Discontinuous Trajectories in Quantum Mechanics

“It’s 10:00 pm, do you know where your photons are?”

As parents, we try to know where our kids are at all times.  We teach them that, when they want to play outside or at a friend’s house, they need to let us know where they will be.  If we were to ever find out that they were not where we expected them to be, we would go ballistic!  Well, could you imagine being the parent of a photon?

A group of scientists at Tel-Aviv University performed an experiment that shows you may not get a sensible answer when you ask a photon where it has been.  Soon to appear in Physical Review Letters, Asking photons where they have been,” demonstrates that the past of a photon cannot be represented by a continuous trajectory, or even by the superposition of continuous trajectories.  To quote from their paper:

“The photons tell us that they have been in the parts of the interferometer through which they could not pass!”

Interrogating Discontinuous Trajectories in Quantum Mechanics

I have previously mentioned two puzzling aspects of the quantum universe: non-locality and the intrinsic, probabilistic nature of outcomes.  I typically caveat these two properties with “apparent”, i.e. apparent non-locality and apparent probabilistic nature.  That is because it is still possible (although not certain) that there could be some underlying causal and realist explanation.  In fact, explanations may already be available, we just do not know how to validate them.

Another mystery that our classical brains struggle with is the apparent discontinuous nature of the trajectories of quantum particles.  What I love about this new result from the Tel-Aviv group (Danan, Farfurnik, Bar-Ad, and Vaidman) is that, not only does it demonstrate a unique and important property of quantum physics, it does so with a straight-forward and conceptually easy to understand experiment.

The scientists used a nested Mach-Zehnder interferometer (MZI).  You may recall that I discussed a MZI in “The Transactional Interpretation of Quantum Mechanics.”  However, in this case, they nested two MZI together – one leg of a MZI includes another MZI nested within one of it’s legs (see the figure below).

Discontinuous Trajectories in Quantum Mechanics: nested Mach-Zender interferometer: image

Schematic of the nested Mach-Zehnder interferometer used to interrogate photons as to their whereabouts. From “Asking photons where they have been,” http://arxiv.org/pdf/1304.7469.pdf.

Photons enter the apparatus from the source in the upper left corner of the figure.  The unlabeled squares represent beam splitters.  After passing through either the lower leg or the nested interferometer in the upper leg, photons are detected by a quad-cell photo-detector (D).  The unique and essential feature of this experiment is that the mirrors (A, B, C, E, and F) vibrate around their horizontal axes at different frequencies fA, fB, fC, fE, and fF, with very small amplitudes.  This induces oscillations of the vertical positions of photons after they encounter each particular mirror along their path.  Hence, each photon carries a record that describes which mirrors they encountered, and thus which path they took through the apparatus.

Each photon’s signature is extracted by measuring (at D) its position coming out of the interferometer. This data is then Fourier-analyzed to produce a power spectrum of the different frequencies present in the output signal.  When the vibration frequency of a certain mirror appears in the power spectrum, the scientists logically conclude that at least some of the photons have been near that particular mirror.

Trajectories That Appear to be Continuous

The first run of the experiment that we will consider is the one depicted above.  One third of the beam power was sent into the lower arm and two thirds of the beam power was sent to the nested MZI in the upper arm (the beam splitter used to split the two legs of the outer interferometer was specifically designed for this one-third, two-thirds spilt; other beam splitters produced a 1:1 split).

The interferometers were aligned to ensure that all the photons ended up at the detector. The power spectrum showed peaks at all frequencies, as intuitively expected.  The peaks at fE and fF were higher due to the larger fraction of photons in contact with them.  The power spectrum at the output of the experiment shows, unsurprisingly, frequencies from all five mirrors.

So Much for Common Sense: Discontinuous Trajectories in Quantum Mechanics

The surprising result was obtained when the interferometer was modified to be a “which-way” experiment.  By slightly shifting mirror B, the nested MZI was aligned so that there would be complete destructive interference between the light reaching mirror F from A and the light reaching mirror F from B (see the figure below but ignore the red and green lines for now).

So, in effect, there were no photons at F.  Hence, there were no photons that could possibly reach the detector D from the upper leg, right?  By that reasonable bit of logic, any photons detected at D should have come from the lower arm of the interferometer.  We would therefore expect that any photons reaching the detector would have interacted only with mirror C.  The punch-line is that the scientists observed three peaks in the power spectrum: the expected one at frequency fC, and two more peaks at frequencies fA and fB.

