Spectroscopy of elementary excitations from quench dynamics in a dipolar XY Rydberg simulator
Abstract
We use a Rydberg quantum simulator to demonstrate a new form of spectroscopy, called quench spectroscopy, which probes the low-energy excitations of a many-body system. We illustrate the method on a two-dimensional simulation of the spin-1/2 dipolar XY model. Through micro- scopic measurements of the spatial spin correlation dynamics following a quench, we extract the dispersion relation of the elementary excitations for both ferro- and anti-ferromagnetic cou- plings. We observe qualitatively different behaviors between the two cases that result from the long-range nature of the interactions, and the frustration inherent in the antiferromagnet. In particular, the ferromagnet exhibits elementary excitations behaving as linear spin waves. In the anti-ferromagnet, spin waves appear to decay, suggesting the presence of strong nonlinear- ities. Our demonstration highlights the importance of power-law interactions on the excitation spectrum of a many-body system.
Dataset DOI: https://doi.org/10.5061/dryad.q83bk3jts
Description of the data and file structure
These files contain all the data for the figures in the article, including Figure 1.xlsx, Figure 2.xlsx, Figure 3.xlsx and Figure 4.xlsx.
Data Organization in Excel Files:
For clarity and consistency, data in Excel files (e.g., Figure 1.xlsx) are structured to align with figures in the paper. Each file contains multiple labeled sheets (e.g., panel C, panel D), with distinct naming conventions to differentiate data types:
- Experimental data: Sheets or variables include the label "data" (e.g., panel E_data_(nx,ny)=(1,0)). This denotes experimental results for panel E of Figure 1 under the condition (nx, ny) = (1, 0).
- Theoretical methods: Sheets or variables use labels like "tVMC" (time-dependent variational Monte Carlo) or "LSW OBC" (linear spin-wave theory with open boundary conditions).
Example: In the sheet panel E_data_(nx,ny)=(1,0), the Variables include:
- Time(hbar/J)_data: Evolution time for experimental data.
- Structure factor: Measured structure factors.
- lower_errors and upper_errors: Error bounds for plotting.
Consistency Across Files:
All Excel files follow this labeling logic: Sheet names map directly to figure panels (e.g., panel X corresponds to panel X in the paper). Variable names (e.g., Time, Structure factor, errors) correspond to axis or colorbar labels in the figures. Method-specific labels (e.g., tVMC, LSW OBC) distinguish theoretical predictions from experimental results. This structured approach ensures alignment between the data sheets and their graphical representations in the paper.
Files and variables
Figure 1.xlsx\panel C
This data sheet provides the data of space-time dynamics of the spin correlations following the quench into the coherent spin state (FM case, Figure 1C). Variable list:
- "data: d (sites)": distances between the sites of the experimental data in the upper left plot of Figure 1C; The data range spans from A2 to A51.
- "data: Corr_zz": zz correlations of the experimental data in the upper left plot of Figure 1C; The data range spans from cell C2 to AQ51.
- "Time (hbar/J)": Evolution time of the experimental data in the upper left plot of Figure 1C; The data range spans from C53 to AQ53.
- "tVMC: d (sites)": distances between the sites of the tVMC data in the lower left plot of Figure 1C; The data range spans from A60 to A109.
- "tVMC: Corr_zz": zz correlations of the tVMC data in the lower left plot of Figure 1C; The data range spans from cell C60 to BN109.
- "Time (hbar/J)": Evolution time of the tVMC data in the lower left plot of Figure 1C; The data range spans from C111 to BN111.
Figure 1.xlsx\panel D
This data sheet provides the data of space-time dynamics of the spin correlations following the quench into the staggered coherent spin state (AFM case, Figure 1D). Variable list:
- "data: d (sites)": distances between the sites of the experimental data in the upper plot of Figure 1D; The data range spans from A2 to A51.
- "data: Corr_zz": zz correlations of the experimental data in the upper plot of Figure 1D; The data range spans from cell C2 to AI51.
- "Time (hbar/J)": Evolution time of the experimental data in the upper plot of Figure 1D; The data range spans from C53 to AI53.
- "tVMC: d (sites)": distances between the sites for the tVMC data in the lower right plot of Figure 1D; The data range spans from A60 to A77.
- "tVMC: Corr_zz": zz correlations of the tVMC data in the lower right plot of Figure 1D; The data range spans from cell C60 to AQ77.
