1. 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)

2. 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)

3. 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)

4. ‘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.

6. 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

7. 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, (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

8. 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, (2004)

9. 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

10. Circuit QED Vacuum fields: mode volume
atom artificial atom: SC qubit
cavity D 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

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

12. 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, (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)

13. 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:

14. Emergence of √n multiphoton peaks

15. 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

16. 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, (2006) (theory)
D. Schuster et al., Nature 445, 515 (2007) (experiment)
(c.f. Haroche group in time domain)

17. 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.

18. 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/ ; Nature (2007)

19. “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!

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

21. M. Hofheinz et al. (Martinis group UCSB)

22. 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:
Nature (in press, 2009)

23. 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)

24. 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.

25. 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)

26. 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

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

28. 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:
Earlier entanglement work:
Steffen et al., Science (2006)
Leek et al., cond-mat (2008)

29. 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!”

30. 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

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

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

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

34. 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

35. 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!

36. 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!

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

38. 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.

39. 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

40. 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

41. 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:

42. 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)

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

44. 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

45. 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

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

47. 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

48. 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 = EC
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/ (2004) J. Q. You and F. Nori, Phys. Today, Nov. 2005, 42