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 A. Blais et al., Phys. Rev. A 69, (2004)

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 The qubit 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 The resonator 3D Optical cavities Superconducting LC resonators Large mode volume (3D!) Small mode volume (V m 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 The Jaynes-Cummings Hamiltonian for light-matter interaction Dipole interaction between atom and cavity field Interaction energy ħg (Vacuum field amplitude) x (Atomic dipole moment) H qr = ħω r a a + ħω q 2 σ z + ħg a + a σ + σ + RWA g ω r, ω q Jaynes-Cummings Hamiltonian H JC = ħω r a a + ħω q 2 σ z + ħg a σ + a σ + Interaction picture, U = e it(ω r a a+ ω q 2 σ z) I H JC = ħg a σ e iδt + a σ + e iδt, Detuning δ ω q ω r Two regimes δ = 0 Resonant regime δ 0 Dispersive regime

15 Energy (arb. units) Diagonalizing the Jaynes-Cummings Hamiltonian H JC = ħω r a a + ħω q 2 σ z + ħg a σ + a σ + 3, g 2, e Intuition a n = n + 1 n + 1 a n = n n 1 2, g 1, g 0, g 1, e 0, e Uncoupled Hamiltonian H JC = ħω r a a + ħω q 2 σ z Resonant regime (δ = 0) Level pairs n + 1, g, n, e Effect of the interaction term H JC,int = ħg a σ + a σ + Start in n + 1, g a σ + a σ + n + 1, g = n + 1 n, e Start in n, e a σ + a σ + n, e = n + 1 n + 1, g H JC,int couples only states within the same level pair ( JC doublet ) H JC decouples into infinite direct product of 2 2-matrices H JC,n n, e n + 1, g = ħ nω r + ω q 2 g n + 1 Result is general (only inspired by intuition!) g n + 1 n + 1 ω r ω q 2 n, e n + 1, g AS-Chap

16 Energy (arb. units) The Jaynes-Cummings Hamiltonian for light-matter interaction 3, g 2, e H JC,n n, e n + 1, g = ħ nω r + ω q 2 g n + 1 g n + 1 n + 1 ω r ω q 2 n, e n + 1, g 2, g 1, e δ ω q ω r H TLS = ε 2 σ z + Δ 2 σ x E ± = ± 1 2 ε2 + Δ 2 1, g 0, g 0, e Θ n 1 2 tan 1 2g n + 1 δ H JC,n = ħ nω r + ω q 2 g n + 1 g n + 1 n + 1 ω r ω q 2 = ħω r n r + ħ δ 2 σ z + ħg n + 1 σ x Analytic diagonalization E n,± = ħω r n + 1 ± ħ 2 2 E 0,g = ħ ω q 2 δ 2 + 4g 2 n + 1 Eigenstates n, + = cos Θ n n + 1, g sin Θ n n, e n, = sin Θ n n + 1, g + cos Θ n n, e AS-Chap

17 Energy (arb. units) Dispersive regime of the JC Hamiltonian H int JC = ħg σ a e iδt + σ + ae iδt 3, g 2, g 1, g 0, g 2, e 1, e 0, e Detuning δ g Qualitative discussion δ ω q ω r No transitions (energy mismatch) Heisenberg uncertainty allows for the creation of excitations from the vacuum for time δt ħ δe Excitation needs to jump back Level shifts Second-order effect (enery scale g 2 δ) Eigenstates still predominantly resonatorlike or qubit-like with small admixture of other component A. Blais et al., Phys. Rev. A 69, (2004) AS-Chap