- "Time (hbar/J)": Evolution time of the tVMC data in the lower right of Figure 1D; The data range spans from C79 to AQ79.
Figure 1.xlsx\panel E_data_(nx,ny)=(1,0)
Figure 1.xlsx\panel E_data_(nx,ny)=(3,0)
Figure 1.xlsx\panel E_data_(nx,ny)=(3,3)
Figure 1.xlsx\panel E_data_(nx,ny)=(5,1)
Figure 1.xlsx\panel E_data_(nx,ny)=(5,5)
These sheets provide experimental data of structure factor dynamics in the ferromagnetic case with different (nx, ny). Variable list:
- "Time(hbar/J)_data": Evolution time of the experimental data in Figure 1E; The data range spans from A2 to A44.
- "Structure factor": Structure factor of the experimental data in Figure 1E; The data range spans from B2 to B44.
- "lower_errors": lower bound of the error bar for experimental structure factor in Figure 1E; The data range spans from C2 to C44.
- "upper_errors": upper bound of the error bar for experimental structure factor in Figure 1E; The data range spans from D2 to D44.
Figure 1.xlsx\panel E_LSW_OBC_(nx,ny)=(1,0)
Figure 1.xlsx\panel E_LSW_OBC_(nx,ny)=(3,0)
Figure 1.xlsx\panel E_LSW_OBC_(nx,ny)=(3,3)
Figure 1.xlsx\panel E_LSW_OBC_(nx,ny)=(5,1)
Figure 1.xlsx\panel E_LSW_OBC_(nx,ny)=(5,5)
These sheets provide the structure factor dynamics of linear spin-wave theory with open boundary conditions in the ferromagnetic case with different (nx, ny). Variable list:
- "Time(hbar/J)_LSW_OBC": Evolution time of the LSW OBC data in Figure 1E; The data range spans from A2 to A631.
- "Structure factor": Structure factor of the LSW OBC data in Figure 1E; The data range spans from B2 to B631.
Figure 1.xlsx\panel E_tVMC_(nx,ny)=(1,0)
Figure 1.xlsx\panel E_tVMC_(nx,ny)=(3,0)
Figure 1.xlsx\panel E_tVMC_(nx,ny)=(3,3)
Figure 1.xlsx\panel E_tVMC_(nx,ny)=(5,1)
Figure 1.xlsx\panel E_tVMC_(nx,ny)=(5,5)
These sheets provide the structure factor dynamics of time-dependent variational Monte Carlo in the ferromagnetic case with different (nx, ny). Variable list:
- "Time(hbar/J)_tVMC": Evolution time of the tVMC data in Figure 1E; The data range spans from A2 to A63.
- "Structure factor": Structure factor of the tVMC data in Figure 1E; The data range spans from B2 to B63.
Figure 1.xlsx\panel F_data_(nx,ny)=(1,0)
Figure 1.xlsx\panel F_data_(nx,ny)=(3,0)
Figure 1.xlsx\panel F_data_(nx,ny)=(3,3)
Figure 1.xlsx\panel F_data_(nx,ny)=(5,1)
Figure 1.xlsx\panel F_data_(nx,ny)=(5,5)
These sheets provide experimental data of structure factor dynamics in the antiferromagnetic case with different (nx, ny). Variable list:
- "Time(hbar/J)_data": Evolution time of the experimental data in Figure 1F; The data range spans from A2 to A52.
- "Structure factor": Structure factor of the experimental data in Figure 1F; The data range spans from B2 to B52.
- "lower_errors": lower bound of the error bar for experimental structure factor in Figure 1F; The data range spans from C2 to C52.
- "upper_errors": upper bound of the error bar for experimental structure factor in Figure 1F; The data range spans from D2 to D52.
Figure 1.xlsx\panel F_LSW_OBC_(nx,ny)=(1,0)
Figure 1.xlsx\panel F_LSW_OBC_(nx,ny)=(3,0)
Figure 1.xlsx\panel F_LSW_OBC_(nx,ny)=(3,3)
Figure 1.xlsx\panel F_LSW_OBC_(nx,ny)=(5,1)
Figure 1.xlsx\panel F_LSW_OBC_(nx,ny)=(5,5)
These sheets provide the structure factor dynamics of linear spin-wave theory with open boundary conditions in the antiferromagnetic case with different (nx, ny). Variable list:
- "Time(hbar/J)_LSW_OBC": Evolution time of the LSW OBC data in Figure 1F; The data range spans from A2 to A1601.
