Exploring the Dynamics of a Quantum-Mechanical Compton Generator Marty Kandes High Performance Computing User Services Group San Diego Supercomputer Center University of California, San Diego XSEDE ECSS Symposium Tuesday, October 15th, 2019 Classical Compton Generator In 1913, while he was still an undergraduate, American physicist Arthur Compton invented a simple way to measure the rotation rate of the Earth with a tabletop-sized experiment, independent of any astronomical observation. Arthur Compton (1927). Image Credit: Wikipedia. Classical Compton Generator The experiment consisted of a large diameter circular ring of thin glass tubing filled with water and oil droplets. A. H. Compton. Science, 37(960):803806, 1913. A. H. Compton. Sci. Am. Supp., (2047):196-197, 1915. Classical Compton Generator After placing the ring in a plane perpendicular to the surface of the Earth and allowing the fluid mixture of oil and water to come to rest, Compton then abruptly rotated the ring, flipping it 180◦ about an axis passing through its own plane. Classical Compton Generator The result of the experiment was that the water acquired a measurable drift velocity due to the Coriolis effect arising from the daily rotation of the Earth about its own axis. v = 2RΩE sin λ, where R is the radius of the ring, ΩE is the rotation rate of the Earth, and λ is the latitude of the experiment. Compton measured this induced drift A. H. Compton. Sci. Am. Supp., (2047):196-197, 1915. velocity by observing the motion of the oil droplets in the water with a microscope. Quantum Compton Generator Classical drift velocity should hold in the correspondence limit. hvi = 2RΩE sin λ Angular momentum about z-axis after \Compton flip”: 2 hLz i = mRhvi = 2mR ΩE sin λ −25 For hLz i = ~, if m ≈ 10 kg, −4 ◦ ΩE ≈ 10 rad/s, and λ = 45 , then R ≈ 10−3 m. Inertial Forces P F a = ext inertial m P F P F a = ext + fict non-inertial m m Image Credit: D. Davis, Eastern Illinois University An inertial (or “ficticious”) force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as an accelerating or rotating reference frame. Centrifugal Force Fcentrifugal = −mΩ × (Ω × x) I The centrifugal force is an inertial force which appears to act on all objects in a rotating reference frame that is directed outward from the axis of rotation. I Examples: merry-go-rounds; roller coasters; turning your car around a bend; uranium enrichment Image Credit: B. Lawdermilk, Getty Images Coriolis Force FCoriolis = −2mΩ × v I The Coriolis force is an inertial force which appears to act on all moving objects in a rotating reference frame that manifests itself as an apparent deflection of an objects trajectory. I Examples: ballistic trajectories; flight paths; hurricanes; ocean currents Image Credit: M. Trenchard, NASA Euler Force dΩ F = −m × x Euler dt I The Euler force is an inertial force which appears to act on all objects in a non-uniformly rotating reference frame due to the change in angular velocity of the frame. I Examples: merry-go-round; spinning tops; gyroscopes Image Credit: The Bucket Problem Schr¨odingerEquation in an Inertial Frame of Reference @ 2 i (x; t) = − ~ r2 + V (x; t) (x; t) ~@t 2m ext I A linear (complex-valued) partial differential equation that describes the time-dependent evolution of the wavefunction of a quantum-mechanical system, (x; t). 2 ~ 2 I − 2m r represents the kinetic energy of the system. I Vext(x; t) represents the potential energy of the system that arises due to the influence of a time-dependent external potential that may be acting on it. 2 I ρ(x; t) = j (x; t)j is the probability density, which represents the probability that the quantum system is located at particular point in space and time. R j (x; t)j2dx = 1 Schr¨odingerEquation in an Rotating Frame of Reference @ 2 i (x; t) = − ~ r2 − Ω · L + V (x; t) (x; t) ~@t 2m ext I Ω is the angular velocity vector that defines the magnitude and orientation of the