18 Energy (arb. units) Dispersive regime of the JC Hamiltonian 3, g 2, g 1, g 0, g ω r + ω r g 2 g 2 δ δ 2, e 1, e 0, e δ H JC = ħω r a a + ħω q 2 σ z + ħg a σ + a σ + H int JC = ħg σ a e iδt + σ + ae iδt Detuning δ g Quantitative discussion δ ω q ω r H int JC has explicit time dependence Transform H JC via U = e g δ σ+ a σ a U cancels first-order photon exchange terms ħg a σ + a σ + Develop up to second order in g δ H (2) = U H JC U = n U n n! H JC m U = H 00 + H 01 + H 10 + H 02 + H 11 + H 02 Neglect terms with g 2 δ H (2) = ħ ω r + g2 σ Δ z a a + ħ ω 2 q + g2 Δ m! σ z m n,m H nm Qubit readout, dispersive gates, photon number calibration A. Blais et al., Phys. Rev. A 69, (2004) Ac Stark / Zeeman shift Lamb shift AS-Chap

19 Transmission magnitude Transm. phase 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 r + g2 δ ω r g2 δ Frequency 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 magnitude will drop drastically from blue to green curve Mixed state Reduced shift, but still ok Transmission phase can also be used! Measurement operator commutes with qubit Hamiltonian Frequency Quantum nondemoliton measurement Qubit wave function may be projected, but neither destroyed nor unknown AS-Chap

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

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

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

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

24 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

25 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 2π g 2π 105 MHz few 10s of MHz 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

26 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

27 Energy (arb. units) Resonant regime of the JC Hamiltonian H int JC = ħg σ a e iδt + σ + ae iδt 3, g 2, g 1, g 0, g 2g 3 2g 2 2g 2, e 1, e 0, e Detuning δ = 0 H JC = ħω r a a + ħω q 2 σ z + ħg a σ + a σ + Degenerate levels n, e, n + 1, g split into JC doublets ±, n due to coupling g n + 1 Ground state is the vacuum g, 0 δ ω q ω r, n x z e, n θ n g, n + 1 +, n y Dynamics? Create non-eigenstate, e.g., n, e by adiabatically detuning the qubit, sending a πpulse and tuning it back to resonance Coherent population exchange between qubit and resonator n, e n + 1, g Vacuum Rabi oscillations with n-photon Rabi frequency Ω n 2g n + 1 A. Blais et al., Phys. Rev. A 69, (2004) AS-Chap

28 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!

29 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.

30 Experimental demonstration of vaccum Rabi oscillations Phase qubit capacitively coupled to a coplanar 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

31 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

32 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

33 Strong coupling regime Definition motivated goal of observing quantum coherent dynamics Vacuum Rabi dynamics happens at rate g Qubit decays at rate γ Γ 1 + Γ φ, resonator at rate κ g > κ + γ required 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

34 Selection rules & Symmetry breaking H TLS = ε 2 σ z + Δ 2 σ x Selection rules Concept of parity Interaction can be dipolar, quadrupolar, Dipolar interaction Only odd transitions (between levels of different parity) allowed Natural atoms Symmetry of atomic potential Strict selection rules Dipolar EM field naturally couples to dipole moment Quadrupolar Requires strong field gradient (crystal fields) Quantum circuits Normally dipolar, quadrupolar for gradiometric designs Well-defined parity only for symmetric qubit potential Selection rules not always present Experimental access to selection rules Multi-photon excitations 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

35 AS-Chap Selection rules & Symmetry breaking Parity operator Π Eigenvalues +1 (even parity) and 1 (odd parity) Qubit parity operator is Π = σ z Π g = g and Π e = e Parity related to symmetry of the potential Resonator potential symmetric (parabola!) Parity well defined, Π r = e iπ a a Flux qubit potential only symmetric at degeneracy point ε = 0 For ε 0, parity is no longer well defined Parity gives rise to selection rules Interaction operators B ± exhibit symmetries! sin θ = H TLS = ε 2 σ z + Δ 2 σ x ω q = ε 2 + Δ 2 Δ ħω q, cos θ = Even operator B + Π B + Π = + B + Commutator Π, B + = 0 Odd operator B Π B Π = B Anticommutator Π, B + = 0 Ψ odd B + Ψ even = Ψ odd Π B + Π Ψ even = Ψ odd B + Ψ even = 0 Ψ even B + Ψ odd = Ψ even Π B + Π Ψ odd = Ψ even B + Ψ odd = 0 Transitions between levels of different parity forbidden for even operators Investigate qubit-driving-field coupling operators ε ħω q

