6.4 Physics of superconducting quantum circuits

Transcription

1 AS-Chap Physics of superconducting quantum circuits

2 6.4 Physics of Superconducting Quantum Circuits AS-Chap General Hamiltonian H HO = E kin + E pot = p m + 1 mω r x k x e.g., 1 k x for mass on spring m LC resonant circuit H LC = E kin + E pot = q C + Φ L = q C + 1 C 1 LC Φ L C Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC Completely analogous treatment

3 6.4 Physics of Superconducting Quantum Circuits E pot ħ Continuous-variable operators Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC L C Parabolic potential q and Φ form a conjugate pair such as x and p Heisenberg uncertainty: ΔqΔΦ ħ Commutation relation Φ, q = iħ q, Φ AS-Chap

4 6.4 Physics of Superconducting Quantum Circuits E pot ħ E n ħ, Discrete Basis (Fock states) Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC L C a ω rc Φ+i q ω r Cħ a ω rc Φ i q ω r Cħ H LC = ħω r a a + 1 is the annihilation operator is the creation operator 3 Photon number operator n a a Eigenstates are Fock states H LC n = E n n 1 0 ω r q ω r Eigenvalues E n = ħω r n + 1 Linear system Equidistant level spacing n = 0 Finite vacuum energy E 0 = ħω r n is the Fock or number basis AS-Chap

5 E pot ħ E n ħ, Annihilation and creation operator Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC L C a ω rc Φ+i q ω r Cħ a ω rc Φ i q ω r Cħ H LC = ħω r a a + 1 is the annihilation operator is the creation operator a When applied to a Fock state, a annihilates a photon inside the oscillator a n = n n 1 a ω r When applied to a Fock state, a creates a photon inside the oscillator q ω r a n = n + 1 n + 1 AS-Chap

6 E pot ħ E n ħ, Coherent states Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC L C a ω rc Φ+i q ω r Cħ a ω rc Φ i q ω r Cħ H LC = ħω r a a + 1 is the annihilation operator is the creation operator a, a = 1 Bosonic commutation relation a Eigenstates of a a α = α α, α C a ω r Coherent states α α = e α n α n n! n q ω r { α } normal but not orthogonal Overcomplete AS-Chap

7 E pot ħ E n ħ, Practical importance of coherent states Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC L C H LC = ħω r a a + 1 a α = α α, α C Bosonic field amplitude operator A(t) 1 ( a + a ) a For intuitive understanding Move to interaction picture a ω r U = e ( )iω rt a a A I t U A U = 1 ae iω rt + a e +iω rt q ω r AS-Chap

8 E pot ħ E n ħ, Practical importance of coherent states Momentum p Charge q Position x Flux Φ Mass m Capacitance C Resonance frequency ω r ω r = 1 LC L C A(t) 1 ae iω rt + a e +iω rt a α = α α, α C Classical limit α A(t) α = α α α e iω rt + α α α e+iω rt a = 1 = 1 = α e i ω rt+φ + e +i ω rt+φ a ω r = α cos ω r t + φ Oscillating field with amplitude α and phase φ = arg α q ω r Coherent state Most classical quantum state (expectation values obey classical equations of motion) AS-Chap

9 AS-Chap λ/ coplanar waveguide resonator (quasi-1d) H LC = ħω r a a µm Each mode n k B T ħω n LC quantum harmonic oscillator Standing waves ( modes ) H TL = ħ n ω n a n a n

10 Spectrum of H LC H LC = ħω r a a + 1 γ 1 γ Must consider coupling to external channel Loss rates γ 1 and γ Simplification γ γ 1 = γ a in L C a out Challenge Describe interaction between a single mode field and a continuum of modes! Measurement requires two components Input probe field a in Detected output field a out Properties of a in and a out free (propagating) multimode fields Field a inside the resonator is a single mode-field Transition mediated by coupling capacitors Borrow input-output formalism from quantum optics! D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

11 Input-output fundamentals Propagating on-chip microwave fields are quasi-1d No polarization Electric field operator of a free multi-mode field propagating along the +x-direction E + x, t = i Mode sum n=0 ħω n ε 0 V b ne iω n t vacuum Amplitude x c Propagation axis Frequency Mode spectrum centered on carrier frequency Ω ω & Continuum limit Dimensionless Fourier amplitudes of mode n Mode volume Bosonic field operator of dimensions b x, t = e iω t x 1 c dω b ω e iω t x c b ω 1, b ω = δ(ω 1 ω ) π s 1 dt e i ω 1 ω t = π δ ω 1 ω Propagation phase often irrelevant & frame rotating with Ω b t = b x > 0, t = 1 π dω b ω e iωt D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

