Circuit Field Theory as a Research Platform for Modeling Quantum Systems Across Scales
From Capacitive Sensing to Cosmic Oscillations — Circuit QED as a “Miniature Resonant Universe”
Circuit QED functions as a "miniaturized cosmic resonant cavity" where we can construct arrays of inductors (L) and capacitors (C) in the laboratory, using these components to create controllable resonant modes. Surprisingly, the mathematical descriptions of these resonant modes exhibit remarkable consistency with the mathematical structures of quantum fields in the universe, displaying profound formal correspondences from field equations to frequency modes.
- Voltage fields (V) correspond to quantum fields (φ)
- Current fields (I) correspond to the canonical conjugate momentum of fields (π)
- Resonant modes in LC circuit arrays correspond to excitation modes of quantum fields
This means we can use circuits to simulate universal behaviors. These are not fantasies but the foundations of quantum technology that have already been realized. Superconducting quantum circuits, microwave resonant cavities, qubits—they are all components of this "scaled universe."
This "scaled universe" has already demonstrated many astonishing quantum phenomena, including quantum phase transitions, quantum entanglement, and quantum non-locality. By introducing quantum computing concepts, we can not only simulate the behavior of these fundamental fields but also perform quantum information processing, transforming profound theoretical concepts into measurable and controllable systems.
The Underrated Microcosm of the Universe — Rethinking the Capacitor
Introduction: Water Flow, Voltage, and the Quantum Ocean
• How capacitors describe changes in water level and flow rate
• Why electrical circuits can serve as a visual and intuitive model for understanding quantum fields
• Purpose of this paper: to map from classical capacitors step by step to quantum field theory and superconducting quantum circuits
Section 1: Capacitors and Electric Field Waves through the Eyes of Fourier
• Review of the capacitor relationship: 𝐼(𝑡) = 𝐶 ⋅ 𝑑𝑉(𝑡)/𝑑𝑡
• Reservoir analogy (voltage as water level, current as water flow)
• How Fourier transforms convert complex water level variations into single-frequency waves
• Euler’s formula: 𝑒^{𝑗𝜃} describes wave phase and direction
• Leading by 90 degrees: why current precedes voltage
"The current through a capacitor can be viewed as 'measuring' the future trend of voltage."
Section 2: Impedance in the Frequency Domain vs. Energy Eigenstates
• Meaning of capacitive impedance: 𝑍_C = 1⁄(𝑗𝜔𝐶)
• 𝜔 corresponds to oscillation frequency, the “rhythm” of energy
• Introducing Planck relation: 𝐸 = ℏ𝜔, establishing frequency-energy mapping
• Frequency response of circuits ≈ mode expansion of quantum fields
• Capacitor’s response to different frequencies resembles field’s response to different energies
Section 3: Full Comparison — Circuit Fields vs. Quantum Fields
• Include full comparison table (can be visualized as a chart)
• Voltage/Current vs. Field values/Field derivatives
• LC oscillator vs. Resonant modes of a field
• Noise vs. Vacuum fluctuations
• Phase difference vs. Wavefunction interference
• Conclusion: Circuits are actually “classicalized” field models
"Circuit theory can be viewed as the classical limit or low-energy approximation of quantum field theory. When we study circuit behavior, we are actually observing the averaged results of numerous quantum effects. Circuit components provide us with a practical way to manipulate and utilize these field behaviors without directly dealing with complex quantum theory."
Section 4: From Analogy to Experiment — The Birth of Quantum Circuits
Traditional circuit elements exhibit entirely new behaviors when entering the quantum domain. At the core of this transformation is the Josephson junction, which consists of two superconductors separated by a thin insulating layer.