36 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 Qubit 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 for ε = 0 Two-photon Rabi drive can excite qubit only for ε 0 AS-Chap

37 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 Both 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

38 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

39 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

40 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

41 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)

42 AS-Chap Ultrastrong light-matter coupling Linear LC-resonator (quant. HO) Nonlinear TLS/qubit Artifical atom 75 µm S I S L J (Φ) Josephson junction

43 AS-Chap Ultrastrong light-matter coupling Qubit-resonator coupling naturally determined by quantum Rabi Hamiltonian H qr = ħω q σ 2 z + ħω r a a + ħg σ + a + σ a + σ + a + σ a Maximum coupling strengths Linear coupling Mutual inductance (capacitance) limited by self inductance (capacitance) g ω q ω r Simplified discussion ω q = ω r ω Dimensionless coupling strength g/ω 1 Superconducting circuits Nonlinear coupling via kinetic inductance or Josephson inductance g/ω > 1 expected to be possible

44 Ultrastrong light-matter coupling Regimes of light-matter coupling Jaynes-Cummings regime ( g ω 1) Coupling is small perturbation to self energy terms Excitation exchange dynamics (vacuum Rabi oscillations) Counterrotating terms can be neglected by RWA Number of excitation is a conserved quantity Ground state 0, g Ultrastrong coupling (USC) regime Deviations from JC Hamiltonian observable in experiment Ground state has contributions with n > 0 Perturbative treatment still possible (modified population oscillation physics) In practice typically for g ω 0.1 Deep strong coupling regime g ω > 1.5 Parity chain physics (coherent dynamics happens within two subgoups of states characterized by odd/even parity) The dark regime of USC ( g ω 1) No intuition/analytics, only numerics AS-Chap

45 AS-Chap Ultrastrong light-matter coupling H JC = ħω q 2 σ z + ħω r a a + ħg σ + a + σ a RWA breaks down Ultrastrong coupling regime Experimental strategies H qr = ħω q 2 σ z + ħω r a a + ħg σ + a + σ a + σ + a + σ a Counterrotating term effects are observable in experiment In practice typically for g ω 10% Dynamics (no more vacuum Rabi oscillations) Complicated & demanding Ground state photons Demanding, few photons Spectroscopy (anticrossing) Only quantitative change Nonconservation of excitation number Multimode setup, natural!

46 Ultrastrong light-matter coupling Niobium CPW resonator ω 1 ω 2 ω 3 2π = 2.70 GHz 2π = 5.35 GHz 2π = 7.78 GHz Qubit position 4th harmonic has node 5th harmonic quite detuned H ħ = ω q 2 σ z + 3 k=1 ω k a k a k + 3 k=1 g k σ x a k + a k 3JJ flux qubit (aluminum) Measure frequency-dependent transmission through resonator! Use nonlinear L J as mutual inductance between qubit an resonator! AS-Chap

47 Ultrastrong light-matter coupling ω rf = probe frequency Color code Transmission magnitude Large anticrossings (expected for large g) Almost unaffected, no qualitative change, could well be JC Reason Only single excitation involved! This experiment g/ω up to 0.12 (recently g/ω up to 1.4 has been demonstrated with flux qubits) T. Niemczyk et al., Nature Phys. 6, 772 (2010). AS-Chap

48 Ultrastrong light-matter coupling ω rf = probe frequency Color code Transmission magnitude Small anticrossings Qualitative difference to JC observable (anticrossing vs. crossing) Reason: coupling of states with different number of excitations Experimental proof for existence of counterrotating wave terms T. Niemczyk et al., Nature Phys. 6, 772 (2010). AS-Chap