Discontinuous Trajectories in Quantum Mechanics: tuned Mach-Zender interferometer: image

Nested Mach-Zender interferometer, tuned so that photons arriving at mirror F interfere destructively. Red and green (dashed) lines are explained in the text. From “Asking photons where they have been,” http://arxiv.org/pdf/1304.7469.pdf.

Common sense tells us that any photons passing through the inner interferometer (so that they could encounter mirror A or B and pickup oscillations at frequency fA and fB) must by necessity have also encountered mirrors E and F.  However, frequencies fE, and fF were not seen in the output power spectrum.  How did photons pick up oscillations at frequencies fA or fB, associated with mirrors A or B, and make it to the detector without also encountering mirror E or F?

Interpreting Discontinuous Trajectories in Quantum Mechanics

Although the conventional interpretation of quantum mechanics can predict the correct outcome for this experiment, it offers little insight into what is going on.  The authors offer an alternative that provides an improved conceptual understanding.  The interpretation they prefer is the two-state vector formalism.  This is a time-symmetric interpretation of quantum mechanics; both forwards and backwards evolving quantum states are required to describe a quantum system. This includes a state vector that evolves from the initial conditions towards the future, and a second state vector that evolves backwards in time from the final conditions of the experiment.  That is to say, the state vector describing the pre-selected state as well as the state vector for the post-selected state are both required to fully describe the system.  This highlights another intrinsic aspect of QM that makes it distinct from classical physics: the past of a quantum particle does not uniquely determine its future.  Past and future measurements, taken together, provide complete information about the system.

In the present experiment (see the above figure), a standard forward evolving quantum state is depicted by the red line and a backward evolving quantum state is depicted by the green dashed line. There is no continuous path for the forward evolving state to proceed through the inner MZI and reach the detector.  However, there is a non-zero probability for the photon to have existed anywhere that both forward and backward quantum wave functions are present.  Hence, this includes the nested MZI in the upper leg, inside mirrors E and F, but in the region of mirrors A and B.

The transactional interpretation (TI) also provides a conceptual explanation of this experiment.  Additionally, at least in my opinion, the TI provides a more straight-forward way of calculating the probability for the photon to be in the inner interferometer, and hence simplifies the prediction of the power spectrum at the output.

Will We Ever Understand Particle Trajectories, Much Less Quantum Physics?

Making progress towards the goal of fully understanding what nature is up to in the quantum world requires that you have a full grasp of the variety of experimental evidence and theoretical results.  If you have been reading my posts up to now, hopefully I have been filling in some gaps.  Don’t get too comfortable, yet.  There is a lot more to the story about trajectories, a story that is being told through “weak measurements”. 

Considerations of pre- and post-selected systems lead to the theory and practice of weak measurements.  In an upcoming article, I plan to discuss what I mean by weak measurements, and how they are being used to survey and reconstruct the properties of quantum particles between pre-selection and post-selection measurements.  The results of these measurements are amazing just due to the fact that they are possible, as well as due to the enlightening results that they provide.  These experiments give me confidence that, as a result of the amazing work of skilled quantum physicists, we are making steady progress along the road to a proper conceptual understanding of our quantum universe.


About Warren Huelsnitz

Former submariner, now high energy physicist. Neutrino physics and astrophysics, quantum physics and other mysteries of the universe. Engaged in life-long learning and pursuit of knowledge. Blogging about quantum mechanics and other issues at www.thefunisreal.com.
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A violation of chronology does not necessarily also violate causality. Instead of the final state sending something back to the initial state, it could be that a "time finger" forms. That is, the apparatus itself rewinds back to the initial state. Until the measurement is complete and the apparatus "catches up" with the surrounding space, the laboratory frame has a bump, and might even be non-Cauchy.


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So has this generated any serious discussions anywhere? Because this is the strangest thing I've ever heard of. I almost don't want to believe it until I spend some time in the room with the device, myself. The PRL is painfully short.

