Introduction to Circuit QED

Transcription

1 Introduction to Circuit QED Michael Goerz ARL Quantum Seminar November 10, 2015 Michael Goerz Intro to cqed 1 / 20

2 Jaynes-Cummings model g κ γ [from Schuster. Phd Thesis. Yale (2007)] Jaynes-Cumming Hamiltonian Ĥ = ω a 2 ˆσ z + ω c â â + g ( âˆσ + + â ˆσ ) Michael Goerz Intro to cqed 2 / 20

3 outline g κ γ Outline [from Schuster. Phd Thesis. Yale (2007)] 1 Superconducting qubits 2 Coplanar waveguide resonators 3 Combined system Michael Goerz Intro to cqed 3 / 20

4 outline g κ γ Outline [from Schuster. Phd Thesis. Yale (2007)] 1 Superconducting qubits 2 Coplanar waveguide resonators 3 Combined system 4 Towards a network description of superconducting circuits Michael Goerz Intro to cqed 3 / 20

5 superconducting circuits and qubits Michael Goerz Intro to cqed 4 / 20

6 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Michael Goerz Intro to cqed 5 / 20

7 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Josephson junction [from Shalibo. Phd Thesis. H. U. Jerusalem (2012)] Michael Goerz Intro to cqed 5 / 20

8 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Josephson junction capacitance [from Shalibo. Phd Thesis. H. U. Jerusalem (2012)] Michael Goerz Intro to cqed 5 / 20

9 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Josephson junction capacitance tunneling [from Shalibo. Phd Thesis. H. U. Jerusalem (2012)] I (t) = I C sin(φ(t)); U(t) = 2e φ t Michael Goerz Intro to cqed 5 / 20

10 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Josephson junction capacitance tunneling [from Shalibo. Phd Thesis. H. U. Jerusalem (2012)] Ĥ = (2e)2 2C J I (t) = I C sin(φ(t)); U(t) = 2e ( ˆn Q ) 2 r φ2 0 cos ˆθ 2e L J φ t Michael Goerz Intro to cqed 5 / 20

11 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Josephson junction capacitance tunneling [from Shalibo. Phd Thesis. H. U. Jerusalem (2012)] Ĥ = (2e)2 2C J I (t) = I C sin(φ(t)); U(t) = 2e ( ˆn Q ) 2 r φ2 0 cos ˆθ 2e L J φ t cf. LC resonator H = q2 2C + φ2 2L Michael Goerz Intro to cqed 5 / 20

12 superconducting circuits SC circuit toolbox: capacitors, inductors, Josephson elements Josephson junction capacitance tunneling [from Shalibo. Phd Thesis. H. U. Jerusalem (2012)] Ĥ = (2e)2 2C J I (t) = I C sin(φ(t)); U(t) = 2e ( ˆn Q ) 2 r φ2 0 cos ˆθ 2e L J φ t cf. LC resonator H = q2 2C + φ2 2L superconductivity: macroscopic quantum coherence Josephson effect: anharmonic oscillator Michael Goerz Intro to cqed 5 / 20

13 types of superconducting qubits standard SC qubits: charge qubit, flux qubit, phase qubit Φ E C > E J E J > E C E J E C [from Devoret et al. arxiv: (2004)] θ Φ Φ δ π Michael Goerz Intro to cqed 6 / 20

14 transmon qubit E/EC E J /E C = n g E J /E C = n g [from Goerz. Phd Thesis. Kassel (2015)] E J /E C = n g Michael Goerz Intro to cqed 7 / 20

15 transmon qubit E/EC E J /E C = n g E J /E C = n g [from Goerz. Phd Thesis. Kassel (2015)] E J /E C = n g 5 U µm Michael Goerz Intro to cqed 7 / 20

16 transmon qubit E/EC E J /E C = E J /E C = E J /E C = CHAPTER 4. DECOHERENCE n g IN THE COOPER n g PAIR BOX n g 108 [from Goerz. Phd Thesis. Kassel (2015)] a) b) c g1 v c s v c g2 [from Schuster. Phd Thesis. Yale (2007)] Figure 4.13: a) Sketch of the transmon inside of a resonator. The transmon is essentially a CPB Michael Goerz with a large Intro to geometric cqed gate capacitance which lowers E while maintaining β close to unity. The 7 / 20

17 transmon Hamiltonian Anharmonic Oscillator Ĥ = 4E C (ˆn n g ) 2 E J cos ˆφ fore J E C Michael Goerz Intro to cqed 8 / 20

