⚛️ general relativity

Sound Wave in the Backreaction Affected Spacetime in Analogue Gravity Based on Number-Conserving Approach

Using a number-conserving approach, this paper demonstrates that sound waves in backreaction-affected dynamical spacetime behave as massive scalar fields in an analogue metric with corrected fluid parameters, leading to a significant enhancement in local correlation functions in finite Bose gases.

Original authors: Sang-Shin Baak, Hyun-Yong Lee

Published 2026-10-14

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CC BY 4.0

Original authors: Sang-Shin Baak, Hyun-Yong Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of modern physics, researchers are building miniature universes inside laboratories to test ideas that are otherwise impossible to reach. These are not universes made of stars and galaxies, but of ultra-cold clouds of atoms, trapped and cooled until they behave as a single, unified quantum entity. Within these clouds, sound waves do not just travel through air; they ripple through a fluid that mimics the fabric of space and time itself. This field, known as analogue gravity, allows scientists to study how gravity might work by observing how sound behaves in these special fluids. For decades, a central question has lingered: if the sound waves themselves are powerful enough to slightly change the fluid they travel through, does the fluid still look like a smooth, curved space? Or does the interaction between the wave and the fluid create a more complex, messy reality that breaks the simple picture? Understanding this feedback loop is crucial because it mirrors a deep mystery in our own universe: how the energy of quantum particles might warp the geometry of space around them, a process that is too subtle to see in real gravity but might be visible in a lab.

A team of researchers has now taken a significant step toward answering this question by simulating these interactions with extreme precision. They focused on a cloud of atoms known as a Bose-Einstein condensate, a state of matter where atoms lose their individual identities and move in perfect unison. In their study, they treated the sound waves in this cloud not just as passive ripples, but as active participants that push back against the fluid. Using a rigorous mathematical approach that strictly conserves the number of atoms in the system, they calculated how the fluid's density and flow speed would adjust in response to the waves. Their work reveals that even when these adjustments are included, the fluid still behaves like a curved spacetime, much like a river that changes its current slightly but still flows in the same general direction. However, the waves do not travel exactly as they would in a static, unchanging environment. The researchers found that the waves acquire a property similar to mass, a kind of heaviness that depends on where they are and when they are measured. This mass-like effect arises directly from the way the fluid corrects itself in response to the passing sound, adding a new layer of complexity to the simple picture of massless sound waves.

To see these subtle effects in action, the team ran detailed computer simulations of a one-dimensional tube of atoms, roughly the size of a small box used in current experiments. They started the system in a calm, non-interacting state and then suddenly turned on the interactions, watching how the system evolved over time. By comparing the behavior of the sound waves with and without the feedback from the fluid, they could isolate the specific influence of the backreaction. The results showed a clear, measurable difference. In the middle of the cloud, the connection between the phase of the wave at one point and the same point a moment later increased by about ten percent when the feedback was included. Near the edges of the cloud, this effect was even stronger, boosting the connection by up to twenty percent. These numbers are not just theoretical guesses; they are the direct output of a simulation that tracked thousands of atomic modes, ensuring that the tiny corrections were not lost in the noise.

The researchers were careful to ensure their findings would hold up in a real-world setting. They modeled the limitations of actual laboratory cameras, which cannot see every single atom but instead see a blurry average over a small area. Even when they accounted for this blurring, the ten percent increase in the middle of the cloud remained visible. This suggests that the effect is robust enough to be detected in future experiments, provided the atoms are cold enough and the interactions are tuned precisely. The team noted that achieving the necessary conditions would require using a specific type of atom and adjusting its interactions to be about twenty times weaker than usual, a feat that is challenging but within the reach of modern technology. They also pointed out that the system must remain stable, with very few atoms leaving the main condensate, a condition that their simulations confirmed was met for the time scales they studied.

This work does not claim to have solved the deepest mysteries of gravity, but it has mapped a clear path forward. It demonstrates that the simple analogy of sound in a fluid remains valid even when the fluid reacts to the sound, provided we add a specific correction: a mass-like term that changes depending on the local conditions. The study confirms that the feedback between the quantum field and the background is not a chaotic disruption but a structured modification that can be predicted and measured. By showing that these effects are large enough to be seen with current imaging techniques, the researchers have turned a theoretical possibility into an experimental target. The next step for the scientific community is to build the specific setup required to observe this ten percent shift, bridging the gap between the equations on a page and the reality of a quantum fluid in a lab.

Technical Summary: Sound Wave in the Backreaction-Affected Spacetime in Analogue Gravity Based on Number-Conserving Approach

Problem Statement
Analogue gravity posits that sound waves in inviscid, barotropic, and irrotational fluids obey the equation of a massless scalar field on an effective curved spacetime. While this correspondence has enabled laboratory observations of analogue Hawking radiation and quantum simulations, most theoretical treatments assume a prescribed background. A complete description requires accounting for backreaction: the modification of the medium by quantum fluctuations, which in turn alters the propagation of those fluctuations. Previous studies have addressed backreaction in various contexts (water waves, light, acoustic black holes, and Bose gases), but often without rigorously separating the condensate from its fluctuations or without describing the resulting metric. Specifically, it remains an open question whether the quantum phase field still propagates on an acoustic metric of the corrected background and what additional terms arise beyond the standard massless equation.

