10.5 Circuit quantum electrodynamics

Transcription

1 AS-Chap Circuit quantum electrodynamics

2 AS-Chap Analogy to quantum optics Superconducting quantum circuits (SQC) Nonlinear circuits Qubits, multilevel systems Linear circuits Waveguides, resonators Artificial atoms with engineered level spectrum Quantum light (microwaves) In quantum optics for 40+ years Natural atoms/ions/molecules Probed with laser light Quantum light in cavities Cavity quantum electrodynamics (QED) QIP, Qsim, Qcomm Light-matter interaction Test quantum mechanics

3 AS-Chap Analogy to quantum optics (single) natural atom supercond. qubit 3D optical cavity resonator

4 AS-Chap Analogy to quantum optics (single) natural atom single artificial atom 3D optical cavity resonator quasi 1D transmission line resonator

5 AS-Chap Analogy to quantum optics

6 AS-Chap Analogy to quantum optics Cavity QED SQC Circuit QED Quantum electrodynmaics with superconducting circuits

7 AS-Chap Quantum light and Artificial atoms Linear circuits Quantum harmonic oscillator Box for microwave photons Nonlinear circuits Quantum bit Artificial atom 75 µm Quantum light Artificially engineerable quantum matter

8 AS-Chap Natural vs. artificial atoms Size Transition frequencies Design flexibility Tunability Selection rules Lasing or masing Effective dipole moment Coherence times Natural atoms Å IR UV None Small Strict Multiple atoms Small Long (min, s) Artificial solid-state atoms nm-cm µ-wave Large Large Relaxed, controllable Single atom Large Short (ns-ms) Interaction strengths Small (Hz-kHz) Large (MHz-GHz)

9 second order first order Large interaction strengths in superconducting circuits Large effective dipole moment 2 nd -order susceptibility still appreciable! T. Niemczyk et al., Supercond. Sci. Technol. 22, (2009). AS-Chap. 10-9

10 Large interaction strengths in superconducting circuits 3D Optical cavities Superconducting LC resonators Large mode volume (3D!) Small mode volume (V m 10 3 m 3 ) Vacuum field amplitudes E 0 = ħω r 2ε 0 V m, B 0 = μ 0ħω r 2V m E 0 2 /ω r, B 0 2 /ω r small Small interaction strengths E 0 2 /ω r, B 0 2 /ω r large Large interaction strength T. Niemczyk et al., Supercond. Sci. Technol. 22, (2009). AS-Chap

11 AS-Chap Circuit QED vs. cavity QED Key advantages Better prospect of scalability Larger interaction strengths outweigh inferior quantum coherence More quantum operations per coherence time Fundamental quantum effects observable on single-photon level

12 AS-Chap The light-matter interaction Hamiltonian Dipole interaction between atom and cavity field Interaction energy ħg (Vacuum field amplitude) x (Atomic dipole moment) H qr = ħω q 2 σ z + ħω r a a + ħg a + a σ + σ + = σ x Known as the quantum Rabi model because of its relation to the Rabi Hamiltonian

13 Vaccum Rabi oscillations Dipole interaction between atom and cavity field Interaction energy ħg (Vacuum field amplitude) x (Atomic dipole moment) H qr = ħω q 2 σ z + ħω r a a + ħg a + a σ + σ + Classical driving field cos ωt e iωt + e +iωt Quantum driving field a + a RWA Coherent population (Rabi) oscillations for ω = ω q Driving field in the ground state ( g min = 0) No oscillations Full quantum version of Rabi Hamiltonian RWA Quantum oscillations for ω r = ω q Driving field ground state is the vacuum ( g min = g vac > 0) Vacuum Rabi oscillations Vacuum Rabi oscillations describe the coherent exchange of an excitation between qubit and resoantor Signature of a quantum system AS-Chap

14 AS-Chap Vaccum Rabi oscillations Dipole interaction between atom and cavity field Interaction energy ħg (Vacuum field amplitude) x (Atomic dipole moment) H qr = ħω q 2 σ z + ħω r a a + ħg σ + σ + a + a RWA g ω r, ω q Jaynes-Cummings Hamiltonian H JC = ħω r a a + ħω q 2 σ z + ħg σ a + σ + a Interaction picture, U = e it(ω r a a+ ω q 2 σ z) H int JC = ħg σ a e iδt + σ + ae iδt, Detuning δ ω q ω r Two regimes δ = 0 Resonant regime δ 0 Dispersive regime

