Tony Lee 7 December 2005

1. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics Tony Lee 7 December 2005

2. Papers • “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics” A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin and R. J. Schoelkopf. Nature (London) 431, 162 (2004) • “Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation” A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin and R. J. Schoelkopf Physical Review A 69, 062320 (2004)

3. Table of Contents • Cavity QED • Transmission lines • Charge qubits • Experiment: coupling of charge qubit to transmission- line cavity

4. Cavity QED • Place atom in cavity: • Important parameters: • wr = cavity resonance frequency • g = coupling between atom and cavity • k= cavity decay rate • g= decay rate of atom’s excited state out of cavity • Strong coupling: g >> k,g

5. Jaynes-Cummings • Hamiltonian includes cavity, atom, coupling • Cavity basis: |n>, for n photons (n = 0,1,2,…) • Atomic basis: |↑>, |↓> • ground, excited • Without coupling, eigenstates: |↑,n>,|↓,n> (bare states) • Due to coupling term, new eigenstates (dressed states) • Angle q gives degree of mixing • When D = 0, maximal mixing • When |D| >> g, almost no mixing • State keeps its identity Dressed States

6. New Energy Levels • For D = 0, bare states are degenerate • Coupling lifts degeneracy • Each level splits into two • For |D| >> g, energy of transition |0> → |1> depends on whether atom was in |↑> or |↓> • Coupling shifts energy levels

7. Amplitude of Transmitted Light • Suppose we shine light of frequency w onto cavity • Scan w • Transmission is maximum when on resonance: • Without atom, max transmission @ w = wr • With atom inside • If system in |↑,0>, max transmission @ • wr ± g for D = 0 • If cavity in |0>, max transmission @ • wr ± g2/D for |D| >> g • This is one way of finding resonance frequencies of system |D| >> g D = 0

8. Transmission Lines • Equivalent to infinite series of inductors and capacitors

9. Transmission Lines • Boundary conditions  discrete longitudinal modes • Yale group used k = 2 mode • wr / 2p = 6.04 GHz • Quantize EM field • a†, a create, annihilate photons in transmission line • Voltage V = Vrms (a + a†) • Transmission line equivalent to a cavity • Microwave vs. optical frequencies wr = k p v / L k=1,2,3…

10. Josephson Junctions • At low temperatures, metals have no DC resistance • In a superconductor, current carried by pairs of electrons (Cooper pairs) • Josephson junction is two superconductors separated by a weak link • Eg. Al / AlOx / Al • Cooper pairs can tunnel through insulator! • Supercurrent goes through insulator Symbol:

11. Cooper Pair Box • When you bias a JJ with a voltage and capacitor, you get a Cooper Pair Box • Voltage allows control of quantum properties of CPB • “Gate voltage”, “gate capacitor” • People usually use two JJs in parallel, which is equivalent to a single JJ • Can control CPB using magnetic flux inside loop

12. Hamiltonian of CPB • Work in basis |N> • N = # Cooper pairs that tunneled out of island • Need two terms in Hamiltonian: • Q2/2C for charge on island • Coupling between |N> and |N+1> for tunneling • 4Ec = (2e)2/(2C) • Ng = CgVg/(2e) • EJ ~ cos(pFext/Fo)

13. Two-level System • If Ng ~ half integer and temperature is low, only the two states with lowest energy are relevant • Call them |N>, |N+1> • Two level system: • Eel = 4 Ec (1- Ng) • |N>, |N+1> are NOT eigenstates of Hamiltonian • Real eigenstates |↑>, |↓> are superpositions of |N>,|N+1> • “Charge qubit” (N - Ng)^2

14. Artificial Atom • Ground state |↑> • Excited state |↓> • Energy separation between ~ 10 GHz: • Can drive Rabi oscillations (single qubit rotations) with microwave pulses Ng = CgVg/(2e) EJ ~ cos(pFext/Fo)

15. Coupling to cavity Ng=CgVg/(2e) • Capacitive coupling • Voltage fluctuations from transmission line mode change Vg: • Vg=Vgdc + Vrms (a + a†) • End up with Jaynes-Cummings Hamiltonian • Equivalent to atom in a cavity • “Circuit QED” 

16. Cavity + Cooper Pair Box • Meandering resonator • L = 24 mm • Niobium • Silicon substrate • Resonator capacitively coupled to external transmission lines • Waves can enter and leave • Q  104 • T < 100 mK • <n> < 0.06

17. Measurement Technique • Send in microwaves at frequency w • Resonator fixed at wr • Change transition frequency of CPB • ng ~ Vg (DC Voltage) • Flux inside loop • Look at amplitude and phase of transmitted waves

18. Strong coupling • Start in |↓,0> ground state of cavity and qubit • wr / 2p = 6 GHz • <n> << 1 for T < 100 mK • “Cool” to ground state by waiting for thermal equilibrium • |D| >> g: • Single peak at wr – g2/D • D = 0: • Peaks at wr ± g • g = 5.8 MHz • k = 0.8 MHz • g = 0.7 MHz • g > k, g  Strong coupling!

19. Applications • New architecture for quantum computing with superconducting qubits • New way of reading out charge qubits • |↑>, |↓> produce transmission peaks at different frequencies • Look at transmission at wr ± g2/D • No direct connections to environment • May help increase coherence time • Use transmission line as a bus • Put qubits at different anti-nodes in transmission line • Equivalent to many atoms in an optical cavity • Scalability • Single microwave photon source

20. Summary • Transmission lines are equivalent to cavities • Cooper pair boxes can be two-level systems • CPB + transmission line ↔ atom + cavity • CPB can be strongly coupled to a cavity • Circuit QED is a promising new architecture for superconducting quantum computing