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

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

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

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

Remarkable Progress in Coherence Progress = better designs & materials

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

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

Diagonalize quadratic Hamiltonian to obtain normal modes

dressed resonator dressed qubit

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

(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

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

‘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

‘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

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

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

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

Coherent state is closest thing to a classical sinusoidal RF signal

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.

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.

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., 2009 (UCSB – Martinis/Cleland) ~ 10 photons ~ 10 photons

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

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

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

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

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

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

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

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…

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…

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

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?

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

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

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

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)

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)

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

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

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

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 -2 0 2 -2 0 2 -2 0 2 -2 0 2 Most macroscopic superposition ever created? determined by fringe frequency

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 -2 0 2 -2 0 2 -2 0 2 -2 0 2 determined by fringe frequency

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

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:

-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, 243604 (2013) Dynamically protected cat-qubits: a new paradigm for universal quantum computation Zaki Leghtas et al. New J. Phys. 16, 045014 (2014) Four-component cat:

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:

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

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

Map multi-qubit parity onto cavity state using new toolbox arXiv:1212.4000 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

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

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

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

Parity 100 200 300 400 400 consecutive parity measurements

N-way stabilizer measurements/pumping for QEC arXiv:1212.4000 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

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