### Edwards Research Hompage

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# Hysteresis in a quantized superfluid 'atomtronic' circuit

### Work of Murray and Edwards featured on the cover of the journal Nature

 Mark Edwards
 Noel Murray
Noel Murray, a student in the joint Physics/Chemistry Masters of Science in Physical Science degree program in the College of Science and Mathematics at Georgia Southern University, and Mark Edwards, Fuller E. Callaway Professor of Physics, will be among the authors of an article featured on the cover of the February 13, 2014 issue of the journal Nature. Nature is the flagship journal of the Nature Publishing Group and is widely regarded as the world's premier interdisciplinary science journal. The article is also profiled by a Nature "News and Views" essay and featured on the weekly Nature Podcast.

Entitled "Hysteresis in a quantized superfluid 'atomtronic' circuit," the article reports on collaborative work between a team of experimental researchers at the Joint Quantum Institute (JQI) and the Georgia Southern University theoretical team consisting of Murray and Edwards. The JQI is an institute run by the University of Maryland (UMD) and the National Institute of Standards and Technology (NIST). NIST is a national laboratory run by the US Department of Commerce. The JQI research team consists of Steve Eckel, Jeff Lee, Fred Jendrzejewski, Charles Clark, Chris Lobb, Bill Phillips, and Gretchen Campbell. One of the co-authors, Bill Phillips, is a co-winner of the 1997 Physics Nobel Prize.

The article reports the results of experiments performed on a ring-shaped gas of about one-half million sodium atoms that was cooled down to a temperature of just a few hundred billionths of a degree above absolute zero. At these temperatures the "quantum" nature of the atoms is enhanced. The hallmark of the quantum nature of atoms, normally thought of as particles, is that they begin to take on wave-like properties. When the gas is made cold (that is, the atoms move slower) and dense (that is, atoms are packed tighter together) enough, the matter-wave shapes of all one-million atoms become exactly the same. Each atom becomes the size of the entire gas. Thus the gas of one-million atoms now behaves like a single atom and the quantum (wave-like) nature of a single atom is magnified up to the macroscopic scale. This phenomenon where the whole gas starts to act like a single atom was first predicted in 1925 by Satyendranath Bose and Albert Einstein and is called Bose-Einstein condensation. The first Bose-Einstein condensate (BEC) in atomic gases was observed at the University of Colorado in 1995 by Eric Cornell and Carl Wieman. For this work, Cornell and Wieman were awarded the 2001 Nobel Prize for Physics along Wolfgang Ketterle of MIT.

The response of a ring-shaped atomic BEC to being stirred with laser beams was studied both experimentally by the JQI team and theoretically by members of the Edwards Ultra-cold Atom Research Group at Georgia Southern . The flow speed of the gas around the ring suddenly jumps up from no-flow to flow and jumps down from flow to no-flow in response to the stirring. These jumps happen at different, well-defined stir speeds in a stationary reference. In a rotating reference frame these speeds will be different. Thus this system has the potential to be used as an ultra-precise rotation sensor.

The experiment where a ring Bose-Einstein condensate is stirred is an example of an "atomtronic" circuit. Atomtronics is the name of an emergent field that seeks to develop new functional methods by creating devices and circuits where ultracold atoms, often superfluids, hava a role analogous to that of electrons in electronic devices and circuits. Atomtronic circuits have the potential to drive many new applications. In contrast to ordinary circuits where the current is carried by electrons, atomtronic-circuit currents are carried by neutral atoms. Neutral atoms can be manipulated in a variety of ways including their motion, their internal energy state, and properties associated with the atoms being part of a superfluid. The work done here enhances the fundamental understanding of these new atomtronic circuit sytems and paves the way for the design of practical devices based on this new technology.

This article is only the fourth time that a regular article or letter has appeared in Nature with authors having a Georgia Southern affiliation (according to Thomson-Reuters "Web of Knowledge" database). The last time was in 1999.

