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Work and heat exchanged during sudden quenches of strongly coupled quantum systems

This paper investigates three definitions of internal energy for strongly coupled quantum systems undergoing sudden quenches, demonstrating that while all satisfy the first law of thermodynamics, only two comply with the second law.

Original authors: Zohreh Davoudi, Christopher Jarzynski, Niklas Mueller, Greeshma Oruganti, Connor Powers, Nicole Yunger Halpern

Published 2026-07-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zohreh Davoudi, Christopher Jarzynski, Niklas Mueller, Greeshma Oruganti, Connor Powers, Nicole Yunger Halpern

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

Imagine you are trying to keep score in a game of hot potato, but the potato is so hot it melts the hands of the players holding it. In the world of everyday physics, we are used to a "weak coupling" game: a system (like a cup of coffee) sits next to a reservoir (the room air). They touch, they exchange heat, but they remain distinct. The coffee has its own energy, the air has its own energy, and the tiny bit of energy where they touch is so small we can ignore it. This makes calculating the rules of the game—thermodynamics—pretty straightforward.

But what happens when the players are tiny, like atoms, and the "hot potato" is a quantum particle that is so deeply entangled with its surroundings that you can't tell where the particle ends and the room begins? This is the realm of "strong coupling." Here, the energy of the interaction is huge, and the old rulebook breaks down. Scientists have been arguing about how to define the most basic scorecard items: What is the system's internal energy? How much work did we do on it? How much heat did it absorb? If you get these definitions wrong, the fundamental laws of physics—specifically the rule that says disorder (entropy) must always increase—might seem to vanish. This paper dives into that messy, melting-hot corner of quantum science to see if we can rewrite the rules without breaking the game.

The authors of this paper, a team of physicists from the University of Maryland and the University of Washington, set out to solve a specific puzzle: When a quantum system is strongly linked to its environment, which definition of "energy" actually plays fair with the laws of thermodynamics? They focused on a dramatic event called a "quench." Imagine you have a quantum system in a calm, balanced state, and then—snap!—you suddenly change the rules of the game (the Hamiltonian) in an instant. This sudden jolt forces the system to react, and in that reaction, energy is shuffled around.

To figure out the score, the team tested three different ways to define the system's internal energy. Think of these three definitions as three different ways to split a bill at a restaurant where the waiter (the interaction) is also eating a slice of the pizza.

The "Difference" Method: You take the total bill for the whole table (system + environment) and subtract the cost of the environment's meal if the system hadn't been there. The rest is the system's bill.
The "Mean Force" Method: You use a special, modified menu (called the Hamiltonian of mean force) that already accounts for the fact that the system and environment are glued together. You calculate the bill based on this new, adjusted menu.
The "Energy Derivative" Method: You use a third, slightly different mathematical recipe to calculate the energy based on how the modified menu changes with temperature.

In the old days of classical physics, all three of these methods gave the exact same answer. They were equivalent. But in the quantum world, the authors found that this equivalence shatters. They proved mathematically and confirmed with a simple computer simulation of two interacting spins (tiny magnets) that only two of these methods obey the Second Law of Thermodynamics.

The Second Law is the universe's way of saying that you can't win, you can't break even, and you can't get out of the game; it basically means that in any real process, some energy is always lost to waste (dissipated work), and the total disorder must go up. The authors showed that if you use the "Difference" method or the "Mean Force" method, the math holds up: the dissipated work is always positive, and the Second Law is safe. However, if you use the third method (the "Energy Derivative" one), the math breaks. In their simulations, this third definition sometimes predicted that the system could create order out of chaos for free, effectively violating the Second Law.

The team illustrated this with a model of two spins (like tiny compass needles) that were either shaken by changing their own magnetic field (a "system quench") or by suddenly turning on their connection to each other (an "interaction quench"). In every scenario where the connection was strong, the third definition failed the test, while the other two passed.

This isn't just a theoretical squabble. The paper suggests that for future experiments in quantum computing, nuclear physics, and materials science—where particles are often deeply entangled with their surroundings—scientists must be careful about which "energy" they measure. If they use the wrong definition, they might think they've discovered a miracle machine that violates the laws of physics, when in reality, they just used the wrong calculator. The authors conclude that while the "Mean Force" and "Difference" definitions are the safe bets for now, the quantum world of strong coupling still holds many surprises, and figuring out how to measure work and heat in these tangled systems is a crucial step toward building the quantum technologies of tomorrow.

Technical Summary: Work and Heat Exchanged During Sudden Quenches of Strongly Coupled Quantum Systems

Problem Statement
The paper addresses a fundamental open question in quantum thermodynamics: how to consistently define thermodynamic quantities—specifically internal energy, work, and heat—for quantum systems strongly coupled to their environments. In the standard weak-coupling regime, the interaction energy is negligible, allowing the total internal energy to be decomposed into separate system and reservoir contributions. However, in the strong-coupling regime (relevant for microscopic systems or those with long-range interactions), the interaction energy is significant and cannot be neatly partitioned. Consequently, it is unclear how to attribute energy to the system versus the reservoir, leading to ambiguity in defining the first and second laws of thermodynamics for such systems. While classical strong-coupling thermodynamics offers consistent definitions, their quantum extensions remain debated.

