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!) EXPERIMENT Rob Schoelkopf, Michel Devoret Luigi Frunzio THEORY SMG, L. Glazman, Liang Jiang Simon Nigg M. Mirrahimi Z. Leghtas M. Hatridge Shyam Shankar G. Kirchmair Brian Vlastakis Andrei Petrenko Claudia degrandi Uri Vool Huaixui Zheng Richard Brierley Matti Silveri
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Coherent state is closest thing to a classical sinusoidal RF signal ψ( Φ ) = ψ ( Φ α) 0 3
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5 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.
The Josephson tunnel junction is the only known non-linear non-dissipative circuit element 2 Qˆ 2π ˆ H = EJ cos hωb b λb b bb 2C Φ0 Φ + Superconductor ω ω 01 12 1 nm Insulating barrier Superconductor (Al) Energy ω 12 ω 01 1 0 Josephson plasma oscillation of ~3-4 Cooper pairs Non-linear electromagnetic oscillator ω 01 ~5 10GHz 6
Transmon Qubit Energy ω ω 01 12 ω 12 ω 01 1 0 ~ mm Josephson tunnel junction ω 01 ~5 10GHz 11 10 mobile electrons Superconductivity gaps out single-particle excitations Quantized energy level spectrum is simpler than hydrogen Quality factor Q= ωt 1 exceeds that of hydrogen
Remarkable Progress in Coherence Progress = better designs & materials
Transmon Qubit Energy ω ω 01 12 ω 12 ω 01 1 0 Anharmonicity allows us to approximately treat oscillator as a two-level spin. 1 = ω 01 ~5 10GHz 0 = Examples of coherent superpositions 1 2 1 2 ( + ) ( ) = =
Transmon Qubit in 3D Cavity Josephson junction ~ mm spi n Spin flip 50 mm r r r 7 d = 2 e 1 mm 10 Debye!! Vdipole d Erms g= h Huge dipole moment: strong coupling x = gσ (a + a ) g 100 MHz
Quantum op<cs at the single photon level Photon state engineering Goal: arbitrary photon Fock state superpositions ψ = a 0 + a 1 + a 2 + a 3 +K 0 1 2 3 Use the coupling between the cavity (harmonic oscillator) and the two-level qubit (anharmonic oscillator) to achieve this goal. 11
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 Q Φ
Quantum op<cs 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) ω q z z H = ωra a+ σ + χσ a a+ Hdamping 2 resonator qubit Dispersive coupling ω r ω q 13
Dispersive Hamiltonian ω q z z H = ωra a+ σ + χσ a a+ Hdamping 2 resonator qubit dispersive coupling cavity frequency = ω + r z χσ g strong-dispersive limit e 2 χ ~ 2 10 3 κ κ ω ωr χ ωr + χ 14
Strong- Dispersive Limit yields a powerful toolbox g e Cavity frequency depends on qubit state ω ωr χ ωr + χ 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 g D α Engineer s tool #1: Conditional displacement of cavity 15
Making a cat: the experiment Q 0i cavity gi C π M qubit P Φ (*fine print for the experts: this is the Husimi Q function not Wigner)
Making a cat: the experiment Q 0i cavity gi C π M qubit P Φ
Making a cat: the experiment Q 0i cavity gi C π M qubit P Φ
Making a cat: the experiment Q 0i cavity gi C π M qubit P Φ
Making a cat: Q Φ 0i cavity gi qubit after time: C π M P t = π χ qubit acquires π phase per photon
Making a cat: Qubit fully entangled with cavity cat is dead; poison bottle open cat is alive; poison bottle closed Q Φ 0i cavity gi qubit after time: C π M P t = π χ qubit acquires π phase per photon
π n Engineer s tool #2: Conditional flip of qubit if exactly n photons ω q z z H = ωra a+ σ + χσ a a+ Hdamping 2 resonator qubit dispersive coupling Reinterpret dispersive term: - quantized light shift of qubit frequency ωq + 2χaa 2 z σ 22
- quantized light shift of qubit frequency (coherent microwave state) ωq + 2χaa 2 z σ N.B. power broadened 100X 4 3 2 1 2χ Microwaves are par<cles! New low-noise way to do axion dark matter detection?