### Making a Bose-Einstein Condensate

The trapping and cooling of the gas was carried out using laser beams. Single atoms can be manipulated with laser light. A laser beam whose photon energy is smaller than the energy difference between two internal electronic states of the atom (a "red-detuned" laser) will exert a force on the atom pushing the atom towards regions of high laser intensity. That is atoms will be sucked into the center of a red-detuned laser beam. A laser beam whose photon energy is larger than the energy difference between two internal electronic states (a "blue-detuned" laser) will repel atoms from the high laser intensity regions. The brighter the laser is, the stronger the force that it will exert on an atom.

Ring-shaped Bose-Einstein condensates (BECs) were confined in this experiment using a combination of red- and blue-detuned lasers. Red-detuned laser light in the shape of a horizontal sheet was intersected with a vertically traveling blue-detuned laser beam. Since red-detuned light pulls atoms into the center of the beam, the laser sheet confines the gas atoms in a horizontal plane. The vertical laser beam was shone through a mask which blocks light within the area between two circles (an "annulus") and imaged onto the sheet with a lens. Thus atoms were pushed into a region where the red-detuned sheet was brightest and where the blue-detuned beam was absent. The blue-detuned laser was absent inside the ring-shaped shadow of the mask. Atoms were thus trapped inside the ring-shaped volume formed by the intersection of the red-detuned sheet laser and the shadow of the annulus-shaped mask that blocked the blue-detuned light.

A sample of sodium gas trapped in this way was then cooled by slowly dimming the blue-detuned beam. Atoms in a thermal-equilibrium gas have a range of speeds that follow a bell-curve-shaped distribution whose average is set by the temperature of the gas. When the blue-detuned laser is dimmed, this enables some of the atoms at the high-speed end of the distribution to escape from the trap. This creates a non-equilibrium situation. Equilibrium is restored after atom redistribute their speeds by colliding with one another. The resulting temperature of the new equilibrium state is lower than the previous state because some fast-moving atoms have escaped and the average atom speed is now lower. The cycle can be repeated by again dimming the blue-detuned laser and then allowing the atoms to again reach thermal equilibrium. By repeating this cycle over and over, the temperature of gas is reduced. Because the atoms move more slowly (on average) at the end of each cycle, they also take up less volume because of their confinement. Eventually the gas reaches the conditions of being cold enough and dense enough so that the Bose-Einstein condensate begins to form. Further cooling causes more and more atoms to join the condensate until essentially all of the remaining gas atoms are members of the condensate.

### Stirring the Condensate

The ring condensate is stirred with another blue-detuned laser beam. This beam travels vertically downward and pierces the annulus of the ring. This beam can then be rotated around the ring and thus acts like a stick (because it is blue-detuned, it pushes atoms out of the volume that the beam occupies) that can be used to stir the condensate. When the condensate is initially formed, the gas does not flow around the ring. However flow can be induced by stirring with the laser.

The flow speed of the condensate around the ring can only take on discrete values. Thus, if a stationary condensate is stirred very slowly by the laser beam, the gas will remain stationary. This is a consequence of the condensate's "superfluid" nature --- it presents no resistance to the gas flow at low stirring speeds. However, when the condensate is stirred more rapidly, a pair of vortices are formed. One can think of a "vortex" as a small tornado created within the annulus. Stirring the condensate fast enough will create a pair of vortices each spinning in the opposite direction of the other. This will also happen if one moves a stick through water in the bathtub. When the condensate is stirred at any speed above a critical value, call it $$v_{up}$$, vortices form and the the condensate flow speed jumps from zero up to its lowest possible non-zero value. It is also possible to stir a flowing condensate more slowly than its lowest flow speed to make the flow speed jump down to zero. When a flowing condensate is stirred at any speed slower than another critical value, call it $$v_{down}$$, the condensate flow speed will jump down to zero.

### The NIST Experiment

In this work, ring-shaped condensates were repeatedly created with zero flow and then stirred at successively faster and faster speeds. After the end of the stirring in each trial, the flow speed of the condensate was measured. In this way, the critical stir speed, $$v_{up}$$, above which the flow jumps from zero to non-zero was measured. Additionally, condensates were repeated created with non-zero flow and then stirred at successively slower and slower speeds. After the end of the stirring in each of these trials, the flow speed was again measured. In this way the critical stir speed, $$v_{down}$$, below which the flow jumps from non-zero to zero was measured.