- "Structure factor": Structure factor of the LSW OBC data in Figure 1F; The data range spans from B2 to B1601.
Figure 1.xlsx\panel F_tVMC_(nx,ny)=(1,0)
Figure 1.xlsx\panel F_tVMC_(nx,ny)=(3,0)
Figure 1.xlsx\panel F_tVMC_(nx,ny)=(3,3)
Figure 1.xlsx\panel F_tVMC_(nx,ny)=(5,1)
Figure 1.xlsx\panel F_tVMC_(nx,ny)=(5,5)
These sheets provide the structure factor dynamics of time-dependent variational Monte Carlo in the antiferromagnetic case with different (nx, ny). Variable list:
- "Time(hbar/J)_tVMC": Evolution time of the tVMC data in Figure 1F; The data range spans from A2 to A21.
- "Structure factor": Structure factor of the tVMC data in Figure 1F; The data range spans from B2 to B21.
Figure 2.xlsx\panel A_data
This data sheet provide the data of dispersion relation from the experiment in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A18.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B18.
- "error bars": Error bars. The data range spans from C2 to C18.
Figure 2.xlsx\panel A_tVMC
This data sheet provide the data of dispersion relation from the tVMC in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A22.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B22.
- "error bars": Error bars. The data range spans from C2 to C22.
Figure 2.xlsx\panel A_LSW_OBC
This data sheet provide the data of dispersion relation from the LSW OBC in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A21.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B21.
- "error bars": Error bars. The data range spans from C2 to C21.
Figure 2.xlsx\panel A_LSW_mPBC
This data sheet provide the data of dispersion relation from the LSW mPBC in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A101.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B101.
Figure 2.xlsx\panel B_data
This data sheet provide the data of dispersion relation from the experiment in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A18.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B18.
- "error bars": Error bars. The data range spans from C2 to C18.
Figure 2.xlsx\panel B_tVMC
This data sheet provide the data of dispersion relation from the tVMC in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A22.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B22.
- "error bars": Error bars. The data range spans from C2 to C22.
Figure 2.xlsx\panel B_LSW_OBC
This data sheet provide the data of dispersion relation from the LSW OBC in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A21.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B21.
- "error bars": Error bars. The data range spans from C2 to C21.
Figure 2.xlsx\panel B_LSW_mPBC
This data sheet provide the data of dispersion relation from the LSW mPBC in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A101.
- "hbar*omega/J": Frequency in the unit of interaction J. The data range spans from B2 to B101.
Figure 2.xlsx\panel C_data
This data sheet provide the data of dispersion relation from the experiment in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A18.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B18.
- "error bars": Error bars. The data range spans from C2 to C18.
Figure 2.xlsx\panel C_tVMC
This data sheet provide the data of dispersion relation from the tVMC in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A22.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B22.
- "error bars": Error bars. The data range spans from C2 to C22.
Figure 2.xlsx\panel C_LSW_OBC
This data sheet provide the data of dispersion relation from the LSW OBC in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A21.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B21.
- "error bars": Error bars. The data range spans from C2 to C21.
Figure 2.xlsx\panel C_LSW_mPBC
This data sheet provide the data of dispersion relation from the LSW mPBC in the ferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A101.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B101.
Figure 2.xlsx\panel D_data
This data sheet provide the data of dispersion relation from the experiment in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A18.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B18.
- "error bars": Error bars. The data range spans from C2 to C18.
Figure 2.xlsx\panel D_tVMC
This data sheet provide the data of dispersion relation from the tVMC in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A22.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B22.
- "error bars": Error bars. The data range spans from C2 to C22.
Figure 2.xlsx\panel D_LSW_OBC
This data sheet provide the data of dispersion relation from the LSW OBC in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A21.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B21.
- "error bars": Error bars. The data range spans from C2 to C21.
Figure 2.xlsx\panel D_LSW_mPBC
This data sheet provide the data of dispersion relation from the LSW mPBC in the antiferromagnetic case. Variable list:
- "k": modulus of wave-vector. The data range spans from A2 to A101.
- "gamma*sqrt(2C-1)": Frequency obtained from the offset Ck. The data range spans from B2 to B101.
Figure 3.xlsx\panel A_data_(nx,ny)=(0,1)
Figure 3.xlsx\panel A_data_(nx,ny)=(0,2)
Figure 3.xlsx\panel A_data_(nx,ny)=(1,1)
Figure 3.xlsx\panel A_data_(nx,ny)=(1,2)
Figure 3.xlsx\panel A_data_(nx,ny)=(2,2)
These sheets provide experimental data of structure factor dynamics in the ferromagnetic 4 by 4 case with different (nx, ny). Variable list:
- "Time(hbar/J)_data": Evolution time of the experimental data in Figure 3A; The data range spans from A2 to A102.
