Department of Physics and Astronomy

The Forbes Group

Negative Mass Hydrodynamics

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Negative-Mass Hydrodynamics

Negative mass is a peculiar concept. Counter to everyday experience, an object with negative effective mass will accelerate backward when pushed forward. This effect is known to play a crucial role in many condensed matter contexts, where a particle's dispersion can have a rather complicated shape as a function of lattice geometry and doping. In our work we show that negative mass hydrodynamics can also be investigated in ultracold atoms in free space and that these systems offer powerful and unique controls. They allow one to cleanly engineer the dispersion of the system in a flexible way by "dressing" the atoms with suitably chosen laser fields. The ability to precisely position ultracold atoms into well defined regions of the dispersion differentiates our system from condensed-matter systems.

In our paper Khamehchi:2017 published in Physical Review Letters, we describe our new platform for detailed and precise studies of negative mass hydrodynamics. Our system is based on a spin-orbit coupled BEC, a topic that is currently the focus of intensive research interest from many leading groups worldwide. In this work we present both experimental measurements and detailed calculations that display surprising dynamics due to the modified dispersion relation. In particular, when atoms are pushed into the negative mass regime of the dispersion, we observe self-trapping, dynamical instabilities, and the formation of solitons and dispersive shocks. We introduce a new theoretical framework that accurately describes these phenomena and connects them to the underlying dispersion.

Compared with previous studies that consider negative mass effects in atoms confined to optical lattices, our system offers the distinct advantage of preserving translation invariance. We have no underlying periodic structure from the lattice, so we can clearly associate these effects with the dispersion relation. This allows us to resolve the controversy about the origin of self-trapping seen in optical lattices where a variety of explanations have been suggested based on spatial properties of the lattice. The lack of a lattice in our system allows us to clearly demonstrate the unified explanation that self-trapping results from the negative effective mass of the modified dispersion.


M.A. Khamehchi, Khalid Hossain, M.E. Mossman, Yongping Zhang, Thomas Busch, Michael McNeil Forbes, and Peter Engels, "Negative mass hydrodynamics in a spin-orbit--coupled Bose-Einstein condensate", Phys. Rev. Lett. 118(15), 155301 (2017). (Preprint: arXiv:1612.04055.)

What system do you study?

In the Engels Lab, they make a dilute vapor of Rubidium 87 (Rb), which they then cool using a variety of techniques to one of the lowest temperatures found anywhere in the universe.

Optical bench in the Engels lab.
One of the optical tables in the [Engels Lab]( where they make some of the coldest matter in the universe.

At these temperatures the quantum wave-like nature of these atoms dominates their behaviour, and they all start moving in synchrony as a quantum superfluid called a Bose-Einstein Condensate (BEC). These superfluids exhibit some very counter-intuitive properties. For example, if you stir a regular fluid such as a cup of coffee, you form a single large vortex. In contrast, if you stir a superfluid, you typically form a lattice of individually quantized vortices that repel rather than clumping together as they do in your coffee.

This BEC is the starting point for our study, but so far still behaves as expected with a positive mass: if you push it, it accelerates in the same direction according to Newton's second law $F=ma$.

We now turn on a set of counter propagating lasers the give the system what is called a spin-orbit coupling (SOC). These lasers simultaneously convert one state of Rb to another state while giving the atoms a kick. These states are referred to as a "spin" states although they really are different hyperfine states of the system. This coupling of the spin states to the momentum is what the term spin-orbit coupling refers to. Thus, at a microscopic level, the atoms are being kicked back and forth by these lasers while simultaneously changing their spin. It is these lasers that are responsible for the negative mass effect, not some violation of a fundamental law of physics.

So what is "effective mass"?

At this level it is similar to describing a cup of coffee in terms of the water molecules which are bouncing around. However, keeping track of the $10^{24}$ different water molecules is not a particularly useful way of describing the cup of water, so instead we describe it as a fluid in terms of the local daensity, pressure, flow velocity, etc. In these hydrodynamic equations, one has a term of the form $F=ma$ which embodies Newton's second law with a property of the fluid called the "effective mass" which, in the case of water, is directly related to the underlying mass of the water molecules. In this case, the effective mass is positive: applying a force to the fluid will cause it to accelerate in the same direction as the force.