strong dispersive coupling I V DISPERSIVE z χa aσ Qubit Spectroscopy n = 2 n = 1 n = 0 Coherent state in the cavity Qubit excitation 2χ Conditional bit flip π n 7.83 7.84 7.85 Spectroscopy frequency (GHz) 24
Strong Dispersive Coupling Gives Powerful Tool Set Cavity conditioned bit flip π n Qubit-conditioned cavity displacement g D α 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 25
Combining conditional cavity displacements with conditional qubit flips, one can disentangle the qubit from the photons ψ = 1 g α + e 2 ( α ) D g π α 0 g D α ψ = 1 g + α 2 ( α ) Qubit in ground state; cavity in photon cat state 26
Does it work in prac<ce? Vlastakis et al. Science 342, 607 (2013) ψ = 1 g ± α 2 ( α ) 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 P = e iπa a = ( 1) n P = p n ( 1) n Coherent state: Mean photon number: 4 Even parity cat state: ψ n = β = 2 ψ = β + β Readout signal 10" Photon number 8" 6" 4" 2" 0" Only photon numbers: 0, 2, 4, ˆP ψ =+ ψ Odd parity cat state: Only photon numbers: 1, 3, 5, ˆP ψ = ψ ψ = β β Spectroscopy frequency (GHz)
Wigner Func<on Measurement Vlastakis, Kirchmair, et al., Science (2013) Density Matrix: ρ( Φʹ, Φ) Wigner Function: W ψ ψ ( Q, Φ ) = dψe iqψ ρ( Φ, Φ+ ) 2 2 Handy identity: β =Φ+iQ W ( β) = Ψ D( + β) Pˆ D( β) Ψ ˆ ˆ N ( 1) parity P = = (will explain parity measurement later)
Wigner Func<on Measurement Vlastakis, Kirchmair, et al., Science (2013) Interference fringes prove cat is coherent: - 4 0 4 4 Im β 0 = Q - 4 1 2 α + α Reβ =Φ Rapid parity oscillations With small displacements
Determinis<c Cat State Produc<on Vlastakis, Kirchmair, et al., Science (2013) - 4 4 0 4 Data! 0 Expt l Wigner function - 4
Determinis<c Cat State Produc<on Vlastakis, Kirchmair, et al., Science (in press 2013) - 4 4 0 4 Data! 0 Expt l Wigner function - 4 0.8 0.4 18.7 photons 32.0 photons 38.5 photons 111 photons 111 photons 0.0-0.4-0.8-2 0 2-2 0 2-2 0 2-2 0 2 determined by fringe frequency Most macroscopic superposition ever created?
Determinis<c Photon Cat Produc<on Vlastakis, Kirchmair, et al., Science (2013) - 4 4 0 4 Three-component cat: Four-component cat: 0 Zurek compass state for sub-heisenberg metrology - 4 0.8 0.4 18.7 photons 32.0 photons 38.5 photons 111 photons 111 photons 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 ωq + 2χaa 2 z σ e z σ σ i2χnt ˆ iπnˆ 2 = e 2 z z ˆ 1,3,5,... n = x ˆ 0,2,4,... n = 34
Coherent State = even cat + odd cat 1 1 1 α ( α α ) ( α α ) = 2 + 2 + 2 144424443 144424443 even cat odd cat Measure of parity of a coherent randomly projects onto even or odd cat state 35
Cat = Coherent State Projected onto Parity! L. Sun et al., Nature (July 2014) qubit is in +x> Fidelity of produced cats: qubit is in -x>
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 Four-component cat: Phys. Rev. Lett. 110, 243604 (2013) Z Z Z Z X X X X Dynamically protected cat-qubits: a new paradigm for universal quantum computation Zaki Leghtas et al. New J. Phys. 16, 045014 (2014)
Circuit QED Team Members 2013 Chen Wang Phillip Reinhold Matt Reagor Teresa Brecht Nissim Ofek Jacob Blumoff Luyan Sun Brian Vlastakis Steve Girvin Kevin Chou Chris Axline Leonid Glazman Luigi Frunzio Eric Holland Reinier Heeres Yvonne Gao Andrei Petrenko Gerhard Kirchmair Z. Leghtas M. Mirrahimi Michel Devoret Liang Jiang Funding:
Extra slides beyond here
Tracking photon jumps with repeated quantum non- demoli<on parity measurements L. Sun et al., Nature (July 2014) odd even odd even odd even 400 consecutive parity measurements
0 2 4 6 8 10 Number of parity jumps 12 0 2 4 6 8 10 12 Number of parity jumps Probability (%) Probability (%) ODD CAT α = 2.0 0 2 4 6 8 10 Number of parity jumps ODD CAT 12 Probability (%) Probability (%) EVEN CAT α = 2.0 0 2 4 6 8 10 Number of parity jumps EVEN CAT α = 1.0 α = 1.0 12
arxiv:1212.4000 Stabilizer Quantum Error Correc<on Toolbox for superconduc<ng qubits Simon Nigg and SMG H a a ω a a qj z z = ωr + σ j + χ jσ j j 2 j Kitaev Toric/Surface Topological QEC Code χ j = χ j (fine tuning) Z Z Z Z X X X Stabilizers ZZZZ =+1 XXXX =+1 X Map mul<- qubit parity onto cavity state using new toolbox 42
arxiv:1212.4000 Stabilizer Quantum Error Correc<on Toolbox for superconduc<ng qubits Simon Nigg and SMG H = ω a a+ ω σ + a a χσ r qj z z χ = χ j j j j j 2 j (fine tuning) π i 2 j= 1 Magic iden<ty: N z j σ N N e = ( i) Z j= 1 j Z j σ z j 43
quan<zed light shia in even cat even cat state: Ψ = 1 2 β + β ( ) P = p n ( 1) n =1 n 4 3 2 1
quan<zed light shia in odd cat odd cat state: Ψ = 1 ( 2 β β ) P = p n ( 1) n n = 1 4 3 2 1
Parity 0 100 200 300 Time µ s 400 400 consecutive parity measurements
arxiv:1212.4000 Stabilizer Quantum Error Correc<on Toolbox for superconduc<ng qubits Simon Nigg and SMG N- way stabilizer measurements/pumping for QEC Kitaev Toric/Surface Topological QEC Code Z Z Z Z X X X Stabilizers ZZZZ =+1 XXXX =+1 X 47
Time evolu<on of a coherent state in cavity for <me T = produces an entangled parity cat 2 π χ π i 2 j= 1 Magic iden<ty: N z j σ N N e = ( i) Z j= 1 j Z j σ z j α P ψ + α P + ψ ZZZZ = 1 ZZZZ =+ 1 Measurement of cavity yields N- way parity of qubits 48