- "Structure factor": Structure factor of the experimental data in Figure 3A; The data range spans from B2 to B102.
- "lower_errors": lower bound of the error bar for experimental structure factor in Figure 3A; The data range spans from C2 to C102.
- "upper_errors": upper bound of the error bar for experimental structure factor in Figure 3A; The data range spans from D2 to D102.
Figure 3.xlsx\panel A_ED_(nx,ny)=(0,1)
Figure 3.xlsx\panel A_ED_(nx,ny)=(0,2)
Figure 3.xlsx\panel A_ED_(nx,ny)=(1,1)
Figure 3.xlsx\panel A_ED_(nx,ny)=(1,2)
Figure 3.xlsx\panel A_ED_(nx,ny)=(2,2)
These sheets provide the results of structure factor dynamics from exact diagonalization simulations in the ferromagnetic 4 by 4 case with different (nx, ny). Variable list:
- "Time(hbar/J)_ED": Evolution time of the exact diagonalization data in Figure 3A; The data range spans from A2 to A201.
- "Structure factor": Structure factor of the exact diagonalization data in Figure 3A; The data range spans from B2 to B201.
Figure 3.xlsx\panel A_ED_err_(nx,ny)=(0,1)
Figure 3.xlsx\panel A_ED_err_(nx,ny)=(0,2)
Figure 3.xlsx\panel A_ED_err_(nx,ny)=(1,1)
Figure 3.xlsx\panel A_ED_err_(nx,ny)=(1,2)
Figure 3.xlsx\panel A_ED_err_(nx,ny)=(2,2)
These sheets provide the results of structure factor dynamics from exact diagonalization simulations with imperfections taken into account in the ferromagnetic 4 by 4 case with different (nx, ny). Variable list:
- "Time(hbar/J)_ED_imperfections": Evolution time of the results from exact diagonalization simulations with imperfections taken into account in Figure 3A; The data range spans from A2 to A597.
- "lower_bound": lower bound of the shadow area for the exact diagonalization simulations with imperfections taken into account in Figure 3A; The data range spans from B2 to B597.
- "upper_bound": upper bound of the shadow area for the exact diagonalization simulations with imperfections taken into account in Figure 3A; The data range spans from C2 to C597.
Figure 3.xlsx\panel B_data_(nx,ny)=(0,1)
Figure 3.xlsx\panel B_data_(nx,ny)=(0,2)
Figure 3.xlsx\panel B_data_(nx,ny)=(1,1)
Figure 3.xlsx\panel B_data_(nx,ny)=(1,2)
Figure 3.xlsx\panel B_data_(nx,ny)=(2,2)
These sheets provide experimental data of structure factor dynamics in the antiferromagnetic 4 by 4 case with different (nx, ny). Variable list:
- "Time(hbar/J)_data": Evolution time of the experimental data in Figure 3B; The data range spans from A2 to A101.
- "Structure factor": Structure factor of the experimental data in Figure 3B; The data range spans from B2 to B101.
- "lower_errors": lower bound of the error bar for experimental structure factor in Figure 3B; The data range spans from C2 to C101.
- "upper_errors": upper bound of the error bar for experimental structure factor in Figure 3B; The data range spans from D2 to D101.
Figure 3.xlsx\panel B_ED_(nx,ny)=(0,1)
Figure 3.xlsx\panel B_ED_(nx,ny)=(0,2)
Figure 3.xlsx\panel B_ED_(nx,ny)=(1,1)
Figure 3.xlsx\panel B_ED_(nx,ny)=(1,2)
Figure 3.xlsx\panel B_ED_(nx,ny)=(2,2)
These sheets provide the results of structure factor dynamics from exact diagonalization simulations in the antiferromagnetic 4 by 4 case with different (nx, ny). Variable list:
- "Time(hbar/J)_ED": Evolution time of the exact diagonalization data in Figure 3B; The data range spans from A2 to A201.
- "Structure factor": Structure factor of the exact diagonalization data in Figure 3B; The data range spans from B2 to B201.
