⚛️ quantum physics

Dicke States for Accelerated Two Two-Level Atoms

This paper investigates the formation of Dicke states for accelerated two-level atoms in the Rindler wedge, deriving analytical expressions for joint excitation probabilities that reveal interference effects and clarify the relationship between single-atom and collective excitation dynamics in non-inertial frames.

Original authors: Muzzamal I. Shaukat, Charles A. Wallace, Anatoly A. Svidzinsky, Marlan O. Scully

Published 2026-02-02

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CC BY 4.0

Original authors: Muzzamal I. Shaukat, Charles A. Wallace, Anatoly A. Svidzinsky, Marlan O. Scully

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 floating in deep space, perfectly still. To you, the universe is empty and cold—a true vacuum. Now, imagine you strap yourself into a rocket and blast off, accelerating at a constant, high speed. According to the laws of physics described in this paper, your experience of the universe changes dramatically. Even though you are still in a "vacuum," your rapid acceleration makes the empty space feel like a warm, bubbling bath of particles. This is known as the Unruh effect.

This paper explores what happens when you put two tiny, simple quantum "switches" (called two-level atoms) into this accelerating rocket and see how they interact with that warm, particle-filled space.

Here is a breakdown of their findings using everyday analogies:

The Setup: Two Atoms in a Rocket

The researchers imagined two identical atoms traveling side-by-side in a rocket ship, accelerating constantly. They are interacting with a "massless scalar field," which you can think of as an invisible ocean of waves filling the universe.

Because the atoms are accelerating, the "empty" ocean looks like a stormy sea full of thermal waves to them. The paper asks: If these two atoms start in their lowest energy state (the "off" switch), can they spontaneously get excited (flip to the "on" switch) just by riding this storm?

The Dance of the Atoms: Symmetry vs. Anti-Symmetry

When the two atoms get excited, they don't just act independently; they act as a team. The paper focuses on two specific ways they can team up, known as Dicke states:

The Symmetric State (The "High-Five"): Imagine the two atoms are dancers. In this state, they move in perfect unison. If one jumps, the other jumps at the exact same time and in the exact same way. They are in sync.
The Anti-Symmetric State (The "Mirror Image"): Here, the atoms move in opposition. If one jumps up, the other jumps down. They are perfectly out of sync, like a mirror image.

The Interference: Constructive vs. Destructive

The most interesting part of the paper is how the distance between the atoms changes the outcome. The authors found that the atoms interfere with each other like ripples in a pond.

Constructive Interference (The "Loud" State): If the atoms are spaced at a specific distance, their "ripples" line up perfectly. This makes it much more likely for the atoms to get excited together in the Symmetric State. It's like two people clapping in rhythm; the sound gets louder.
Destructive Interference (The "Silent" State): If the atoms are spaced at a different distance, their ripples cancel each other out. This suppresses the Anti-Symmetric State, making it very hard for them to get excited in that specific way. It's like two people clapping out of rhythm; the sound disappears.

The paper provides a mathematical formula showing that the chance of the atoms getting excited depends on this distance and the "temperature" of the vacuum created by their acceleration.

Scaling Up: From Two to Many

The researchers didn't stop at two atoms. They asked, "What if we have a whole crowd of $N$ atoms?"

They discovered a simple rule: If you have a crowd of $N$ atoms all accelerating together, the probability of any one of them getting excited is exactly $N$ times the probability of a single atom getting excited on its own. It's as if the crowd amplifies the effect, making it easier for the group to react to the "warm vacuum" than for a lone atom to do so.

The Double Excitation

Finally, the paper looks at the rare event where both atoms get excited at the same time, emitting two particles. They found that this process is also influenced by the distance between the atoms. The math shows a complex interference pattern here too, where the "warmth" of the vacuum (the Unruh effect) and the spacing of the atoms combine to determine how likely this double-excitation is.

The Bottom Line

In simple terms, this paper shows that acceleration changes the rules of the game for quantum particles. By accelerating, atoms can "feel" the vacuum as a thermal bath. Depending on how far apart they are, they can either work together to get excited (symmetric state) or cancel each other out (anti-symmetric state). The study confirms that these collective behaviors, known as Dicke states, exist even in the strange, non-stationary world of accelerating frames, and the likelihood of these events is directly tied to the distance between the atoms and the intensity of their acceleration.

