Departments of Physics and Applied Physics, Yale University Circuit QED
THEORY
Steve Girvin Jens Koch Lev Bishop Terri Yu EXPERIMENT Lars Tornberg (Göteborg)
Rob Schoelkopf, Michel Devoret
Andrew Houck David Schuster Luigi Frunzio Leonardo Di Carlo Jerry Chow Joseph Schreier Blake Johnson Adam Sears Alexandre Blais (Univ. Sherbrooke) Jay Gambetta (IQC, Univ. Waterloo) Florian Marquardt (LMU Munich)
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.
Single Spin ½ Quantum Measurement
Vds C C C gb Box c ge Vgb Vge SET
2 (After Konrad Lehnert) 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, C Saclay, Yale charge Nakamura et al., NEC Labs qubit Vion et al., Saclay = E
J (CPB) Devoret et al., Schoelkopf et al., Yale, E Delsing et al., Chalmers
C TU Delft,UCB Lukens et al., SUNY flux Mooij et al., Delft qubit Orlando et al., MIT = 40-100E
J Clarke, UC Berkeley E C NIST,UCSB phase Martinis et al., UCSB qubit Simmonds et al., NIST
= 10,000E Wellstood et al., U Maryland J E …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
‘Transmon’ Cooper Pair Box: Charge Qubit that Works!
Josephson junction: EJ >> EC
300 μm e
g
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: e
g 300 μm EJ >> EC
-4 -3 -2 -1 0 +1 +2 +3 +4
n = Number of pairs that have tunneled 2 2 5.8 GHz HE=− JC[tunneling]+ 4 Enˆ 1 6.0 GHz Almost a harmonic oscillator except coordinate n is integer valued. 0 6 7 Coherence in Transmon Qubit at flux sweet spot (7.35 GHz) below the cavity (5.9 GHz)
T1 =1.5μs T1 = 2.2μs
* * TT21==23.0μs T2 =1.1μs Randomized Benchmarking Results for SC Qubit 3 ns σ gaussian with 2 σ truncation and 8 ns buffer
Transmon
Error per gate = 1.2 %
Average randomized fidelity for four different Clifford sequences, truncated at 17 different lengths 9 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)
g κ circuit QED A. Blais et al., Phys. Rev. A 69, 062320 (2004) A. Wallraff et al., Nature 431,162 (2004) γ R. J. Schoelkopf, S.M. Girvin, Nature 451, 664 (2008)
2g = vacuum Rabi freq. κ = cavity decay rate γ = “transverse” decay rate
Goal: strong coupling limit: gt {}κγ,,1/transit
Jaynes-CummingsNeed: small cavity Hamiltonian and big atom so photons collide with atom frequently.
atom/qubit resonator coupling A Circuit Analog for Cavity QED ‘Circuit Quantum Electrodynamics’ λ ≈ 2.5 cm out cm 2.5 λ ~ L = transmission Artificial ‘atom’ line “cavity” Q =−1036 10 DC + Cross-section 5 μm of mode: 6 GHz in photons travel ≥10E kilometB ers while- in the+ +resonator!- CPW ≈ optical fiber 10 μm (if superconducting) 11 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 ide gu Vacuum fields: ve wa mode volume 10−63λ zero-point energy density R ≥ λ +6 e enhanced by abl 10 l c xia coa EVRMS ∼∼0.25 V/m RMS 1 μ V
R λ cm 2.5 λ ~ L = Cross-section of mode:
E B - + + -
5 μm 10 μm 12 Circuit QED
atom artificial atom: SC qubit integrated on cavity 2D transmission line resonator microchip
Vacuum fields: mode volume10−63λ zero-point energy density +6 enhanced by 10
Coupling reaches limit set by fine structure constant
g α g 200MHz ∼∼0.04 ∼∼0.04 ωε ω 5GHz Jaynes Cummings Hamiltonian: “dressed atom” picture
cavity qubit dipole coupling ω H =++ ωσσσaa††01 z g() a−+ +++ a H H r 2 κ γ ↑ 3
↓ 3 32g ↑ 2
↓ 2 22g ↑1
↓1 12g ↑ 0
ωr ωa ↓ 0 Vacuum Rabi splitting Degenerate case: ω01= ω r 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) on resonance 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) 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
strong coupling limit: first-order atom-cavity coupling gt {κγ,,1/transit } excees cavity linewidth
Ultra Strong Coupling Limit
‘strong dispersive’ limit: g 2 second-order atom-cavity coupling > κ exceeds cavity linewidth Δ atom-cavity detuning 17 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) ω → Dispersive coupling in second order p.t.
