Dataset DOI: 10.5061/dryad.02v6wwqjg

Description of the data and file structure

Files and variables

File: fig3a.csv

Description: The measured actions for free space in three specific boundary conditions.

Variables

•  j: Index of paths sorted by their associated action in ascending order. It is represented as the horizontal axis in Fig. 3(a).
• S_th(0,0): Theoretical results for the action of the paths with boundary conditions x_i = 0 and x_f = 0, presented as a blue solid line in Fig. 3(a) with the results given in units of πħ.
• S_exp(0,0)-M: Mean results for the measured action of the paths with boundary conditions x_i = 0 and x_f = 0, presented as blue circles in Fig. 3(a) with the results given in units of πħ.
• S_exp(0,0)-SD: Standard deviations for the measured action of the paths with boundary conditions x_i = 0 and x_f = 0, presented as a blue error bars in Fig. 3(a) with the results given in units of πħ.
• S_th(-10,10): Theoretical results for the action of the paths with boundary conditions x_i = -10 and x_f = 10, presented as a orange solid line in Fig. 3(a) with the results given in units of πħ.
•  S_exp(-10,10)-M: Mean results for the measured action of the paths with boundary conditions x_i = -10 and x_f = 10, presented as orange squares in Fig. 3(a) with the results given in units of πħ.
• S_exp(-10,10)-SD: Standard deviations for the measured action of the paths with boundary conditions x_i = -10 and x_f = 10, presented as orange error bars in Fig. 3(a) with the results given in units of πħ.
• S_th(15,-15): Theoretical results for the action of the paths with boundary conditions x_i = 15 and x_f = -15, presented as a purple solid line in Fig. 3(a) with the results given in units of πħ.
• S_exp(15,-15)-M: Mean results for the measured action of the paths with boundary conditions x_i = 15 and x_f = -15, presented as purple triangles in Fig. 3(a) with the results given in units of πħ.
• S_exp(15,-15)-SD: Standard deviations for the measured action of the paths with boundary conditions x_i = 15 and x_f = -15, presented as purple error bars in Fig. 3(a) with the results given in units of πħ.
• j_subfigure: Index of 1000 least-action paths, sorted by their associated action in ascending order. They are represented as the horizontal axis in subfigure of Fig. 3(a).
• S_subfigure(0,0): Measured actions of 1000 least-action paths under boundary conditions x_i = 0, x_f = 0 from one experiment repetition. They are presented as blue circles in subfigure of Fig. 3(a) with the results given in units of πħ.
• S_subfigure(-10,10): Measured actions of 1000 least-action paths under boundary conditions x_i = -10, x_f = 10 from one experiment repetition. They are presented as orange squares in subfigure of Fig. 3(a) with the results given in units of πħ.
• S_subfigure(15,-15): Measured actions of 1000 least-action paths under boundary conditions x_i = 15, x_f = -15 from one experiment repetition. They are presented as purple triangles in subfigure of Fig. 3(a) with the results given in units of πħ.

File: fig3b.csv

Description: The identified classical paths with minimum action for free space in three specific boundary conditions.