Technical Summary: Dicke States for Accelerated Two Two-Level Atoms

Problem Statement
The paper addresses the formation and properties of Dicke states—collective quantum states characterized by symmetric and antisymmetric superpositions of atomic excitations—within non-inertial frames. While Dicke states are well-understood in inertial settings regarding superradiant and subradiant behaviors, their dynamics in Rindler space (specifically for uniformly accelerated atoms) remain an open area of research. The authors investigate a system of two identical two-level atoms undergoing constant acceleration in the right Rindler wedge, interacting with a massless scalar field. The primary goal is to determine the conditions under which superradiant (symmetric) or subradiant (antisymmetric) states can be formed and to analyze the joint excitation probabilities influenced by the Unruh effect.

Methodology
The study employs a theoretical framework based on quantum field theory in curved spacetime (specifically Rindler coordinates) and perturbation theory.

System Setup: Two identical two-level atoms with ground states $|b_i\rangle$ and excited states $|a_i\rangle$ move with constant acceleration $a$ along the $z$ -axis. Their trajectories are defined by Rindler coordinates, with a constant coordinate separation $d$ .
Interaction Model: The atoms interact with a massless scalar field satisfying the Klein-Gordon equation. The interaction Hamiltonian is formulated in terms of the atoms' proper time $\tau$ .
Perturbative Analysis: The evolution of the system is analyzed using the Dyson series expansion of the evolution operator up to the second order ( $U_I$ $U_{I}$ and $U_{II}$ $U_{I I}$ ).
- First Order ( $U_I$ ): Calculates the probability amplitudes for the system to transition from the initial ground state ( $|b_1b_2\rangle$ ) and field vacuum to states where exactly one atom is excited and one photon is emitted. This involves evaluating integrals over proper time, resulting in expressions containing Gamma functions and phase factors dependent on acceleration and frequency.
- Second Order ( $U_{II}$ ): Analyzes the probability amplitude for the doubly excited state ( $|a_1a_2\rangle$ ) accompanied by the emission of two photons. This requires evaluating double integrals involving counter-rotating atomic operators and field creation operators.
Mathematical Tools: The authors utilize variable substitutions to solve the time integrals, expressing results in terms of the Gamma function $\Gamma(z)$ , the incomplete Gamma function, and Hypergeometric functions ( $_2F_1$ , $_1F_2$ ).

Key Contributions and Results

Analytical Expressions for Excitation Probabilities:
- The authors derived analytical expressions for the probability amplitudes of forming symmetric ( $|s\rangle$ ) and antisymmetric ( $|a\rangle$ ) Dicke states.
- The probability of exciting the symmetric state ( $P_s$ ) is proportional to $\cos^2(kd/2)$ , while the antisymmetric state ( $P_a$ ) is proportional to $\sin^2(kd/2)$ .
- Both probabilities depend on the Planck factor associated with the Unruh temperature ( $T_U = \hbar a / 2\pi k_B c$ ) and the atomic transition frequency $\omega$ .
- The results demonstrate that the excitation probability is a function of the inter-atomic separation $d$ , encoding constructive interference for the symmetric state and destructive interference for the antisymmetric state.
Scaling for $N$ Atoms:
- The model was extended to a collection of $N$ two-level atoms. The analysis shows that the probability to excite exactly one atom in a collection of $N$ atoms is $N$ times the probability of exciting a single atom executing the same trajectory ( $P_r(1 \text{ of } N) = N P_r(\text{lone atom})$ ).
Joint Excitation and Interference:
- The probability for the doubly excited state ( $P_{e_1e_2}$ $P_{e_{1} e_{2}}$ ) was derived. The expression contains two distinct terms:
  - A term representing the thermal nature of the vacuum due to the Unruh effect (dependent on acceleration).
  - A term containing a cosine function of the spatial separation $d$ and a phase difference, explicitly rendering the interference effect between the atoms.
Formation of Dicke Bases:
- The study establishes that the final state of the system forms a set of Dicke bases, comprising the ground state, the symmetric and antisymmetric single-excitation states, and the doubly excited state. The specific superposition (symmetric vs. antisymmetric) is determined by the phase relationship $kd$ (where $d$ is the separation and $k$ is the wave number).

Significance
The paper claims to provide new insights into the behavior of quantum systems in non-inertial frames. Specifically, it elucidates how acceleration influences the formation of symmetric states and joint excitation probabilities for $N$ two-level atoms. By deriving analytical expressions that link the Unruh effect (thermal bath perception) with collective quantum phenomena like superradiance and subradiance, the work contributes to the broader understanding of relativistic quantum information theory. The findings highlight the unique role of Rindler spacetime features, such as horizons and thermal vacuum properties, in shaping the dynamics of entangled atomic states, offering a theoretical foundation for understanding quantum entanglement and coherence in accelerated reference frames.