18 transmon Hamiltonian Anharmonic Oscillator Ĥ = 4E C (ˆn n g ) 2 E J cos ˆφ fore J E C Expand cos ˆφ to 1 φ2 2 + φ4 24, using HO ˆb, ˆb (Duffing Oscillator) Ĥ = 8E C E J ˆb ˆb E ) C (ˆb 4 + ˆb + const. 12 Michael Goerz Intro to cqed 8 / 20

19 transmon Hamiltonian Anharmonic Oscillator Ĥ = 4E C (ˆn n g ) 2 E J cos ˆφ fore J E C Expand cos ˆφ to 1 φ2 2 + φ4 24, using HO ˆb, ˆb (Duffing Oscillator) Ĥ = 8E C E J ˆb ˆb E ) C (ˆb 4 + ˆb + const. 12 Leading order perturbation theory on quartic term: Ĥ = ω ˆb qˆb + α ˆb ˆb ˆbˆb 2 with ω q 8E J E C, α E c. Michael Goerz Intro to cqed 8 / 20

20 transmon Hamiltonian Anharmonic Oscillator Ĥ = 4E C (ˆn n g ) 2 E J cos ˆφ fore J E C Expand cos ˆφ to 1 φ2 2 + φ4 24, using HO ˆb, ˆb (Duffing Oscillator) Ĥ = 8E C E J ˆb ˆb E ) C (ˆb 4 + ˆb + const. 12 Leading order perturbation theory on quartic term: Ĥ = ω ˆb qˆb + α ˆb ˆb ˆbˆb 2 with ω q 8E J E C, α E c. Example: α = 300 MHz, E J /E C = 50, ω q = 6 GHz Michael Goerz Intro to cqed 8 / 20

21 Coplanar Waveguide Resonantors Michael Goerz Intro to cqed 9 / 20

22 coplanar waveguide resonator CHAPTER 1. INTRODUCTION 31 a 1 µm b c 5 µm 50 µm [from Schuster. Phd Thesis. Yale (2007)] Figure 1.8: a. Optical image of transmission line resonator. The metal is superconducting niobium, and the substrate is silicon. b. Optical microscope image of one of the coupling capacitors (metal is beige, substrate is green). c. False colored scanning electron micrograph of Cooper pair box (blue) inside the cavity (beige) on the silicon substrate (green). The box consists of two aluminum islands Michael (blue) Goerz connected Intro to bycqed a small tunnel junction (the overlap of the two fingers on the thin island). 10 / 20

23 distributed element description CHAPTER 1. INTRODUCTION 30 L = λ = 25 mm 2 µm 5 µm [from Schuster. Phd Thesis. Yale (2007)] Figure 1.7: Schematic representation of cavity QED with superconducting circuits. Cooper pair box (green) microwave at the center pulses of a coplanar waveguide lump element resonator (blue). description The cavity resonant inaccurate frequency ( 5GHz)issetbythe lengthbetweentwogapcapacitorsattheends, which act as highly reflective microwave mirrors. The cavity is driven from one side and the transmitted signal is measured on the other side. The box is placed at an antinode in the spatial voltage profile for the first harmonic (pink). The system can be accurately modeled as a series of lumped element circuits. advent of cavity QED. The primary contribution of this thesis istoapplythenotionsofcavityqed Michael to the study Goerz of quantum Intro tocircuits, cqed a notion we dub Circuit Quantum Electrodynamics. 11 / 20

24 distributed element description CHAPTER 1. INTRODUCTION 30 L = λ = 25 mm 2 µm 5 µm [from Schuster. Phd Thesis. Yale (2007)] Figure 1.7: Schematic representation of cavity QED with superconducting circuits. Cooper pair box (green) microwave at the center pulses of a coplanar waveguide lump element resonator (blue). description The cavity resonant inaccurate frequency ( 5GHz)issetbythe lengthbetweentwogapcapacitorsattheends, which act as highly reflective microwave mirrors. The cavity is driven from one side and the transmitted signal is measured on the other side. The box is placed at an antinode in the spatial voltage profile for the first harmonic series of infinitessimal LC circuits (pink). The system can be accurately modeled as a series of lumped element circuits. see Blais et al, PRA 69, (2004) advent of cavity QED. The primary contribution of this thesis istoapplythenotionsofcavityqed Michael to the study Goerz of quantum Intro tocircuits, cqed a notion we dub Circuit Quantum Electrodynamics. 11 / 20