Figure 3.xlsx\panel B_ED_err_(nx,ny)=(0,1)
Figure 3.xlsx\panel B_ED_err_(nx,ny)=(0,2)
Figure 3.xlsx\panel B_ED_err_(nx,ny)=(1,1)
Figure 3.xlsx\panel B_ED_err_(nx,ny)=(1,2)
Figure 3.xlsx\panel B_ED_err_(nx,ny)=(2,2)
These sheets provide the results of structure factor dynamics from exact diagonalization simulations with imperfections taken into account in the antiferromagnetic 4 by 4 case with different (nx, ny). Variable list:
- "Time(hbar/J)_ED_imperfections": Evolution time of the results from exact diagonalization simulations with imperfections taken into account in Figure 3B; The data range spans from A2 to A567.
- "lower_bound": lower bound of the shadow area for the exact diagonalization simulations with imperfections taken into account in Figure 3B; The data range spans from B2 to B567.
- "upper_bound": upper bound of the shadow area for the exact diagonalization simulations with imperfections taken into account in Figure 3B; The data range spans from C2 to C567.
Figure 4.xlsx\panel A
This data sheet provides the data of time-averaged structure factor of the experimental data (FM case, Figure 4A). Variable list:
- "kx ": The data range spans from C14 to M14.
- "ky ": The data range spans from cell A2 to A12.
- "Time-averaged structure factor ": Time-averaged structure factor of the experimental data in Figure 4A; The data range spans from C2 to M12.
Figure 4.xlsx\panel B
This data sheet provides the structure factor of the Quantum Monte Carlo simulations (FM case, Figure 4B). Variable list:
- "kx ": The data range spans from C14 to M14.
- "ky ": The data range spans from cell A2 to A12.
- "Time-averaged structure factor ": Time-averaged structure factor of the Quantum Monte Carlo (QMC) simulations in Figure 4B; The data range spans from C2 to M12.
Figure 4.xlsx\panel C
This data sheet provides the time-averaged structure factor along the path connecting the high-symmetry
points of the Brillouin zone (FM case, Figure 4C). Variable list:
- "k ": The wave-vector. The data range spans from A3 to A16.
- "data": "Time-averaged SF": Structure factor of the experimental data. The data range spans from cell C3 to C16.
- "data": "error bars": error bars of the experimental data. The data range spans from cell D3 to D16.
- "tVMC": "Time-averaged SF": Structure factor of the tVMC data. The data range spans from cell F3 to F16.
- "QMC ": "Time-averaged SF": Structure factor of the QMC data. The data range spans from cell H3 to H16.
- "QMC ": "error bars": error bar of the QMC data. The data range spans from cell I3 to I16.
Figure 4.xlsx\panel D
This data sheet provides the data of time-averaged structure factor of the experimental data (AFM case, Figure 4D). Variable list:
- "kx ": The data range spans from C14 to M14.
- "ky ": The data range spans from cell A2 to A12.
- "Time-averaged structure factor ": Time-averaged structure factor of the experimental data in Figure 4D; The data range spans from C2 to M12.
Figure 4.xlsx\panel E
This data sheet provides the structure factor of the Quantum Monte Carlo simulations (AFM NN case, Figure 4E). Variable list:
- "kx ": The data range spans from C14 to M14.
- "ky ": The data range spans from cell A2 to A12.
- "Time-averaged structure factor ": Time-averaged structure factor of the Quantum Monte Carlo (QMC) simulations with only nearest-neighbor coupling in Figure 4E; The data range spans from C2 to M12.
Figure 4.xlsx\panel F
This data sheet provides the time-averaged structure factor along the path connecting the high-symmetry
points of the Brillouin zone (AFM case, Figure 4F). Variable list:
- "k ": The wave-vector. The data range spans from A3 to A16.
- "data": "Time-averaged SF": Structure factor of the experimental data. The data range spans from cell C3 to C16.
- "data": "error bars": error bars of the experimental data. The data range spans from cell D3 to D16.
- "tVMC": "Time-averaged SF": Structure factor of the tVMC data . The data range spans from cell F3 to F16.
- "LSW": "Time-averaged SF": Structure factor of the LSW data. The data range spans from cell H3 to H16.
- "QMC NN": "Time-averaged SF": Structure factor of the QMC data. The data range spans from cell J3 to J16.
- "QMC NN ": "error bars": error bar of the QMC data. The data range spans from cell K3 to K16.
Code/software
You can use the following software to view the data:
(1) Python 3.12.3 + pandas 2.2.2
(2) LibreOffice
(3) Microsoft Excel.