Superconducting Qubit Technology

1. Quantized Energy Levels and Systems

Small physical systems (atomic, electronic, or optical) exist in discrete energy levels known as a spectrum. The arrangement and spacing of these levels reveal the fundamental nature of the system.

In qubits, which only use states $|0\rangle$ and $|1\rangle$, many physical systems have a much larger range of possible states

Types of Energy Spacings

• Harmonic (Evenly Spaced): The energy difference between any two adjacent levels is identical.

• Anharmonic (Unevenly Spaced): The energy gaps vary between levels. This is critical for isolating specific states.

• Degenerate: Multiple distinct quantum states exist at the exact same energy level.

• Chaotic: Complex systems where levels follow specific mathematical relationships.

2. Harmonic Systems

Harmonic systems are common in optical and electronic domains (e.g., photons).

• Data Analogy: They are the quantum version of an “unsigned int” because they can occupy infinitely many possible states ($|0\rangle, |1\rangle, |2\rangle, \dots$).

• Operators: These systems are manipulated using Raising ($\hat{a}^*$) and Lowering ($\hat{a}$) operators.

  • The lowering operator reduces the state: $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$.
  • The raising operator increases the state: $\hat{a}^*|n\rangle = \sqrt{n+1}|n+1\rangle$.

The Operator $\hat{a}^*\hat{a}$: This is defined as the number operator ($\hat{n}$). When it acts on a state $|n\rangle$, it “measures” the energy level and returns the number $n$. $\hat{a}\hat{a}^*|n\rangle = (n+1)|n\rangle = |n\rangle + n|n\rangle = (I + \hat{n})|n\rangle$ if you raise a state and then immediately lower it, you get a term proportional to the energy level plus one

• Position ($\hat{x}$): Represented as the sum of the raising and lowering operators multiplied by a scaling factor involving Planck’s constant ($\hbar$), mass ($m$), and frequency ($\omega$).

  $\hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}+\hat{a}^*)$

• Momentum ($\hat{p}$): Represented as the difference between the operators10. Notice the $i$ (imaginary unit), which is a hallmark of quantum momentum.

  $\hat{p}=-i\sqrt{\frac{\hbar m\omega}{2}}(\hat{a}-\hat{a}^*)$

• Linearity: Because the levels are evenly spaced ($E_n = n\hbar\omega$), Energy of each state is: $\hat{H}|n\rangle = \hbar\omega\hat{a}^*\hat{a}|n\rangle = n\hbar\omega|n\rangle$

• a frequency that drives a system from $0 \rightarrow 1$ will also drive it from $1 \rightarrow 2$, causing it to “climb the ladder” rather than staying in two specific states.

3. Anharmonic Systems and Qubits

To create a qubit (a “bool” data type), a system must be anharmonic.

• Isolation: In an anharmonic system, the frequency required to move from $0 \rightarrow 1$ is different from $1 \rightarrow 2$. This allows researchers to isolate the bottom two levels to function as a qubit.

• Anharmonicity Formula: It is measured by the difference in energy gaps:

  $\text{Anharmonicity} = (E_2 - E_1) - (E_1 - E_0)$

4. The Josephson Junction

The Josephson junction is the “key element” used to create artificial atoms called transmon qubits. It consists of two superconducting reservoirs separated by a thin insulating barrier.

• Superconductivity: Electrons form charge-pairs (Cooper pairs) with a charge of $-2e$. Superconductors are used because they allow electrons to move across the junction without dissipating energy.

• Tunneling: Charge pairs can tunnel across the barrier, changing the distribution of charge between the two sides.

• The Phase ($\phi$): The quantum state is often expressed using a phase variable $\phi$, which is the Fourier Transform of the charge states.

The Quantum Hamiltonian

The energy of the junction is described by an equation resembling the Schrödinger equation:

$\hat{H}\psi(\phi) = -\frac{(2e)^2}{2C}\frac{\partial^2\psi(\phi)}{\partial\phi^2} - E_j \cos\phi \, \psi(\phi)$

• Kinetic Energy: The junction’s capacitance ($C$) acts like mass in this analogy.

• Potential Energy: The tunneling acts like a cosine potential.

5. The RCSJ Model (Classical Limit)

When a junction is large, it can be treated classically using the Resistively and Capacitively Shunted Junction (RCSJ) model.

The “Tilted Washboard” Analogy

The RCSJ model is visualized as a particle in a tilted washboard potential:

• The Washboard: The cosine term creates the “bumps” (minima)

• The Tilt: An external current bias ($I_{ext}$) tilts the potential.

• Damping: A resistive shunt ($R$) acts like viscous damping (friction).

Behavior

1. Stable (Zero Voltage): If the tilt is small, the “particle” (phase) sits in a minimum. There is a supercurrent but no average voltage.

2. Running State (Voltage): If the tilt is too steep (high current), the particle rolls down the hills forever. This represents a constant average voltage with small AC oscillations as it bumps over each crest.