25 combined system Michael Goerz Intro to cqed 12 / 20

26 coupling the transmon to a cavity from: J. Koch. PRA 76, (2007) Jaynes-Cummings Hamiltonian TLS in an optical cavity Ĥ = ω c â â + ω q 2 ˆσ+ ˆσ + g (â + â ) (ˆσ + ˆσ +) RWA if ω c ω q ω c + ω q : Ĥ = ω c â â + ω q 2 ˆσ+ ˆσ + g (âˆσ + + â ˆσ ) Michael Goerz Intro to cqed 13 / 20

27 coupling the transmon to a cavity from: J. Koch. PRA 76, (2007) Jaynes-Cummings Hamiltonian TLS in an optical cavity Ĥ = ω c â â + ω q 2 ˆσ+ ˆσ + g RWA if ω c ω q ω c + ω q : Ĥ = ω c â â + ω qˆb ˆb + α 2 ˆb ˆb ˆbˆb + g (â + â ) (ˆσ + ˆσ +) ( âˆb + â ˆb ) Michael Goerz Intro to cqed 13 / 20

28 two coupled transmon qubits A. Blais et al. PRA 75, (2007) Full Hamiltonian Cavity mediates driven excitation of qubit interaction between left and right qubit Ĥ = ω c â â + ω }{{} 1ˆb 1ˆb 1 + ω 2ˆb 2ˆb 2 1 }{{} 2 (α 1ˆb 1ˆb 1ˆb 1ˆb 1 + α 2ˆb 2ˆb 2ˆb 2ˆb 2 ) + 1 }{{} g 1 (ˆb 1â + ˆb 1 â ) + g 2 (ˆb 2â + ˆb 2 â ) + ɛ (t)â + ɛ(t)â }{{}}{{} 4 5 Michael Goerz Intro to cqed 14 / 20

29 parameter regimes g / Γ Resonant Strong g= g 4 / 3 =Γ Rydberg atoms Anharmonic Dispersive Strong Alkali atoms Quasi-Dispersive Strong g= /10 circuit QED Dispersive Strong g 2 / =Γ Dispersive Weak 10 Quantum Dots / Γ [from Schuster. Phd Thesis. Yale (2007)] e 2.4: A phase diagram for cavity QED. The parameter 3Dspace optical is described 1Dby circuit the atom-ph ing strength, dispersive: g, andthedetuning,, ω r /2πbetweentheatomandcavityfrequencies, 350 THz 10 GHz normal e rates ofgdecay represented by Γ g/2π, =max[γ, g/ωκ, 1/T ]. Different cavity QED systems, inclu r 220 MHz, MHz, 10 2 erg atoms, alkali atoms, quantum dots, and circuit QED, are represented by dashed horizo strong coupling: 1/κ, Q = ωr κ 10 ns, µs, 10 4 In the blue region the qubit and cavity are resonant, and undergo vacuum Rabi oscillati e red, weak g dispersive, Γ 1/γ region the ac Stark shift g 2 50 ns 10 µs / < Γ is too small to dispersively res idual photons, but a QND measurement of the qubit can still be realized by using many phot Michael Goerz Intro to cqed 15 / 20

30 CTRODYNAMICS 36 dispersive frame -1 n+1 n (ω a +(2n+1)g 2 / ) (ω r -g 2 / ) ω r g (ω a +3g 2 / ) ω a -ω r = > g (ω a +g 2 / ) (ω r +g 2 / ) e [from Schuster. Phd Thesis. Yale (2007)] Jaynes-Cummings Hamiltonian 1. The dashed lines are n, where left is qubit in the g state and right in the e number. The solid lines are the energies in the presence upled qubit and resonator are resonant (ω a ω r g) t have both photon and qubit character. These levels g strength, and to the square root of the number of ω a n dispersive shift: χ = g 2 Michael Goerz Intro to cqed 16 / 20

