Unveiling emergent internal time from entropy exchange in a cold-atom system

This paper experimentally demonstrates that a well-isolated cold-atom system can realize relational time by constructing an entropic clock from internal degrees of freedom, which successfully orders dynamical events and reproduces the system's evolution via an effective Schrödinger equation.

The Gist

Imagine you are locked inside a room with no windows, no clocks, and no way to tell time. You can't see the sun rise or set. The only thing you have is a large, bouncing ball in the center of the room.

In the world of physics, specifically in theories trying to combine gravity and quantum mechanics (like the Wheeler-DeWitt equation), the entire universe is like that room. The math says the universe is a "frozen" block where nothing truly changes because there is no "clock" ticking outside of it. This creates a huge puzzle: If there is no external clock, how do we experience time flowing? How do we know what happened before what?

This paper, by Giovanni Barontini, is like a magic trick performed in a laboratory to solve this puzzle. The team built a tiny, controlled "mini-universe" using cold atoms to test if time can be created from the inside out, using nothing but the system's own internal chaos and order.

Here is the story of how they did it, explained simply:

1. The Setup: A Room Divided in Two

The scientists took a cloud of super-cold atoms (a Bose-Einstein Condensate) and trapped it in a bowl-shaped laser field. Then, they built a very thin, invisible "wall" right in the middle of the bowl using light.

This split their mini-universe into two rooms:

• The Dark Room (Unobserved): One side of the wall where no one is looking.
• The Bright Room (Observed): The other side, which they can watch and measure.

The atoms are like a bouncing ball that can jump over the wall. Sometimes they are in the Dark Room, sometimes they jump into the Bright Room, and sometimes they jump back.

2. The Problem: No Clocks Allowed

In their experiment, they wanted to describe the movement of the atoms in the Bright Room without using the "lab clock" (the time kept by the scientists outside). They wanted to know: Can the atoms tell time using only their own movements?

Usually, if you watch a ball bounce up and down, you can say "it went up, then it came down." But if the ball bounces back and forth forever, how do you know if it's the first time it bounced or the hundredth time? You need a way to distinguish the moments.

3. The Solution: Time as a Measure of "Messiness" (Entropy)

The scientists realized that while the total amount of "stuff" in the system stays the same, the arrangement of the atoms changes. They used a concept called Entropy, which is basically a measure of "messiness" or how much information is exchanged between the two rooms.

Think of it like this:

• Imagine the Dark Room is a messy closet, and the Bright Room is a clean desk.
• When atoms jump from the closet to the desk, they bring some "mess" with them.
• When they jump back, they take some mess with them.

The scientists defined a new kind of time, which they called "Entropic Time."

• The Rule: Time only "ticks" when atoms jump across the wall and swap messiness (entropy) between the rooms.
• The Result: If the atoms are just sitting still in the Bright Room and not jumping, time stops. If they are frantically jumping back and forth, time flies.

4. The Big Bang and Big Crunch

In their experiment, the atoms would suddenly flood into the Bright Room (like a Big Bang), spread out, and then get pulled back together into the Dark Room (like a Big Crunch).

In normal time, this cycle happens over and over. But when they looked at their data using Entropic Time, something amazing happened:

• The "Big Bang" to "Big Crunch" cycle happened smoothly.
• But the moment the atoms stopped jumping and the entropy exchange stopped, time froze.
• This created a one-way arrow. Even though the atoms were bouncing back and forth, the "Entropic Time" kept moving forward because it was counting the exchange of messiness, not just the position of the atoms.

5. The Magic Equation

The team didn't just watch this; they wrote a new version of the famous Schrödinger equation (the rulebook for how quantum things move). But instead of using "seconds" as the time variable, they used their new Entropic Time.

When they ran their computer simulations with this new equation, it perfectly matched what they saw in the lab. This proved that you don't need an external clock to describe how a system evolves; you can build time from the flow of entropy inside the system itself.

Why This Matters

This is a big deal for physics because:

1. It solves a philosophical puzzle: It shows that "time" might not be a fundamental thing that exists everywhere. Instead, it might be an emergent property—something that appears only when parts of a system interact and exchange information.
2. It's a testbed: They created a "playground" where we can test ideas about the beginning of the universe (the Big Bang) and black holes in a safe, controlled lab setting.
3. It's practical: It suggests that in the future, we might be able to design quantum computers or sensors that operate based on these internal "clocks" rather than relying on external signals.

In a nutshell: The paper shows that if you have a closed system (like our universe), you don't need a grandfather clock to tell time. You can just watch how the system gets messier or cleaner as its parts interact. That flow of change is time.

Technical Summary

1. Problem Statement

The paper addresses two fundamental challenges in theoretical physics:

• The Problem of Time in Quantum Gravity: In the Wheeler–DeWitt (WDW) framework, the equation $\hat{H}\Psi = 0$ implies a static universe with no external time parameter to sequence physical changes. This contrasts with the observed flow of time.
• The Arrow of Time: Fundamental laws (Newtonian, Quantum, Relativistic) are time-symmetric. The only robust asymmetry is the Second Law of Thermodynamics (entropy increase). However, in a closed quantum system described by WDW, the global state is pure, and fine-grained entropy is conserved, seemingly contradicting the observed growth of coarse-grained entropy.

The authors aim to experimentally test relational time constructions, specifically whether an "internal time" can emerge from the entropy exchange between subsystems within a closed, time-independent quantum system, thereby providing a quantitative benchmark for WDW-inspired models.

