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

2. Outline 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: charge qubit - Chalmers phase qubit - UCSB flux qubit - Delft

3. state Quantum Bits and all that jazz 2-level quantum system (two distinct states ) can exist in an infinite number of physical states intermediate between and. superposition of AND quantum cryptography N. Gisin et al., RMP 74, 145 (2002) computational speedup P.W. Shor, SIAM J. Comp. 26, 1484 (1997) fundamental questions 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 …

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

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

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

7. Relaxation and dephasing relaxation – time scale T 1 dephasing – time scale T 2 qubit transition ► phase randomization► random switching ► fast parameter changes: sudden approx, transitions ► slow parameter changes: adiabatic approx, energy modulation

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

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

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

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

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

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

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

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

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

17. Circuit quantization – a quick survival guide ► Step 1: set up Lagrangian - determine the circuit's independent coordinates branch node ► 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  capacitive energies  inductive 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: position momentum ? Peierls: leads to contradiction -- phase operator is ill-defined! charge basis

21. ► is periodic! 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!

22. Different types of SC qubits charge qubit flux qubit phase qubit 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) Nakamura et al., NEC Labs Vion et al., Saclay Devoret et al., Schoelkopf et al., Yale, Delsing et al., Chalmers Lukens et al., SUNY Mooij et al., Delft Orlando et al., MIT Clarke, UC Berkeley Martinis et al., UCSB Simmonds et al., NIST Wellstood et al., U Maryland NEC, Chalmers, Saclay, Yale E J = E C, E J =50E C NIST,UCSB TU Delft,UCB E J = 10,000E C E J = 40-100E C ► Nonlinearity from Josephson junctions

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

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