Elementary Tutorial on Loss Mechanisms in Circuit QED. Steven M. Girvin Yale University

Transcription

1 Elementary Tutorial on Loss Mechanisms in Circuit QED Steven M. Girvin Yale University 1

2 Hard to believe the JJ losses are so small 2

3 Qubits, Cavities, Lumped-Element Oscillators G C tot L Geometric, kinetic, Josephson Inductance Dielectric Loss SC quasi-particles in wires and JJ s Normal Fluid Better Model for SC: (Thevenin/Norton equivalence) G L Superfluid 3

4 Qubits, Cavities, Lumped-Element Oscillators G C L tot eff For weak damping: V = V cosω t 0 0 I = I sinω t Stored Energy: Characteristic Impedance: 1 1 V E = CV = LI = Z, Z I0 L C 4

5 Qubits, Cavities, Lumped-Element Oscillators G C L tot eff For weak damping: V = V cosω t 0 0 I = I sinω t Stored Energy: Characteristic Impedance: 1 1 V E = CV = LI = Z, Z I0 L C If GZ = 1 and / Z = 1 then resonator with impedance Z drives G as a nearly perfect voltage source, and drives as a nearly perfect curren t sourc e. 5

6 Qubits, Cavities, Lumped-Element Oscillators G C L tot eff For weak damping: V = V cosω t 0 0 I = I sinω t Power dissipation: P = GV0 cos ωt+ I0sin ωt = I 0 GZ + 2 If GZ = 1 and / Z = 1 then resonator with impedance Z drives G as a nearly perfect voltage source, and drives as a nearly perfect curren t sourc e. 6

7 Qubits, Cavities, Lumped-Element Oscillators G C tot L eff de dt Q P = κe, κ = E P = = = GZ + ω ω E 1 κ 1 Z Power dissipation: P = GV0 cos ωt+ I0sin ωt = I0Z GZ + 2 Z Stored Energy: E = 1 CV = 1 LI = ω I Z 7

8 Qubits, Cavities, Lumped-Element Oscillators G C tot L eff de dt Q 1 = κe, κ = κ = = GZ + ω P E Z Dielectric Loss: ( j i) G y( ω) = jωc+ G = jω C j = jωc j ω [ Ú Ú] tanδ C Loss tangent: Ú Ú = 2 = = 1 G Cω GZ Ú 2 δ C 1Ú 8

9 Qubits, Cavities, Lumped-Element Oscillators G C tot L eff de dt Q 1 = κe, κ = κ = = GZ + ω P E Z Inductive Loss: ( j i) z( ω) = jωl+ = jω L j ω tanδ L Loss tangent: = Lω = Z 9

10 Qubits, Cavities, Lumped-Element Oscillators G C L tot eff de P G = κe, κ = = + dt E C L 1 κ Q = = GZ + ω Z 1 Q = tanδc + tanδl = Q + Q 1 1 C L 10

11 5 CONTIBUTIONS TO QUBIT DISSIPATION environment capacitance junction capacitance junction environment inductance inductance C tot L eff CE C J LJ LE G cap G E tls? cap G J tls? ind G J qp? ind G E qp? G rad radiation SCALING! G cap E C ω C ω E 01 cap J 01 = G cap J = cap QE QJ G ind J = 1 L ω Q J 01 ind J G ind E = 1 L ω Q E 01 ind J 11

12 PATICIPATION ATIO: CAPACITANCES Definition of participation ratio of [linear] element: fraction of energy stored in element largest capacitance matters most! G E C E G J C J 1 Z p = C ω = C + C cap J ( ) ω tot 01 J E 01 CJ cap = ; pe = C tot cap 1 pe p = + Q Q Q C C cap J cap cap cap E J E tot example: if junction participation is 10% with Q=100,000, expect overall Q=1,000,

13 Surface Dielectric Participation atio G C tot L 3D Cavity 110 mode Dielectric layer 13

14 Surface Dielectric Participation atio and Geometric Dimensions Parallel Plate (t << d) Surface participation ratio: P S t ds dv r r E D r r E D 2t 1 d ε S ( D continuous) Example V2 cavity (TE101 mode): Assuming t = 3 nm, ε s = 10, d = 5 mm, P s = 1.2x10-7!! 1 = tan S S S 14 Q P δ