12 λ/4-resonator (single port) H LC = ħω r a a + 1 γ = 0 γ 1 a in Frame rotating with Ω L a C a out V t = iħ Interaction energy dωg ω b ω a b ω a Heisenberg picture Wavefunction constant Operators time-dependent Explicit time dependence of frequency space operators b ω b t, ω H = H LC + ħω b t, ω b t, ω + iħg ω b t, ω a b t, ω a d dt b t, ω = i ħ H, b t, ω + t b t, ω d dt b t, ω = iω b ω + g ω a D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

13 λ/4-resonator (single port) γ = 0 γ 1 a in d dt b t, ω = iω b t, ω + g ω a L a C a out Initial condition b 0 ω b t 0, ω Final condition b 1 ω b t 1, ω Usually t 0 = and t 1 = (free field) Solutions Input (t 0 < t) b t, ω = e iω t t 0 b 0 ω + g ω t dt e iω t t a t t 0 Output (t < t 1 ) b t, ω = e iω t t 1 b 1 ω g ω t 0 t dt e iω t t a t D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

14 λ/4-resonator (single port) γ = 0 γ 1 dωe iω t t = πδ t t d dt b t, ω = iω b t, ω + g ω a a in Resonator field Equation of motion L a C a out d dt a t = i ħ H LC, a dωg ω b t, ω Coupling to multi-mode field! d dt a t = i ħ H LC, a(t) dωg ω e iω t t 0 b 0 ω Input field a in t 1 γ 1 πg ω Frequencyindependent & π well-behaved functions dωg ω t t 0 dt e iω t t a t dωe iω t t 0 b 0 ω a in t, a in t = δ t t d dt a t = i ħ H LC, a t + γ 1 a t + γ 1 a in t D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

15 λ/4-resonator (single port) γ = 0 γ 1 γ 1 πg ω a in t 1 π dωe iω t t 0 b 0 ω a in Output field Analogous result L a C a out t 1 π dωe iω t t 1 b 1 ω a out a out t, a out t = δ t t d dt a t = i ħ H LC, a t γ 1 a t + γ 1 a in t d dt a t = i ħ H LC, a t + γ 1 a t γ 1 a out t a in t + a out t = γ 1 a t Fourier integral a t = 1 π dωe iω t t 0 a ω Solving differential equations simplifies to root finding! D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

16 λ/4-resonator (single port) γ = 0 γ 1 a t = 1 π dωe iω t t 0 a ω a in H LC = ħω r a a + 1 a in t + a out t = γ 1 a t L a C a out d dt a t = i ħ H LC, a t γ 1 a t + γ 1 a in t iω a ω = iω r a ω γ 1 a ω + γ 1 a in ω a in ω + a out ω = γ 1 a ω D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

17 λ/4-resonator (single port) γ = 0 γ 1 a in H LC = ħω r a a + 1 a in t + a out t = γ 1 a t L a C a out Reflection coefficient Γ a out a in = γ1 +i ω ω r γ1 i ω ω r = γ1 +i ω ωr γ 1 ω ω r γ1 + ω ωr What goes in, must go out Γ = 1 Resonator field Γ r a a in = γ 1 γ1 i ω ω r Γ r is Lorentzian Field enhancement inside resonator D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

18 λ/-resonator (two ports) a t = 1 π dωe iω t t 0 a ω γ γ 1 b in a in H LC = ħω r a a + 1 a in ω + a out ω = γ 1 a ω L a C a out b out d dt a t = i ħ H LC, a t γ 1 + γ a t + γ 1 a in t + γ b in t iω a t = iω r a t γ 1 + γ a t + γ 1 a in t + γ b in t a in t + a out t = γ 1 a t b in ω + b out ω = γ a ω D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) AS-Chap

19 λ/-resonator (two ports) a t = 1 π dωe iω t t 0 a ω γ γ 1 b in a in H LC = ħω r a a + 1 a in ω + a out ω = γ 1 a ω L a C a out Transmission coefficient ( b in = 0, γ 1 = γ γ) b out Now reflection and transmission (two ports) Γ + T = 1 T b out a in = γ γ i ω ω r Transmitted power T is Lorentzian! (equals the classical result) T Δω = γ D. F. Walls & G. Milburn, Quantum Optics (Springer, Berlin-Heidelberg, 008) ω r ω AS-Chap