15 Dispersive regime of the JC Hamiltonian H int JC = ħg σ a e iδt + σ + ae iδt Detuning δ g ( dispersive ) H int JC has explicit time dependence Transform H JC via U = e g δ σ+ a σ a Develop up to second order in g δ Approximate diagonalization H (2) = ħ ω r + g2 σ Δ z a a + ħ ω 2 q + g2 Δ δ ω q ω r σ z g A. Blais et al., Phys. Rev. A 69, (2004) e Ac Stark / Zeeman shift No transitions (energy mismatch) Lamb shift Heisenberg uncertainty allows for the creation of excitations from the vacuum for time δt ħ δe Excitation needs to jump back Level shifts Resonator shift depends on qubit state σ z Dispersive readout, gates Qubit shift depends on resonator population a a Photon number calibration AS-Chap

16 Transmission magnitude AS-Chap Dispersive readout H int JC = ħ g2 e g δ σ z a a + ħ 2 g 2 δ σ z δ ω q ω r ω q + g2 δ Two-tone spectroscopy Send probe tone ω rf = ω r + g2 δ ω r + g2 δ ω r g2 δ Sweep spectroscopy tone ω s When the qubit is in e, the transmission will drop drastically from blue to green curve Mixed state Reduced shift, but still ok Measurement operator commutes with qubit Hamiltonian Frequency Quantum nondemoliton measurement Qubit wave function may be projected, but neither destroyed nor unknown

17 Dispersive readout Spectroscopy tone Readout tone Resonator CPB qubit A. Wallraff et al., Phys. Rev. Lett 95, (2005) AS-Chap

18 Energy relaxation and driven Rabi oscillations A. Wallraff et al., Phys. Rev. Lett 95, (2005) AS-Chap

19 Energy relaxation and driven Rabi oscillations A. Wallraff et al., Phys. Rev. Lett 95, (2005) AS-Chap

20 Ramsey fringes A. Wallraff et al., Phys. Rev. Lett 95, (2005) AS-Chap

21 Dispersive protocol for quantum state transfer δ A,B ω A,B ω r U = e g δ σ A + a σ A a + σ B + a σb a Couple two transmon qubits A nd B to the same resonator Use resonator as quantum bus for state transfer H = ħω A σ 2 z A + ħω B σ 2 z B + ħω r a a + ħg σ A a + + σ A a ħg( σ B a + + σ B a) Dispersive regime g δ H 2 = ħω A 2 σ z A + ħω B 2 σ z B + ħ ω r + g2 δ A σ z A + g2 δ B σ z B a a + ħj σ A σ B + + σ A + σ B Dispersive (2nd order) qubit- qubit coupling J = g δ A δ B σ A σ + + B + σ A σ B can be understood from σ z = σ + σ σ σ + Virtual from qubit A to the resonator and back to qubit B Minus sign from different sign of the mode voltages at the qubit positions J. Majer et al., Nature 449, (2007) AS-Chap

22 Dispersive protocol for quantum state transfer δ A,B ω A,B ω r U = e g δ σ A + a σ A a + σ B + a σb a H 2 = ħω A 2 σ z A + ħω B 2 σ z B + ħ ω r + g2 δ A σ z A + g2 δ B σ z B a a + ħj σ A σ B + + σ A + σ B g 105 MHz 2π J few 10s of MHz 2π Both qubits dispersively shift the resonator Only virtual photons exchanged with the bus Bus can also be used for readout without degrading qubit coherence J. Majer et al., Nature 449, (2007) AS-Chap

23 Dispersive protocol for quantum state transfer δ A,B ω A,B ω r U = e g δ σ A + a σ A a + σ B + a σb a H 2 = ħω A 2 σ z A + ħω B 2 σ z B + ħ ω r + g2 δ A σ z A + g2 δ B σ z B a a + ħj σ A σ B + + σ A + σ B J. Majer et al., Nature 449, (2007) Protocol Qubits detuned at different frequencies Coupling off, J 0 Bring one qubit to e using a π-pulse Apply strong & detuned Stark shift pulse to bring qubits into resonance J 2π 23 MHz for time Δt After the pulse (J 0 again) send readout pulse to resonator Results Qubits coherently exchange population Curved slope due to residual Rabi drive by the off-resonant Stark tone AS-Chap