15 Surface Dielectric Participation atio and Geometric Dimensions Coplanar Waveguide Selected simulation results from Martinis group: (J. Wenner, et al., APL 2011) Assuming t = 3 nm, ε s = 10 g (µm) w (µm) Metal-Substrate P s Effective d (µm) x x x x Effective Parallel Plate Distance 15

16 T1 vs. surface dielectric participation ratio: arxiv: C. Wang et al., Appl. Phys. Lett. 107, (2015) 16

17 Surface current losses G C tot L Surface Impedance encapsulates boundary condition that allows us to ignore details of skin depth, London length, solving Maxwell s equations inside the cavity walls, etc. 3D Cavity 110 mode Surface currents 17

18 Qubits, Cavities, Lumped-Element Oscillators G C tot L Surface Impedance encapsulates boundary condx. Allows us to ignore details of skin depth, London length, etc. 1D toy example Z S = Z C Surface impedance Surface impedance Z S = 0 Z C = l ZS = jωls + S l dx c L S V( x= 0) I( x= 0) c dx = Z = S 0 V( x= D) I( x= D) = Z = jω L + S S S S 18

19 I( x, t) = I cos( kx)sin( ω t) P = I D 1 1 U = dx li cos ( kx) = ldi κ = 2 Q = 2 ld ldω Surface participation ratio p S tanδ Q LS = ( 1/ 2) ld+ S = L S S ω 1 S S S L S = p tanδ = 2 l LS 2 ld S Dω Surface impedance Surface impedance Z S = 0 Z C = l ZS = jωls + S c L S V( x= 0) I( x= 0) = Z = S 0 V( x= D) I( x= D) = Z = jω L + S S S S 19

20 Alternative derivation in terms of reflection coefficient due to impedance mismatch: ound trip time: ZS ZC 1 ZS / ZC r = = 1 2 ZS / Z Z + Z 1 + Z / Z 2 r 1 4 / t 2 κ = t S C S C 4/ Z Z 1 C 2 1 l c2d C = 2 Q = 2 ld l C Dω Surface impedance Surface impedance Z S = 0 Z C = l ZS = jωls + S c L S S 20

21 3D Cavity 110 mode Surface currents Surface current losses Surface Impedance encapsulates boundary condition that allows us to ignore details of skin depth, London length, solving Maxwell s equations inside the cavity walls, etc. For semi-infinite slab of material: 21

22 3D Cavity 110 mode Surface currents Surface current losses Surface Impedance encapsulates boundary condition that allows us to ignore details of skin depth, London length, solving Maxwell s equations inside the cavity walls, etc. For normal metal current penetrates a distance = skin depth d S : Z 1/2 Ex (0) 1 j 1 j µω 0 = = µω 0 ds = = (1 j) H y(0) 2 σds 2σ Q tanδ 1 (!!) Participation ratio: d D = 1 S S = S = PS ~ 1 22

23 3D Cavity 110 mode Surface currents Surface current losses Surface Impedance encapsulates boundary condition that allows us to ignore details of skin depth, London length, solving Maxwell s equations inside the cavity walls, etc. For superconductors, current penetrates a distance = London length. esponse of walls is mostly reactive. λ L Two-fluid model: Normal Fluid n = (1 x ) n, n = x n S qp N qp G L Superfluid 23

24 3D Cavity 110 mode Surface currents Surface current losses Surface Impedance encapsulates boundary condition that allows us to ignore details of skin depth, London length, solving Maxwell s equations inside the cavity walls, etc. For superconductors, current penetrates a distance = London length. esponse of walls is mostly reactive. λ L Two-fluid model: n = (1 x ) n, n = x n S qp N qp Q = Participation tan δ ~ x ~ λ L P ratio: 1 S = D 1 S S qp 24

25 No time to discuss: Quasiparticles in junctions The Purcell Effect Single-qubit, single mode Single-qubit, multi-mode Multi-qubit, single mode Purcell filters Inverse Purcell Effect 25