• Josephson junctions and quantum tunneling: making inductors/capacitors quantum devices
• LC circuits → superconducting qubits (e.g., transmon)
• Quantum energy levels in a capacitor: 𝐸ₙ ≠ (𝑛 + ½)ℏ𝜔
• Microwave cavities and field quantization: cQED realizes particle–field interactions
• Operating quantum fields: how microwave waveguides excite field modes, creating/annihilating photons
Quantum circuit experiments demonstrate an astonishing fact: theories originally developed to describe macroscopic circuit behavior, after appropriate quantization, can precisely describe microscopic quantum systems."
This translation captures the key insight about how classical circuit theory, when properly quantized, successfully describes quantum behavior - highlighting the remarkable connection between macroscopic electrical engineering concepts and microscopic quantum phenomena.
Section 5: Is the Universe a Quantum Resonator?
• Are cosmic fields (like the Higgs field, EM field) analogous to a gigantic circuit?
• Dark energy, vacuum energy density and the similarity to zero-point energy in superconducting circuits
• Extending the LC model as an intuitive framework for cosmic field models
• “The universe is a resonant space of quantum fields” — valid both metaphorically and theoretically
From the perspective of theoretical physics, this analogy has a rigorous mathematical foundation:
Quantum field theory describes the possible modes of vibration of fields in the universe.
Field equations (such as the Klein-Gordon equation) are analogous to wave equations.
The process of field quantization is highly similar to the quantization process in electrical circuits.
Conclusion: A Capacitor Is Not Just an Energy Storage Device — It’s the Starting Point for Understanding the Universe
• Translating circuit language into quantum field language
• Inspiration for education, science communication, and cross-disciplinary innovation
• A call for building more bridges between engineering and theoretical physics
Appendix & Figures
• [Figure 1] Capacitor: 𝑉(𝑡), 𝐼(𝑡) — Time Domain vs. Frequency Domain Waveforms
• [Figure 2] Fourier Decomposition of Water-Level Waves and Frequency Components
• [Figure 3] Euler’s Formula and Phase Rotation Animation: 𝑒^{𝑗𝜃}
• [Figure 4] LC Resonance vs. Field Mode Energy Distribution
• [Figure 5] Complete Comparison Diagram: Quantum Fields vs. Circuit Fields (Tabular Form)
• [Figure 6] Superconducting Qubit and Quantum Energy Level Structure Diagram
Python Script:
The Six_Plots_Quantum_Circuit.py script for visualizing the principles of circuit field theory has been published on GitHub:
https://github.com/fullyloaded/Python-Code
A Capacitor Is Not Just an Energy Storage Device — It’s the Starting Point for Understanding the Universe
In many engineering textbooks, a capacitor is defined as a component that “stores electrical energy,” holding energy in the form of an electric field between two plates. Yet, this definition only scratches the surface of its function. When we re-express the behavior of a capacitor through the lens of Fourier transforms and Euler’s formula, we begin to perceive that it is not just about energy storage — it is about sensing change: it responds to the rate of change of voltage, the derivative of future trends. In the language of physics, it is no longer merely a container of energy, but a sensor of time-varying fields.
More profoundly, when we translate circuits made of capacitors and inductors into the frequency domain, and link them with the quantum mechanical relation 𝐸 = ℏ𝜔, the circuit ceases to be a mere network of voltages and currents. Instead, it becomes a metaphorical space of quantum field oscillations. We are astonished to realize that the essence of a capacitor’s behavior is a differential response to a field. It is not merely a human-made part, but a physical manifestation of “field and change,” the fundamental structure of the universe.
In superconducting quantum circuits, a capacitor is no longer just a component — it becomes part of a quantum resonator, within which phenomena akin to vacuum fluctuations of the universe can be observed. When we manipulate these circuits in the laboratory, we are, in miniature, manipulating quantum fields themselves. The quantized energy levels of a capacitor, its selectivity toward frequencies, and its coupling to microwave fields, all philosophically echo our imagination of the Higgs field, the photon field, and even dark energy.
