Section 7: Phase Control and New Structures
We have mentioned several times that laser light and the bosonic atoms in a gaseous BEC are coherent: The quantum phase of each particle is locked in phase with that of every other particle. As this phase coherence is lost, the condensate is lost, too. What is this quantum phase? Here, we must expand on our earlier discussion of standing waves and quantum wavefunctions. We will discuss what is actually waving in a quantum wave as opposed to a classical one, and acknowledge where this waving shows up in predictions made by quantum mechanics and in actual experimental data. Unlike poetry or pure mathematics, physics is always grounded when faced with experimental fact.
Like vibrating cello strings or ripples in the ocean, quantum wavefunctions are waves in both space and time. We learned in Unit 5 that quantum mechanics is a theory of probabilities. The full story is that quantum wavefunctions describe complex waves. These waves are sometimes also called "probability amplitudes," to make clear that it is not the wavefunction itself which is the probability. Mathematically, they are akin to the sines and cosines of high school trigonometry, but with the addition of i, the square root of -1, to the sine part. So, the wave has both a real part and an imaginary part. This is represented mathematically as (see sidebar), where the i is called the "complex phase" of the wavefunction.
Figure 25: While the complex part of a quantum wavefunction "waves," the probability density does not.
Source: © William P. Reinhardt. More info
The probability density (or probability distribution), which tells us how likely we are to detect the particle in any location in space, is the absolute value squared of the complex probability amplitude, and as probabilities should be, they are real and positive. What is this odd-sounding phrase "absolute value squared"? A particle's probability density, proportional to the absolute value of the wavefunction squared, is something we can observe in experiments, and is always measured to be a positive real number. Our detectors cannot detect imaginary things. One of the cardinal rules of quantum mechanics is that although it can make predictions that seem strange, its mathematical description of physics must match what we observe, and the absolute value squared of a complex number is always positive and real. See the math
What about the stationary states such as the atomic energy levels discussed both in this unit and in Unit 5? If a quantum system is in a stationary state, nothing happens, as time passes, to the observed probability density: That's, of course, why it's called a stationary state. It might seem that nothing is waving at all, but we now know that what is waving is the complex phase.
For a stationary state, there is nothing left of the underlying complex waving after taking the absolute value squared of the wavefunction. But if two waves meet and are displaced from one another, the phases don't match. The result is quantum interference, as we have already seen for light waves, and will soon see for coherent matter waves. Experimentalists have learned to control the imaginary quantum phase of the wavefunction, which then determines the properties of the quantum probability densities. Phase control generates real time-dependent quantum phenomena that are characteristic of both BECs and superconductors, which we will explore below.
Phase imprinting and vortices
When does this hidden phase that we don't see in the experimentally measurable probability density become manifest? The simplest example is in interference patterns such as those shown in Figure 26. Here, the phase difference between two macroscopic quantum BECs determines the locations of the regions of positive and negative interference. So, situations do exist in which this mysterious phase becomes actual and measurable: when interference patterns are generated.
Figure 26: Interference of two coherent BECs, separated and allowed to recombine.
Source: © William P. Reinhardt (left image) and Wolfgang Ketterle (right image). More info
But, there is more if we fiddle with the overall phase of a whole superfluid or superconductor with its many-particle macroscopic wavefunction. Imprinting a phase on such a system can create moving dynamical structures called vortices and solitons. Both signify the existence of a fully coherent, macroscopic many-particle quantum system.
A phase can be "imprinted" by rotating such a macroscopic quantum system. This creates vortices as we see in the density profile calculated in Figure 27. Physicists have observed such vortices in liquid superfluids, gaseous BECs, and superconductors. All have the same origin.
Figure 27: Quantum vortices in a BEC (top) and the corresponding phase of the quantum wavefunction (bottom).
Source: © William P. Reinhardt. More info
We can obtain a physical picture of what's happening if we examine the phase of the underlying macroscopic wavefunction. That's difficult to do experimentally, but easy if we use a computer calculation to generate the coherent many-body macroscopic wavefunction, as Figure 27 illustrates. Here, unlike in an experimental situation, we fully know the phase of the wavefunction because we have programmed it ourselves. We can then calculate its absolute value squared if we like, which gives the resulting probability density that an experiment would observe. If we calculate the phase in a model of a circulating BEC, superfluid, or superconducting current, and compare it to the observable probability density, we find the phase changes periodically as you trace a circle around what appear to be holes in the probability density, as you can see in Figure 27. Each hole is a tiny quantum whirlpool, or vortex; the lines of phase radiating outward indicate that the quantum fluid is circulating around these holes.
Solitons: waves that do not decay
Clearly, then, altering the phase of a macroscopic wavefunction may have dramatic effects. Another such effect is the creation of the special nonlinear waves called "solitons," which physicists have observed in BECs following a phase imprinting. Solitons are a very special type of wave. They can, and do, pass right though each other without dissipating. And if they are in a superfluid (a gaseous atomic BEC in this case), they will carry on this oscillatory motion forever. These are called "density notch solitons," and are created by phase imprinting on part of a condensate. The process both creates and drives the motion of the solitons, just as circular phase imprints both create and maintain superfluid vortices.
Figure 28: Density notch soliton.
Source: © William P. Reinhardt. More info
The vortices and solitons are both highly unusual persistent defects in the quantum wavefunction. Were these systems not macroscopic superfluids, such defects would just disappear and dissipate of their own accord. Try to make a hole in a tub of water and watch it fill in. (On the other hand, lumps of water can propagate without dispersion: Think of tsunamis, where, in exactly the right circumstances, lumps of water can move across a whole ocean.) The same would happen to a quantum bump or dip in the density corresponding to one or a few quantum particles in free space: The wavefunction would spontaneously disperse. It is the collective and coherent nature of the many-particle BEC that allows these nonlinear wave structures to persist. Of course, if you can create a whirlpool as the bath tub drains, it will persist; but such a classical whirlpool has nothing to do with the phase of the liquid water, phase being a purely quantum concept as we apply it here.
Figure 29: The formation of vortices in this BEC shows that it is a superfluid.
Source: © Martin Zwierlein. More info
The vortices and solitons we describe are all consistent with the idea that all the particles are in a single quantum state. These highly dilute gaseous condensates are essentially 100 percent pure, in the sense that all the particles occupy a single quantum state in the trap—although, to be sure, the quantum state wavefunction distorts as the trap is filled. The simple tools developed here have their limitations. They apply to dilute atomic gases, but cannot deal with a more traditional liquid such as 4He in its superfluid state. Owing to the very strong interactions between the helium atoms at the much higher density of the liquid, the condensate is only about 10 percent pure condensate. This does not affect its remarkable properties as a superfluid, but certainly makes its theoretical description more difficult. In fact, it makes the quantitative description of the coherent quantum wavefunction a tremendously exciting exercise in computational physics.