26 Circuit QED Team Members 2013 Chen Wang Phillip einhold Matt eagor Teresa Brecht Nissim Ofek Jacob Blumoff Luyan Sun Brian Vlastakis Steve Girvin Michel Devoret Kevin Chou Chris Axline Leonid Glazman Luigi Frunzio Eric Holland Z. Leghtas einier Heeres Yvonne Gao M. Mirrahimi Andrei Petrenko Gerhard Kirchmair Liang Funding: Jiang 26

27 Thanks! 27

28 Δ Quantum Participation atio Strong Qubit-Cavity Coupling: Good Cavity Limit cavity Qubit narrower than the cavity. Hybridization of with cavity broadens the qubit. cavity 2g : 250 MHz κ With proper engineering and detuning from cavity resonances, spontaneous fluorescence can be made small. In limit of large detuning: Δ? g 1 1 g + T T Δ N κ 28

29 Cavity Quantum Electrodynamics (cqed) 2g = vacuum abi freq. κ = cavity decay rate γ = transverse decay rate t = transit time Strong Coupling= g > κ, γ, 1/t ˆ 1 + H = hω ( 2 ) ˆ r aa+ h ωσ a z hg( aσ + σ a) 2 Quantized Field Jaynes-Cummings Hamiltonian 2-level system Electric dipole Interaction 29

30 Cavity QED: esonant Case ω r = ω a # of photons vacuum abi oscillations g e + = e,0 + g,1 qubit state dressed state ladders - = e,0 g,1 (e.g. Haroche et al., Les Houches notes) 30

31 Vacuum abi Mode Splitting by an Artificial Atom 2g quton = e,0 g,1 γ κ 2 +phobit = e,0 + g,1 Decay rate of the new e-states: γ + κ 2 31

32 Dispersive egime of Cavity QED Large detuning: Δ= ω? a ω g Δ,0,0 { +,0 g 2 /Δ,0 H eff g h g = h + + Δ 2 Δ 2 2 ω ˆ r σz a a ωa σz See e.g. Haroche, Les Houches 1990 Level shift: 2 ~ g δω Δ 32

33 Lifetime Enhancement: Mixing of Qubit and Photon ω = ω +Δ 01 +,0 = cos θ e,0 + sin θ g,1,0 = sin θ e,0 + cos θ g,1 Δ { +,0,0 e,0 Γ +,0 = cos + sin 2 2 γ θ κ θ Γ,0 = sin + cos 2 2 γ θ κ θ g For Δ? g : θ = 1 Δ g,0 See e.g. Haroche, Les Houches 1990 γ 2 κ γ g Δ 2 κ κγ g Δ Atom/photon acquire a small additional decay from each other! γ 33

34 Purcell Effect Dominant qubit relaxation channel is spontaneous 2 g emission: γ κ = κ > γ Δ 2 N 34

35 Multimode Purcell Theory 35

36 Purcell Filter eed et al., APL 96, (2010). 36

37 Purcell Filter 50 x better than single-mode eed et al., APL 96, (2010). 37

38 ω 1 ω 2 ω meas ω Stark Two Qubits in a Cavity Homodyne Detection Qubit frequencies tunable with magnetic field 38

39 Two Qubit Spectroscopy Frequency (GHz) Magnetic Field (Gauss) 39

40 Qubit-Qubit Avoided Crossing Experiment Hannes Majer & Jerry Chow, st gen. transmon/cqed Frequency (GHz) J T 1 ~ 50 ns κ = 33 MHz ~ 50 ns ( ) Magnetic Field (Gauss) ( + + ) H = J σσ + σσ int Coupling by virtual photons: J = gg 1 2 ~ 26 MHz Δ 40

41 Qubit-Qubit Avoided Crossing Experiment Theory 6.60 Hannes Majer & Jerry Chow Jay Gambetta & Jens Koch Frequency (GHz) J Magnetic Field (Gauss) Coupling by virtual photons: J = gg 1 2 ~ 26 MHz Δ 41