## Combined Chart: Atom Number Distribution ### Overview The image presents two plots, (a) and (b), showing the probability distribution P(NΩ) of detected atom numbers NΩ under different conditions. Plot (a) compares a Mott insulator distribution with thermal and Poisson distributions on a semi-log scale. Plot (b) compares a BEC mode distribution with thermal and Poisson distributions on a linear scale. ### Components/Axes **Plot (a):** * **Title:** (a) (top-left) * **X-axis:** Detected atom number NΩ, ranging from 0 to 7. * **Y-axis:** Probability P(NΩ), ranging from 10^-3 to 10^0 (logarithmic scale). * **Legend (top-right):** * Thermal (black dashed-dotted line) * Poisson (blue dashed line) * Mott insulator (gray squares with error bars) **Plot (b):** * **Title:** (b) (top-left) * **X-axis:** Detected atom number NΩ, ranging from 0 to 19. * **Y-axis:** Probability P(NΩ), ranging from 0.00 to 0.15 (linear scale). * **Legend (top-right):** * Thermal (black dashed-dotted line) * Poisson (blue dashed line) * BEC mode (blue circles with error bars) ### Detailed Analysis **Plot (a): Mott Insulator, Thermal, and Poisson Distributions** * **Mott insulator (gray squares):** The data points are approximately: * (0, 0.8) * (1, 0.3) * (2, 0.08) * (3, 0.03) * (4, 0.01) * (5, 0.004) * (6, 0.001) * (7, 0.0003) The Mott insulator distribution shows a decreasing probability with increasing atom number. The gray shaded area around the Mott insulator data points represents the uncertainty. * **Thermal (black dashed-dotted line):** The thermal distribution follows a similar decreasing trend as the Mott insulator, but is slightly higher at larger atom numbers. * **Poisson (blue dashed line):** The Poisson distribution decreases more rapidly than the Mott insulator and thermal distributions. It falls below the Mott insulator data after NΩ = 2. The blue shaded area around the Poisson line represents the uncertainty. **Plot (b): BEC Mode, Thermal, and Poisson Distributions** * **BEC mode (blue circles):** The data points are approximately: * (0, 0.02) * (2, 0.08) * (4, 0.14) * (5, 0.15) * (6, 0.14) * (8, 0.08) * (10, 0.05) * (12, 0.02) * (14, 0.01) * (16, 0.005) * (18, 0.002) The BEC mode distribution shows a peak around NΩ = 5. The blue shaded area around the Poisson line represents the uncertainty. * **Thermal (black dashed-dotted line):** The thermal distribution starts high and decreases monotonically with increasing atom number. * **Poisson (blue dashed line):** The Poisson distribution shows a peak around NΩ = 5, similar to the BEC mode, but the peak is sharper and the distribution decays faster. ### Key Observations * In plot (a), the Mott insulator distribution is broader than the Poisson distribution. * In plot (b), the BEC mode distribution is similar to the Poisson distribution but has a slightly broader peak. * The thermal distribution decreases monotonically in both plots. ### Interpretation The plots illustrate the probability distributions of detected atom numbers for different quantum states. Plot (a) shows that the Mott insulator distribution is broader than the Poisson distribution, indicating a more uniform distribution of atoms across lattice sites. Plot (b) shows that the BEC mode distribution is similar to the Poisson distribution, but with a broader peak, suggesting a less coherent state than a pure Bose-Einstein condensate. The thermal distribution represents a classical, incoherent state, which is characterized by a monotonically decreasing probability with increasing atom number. The shaded regions around the Poisson distribution in both plots, and the Mott insulator data in plot (a), represent the uncertainty in the measurements or theoretical calculations.

