Lecture 9 Superconducting qubits Ref: Clarke and Wilhelm, Nature 453, 1031 (2008). Newcomer in the quantum computation area ( 2000, following experimental demonstration of coherence in charge + flux qubits). Advantages: long experience (> 50 year) of behavior at classical level. versatile and easily tunable (hopefully) relatively easily scalable Main disadvantage: decoherence typically much stronger than in microscopic qubits (nuclear spins, trapped ions,...) (but situation orders of magnitude more favorable than widely thought before 2000). Principal types of superconducting qubits: Type Natural variable First demonstration of coherent 2-states behavior 1. Charge Q (charge on box) Nakamura et al. 1997 2. Flux Φ (flux through loop) Friedman et al. 2000 van der Wal et al. 2000 3. Phase φ (phase drop across CBJ) Martinis et al. 2002 4. Mixed Q, Φ Vion et al. 2002 (e.g. quantronium) 1
Charge qubit (Cooper-pair box) 2D Hilbert space corresponds to n and n+1 Cooper pairs on island: bulk V g island SET Ĥ = E C (V g ) ˆσ z E J ˆσ x where junction E C (V g ) = (2e)2 ( (n+1) 2 n 2) 2eV g 2e(V g V 0 ), V 0 2n + 1 2C 2C (2e)2. As f(v g ) has relevant energy levels showing typical level-crossing behavior. In practice, junction is usually of dc SQUID type: and so can be tuned by external flux Φ (c-number!) which modulates E J. Measurement: usually of ˆn, by coupling E island capacitively to an SET (single- electron-tunneling) device: SET is itself a sort of single-electron box whose conductance is highly sensitive (bec. of Coulomb blockade) to the total gate voltage V g applied V g E to it. V g n/c s + Ṽg: when Ṽg bulk Φ is tuned away from resonance, no current island flows for either value of n no measurement. When Ṽg is tuned close to resonance, current I depends on V g, hence I C s SET on n measurement. So measurement island can be turned on or off (essential for operation as Ṽ g qubit!). 2
Typical parameter values of charge qubit 1 : island size: 700 50 15 nm (Al on SiN x ) (bulk also Al) island capacitance: C 6 af E C 500 µev superconducting gap 250 µev (so E C 2 : doesn t matter!) junction N-state resistance 10 kω which by AB relation E J 75 µev ( 0.3 ) temperature of experiment: 30 mk 3 µev (so k B T E C, E J ) 1 Nakamura et al., Nature 398, 786 (1999) 3
Phase qubit Recall that for a CBJ ( current-biased junction) potential is washboard ω 0 V (ϕ) V 0 ϕ V (ϕ) = E J cos ϕ I(Φ 0 /2π)ϕ E J ( cos ϕ+(i/ic )ϕ ). Near lability (I I C ) can expand as function of deviation ɛ of ϕ from equilibrium value sin 1 I/I C (cf. lecture 6): V (ϕ) V (ɛ) = const. + E J (Aɛ 2 Bɛ 3 ), A = 1 ( 1 I ) 1/2, B = 1/6. 2 I C Thus the barrier height V 0 is (4/27)(A 3 /B 2 )E J = E J (4 2/3)(1 I/I C ) 3/2 Also, since mass of equivalent particle is C(Φ 0 /2π) 2, ω 0 = ( ) [ 1/2 2A = 2 1/4 2πI C C(Φ 0 /2π) CΦ 0 ( 1 I ) ] 1/2 1/2 I C ( 2 1/4 1 I ) 1/4 ω J, I C ω 0 : small oscillation frequency around metastable state ω J (2πI C /CΦ 0 ) 1/2 : Josephson plasma frequency of junction at zero bias From lecture 7, for quadratic-plus-cubic potential, the tunneling rate at T = 0 out of GS is Γ 0 const. ω 0 e B, B = 36 5 4 V 0 ω 0 (1 I/I C ) 5/4