Methodology
The authors employ a number-conserving expansion to rigorously analyze the system beyond the linear (Bogoliubov) approximation. This approach organizes feedback in powers of $1/N$ , where $N$ is the total particle number, ensuring the conservation of particle number and avoiding the symmetry-breaking issues of standard mean-field theories.

Formalism: Starting from the Hamiltonian of a Bose gas, the field operator is expanded using a number-conserving ansatz: $\hat{\Psi} = \phi_c(1 + \hat{\psi})\hat{A}/\sqrt{\hat{N}}$ . Here, $\phi_c$ is the $U(1)$ -conserving order parameter and $\hat{\psi}$ is the quantum fluctuation.
Modified Fluid Dynamics: By deriving the equations of motion for the classical density and current, the authors obtain amended Gross-Pitaevskii and Bogoliubov-de Gennes (BdG) equations. These include $O(1/N)$ correction terms ( $\Delta_C$ and $\Delta_E$ ) arising from quantum fluctuations, which modify the continuity and Euler equations.
Derivation of the Wave Equation: The authors derive the equation of motion for the phase fluctuation $\delta\hat{\theta}$ on this backreaction-corrected background. They demonstrate that in the long-wavelength limit, this equation takes the form of a massive Klein-Gordon equation in an analogue spacetime.
Numerical Simulation: To quantify these effects, the authors simulate a finite-size, homogeneous, quasi-one-dimensional Bose gas quenched from a non-interacting state at $t=0$ . They solve the coupled condensate and BdG equations numerically, truncating the mode expansion at $n_{max}=1000$ , and compute the canonically normal-ordered correlation functions.

Key Contributions

Massive Scalar Field in Analogue Spacetime: The paper establishes that sound waves in a backreaction-affected dynamical spacetime obey the equations of motion for a massive scalar field.
Structure of Corrections: The authors show that the analogue metric retains its standard functional form (dependent on sound velocity, fluid density, and flow velocity), but these parameters incorporate small $O(1/N)$ corrections. Crucially, the backreaction introduces a spacetime-dependent mass term ( $m^2$ ) as an additional correction, rather than merely modifying the metric coefficients. This mass term arises from the same sources that modify the continuity and Euler equations.
Linearized BdG Equation: A linear form of the modified BdG equation with number conservation is presented, enabling numerical solutions for the coupled dynamics of the condensate and fluctuations.

Results
Using a 1D Bose gas model with parameters corresponding to a box of 46 $\mu$ m and $N \approx 6000$ atoms:

Correlation Enhancement: At $t=5$ $t = 5$ (where depletion remains below 2%), backreaction enhances the canonically normal-ordered equal-point phase correlation function.
- In the bulk of the condensate, the correlation increases by approximately 10%.
- Near the edges (within a few healing lengths of the walls), the enhancement reaches up to 20%.
Spatial Dependence: The difference between the backreaction-included and linear-theory correlation functions is negligible for distant points, where the correlation itself is small.
Imaging Resolution: The bulk enhancement persists even when accounting for finite imaging resolution (modeled as Gaussian convolution and pixel averaging), remaining around 10% for widths up to 9 $\mu$ m. The edge enhancement is also visible in the outermost pixels.
Validity: The expansion remains valid (depletion $< 2\%$ ) for the chosen parameters up to $t=5$ . The authors note that for larger times or smaller particle numbers, the depletion grows, limiting the validity of the approximation.

Significance and Claims
The paper claims to provide a rigorous framework for studying quantum fields on backreaction-affected analogue spacetimes. By demonstrating that the field propagates as a massive scalar field with a spacetime-dependent mass, the authors show that the analogue-spacetime framework remains applicable even when backreaction is included.

The work suggests a potential route to probing feedback mechanisms through quantum correlations in cold-atom experiments. The authors note that observing this effect experimentally would require:

Control of thermal fluctuations and contact contributions.
Tunable interaction strengths (e.g., via Feshbach resonances) to achieve the specific density and scattering length regimes required for the $O(1/N)$ effects to be measurable while keeping depletion low.
Appropriate particle numbers and imaging resolutions.

The authors remain modest regarding immediate experimental realization, noting that while the required parameters (box size, density) are within reach of current technology, the specific combination of low scattering length and high density required for the calculated regime has not yet been demonstrated. The study serves as a theoretical proof-of-concept that backreaction introduces a measurable mass term and modifies correlations in a predictable, calculable manner.

Technical Summary: Sound Wave in the Backreaction-Affected Spacetime in Analogue Gravity Based on Number-Conserving Approach

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