\n ## Chart: Atom Number Distribution ### Overview The image presents two charts (labeled (a) and (b)) displaying the probability distribution of detected atom numbers. Both charts share the same x-axis label ("Detected atom number NΩ") and y-axis label ("Probability P(NΩ)"), but differ in their y-axis scale (logarithmic in (a) and linear in (b)). Each chart compares experimental data (represented by markers with error bars) to theoretical distributions: Thermal and Poisson. Chart (a) focuses on the Mott insulator state, while chart (b) focuses on the BEC mode. ### Components/Axes * **X-axis:** "Detected atom number NΩ", ranging from approximately 0 to 6 in (a) and 0 to 12 in (b). * **Y-axis (a):** "Probability P(NΩ)", logarithmic scale, ranging from approximately 10⁻³ to 10⁰. * **Y-axis (b):** "Probability P(NΩ)", linear scale, ranging from approximately 0 to 0.16. * **Legend:** * **Chart (a):** * Black dashed line: "Thermal" * Blue dashed line: "Poisson" * Gray square markers with error bars: "Mott insulator" * **Chart (b):** * Black dashed line: "Thermal" * Blue dashed line: "Poisson" * Blue circle markers with error bars: "BEC mode" * **Grid:** Both charts have a grid for easier value reading. ### Detailed Analysis or Content Details **Chart (a): Mott Insulator** * **Mott Insulator (Gray Squares):** The data points show a decreasing probability as the detected atom number increases. * NΩ = 0: P(NΩ) ≈ 0.95 ± 0.05 * NΩ = 1: P(NΩ) ≈ 0.30 ± 0.05 * NΩ = 2: P(NΩ) ≈ 0.10 ± 0.02 * NΩ = 3: P(NΩ) ≈ 0.03 ± 0.01 * NΩ = 4: P(NΩ) ≈ 0.01 ± 0.005 * NΩ = 5: P(NΩ) ≈ 0.003 ± 0.002 * NΩ = 6: P(NΩ) ≈ 0.001 ± 0.001 * **Thermal (Black Dashed):** Starts at approximately 1 and decreases slowly with increasing NΩ. * **Poisson (Blue Dashed):** Starts at approximately 1 and decreases more rapidly than the Thermal distribution. **Chart (b): BEC Mode** * **BEC Mode (Blue Circles):** The data points show an initial increase in probability, reaching a maximum around NΩ = 4, and then decreasing. * NΩ = 0: P(NΩ) ≈ 0.02 ± 0.01 * NΩ = 1: P(NΩ) ≈ 0.06 ± 0.01 * NΩ = 2: P(NΩ) ≈ 0.10 ± 0.01 * NΩ = 3: P(NΩ) ≈ 0.14 ± 0.01 * NΩ = 4: P(NΩ) ≈ 0.15 ± 0.01 * NΩ = 5: P(NΩ) ≈ 0.11 ± 0.01 * NΩ = 6: P(NΩ) ≈ 0.07 ± 0.01 * NΩ = 7: P(NΩ) ≈ 0.04 ± 0.01 * NΩ = 8: P(NΩ) ≈ 0.02 ± 0.01 * NΩ = 9: P(NΩ) ≈ 0.015 ± 0.005 * NΩ = 10: P(NΩ) ≈ 0.01 ± 0.005 * NΩ = 11: P(NΩ) ≈ 0.005 ± 0.003 * NΩ = 12: P(NΩ) ≈ 0.003 ± 0.002 * **Thermal (Black Dashed):** Decreases monotonically with increasing NΩ. * **Poisson (Blue Dashed):** Increases initially, reaches a maximum around NΩ = 2, and then decreases. ### Key Observations * In Chart (a), the Mott insulator data deviates significantly from both the Thermal and Poisson distributions, especially at higher atom numbers. The Mott insulator distribution decays much faster than the Poisson distribution. * In Chart (b), the BEC mode data shows a clear peak, indicating a preferred atom number. The BEC mode distribution is significantly different from both the Thermal and Poisson distributions. * The logarithmic scale in Chart (a) emphasizes the differences in the tail of the distribution, while the linear scale in Chart (b) highlights the peak. ### Interpretation These charts demonstrate the distinct statistical properties of different quantum states of atoms. Chart (a) shows that the Mott insulator state exhibits a strong suppression of atom number fluctuations, deviating from the predictions of both thermal and Poisson statistics. This is consistent with the theoretical expectation that Mott insulators are characterized by a fixed number of atoms per site. Chart (b) shows that the BEC mode exhibits a peak in the probability distribution, indicating a tendency for atoms to condense into the same quantum state. The shape of the BEC mode distribution is also different from both thermal and Poisson statistics, reflecting the unique quantum mechanical properties of Bose-Einstein condensates. The comparison to the Poisson distribution is particularly relevant, as the Poisson distribution describes the statistics of independent events, while the Mott insulator and BEC mode represent correlated quantum states. The error bars on the experimental data indicate the uncertainty in the measurements, and the agreement between the experimental data and the theoretical predictions provides strong evidence for the validity of the theoretical models. The use of two different y-axis scales allows for a more comprehensive comparison of the distributions, highlighting both the overall shape and the details of the tails.

