Superconducting Qubits in 1 hour

Based off the CMC Superconducting Qubits Workshop

Zhi Han

Overview

Review of superconductivity
The LC Circuit
The Josephson effect
Transmons
- Jaynes-Cumming Hamiltonian
SQUIDs

Main goal is give an overview to superconductors and provide reference

What is a superconductor?

Meissner effect
Zero resistivity under \(Tc\)
Cooper pairs
One macroscopic wavefunction

Image: Reddit

Cooper pairs

Electrons pair up to form cooper pairs.
These cooper pairs behave as bosons rather than photons.
Bosons do not obey the Pauli exclusion principle.
Therefore, all the cooper pairs can simultaneously occupy the ground state, and behave as one wavefunction.
We will make this ansatz later: \[ \psi = \sqrt{n}e^{i\varphi}\]

Image: Higgsino Physics

\displaystyle

LC Circuit

\omega_r = \frac{1}{\sqrt{LC}}

\displaystyle

Looks like SHO:

\[E=\frac{1}{2}m\omega^2x^2 + \frac{1}{2}m\dot{x}^2\]

Total energy:

\[ E_m + E_e = \frac{1}{2}C\omega_r^2\Phi^2(t) + \frac{1}{2}C\dot{\Phi}^2(t)\]

x \to \Phi, m \to C

LC Circuit

Image: Wikipedia - LC Circuit

\omega_r = \frac{1}{\sqrt{LC}}

E_m = \frac{1}{2L} \Phi^2(t) = \frac{1}{2}C\omega_r^2 \Phi^2(t)

E_e = \frac{1}{2} C V(t)^2 = \frac{1}{2}C\dot{\Phi}^2(t)

Electric and Magnetic Energy

Lagrangian Formulation

\displaystyle

\displaystyle \mathcal{L} = E_e - E_m = \frac{1}{2}C\dot{\Phi}^2(t) -\frac{1}{2}C\omega_r^2\Phi^2(t)

\begin{gathered} \frac{d}{d t} \frac{\partial \mathscr{L}}{\partial \dot{\Phi}}-\frac{\partial \mathscr{L}}{\partial \Phi}=0 \end{gathered}

\implies \ddot{\Phi}(t)+\omega_{r}^{2} \Phi(t)=0

\Phi(t) = \Phi_0 \cos(\omega_r t)

Solution

\frac{d}{d t} \frac{\partial \mathscr{L}}{\partial \dot{\Phi}}=C \ddot{\Phi} \\ \frac{\partial \mathscr{L}}{\partial \Phi}=\frac{1}{L} \Phi

Euler Lagrange Equations

Equation of motion

\omega_r = \frac{1}{\sqrt{LC}}

\displaystyle

\displaystyle Q=\frac{\partial\mathscr{L}}{\partial\dot{\Phi}}=C \dot{\Phi}

Conjugate momentum

to flux is charge

E_m = \frac{1}{2L}\Phi^2(t) = \frac{1}{2L}\Phi_0 \cos^2(\omega_rt)

E_e = \frac{C}{2}\dot{\Phi}^2(t) = \frac{1}{2L}\Phi_0^2\sin^2(\omega_r t)

E_t = \frac{1}{2L}\Phi^2_0

Oscillating between electric and magnetic

LC Hamiltonian

H_{L C}=Q \dot{\Phi}-\mathscr{L}=\frac{1}{2 C} Q^{2}+\frac{1}{2 L} \Phi^{2}

Promote flux and charge to quantum operators (canonical/dirac quantization)

\{A,B\} \longmapsto \tfrac{1}{i \hbar} [\hat{A},\hat{B}]

Since flux and charge are already canonical coordinates we just promote

\hat H_{L C} = \frac{1}{2 C} \hat Q^{2}+\frac{1}{2 L} \hat \Phi^{2}

[\hat \Phi, \hat Q] = i\hbar

LC Hamiltonian

\hat H_{L C} = \frac{1}{2 C} \hat Q^{2}+\frac{1}{2 L} \hat \Phi^{2}

\hat{\Phi}=\Phi_{z p f}\left(\hat{a}^{\dagger}+\hat{a}\right) \quad \hat{Q}=i Q_{z p f}\left(\hat{a}^{\dagger}-\hat{a}\right)

\hat{H}_{L C}=\hbar \omega_{r}\left(\hat{a}^{\dagger} \hat{a}+1 / 2\right)

\begin{aligned} \Phi_{z p f} &=\sqrt{\hbar Z_{r} / 2} \\ Q_{z p f} &=\sqrt{\hbar / 2 Z_{r}} \end{aligned}

Original Hamiltonian

Raising/lowering operators

LC circuit as QHO

Eigenstates of LC circuit satisfies \( a^\dagger a | n \rangle = n |n\rangle \)
\(a^\dagger \) creates a quantized excitation of flux and charge. In other words, a photon of frequency \( \omega_r\) is created.

Coplanar waveguide

Resonant Modes

Transmission line LC circuit

Conductors

Dielectric

Coplanar Waveguides

Transmission line.
Behave as coupled LC circuits.
The solution for a chain of 1d harmonic oscillators is given by a wave equation.
Described by the Telegrapher's equations.