These measurements of the critical speeds $$v_{up}$$ and $$v_{down}$$ were carried out for a given brightness of the stirring laser beam. The measurement of these critical speeds was repeated for a total of six different stirring beam brightnesses. The brighter the stirring laser beam, the stronger the force it exerts on the condensate atoms. The experimental data showed that the values of $$v_{up}$$ and $$v_{down}$$ where different. The critical stirring speed to cause a jump up from zero to non-zero flow was faster than the critical speed to cause a jump down from non-zero to zero flow. Thus $$v_{up}$$ was measured to be greater than $$v_{down}$$. Furthermore, the difference $$\delta v = v_{up}-v_{down}$$ was found to depend on the brightness of the stirring laser beam. For the least bright stirring beam the value of $$\delta v$$ was large. As the brightness of the beam was decreased, the value of $$\delta v$$ decreased until it went to zero. In that case, $$v_{up}$$ and $$v_{down}$$ had the same value.

### The Phenomenon of "Hysteresis"

These experimental results are an example of the phenomenon of "hysteresis". A system exhibits hysteresis when its response to a variable external "force" depends on the history of how the "force" is applied. We use the word "force" here to mean any kind of externally applied change in the environment of the system.

One everyday example of hysteresis is an iron rod (like a nail) which has an electrical wire wrapped around it. If the wire ends are connected to a battery, the current flowing through the wire will cause the iron rod to become a magnet. The usual way we describe this is to say that the current flowing through the wire around the rod creates a magnetic field inside the rod and pointing along the rod's length. This external (to the rod) magnetic field is the "force" referred to above. If the voltage of the battery is slowly increased, the external magnetic field inside the rod will increase and the magnetization of the rod will increase along with it. Eventually, the magnetization of the rod will stop increasing as the external magnetic field increases (this is called "saturation"). If then the external magnetic field is now decreased, the magnetization of the rod will also begin to decrease, however, when the external magnetic field is turned all the way off, the rod still has some magnetization. If the external magnetic field direction is reversed (by, say, reversing the polarity of the battery,) and then increased, this will decrease the magnetization of the rod further. At some point, this reversed external magnetic field will make the magnetization of the rod zero again. This is an example of hysteresis: initially the magnetization of the rod was zero when there was no external magnetic field present, later the magnetization was again zero but now there is a non-zero external magnetic field. The magnetization of the rod depended on the history of the external "force" (the magnetic field produced by the current through the wire.)

The NIST experiment found that an analogous effect occuring in stirring a ring condensate. The stir speed is analogous to the external magnetic field in the previous example and the condensate flow is analogous to the magnetization of the rod.If the condensate with zero flow is stirred faster and faster, it will jump to non-zero flow at stir speed $$v_{up}$$. If the stir speed is now decreased, the flow will jump back down to zero but the stir speed will be at the lower value, $$v_{down}$$. One can view the hysteretic response of a system as a lag between the "force" exerted on the system and the system's response.

Hysteresis is an expected property of a superfluid system and one that is important for applications. In this case, the speeds $$v_{up}$$ and $$v_{down}$$ in the reference frame of the laboratory can be measured (as was done in this experiment). If the same experiment is carried out in a rotating reference frame (imagine that the whole laboratory is placed on a rotating platform,) the speeds $$v_{up}$$ and $$v_{down}$$ will be shifted by the rate at which the platform is rotating. Thus the measurement can serve as a sensor of the rotation speed of the platform.

Ultra-precise rotation sensors are important in military applications. On the battlefield it is essential for soldiers to determine their positions. This most likely can't be done using the Global Positioning Systems (GPS) because the enemy will be jamming it. Soldiers can located their position in real time in battle situations by first determining their position away from the battlefield using GPS and then using devices that sense their acceleration and rotation. The soldiers' position relative to the point determined by GPS can then be found using a technique call "dead reckoning". Hence it is import- ant to have ultra-precise acceleration and rotating sensing devices. The ring condensate system is a prime candidate for being the heart of a rotation- sensing device.