For qubit operation, choose I/I C so that Γ 0 is negligibly small. Crucial point: also excited states in metastable well, some of which have Γ Γ 0. Moreover since potential anharmonic ω 12 ω 01. Hence can use states 0 and 1 as qubit basis. If τ is typical lifetime of experimental need Γ 0 τ, Γ 1 τ 1. V (ϕ) ω 12 ω 01 ϕ However, want to use state 2 as readout, thus should have Γ 2 τ 1. To implement readout, irradiate with microwaves of frequency ω 12. Then no effect on state 0, (offresonance), but 1 2 escape junction in voltage state. If we represent the Hamiltonian in the 2D Hilbert space of 0 and 1 in the standard Bloch-sphere notation, Ĥ = σ H, ω 12 2 0 1 thus evidently H z = ω 01 /2. But what are H x and H y? Answer: Add extra terms to bias current I of the form δi(t) = δi dc + δi x cos ω 01 t + δi y sin ω 01 t δv (ϕ) = δi(t)ϕ (where δi dc, δi x, δi y slowly varying over τ ω01 1 ). Then since 0 ϕ 1 = ( /2Cω 01 ) 1/2, Hamiltonian in rotating frame (rotating with frequency 5
ω 01 ) is with H x ( ) Ĥ = 1 ɛ x + i y σ H, 2 x i y ɛ ( ) 1/2 ( ) 1/2 δi x, H y δi y, H z 2Cω 01 2Cω 01 ( ω01 I ). Typical parameters 2 : Area of junction (Nb-AlO-Nb): 10 10 µm 2 I C = 21 µa C = 6 pf, so quantum parameter (lecture 7) 2 /CI C Φ 3 0 10 8 (or K 6 10 4 ) Q-factor of circuit 350 relative frequency separation (ω 12 ω 01 )/ω 01 0.1 Γ 2 /Γ 1 Γ 1 /Γ 0 10 3. In original experiment fidelity of readout 85%: theoretical estimate 3 indicates 97% possible. 2 Martinis et al. PRL 89, 117901 (2002) 3 Revin et al., APL 98, 162501 (2011) 6
Flux qubits Recall (lecture 6): for (single-junction) flux qubit in external flux Φ ext potential energy is of form ( 2πΦ V (Φ) = E J cos )+ (Φ Φ ext) 2 2L Φ 0 C Φ ext E J (and kinetic energy = 1 2 C Φ 2 ). L For β L 2πLI C /Φ 0 > 1 potential can have one or more metastable minima. Consider case 1 β L 1 > 0, and Φ ext close to Φ 0 /2, then the interesting region of Φ is also close to Φ and in terms of q Φ Φ 0 /2 form of potential is V (Φ) ɛ ω 0 V 0 0 q 0 q q 0 V (q) = 1 2 ( Aq2 + B 4 Cq) A (β L 1)/L B (π 2 /3)(LΦ 2 0) 1 q 0 = (6(β L 1) 1/2 /π)φ 0 C (Φ ext Φ 0 /2)L Thus, ω 0 = 2(βL 1) 1/2 (LC) 1/2 2(β L 1) 1/2 ω LC freq. of circuit for E J 0 V 0 = 3 8π 2 Φ 2 0 L (β L 1) 2. 7
Note if we assume β L 1 1 we can replace, in order-of-magnitude estimates, L 1 by (2π/Φ 0 )I C. So define quantum parameter S by ( π 3 2 ) 1/2 S 8CI C Φ 3 ( const. K 5/4 ). 0 Then a straightforward WKB calculation gives for tunneling amplitude between wells (ind. of bias ɛ for ɛ ω 0 ) = const. ω 0 e (16/3)(V 0/ ω 0 ) = const. ω 0 e 2 1/2 (β L 1) 3/2 /S, so even if S is small, an get appreciable tunneling by taking β 1 (cf. I I C in phase-qubit case). In this case qubit basis is ground states in L and R wells: Ĥ = ɛ ˆσ z + ˆσ x, ɛ = 2q 0 (Φ ext Φ 0 /2) Readout: normally measure with dc SQUID. However a problem: what does this actually measure? Prima facie, since flux qubit - dc SQUID interaction (M/(L FQ L dc ))ΦΦ dc, and escape of dc SQUID is f(φ dc ), measures flux Φ. However, need to do detailed analysis of coupled system. (M-C. Yeh): When we want measurement off, need (a) I dc = 0 (bias current on SQUID) (or at least I C ) (b) compensate any effect of SQUID by external flux so as to get total ɛ = 0. But to make measurement, need to sweep bias flux qubit L M R dc SQUID current I dc up towards critical value I C (Φ) and measure escape statistics. If sweep is slow, this will project on to eigenstates of energy, not of Φ; if fast, on to eigenstates of Φ. 8 I dc
(What is fast/slow? Crudely, depends whether time taken to reach I dc s.t. energy bias ɛ is or / ). Typical parameters 4 I C (of flux qubit) = 0.5 µa E J 200 GHz E C e 2 /2C = 7.4 GHz (so E J /E C 35) (tunneling frequency) = 3.4 GHz T (temperature of experiment) 25 mk ( 0.5 GHz). 4 Chiorescu et al., Science 299, 1869 (2003) 9
Decoherence in superconducting qubits Describe interaction with dissipation/decoherent environment by truncating the OB Hamiltonian of lecture 8 to the relevant Hilbert space. E.g. for flux qubit, if σ z labels eigenstates of flux, Ĥ Ĥ(Φ{x i}) = Ĥ(Φ) + H SHO + q 0 σ z C i x i π 2 ( Ci 2 ) δ(ω ω i ) J(ω) fixed e.g. from expts. in classical regime m i ω i i (Note: truncated counterterm is prop. q 2 0σ 2 z = q 2 0 = const. so can be omitted). Note: in more general case (e.g. phase qubit) may get (with axis choice s.t. Ĥ 0 diagonal) terms not only in ˆσ z but also in ˆσ y and ˆσ x. For practical qubits, usually have q 0 J(ω)/ω 1, so effect of environment q 2 0J(ω). Traditional NMR-language description (note σ z σ x!): dσ z dt = (σ z σ z eq ), dσ dt T 1 i = σ T 2, where T 2 is the intrinsic dephasing time (Distinguish the ensembleaveraged dephasing time T, due to inhomogeneity between runs of H z. Can eliminate this effect by spin-echo technique). Can calculate T 1, T 2 by golden-rule perturbation theory: T 1 1 = q 2 0J(ω) coth(β ω/2) T 1 2 = 1/(2T 1 ) + 1/T ϕ pure dephasing rate Mechanisms different: T1 1 and non-pure-dephasing contribution to T2 1 from exchange of energy with E, Tϕ 1 from fluctuations of relative energy of eigenstates of Ĥ 0 due to contact with E (e.g. in phase qubit, random fluctuations of I dc ). Generally speaking, pure dephasing fluctuations much more dangerous T 2 often T 1. 10
Formal expression for T 1 ϕ T 1 ϕ in terms of J(ω) is = lim ω 0 q 2 0J(ω) coth(β ω/2). (1) For ohmic dissipation (J(ω) = ηω) this gives 2ηq 2 0k B T, thus 0 for T 0. For J(ω) ω s, s > 1, (e.g. 1/f noise), (1) divergent need to cut off at ω inverse timescale of experiment (exponential Gaussian). Physical mechanisms of decoherence: charge qubit: mainly fluctuations of charge of environment phase qubit: fluctuations in I dc (likely due to motion of TLS (twolevel systems) in junction) flux qubit: fluctuations in ambient flux In the case of charge qubits, the effect of charge fluctuations much reduced by quantronium design. : values of T 1, T 2 measured for t T 1, T 2 may not be ones relevant to quantum computation! 11