reference frame's rotation with respect to a fixed inertial frame. I L is the (orbital) angular momentum operator. I Inertial forces arise from an effective coupling between the rotational motion (Ω) of the reference frame and the angular momentum (L) of the quantum-mechanical particle. Rotation and Quantized Vortices Motion of a quantum-mechanical particle is \more restricted" than a classical fluid. Velocity field of the particle's probability density must be irrotational. ∇×v(x) = 0 where v(x) = ~ r'(x) m and ' is the phase of the wave function. I I v(x)·dx = ~ r'(x)·dx = ~ n C m C m Vortex Lattice in Bose-Einstein Condensate. Image Credit: Cornell Group. JILA/NIST/Boulder. gpse [jip-see] I Written in Fortran I O(5K) Lines of code I GRK4 + CDX algorithm I Hybrid MPI + OpenMP I 1D Domain Decomposition on a Regular 3D Grid I Limited I/O: hOi I Serial I/O: hOi + VTK I Parallel I/O: hOi + VTK hOi + MPI-I/O I Imaginary Time Propagation https://github.com/mkandes/gpse Generalized 4th-Order Runge-Kutta dy = f (y; t) y(t ) = y dt 0 0 ∆t y = y + [k + (4 − λ) k + λk + k ] n+1 n 6 1 2 3 4 k1 = f (yn; tn) ∆t ∆t k = f y + k ; t + 2 n 2 1 n 2 1 1 ∆t ∆t k = f y + − ∆tk + k ; t + 3 n 2 λ 1 λ 2 n 2 λ λ∆t k = f y + 1 − ∆tk + k ; t + ∆t 4 n 2 2 2 3 n GRK4+CDX ! @ 1 δ2 δ2 δ2 i (t) = − x + y + z (t) @t jkl 2 ∆x2 ∆y 2 ∆z2 jkl δ δ + iΩ (t) y z − z y (t) x k 2∆z l 2∆y jkl δ δ + iΩ (t) z x − x z (t) y l 2∆x j 2∆z jkl δy δx + iΩz (t) xj − yk jkl (t) 2∆y 2∆x 2 + Vjkl (t) jkl (t) + gs j jkl (t)j jkl (t) 1D (Slab) Domain Decomposition 3 2 z 1 0 y x Two-Round MPI Communication Pattern (per ∆t) SEND SEND SEND NO SEND 0 1 2 3 P−2 P−1 P RECV RECV RECV SEND SEND SEND 0 2 P−2 0 1 2 3 P−2 P−1 P NO RECV RECV RECV RECV NO SEND SEND SEND SEND 0 1 2 P−3 P−2 P−1 0 1 2 3 P−2 P−1 P RECV RECV RECV SEND SEND SEND 02 1 2 4 P−3 P−2P P−1 0 1 2 3 P−2 P−1 P RECV RECV RECV NO RECV Strong Parallel Scalability 400 Ideal Linear Speedup = Ts T1 350 1 MPI x 24 OpenMP 2 MPI x 12 OpenMP 4 MPI x 6 OpenMP fsTT1 fp 1 300 Amdahl: fp = 0.997, S∞ = 333 SERIAL PARALLELIZABLE 250 P = 1 p 200 Speedup, S P = 2 150 100 50 GRK4 : λ = 2 + CD2 -real-size 64 MPI + OpenMP + LIMITED I/O P = 4 0 0 200 400 600 800 1000 1200 1400 1600 Number of CPU cores, P 1 Ideal Efficiency 1 MPI x 24 OpenMP 0.8 2 MPI x 12 OpenMP 4 MPI x 6 OpenMP p Amdahl: f = 0.997, S = 333 P = 8 p ∞ 0.6 GRK4 : λ = 2 + CD2 -real-size 64 MPI + OpenMP + LIMITED I/O Efficiency, E 0.4 0.2 fpT1 / P T p 0 200 400 600 800 1000 1200 1400 1600 Number of CPU cores, P Weak Parallel Scalability Ideal Speedup 1024 Dugong: 4 MPI x 8 OpenMP Gustafson: f * = 0.333 T1 s 256 fsT1 fpT1 SERIAL PARALLELIZABLE p 64 = + P = 1 Ts fsT1 fpT1 Speedup, S 16 = + P = 2 Ts fsT1 2fpT1 4 GRK4 : λ = 2 + CD2 -real-size 64 MPI + OpenMP + LIMITED I/O P = 4 Ts = fsT1 + 4fpT 1 1 1 4 16 64 256 1024 Number of CPU cores, P 1 0.8 P = 8 Ts=fsT1 + 8 fpT 1 p 0.6 Efficiency, E 0.4 Ideal Efficiency 0.2 Dugong: 4 MPI x 8 OpenMP Tp α Gustafson: = 0.333 GRK4 : λ = 2 + CD2 -real-size 64 MPI + OpenMP + LIMITED I/O 0 1 4 16 64 256 1024 Number of CPU cores, P Numerical Verification of Temporal Order-of-Accuracy 0.001 ∆t4 GRK4 : λ = 2 + CD2 -real-size 64 Ω = Ω = 0.03536, Ω = 0.0866 0.0001 t = 1 x y z t = 2 gs = 1000.0 2 || N = N = N = 512 2 t = 5 x y z 1e-05 ∆x = ∆y = ∆z = 0.03125 t = 10 Ψ - 2 t 1e-06 ∆ x, ∆ 1e-07 Ψ 1e-08 1e-09 Density Error, || Axisymmetric SHO : nr = 0 , ml = 1 , nz = 0 1e-10 κr = 1.0, κz = 1.0 xo = yo = 1.0, xo = 0.0 1e-11 1e-05 0.0001 0.001 Interval of a Time Step, ∆t ( osc. units ) Numerical Verification of Spatial Order-of-Accuracy ) z 0.01 N 2 y ∆x GRK4 : λ = 2 + CD2 -real-size 64 N x t = 1 Ωx = Ωy = 0.03536, Ωz = 0.0866 /(N 2 0.001 t = 2 g = 1000.0 || s 2 t = 5 Nt = 800000 Ψ - t = 10 ∆t = 0.0000125 2 0.0001 t ∆ x, ∆ Ψ 1e-05 1e-06 Axisymmetric SHO : nr = 0 , ml = 1 , nz = 0 1e-07 κr = 1.0, κz = 1.0 xo = yo = 1.0, xo = 0.0 Normalized Density Error, || 1e-08 0.01 0.1 1 10 Distance Between Grid Points , ∆x ( osc.