24 Resonant regime of the JC Hamiltonian H int JC = ħg σ a e iδt + σ + ae iδt, n x g z e, n θ n g, n + 1 A. Blais et al., Phys. Rev. A 69, (2004) e +, n y Detuning δ = 0 H JC = ħω r a a + ħω q 2 σ z + ħg a σ + a σ + Degenerate levels e, n and g, n + 1 Split into JC doublets ±, n due to coupling g +, n = + cos θ n e, n + sin θ n g, n + 1, n = sin θ n e, n + cos θ n g, n + 1 Eigenenergies E ±,n = n + 1 ħω r ± 4g 2 n Δ 2 Resonator state n Splitting 2g n + 1 g ω required for RWA Each doublet is an effective TLS Bloch sphere picture with tan θ n Phase global φ n = 0 δ ω q ω r 2g n+1 Δ AS-Chap

25 Resonant regime of the JC Hamiltonian H int JC = ħg σ a e iδt + σ + ae iδt δ ω q ω r Ground state is the vacuum g, 0 with E g,0 = ħδ 2 Dynamics? Create non-eigenstate, e.g., e, 0 by adiabatically detuning the qubit, sending a π-pulse and tuning it back to resonance, n x g z e, n θ n g, n + 1 e +, n y Coherent population exchange between qubit and resonator Vacuum Rabi oscillations with frequency 2g Intuition σ a swaps excitation from qubit to resonator σ + a swaps excitation back from resonator to qubit Hermitian! Rigorous calculations similar to Rabi oscillations A. Blais et al., Phys. Rev. A 69, (2004) AS-Chap

26 AS-Chap Coherent dynamics Dynamics goverend by interaction Hamiltonian and intial state Ψ 0 Ψ q t = 0, Ψ r t = 0 On resonance (ω q = ω r ) H int JC = ħg σ a + σ + a time-independent Ψ Ψ0 t Ψ q t, Ψ r t = e igt σ a + σ + a igt Ψ 0 = n n σ a + σ + a n Ψ 0 Initial state is ground state, Ψ 0 = g, 0 Ψ g,0 t = n igt n n! σ a + σ + a n g, 0 σ a + σ + a n g, 0 for n = 0 g, 0 = 0 for n > 0 Ψ g,0 t = g, 0 No time evolution when starting from an eigenstate! n!

27 AS-Chap Coherent dynamics Ψ Ψ0 t = n igt n n! σ a + σ + a n Ψ 0 Initial state is e, n Coherent dynamics in doublet ±, n igt Ψ e,n t = n n σ a + σ + a n e, n n! σ a + σ + a m e, n = Ψ e,n t = n 1 n g n+1t 2n n + 1 2n e, n for m = 2n n + 1 2n+1 g, n + 1 for m = 2n + 1 2n! e, n + i n 1 n g n+1t 2n+1 2n+1! Ψ e,n t = cos g n + 1t e, n + i sin g n + 1t g, n + 1 Qubit and resonator exchange an excitation at rate g n + 1 Vaccum Rabi oscillations Initial state has arbitrary cavity component, Ψ e,ψr a n = n + 1 n + 1 a n = n n 1 g, n + 1 e n c n n = n c n e, n Ψ e,n t = n cos g n + 1t e, n + i sin g n + 1t g, n + 1 Superposition of noncommensurate oscillations Beatings, collapse and revivals etc.

28 Experimental demonstration of vaccum Rabi oscillations Phase qubit capacitively coupled to a cploanar waveguide resonator Readout method: Switching method using a dedicated readout SQUID Coupling strength g 2π = 36 MHz Coherence times T 1 q 550 ns T 2 q 100 ns T 1 r 1 μs T 2 r 2 μs M. Hofheinz et al., Nature 454, (2004) AS-Chap

29 Experimental demonstration of vaccum Rabi oscillations Sequential generation of Fock states putting one excitation after another Confirms n + 1-dependence of the vacuum Rabi frequency M. Hofheinz et al., Nature 454, (2004) AS-Chap

30 Experimental demonstration of vaccum Rabi oscillations Initial state g, α Beatings and collapse & revival features observed M. Hofheinz et al., Nature 454, (2004) AS-Chap

31 Strong coupling regime Definition motivated goal of observing quantum coherent dynmaics Vacuum Rabi dynamics happens at rate g Qubit decays at rate γ Γ 1 + Γ φ, resonator at rate κ g > κ + γ rquired to see at leat one Vacuum Rabi flop Strong coupling regime Order-of-magnitude citerion A. Blais et al., Phys. Rev. A 69, (2004) Large coupling strength outweighs restricted coherence of superconducting circuits AS-Chap