So, for example, if one puts a piece of paper in the F arm, presumably, the effect goes away, right?  Which is bizarre, because there's not supposed to be any light in the F arm anyway ... I mean, If I spray some dust or whatever, to see where the lasers are going, then there should be nothing in the F arm, right?  Yet blocking the F arm must surely make the effect go away, as F is needed for the TIQM explanation to make sense.  I would really really have liked it for someone to state this very explicitly, because it is so .. unbelievable ... 

huelsnitz moderator

@linasvepstas You are correct that, if something was placed in the vicinity of F, it would block the confirmation wave.  That nested branch of the interferometer would then not be available to the photons.  I agree it would have been interesting to see that specifically done.

There have been other experiments that imply quantum trajectories are not continuous.  But, at least to me, this one is a striking and easily visualize-able example.  Here is another paper by Vaidman that discusses it:  http://arxiv.org/pdf/1304.7474.pdf


@huelsnitz Well, there is one old and simplistic, yet appealing interpretation: namely that light is always a wave, and photons are merely the units by which one can add to and remove from this wave. (I don't know what this interpretation is called) Its nice because it "explains" most interferometric experiments, (since its "always a wave") and reduces welcher-weg arguments to silliness ("its always a wave, what did you think would happen?")  Its also nicely consistent with second quantization, where the ladder operators merely add and subtract from the field; nothing that I know of in second quantization ever says that the field itself comes in quanta or travels in quanta, only that one must add/subtract to it in quanta.  So its a fairly comfortable interpretation.

This is the first experiment I've heard of that explicitly denies that interpretation. So I would really would like to have seen something like "we placed a photometer in arm F, and explicitly measured the light intensity to be 30dB down (or 40 dB down or 60 or 120 or whatever), from when there is constructive interference (i.e. when there is light there). Of course placing a photodetector there destroys the effect, but as long as it *is* in place, we can confirm that the intensity of the EM field in arm F is xxx dB down from full intensity."  Because if they publish this, and xxx is just 10dB, then one can quibble. If its 30dB, you can still ask "gee are you sure about your knob settings?". But if its 120dB down, ... well, that's really hard to argue with, its hard to screw up an experiment that bad, and still get 120dB down, and its hard to hand-wave some alternative explanation at that level.

I guess I'll have to hunt down the other discontinuous experiments; perhaps I'll find them on this website?

huelsnitz moderator

@linasvepstas I think so-called "Interaction Free Measurements" (IFM) may be related to the phenomenon behind apparent discontinuous trajectories.  (of course, I think the "trajectory" "happens" the same way, whether it appears to be continuous or discontinuous; the trick is figuring out what exactly a "trajectory" is).  There are a lot of papers on IFM at arxiv.org. 

 My next blog article, which I should get around to writing in the next couple days, will be on the "Quantum Cheshire Cat" thought experiments that have been around for a couple years - ideas to do experiments to demonstrate how a particle can be separated from it's properties.


@linasvepstas @huelsnitz  

Nice article, nice experiment.

Why does this experiment not allow a 'light is a wave' interpretation? 

The point at F is a node in a standing wave. The mechanical equivalent on a string is to put a piece of wood with a small hole that would block waves, yet the standing wave obviously passes right through the hole. If you block the hole completely then you block the standing wave.  


@tomandersen Oops, reply went to wrong spot: so I repost:

Curious idea, made me stop for a moment.  But I see no reason why there would be any standing waves anywhere, or a node at F.   

However, I am utterly frustrated that the paper does not state the "obvious" about the F arm, like "we put a piece of cardboard in the F arm and the effect persists/goes away", or "we put a photodetector in the F arm and can confirm that its really dark there". Little statements liek that would help with the WTF aspect of this experiment.


@huelsnitz Since the maths behind both CI and TI are the same, can these discontinuous trajectory effects be calculated from Schrodinger's equation while being blind to the interpretation? That way @linasvepstas could get the measurements he wants, just in a theoretical setting.

I understand that the maths of QM and what it predicts are not questioned by this experiment, so one could in theory derive these effects. But the scientists at Tel-Aviv would rather do an experiment instead of a pen-and-paper derivation. Is that because it would be too difficult?


  1. […] weak measurements have become vital tools in this quest.  The article I wrote a couple weeks ago, Discontinuous Trajectories in Quantum Mechanics, was an example of weak measurements.  Today, I discuss weak measurements used to reconstruct […]

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