What's super about superconducting qubits?

1. Departments of Physics and Applied Physics, Yale University Chalmers University of Technology,Feb. 2009 What's super about superconducting qubits? Jens Koch

2. Outline charge qubit - Chalmers Introduction Superconducting qubits ► overview, challenges ► circuit quantization ► the Cooper pair box Transmon qubit ► from the CPB to the transmon ► advantages of the transmon ► experimental confirmation Circuit QED with the transmon: examples next lecture: flux qubit - Delft phase qubit - UCSB

3. Quantum Bits and all that jazz superposition of AND 2-level quantum system (two distinct states ) can exist in an infinite number of physical states intermediate between and . computational speedup P.W. Shor, SIAM J. Comp. 26, 1484 (1997) quantum cryptography N. Gisin et al., RMP 74, 145 (2002) fundamental questions state What makes quantum information more powerful than classical information? Entanglement – how to create it? How to quantify it? Mechanisms of decoherence? Measurement theory, evolution under continuous measurement … state

4. 2-level systems Nature provides a few true 2-level systems: Spin-1/2 systems, e.g. electron (→ Loss-DiVincenzo proposal) nuclei (→ NMR) Polarization of electromagnetic waves (→ linear optics quantum computing)

5. 2-level systems … … Using multi-level systems as 2-level systems • Requirements: • anharmonicity • long-lived states • good coupling to EM field • preparation, trapping etc. e.g. atoms and molecules (→ cavity QED, → trapped ions → liquid-state NMR) R. Schoelkopf artificial atoms: superconducting qubits, quantum dots (→ cavity QED, → circuit QED…) C. Schönenberger

6. The crux of designing qubits environment environment qubit protection against decoherence control measurement ►need good coupling! ►need to be uncoupled!

7. Relaxation and dephasing relaxation – time scale T1 dephasing – time scale T2 qubit ► fast parameter changes: sudden approx, transitions ► slow parameter changes: adiabatic approx, energy modulation transition ► random switching ► phase randomization

8. Bringing the  into electrical circuits Idea of superconducting qubits: Electrical circuits can behave quantum mechanically! 

9. Bringing the  into electrical circuits Idea of superconducting qubits: Electrical circuits can behave quantum mechanically! • What's good about circuits? • Circuits are like LEGOs! a few elementary building blocks, gazillions of possibilities!

10. Bringing the  into electrical circuits Idea of superconducting qubits: Electrical circuits can behave quantum mechanically! • What's good about circuits? • Circuits are like LEGOs! a few elementary building blocks, gazillions of possibilities! • Chip fabrication: • well-established techniques • hope: possibility of scaling

11. Why use superconductors? Wanted:► electrical circuit as artificial atom ► atom should not spontaneously lose energy ► anharmonic spectrum superconductor E “forest” of states Superconductor ► dissipationless! ► provides nonlinearity via Josephson effect ► can use dirty materials for superconductors 2D ~ 1meV superconducting gap

12. Building Quantum Electrical Circuits circuit elements SC qubits: macroscopic articifical atoms ( ) • ingredients: • nonlinearities • low temperatures • small dissipation Two-level system: fake spin 1/2

13. Review: Josephson Tunneling • couple two superconductors • via oxide layer → acts as tunneling barrier • superconducting gap inhibits e- tunneling • Cooper pairs CAN tunnel!► Josephson tunneling (2nd order with virtual intermediate state) normal state conductance Tunneling operator for Cooper pairs: SC gap Josephson energy Tight binding model: hopping on a 1D lattice!

14. Review: Josephson Tunneling II … … Tight binding model: Diagonalization: ‘position’ ‘wave vector’ (compact!) ‘plane wave eigenstate’

15. Junction capacitance: charging energy Transfer of Cooper pairs across junction -2en +2en charging of SCs ► junction also acts as capacitor! charging energy with quadratic in n

16. Circuit quantization Best reference that I know: (beware of a few typos though)

17. Circuit quantization – a quick survival guide branch node • Step 1: set up Lagrangian - determine the circuit's independent coordinates • use generalized node fluxes as position variables also: ideal current sources, ideal voltage sources, resistors

18. Circuit quantization – a quick survival guide • Step 1: set up Lagrangian S inductive energies S capacitive energies • Step 2: Legendre transform  Hamiltonian conjugate momenta: charges

19. Circuit quantization – a quick survival guide • Final Step 3: canonical quantization Canonical quantization makes NO statement about boundary conditions! Usually, assume Works if each node is connected to an inductor ( confining potential). This does NOT work if SC islands are present! • charge transfer between island and rest of circuit: only whole Cooper pairs! • canonical quantization • is blind to the • quantization of electric charge!

20. Circuit quantization – a quick survival guide • Final Step 3: quantization in the presence of SC islands island charge operator has discrete spectrum: charge basis position momentum ? Peierls: leads to contradiction -- phase operator is ill-defined!

21. Circuit quantization – a quick survival guide • Final Step 3: quantization in the presence of SC islands Have already defined charge operator What about ? ► should define this in phase basis! usually: now: ► lives on circle! ► is periodic!

22. Different types of SC qubits ► Nonlinearity from Josephson junctions NEC, Chalmers, Saclay, Yale charge qubit Nakamura et al., NEC Labs Vion et al., Saclay Devoret et al., Schoelkopf et al., Yale, Delsing et al., Chalmers EJ = EC, EJ =50EC TUDelft,UCB Lukens et al., SUNY Mooij et al., Delft Orlando et al., MIT Clarke, UC Berkeley flux qubit EJ = 40-100EC phase qubit NIST,UCSB Martinis et al., UCSB Simmonds et al., NIST Wellstood et al., U Maryland EJ = 10,000EC Reviews: Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001) M. H. Devoret, A. Wallraff and J. M. Martinis, cond-mat/0411172 (2004) J. Q. You and F. Nori, Phys. Today, Nov. 2005, 42 J. Clarke, F. K. Wilhelm, Nature 453, 1031 (2008)

23. CPB Hamiltonian 3 parameters: offset charge (tunable by gate) Josephson energy charging energy (fixed by geometry) charge basis: numerical diagonalization phase basis: exact solution with Mathieu functions

24. CPB as a charge qubit Charge limit: big small perturbation

25. CPB as a charge qubit Charge limit: big small perturbation Next lecture: from the charge regime to the transmon regime