We can apply a similar fluid dynamical description to the spin-orbit coupled BEC, however, now we find that if the fluid is moving with a certain velocity (to the left in the images and moves below), then the effective mass in the hydrodynamic equations becomes negative. Consequently, when a force is applied to the fluid in this regime, it accelerates in the opposite direction, towards the applied force.

Mathematically, this results from the form of the "dispersion relationship" which describes how the kinetic energy of the system depends on the velocity of the fluid. For free particles (moving substantially less than the speed of light), the kinetic energy has the familiar form $E_k = mv^2/2$, or when expressed in terms of the particles momentum $p=mv$: $E_k(p) = p^2/2m$. The (inverse) effective mass of the system comes from the second derivative of this expression $E_k''(p) = 1/m$. It is this dispersion relationship that enters the hydrodynamic equations, and therefore which governs the general behaviour of the fluid.

In this spin-orbit coupled BEC, we find that the dispersion relationship $E_k(p)$ has a different behavior. In particular, for certain speeds (moving in the videos below), the curvature of $E_k(p)$ becomes negative: $E_k''(p) < 0$. This is the region of negative effective mass.

How do you see this negative effective mass?

To observe this negative mass effect, we start with a fairly tightly confined cloud. In these systems, the atoms are trapped by lasers. The atoms are attracted to the light, so by carefully aiming the laser beams, one can create an optical "cup" to hold the atoms close together. The trap is then turned off in one direction, allowing the gas to expand. The Rb atoms have a repulsive interaction, so the gas has an outward pressure which provides the force $F$ in this experiment.

Initially, the cloud is at rest, and with low velocities, the effective mass is positive. The outward force from the pressure of the gas thus causes the cloud to expand outward - accelerating in the same direction as the force - as can be seen in the initial times of the video below.

After about $t=6$ms of expansion, the right side of the cloud is now moving fast enough to enter the regime of negative effective mass. From this time forward, the outward pressure of the gas causes the right edge of the cloud to decelerate - accelerating in back towards the center, and against the outward force. This causes the atoms to slow down, and they pile up forming a shockwave. Associated with the negative mass region are instabilities where small amplitude oscillations - sound waves - grow exponentially. This is seen as a bunch of rapid wiggles at the right edge of the cloud. As is typical with shockwaves, energy radiates away from the shock front in the form of sound waves (phonons) and in this superfluid system, as "solitons". These can be seen moving both left and right from the shock front.

The boundary of this system is not perfect, and some particle "leak" out to the right. Note that these are moving very quickly - quickly enough that they are no-longer in the negative mass regime of the dispersion relationship.

Is "effective mass" a real mass?

In her BackRe(Action) Blog post, Sabine Hossenfelder quotes David Abergel who refers to the term "effective mass" as a "historical artefact". Although the distinction between "effective mass" and "gravitational mass" is valid - we certainly have not created anything with negative gravitational mass, the argument that "effective mass" is somehow not real is flawed.

To define something like "mass" you must provide an operational definition, and then check that this is consistent when applied in various contexts. As of 2017, the (1kg) still depends on comparison with a particular block of platinum-iridium. Although it would be nice to define the kg in terms of more fundamental properties, such as the mass of a certain number of carbon molecules, experimental uncertainties make such an operational definition less accurate than the current standard. This will eventually change to provide a more fundamental definition once experimental procedures reach the appropriate level of precision.

Thus, an operational definition for the inertial mass of an object might be to place your object next to the platinum-iridium block so they both start at rest. Then explode a small charge between the two so the block and the object start moving away from each other. The mass of the object in kg may then be determined as the ratio of the relative speeds of the block and the object. (Of course, you would not want to do this with the kg standard mass as the explosion might damage it, but in principle this definition works).

This is in essence what we have done in our experiment. The repulsive interactions in the atomic cloud produce an outward force that leads to an acceleration of the atoms. Although the situation is more complicated than the standard case due to the fact that the effective mass in our system depends on velocity, the resulting motion is described by a set of equations where the effective mass is negative. In this operational sense, the negative effective mass we observed is in every sense as "real" as the mass of a macroscopic objects measured in comparison with the block in France.

(To emphasize how an operational definition might fail to capture the desired property, consider placing the object on a spring scale, and defining the "mass" as the number on the display. This definition would change if you repeated the experiment at different places on earth, or on the moon, or in a falling elevator. Here we have instead an operational definition for a different quantity – the "weight of the object rather than the mass.)

Does this bring us closer to a "Warp Drive"?

Unfortunately, no. Several popular news sources have attempted to sensationalize our results by linking them to the possibility of faster-than-light (FTL) travel with wormholes or Alcubierre drive which require some form of exotic matter with "negative mass".

The confusion comes from the two different roles that mass plays in physics. What we have created is a fluid that displays negative inertial mass. This is the mass $m$ that enters Newton's second law $F=ma$. Wormholes etc. require a negative gravitational mass - the mass that enters Newton's law of gravitation $F=-mM/r^2$ (and which causes space-time to curve in Einstein's theory of general relativity). For standard particles, the inertial mass and gravitational mass are the same - something called the "equivalence principle", which is a prediction of Einstein's theory and has been tested to high precision. It is this principle which causes objects to fall at the same rate in the absence of air resistance, such as on the moon:

Feather and hammer falling at the same rate on the moon.

Feathers and bowling balls falling at the same rate in the largest vacuum chamber.

In our experiment, the presence of the spin-orbit coupling changes the effective inertial mass, causing the strange behaviour, but not the gravitational mass, which remains positive. Thus, our research does not directly aid the development of FTL technology.

Why is this interesting?

The phenomenon of negative effective mass is interesting in its own right, and can be seen in several systems, including optical lattices, and non-linear optics (see the references give in the paper). What is new here is the ability to manipulate the dispersion relationship to produce a negative effective mass without any complications of an underlying lattice potential. We were thus able to demonstrate that the self-trapping phenomenon seen in optical lattices results purely from the negative effective mass. There are still many properties of these systems that are not well understood, such as the nature and stability of the shockwave.

Slightly further along are application where this effect might be able to be used to manipulate cold atoms for studying other phenomena. For example, it turns out that another form of trapped cold atoms - lithium (Li) - can be made to behave almost exactly like the neutrons found in the crust of neutron stars, which also form a superfluid. Thus, by trapping lithium, one can probe properties of the superfluid neutrons in a controlled setting, thereby testing the theories used to understand the physics of neutron stars.

Universality: Neutron stars can be studied with Cold Atoms?

Yes! Through a beautiful coincidence, it turns out that the neutron-neutron interaction is characterized by a length (called the s-wave scattering length) that is unnaturally large. This means neutrons that are not to crowded together can be almost completely described simply by an interaction that does not depend on all of the complications of the underlying nuclear physics. These conditions are realized in the crust of neutron stars. (Further toward the core of a neutron star, the neutrons get so close together than one cannot neglect the full structure of the nuclear interactions, and the description becomes much more complicated.) Using magnetic fields and a property called a Feshbach resonance, dilute lithium atoms can be made to interact with a large scattering limit (something called the "unitary Fermi gas") in cold atom experiments.

Thus, some of the densest objects in the universe - neutron stars - can be described by some of the most dilute matter in the universe - trapped clouds of lithium. Just think about this for a moment: a teaspoon of neutron star matter would weight as much as a small mountain, yet it shares many properties with clouds of lithium that are more dilute than the vacuum of space! As perplexing as this might sound, it is true, and provides a unique opportunity to study the properties of neutron stars in a laboratory where we can probe and observe the behaviour of the matter in conditions that are impossible with neutron stars which are located thousands of light years away.