Variables

• t: Evolution time, in units of ϵ= 15 mm/c, is represented as the horizontal axis in Fig. 3(b).
• xcl_th(15,-15): Theoretical results for a classical path under the boundary conditions x_i = 15, x_f = -15, presented as a purple solid line in Fig. 3(b) with the results given in units of δx = 5.73 μm.
• xcl_exp(15,-15)-M: Mean results for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=15, x_f=-15. The results are given in units of δx=5.73μm and are presented as purple triangles in Fig. 3(b).
• xcl_exp(15,-15)-SD: Standard deviations for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=15, x_f=-15. The results are given in units of δx=5.73μm** and are presented as purple error bars in Fig. 3(b).
• xcl_th(-10,10): Theoretical results for a classical path under the boundary conditions x_i = -10, x_f = 10, presented as a orange solid line in Fig. 3(b) with the results given in units of δx = 5.73 μm.
• xcl_exp(-10,10)-M: Mean results for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=-10, x_f=10. The results are given in units of δx=5.73μm and are presented as orange squares in Fig. 3(b).
• xcl_exp(-10,10)-SD: Standard deviations for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=-10, x_f=10. The results are given in units of δx=5.73μm** and are presented as orange error bars in Fig. 3(b).
• xcl_th(0,0): Theoretical results for a classical path under the boundary conditions x_i = 0, x_f = 0, presented as a blue solid line in Fig. 3(b) with the results given in units of δx = 5.73 μm.
• xcl_exp(0,0)-M: Mean results for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=0, x_f=0. The results are given in units of δx=5.73μm and are presented as blue circles in Fig. 3(b).
• xcl_exp(0,0)-SD: Standard deviations for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=0, x_f=0. The results are given in units of δx=5.73μm and are presented as blue error bars in Fig. 3(b).
• xcl_exp(15,-15)-Smin: Paths derived with minimum of the measured action under the boundary conditions x_i = 15, x_f = -15, presented as a purple dashed line in Fig. 3(b) with the results given in units of δx = 5.73 μm.
• xcl_exp(-10,10)-Smin: Paths derived with minimum of the measured action under the boundary conditions x_i =-10, x_f = 10, presented as a orange dashed line in Fig. 3(b) with the results given in units of δx = 5.73 μm. 
• xcl_exp(0,0)-Smin: Paths derived with minimum of the measured action under the boundary conditions x_i =0, x_f = 0, presented as a blue dashed line in Fig. 3(b) with the results given in units of δx = 5.73 μm.

File: fig3c.csv

Description: The norm of action gradient spectrum for free space in three specific boundary conditions.

Variables

• S(0,0): Measured actions of 10,000 paths with the smallest norm of action gradient (NAG) under boundary conditions x_i = 0 and x_f = 0. The results are presented Fig. 3(c1) and are given in units of πħ.
• S_NAG(0,0): 10,000 smallest NAG values of paths under boundary conditions x_i = 0 and x_f = 0. The results are presented Fig. 3(c1) and are given in units of πħ/δx.
• S(-10,10): Measured actions of 10,000 paths with the smallest norm of action gradient (NAG) under boundary conditions x_i = -10 and x_f = 10. The results are presented Fig. 3(c2) and are given in units of πħ.
• S_NAG(-10.10): 10,000 smallest NAG values of paths under boundary conditions x_i = -10 and x_f = 10. The results are presented Fig. 3(c2) and are given in units of πħ/δx.
• S(15,-15): Measured actions of 10,000 paths with the smallest norm of action gradient (NAG) under boundary conditions x_i = 15 and x_f = -15. The results are presented Fig. 3(c3) and are given in units of πħ.
• S_NAG(15,-15): 10,000 smallest NAG values of paths under boundary conditions x_i = 15 and x_f = -15. The results are presented Fig. 3(c3) and are given in units of πħ/δx.

File: fig3d.csv

Description: Identified paths from Fig.3(c).

Variables

• t: Evolution time, in units of ϵ= 15 mm/c, is represented as the horizontal axis in Fig. 3(d).
• xcl_th(15,-15): Theoretical results for a classical path under the boundary conditions x_i = 15, x_f = -15, presented as purple solid line in Fig. 3(d) with the results given in units of δx = 5.73 μm.
• xcl_NAG(15,-15)-R1: Identified path from the NAG spectrum under the boundary conditions x_i = 15, x_f = -15, presented as purple triangles in Fig. 3(d) with the results given in units of δx = 5.73 μm.
• xcl_th(-10,10): Theoretical results for a classical path under the boundary conditions x_i = -10, x_f = 10, presented as orange solid line in Fig. 3(d) with the results given in units of δx = 5.73 μm.
• xcl_NAG(-10,10)-R2: Identified path from the NAG spectrum under the boundary conditions x_i = -10, x_f = 10, presented as orange squares in Fig. 3(d) with the results given in units of δx = 5.73 μm.
• xcl_th(0,0): Theoretical results for a classical path under the boundary conditions x_i = 0, x_f = 0, presented as a blue solid line in Fig. 3(d) with the results given in units of δx = 5.73 μm.
• xcl_NAG(0,0)-R3-M: Mean values of identified paths from the NAG spectrum under the boundary conditions x_i = 0, x_f = 0, presented as blue circles in Fig. 3(d) with the results given in units of δx = 5.73 μm.
• xcl_NAG(0,0)-R3-SD: Standard deviations of identified paths from the NAG spectrum under the boundary conditions x_i = 0, x_f = 0, presented as blue circles in Fig. 3(d) with the results given in units of δx = 5.73 μm.

File: fig4a1.csv

Description: Experimental NAG spectra for an impulsive potential with boundary conditions x_i = 0 and x_f = 0.

Variables

• S(0,0): Measured actions of 10,000 paths with the smallest norm of action gradient (NAG) under boundary conditions x_i = 0 and x_f = 0. The results are presented Fig. 4(a1) and are given in units of πħ.
• S_NAG(0,0): 10,000 smallest NAG values of paths under boundary conditions x_i = 0 and x_f = 0. The results are presented Fig. 4(a1) and are given in units of πħ/δx.

File: fig4a2.csv

Description: Experimental NAG spectra for an impulsive potential with boundary conditions x_i = -7 and x_f = 5.

Variables

• S(-7,5): Measured actions of 10,000 paths with the smallest norm of action gradient (NAG) under boundary conditions x_i = -7 and x_f = 5. The results are presented Fig. 4(2) and are given in units of πħ.
• S_NAG(-7,5): 10,000 smallest NAG values of paths under boundary conditions x_i = -7 and x_f = 5. The results are presented Fig. 4(a2) and are given in units of πħ/δx.

File: fig4b1.csv

Description: Corresponding classical paths derived from the data in Fig.4(a1).

Variables

• t: Evolution time, in units of ϵ= 15 mm/c, is represented as the horizontal axis in Fig. 4(b1).
• xcl_th(0,0)-s1: Theoretical result for the first classical path solution under boundary conditions x_i = 0, x_f = 0, shown as the lower blue solid line in Fig. 4(b1) in units of δx = 5.73 μm.
• xcl_NAG(0,0)-R1: Identified path from region R1 of the NAG spectrum under the boundary conditions x_i = 0, x_f = 0, presented as green squares in Fig. 4(b1) with the results given in units of δx = 5.73 μm.
• xcl_th(0,0)-s2: Theoretical result for the second classical path solution under boundary conditions x_i = 0, x_f = 0, shown as the upper blue solid line in Fig. 4(b1) in units of δx = 5.73 μm.
• xcl_NAG(0,0)-R2: Identified path from region R2 of the NAG spectrum under the boundary conditions x_i = 0, x_f = 0, presented as purple triangles in Fig. 4(b1) with the results given in units of δx = 5.73 μm.
• xcl_NAG(0,0)-R3_1: The first identified path from region R3 of the NAG spectrum under the boundary conditions x_i = 0, x_f = 0, presented as pink crosses in Fig. 4(b1) with the results given in units of δx = 5.73 μm.
• xcl_NAG(0,0)-R3_2: The second identified path from region R3 of the NAG spectrum under the boundary conditions x_i = 0, x_f = 0, presented as pink diagonal crosses in Fig. 4(b1) with the results given in units of δx = 5.73 μm.
• xcl_Smin(0,0)-M: Mean results for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=0, x_f=0. The results are given in units of δx=5.73μm and are presented as orange circles in Fig. 4(b1).
• xcl_Smin(0,0)-SD: Standard deviations for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=0, x_f=0. The results are given in units of δx=5.73μm and are presented as orange error bars in Fig. 4(b1).

File: fig4b2.csv

Description: Corresponding classical paths derived from the data in Fig.4(a2).

Variables

• t: Evolution time, in units of ϵ= 15 mm/c, is represented as the horizontal axis in Fig. 4(b2).
• xcl_th(-7,5)-s1: Theoretical result for the first classical path solution under boundary conditions x_i = -7, x_f = 5, shown as the lower blue solid line in Fig. 4(b2) in units of δx = 5.73 μm.
• xcl_NAG(-7,5)-R1_1: The first identified path from region R1 of the NAG spectrum under the boundary conditions x_i = -7, x_f = 5, presented as green squares in Fig. 4(b2) with the results given in units of δx = 5.73 μm.
• xcl_NAG(-7,5)-R1_2: The second identified path from region R1 of the NAG spectrum under the boundary conditions x_i = -7, x_f = 5, presented as green diamonds in Fig. 4(b2) with the results given in units of δx = 5.73 μm.