Thus, we may say that a capacitor is not merely a container of energy, but a microcosm of cosmic structure. From it, we do not merely learn electronics — we gain deep intuitions about fields, frequencies, energy, and states of being. The capacitor might be one of the most accessible yet underestimated “cosmic microcosms” in our everyday lives.
Quantum Field vs Electrical Circuit Field: Core Analogy Table
|
Conceptual Category |
Classical Circuit Theory |
Quantum Field Theory (QFT) |
Analogy Explanation |
|
Nature of Fields |
Voltage field 𝑉(𝑥,𝑡), Current field 𝐼(𝑥,𝑡) |
Field operators 𝜙(𝑥,𝑡), 𝜓(𝑥,𝑡) (e.g., scalar, fermion fields) |
Both describe spatial-temporal distributions of field values |
|
Excitations & Particles |
Signal spikes in voltage/current |
Particles as field excitations (e.g., photons, electrons) |
Current pulses ≈ Particle excitations; carriers of energy |
|
Energy & Frequency |
Energy-frequency link is implicit (unless in AC analysis) |
𝐸 = ℏ·𝜔 (Planck relation) |
In frequency domain, both use frequency to encode energy |
|
Fourier Transform |
Converts signals to frequency domain |
Expands fields into momentum eigenstates (Fourier modes) |
Fourier = bridge: time ↔ frequency, space ↔ momentum |
|
Phase & Interference |
Euler’s formula 𝑒^{𝑗𝜃} describes phase difference (e.g., 90° lead) |
Wavefunction phase 𝑒^{𝑖𝑆}, operator interference |
Phase governs interference and superposition in both realms |
|
Uncertainty |
No explicit uncertainty; but frequency vs amplitude has trade-off |
Δ𝐸·Δ𝑡 ≥ ℏ⁄2, Δ𝑥·Δ𝑝 ≥ ℏ⁄2 (uncertainty relations) |
Frequency-time resolution is bounded by Fourier-type inequalities |
|
Noise & Vacuum Fluctuations |
Thermal noise (white noise, 1⁄𝑓 noise) |
Vacuum fluctuations (virtual particle pairs) |
Circuit noise ≈ Quantum field fluctuations, especially in cryogenic QED circuits |
|
Discretization & Quantization |
Sampling and quantizing signals in simulation |
Fields are quantized; excitations are particles, operators obey commutation rules |
Digital signals ≈ Discretized quantum fields |
|
Propagation |
EM wave propagation in transmission lines, waveguides |
Particles (e.g., photons) propagate via fields |
Wavepackets carry energy; exhibit wave and interference behaviors |
|
Units & Constants |
Capacitance 𝐶, Inductance 𝐿, Impedance 𝑍 = 1⁄𝑗𝜔𝐶 |
Planck constant ℏ, speed of light 𝑐, mass 𝑚 |
𝐶, 𝐿 are classical response constants; ℏ governs quantum response |
Deepening Perspective: Revealing the Core Nature of Quantum Fields Through the Capacitor
Let us take a closer look at the differential relationship governing a capacitor:
• I(t) = C · dV(t)/dt
After applying the Fourier transform:
• I(ω) = jωC · V(ω)
Here, jω represents sensitivity to frequency variation, and C acts like a field response constant. This strongly resembles how, in quantum field theory:
• The time derivative of φ(x, t) corresponds to the energy operator;
• The spatial derivative corresponds to the momentum operator.
Just as electric current responds to the rate of change in voltage, particle excitations in a quantum field arise from variations in the field values over time and space.
From Electron Fields to Photon Fields: A Broader Generalization
In a typical LC oscillator, the natural frequency is given by:
• ω₀ = 1/√(LC)
This mirrors the concept of mode frequency in quantum fields, where:
• E = ℏ·ω₀
Here, inductance L and capacitance C form an effective structure analogous to mass and elasticity—precisely the core components found in the Lagrangian density of a field theory (i.e., kinetic term minus potential term). It implies that an electrical circuit is, in essence, “oscillating” a field.
