Researchers at Seoul National University and the University of Texas at El Paso have developed a new method for analyzing quantum phase transitions in one-dimensional Rydberg atom arrays. The work centers on characterizing Z_n-ordered phases by examining the structure of pairwise entanglement across the atomic chain, measured using two-qubit concurrence. This approach defines a through Fourier analysis, offering a potentially powerful tool for understanding symmetry-breaking transitions alongside conventional magnetization-based methods. The team also details how this concurrence could be measured using site-selective erasure and precisely tuned laser pulses within analog Rydberg atom arrays.

Entanglement-Structure Factor Reveals ℤn Phase Transitions

An analysis of quantum entanglement, using a method called an entanglement-structure factor, directly correlates with the established order parameters of $Z_{n}$ phases in one-dimensional Rydberg atom chains. This new approach offers a distinct way to identify translational symmetry breaking, moving beyond reliance on traditional magnetization measurements. Researchers detail how this factor, derived through Fourier analysis of two-qubit concurrence across the chain, reveals characteristics of these phase transitions.

As the range of Rydberg blockade, the interaction preventing nearby atoms from simultaneously being excited, increases, so too does the period of this density wave. A pronounced peak in $F_{M} (k)$ at k = 2π/n signifies a $Z_{n}$ ordered phase; for example, a peak at $F_{M} (π)$ indicates an antiferromagnetic structure. Finite-size scaling analysis of $Z_{3}$ and $Z_{4}$ ordered phases demonstrates the effectiveness of this entanglement-based approach.

The team also proposes a method for experimentally measuring the required two-qubit concurrence in analog Rydberg atom arrays. The analysis separates the total concurrence scale from its Fourier amplitude, providing a clearer signal for identifying phase transitions.

The investigation suggests that this entanglement-structure-based approach can provide a powerful tool in analyzing symmetry-breaking in quantum phase transitions, offering a complementary perspective to conventional methods. By directly linking entanglement structure to the order parameter, the research provides a new lens through which to understand the emergence of quantum phases in these systems.

Two-Qubit Concurrence Measures Rydberg Atom Entanglement

The distribution of two-qubit concurrence across a one-dimensional Rydberg atom lattice provides a new method for identifying quantum phases and their transitions, offering a complementary approach to traditional order parameter analysis. Researchers demonstrated that examining the total concurrence at each site reveals information about the system’s quantum state, specifically in relation to $Z_{n}$ ordered phases, which emerge from interactions between Rydberg atoms. This analysis moves beyond simply detecting entanglement to mapping its structure across the atomic chain, potentially unlocking deeper insights into complex quantum systems.

The study introduces an entanglement-structure factor, $F_{C} (k)$ , derived from the Fourier analysis of total concurrence, and directly compares it to the density order parameter, $F_{M} (k)$ . This comparison allows for investigation of $Z_{n}$ phase transitions in the one-dimensional Rydberg chain, revealing how entanglement patterns correlate with the emergence of specific quantum orders.

The team’s work highlights the importance of understanding how entanglement is distributed, not just its presence or absence, in characterizing these transitions. In ideal $Z_{n}$ ordered phases, quantum states repeat after n atoms, a characteristic reflected in the observed concurrence profiles. Visualizations of magnetization alongside concurrence profiles reveal a direct link between the occupation of Rydberg atoms and the strength of entanglement between neighboring atoms.

The researchers evaluated two-qubit concurrence with respect to different reference atoms within each phase, using color-coding to represent entanglement strength. Arrows indicate the reference atom against which entanglement is evaluated, demonstrating how the entanglement structure mirrors the underlying order. This supports verifying the theoretical predictions and further exploring the relationship between entanglement structure and quantum phase transitions. The researchers emphasize the need to develop frameworks identifying the role and characteristics of quantum entanglement in these transitions, and suggest their entanglement structure approach serves as both a theoretical and experimental tool.

Rydberg Blockade Drives Symmetry Breaking in 1D Arrays

The emergence of ordered phases in one-dimensional Rydberg atom arrays is intimately linked to the distribution of quantum entanglement across the chain, according to new analysis. Researchers detail how the structure of pairwise entanglement, quantified by two-qubit concurrence, reveals characteristics of $Z_{n}$ phases, density wave states where quantum properties repeat after n atoms, and provides a complementary approach to identifying phase transitions compared to conventional methods based on local density.

This work focuses on characterizing these phases not just by what order exists, but how entanglement manifests as that order. The analysis reveals that next-nearest-neighbor concurrence, a measure of entanglement between atoms separated by one intervening atom, exists only between Rydberg atoms with higher densities, directly reflecting the broken $Z_{2}$ symmetry.

This concurrence profile, evaluated at a specific energy scale, demonstrates a clear signature of this symmetry breaking, becoming smaller near the transition point estimated from the magnetization structure. The team’s investigation extends to Z_4 transitions, examining two distinct transition lines, the Ashkin-Teller and incommensurate-commensurate transitions, to further refine the understanding of how entanglement patterns evolve.

The ability to experimentally measure this two-qubit concurrence is also addressed, with a proposed method using parametrized laser pulses in Rydberg atom arrays. “In order to identify period for each translational symmetry breaking, we can introduce a structure factor $F_{M} (k)$ based on the Fourier transform of Rydberg densities,” the paper states.