• xcl_th(-7,5)-s2: Theoretical result for the second classical path solution under boundary conditions x_i = -7, x_f = 5, shown as the upper blue solid line in Fig. 4(b2) in units of δx = 5.73 μm.

• xcl_NAG(-7,5)-R2_1: The first identified path from region R2 of the NAG spectrum under the boundary conditions x_i = -7, x_f = 5, presented as upward triangles in Fig. 4(b2) with the results given in units of δx = 5.73 μm.
• xcl_NAG(-7,5)-R2_2: The second identified path from region R2 of the NAG spectrum under the boundary conditions x_i = -7, x_f = 5, presented as downward triangles in Fig. 4(b2) with the results given in units of δx = 5.73 μm.
• xcl_NAG(-7,5)-R3_1: The first identified path from region R3 of the NAG spectrum under the boundary conditions x_i = -7, x_f = 5, presented as pink crosses in Fig. 4(b2) with the results given in units of δx = 5.73 μm.

• xcl_NAG(-7,5)-R3_2: The second identified path from region R3 of the NAG spectrum under the boundary conditions x_i = -7, x_f = 5, presented as pink diagonal crosses in Fig. 4(b2) with the results given in units of δx = 5.73 μm.
• xcl_Smin(-7,5)-M: Mean results for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=-7, x_f=5. The results are given in units of δx=5.73μm and are presented as orange circles in Fig. 4(b2).
• xcl_Smin(-7,5)-SD: Standard deviations for paths with measured action within the range [S_min,S_min+ε], under the boundary conditions x_i=-7, x_f=5. The results are given in units of δx=5.73μm and are presented as orange error bars in Fig. 4(b2).

File: figS1.csv

Description: Experimental produced potential.

Variables

• x: Position at moment t_m, in units of δx=5.73μm, is represented as the horizontal axis in Fig. S1.

• V_m_th: Theoretical potential curve in units of πħ, is represented as solid lines in Fig. S1.
• V_m_exp: Measured results of potential curve in units of πħ, is represented as circles with dashed lines in Fig. S1.

File: figS2(a1).xlsx

Description: The first column and the first row of the table represent the coordinates x_k and y_k, respectively, in units of δx. The remaining values indicate the grayscale values of the grayscale image obtained by projecting onto the polarization basis |D⟩.

File: figS2(a2).xlsx

Description: The first column and the first row of the table represent the coordinates x_k and y_k, respectively, in units of δx. The remaining values indicate the grayscale values of the grayscale image obtained by projecting onto the polarization basis |A⟩.

File: figS2(a3).xlsx

Description: The first column and the first row of the table represent the coordinates x_k and y_k, respectively, in units of δx. The remaining values indicate the grayscale values of the grayscale image obtained by projecting onto the polarization basis |R⟩.

File: figS2(a4).xlsx

Description: The first column and the first row of the table represent the coordinates x_k and y_k, respectively, in units of δx. The remaining values indicate the grayscale values of the grayscale image obtained by projecting onto the polarization basis |L⟩.

File: figS2(b)-(d).csv

Description: Sample results from the action measurements.

Variables

• x_k: Position at moment t_k, in units of δx=5.73μm, is represented as the horizontal axis in Fig. S2(b)-(d).

• P_A: Projected intensities on polarization basis |A⟩ after integration along the y-direction. The results are presented Fig. 4(b2) and are given in arbitrary units.
• P_D: Projected intensities on polarization basis |D⟩ after integration along the y-direction. The results are presented Fig. 4(b1) and are given in arbitrary units.
• P_L: Projected intensities on polarization basis |L⟩ after integration along the y-direction. The results are presented Fig. 4(b4) and are given in arbitrary units.
• P_R: Projected intensities on polarization basis |R⟩ after integration along the y-direction. The results are presented Fig. 4(b3) and are given in arbitrary units.
• Sigma_x: Expectation values of operator σ_x. The results are presented Fig. 4(c1) and are given in arbitrary units.
• Sigma_y: Expectation values of operator σ_y. The results are presented Fig. 4(c1) and are given in arbitrary units.
• S(x_k): The measured incremental actions as a function of x_k. The results are presented Fig. 4(d) and are given in unit of πħ.