Interactive Macroscopic Quantum Tunneling Simulator: Josephson Junctions and Washboard Potential

Macroscopic Quantum Tunneling (MQMT)

Developed By: Ir. MD Nursyazwi

A Superconducting Phase State Simulation and Analysis Tool

I. Instructions on How To Use

This simulation is designed to visualize the quantum tunneling phenomenon of the superconducting phase difference (Phi) within a current-biased Josephson junction, modeled by the Washboard Potential.

1. Parameter Specification: Adjust the slider controls in Section II to define the Critical Current (Ic) and Bias Current (Ib). These parameters directly govern the barrier height (E_B) and thickness of the potential barrier.
2. System Initialization: Select the Start Observation button to commence the simulation. The phase state (represented by the particle) is initialized at the local potential minimum (Phi_min).
3. Observation: A Macroscopic Quantum Tunneling (MQMT) event is registered when the particle spontaneously penetrates the barrier and transitions to the next well, regardless of insufficient classical energy.
4. Rate Analysis: Observe the Quantum Tunneling Probability (Gamma_Q) in Section IV, which is derived from the WKB approximation and dictates the frequency of tunneling events.
5. Camera Control: Utilize the 3D visualization by dragging the mouse to rotate and using the scroll wheel to zoom for comprehensive analysis of the potential landscape.

II. Data Input: Critical Parameters

Critical Current (Ic) [μA]

51 μA

Bias Current (Ib) [μA]

49 μA

III. Graphical Simulation: Phase-Space Visualization

3D Legend

Phase State (Phi)

Barrier Region

Barrier Height (E_B)

Potential Energy U(Phi)

Controls: Drag to rotate, Scroll to zoom.

IV. Data Output: Quantifiable Results

Potential Barrier Height (E_B) 1.04 [Normalized Units]

Classical Thermal Escape Rate (Gamma_th) 0.0010 [Normalized Units]

Quantum Tunneling Probability (Gamma_Q) (WKB Approx.) 91.080 %

Observation Status Best Tunneling Setup Applied. Ready to run.

V. Graphs and Charts: The Washboard Potential (2D Projection)

The governing potential energy function is defined by the tilted washboard:
U(Phi) is proportional to: -E_j * cos(Phi) - Ib * Phi
(Where E_j is the Josephson Energy and Ib is the Bias Current.)

VI. Science Explanations: Macroscopic Quantum Coherence

Macroscopic Quantum Tunneling (MQMT) is the definitive evidence that quantum mechanics governs the dynamics of collective variables, such as the superconducting phase difference (Phi), despite the involvement of about $10^{15}$ electrons. This phenomenon allows the entire macroscopic system to penetrate an energy barrier via quantum means, not thermal activation.

The Washboard Potential and State Definition

The potential energy of a current-biased Josephson Junction takes the form of a periodic, tilted "washboard." The critical ratio Ib / Ic governs this potential:

• Ic (Critical Current): Scales the depth of the wells (related to the Josephson Energy, E_j).
• Ib (Bias Current): Introduces a linear tilt, reducing the barrier height (E_B) and width. MQMT is only observable when Ib / Ic is near 1, causing the barrier to become sufficiently thin and low.

Real-World Application in Quantum Technology

MQMT is not merely a theoretical curiosity; it is a critical operational factor in cryogenic superconducting circuits:

• Flux Qubits: In a superconducting quantum bit (qubit), the two distinct states, State 0 and State 1, correspond to different circulating currents. The transition between these states occurs via controlled MQMT, enabling the qubit to enter a superposition state. Controlling the tunneling rate is essential for maintaining quantum coherence.
• SQUIDs (Superconducting Quantum Interference Devices): These ultra-sensitive magnetometers are limited by internal noise sources. At extremely low temperatures, unwanted, spontaneous MQMT events serve as a primary source of intrinsic quantum noise, reducing the device's ultimate sensitivity.

VII. References: Key Scholarly Sources

1. T. Fulton, P. W. Anderson. "Junction Tunneling and Critical Supercurrents." Physical Review, 147(1):173–185, 1966. (Foundational work on Josephson junction physics and the potential model.)
2. A. O. Caldeira, A. J. Leggett. "Influence of Dissipation on Quantum Tunneling in Macroscopic Systems." Physical Review Letters, 46(4):211–214, 1981. (Established the theoretical framework for dissipative quantum mechanics applied to MQMT.)
3. J. Clarke, A. Braginski (Eds.). "The SQUID Handbook: Applications of SQUIDs and SQUID Systems." Wiley-VCH, 2004. (Detailed analysis of SQUID noise, including the limits imposed by Macroscopic Quantum Tunneling.)

VIII. Further Learning Resources and Related Educational Platforms

To support continued professional development and the integration of simulation tools into broader educational contexts, the following platforms and resources are provided for complementary study and training:

• The Transformative Role of Online Simulators in Modern Pedagogy

  An analytical article examining the integration of digital tools, such as advanced physics simulators, into contemporary educational frameworks to enhance accessibility and student engagement with complex scientific concepts.

• Certificate Courses for Professional Skill Enhancement

  A gateway to various certificate programs, offering formalized training in technical disciplines, including mathematics, engineering, and programming skills essential for developing and utilizing advanced scientific simulation environments.