g 2 Quantized ‘light shift’: atom † z and cavity pull each other by Vaaeff ≈ σ Δ many line widths
J. Gambetta et al., PRA 74, 042318 (2006) (theory) D. Schuster et al., Nature 445, 515 (2007) (experiment) 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.
(αβge+⇒) 0 photons g (α 0 βphotons + 1 photon ) “Fluorescence Tomography” • Apply pulse about arbitrary qubit axis
• Qubit state mapped on to photon superposition
† † σˆ z aa+ aa− Qubit
Houck et al. allNature of the above are449 data!, 328 (2007)
Up to n=15 Photon Fock States with High Fidelity
Martinis UCSB group 2008 readout via qubit Rabi oscillations 21
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)
22 A two-qubit quantum processor
1 ns resolution DC - 2 GHz
cavity: “entanglement bus” driver & detector
transmon qubits 23 Several earlier instances of 2-qubit interactions: Majer et al., Nature (2007) Nakamura/NEC (2003), Martinis/UCSB (2006), Mooij/Delft (2007) Chow et al., PRL (2009)
Flux-bias lines: local, fast and simple
1 ns • Independent qubit tuning
• dc to 2 GHz
• Simplicity to realization of algorithms Φ
Fast qubit tuning with flux bias line implements a controlled 24 phase gate 100:1 on-off ratio. Flux-bias lines: local, fast and simple
Fast qubit tuning with flux bias ⎛⎞100 0 ⎜ ⎟ line implements a controlled 010 0 cU = ⎜⎟ phase gate 100:1 on-off ratio 11 ⎜⎟001 0 ⎜⎟ Uses higher level of one qubit. ⎝⎠0 00−1
• Avoided crossing (160 MHz)
• A frequency shift
Related earlier idea: Strauch et al., PRL (2003)25
EntanglementBell states on demand on demand
Leo DiCarlo et al., Nature (in press, 2009) arXiv:0903.2030
Earlier entanglement work: Steffen et al., Science26 (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: 0 I II III
“Find the surprise!” 27 GroverTwo-qubit in Grover action Algorithm
Challenge: “oracle” Find the location “unknown” of the -1 !!! unitary operation:
ORACLE
28 10 pulses w/ nanosecond resolution, total 104 ns duration GroverA Grover in step-by-step action movie “quantum debugger”
Begin in ground state:
29 GroverA Grover in step-by-step action movie
Create a maximal superposition: look everywhere at once!
30 GroverA Grover in step-by-step action movie
Apply the “unknown” function, and mark the solution
31 GroverA Grover search in step-by-step action in action movie
Some more 1-qubit rotations…
Now we arrive in one of the four Bell states
32 GroverA Grover search in step-by-step action in action movie
Another (but known) 2-qubit operation now undoes the entanglement and makes an interference pattern that holds the answer!
33 GroverA Grover search in step-by-step action in action movie
Final 1-qubit rotations reveal the answer:
The binary representation of “2”!
The correct answer is found >80% of the time!
34 Grover searchwith in action other in action oracles
Oracle
Fidelity to ideal output (average over 10 repetitions)
35 Deutsch-Jozsa Algorithm
Constant functions
Answer is encoded in the state of left qubit
Balanced functions The correct answer
is found 36 >84% of the time. Summary
Local and fast qubit control Cavity-mediated 2-qubit state tomography interaction tunable by 2 orders of magnitude
• Entanglement on demand Fidelity Concurrence
• Grover algorithm with Fidelity
• Deutsch-Jozsa with Fidelity
37 ArXiv: cond-mat 0903.2030 Circuit QED Team Members: 2007 Jared Steve Blake Schwede Jens Jay Girvin Joe Johnson Koch Gambetta Schreier
Hannes Majer
David Schuster Jerry Chow Luigi Andrew Michel Emily Frunzio Houck Devoret Chan
Funding: