Direct energy dissipation measurements for a driven superfluid via the harmonic-potential theorem

This paper proposes and experimentally demonstrates a method to directly measure energy dissipation in a linearly driven superfluid by leveraging a perturbed harmonic-potential theorem to convert center-of-mass motion into internal energy, thereby enabling the quantitative determination of dissipation and the observation of critical velocity behavior in a driven Bose-Einstein condensate.

The Gist

Imagine you are riding a bicycle down a smooth, empty road. You stop pedaling, and the bike coasts. If the road is perfectly frictionless, the bike would coast forever at the same speed. But in the real world, air resistance and friction slow you down. That lost speed? That's energy turning into heat.

Now, imagine you are riding that bike, but instead of a flat road, you are riding inside a giant, invisible bowl that constantly pushes you back toward the center. This is a harmonic trap, and the "bike" is a cloud of atoms so cold they act like a single, giant quantum wave. This state of matter is called a Bose-Einstein Condensate (BEC), and it behaves like a superfluid—a liquid with zero friction.

The scientists in this paper wanted to answer a simple question: How much energy does it take to stir this superfluid?

The Problem: Measuring the Invisible

Usually, to measure how much energy a fluid loses to friction, you have to wait until the fluid heats up. But in the quantum world, things are tricky.

1. It's too cold: The atoms are so cold that the tiny amount of heat generated by friction is hard to detect.
2. It takes too long: The atoms might not "settle down" and turn that friction into heat quickly enough for you to measure it before the experiment ends.

It's like trying to measure how much a bike slows down by checking if the tires are warm. If the ride is short, the tires might not feel warm yet, even if friction is happening.

The Solution: The "Bicycle Theorem"

The authors came up with a clever new way to measure this energy loss without waiting for the heat. They used a rule from physics called the Harmonic-Potential Theorem (HPT).

Here is the analogy:
Imagine the entire cloud of atoms is a single, giant "super-bike."

• The Center of Mass (COM): This is the average position of the whole bike.
• The Internal Energy: This is the wiggling, spinning, and vibrating of the atoms inside the bike.

The HPT says that if you push this "super-bike" with a specific, rhythmic force (like a parent pushing a child on a swing), the bike will swing back and forth perfectly. Unless there is something inside the bike causing friction.

If the bike slows down (damps), the HPT tells us exactly where that energy went. It didn't disappear; it was transferred from the swinging motion (the bike moving back and forth) into the internal wiggling (the atoms getting excited).

By measuring exactly how much the "swing" slowed down, the scientists could calculate exactly how much energy was stolen by the friction, instantly. No waiting for heat. No guessing.

The Experiment: Stirring the Quantum Soup

Here is what they actually did in the lab:

1. The Setup: They trapped about 55,000 Rubidium atoms in a magnetic "bowl."
2. The Stirrer: They used a laser beam (like a tiny, invisible spoon) to poke the atoms.
3. The Shake: They shook the whole bowl back and forth at just the right rhythm to make the whole cloud of atoms swing.
4. The Measurement: They watched the cloud swing.
   • Without the laser spoon: The cloud swung perfectly, like a frictionless pendulum.
   • With the laser spoon: The cloud still swung, but it started to slow down faster than expected.

The Discovery: The "Critical Speed"

The most exciting part of their discovery is what happened when they changed how fast they shook the bowl.

• Slow Shaking: When they shook the bowl slowly, the laser spoon couldn't "grab" the superfluid. The atoms flowed around the spoon like water around a rock in a stream. Zero energy loss. The superfluid remained perfect.
• Fast Shaking: Once they crossed a specific speed (the Critical Velocity), the superfluid broke. The atoms started to get "stuck" on the spoon, creating little ripples and waves (called solitons). Suddenly, the swinging stopped, and the energy was dumped into the atoms.

This confirmed a famous prediction in physics: Superfluids have a speed limit. Below that limit, they flow without friction. Above it, they act like normal, sticky fluids.

Why This Matters

This paper is a big deal because it gives scientists a direct, instant ruler to measure friction in quantum fluids.

• Before: Scientists had to guess or wait for the fluid to heat up.
• Now: They can look at the "swing" of the atoms and immediately know how much energy is being lost.

It's like having a speedometer that doesn't just tell you how fast you are going, but also instantly tells you how much fuel you are burning due to air resistance, just by looking at how the car's suspension bounces.

This new tool will help researchers understand everything from how superconductors (electricity without resistance) work to how black holes might behave, by letting them "stir" the quantum world and measure the ripples in real-time.

Technical Summary

1. Problem Statement

Superfluidity is typically diagnosed by observing the response of a quantum system to a moving perturbation (a "stirrer"). Historically, measuring energy dissipation in these systems has relied on indirect methods or calorimetric approaches:

• Indirect Proxies: Observing changes in condensate fraction, density profiles, or the excitation of vortices and solitons.
• Calorimetry: Measuring the temperature increase of the cloud after a hold time. This method assumes the system equilibrates to a thermal state during the hold period. However, for dilute, weakly interacting systems, thermalization can be slow, making this assumption unreliable (as noted in recent re-analyses of historic BEC experiments).

There was a lack of a method to directly and quantitatively measure the instantaneous energy dissipation into internal degrees of freedom without relying on thermalization assumptions.

2. Methodology

The authors propose and experimentally demonstrate a method based on a perturbed version of the Harmonic-Potential Theorem (HPT).

Theoretical Framework

• The HPT: In a harmonic trap, the center-of-mass (COM) motion of a many-body system is decoupled from internal dynamics (generalized Kohn theorem). If the trap is perturbed, this decoupling breaks.
• Energy Decomposition: The authors derive that the total energy per particle ($e$) in a linearly driven system can be decomposed into:
  $$e = e_{COM} + e_{int}$$
  Where $e_{COM}$ is the energy associated with the macroscopic COM motion, and $e_{int}$ is the internal energy in the COM rest frame.
• Dissipation Mechanism: The perturbation $\delta V$ acts as a moving stirrer in the COM rest frame. Frictional forces arising from this relative motion excite internal degrees of freedom.
• Direct Measurement: By measuring the damping of the COM motion (position $y$ and velocity $v$), one can calculate the change in $e_{COM}$. Using energy conservation ($\dot{e} = -\dot{f} \cdot r$), the dissipated energy ($\Delta e_{int}$) is directly inferred from the difference between the total work done and the change in COM energy, without needing to measure temperature or assume thermal equilibrium.

Experimental Setup

• System: A nearly pure Bose-Einstein Condensate (BEC) of $\approx 55,000$ $^{87}$Rb atoms in an elongated magnetic trap (cigar-shaped).
• Stirrer: A blue-detuned laser beam shaped by acousto-optical deflectors (AODs) creates a static optical potential perturbation $\delta V$ crossing the trap.
• Driving Protocol:
  1. Ramp: The optical perturbation is ramped up linearly over 20 ms.
  2. Drive: A resonant linear magnetic gradient oscillates at the longitudinal trap frequency ($\omega_y$), driving the COM motion.
  3. Hold & Imaging: The drive and perturbation are switched off. The cloud is held for variable times ($t_{hold}$) and then imaged via Time-of-Flight (TOF).
• Data Extraction: By analyzing TOF images at different hold times, the authors reconstruct the in-situ COM position and velocity at the moment the drive stopped. This allows the calculation of $e_{COM}(t)$ and subsequently $\Delta e_{int}(t)$.

3. Key Contributions

1. Novel Measurement Technique: The first experimental demonstration of a method to directly quantify energy dissipation in a driven superfluid using the HPT, bypassing the need for thermalization assumptions.
2. Theoretical Extension: Application of the HPT to a perturbed and linearly driven system, explicitly separating COM and internal energy dynamics in the COM rest frame.
3. Experimental Validation: Successful application of the method to a BEC, observing characteristic superfluid dissipation curves.
4. Validation via Simulation: Corroboration of experimental findings with 3D Gross-Pitaevskii equation (GPE) simulations, confirming the theoretical decomposition of energy.

4. Key Results

• Dissipation Curves: The experiment successfully measured the internal energy evolution ($\Delta e_{int}$).
  • Unperturbed Case ($\delta V = 0$): As predicted by the HPT, the internal energy remained zero ($\Delta e_{int} \approx 0$), confirming undamped harmonic motion.
  • Perturbed Case ($\delta V > 0$): A clear onset of dissipation was observed. The internal energy remained near zero for an initial period ($t_{onset} \approx 50$ ms) and then grew, indicating the onset of energy transfer to internal modes.
• Critical Velocity: The onset time of dissipation depended on the stirrer strength ($\delta V_0$). Stronger perturbations led to earlier onset times, corresponding to lower critical velocities. This behavior aligns with the local Landau criterion:
  $$v_c = \left(1 - \frac{\delta V_0}{\mu_0}\right)^{5/2} c_0$$
  where $c_0$ is the sound velocity and $\mu_0$ is the chemical potential.
• Quasi-Steady State: For larger stirrer strengths, the system entered a quasi-steady state where the energy injection rate balanced the dissipation rate, resulting in a constant internal energy growth rate. For weaker stirrers, this regime was not reached within the experimental timeframe.
• Excitation Types: Mean-field simulations revealed that the dissipation was mediated by the production of dark solitons (large density dips, phase jumps of $\pi$) and sound waves (phonons), rather than vortices, consistent with the 1D-like geometry of the trap.

5. Significance

• Fundamental Physics: This work provides a rigorous, model-independent tool to study non-equilibrium dynamics in quantum fluids. It resolves ambiguities regarding thermalization assumptions in previous calorimetric studies.
• Diagnostic Power: The method allows for the direct mapping of friction forces and dissipation rates as a function of drive parameters, offering a precise way to probe the superfluid-to-normal transition.
• Future Applications: The technique opens avenues for:
  • Spectroscopic studies of excitations (phonons, solitons, vortices).
  • Investigating energy transport in quantum turbulence.
  • Studying two-component dynamics in finite-temperature BECs.
  • Exploring non-thermal fixed points in internal momentum distributions.

In summary, the paper establishes a robust experimental protocol to "weigh" the energy dissipation of a superfluid in real-time by observing the deceleration of its center of mass, effectively treating the quantum fluid as a classical object subject to friction, but with the friction arising from quantum many-body effects.