31 CTRODYNAMICS 36 dispersive frame -1 n+1 n (ω a +(2n+1)g 2 / ) (ω r -g 2 / ) ω r g (ω a +3g 2 / ) ω a -ω r = > g (ω a +g 2 / ) (ω r +g 2 / ) Jaynes-Cummings Hamiltonian 1. The dashed[ lines g are ) n, where left isdispersive qubit in the frame: g stateû and = right exp in the (âˆσ+ ] e â ˆσ number. The solid lines are the energies in the presence upled qubit and resonator are resonant (ω a ω r g) ÛĤÛ = (ω r χˆσ z )â â + 1 t have both photon and qubit character. These levels 2 (ω a + χ) ˆσ z g strength, and to the square root of the number of e [from Schuster. Phd Thesis. Yale (2007)] ω a n dispersive shift: χ = g 2 Michael Goerz Intro to cqed 16 / 20

32 towards a network description of superconducting circuits Michael Goerz Intro to cqed 17 / 20

33 transmission properties of resonator HAPTER 3. CAVITY QED WITH SUPERCONDUCTING CIRCUITS 53 a) b) 1 Exact Z 0 C in Z 0, L λ/2 C out 1 2 q in =ωc in Z 0 q out =ωc out Z 0 Transmission ( S 21 ) q q T = 2 in out 0 q 2 + q 2 in out Lorentzian 2 κ = ( q 2 + q 2 ) ω n π in out n δω = (q + q )ω in out n [from Schuster. Phd Thesis. Yale (2007)] 0 0 ω n nω λ/2 igure 3.5: Transmission of asymmetric cavity near one of its resonances(ω n ), using an exact nalytic calculation (solid red) and a lorentzian model (dashed blue). The lorentzian is a very good pproximation toimpedance the full transmission mismatch spectrum at near capacitor resonance. acts Theasinput mirror and output capacitor ct as extra stray capacitance to ground shifting the center frequency of the resonance, ω n,downby ω n from the unloaded half-wave resonance nω λ/2.thewidthoftheresonanceisproportionaltothe Input/Output behavior given by scattering matrix quare of the capacitances. If the input and output couplings arethesameandthelineislossless ransmission will be unity. This plot (transmission was generated+ using reflection) q out /q in =5andQ =2/πqout+2/πq 2 in 2 =20. Michael Goerz Intro to cqed 18 / 20

34 superconducting networks arrays of cavities? Michael Goerz Intro to cqed 19 / 20

35 superconducting networks arrays of cavities? beam splitters capacitive junctions (back-reflection!) Michael Goerz Intro to cqed 19 / 20

36 superconducting networks arrays of cavities? beam splitters capacitive junctions (back-reflection!) phase shifters? Michael Goerz Intro to cqed 19 / 20

37 superconducting networks arrays of cavities? beam splitters capacitive junctions (back-reflection!) phase shifters? T-junctions Michael Goerz Intro to cqed 19 / 20

38 superconducting networks arrays of cavities? beam splitters capacitive junctions (back-reflection!) phase shifters? T-junctions... Michael Goerz Intro to cqed 19 / 20

39 superconducting networks arrays of cavities? beam splitters capacitive junctions (back-reflection!) phase shifters? T-junctions... electrical engineering methods for microwave engineering Book: D. Pozar. Microwave Engineering 4th Ed. Wiley (2012). Michael Goerz Intro to cqed 19 / 20

40 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements Michael Goerz Intro to cqed 20 / 20

41 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements circuits yield Ĥ in canonical variables charge, flux Michael Goerz Intro to cqed 20 / 20

42 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements circuits yield Ĥ in canonical variables charge, flux transmon qubit: Duffing oscillator, robust to noise Michael Goerz Intro to cqed 20 / 20

43 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements circuits yield Ĥ in canonical variables charge, flux transmon qubit: Duffing oscillator, robust to noise cavity: coplanar waveguide oscillator Michael Goerz Intro to cqed 20 / 20

44 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements circuits yield Ĥ in canonical variables charge, flux transmon qubit: Duffing oscillator, robust to noise cavity: coplanar waveguide oscillator transmission lines: distributed element description Michael Goerz Intro to cqed 20 / 20

45 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements circuits yield Ĥ in canonical variables charge, flux transmon qubit: Duffing oscillator, robust to noise cavity: coplanar waveguide oscillator transmission lines: distributed element description Outlook: theory of microwave engineering may provide network description Michael Goerz Intro to cqed 20 / 20

46 summary & outlook SC circuit toolbox: capacitances, inductors, Josephson elements circuits yield Ĥ in canonical variables charge, flux transmon qubit: Duffing oscillator, robust to noise cavity: coplanar waveguide oscillator transmission lines: distributed element description Outlook: theory of microwave engineering may provide network description Thank you! Michael Goerz Intro to cqed 20 / 20