Methodology
The authors investigate three candidate definitions for the internal energy ( $U_S$ ) of a strongly coupled quantum system $S$ interacting with a reservoir $R$ . The total Hamiltonian is $\hat{H}_{S\cup R} = \hat{H}_S + \hat{H}_R + \hat{V}_{S\cup R}$ . The three definitions are:

$U_{\text{diff}}$ (Difference): The total internal energy minus the equilibrium internal energy of an isolated reservoir ( $U_{S\cup R} - U_R^0$ ).
$U_{H^*}$ (Mean Force Hamiltonian): The expectation value of the Hamiltonian of mean force, $\hat{H}_S^* = -\beta^{-1} \ln(\text{Tr}_R[e^{-\beta \hat{H}_{S\cup R}}] / \text{Tr}_R[e^{-\beta \hat{H}_R}])$ .
$U_{E^*}$ (Energy Operator): The expectation value of the operator $\hat{E}_S^* = \partial_\beta (\beta \hat{H}_S^*)$ .

The study focuses on quench processes, where the Hamiltonian changes abruptly at $t=0$ . This setup allows for a natural partitioning of the internal energy change ( $\Delta U_S$ ) into work ( $W$ , performed during the instantaneous change) and heat ( $Q$ , absorbed during subsequent evolution). The authors analyze two scenarios:

System Quench: The system Hamiltonian $\hat{H}_S$ changes abruptly.
Interaction Quench: The interaction term $\hat{V}_{S\cup R}$ is turned on abruptly.

For each internal energy definition, the authors derive corresponding definitions for work and heat. They then test these definitions against the Second Law of Thermodynamics, specifically the inequality $W \geq \Delta F_S$ (where $\Delta F_S$ is the change in free energy), which must hold for processes starting from an equilibrium state.

Key Contributions and Results

First Law Consistency: By construction, all three definitions of internal energy satisfy the First Law of Thermodynamics ( $\Delta U_S = W + Q$ ) for the quench processes considered.
Second Law Violation: The authors prove that only two of the three definitions satisfy the Second Law ( $W \geq \Delta F_S$ $W \geq Δ F_{S}$ ) for general quench processes:
- The definitions based on $U_{\text{diff}}$ and $U_{H^*}$ satisfy the inequality. The authors demonstrate this by showing that the dissipated work ( $W - \Delta F_S$ ) corresponds to the non-negative quantum relative entropy between the initial and final global (or reduced) states.
- The definition based on $U_{E^*}$ (and its associated work $W_{E^*}$ ) fails to satisfy the Second Law. The authors provide a numerical counterexample using a two-spin model where $W_{E^*} < \Delta F_S$ for certain parameter values, indicating a violation of the second law.
Equivalence Conditions: The paper identifies specific conditions under which the three definitions coincide. For system quenches where the initial and final system Hamiltonians commute with the interaction term ( $[\hat{H}_S, \hat{V}_{S\cup R}] = 0$ ), all three work definitions ( $W_{\text{diff}}$ , $W_{H^*}$ , and $W_{E^*}$ ) are identical and satisfy the Second Law.
Heat and Entropy: When the system equilibrates after the quench, the Second Law can be expressed in terms of heat and entropy ( $Q \leq \beta^{-1} \Delta S$ ). The results mirror the work analysis: definitions derived from $U_{\text{diff}}$ and $U_{H^*}$ satisfy this bound, while the $U_{E^*}$ -based definition can violate it.

Significance
The paper claims to advance the field of quantum thermodynamics by rigorously distinguishing which definitions of thermodynamic quantities are consistent with the Second Law in the strong-coupling regime. The findings highlight a divergence between classical and quantum thermodynamics: while the three definitions are classically equivalent and all satisfy the Second Law, in the quantum regime, only the definitions based on the total energy difference ( $U_{\text{diff}}$ ) and the Hamiltonian of mean force ( $U_{H^*}$ ) are valid.

The authors suggest these results are significant for studying strongly coupled systems in condensed matter, high-energy and nuclear physics (specifically lattice gauge theories), quantum chemistry, and quantum error correction. The work provides a guide for selecting appropriate thermodynamic definitions in these contexts, particularly noting that the Hamiltonian of mean force ( $\hat{H}_S^*$ ) is physically motivated by its relation to the system's entanglement Hamiltonian in equilibrium, potentially linking thermodynamic measurements to entanglement tomography techniques. The study is modest in scope, focusing on quench processes and average quantities, and explicitly notes that extending these findings to fluctuating quantities or non-quench processes (like adiabatic processes) remains an open direction for future research.

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