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 ψ(x, t) = − ~ ∇2 + V (x, t) ψ(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, ψ(x, t). 2 ~ 2 I − 2m ∇ 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) = |ψ(x, t)| is the probability density, which represents the probability that the quantum system is located at particular point in space and time. R |ψ(x, t)|2dx = 1 Schr¨odingerEquation in an Rotating Frame of Reference
∂ 2 i ψ(x, t) = − ~ ∇2 − Ω · L + V (x, t) ψ(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) = ~ ∇ϕ(x) m and ϕ is the phase of the wave function. I I v(x)·dx = ~ ∇ϕ(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 ψ (t) = − x + y + z ψ (t) ∂t jkl 2 ∆x2 ∆y 2 ∆z2 jkl δ δ + iΩ (t) y z − z y ψ (t) x k 2∆z l 2∆y jkl δ δ + iΩ (t) z x − x z ψ (t) y l 2∆x j 2∆z jkl δy δx + iΩz (t) xj − yk ψjkl (t) 2∆y 2∆x 2 + Vjkl (t)ψjkl (t) + gs |ψjkl (t)| ψjkl (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. units ) Numerical Verification of Solutions
1000 Variational Approximation Thomas-Fermi Approximation λ = 1 : Before ITP @ t = 0 λ = 2 @ t = 0 100 λ = 3 @ t = 0
(osc. units) λ = 4 @ t = 0 µ λ = 5 @ t = 0 λ = 1 : After ITP @ t = 50 λ = 2 @ t = 50 10 λ = 3 @ t = 50 λ = 4 @ t = 50 λ = 5 @ t = 50 Chemical Potential, GRK4 : λ + CD2 -real-size 64
Axisymmetric SHO : nr = 0 , ml = 0 , nz = 0 κr = 1.0, κz = 1.0 1 1 10 100 1000 10000
Mean-Field Coupling Constant, gs (osc. units) 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. new-load.mp4 Simple Harmonic Oscillator Ring Potential
1 1 V (r, θ, z) = κ2(r − r )2 + κ2z2 SHOR 2 r o 2 z Non-Uniformly Rotating Reference Frame
∆β σT Ω(t) = R [β(t)] Ω β(t) = coth tanh [σ (t − τ )] + 1 y 0 2 2 σ Preliminary Results: Angular Momentum vs. Time
κr = κz = 1.0; Ω0 = 0.2ˆz, ml = 0, gs = 0 Preliminary Results: Angular Momentum vs. Time
κr = κz = 1.0; Ω0 = 0.5ˆz, ml = 0, gs = 0 Preliminary Results: Angular Momentum vs. Time
κr = κz = 1.0; Ω0 = 0.5ˆz, ml = 1, gs = 0 Preliminary Results: Quantized Vortices?
movies/psi2xy.mp4 movies/psi2xz.mp4 XSEDE Allocation (Awarded Fall 2016) Compute Resources by the Numbers
Resource Requested SUs Allocated SUs Spent SUs Use Case Comet 1.5 M 503 K 389 K Simulations on the dynamics of a quantum Compton generator 300 K Imaginary time propagation calculations; initial condition generation 50 K Parallel I/O development and testing 25 K GRK4+CD4 verification and parallel scalability tests 25 K CPU + GPU hybridization; MPI + CUDA development and testing Stampede(2) 50 K 51 (1.6) K 51 (1.6) K Parallel I/O development and testing 25 K GRK4+CD4 verification and parallel scalability tests 25 K GRK4+CD4 verification and parallel scalability tests with MIC offloading Maverick 375 K 375K 321 K Visualization and analysis of simulation data via Paraview Storage Resources by the Numbers
Resource Requested GB Allocated GB Use Case Data Oasis 32 K 13 K Intermediate-term storage for data awaiting post-processing on Maverick Ranch 500 500 Archival storage for visualization files produced via Paraview on Maverick Latest Results: Angular Momentum vs. Time
〈 Lx 〉 0.2 〈 Ly 〉 〈 Lz 〉
0.1 ( osc. units ) 〉
i psi2-xyz.mp4 L 〈 0 psi2-xzy.mp4
-0.1 psi2-phi-v-xyz.mp4 Angular Momentum, -0.2
0 5 10 15 20 25 30 Time, t ( osc. units )
κr = 1.0, κz = 1.0; Ω0 = 0.5ˆz, ml = 0, gs = 0 Latest Results: Angular Momentum vs. Time
〈 Lx 〉 0.2 〈 Ly 〉 〈 Lz 〉
0.1 ( osc. units ) 〉
i psi2-xyz.mp4 L 〈 0 psi2-xzy.mp4
-0.1 psi2-phi-v-xyz.mp4 Angular Momentum, -0.2
0 5 10 15 20 25 30 Time, t ( osc. units )
κr = 1.0, κz = 2.0; Ω0 = 0.5ˆz, ml = 0, gs = 0 Latest Results: Angular Momentum vs. Time
〈 Lx 〉 0.2 〈 Ly 〉 〈 Lz 〉
0.1 ( osc. units ) 〉
i psi2-xyz.mp4 L 〈 0 psi2-xzy.mp4
-0.1 psi2-phi-v-xyz.mp4 Angular Momentum, -0.2
0 5 10 15 20 25 30 Time, t ( osc. units )
κr = 1.0, κz = 5.0; Ω0 = 0.5ˆz, ml = 0, gs = 0 Latest Results: Angular Momentum vs. Time
〈 Lx 〉 0.2 〈 Ly 〉 〈 Lz 〉
0.1 ( osc. units ) 〉
i psi2-xyz.mp4 L 〈 0 psi2-xzy.mp4
-0.1 psi2-phi-v-xyz.mp4 Angular Momentum, -0.2
0 5 10 15 20 25 30 Time, t ( osc. units )
κr = 1.0, κz = 10.0; Ω0 = 0.5ˆz, ml = 0, gs = 0 Summary and Future Work
I Observe nucleation of quantized vortices
I Drift velocity does not (yet) match classical R I Compute probability currents, I = S j · dσ
I Compute average velocities, j = ρv
I Parameter sweep in R (and Ω)
I Nonlinear dynamics of BEC-based Compton generator Acknowledgments
Dr. Ricardo Carretero-Gonz´alez Department of Mathematics and Statistics San Diego State University
Dr. Michael W. J. Bromley Centre for Quantum-Atom Optics School of Mathematics and Physics The University of Queensland, Australia