20 6.4 Physics of Superconducting Quantum Circuits.. Resonator lifetime T 1 λ/-resonator (two ports) γ γ T = γ γ i ω ω r b in a in T L b out a C a out ω r Δω κ = γ ω Storage time of HO characterized by its quality factor Q f r Δf = ω r Δω = ω r κ, where ω r πf r Energy-time uncertainty ΔEΔt ħ ΔωΔt 1 E = ħω Identify Δt with dephasing time T T = 1 πδf Typically, ω, γ, κ are angular frequencies, but f, T 1, T, T, T φ are normal frequencies AS-Chap

21 6.4 Physics of Superconducting Quantum Circuits.. Resonator lifetime T 1 AS-Chap λ/-resonator (two ports) γ γ T = γ γ i ω ω r b in a in T L b out a C a out ω r Δω κ = γ ω Q γ 1 Quality factor determined by loss ports Many types of loss ports Coupling capacitors Q c Internal dissipative/dielectric losses Q i Radiation losses Q rad Loaded quality factor 1 Q = 1 Q c + 1 Q i + 1 Q rad +

22 AS-Chap D resonators T = 1 πδf Niobium on SiO -coated high-resistivity Si temperatures f 0 = few GHz Q i 10 5 T 10 μs MBE grown (epitaxially) Al on sapphire mk temperatures f 0 = 6.11 GHz Q i = , Q c = T 0.1 ms Megrant et al., APL 100, (01)

23 3D resonators Loss sources for planar superconducting resonators Resistive or QP losses Superconductivity & low temperatures Radiation losses Clever design Problem: Dielectric losses from material defects (spurious TLS) TLS in bulk substrate Use clean single crystal (sapphire, intrinsic Si) TLS at substrate-metal interface Need clean materials & growth processes! Alternative: 3D cavity resonators No more dielectric No TLS in cavity Q i Reduce participation ratio of interace losses for embedded circuits 3D transmon qubit T ms Paik et al., PRL 107, (011) T = 1 πδf AS-Chap

24 AS-Chap Applications of quantum harmonic oscillators Quantum HO is linear Not a qubit Not directly useful for quantum computation! Quantum simulation of many-body Hamiltonians Nevertheless indirect use: Mediate coupling between qubits Quantum bus Qubit readout ( dispersive readout ) L C Typically long coherence times Quantum memory Ancilla qubit/nonlinearity Explore quantum physics (Fock states, squeezing etc.) Identify decoherence sources in superconducting quantum circuits

25 E pot ħ E pot ħ AS-Chap Josephson nonlinearity Linear system (parabolic potential) Nonlinear system (general potential) ω r ω r ω r ω r ω 1 ω 01 q q L C L C

26 How much anharmonicity is required? Coherence time: T dec 1 δν 01 = ν 01 δν 01 1 ν 01 = Q 1 ν 01 1 bit operation time: t op > 1 Δν (otherwise 1 -transitions are induced!) # of 1 bit operations: T dec t op = Q ν 01 t op < Q Δν ν 01 Anharmonicity AS-Chap

27 AS-Chap Repetition -- Quantum description of JJ C Charging energy ``kinetic E kin = en C = 4E C N L J (I c, φ) Josephson energy ``potential E pot = I cφ 0 π = E J (1 cos φ) R Quantization Hamiltonian N N = i φ H J = 4E C φ φ φ (ΔN) Δφ 1 + E J (1 cos φ) q = e N Φ = Φ 0 π φ Characteristic energies Typical parameters Al/AlO x /Al junction with A = nm C 1 ff and I c 300 na E C 60 μev and E J 600 μev Quantum effects observable only at T 0.5 K (100 μev 1 K)

28 phase qubit (E J >> E C ) current biased JJ flux qubit (E J > E C ) fluxon boxes charge qubit (E J < E C ) Cooper pair boxes I I V Nowadays superconducting qubit zoo is larger Transmon, camel-back, capacitively I shunted 3JJ-FQB, fluxonium Traditional classification via E J /E C increasingly difficult J. Martinis (NIST) H. Mooij (Delft) V. Bouchiat (Quantronics) AS-Chap

29 C Engineer potential H J = 4E C φ + E J (1 cos φ) Unsuitable for TLS! L J (I c, φ) Flux/phase engineering Add junctions (3JJ flux qubit) Add inductance (rf SQUID &phase qubit) Add bias current (phase qubit) Charge engineering Add gate capacitor (charge qubit) Add shunt capacitor (transmon qubit) E J naturally induces anticrossings Slightly delocalized charge states with residual anharmonicity AS-Chap

30 I L Flux bias Φ x E J E C N = i φ H = 4E C N φ πf + E J 1 cos φ + E L Inductive energy E L Φ 0 L Circulating current due to external flux I L = Φ x = fφ 0 L L Frustration f Φ x Φ 0 f = 1 E J E C (phase/flux regime) ħω p k B T (ω p = 8E J E C ) I C L Φ 0 Requires large L (otherwise flux quantization kills our quantum variable) 1 0 Δ MQT causes level splitting Δ 0, 1 are symmetric and antisymmetric superpositions of +I L, I L +I L I L φ + πf = 0 φ + πf = π π Φ Φ 0 Theoretical prediction: Leggett (1984) Experimental realization: Friedman et al. (000) AS-Chap

31 Current bias I x E J E C H = 4E C N + E J 1 cos φ + ħ e I xφ Additional force term due to current source Tilted washboard potential Significant anharmonicity 1 0 U m A > 1µm x 1µm E J / E C ~ A² >> 10 4 φ classical arcsin(i e /I c ) G G 1 G 0 φ levels 0, 1 form the qubit oscillator states differ in phase phase qubit Γ Γ 1, Γ 0 pump ω 1 for readout readout detects running phase (voltage) AS-Chap

32 AS-Chap First Rabi oscillations in phase qubit

33 Construction principle current bias I x I L Flux bias Φ x E J E C E J E C parameters similar to RF SQUID qubit E J >> E C ħw p > k B T I c L F 0 opposite circulating current dc SQUID readout Better decoupling from readout electronics significantly longer decoherence times Preferred over current-biased version AS-Chap

34 3-Josephson-junction persistent current flux qubit single JJ in a loop (RF-SQUID flux & flux biased phase qubit) Single quantum degree of freedom Flux quantization fixes JJ phase for β L 1 Requires β L 1 Alternative: More than one JJ in the loop φ 1 φ +I p α Φ x I p J.E. Mooij et al., Science 85, 1036 (1999) T. P. Orlando et al., PRB 60, (1999) Persistent current flux qubit 3 JJ in superconducting loop 1 (L loop L J β L 1) Two junctions have identical size Third junction smaller by factor α E J E J1 = E J and E C E C1 = E C E J3 = αe J and E C3 = E C α E J > E C (phase regime) & ħω p k B T Control knob External flux Φ x applied to loop Still quantum degrees of freedom left after flux quantization AS-Chap

35 φ /π φ /π Double-well potential U J = E J 1 cos φ U φ 1, φ, φ 3 = E J cos φ 1 cos φ + α 1 cos φ 3 Flux quantization φ 1 φ + φ 3 = πf with frustration f Φ x Φ 0 Signs are mere convention! U φ 1, φ = E J + α cos φ 1 cos φ α cos πf + φ 1 φ φ 1 /π Color code: U φ 1,φ E J Φ x = Φ 0 α = 0.8 Double-well potential in each unit cell! φ 1 /π AS-Chap

36 φ /π φ /π Double-well potential φ 1 /π φ + /π Two stable minima at φ, φ and ( φ, φ ), where cos φ 1 Double well rotated by 45 in the φ 1 φ -plane Variable transformation α φ + 1 φ 1 + φ φ 1 U φ +, φ = E J [ + α cos φ + cos φ α cos πf + φ φ 1 φ Variable relevant for qubit dynamics φ AS-Chap

37 φ /π Double-well potential 0.5 U φ +, φ = E J [ + α cos φ + cos φ α cos πf + φ 0 U E J Φ x = n + 1 Φ 0 Symmetric double-well potential Thermodynamics: I = U Φ φ + /π No tunneling Degenerate ground state Left/right well correspond to clockwise/anticlockwise circulating current Persistent current ±I p = U(φ = φ ) = E J α sin πf φ Φ x Φ x =Φ 0 / +I p φ I p φ φ π Φ 0 Φx =Φ 0 / = I = I c α sin φ c α sin π φ cos π = I c α sin φ cos φ = I c 1 1 α sin(x + y) = sin x cos y + cos x sin y sin x = sin x cos y cos φ 1 α AS-Chap