## [Two-Panel Line Graph]: Probability Distributions of Detected Atom Numbers for Different Quantum States ### Overview The image contains two vertically stacked line graphs, labeled (a) and (b), comparing the probability distributions of detected atom numbers (`N_Ω`) for different physical states: Thermal, Poisson, Mott insulator, and BEC (Bose-Einstein Condensate) mode. Plot (a) uses a logarithmic probability scale, while plot (b) uses a linear scale. ### Components/Axes **Common Elements:** * **X-axis (both plots):** Label: `Detected atom number N_Ω`. Plot (a) range: 0 to 7. Plot (b) range: 0 to 18. * **Y-axis (both plots):** Label: `Probability P(N_Ω)`. * Plot (a): Logarithmic scale from `10^-3` to `10^0`. * Plot (b): Linear scale from `0.00` to `0.15`. * **Legends:** * Plot (a) - Top-right corner: * `Thermal`: Black dash-dotted line (`-.-`). * `Poisson`: Blue dashed line (`--`) with a light blue shaded uncertainty band. * `Mott insulator`: Black square markers (`□`) with vertical error bars. * Plot (b) - Top-right corner: * `Thermal`: Black dash-dotted line (`-.-`). * `Poisson`: Blue dashed line (`--`). * `BEC mode`: Blue circle markers (`○`) with vertical error bars. ### Detailed Analysis **Plot (a) - Logarithmic Scale:** * **Trend Verification:** All three series show a decreasing trend as the detected atom number `N_Ω` increases. The `Poisson` distribution decreases more steeply than the `Thermal` distribution. * **Data Points & Values (Approximate):** * At `N_Ω = 0`: All three series start at a high probability, ~`0.8`. * At `N_Ω = 2`: `Thermal` and `Mott insulator` are ~`0.07`. `Poisson` is slightly lower, ~`0.06`. * At `N_Ω = 4`: `Thermal` and `Mott insulator` are ~`0.006`. `Poisson` is ~`0.001`. * At `N_Ω = 6`: `Thermal` is ~`0.001`. `Mott insulator` is ~`0.001` (with error bar spanning ~`0.0005` to `0.002`). `Poisson` is below `10^-3` (off the plotted line trend). * **Component Isolation:** The `Mott insulator` data points (black squares) closely follow the `Thermal` distribution line across the entire range, with slight deviations at higher `N_Ω`. **Plot (b) - Linear Scale:** * **Trend Verification:** The `Thermal` distribution decreases monotonically. The `Poisson` distribution is bell-shaped, peaking around `N_Ω = 5`. The `BEC mode` data points follow the `Poisson` curve. * **Data Points & Values (Approximate):** * **Poisson/BEC Mode Peak:** At `N_Ω = 5`, the `Poisson` line peaks at ~`0.17`. The `BEC mode` data point at `N_Ω = 5` is ~`0.155`. * **Thermal Line:** At `N_Ω = 0`, ~`0.16`. At `N_Ω = 5`, ~`0.08`. At `N_Ω = 10`, ~`0.02`. * **BEC Mode Data:** Closely matches the `Poisson` curve. For example, at `N_Ω = 2`, ~`0.04`; at `N_Ω = 8`, ~`0.09`; at `N_Ω = 12`, ~`0.01`. * At high atom numbers (`N_Ω > 15`), both the `Thermal` and `Poisson` distributions, as well as the `BEC mode` data, approach a probability of `0.00`. ### Key Observations 1. **State-Dependent Distributions:** The `Mott insulator` state (plot a) exhibits a probability distribution nearly identical to the `Thermal` distribution. In contrast, the `BEC mode` state (plot b) exhibits a distribution nearly identical to the `Poisson` distribution. 2. **Scale Revelation:** The logarithmic scale in (a) clearly shows the exponential decay of the Thermal and Mott insulator distributions and highlights the divergence of the Poisson distribution at higher atom numbers. The linear scale in (b) clearly reveals the characteristic peak of the Poisson/BEC mode distribution. 3. **Data Agreement:** The experimental data points (`Mott insulator` and `BEC mode`) show strong agreement with their respective theoretical distribution lines (`Thermal` and `Poisson`), within the shown error bars. ### Interpretation This figure demonstrates a fundamental difference in the atom number statistics of two quantum phases of matter. * The **Mott insulator** phase, characterized by a fixed number of atoms per lattice site, shows **super-Poissonian** or thermal-like number fluctuations, as its distribution matches the `Thermal` curve. This suggests a higher degree of randomness or "bunching" in the detected atoms. * The **BEC mode**, representing a coherent quantum state, shows **Poissonian** number fluctuations. This is the hallmark of a coherent state (like a laser), where the variance in particle number equals the mean. The data confirms that the BEC mode's statistics are consistent with this fundamental quantum optical model. * The comparison underscores how measuring atom number distributions can serve as a diagnostic tool to identify and characterize different quantum phases in ultracold atomic systems. The clear separation between the Thermal and Poisson curves in plot (b) provides a distinct signature for identifying a coherent BEC versus a thermal gas.

