      = 3 + 1 + (-1 + 1) + (-1 + 1) + ... 
      = 4 + 0 + 0 + 0 + ...
      = 4

    The story is about a guy (Borges’ friend Sir William Alexander Craigie, the editor of the Oxford English Dictionary—but in this story he’s a logician) who goes to India for a love of tigers, and then hears of a ‘blue’ tiger and goes to hunt it.

    The locals at first are very concerned, but when they hear he wants to catch an actual tiger, they’re relieved. They forbid him to go up on a plateau, though, saying it’s sacred. He sneaks up in the middle of the night and finds small blue stones filling the cracks in the mud of the plateau.

    Back in my hut, I took off my jacket. I lay down and dreamt once more of the tiger. In my dream I took a special note of its colour; it was the colour of the tiger I had dreamt of, and also of the little stones from the plateau. The late-morning sun in my face woke me. I got up.

    The scissors and the letter made it hard to take the discs out of the pocket; they kept getting in the way. I pulled out a handful, but felt that there were still two or three I had missed. A tickling sensation, the slightest sort of quivering, imparted a soft warmth to my palm. When I opened my hand, I saw that it held 30 or 40 discs; I’d have sworn I’d picked up no more than 10. I left them on the table and turned back to get the rest out of the pocket. I didn’t need to count them to see that they had multiplied. I pushed them together into a single pile, and tried to count them out one by one.

    That simple operation turned out to be impossible. I would look fixedly at any one of them, pick it up with my thumb and index finger, yet when I had done that, when that one disc was separated from the rest, it would have become many. I checked to see that I didn’t have a fever (which I did not), and then I performed the same experiment, over and over again. The obscene miracle kept happening. I felt my feet go clammy and my bowels turn to ice; my knees began to shake. I do not know how much time passed…

    I was, I believe, no less terrified than he, but I closed my eyes and picked up a handful of stones with my left hand. I tucked the pistol in my belt and dropped the stones one by one into the open palm of my right hand. Their number had grown considerably.

    I had unwittingly become accustomed to these transformations. They now surprised me less than Bhagwan Dass’s cries.

    “These are the stones that spawn!” he exclaimed. “There are many of them now, but they can change. Their shape is that of the moon when it is full, and their colour is the blue that we are permitted to see only in our dreams. My father’s father spoke the truth when he told men of their power.”

    The entire village crowded around us.

    I felt myself to be the magical possessor of those wondrous objects. To the astonishment of all, I picked up the discs, raised them high, dropped them, scattered them, watched them grow and multiply or mysteriously dwindle…

    Rereading what I have written, I see that I have committed a fundamental error. Led astray by the habit of that good or bad literature wrongly called psychology, I have attempted to recover – I don’t know why – the linear chronology of my find. Instead, I should have stressed the monstrousness of the discs.

    If someone were to tell me that there are unicorns on the moon, I could accept or reject the report, or suspend judgement, but it is something I could imagine. If, on the other hand, I were told that six or seven unicorns on the moon could be three, I would declare a priori that such a thing was impossible. The man who has learnt that three plus one are four doesn’t have to go through a proof of that assertion with coins, or dice, or chess pieces, or pencils. He knows it, and that’s that. He cannot conceive a different sum. There are mathematicians who say that three plus one is a tautology for four, a different way of saying “four” … But I, Alexander Craigie, of all men on earth, was fated to discover the only objects that contradict that essential law of the human mind.

    At first I was in a sort of agony, fearing that I’d gone mad; since then, I have come to believe that it would have been better had I been merely insane, for my personal hallucinations would be less disturbing than the discovery that the universe can tolerate disorder. If three plus one can be two, or 14, then reason is madness…

    The properties of the stones seemed to be all the things that make quantum systems seems so weird, so I wrote the thing on lazulinos both in honour of and in response to Borges.

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