38 φ /π Double-well potential 0.5 U φ +, φ = E J [ + α cos φ + cos φ α cos πf + φ 0 U E J Φ x = n + 1 Φ 0 Symmetric double-well potential +I p I p φ + /π No tunneling Degenerate ground state Left/right well correspond to clockwise/anticlockwise circulating current Persistent current ±I p U E J φ φ Φ x < n + 1 Φ 0 φ Φ x n + 1 Φ 0 Tilted double-well potential +I p I p Flux bias induces energy bias ε Φ x ε Φ x Near Φ x = n + 1 Φ 0 ε Φ x = I p Φ 0 f n 1 φ AS-Chap

39 AS-Chap Quantum treatment N 1, i φ 1, E C e C φ + 1 φ 1 + φ φ 1 φ 1 φ H = 4E C N 1 + N + E J [ + α cos φ + cos φ α cos πf + φ Task convert N 1 and N into N + and N N 1 + N i = φ 1 + φ = φ + dφ + dφ 1 + φ dφ dφ 1 + φ + dφ + dφ + φ dφ dφ = 1 4 φ + + φ φ + φ = 1 φ + + φ = 1 N + + N i Flux qubit Hamiltonian H = E C N + + N + E J [ + α cos φ + cos φ α cos πf + φ

40 E n /ħ (GHz) E n /ħ (GHz) AS-Chap Quantum treatment f = Φ x /Φ 0 H = E C N + + N + E J [ + α cos φ + cos φ α cos πf + φ Numerical diagonalization Eigenenergies E n f f Near Φ x = n + 1 Φ 0 Approximate as two-level system with linear energy bias!

41 AS-Chap E 0,1 δφ x Φ 0 f n 1 Energy bias ε Φ x = I p δφ x 1 +I p I p +I p and I p are eigenstates of ε Φ x σ z Tunneling rate Δ/h over potential barrier Tunnel splitting Δ exp E J E C 0 1 +I p + I p Δ H = ε Φ x σ z + Δ σ x E 1 E 0 ħω q Φ x = ε Φ x + Δ I p σ z 0 δφ x /Φ 0 I p Bloch angle θ denotes operation point Φ x Δ sin θ and cos θ ε Φ x ħω q Φ x ħω q Φ x 0 Persistent circulating current I p σ z I p σ z = I p cos θ depends on Φ x I p 0 δφ x /Φ 0

42 Readout requirements L J Φ 0 πi c E 0,1 1 +I p I p 0 D Inductive coupling to readout circuit Goal: E J 50 E C Junction I c 750 na, C 3 ff, α 0.7 Loop: d 10 μm L μ 0 d 10 ph L L J 400 ph Self-induced flux negligible Flux signal LI p LαI c 3 mφ 0 ±I p can be distinguished by an on-chip dc SQUID I p σ z 0 1 +I p + I p 0 δφ x /Φ 0 I p I p 0 δφ x /Φ 0 AS-Chap

43 Continuous-wave readout with dc SQUID E 0,1 1 +I p I p 0 D I p σ z 1 +I p + I p 0 δφ x /Φ 0 I p 0 I p F. Deppe, Ph.D. thesis (Technische Universität München, 009) 0 δφ x /Φ 0 AS-Chap

44 Pulsed dc SQUID readout away from the degeneracy point E readout pulse sequence time 0 Bias here 0 df x /F 0 K. Kakuyanagi et al., Phys. Rev. Lett. 98, (007) F. Deppe et al., Phys. Rev. B 76, (007) Principle Away from Φ x = Φ 0 /, qubit eigenstates coincide with (anti-)clockwise circulating current states Add/remove flux from SQUID loop Increase/decrease I c,squid (Φ x,squid ) Protocol Bias SQUID just below switching point (short switching pulse) Depending on qubit state switching or not Measure voltage or not Hold pulse avoids retrapping More signal AS-Chap

45 Pulsed dc SQUID readout at the degeneracy point E Prepare adiabatic flux shift pulse 0 ħw microwave pulse sequence readout pulse sequence Readout Prepare & Readout 0 df x /F 0 time K. Kakuyanagi et al., Phys. Rev. Lett. 98, (007) F. Deppe et al., Phys. Rev. B 76, (007) AS-Chap