2. Methodology

The researchers realized a controlled cold-atom analogue system using a Bose–Einstein Condensate (BEC) of approximately $2.4 \times 10^4$ $^{87}\text{Rb}$ atoms.

• System Setup:

  • The BEC oscillates in a conservative, radially symmetric optical dipole trap.
  • A thin optical barrier (generated by a Digital Micromirror Device at 675 nm) partitions the trap into two sectors:
    • Dark Sector: Unobserved.
    • Bright Sector: Observed.
  • The system is effectively closed with a time-independent Hamiltonian on the experimental timescale (~100 ms), with negligible dissipation.

• Theoretical Mapping (Mini-superspace Analogue):

  • The system is modeled as a "mini-universe." The bright sector's effective Hamiltonian is structurally analogous to a WDW minisuperspace model.
  • Variables:
    • $X$ (Center of Mass position of the bright sector) $\rightarrow$ Analogous to a uniform massive scalar field $\phi$ (the "clock").
    • $R$ (Radial width of the condensate) $\rightarrow$ Analogous to the scale factor $a$ (the "universe size").
  • The coupling between sectors allows for atom exchange, modeled as a non-Hermitian term ($\hat{H}_{eff} = \hat{H}_{bright} - i\hbar\Gamma/2$), representing gain/loss.

• Defining Entropic Time:

  • Since the scalar field $\phi$ (position) is non-monotonic (it oscillates back and forth), it cannot serve as a global time coordinate.
  • The authors define an entropic time $\tau$ based on the flow of coarse-grained entropy $S$:
    $$\tau(\lambda) = \frac{\sigma}{k_B} \int_{\lambda} dS \frac{|d\phi|}{d\phi}$$
    where $\sigma$ is an arbitrary unit, $k_B$ is Boltzmann's constant, and the integral ensures $\tau$ increases monotonically as long as entropy and the clock variable change in the same direction.

• Experimental Procedure:

  • The system undergoes cycles of expansion ("Big Bang") and recollapse ("Big Crunch").
  • Absorption imaging is performed every 2 ms over 120 ms.
  • For each image, the authors calculate the number of atoms $N$, the center of mass $\phi$, the width $\Sigma$, and the entropy per atom $s$ (using a coarse-grained Shannon entropy method on the density profile).
  • The barrier height is varied to control the rate of entropy exchange between sectors.

3. Key Contributions

1. Experimental Realization of Relational Time: The paper provides the first quantitative experimental test of constructing time solely from internal degrees of freedom (entropy exchange) in a closed quantum system.
2. Operational Definition of Entropic Time: It introduces a robust, monotonic internal time variable $\tau$ derived from thermodynamic entropy flow, resolving the non-monotonicity issue of standard clock variables in recollapsing models.
3. Derivation of an Entropic Schrödinger Equation: The authors derive a time-dependent Schrödinger equation parameterized by $\tau$:
   $$i\hbar\partial_\tau\psi(\tau, a) = \Phi(\tau)\psi(\tau, a) + \Lambda(\tau)\hat{H}_{geom}\psi(\tau, a)$$
   Here, $\Lambda(\tau)$ acts as an entropy-dependent energy pump. When entropy flow stops, time stops ($\partial_\tau \Lambda \approx 0$), recovering standard dynamics only when entropy is exchanged.
4. Validation of the Arrow of Time: The experiment demonstrates that the "arrow of time" aligns with the direction of entropy flow between subsystems, even when the global state remains pure.

4. Results

• Monotonicity of Entropic Time: The calculated entropic time $\tau$ grows monotonically with respect to external lab time, except for minor fluctuations where the sampling of the clock field $\phi$ is coarse.
• Entropy Flow Dynamics:
  • Low Barrier ($V \approx 0$): High entropy exchange occurs. The system cycles from Big Bang to Big Crunch. Entropic time flows rapidly during expansion/contraction and effectively "stops" at the turnaround points where entropy exchange is momentarily zero.
  • High Barrier ($V \approx 1$): Entropy exchange is suppressed. The system evolves toward a "heat death" (stationary state). In this regime, entropic time ceases to flow, while lab time continues.
• Quantitative Reproduction: Numerical simulations solving the derived entropic Schrödinger equation (Eq. 6) using experimental parameters successfully reproduced the measured evolution of the condensate width ($\Sigma$) as a function of entropic time $\tau$. The agreement between theory and experiment was excellent.
• Global Entropy Conservation: The total entropy of the combined dark and bright sectors remained constant within error bars, confirming the system is closed, while the reduced entropy of the bright sector fluctuated, driving the internal time.

5. Significance

• Benchmarks Quantum Gravity Models: This work establishes a controlled laboratory setting to test relational time constructions and the emergence of time from quantum correlations, moving these concepts from theoretical speculation to experimental verification.
• Clarifies the Arrow of Time: It demonstrates that an arrow of time can emerge in a closed quantum system through the partitioning of the system and the tracing over of unobserved degrees of freedom (decoherence/entanglement), linking the Second Law directly to the definition of time.
• Platform for Future Research: The cold-atom platform offers a versatile tool for investigating:
  • The role of singularities (Big Bang/Big Crunch) and quantum bounces.
  • The reversibility of time (Loschmidt echoes).
  • Analogue black holes and Vilenkin-type tunneling scenarios.
  • Generalization to midisuperspace models with more degrees of freedom.

In conclusion, the paper successfully demonstrates that time can be an emergent, relational property defined by the flow of entropy between subsystems, providing a concrete, experimentally verified framework for understanding time in the absence of an external clock.