Advanced Outlook: Superconducting Quantum Circuits as Experimental Quantum Fields
In superconducting quantum circuits (e.g., the Transmon qubit), the interplay between capacitors and Josephson junctions creates a concrete, tunable quantum field system. This enables:
• Observable quantization of the field (with non-equidistant energy levels);
• Quantum interference of microwave fields;
• The creation of artificial atoms and realization of circuit quantum electrodynamics (cQED).
These systems effectively materialize quantum field theory into tangible laboratory platforms, bridging theoretical constructs with controlled physical implementation.
Note: Capacitor’s Voltage-Current Relationship and Its Frequency-Domain Interpretation
In circuit theory, the time-domain relationship between current and voltage in a capacitor is given by:
• I(t) = C · dV(t)/dt
Let’s apply Fourier transform and Euler’s formula to analyze this.
The Fourier transform converts a time-domain signal into its frequency-domain representation:
• V(ω) = 𝔽{V(t)} = ∫₋∞⁺∞ V(t) · e⁻ʲωt dt
There is a key property for the derivative in the Fourier domain:
• 𝔽{dV(t)/dt} = jω · V(ω)
So, the capacitor’s relationship in the frequency domain becomes:
• I(ω) = C · jω · V(ω)
Using Euler’s formula:
• e^{jθ} = cos(θ) + j·sin(θ)
We can represent the capacitor’s impedance as:
• Z₍c₎(ω) = V(ω)/I(ω) = 1/(jωC)
• Applying Euler’s formula further:
Z₍c₎(ω) = −j/(ωC) = (1/ωC) · e⁻ʲπ⁄2
This shows that the magnitude of the impedance is:
• |Z₍c₎(ω)| = 1/(ωC)
And the phase angle is:
• −90° or −π⁄2
This means that in a sinusoidal AC circuit, the current leads the voltage by 90° in a capacitor.
Analogy: Reservoir and Water Flow
• Reservoir width (Capacitance C): Determines how much water the reservoir can store. A larger C is like a wider reservoir—more current (water flow) is needed for the same water level (voltage) change.
• Wave climbing the slope (Voltage change): As the water level rises, water flows into the reservoir—this mirrors how current I(t) responds to the rate of change of voltage dV(t)/dt. Current leads voltage because the water flow responds to the speed of level change, just like current responds to dV/dt.
• Fourier transform: Breaks complex water level patterns into simple sinusoidal waves, allowing us to analyze the flow for each wave individually.
• Euler’s formula: Quantifies the “timing difference” (phase) between water flow and water level, confirming that current leads voltage by 90°.
Reframing the Capacitor as a Water Reservoir
• Voltage V(t) = Water level (height) of the reservoir, varying over time.
• Current I(t) = Rate of water flowing into or out of the reservoir.
• Capacitance C = Width of the reservoir (how sensitive the flow is to level changes).
• The capacitor equation tells us:
Water flow speed depends on the reservoir width and how quickly the water level changes (rising or falling).
Why Use Fourier and Euler?
If the water level V(t) varies in a complicated way (like waves or sudden floods), directly calculating the flow I(t) becomes difficult. This is where Fourier transform and Euler’s formula come in—they act like analytical tools that simplify complex signals into manageable components.
Fourier Transform: Decomposing Water Levels into Waves
Imagine the water level V(t) in a reservoir is not changing smoothly, but fluctuating like the surface of the sea—filled with waves of different sizes and frequencies (big waves, small waves, fast and slow oscillations). Looking at the raw water level signal might seem chaotic, but we can use a kind of “wave decomposition machine” — the Fourier transform.
• Role of the Fourier transform:
It breaks down a complex time-domain signal V(t) into a collection of simple sine waves, each with its own frequency ω (how fast the wave oscillates) and amplitude (wave height).
• It’s like decomposing a complex piece of music into individual notes, where each note has a pitch (frequency) and loudness (amplitude).
