1. Introduction to Circuit QED: Part II Josephson junction qubits
Schrödinger cat states of photons

2. The Josephson tunnel junction is the only known non-linear non-dissipative circuit element
Superconductor
1 nm
Insulating barrier
Energy
Superconductor (Al)
Josephson plasma
oscillation of
~3-4 Cooper pairs
Non-linear
electromagnetic
oscillator

3. Transmon Qubit Anharmonicity allows us to approximately treat
Energy
Anharmonicity allows us to approximately treat
oscillator as a two-level ‘spin’.
Examples of coherent
superpositions

4. Transmon Qubit
Energy
Josephson
tunnel junction
~ mm
Superconductivity gaps out single-particle excitations
Quantized energy level spectrum is simpler than hydrogen
Quality factor exceeds that of hydrogen

5. Remarkable Progress in Coherence
Progress = better designs & materials

6. Transmon Qubit in 3D Cavity
spin
Josephson junction
~ mm
50 mm

7. Transmon Qubit in 3D Cavity
spin
Josephson junction
~ mm
50 mm
g  100 MHz
Huge dipole moment: strong coupling

8. Diagonalize quadratic Hamiltonian to obtain normal modes

9. dressed resonator
dressed qubit

10. Now express quartic term in normal modes:
(large) Dressed qubit anharmonicity
(medium) Qubit-Cavity cross-Kerr
(small) Cavity self-Kerr

11. (large) Dressed qubit anharmonicity
(medium) Qubit-Cavity cross-Kerr
(small) Cavity self-Kerr
Qubit-Cavity cross-Kerr: Frequency of cavity depends on excitation number of the qubit

12. Transmon Qubit Anharmonicity allows us to approximately treat
Energy
Anharmonicity allows us to approximately treat
oscillator as a two-level ‘spin’.
Examples of coherent
superpositions

13. ‘Dispersive’ coupling
Qubit-Cavity cross-Kerr: Frequency of cavity depends on excitation number of the qubit
Qubit-Cavity cross-Kerr for two lowest levels of dressed transmon.
‘Dispersive’ coupling

14. ‘Dispersive’ coupling
Qubit-Cavity cross-Kerr for two lowest levels of dressed transmon.
‘Dispersive’ coupling
Can read out qubit state by measuring cavity resonance frequency
cavity response

15. Can read out qubit state by measuring cavity resonance frequency
cavity response
cavity
circulator
quantum limited amplifier x
reflection phase

16. X Y
State of qubit is entangled with the ‘meter’ (microwave phase) Then ‘meter’ is read with amplifier.
cavity
circulator
quantum limited amplifier x
reflection phase

17. Departments of Physics and Applied Physics, Yale University
Circuit QED: Taming the World’s Largest Schrödinger Cat and Measuring Photon Number Parity (without measuring photon number!)
THEORY
SMG, L. Glazman, Liang Jiang
Simon Nigg
M. Mirrahimi
Z. Leghtas
Claudia deGrandi
Uri Vool
Huaixui Zheng
Richard Brierley
Matti Silveri
EXPERIMENT
Rob Schoelkopf,
Michel Devoret
Luigi Frunzio
M. Hatridge
Shyam Shankar
G. Kirchmair
Brian Vlastakis
Andrei Petrenko

19. Coherent state is closest thing to a classical sinusoidal RF signal

21. How do we create a cat?
‘Classical’ signal generators only displace the vacuum and create coherent states. We need some non-linear coupling to the cavity via a qubit.

22. Quantum optics at the single photon level
Photon state engineering
Goal: arbitrary photon Fock state superpositions
Use the coupling between the cavity (harmonic oscillator)
and the two-level qubit (anharmonic oscillator)
to achieve this goal.