### Theoretical Analysis of the Experiment

To understand these results, Georgia Southern researchers Noel Murray and Mark Edwards modeled the experiment using the generally accepted theory for gaseous Bose-Einstein condensates. Recall that, in a Bose-Einstein condensate, all of the atoms have the same matter-wave shape. In this theory, the matter- wave shape of a condensate atom obeys the Schroedinger equation from quantum mechanics that includes the energies that the atom would have if no other atoms were present and accounts for energy of collisions with other condensate atoms in an average way. The resulting equation for the matter-wave shape of a condensate atom is called the nonlinear Schroedinger equation or the Gross- Pitaevskii equation (GPE).

Analyzing the experiment using the GPE theory resulted in the prediction of the hysteresis phenomenon. The theory also provided theoretical values for $$v_{up}$$ and $$v_{down}$$ for each of the six brightnesses of the stirring laser beam. It also revealed a theoretical picture of the mechanism by which the condensate jumps between zero and non-zero flow via the production of vortices. According to the theory, when a condensate with zero flow is stirred, a pair of oppositely spinning vortices (a vortex/antivortex pair) is formed in the low-density region of the ring where the stirring laser is shining. If the stirring speed is below the critical speed for jumping to non-zero flow, then the vortices collide and annihilate. If the stirring speed is above the critical speed the vortices separate with one moving moving to the interior of the ring and the other moving to the exterior of the ring.

The results of the Georgia Southern Theory Team's calculations are shown in the figure. The figure shows six graphs. In each graph, the horizontal axis is the speed of the stirring laser beam and the vertical axis is the flow at the end of the stirring (0 = "zero flow", 1 = "non-zero flow"). Each graph has a red curve corresponding to a initial flow of zero and a green curve corresponding to a non-zero initial flow. So, the speed at which the red curve jumps from 0 to 1 is $$v_{up}$$ and the speed at which the green curve jumps from 1 to 0 is $$v_{down}$$. The difference of these is the "hysteresis loop width", $$\delta v$$, and is indicated with blue arrows. Each of the six different graphs corresponds to a different brightness in the stirring laser beam. The least bright beam corresponds to the upper left graph and decreasing brightness proceeds from left to right.

The GPE theory failed to predict correctly the values of $$\delta v=v_{up}-v_{down}$$, however. The experimental values of $$\delta v$$ ranged from being as much as a factor of five times smaller for the lowest brightness stirring laser beam to approximately equal for the highest brightness. This is quite surprising because the GPE theory has been highly successful over the past 20 years in predicting the properties and behaviors of Bose-Einstein condensates. This discrepancy makes clear that a more refined microscopic condensate theory needs to be applied here. It also raises intriguing questions about what physical effects needed to be added in to obtain agreement with the experimental data.

To gain a better understanding of the experimental results, a toy model based on the vortex/antivortex picture predicted by the GPE theory was devised. In this model, the energy of a pair of vortices in the low-density stirring- beam region was calculated as a function of the distance between the vortex pair. Researchers found that it was possible to obtain reasonable agreement with the data by fitting parameter appearing in the toy model. However, with an accurate microscopic theory of this phenomenon, it is unlikely that progress can be made towards designing a practical rotation-sensing device.

### Atomtronics

The experiment where a ring Bose-Einstein condensate is stirred is an example of an "atomtronic" circuit. Atomtronics is the name of an emergent field that seeks to develop new functional methods by creating devices and circuits where ultracold atoms, often superfluids, hava a role analogous to that of electrons in electronic devices and circuits. Atomtronic circuits have the potential to drive many new applications. In contrast to ordinary circuits where the current is carried by electrons, atomtronic-circuit currents are carried by neutral atoms. Neutral atoms can be manipulated in a variety of ways including their motion, their internal energy state, and properties associated with the atoms being part of a superfluid. The work done here enhances the fundamental understanding of these new atomtronic circuit sytems and paves the way for the design of practical devices based on this new technology.