## Line Graphs: Probability Distributions of Detected Atom Numbers ### Overview The image contains two line graphs (a) and (b) comparing probability distributions of detected atom numbers (N_Ω) under different theoretical models (Thermal, Poisson) and experimental modes (Mott insulator, BEC mode). Both graphs use logarithmic and linear scales for probability (P(N_Ω)) and linear scales for N_Ω, respectively. --- ### Components/Axes #### Graph (a) - **Y-axis**: Probability P(N_Ω) (logarithmic scale: 10⁻³ to 10⁰) - **X-axis**: Detected atom number N_Ω (linear scale: 0 to 6) - **Legend**: - Dashed black line: Thermal distribution - Dashed blue line: Poisson distribution - Gray squares with error bars: Mott insulator experimental data - **Legend Position**: Top-right corner #### Graph (b) - **Y-axis**: Probability P(N_Ω) (linear scale: 0 to 0.15) - **X-axis**: Detected atom number N_Ω (linear scale: 0 to 15) - **Legend**: - Dashed black line: Thermal distribution - Dashed blue line: Poisson distribution - Blue circles with error bars: BEC mode experimental data - **Legend Position**: Top-right corner --- ### Detailed Analysis #### Graph (a) - **Thermal Distribution**: - Dashed black line decreases monotonically from ~10⁰ at N_Ω=0 to ~10⁻³ at N_Ω=6. - No shaded region. - **Poisson Distribution**: - Dashed blue line decreases similarly to Thermal but with a shaded blue region (likely confidence interval) around it. - Shaded region widens at lower N_Ω (0–2), indicating higher uncertainty. - **Mott Insulator Data**: - Gray squares with error bars align closely with the Poisson line. - Error bars increase in size at lower N_Ω (e.g., ±0.1 at N_Ω=0, ±0.05 at N_Ω=6). #### Graph (b) - **Thermal Distribution**: - Dashed black line decreases monotonically from ~0.15 at N_Ω=0 to ~0.005 at N_Ω=15. - **Poisson Distribution**: - Dashed blue line peaks at N_Ω=5 (~0.12) and decreases afterward. - No shaded region. - **BEC Mode Data**: - Blue circles with error bars align with the Poisson peak at N_Ω=5. - Error bars are smaller at higher N_Ω (e.g., ±0.01 at N_Ω=10, ±0.02 at N_Ω=15). --- ### Key Observations 1. **Graph (a)**: - Mott insulator data closely follows the Poisson distribution, suggesting Poissonian statistics dominate at low N_Ω. - Thermal distribution deviates slightly at higher N_Ω but remains close to Poisson. - Shaded region around Poisson in (a) implies greater uncertainty at low N_Ω. 2. **Graph (b)**: - BEC mode data peaks at N_Ω=5, matching the Poisson distribution’s maximum. - Thermal distribution declines steadily without a peak, contrasting with Poisson/BEC mode. - Error bars in BEC mode are smaller at higher N_Ω, indicating more precise measurements. --- ### Interpretation - **Poisson vs. Thermal**: The Poisson distribution (dashed blue) consistently aligns with experimental data (Mott insulator in (a), BEC mode in (b)), suggesting quantum systems exhibit Poissonian statistics, likely due to coherent state behavior or shot noise. - **Thermal Distribution**: The dashed black line represents a classical thermal model, which diverges from experimental results, indicating quantum effects dominate over thermal noise. - **Shaded Regions**: In (a), the shaded area around Poisson highlights uncertainty in low-N_Ω measurements, possibly due to experimental limitations or quantum fluctuations. - **BEC Mode Peak**: The sharp peak in BEC mode data at N_Ω=5 (graph b) mirrors the Poisson distribution’s maximum, reinforcing the role of quantum coherence in atom number statistics. --- ### Notable Trends - **Low-N_Ω Uncertainty**: Larger error bars at N_Ω=0–2 in graph (a) suggest challenges in detecting small atom numbers. - **High-N_Ω Precision**: Smaller error bars at N_Ω>10 in graph (b) indicate improved measurement accuracy for larger atom numbers. - **Model Agreement**: Experimental data (Mott/BEC) closely follow Poisson predictions, validating the theoretical model for these systems. --- ### Conclusion The graphs demonstrate that quantum systems (Mott insulator, BEC mode) exhibit Poissonian statistics for detected atom numbers, contrasting with classical thermal models. The shaded regions and error bars emphasize experimental uncertainties, particularly at low N_Ω. These results highlight the dominance of quantum coherence over thermal noise in these systems.