46 Flux qubit Relaxation rate fit by Bloch-Redfield model (or Golden rule argument): T 1 1 = π ħ ε Φ x sin θ S Φ ω q Transfer function Actual flux noise This data S Φ Δ ħ 10 0 Φ 0 Hz High frequency noise causing resonant transitions Physical coupling via σ z (flux through loop) σ z cos θ σ z sin θ σ x Stronger near magic point (sin θ 1) Inductive coupling MI p I noise Stronger for large ε Φ x = I p F. Deppe et al., Phys. Rev. B 76, (007) AS-Chap

47 ωt F Ramsey filter F R Spin echo filter F E Flux qubit Dephasing rates Depend on type of low frequency noise! Rates depend on the measurement protocol (filters!) Ramsey Probes low-frequency noise T 1 R = T T φr Spin echo T 1 E = T T φe Filters low-frequency noise ωt G. Ithier et al., Phys. Rev. B 7, (005). A. Shnirman et al., Phys. Rev. Lett. 94, 1700 (005). K. Kakuyanagi, F. Deppe et al., PRL 98, (007) F. Deppe et al., Phys. Rev. B 76, (007) AS-Chap

48 Flux qubit dephasing rates This experiment Clearly cusped 1/f-noise dominates Additional small white noise contribution Predictions White noise (Bloch-Redfield) T 1 φ = π ε cos θ ħ Φ x S BR Φ ω = 0 Low-frequency spectrum important T Φ x is smooth function 1/f-noise S Φ ω = A 1/f ω T 1 φe = 1 ε cos θ A ln ħ Φ x Φ x -dependence cusped G. Ithier et al., Phys. Rev. B 7, (005). A. Shnirman et al., Phys. Rev. Lett. 94, 1700 (005). K. Kakuyanagi, F. Deppe et al., PRL 98, (007) F. Deppe et al., Phys. Rev. B 76, (007) AS-Chap

49 Flux qubit dephasing rates Summary & numbers T E 1 T 1 1 at magic point Relaxation-limited T 1 T φe μs Strong 1/f-contribution A 1/f 10 6 Φ 0 Typical value (known from SQUIDs) Additional white-noise contribution S Φ BR ω 0 = Φ 0 S BR Φ ω 0 S Φ ħ Consistent (white noise should be frequencyindependent) Δ Hz G. Ithier et al., Phys. Rev. B 7, (005). A. Shnirman et al., Phys. Rev. Lett. 94, 1700 (005). K. Kakuyanagi, F. Deppe et al., PRL 98, (007) F. Deppe et al., Phys. Rev. B 76, (007) AS-Chap

50 Z(w) Flux qubit noise sources White noise Attenuator V out White noise from control line Cold attenuators I bias White noise from readout circuit Engineer Z(ω) 1/f noise 1/f noise independent of area of qubit loop local noise sources in close qubit environment Two-level fluctuators Improve fabrication technology AS-Chap

51 AS-Chap Charge engineering The Cooper pair box (CPB) Superconducting island with negligible self-capacitance Hamiltonian (for any E J, E C n E J, C J C g V g e C g +C J, n g ) Charge regime E C E J Charge is good quantum number Schrödinger equation H CPB Ψ k = E k Ψ k Mathieu equation for Ψ k Ψ k e in gφ H CPB = 4E C n n g + EJ 1 cos φ Ψ k α E J E C cos α Ψ k = 4E k E C Ψ k Gate charge n g C gv g induced by source V e g Adds/removes excess CP to/from island (always many equilibrium CP) Classical quantity May assume fractional values! α φ/ Numerical solution Eigenenergies E k

52 The split Cooper pair box V g Tunable JJ (dc SQUID) C J / C g Island C J / Readout E E/E / E C C E E J E C = 0.06 E J E + Dc SQUID with β L 1 Tunable JJ with effective E J and E C Gate voltage V g is control knob Readout with additional JJ Detect number of excess Cooper pairs on island Josephson energy E J cos φ Couples charge states/parabolas Avoided crossings 0.0 E - E CV e / e n g H CPB = 4E C n n g + EJ cos φ More typical parameters E C 5 GHz, E J 5 GHz AS-Chap

53 AS-Chap The split Cooper pair box V g Tunable JJ (dc SQUID) C J / C g Island C J / Readout E E/E / E C C E E J E C = 0.06 E J E + Theory questions Why is coupling exactly E J? Near n g = 1, energy levels look like hyperbola. Correct? 0.0 E - E CV e / e n g H CPB = 4E C n n g + EJ cos φ More typical parameters E C 5 GHz, E J 5 GHz