The work highlights that Rydberg Hamiltonians can lead to exotic phenomena such as chiral transitions, floating phases and glassy or topological behavior, and that translational symmetry breaking may occur within the system. The analysis of these systems is driven by the Rydberg blockade, which prevents two neighboring atoms from simultaneously occupying the Rydberg state, providing a mechanism for rich phase diagrams.

Density Matrix Renormalization Group Analyzes Entanglement Distribution

This work demonstrates how the distribution of entanglement, quantified by concurrence, can provide useful insights into the system’s quantum state, particularly in identifying transitions between different ordered phases. The research team used the density matrix renormalization group to map entanglement structure and compare it with conventional methods based on local density. The study uses concurrence, a measure of entanglement between two qubits, to assess how entanglement is distributed throughout the Rydberg array.

To calculate concurrence, the researchers first obtain a spin-flipped state from the reduced density matrix of two qubits, then determine the eigenvalues of a Hermitian matrix derived from that state. This approach allows for a detailed examination of pairwise entanglement, an essential element in understanding the behavior of these quantum systems.

While concurrence effectively measures entanglement between two qubits, the team also explored negativity as a means to quantify entanglement involving more than two qubits, using the partial transpose criterion. The analysis reveals nuances in the transition between $Z_{3}$ and $Z_{4}$ ordered phases, with the team noting an intermediate profile between these states. Previous work indicated that the transition to the period-4 phase can exhibit Ashkin-Teller-like behavior, and the density structure factor may deviate from the expected period-4 wave vector.

The researchers observed similar features, and also identified additional bright structures between the $Z_{3}$ and $Z_{4}$ phases, potentially indicative of floating phases, states located between ordered phases where the density structure factor evolves continuously. The paper states, referencing features observed in their data. The work highlights the potential of this approach to deepen understanding of quantum many-body physics and to identify subtle differences in phase transitions that might be missed by traditional methods.

Z_n Ordered Phases in Varied Rydberg Interactions

Researchers numerically calculated concurrence using density matrix renormalization group (DMRG) and directly compared the resulting spatial structure with Rydberg density profiles, finding that the concurrence profile mirrored the local period and characteristics of magnetization density in both $Z_{3}$ and $Z_{4}$ ordered phases. This suggests that entanglement patterns can be an alternative order parameter to traditional magnetization measurements.

The team introduced a Fourier analysis of the local entanglement sum, denoted as $F_{C} (k)$ , to characterize quantum phase transitions within the 1D Rydberg array, demonstrating its ability to accurately identify these transitions. This analysis provides a distinct visualization of how entanglement organizes itself within these different phases, complementing the conventional understanding based on Rydberg density.

Rydberg Hamiltonian Defines Atomic Interactions and Blockade Radius

The Rydberg interaction, a force dependent on the distance between atoms, fundamentally limits simultaneous excitation, a phenomenon quantified by the Rydberg blockade radius, R_b/a, where ‘a’ represents the lattice constant. This radius dictates the extent to which adjacent atoms prevent each other from entering the Rydberg state, an essential factor in the emergence of ordered phases within one-dimensional atomic arrays. The system’s Hamiltonian, as described in the work, reveals a landscape of quantum phases arising from this interplay of interaction and blockade.

Varying the strength of this Rydberg interaction leads to distinct phases, transitioning from disordered arrangements to crystalline ordered states as the interaction approaches zero. These ordered phases are categorized by a periodicity, denoted as $Z_{n}$ , which defines the repeating pattern of Rydberg excitation across the chain.

A $Z_{2}$ ordered phase, for example, exhibits a density wave with a period of two, meaning every other atom is excited, while higher-order $Z_{n}$ phases demonstrate correspondingly longer repeating patterns. The classification of these phase transitions has been a substantial focus of study alongside their experimental realization. This offers a means to analyze symmetry-breaking at phase transitions, a critical aspect of understanding the system’s behavior. The researchers state that an entanglement-structure-based approach can provide a powerful tool in analyzing symmetry-breaking in quantum phase transitions.

Entanglement Analysis Complements Magnetization Order Parameter

This alignment suggests that analyzing the Fourier transform of local entanglement-sum, denoted as $F_{C} (k)$ , effectively characterizes these quantum transitions. Finite-size scaling of $F_{C} (k)$ near the $Z_{3}$ and $Z_{4}$ transitions produces results similar to those obtained using the conventional order parameter, the density order parameter and its associated structure factor, $F_{M} (k)$ and the correlation length, ξ/L.

They used the form ξ/L = f_ξ(L^(1/ν)[(Δ/Ω)-(Δ/Ω)_c]) to find the correlation-length exponent ν, with (Δ/Ω)_c representing the detuning at the transition point. This highlights the importance of examining entanglement directly, as it can reveal subtle features missed by conventional order parameters.

👉 More information
🗞 Entanglement structure across Zₙ phase transitions in one-dimensional Rydberg atom arrays
✍️ Hyeonjun Yeo, Kabgyun Jeong and Hyunchul Nha
🧠 DOI: http://link.aps.org/doi/10.1103/8zrf-chts

More like this