In the context of the reservoir analogy:
• The water level V(t) is transformed into V(ω) in the frequency domain, which tells us how strong the waves are at each frequency.
• The equation
I(t) = C · dV(t)/dt
becomes in the frequency domain:
I(ω) = jωC · V(ω)
This means:
For each wave component (at a given frequency ω), the water flow speed I(ω) depends on the wave’s frequency, the reservoir’s width C, and the wave’s amplitude V(ω).
Euler’s Formula: Wave “Direction” and “Rhythm”
Each sine wave not only has a size and frequency but also a phase—a kind of “rhythm” or “timing offset,” indicating whether the wave is moving forward, backward, or somewhere in between.
Euler’s formula:
e^(jθ) = cos(θ) + j·sin(θ)
acts like a wave rhythm meter, helping us describe the motion and direction of these wave components.
In a capacitor:
• The impedance is:
Z₍C₎ = V(ω)/I(ω) = 1/(jωC)
• Applying Euler’s identity:
1/j = e^(–jπ/2)
shows that the current wave leads the voltage wave by 90 degrees in phase.
Just like how water flow (current) reacts before the water level (voltage) visibly rises or falls—because flow is determined by the rate of change of the water level.
Reservoir Width: The Role of Capacitance (C)
Core Analogy: Capacitor = Reservoir; Capacitance C = Width (or storage capacity) of the reservoir
Analogy Explained:
• Imagine a reservoir where water level = voltage V(t), and water flow = current I(t).
• Capacitance C is like the width of the reservoir:
• A wide reservoir (large C) holds more water, so to raise the water level by the same amount, it requires more water (more current).
• A narrow reservoir (small C) fills up faster, so the water level changes more easily with less water (less current).
Equation Correspondence:
In time domain:
I(t) = C · dV(t)/dt
• The water flow (current) depends on how fast the water level (voltage) is rising or falling, multiplied by the reservoir’s width (C).
In frequency domain (via Fourier Transform):
I(ω) = jωC · V(ω)
• Each frequency component of the wave (voltage) generates a water flow (current) scaled by C.
• The larger the capacitance (wider reservoir), the greater the response to each wave.
From the impedance perspective (ease of water flow):
Z₍C₎ = 1 / (jωC)
• A larger C means lower impedance: it’s easier for current to flow through.
• Just like a wider reservoir doesn’t “resist” the wave — even small wave changes produce noticeable flow.
Wave Climbing: Phase Lead of Current over Voltage
Core Analogy: Rising water level = wave climbing; Current leads voltage = water flow rushing in before the peak
Analogy Explained:
• Suppose the voltage is a cosine wave: V(t) = cos(ωt). Its rate of change is: dV(t)/dt = –ω·sin(ωt).
• When the wave is climbing from trough to peak (0 → π/2), the water level is rising most rapidly — this is the “wave climbing” phase.
Current Response:
• According to the equation:
I(t) = C · dV(t)/dt = –ωC·sin(ωt)
• The current waveform I(t) leads the voltage waveform V(t) by 90° (π/2 radians), since sin(ωt) leads cos(ωt) by a quarter cycle.
Euler’s Formula: Quantifying the “Lead”:
• Euler’s Identity: e^(jθ) = cos(θ) + j·sin(θ)
• In frequency domain: Z₍C₎ = 1 / (jωC) = e^(–jπ/2) / (ωC)
• This shows that the current leads the voltage by π/2 radians (90°) — just like the water flow rushes in before the water level peaks.
Full Process Summary: From Chaotic Water Surface to Precise Water Flow
1. Chaotic water level V(t): Like a stormy sea, too messy to analyze directly.
2. Fourier Transform: Decomposes the chaos into clean, single-frequency waveforms V(ω).
3. Frequency Domain Equation I(ω) = jωC · V(ω): Computes the water flow for each frequency component based on wave slope (ω) and reservoir width (C).
