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Departments of Physics and Applied Physics, Yale University Chalmers University of Technology, Feb. 2009 What's super about superconducting qubits? Jens Koch Outline charge qubit - Chalmers Introduction Superconducting qubits ► overview, challenges ► circuit quantization ► the Cooper pair box next lecture: flux qubit - Delft phase qubit - UCSB Transmon qubit ► from the CPB to the transmon ► advantages of the transmon ► experimental confirmation Circuit QED with the transmon: examples Quantum Bits and all that jazz 2-level quantum system (two distinct states ) computational speedup P.W. Shor, SIAM J. Comp. 26, 1484 (1997) quantum cryptography can exist in an infinite number of physical states intermediate between and . state N. Gisin et al., RMP 74, 145 (2002) fundamental questions What makes quantum information more powerful than classical information? Entanglement – how to create it? How to quantify it? Mechanisms of decoherence? superposition of AND state Measurement theory, evolution under continuous measurement … 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) 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 The crux of designing qubits environment environment qubit control measurement ►need good coupling! protection against decoherence ►need to be uncoupled! Relaxation and dephasing relaxation – time scale T1 dephasing – time scale T2 qubit ► fast parameter changes: sudden approx, transitions transition ► random switching ► slow parameter changes: adiabatic approx, energy modulation ► phase randomization Bringing the into electrical circuits Idea of superconducting qubits: Electrical circuits can behave quantum mechanically! 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! 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 Why use superconductors? Wanted: ► electrical circuit as artificial atom superconductor ► atom should not spontaneously lose energy ► anharmonic spectrum E “forest” of states Superconductor 2D ~ 1 meV ► dissipationless! ► provides nonlinearity via Josephson effect ► can use dirty materials for superconductors superconducting gap Building Quantum Electrical Circuits circuit elements ( ) SC qubits: macroscopic articifical atoms ingredients: • nonlinearities • low temperatures • small dissipation Two-level system: fake spin 1/2 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) Tunneling operator for Cooper pairs: normal state conductance SC gap Josephson energy Tight binding model: hopping on a 1D lattice! Review: Josephson Tunneling II … … Tight binding model: Diagonalization: ‘position’ ‘wave vector’ (compact!) ‘plane wave eigenstate’ 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 Circuit quantization Best reference that I know: (beware of a few typos though) Circuit quantization – a quick survival guide ► Step 1: set up Lagrangian - determine the circuit's independent coordinates branch node ► use generalized node fluxes also: ideal current sources, ideal voltage sources, resistors as position variables Circuit quantization – a quick survival guide ► Step 1: set up Lagrangian S capacitive energies S inductive energies ► Step 2: Legendre transform Hamiltonian conjugate momenta: charges 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! 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! ? 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! Different types of SC qubits ► Nonlinearity from Josephson junctions NEC, Chalmers, charge Saclay, Yale qubit EJ = EC, EJ =50EC TU Delft,UCB flux qubit EJ = 40-100EC NIST,UCSB phase qubit EJ = 10,000EC 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 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) 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 CPB as a charge qubit Charge limit: big small perturbation CPB as a charge qubit Charge limit: big small perturbation Next lecture: from the charge regime to the transmon regime