One might also like to use cold-atom systems to study the properties of heavy nuclei to better understand how they might be used to produce energy for example. However, unlike neutron stars which are held together by gravity in a similar way to which cold atoms are held together with laser traps, nuclei are self-bound, which is quite different from cold atoms. There is thus a quest to try to produce self-bound assemblies of cold atoms. One proposal by my collaborator Aurel Bulgac suggests using three-body forces to form Dilute Quantum Droplets (called boselets, fermilets, or ferbolets). The self-trapping effecting of produced by a negative effective mass might provide a different strategy.

Neutron stars are not the only cosmological phenomenon that can be simulated with cold atoms. It might also be possible to directly simulate properties of black holes by producing the superfluid analogue called a sonic black hole or "dumb" hole. In these systems, the event horizon of a black hole is modeled by a superfluid flowing faster and faster down an increasingly steep pipe. Once the local flow velocity exceeds the speed of sound in the superfluid, one has a point beyond which sound cannot flow backwards. It turns out that sound in these systems behaves mathematically like light does in the vacuum (the equations satisfy a form of relativistic invariance) and so the mathematical description of the horizon of a "dumb" hole is the same as that of the event horizon at a black hole.

Another cosmological application is related to dark matter - a mysterious substance that seems necessary to explain cosmological observation, whose nature remains to be discovered. One proposal, a bosonic particle called the Axion, may be light enough that it forms a Bose-Einstein condensate on the scale of the galaxy. This fluid can be described by similar equations to the BEC we study in cold atom systems, and strange quantum properties such as the structure of vortices may be visible in the structure of galaxies.

What about practical applications?

While the applications discussed above are mostly of interest to physicists, quantum phenomena tend to find there way into practical applications in what are referred to as quantum technologies. These are applications of quantum phenomena such as entanglement, superposition, and tunneling, to technologies such as accurate sensing, secure communication, and quantum computing. Atomic clocks and highly sensitive magnetometers called SQUIDS are examples of quantum technology applications resulting from improved control over quantum phenomena. At this stage it is too early to tell if there will be direct applications for spin-orbit coupled BECs and negative mass hydrodynamics, but this field is one of the most rapidly progressing fields in physics, with significant interest from industry. (Microsoft and Google, for example, are both highly invested in quantum computing, which is a part of the quantum technologies field.)

Media Attention

This research received an unusual amount of media attention. Most likely this was due to a couple of factors:

  • A catchy tagline: "Researchers create 'negative mass'...". The term "negative effective mass" would have been more accurate, and probably would have suppressed some of the attention, but would likely have also been passed over by interested people who did not understand what "effective mass" means. This can be seen in the discussion and comments on Sabine Hossenfelder's BackRe(Action) Blog where the term is called a "historical artefact". As Thomas Schaefer discusses in the comments, and as outlined above, all inertial mass is, by definition, "effective". Thus it is highly likely that including the term "effective" in the title would have turned away some otherwise interested readers. In retrospect, the term "negative inertial mass" would probably have been the best compromise since the most outlandish associations results from confusion with gravitational mass.
  • The physics concepts appeal to a broad audience. Most people are familiar with the notion of mass. Many are familiar with Newton's second law $F=ma$. Thus, the article resonates and people feel they have a chance of understanding what is going on.
  • An unfortunate consequence of the wording is that many sites simply equated "negative mass" with negative gravitational mass, which would connect our work to wormholes, faster-than-light travel, etc. This has nothing to do with what we have done, as I discussed above, but captures people's attention.

For a summary of the media attention this article has generated, see:

  • Altmetric Report: As of 31 May 2017, this article is the 11'th most popular PRL article by Altmetric's categorization.

Here are some of the more serious articles. These sources either interviewed me, or contacted other scientists for opinions (or both) and basically got things right:

A couple of sites have posted articles that attempt to downplay the significance of the results - primarily aimed at news sites that did not properly investigate the paper and sensationally connected our work with wormholes, warp drives and the like (which require negative gravitational mass). Reading these articles along with the comments gives a good perspective

Recently, a nice article discussing the nature of mass appeared in Physics Today: This article also discusses the possibility of negative gravitational mass.

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