32 Selection rules & Symmetry breaking H TLS = ε 2 σ z + Δ 2 σ x Natural atoms Strict selection rules (Dipolar for TLS) Selection rules related to the concept of parity Only transitions between levels of different parity allowed Quantum circuits Well-defined parity only for symmetric qubit potential Selection rules not always present Experimental access to selection rules Multi-photon excitations Degeneracy point (ε = 0) g e transitions allowed only for odd-numbered photon excitations Away from degeneracy point (ε 0) Parity not a well-defined quantity No selection rules, coexistence of one- and two-photon excitation Y-X Liu et al., Phys. Rev. Lett. 95, (2005) F. Deppe et al., Nature Phys. 4, (2008) T. Niemczyk et al., ArXiv: (2011) AS-Chap

33 Selection rules & Symmetry breaking Example: Flux qubit inductvely coupled to resonator H = ε σ 2 z + Δ σ 2 x + ħω r a a + ħg σ z a + a + Ω σ 2 z cos ωt Qubit Resonator JC coupling Drive sin θ = H TLS = ε 2 σ z + Δ 2 σ x ω q = ε 2 + Δ 2 Δ ħω q, cos θ = ε ħω q First-order Hamiltonian using ω q = ω H 1 Second-order Hamiltonian using ω q = 2ω H 2 = ħ ω q 2 σ z + ħω r a a ħg sin θ a σ + a σ + Ω 4 sin θ σ e iωt + σ + e iωt = ħ ω q 2 σ z + ħω r a a ħg sin θ a σ + a σ + Ω2 4Δ sin2 θ cos θ σ e i2ωt + σ + e i2ωt One-photon Rabi drive can excite qubit at any operating point Two-photon Rabi drive can excite qubit only away from degeneracy point AS-Chap

34 AS-Chap Selection rules & Symmetry breaking Parity operator Π Eigenvalues +1 (even parity) and 1 (odd parity) Qubit parity operator is Π = σ z In other words, Π g = g and Π e = e Parity related to symmetry of the potential sin θ = Resonator potential symmetric (parabola!) Parity well defined Flux qubit potential only symmetric at degeneracy point ε = 0 For ε 0, parity is no longer well defined H TLS = ε 2 σ z + Δ 2 σ x ω q = ε 2 + Δ 2 Δ ħω q, cos θ = ε ħω q Parity is related to selection rules! Even operator B + Π B + Π = B + Commutator Π, B + = 0 Odd operator B Π B Π = B Anticommutator Π, B + = 0 odd B + even = odd Π B + Π even = odd B + even = 0 even B + odd = even Π B + Π odd = even B + odd = 0 Transitions between levels of different parity Forbidden for even operators Selection rules only exist if parity is well defined!

35 Selection rules & Symmetry breaking H TLS = ε 2 σ z + Δ 2 σ x Qubit degeneracy point, ε = 0, interaction picture One-photon drive operator σ x has odd parity, σ z, Ω 4 σ x = 0 One-photon transitions allowed Two-photon drive operator has even parity, σ z, Ω2 8Δ σ z = 0 Two-photon transitions forbidden Dipolar selection rules! Atrificial atom behaves like natural atom sin θ = ω q = ε 2 + Δ 2 Δ ħω q, cos θ = ε ħω q Away from qubit degeneracy point, ε 0, interaction picture σ z cos θ σ z sin θ σ x Drive operator does not have a well-defined parity One- and two-photon transitions allowed Artificial atom different from natural atom Physics in circuit QED goes beyond the physics of cavity QED AS-Chap

36 Selection rules & Symmetry breaking Example: ω q = ω r Φ x = 1.5Φ 0 ε = 0 g sin θ 2 2 qubit potential symmetric e g sin θ 1 ω r = ω 1 ω q dipolar selection rules no transitions under two-photon driving g Qubit ω 2 = ω q 2 ω 1 Resonator = ω q 0 AS-Chap

37 Selection rules & Symmetry breaking Example: ω q = ω r Φ x 1.5Φ 0 ε 0 g sin θ 2 2 symmetry of qubit potential broken ω q e g sin θ 1 ω r = ω 1 one- and two-photon excitations coexist g Qubit Resonator 0 upconversion dynamics ω 2 = ω q 2 ω 1 = ω q AS-Chap

38 Selection rules & Symmetry breaking H TLS = ε 2 σ z + Δ 2 σ x In two-tone spectroscopy, the one-photon dip is clearly visible at the qubit degeneracy point AS-Chap

39 AS-Chap Selection rules & Symmetry breaking H TLS = ε 2 σ z + Δ 2 σ x Two-photon dip not observed near the qubit degeneracy point Bare qubit resonator anticrossing observable (no drive photons put into resonator)