Departments of Physics and Applied Physics, Yale University Circuit QED THEORY Steve Girvin Jens Koch Lev Bishop Terri Yu Lars Tornberg (Göteborg) Alexandre Blais (Univ. Sherbrooke) Jay Gambetta (IQC, Univ. Waterloo) Florian Marquardt (LMU Munich) EXPERIMENT Rob Schoelkopf, Michel Devoret Andrew Houck David Schuster Luigi Frunzio Leonardo Di Carlo Jerry Chow Joseph Schreier Blake Johnson Adam Sears Johannes Majer (TU Vienna) Andreas Wallraff (ETH Zurich)

Quantum Computation and NMR of a Single ‘Spin’ Electrical circuit with two quantized energy levels is like a spin -1/2. Quantum Measurement Single Spin ½ Box SET Vgb Vge Cgb Cc Cge Vds (After Konrad Lehnert)

plasma oscillation of 2 or 3 Cooper pairs: no static dipole ‘Transmon’ Cooper Pair Box: Charge Qubit that Works! Josephson junction: EJ >> EC 300 mm Added metal = capacitor & antenna plasma oscillation of 2 or 3 Cooper pairs: no static dipole Transmon qubit insensitive to 1/f electric fields * Theory: J. Koch et al., PRA (2007); Expt: J. Schreier et al., PRB (2008) Flux qubit + capacitor: F. You et al., PRB (2006)

‘Transmon’ Cooper Pair Box: Charge Qubit that Works! Josephson junction plasma oscillations are anharmonic: 300 mm EJ >> EC +1 +2 +3 +4 -4 -3 -2 -1 n = Number of pairs that have tunneled 2 6.1 GHz 1 6.5 GHz Almost a harmonic oscillator except coordinate n is integer valued.

Record Coherence in Transmon Qubit at flux sweet spot (7.35 GHz) below the cavity (5.9 GHz) These plots can be found in folder Y:\_Talks\ARO Program Review 2008\Benchmarking\IgorExpmts TimeDomainExpmts.pxp for plots on left TimeDomainDetuned.pxp for plots on right (option of using 2 us T2 figure is in another experiment, HigherT2Detuned.pxp) To do the decaying exponential cosine fits, the procedure file named RabiEnv1.ipf is needed, and it makes a fit routine under Curve Fitting named RabiEnv 7.356 GHz 6.9 Ghz cavity 5.98 GHz

Cavity & circuit quantum electrodynamics ►coupling an atom to discrete mode of EM field cavity QED Haroche (ENS), Kimble (Caltech) J.M. Raimond, M. Brun, S. Haroche, Rev. Mod. Phys. 73, 565 (2001) circuit QED A. Blais et al., Phys. Rev. A 69, 062320 (2004) A. Wallraff et al., Nature 431,162 (2004) R. J. Schoelkopf, S.M. Girvin, Nature 451, 664 (2008) 2g = vacuum Rabi freq. k = cavity decay rate g = “transverse” decay rate Describe diagram Describe each rate Describe each term in Hamiltonian Discuss strong coupling Goal: strong coupling limit: Jaynes-Cummings Hamiltonian atom/qubit resonator coupling

A Circuit Analog for Cavity QED ‘Circuit Quantum Electrodynamics’ 5 mm DC + 6 GHz in out L = l ~ 2.5 cm transmission line “cavity” Artificial ‘atom’ Cross-section of mode: E B 10 mm + - Need: small cavity and big atom so photons collide with atom frequently. A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, PRA 69, 062320 (2004)

World’s smallest microwave cavity: On-chip CPW resonator coaxial cable wave guide Vacuum fields: mode volume zero-point energy density enhanced by L = l ~ 2.5 cm Cross-section of mode: E B 10 mm + - 5 mm Dispersive coupling of atom to photons enhanced by

Circuit QED Vacuum fields: mode volume atom artificial atom: SC qubit cavity 2D transmission line resonator integrated on microchip Vacuum fields: mode volume zero-point energy density enhanced by Describe diagram Describe each rate Describe each term in Hamiltonian Discuss strong coupling Coupling reaches limit set by fine structure constant

Jaynes Cummings Hamiltonian: “dressed atom” picture cavity qubit dipole coupling Vacuum Rabi splitting Degenerate case:

Strong-coupling: Vacuum Rabi splitting Signature for strong coupling: Placing a single resonant atom inside the cavity leads to splitting of transmission peak 2008 vacuum Rabi splitting atom off-resonance observed in: cavity QED R.J. Thompson et al., PRL 68, 1132 (1992) I. Schuster et al. Nature Physics 4, 382-385 (2008) circuit QED A. Wallraff et al., Nature 431, 162 (2004) quantum dot systems J.P. Reithmaier et al., Nature 432, 197 (2004) T. Yoshie et al., Nature 432, 200 (2004) on resonance A. Wallraff et al., Nature 431, 162 (2004)

L. S. Bishop et al. (Yale) Nature Physics 5, 105 (2009). Multiphoton transitions reveal √n nonlinearity of JC ladder Related work on √n nonlinearity: I. Schuster et al., Nature Phys. 4, 382 (2008) A. Wallraff et al. (Nature 2008) J. Martinis et al. (unpublished) R. Simmonds et al. (unpublished) Deppe et al. arXiv:0805.3294

Emergence of √n multiphoton peaks

Ultra-Strong Coupling Limit qubit degenerate with cavity: first-order atom-cavity coupling excees cavity linewidth Ultra-Strong Coupling Limit ‘strong dispersive’ limit, qubit detuned from cavity: second-order atom-cavity coupling exceeds cavity linewidth atom-cavity detuning

Other Circuit QED results with transmons 2006/7 Probing photon states via the Number splitting effect ►transmon as a detector for photon states (qubit detuned from cavity) Dispersive coupling in second order p.t. Quantized ‘light shift’: atom and cavity pull each other by many line widths J. Gambetta et al., PRA 74, 042318 (2006) (theory) D. Schuster et al., Nature 445, 515 (2007) (experiment) (c.f. Haroche group in time domain)

Mapping coherent superposition states of the qubit onto a superposition of 0 and 1 photon: (‘flying qubit’ for quantum communication) Microwave control pulse can be used to place qubit in arbitrary quantum superposition of ground and excited states. Use ‘Purcell effect’ to insure qubit excitation decays by photon emission (out port #2) >90% of the time.

Measured qubit state Measured photon state Mapping the qubit state on to a photon Maximum at p Measured qubit state Zero at p Measured photon state E= Maximum at p/2 Houck, Schuster et. al., cond-mat/0702648; Nature (2007)

“Fluorescence Tomography” Apply pulse about arbitrary qubit axis Qubit state mapped on to photon superposition Qubit Fock state has no average electric field. Superposition of 0 and 1 photon does. Houck et al. Nature 449, 328 (2007) all of the above are data!

M. Hofheinz et al. (Martinis group UCSB) Readout via qubit Rabi oscillations M. Hofheinz et al. (Martinis group UCSB)

M. Hofheinz et al. (Martinis group UCSB)

Demonstration of Two-Qubit Algorithms with a Superconducting Quantum Processor L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, R. J. Schoelkopf arXiv:0903.2030 Nature (in press, 2009)

cavity: “entanglement bus” driver & detector A two-qubit quantum processor 1 ns resolution DC - 2 GHz cavity: “entanglement bus” driver & detector transmon qubits Majer et al., Nature (2007) Chow et al., PRL (2009) Several earlier instances of 2-qubit interactions: Nakamura/NEC (2003), Martinis/UCSB (2006), Mooij/Delft (2007)

F Flux-bias lines: local, fast and simple 1 ns • Independent qubit tuning • dc to 2 GHz • Simplicity to realization of algorithms F Fast qubit tuning with flux bias line implements a controlled phase gate 100:1 on-off ratio.

Flux-bias lines: local, fast and simple Fast qubit tuning with flux bias line implements a controlled phase gate 100:1 on-off ratio Uses higher level of one qubit. • Avoided crossing (160 MHz) • A frequency shift Related earlier idea: Strauch et al., PRL (2003)

We have the two-qubit gate. Now need two-qubit read out. Strong dispersive regime: each qubit pulls the cavity frequency by amount comparable to cavity linewidth

Multiplexed Readout: 2 classical bits of information Can obtain high fidelity two qubit correlators even without single shot readout.

Leo DiCarlo et al., Nature (in press, 2009) arXiv:0903.2030 Entanglement on demand Bell states on demand Leo DiCarlo et al., Nature (in press, 2009) arXiv:0903.2030 Earlier entanglement work: Steffen et al., Science (2006) Leek et al., cond-mat (2008)

The search problem Classically, takes on average 2.25 guesses to succeed… Use QM to “peek” inside all eggs, find the bunny on first try Position: I II III “Find the surprise!”

Grover in action Two-qubit Grover Algorithm Challenge: Find the location of the -1 !!! “oracle” “unknown” unitary operation: ORACLE 10 pulses w/ nanosecond resolution, total 104 ns duration

Grover in action A Grover step-by-step movie “quantum debugger” Begin in ground state:

Grover in action A Grover step-by-step movie “quantum debugger” Create a maximal superposition: look everywhere at once!

Grover in action A Grover step-by-step movie “quantum debugger” Apply the “unknown” function, and mark the solution

Grover in action Grover in action A Grover step-by-step movie Grover search in action “quantum debugger” Some more 1-qubit rotations… Now we arrive in one of the four Bell states

Grover in action Grover in action A Grover step-by-step movie Grover search in action “quantum debugger” Another (but known) 2-qubit operation now undoes the entanglement and makes an interference pattern that holds the answer!

The correct answer is found >80% of the time! Grover in action A Grover step-by-step movie Grover in action Grover search in action “quantum debugger” Final 1-qubit rotations reveal the answer: The binary representation of “2”! The correct answer is found >80% of the time!

Grover in action Grover in action Grover with other oracles Grover search in action Oracle Fidelity to ideal output (average over 10 repetitions)

is found >84% of the time. Deutsch-Jozsa Algorithm Constant functions Answer is encoded in the state of left qubit Balanced functions The correct answer is found >84% of the time.

CIRCUIT QED: QUANTUM OPTICS Summary CIRCUIT QED: QUANTUM OPTICS Ultra strong ‘atom’-photon coupling and photon-photon coupling bring quantum optics to a new regime. New microwave amplifiers that reach the quantum limit (Devoret group: record 20dB of two-mode vacuum squeezing) Future directions: -photomultiplier for detecting single microwave photons (axion search?) -multi-cavity boson Hubbard model for photons: ultra strong dispersive regime: “U >> t” -hybrid SC qubit/ molecule systems

Summary QUANTUM COMPUTATION ArXiv: cond-mat 0903.2030 Local and fast qubit control Cavity-mediated interaction tunable by 2 orders of magnitude 2-qubit state tomography • Entanglement on demand Fidelity Concurrence • Grover algorithm with Fidelity • Deutsch-Jozsa with Fidelity Next up: quantum error correction ArXiv: cond-mat 0903.2030

Circuit QED Team Members: 2007 Jared Schwede Steve Girvin Blake Johnson Jens Koch Jay Gambetta Joe Schreier Hannes Majer David Schuster Jerry Chow Luigi Frunzio Michel Devoret Andrew Houck Emily Chan Funding:

SCHEMATICS OF JOSEPHSON AMPLIFIER BASED ON RING MODULATOR Experimental Reality (Devoret, Schackert, Bergeal, Frunzio, Manucharyan, et al.) SCHEMATICS OF JOSEPHSON AMPLIFIER BASED ON RING MODULATOR SIGNAL (w1) IDLER (w2) PUMP (Wp)

Average System Noise vs. (Input) Effective Temperature T_noise 40x lower than best HEMTs expt. theory JPC: 130mK Quantum limit: 40mK

Interference between Signal and Idler 18 dB ≤ squeezing ≤ 23 dB idler Near perfect quantum correlations between signal and idler lead to destructive interference. Proof that the amplifier adds almost no entropy to the universe. pump

Randomized Benchmarking Results for SC Qubit 3 ns  gaussian with 2  truncation and 8 ns buffer Transmon Error per gate = 1.2 % Y:\_Talks\ARO Program Review 2008\Benchmarking\IgorExpmts Good3nsEPG.pxp Average randomized fidelity for four different Clifford sequences, truncated at 17 different lengths

Up to n=15 Photon Fock States with High Fidelity Martinis UCSB group 2008 readout via qubit Rabi oscillations

Building Quantum Electrical Circuits circuit elements SC qubits: macroscopic articifical atoms ( ) ingredients: nonlinearities low temperatures small dissipation isolation from environment M. H. Devoret, Quantum Fluctuations (Les Houches Session LXIII), Elsevier 1997, pp. 351–386. Two-level system: fake spin 1/2

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