54 Two-level-representation of the CPB Goal Express H CPB = 4E C n n g + EJ cos φ as TLS near n g = 1 Charge states n n n = n n n ng = n ng n n n n g = Commutation relations n, φ = 1 n n n n g n n n = 1 π 0 πdφ exp in φ φ exp ip φ n = 1 π 0 πdφ exp i n + p φ φ = n + p exp ±i φ n = n ± 1 cos φ = 1 exp i φ + exp i φ = = 1 n n exp i φ + exp i φ n n = 1 n n n n + 1 n n n AS-Chap

55 Two-level-representation of the CPB TLS n {0,1} n n n = n n exp ±i φ n = n ± 1 n n n g n n 1 n g σ z HCPB = 4EC n ng + EJ cos φ 1 = 1 n n n exp i φ + exp i φ n n n exp i φ + exp i φ = 1 ( 0 0 exp i φ exp i φ exp i φ exp i φ exp i φ exp i φ 1 1 H CPB = E el σ z + E J σ x E el 4E C 1 n g = 1 ( ) = 1 σ x AS-Chap

56 Two-level-representation of the CPB V g Tunable JJ (dc SQUID) C J / C g Island C J / Readout E E/E / E C C E E J E C = 0.06 E J E + Why is coupling exactly E J? Because of the two-level representation Near n g = 1, energy levels look like hyperbola. Correct? Yes, because E el V g can be linearized for n n g E CV e / e n g H CPB = E el σ z + E J σ x E el 4E C n n g E - Practical devices parameters E C 5 GHz, E J 5 GHz AS-Chap

57 q = 0 q = 1 q = From the Cooper pair box to the transmon qubit Advantages of the CPB Simple design (JJ, β L 1) Coupling element E J I c (Flux qubit: Δ exp E J E C ) Voltages convenient for Coupling to other qubits Coupling to readout circuitry Coupling to control signals Large anharmonicity (few GHz) E / E C 1.0 E J / E C = E + E + E J E J E - E CV e / e In first order insensitive to charge fluctuations at sweet spot n g = n + 1 Big disadvantage Coherence times short due to susceptibility to 1/f charge noise In practice: Coherence times of a few tens of nanoseconds even at the sweet spot! (Typical charge noise magnitude Typical flux noise magnitude) Idea Flatten energy dispersion AS-Chap

58 The transmon qubit (Transmission line shunted plasma oscillation qubit) Take a CPB geometry and increase E J E C E m n g E m n g = 1 4 ε m cos πn g ε m 1 m E C 4m+5 m π E J E C m +3 4 e 8E J E C Charge dispersion decreases exponentially with E J E C Anharmonicity decreases only polynomially with E J E C Optimum trade-off for E J E C 50 Few hundreds of MHz anharmonicity left Charge no longer good quantum number Not tunable via gate voltage anymore Tune via flux (dc SQUID) J. Koch et al., Phys. Rev. A 76, (007). AS-Chap

59 The transmon qubit Embed into a resonator for Readout Filtering Control The transmon is currently most successful qubit with respect to coherence times Coherence of transmons mostly limited by spurious TLS (defects) in substrate and metal-substrate interface D geometries μs 3D geometries up to 00 μs J. Koch et al., Phys. Rev. A 76, (007). AS-Chap

60 AS-Chap The C-shunted flux qubit C sh αe J, αc J Decreasing E J /E C and/or α reduces influence of flux noise by level flattening However, sensitivity to charge noise on islands a,b,c is increased Suppress charge noise by shunt capacitance C sh = β α C J Typically, C sh 100 ff C J 5 ff First promising results T T μs E J, C J n c Φ x n a n b E J, C J J. Q. You et al., Phys. Rev. B 75, (R) (007). M. Steffen et al., Phys. Rev. Lett 105, (010).

61 The C-shunted flux qubit C sh Optimize thin film fabrication as in transmon Anharmonicity 800 MHz slightly larger than for typical transmon qubits Neverthless, transmon-like design Noise sources limiting T / n c αe J, αc n b Resonator loss Ohmic charge noise 1/f flux noise Temporal variations attributed to quasiparticles Noise sources limiting T 85 0 / Photon shot noise from residual thermal photons in the readout resonator E J, C J Φ x n a E J, C J F. Yan et al., arxiv: (015). AS-Chap