\displaystyle \frac{\partial^2 V}{{\partial t}^2} - u^2 \frac{\partial^2 V}{{\partial x}^2} = 0

\displaystyle \frac{\partial^2 I}{{\partial t}^2} - u^2 \frac{\partial^2 I}{{\partial x}^2} = 0

u = \frac{1}{\sqrt{LC}}

Images: Wikipedia

SC Device	Physics	Function in SC circuit
LC circuit/Resonator	Quantum Harmonic Oscillator	Readout, control, couple qubits.
Coplanar Waveguide	Telegrapher's Equations	Transmission of qubits as photons
Josephson Junction	Josephson's Equations	Nonlinear inductor
SQUID	Josephson's Equations, Two state system	Qubit (Magnetic flux)
Transmon	Two state system	Qubit (Charge)
Transmon with LC circuit	Jaynes-Cumming Hamiltonian	Qubit (Charge)

Josephson Junctions

LC circuit is a linear device that cannot be used for a qubit. But it is used for readout, control, and couple qubits.
Non-linearity is needed to process and encode quantum information.
Superconducting qubits and quantum parametric amplifiers are possible only with a non linear element.

Josephson Relations

Obtained by solving the Schrodinger equation with the ansatz \[ \psi = \sqrt{n}e^{i\phi}\] on both sides of the junction.

\boxed{{\displaystyle I(t)=I_{c}\sin(\varphi (t))}}

\boxed{{\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {2eV(t)}{\hbar }}}}

\varphi=2 \pi \Phi / \Phi_{0} = \phi_2 - \phi_1

\Phi_{0}=h / 2 e

Josephson Phase

Quantum flux

Worth memorizing!!!!

{\displaystyle I(t)=I_{c}\sin(\varphi (t))}

L_{J}=\left(\frac{\partial I}{\partial \Phi}\right)^{-1}=\frac{\Phi_{0}}{2 \pi I_{c}} \frac{1}{\cos \left(2 \pi \Phi / \Phi_{0}\right)}

Josephson current equation

Non-linear inductor

Linear inductor

I = \Phi/L

\begin{aligned} \varepsilon_{J J}(t) &=\int V\left(t^{\prime}\right) I\left(t^{\prime}\right) d t^{\prime}=\int \frac{d \Phi\left(t^{\prime}\right)}{d t^{\prime}} I_{c} \sin \left(2 \pi \Phi / \Phi_{0}\right) d t^{\prime} \\ &=\boxed{-E_{J} \cos \left(2 \pi \Phi / \Phi_{0}\right)} \end{aligned}

Energy of a Josephson Junction

E_{J}:=\Phi_{0} I_{c} / 2 \pi

Archetype 1:

Transmon Qubit

Replace linear inductor with a Josephson Junction.

\mathcal{L}=\frac{C_{\Sigma}}{2} \dot{\Phi}^{2}(t)+E_{J} \cos \left(2 \pi \Phi / \Phi_{0}\right)

\hat{H}=\frac{1}{2 C_{\Sigma}}\left(\hat{Q}-Q_{g}\right)^{2}-E_{J} \cos \left(2 \pi \hat{\Phi} / \Phi_{0}\right)

C_\Sigma = C_j + C_c

[\hat{Q}, \hat{\Phi}]=i\hbar

H = Q\dot{\Phi} - \mathcal{L}

Quantization

Hamiltonian of a Transmon

Transmon Qubit

Image Source

LC-Transmon Hamiltonian

Transmon coupled with LC circuit.
Solving this circuit results in the Jaynes-Cumming Hamiltonian.
Qubit readout commutes with the Hamiltonian, so we don't disturb the state.
Interaction is mediated by virtual photons.

Archetype 2: SQUID Qubits

SC Device	Physics	Function in SC circuit
LC circuit/Resonator	Quantum Harmonic Oscillator	Readout, control, couple qubits.
Coplanar Waveguide	Telegrapher's Equations	Transmission of qubits as photons
Josephson Junction	Josephson's Equations	Nonlinear inductor
SQUID	Josephson's Equations, Two state system	Qubit (Magnetic flux)
Transmon	Two state system	Qubit (Charge)
Transmon with LC circuit	Jaynes-Cumming Hamiltonian	Qubit (Charge)

Superconducting loop

\psi(z)=m_{s} e^{i \theta(z)}, z \in[0,2 \pi R)

\frac{1}{2 m^{*}}\left(\frac{\hbar}{i} \nabla-q^{*} \vec{A}\right)^{2} \psi \approx 0

Ginzburg-Landau

\Phi

\implies \frac{1}{2 m^{*}} \frac{\hbar}{i} \nabla \cdot \frac{\hbar}{i} \vec{\nabla} \cdot\left[\psi_0(r) e^{\frac{-iq^{*}}{\hbar} \int_{0}^{z} \mathbf{A} \cdot d \mathbf{l}}\right] \approx 0

\psi(z)=n_{s} e^{\frac{iq^*}{\hbar}\int_0^z\mathbf{A} \cdot d \mathbf{l}}

\Phi = LI_s + \Phi_{ext}

Internal flux

External flux

Source: Wikipedia