23. Previous State of the Art for Complex Oscillator States
Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST – Wineland group)
Rydberg atom cavity QED
Phase qubit circuit QED
Haroche/Raimond, 2008 Rydberg (ENS)
Hofheinz et al., (UCSB – Martinis/Cleland)
~ 10 photons
~ 10 photons

24. Quantum optics at the single-photon level
Large dipole coupling of transmon qubit to cavity permits:
Quantum engineer’s toolbox to make arbitrary states:
‘Dispersive’ Hamiltonian: qubit detuned from cavity
-qubit can only virtually absorb/emit photons
(DOUBLY QND)
resonator
qubit
Dispersive coupling

25. Dispersive Hamiltonian
resonator
qubit
dispersive coupling
‘strong-dispersive’ limit

26. Strong-Dispersive Limit yields a powerful toolbox
Cavity frequency depends on qubit state
Microwave pulse at this frequency excites cavity only if qubit is in ground state
Microwave pulse at this frequency excites cavity only if qubit is in excited state
Engineer’s tool #1:
Conditional displacement of cavity

27. Making a cat: the experiment
cavity
qubit
𝐶 𝜋 P M
(*fine print for the experts: this is the Husimi Q function not Wigner)

28. Making a cat: the experiment
cavity
qubit
𝐶 𝜋 P M

29. Making a cat: the experiment
cavity
qubit
𝐶 𝜋 P M

30. Making a cat: the experiment
cavity
qubit
𝐶 𝜋 P M

31. Making a cat: 𝐶 𝜋 M P after time: qubit acquires p phase per photon…
cavity
qubit
𝐶 𝜋 P M
after time:
qubit acquires p phase per photon…

32. Making a cat: 𝐶 𝜋 M P Qubit fully entangled with cavity
‘cat is dead; poison bottle open’
‘cat is alive; poison bottle closed’
cavity
qubit
𝐶 𝜋 P M
after time:
qubit acquires p phase per photon…

33. Conditional flip of qubit if exactly n photons
Engineer’s tool #2:
Conditional flip of qubit if exactly n photons
resonator
qubit
dispersive coupling
Reinterpret dispersive term:
- quantized light shift of qubit frequency

34. Microwaves are particles!
quantized light shift of qubit frequency
(coherent microwave state)
N.B. power broadened
100X … 2c
Microwaves are particles!
New low-noise way to do axion dark matter detection?

35. strong dispersive coupling I
Qubit Spectroscopy
Coherent state
in the cavity
Conditional bit flip

36. Strong Dispersive Coupling Gives Powerful Tool Set
Cavity conditioned bit flip
Qubit-conditioned cavity displacement
multi-qubit geometric entangling phase gates (Paik et al.)
Schrödinger cats are now ‘easy’ (Kirchmair et al.)
Photon Schrödinger cats on demand
experiment theory G. Kirchmair M. Mirrahimi
B. Vlastakis Z. Leghtas
A. Petrenko

37. Combining conditional cavity displacements with conditional qubit flips, one can disentangle the qubit from the photons
Qubit in ground state; cavity in photon cat state

38. Does it work in practice?
Vlastakis et al. Science 342, 607 (2013)
To prove the cat is not an incoherent mixture:
- measure photon number parity in the cat
measure the Wigner function
(phase space distribution of cat)

39. Proving phase coherence via Parity
Photon number 10 8 6 4 2
Coherent state:
Mean photon number: 4
Readout signal
Even parity cat state:
Only photon numbers: 0, 2, 4, …
Odd parity cat state:
Only photon numbers: 1, 3, 5, …
Spectroscopy frequency (GHz)

40. Wigner Function Measurement
Vlastakis, Kirchmair, et al., Science (2013)
Density Matrix:
Wigner Function:
Handy identity:
(will explain parity
measurement later)

41. Wigner Function Measurement
Vlastakis, Kirchmair, et al., Science (2013)
Interference fringes prove cat is coherent: 4 -4
Rapid parity oscillations
With small displacements