4. Euler’s formula explains phase shift: Water flow (current) is ahead of water level (voltage), because it reacts to the change, not the value itself.
5. Inverse Fourier Transform: Adds up the water flows from each wave to recover the actual current I(t).
Why Does Current Lead Voltage in a Capacitor?
• In a sinusoidal wave, where the water level (voltage) is
V(t) = cos(ωt),
its rate of change is:
dV(t)/dt = –ω·sin(ωt) = ω·cos(ωt + π/2)
This means the current
I(t) = C·dV(t)/dt
has a waveform that leads the voltage V(t) by 90 degrees (π/2 radians).
Wave Analogy:
• When a wave begins to rise (water level starts climbing), the water flow is already rushing into the reservoir at full speed — this is the maximum current.
• When the water level reaches the peak of the wave (maximum height), the rate of change becomes zero — and so does the current.
• This visualizes current leading voltage by 90°: current reacts to how fast voltage changes, not to the voltage itself.
Euler’s Formula
e^{jθ} = cos(θ) + j·sin(θ)
This helps us quantify the phase lead between current and voltage using complex numbers.
In Frequency Domain, the Capacitor’s Impedance is:
Z_C = 1 / (jωC) = (1 / ωC) · e^(–jπ/2)
• The phase of –π/2 means the voltage lags the current by 90°, or equivalently, current leads voltage by 90°.
• It’s like saying that as the wave climbs, the water flow (current) is always “a step ahead” of the water level (voltage).
• Euler’s formula lets us turn this difference in timing into a measurable phase angle.
Visual Analogy:
Imagine you’re standing on the beach, watching the waves.
• As the wave begins to surge toward the shore (the upward slope), the water races across the sand even though the wave hasn’t yet reached its peak.
• When the wave crest finally arrives (maximum height), the water flow slows or stops.
• That’s how a capacitor behaves: the current (water flow) is most active while the voltage (water level) is still rising — always one step ahead.
Visualizing the Bridge Between Classical Circuits and Quantum Mechanics
To visualize the relationship between capacitor I–V dynamics, Fourier transforms, Euler’s formula, and Planck’s constant as the “metronome” of the quantum world, I propose the following visual plan:
1. Waveforms in a Classical Capacitor: Show voltage and current over time, highlighting the 90° phase difference (the rhythm between water level and flow).
2. Fourier Transform: Illustrate how time-domain waveforms are broken into frequency components — like decomposing wave motions into individual ripples.
3. Euler’s Formula: Represent phase differences using the complex plane — linking this idea to quantum wavefunction phase shifts.
4. Planck’s Constant as a Quantum Metronome: Show how it connects energy and frequency (E = ħω), and how this frequency defines the “tempo” of quantum systems.
Related Topic: Using Fourier Transforms, Euler’s Formula, and Planck’s Constant to Connect Circuit Theory and the Quantum World
Visualization Details and Python Script
Figure 1: Capacitor V(t) V(t) V(t), I(t) I(t) I(t) Time Domain vs. Frequency Domain Waveforms
• Description: Show the time-domain waveforms of voltage 𝑉(𝑡) = cos(𝜔𝑡) and current 𝐼(𝑡) = 𝐶𝜔 sin(𝜔𝑡), highlighting the 90° phase lead of current over voltage. Include frequency-domain amplitude spectra via Fourier transform, showing single-frequency peaks.
• Parameters:
• 𝜔 = 1 rad/s
• 𝐶 = 1 F
• Time range: 𝑡 ∈ [0, 4𝜋]
• Visuals:
• Time Domain: Plot 𝑉(𝑡) (blue solid line) and 𝐼(𝑡) (red dashed line), showing current leading voltage by 90°.
• Frequency Domain: Plot amplitude spectra of 𝑉(𝑡) and 𝐼(𝑡), with peaks at 𝜔 = 1 rad/s.
• Inset: Reservoir analogy showing water level (voltage) and flow (current) with phase difference.
• Output: voltage_current.png
Figure 2: Fourier Decomposition of Square Wave and Frequency Components
• Description: Display a square wave 𝑠(𝑡) in the time domain and its Fourier transform, emphasizing odd harmonics and linking to quantum energy levels.
• Parameters:
• Period 𝑇 = 4 s
• Frequencies: 𝑓 = 𝑛⁄𝑇, 𝑛 = 1, 3, 5, …
• Visuals:
• Time Domain: Plot square wave 𝑠(𝑡).
• Frequency Domain: Amplitude spectrum showing peaks at odd harmonics.
• Annotation: Note “Related to quantum energy levels (see Figure 6)” to connect to quantized modes.
• Output: square_wave_fft.png
Figure 3: Euler’s Formula and Phase Rotation Animation
• Description: Visualize Euler’s formula 𝑒^{𝑗𝜃} = cos(𝜃) + 𝑗 sin(𝜃) with rotating vectors in the complex plane.
• Vectors:
• Voltage (blue): cos(𝜔𝑡) + 𝑗 sin(𝜔𝑡), length 1
• Current (red dashed): sin(𝜔𝑡) + 𝑗 cos(𝜔𝑡), leading by 90°, length 1
• Quantum (cyan): Initial phase 45°, length 0.5
• Visuals:
• Animate vectors rotating on the unit circle at 20 fps, looping infinitely
• Inset: Reservoir analogy showing water level vs. flow phase difference
• Output: euler_animation.gif
Figure 4: LC Resonance vs. Field Mode Energy Distribution
• Description: Compare LC circuit resonance with quantum field mode energy levels.
• Parameters:
• 𝐿 = 1 𝐻, 𝐶 = 1 𝐹, resonance frequency 𝑓₀ = 1∕(2π√(𝐿𝐶)) ≈ 0.159 𝐻𝐳
• Quantum modes: 𝐸ₙ = 𝑛 ħ 𝜔, 𝑛 = 0, 1, 2, …
• ħ = 1.0545718 × 10⁻³⁴ 𝐽⋅𝑠
• Visuals:
• Top: LC frequency response (current amplitude vs. frequency)
• Bottom: Discrete quantum energy levels 𝐸ₙ
• Output: lc_resonance.png
Figure 5: Quantum Field vs. Circuit Field Comparison (Table)
• Description: Tabular comparison of circuit and quantum field concepts.
Figure 5: Quantum Field vs. Circuit Field Comparison (Table)
| Physical Quantity | Circuit Field | Quantum Field |
|---|---|---|
| Voltage | 𝑉(𝑡) = 𝑉₀ cos(𝜔𝑡) | ψ ∼ e^(−𝐸𝑡⁄ħ) |
| Current | 𝐼(𝑡) = 𝐶 d𝑉⁄d𝑡 | Photon flux ∝ 𝑎†𝑎 |
| Energy | 𝐸 = (1⁄2)𝐶𝑉² + (1⁄2)𝐿𝐼² | 𝐸ₙ = ħ𝜔 (𝑛 + 1⁄2) |
| Resonance Frequency | 𝑓₀ = 1⁄(2π√(𝐿𝐶)) | Mode frequency: 𝜔 |
|
| ||
• Output: quantum_circuit_table.png
Figure 6: Superconducting Qubit and Energy Level Diagram
• Description: Show energy levels of a superconducting qubit, analogous to a quantum harmonic oscillator, with a simplified circuit diagram.
• Parameters:
・𝜔 = 2π × 5 GHz
・ħ = 1.0545718 × 10⁻³⁴ J·s
・Energy levels: 𝐸ₙ = ħ𝜔 (𝑛 + 1∕2), 𝑛 = 0, 1, 2, …
• Visuals:
・Energy level diagram with ∣0⟩, ∣1⟩, and transition frequency 𝜔₀₁
・Inset: Simplified LC + Josephson junction circuit
• Output: superconducting_qubit.png
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