42. Deterministic Cat State Production
Vlastakis, Kirchmair, et al., Science (2013) 4 -4
Data!
Expt’l Wigner function

43. Deterministic Cat State Production
Vlastakis, Kirchmair, et al., Science (in press 2013) 4 -4
Data!
Expt’l Wigner function
18.7 photons
32.0 photons
38.5 photons
111 photons
0.8
111 photons
0.4
say measured Wigner function
0.0
-0.4
-0.8
Most macroscopic superposition ever created?
determined by fringe frequency

44. Deterministic Photon Cat Production
Vlastakis, Kirchmair, et al., Science (2013)
Three-component cat:
Four-component cat: 4 -4
Zurek ‘compass’ state for
sub-Heisenberg metrology
18.7 photons
32.0 photons
38.5 photons
111 photons
0.8
111 photons
0.4
0.0
-0.4
-0.8
determined by fringe frequency

45. Measuring Photon Number Parity
- use quantized light shift of qubit frequency

46. Cat = Coherent State Projected onto Parity!
L. Sun et al., Nature (July 2014)
“qubit is in |+x>”
“qubit is in |-x>”
Fidelity of produced cats:

47. -Continuous QND monitoring of -photon number parity
No time to talk about:
-Continuous QND monitoring of
-photon number parity
-multi-qubit parity (stabilizers)
-Bell inequality violation between a qubit and a macroscopic cat
-Quantum error correction using cat state encoding
Stabilizer Quantum Error Correction Toolbox for Superconducting Qubits
(topological Kitaev toric code)
Simon Nigg and SMG
Phys. Rev. Lett. 110, (2013)
Dynamically protected cat-qubits:
a new paradigm for universal quantum computation
Zaki Leghtas et al. New J. Phys. 16, (2014)
Four-component cat:

48. Circuit QED Team Members 2013
Chen Wang
Nissim Ofek
Phillip Reinhold
Matt Reagor
Jacob Blumoff
Teresa Brecht
Luyan Sun
Brian Vlastakis
Reinier Heeres
Kevin Chou
Chris Axline
Andrei Petrenko
Eric Holland
Yvonne Gao
Gerhard Kirchmair
Steve
Girvin
Luigi Frunzio
Leonid Glazman
M. Mirrahimi
Z. Leghtas
Liang Jiang
Michel Devoret
Funding:

49. Tracking photon jumps with repeated quantum non-demolition parity measurements
L. Sun et al., Nature (July 2014)
even
odd
400 consecutive parity measurements

50. EVEN CAT ODD CAT Number of parity jumps Number of parity jumps ODD CAT
Probability (%)
Probability (%) 2 4 6 8 10 12 2 4 6 8 10 12
Number of parity jumps
Number of parity jumps
ODD CAT
EVEN CAT
Probability (%)
Probability (%) 2 4 6 8 10 12 2 4 6 8 10 12
Number of parity jumps
Number of parity jumps

51. Map multi-qubit parity onto cavity state using new toolbox
arXiv:
Stabilizer Quantum Error Correction Toolbox for superconducting qubits
Simon Nigg and SMG
(fine tuning)
Kitaev Toric/Surface Topological QEC Code
Stabilizers
Map multi-qubit parity
onto cavity state
using new toolbox

52. (fine tuning) Magic identity: arXiv:1212.4000
Stabilizer Quantum Error Correction Toolbox for superconducting qubits
Simon Nigg and SMG
(fine tuning)
Magic identity:

53. quantized light shift in ‘even’ cat
even cat state: …

54. quantized light shift in ‘odd’ cat
odd cat state: …

55. Parity
100
200
300
400
400 consecutive parity measurements

56. N-way stabilizer measurements/pumping for QEC
arXiv:
Stabilizer Quantum Error Correction Toolbox for superconducting qubits
Simon Nigg and SMG
N-way stabilizer measurements/pumping for QEC
Kitaev Toric/Surface Topological QEC Code
Stabilizers

57. Time evolution of a coherent state in cavity for time
produces an entangled ‘parity cat’
Magic identity:
Measurement of cavity yields N-way parity of qubits