Lecture 10 Superconducting qubits: advanced designs, operation 1 Generic decoherence problem: Λ 0 : intended

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1 Lecture 10 Superconducting qubits: advanced designs, operation 1 Generic decoherence problem: Ĥ = Ĥ(p, q : Λ), Λ: control parameter { e.g. charge qubit Λ = V g gate voltage phase qubit Λ = I bias current flux qubit Λ = Φ ext external flux Λ = Λ 0 + δλ, Λ 0 : intended δλ: effect of fluctuations of E environment Project on to qubit space: Ĥ qb = Ĥ0 + δĥ, so in energy basis Ĥ qubit = 1 2 ( ) ɛ(λ0 ) 0 + δĥ, δĥ σ H, 0 ɛ(λ 0 ) H parametrized by J i (ω), i = x, y, z, δĥ Γ 1, Γ 2 = 1 2 Γ 1 + Γ ϕ (T 1 1, T 1 2 ) Γ 1 : determined by J x,y (ω 10 ), (ω 10 ɛ(λ 0 )/ ) Γ ϕ : determined by J z (ω 0) In most cases, J(ω 0) most dangerous (e.g. 1/f noise) want no ˆσ z -coupling want ɛ/ Λ = 0. 1 General references: Devoret+Schoelkopf, Science 339, 1169 (2013) Zhang et al., Chinese Physics B 22, (2013) (decoherence + readout) Yan et al., Rev. Mod. Phys. 85, 623 (2013) 1

2 e.g. for charge qubits, ɛ/ q should be 0: for flux qubits, ɛ/ Φ = 0, etc. One obvious way to achieve this: H(p, q : Λ) = p2 2m mω 0(q Λ) 2 all energy-level differences independent of Λ, but this is just SHO, so no truncation to 2-state (qubit) system! So generic problem: How to minimize sensitivity to environmental fluctuations while maintaining enough anharmonicity to justify truncation to 2-state Hilbert space? 2

3 Quantronium 2 V g (t) island ˆθ + δ/2 Φ γ ˆθ + δ/2 I b (t) The quantities δ and γ are regarded as classical, and related by δ = γ + Φ/Φ 0, γ is controlled by the bias current I b (t). The quantity ˆθ ( average phase difference between island and rest of circuit) is regarded as QM l. For δ = 0 both the total current across the island and the total current into it are 0; the two energy eigenstates have θ = 0 and θ = π respectively. For nonzero δ the current into the island is still zero, but the current across it (i.e. around the circuit) is E J cos θ sin δ/2, i.e. E J sin(δ/2) ˆσ z /2 (if basis is even/odd combinations of n and n + 1 ). Note that for V g tuned to the degeneracy point, eigenstates in n-basis are 1 2 ( n ± n+1 ), so low-frequency fluctuations (as of gate charge ) which couple to ˆn are particularly ineffective (i.e. they involve, in this (energy) basis, only ˆσ x not ˆσ z so dephase only through the 1/2T 1 term not through 1/T ϕ ). Also for γ = Φ = 0 the energy is insensitive to fluctuations of Φ (E E J cos θ cos δ/2, E/ δ = 0). For readout, increase δ from 0 to finite value, measure current. 2 Vion et al., Science 298, 886 (2002) 3

4 Transmon 3 Simply, charge qubit shunted with large capacitance. For arbitrary values of ˆQ 2eˆn and gate voltage V g, ˆQ-dependent term in Ĥ is ˆQ 2 2C ˆQV g 2e2 C (ˆn n g) 2 + const., n g CV g /2e, so total Hamiltonian is Ĥ(ˆn, ˆϕ) = E C (ˆn n g ) 2 E J cos ˆϕ, In original charge qubit experiments, E C E J can truncate at lowest pair of states n, n + 1. In transmon, C is very large so E C E J In pendulum analogy: Ĥ = 2( ˆL L g ) 2 2I ˆϕ nearly well defined variable. transmon corresponds to 2 /2I mgl, i.e. ω mgl mgl cos ˆθ ( ˆL i ), ˆθ semiclassical limit! Problem: in this limit pendulum SHO! (no use for qubit). However, saving point is precise dependence on ratio E C /E J in limit E C E J : anharmonicity is power-law ( (E C /E J ) n ) sensitivity to n g is exponential ( exp[ const.(e J /E C )]) Why? use pendulum analogy: ( ) θ 2 V (θ) mgl cos θ = const. + mgl 2 θ4 4! +..., 3 Koch et al., Phys. Rev. A 76, (2007) θ 4

5 diference ω 21 ω 10 GSEV (ground state expectation value) of θ 4, but θ 2 0 ( /Iω) θ 4 ( /Iω) 2. In original problem ( 2 /2I E C, mgl E J, ω (E C E J ) 1/2 / ) ϕ 4 (E C /E J ). Sensitivity to n g (L g ): this must be proportional to amplitude for rotation through 2π ( quantization of ˆL). ω 21 ω 10 A(0 2π) exp( IV (θ)/ dθ) exp( const.(v 0 / ω)). But since V 0 E J, ω (E J E C ) 1/2 /, this A(0 2π) exp( const.(e J /E C ) 1/2 ). ω 10(n g ) n g exp( const.(e J /E C ) 1/2 ). More quantitative calculation (Koch et al.): ( )m EJ E m (n g = 1/2) E m (n g = 0) = const. E C exp[ (8EJ /E C ) 1/2 ]. E C Thus, for given tolerance in ratio (ω 21 ω 10 )/ω 10, can find value of ratio E C /E J s. t. anharmonicity still sufficient for qubit operation while charge noise exponentially suppressed V 0 5

6 Fluxonium 4 Digression: simple LC-circuit treated QM ly Ĥ = ˆQ 2 2C + ˆΦ 2 2L [ ˆQ, ˆΦ] = i (so e.g. ˆΦ ˆx, ˆQ ˆp). Introduce dimensionless quantities Φ Φ/Φ 0, Q ˆQ/2e C Q Φ ext L Ĥ = (2e)2 2C Q 2 + Φ2 0 2L Φ 2 E C Q2 + E L Φ2, and [ Q, Φ] = i. What is ratio of mean-square values Q, Φ in GS? E C Q = E L Φ = 1 4 ω LC (LC) 1/2, so Φ / Q = E C /E L (Z/R Q ) 2, where Z L/C impedance of circuit R Q = h/(2e) 2 = 6 kω ( quantum unit of resistance ). General theorem of electrical engineering: for any kind of simple electrical circuit, Z Z 0 (µ 0 /ɛ 0 ) 1/2 = 377 Ω vacuum impedance. (Crudely: C ɛ 0 L max, L µ 0 L max L/C µ 0 /ɛ 0 ). Thus Φ 2 / Q 2 (Z 0 /R Q ) 2 (377/6000) , in words: for any simple LC-circuit, in dimensionless units, we always have charge fluctuations flux fluctuations and since Φ 2 Q 2 1 (uncertainty principle!), Q Manucharyan et al., Science 325, 113 (2009) 6

7 Now, at least for small fluctuations of ϕ and order-of-magnitude estimates, a Josephson junction is equivalent to a nonlinear inductance: E J (ϕ) = I ( ) CΦ 0 2π cos ϕ IC Φ 0 ϕ 2 = const. +, so for ϕ 2π Φ, 2π 2 E J = (const.) + 2πI C Φ 0 ( Φ/2) L 1 eff = 2πI C/Φ 0 L J = Φ 0 /(2πI C ) small I C implies large effective inductance! (thus, charge qubit can have Z J (L J /C) 1/2 R Q fluctuations of Q( n) 1). Fluxonium: hybrid between charge and flux qubits. Standard flux-qubit geometry but with geometrical inductance L J shunted by a number L g of large (hence small-l eff ) Josephson junctions. Since junctions are in series, L tot NL eff = large. The E J /E C ratio is much smaller than for typical flux qubits, on the other hand E L is E J : E L = 0.52 GHz E J = 9.0 GHz E C = 2.5 GHz Thus structure of energy levels different from standard flux qubit, e.g. for Φ ext = 0 only 1 state per well. Φ ext = 0 Φ ext 7

8 Circuit QED 5 Consider atom in (superconducting) cavity with a single (relevant) mode created by a : E E is radially outward L = λ/2, ω c = 2πc/λ ω c cavity mode frequency. In RW approximation, Hamiltonian is Jaynes-Cummings: λ/2 r Ĥ JC = ω c a a + ω 01 2 ˆσ z + g(aσ + + a σ ) ω 01 : atomic transition frequency. Eigenstates are atomic levels Question: how large can g be? Answer: g = d E, d: atomic dipole, E: electric field of single photon. d ea 0 (Bohr radius) E = ( ω01 ɛ 0 V ) 1/2 = r 1 ω 01 (2π 2 ɛ 0 c) 1/2 }{{} since ω 01 ω c V : vol. of cavity α: fine structure constant so g r 1 ω01 e (2α/π)1/2, = a 0 2α ω 01 r π α1/2 0.1 in traditional cavity QED, cannot obtain (g/ ω 01 ) > 1 (to date, max ). However, may be able to obtain g strong coupling κ }{{} cavity loss regime. Note that in a QED cavity (especially a superconducting one) if the 5 Refs. Girvin + Schoelkopf, Nature 351, 664 (2008) Girvin et al., Physica Scripta T 137, (2009) 8

9 atomic transition is not close to resonance with any of cavity modes, spontaneous lifetime of atom much increased in relation to its free-space value. Replace atom by qubit (e.g. charge qubit/transmon) and cavity by (open) transmission line: schematically, qubit is at antinode of E. in λ/2 out r 10 µm E C. pair box Simple charge qubit: take basis as eigenstates of ĤQ, i.e. at degeneracy eigenstates of ˆσ z are 2 1/2 ( n ± n + 1 ). Then dipole interaction with EM field is Ĥ int = d E = d E ˆσ x, with d 2el q 2er, l q : qubit dimension. So in RW approximation recover ĤJC, Ĥ JC = ω c a a + 2 ˆσ z + g(aˆσ + + a σ ), but now much easier to get large values of g (in fact, easily get g κ). Two main regimes: 1. ω 01 ω c g: get vacuum Rabi splitting between 0 q 1 γ and 1 q 0 γ - level-crossing effects. 2. ω 01 ω c g, strongly dispersive regime; by doing 2nd order perturbation theory in g, get effective interaction: ( Ĥ disp = g2 a a + 1 ) ˆσ z (const.+) g2 2 ˆn photˆσ z 9

10 ˆn phot : number of photons in cavity (thus, fluctuations in photon number can lead to qubit dephasing). Alternative design 6 : flux qubit inductively coupled to magnetic field of cavity mode. Very schematically: (actually, 4 junctions) interaction is ΦH magnetic field of cavity mode Φ ext Ĥ ω c a a + ( ˆσ x + ɛˆσ z ) + g(a + a )ˆσ z, at ɛ = 0 (Φ ext = Φ 0 /2) identical to chargequbit case (σ x σ z ) so in RW approx. gives JC Hamiltonian. But g can be much larger, in fact can get g/ ω 01 > 1. Φ ext σ z = +1 σ z = 1 6 Bourassa et al., Phys. Rev. A 80, (2009) 10

11 Improved circuit QED design 7 Paik et al., and Rigetti et al., got considerably improved values of T 1 and T 2 by embedding transmon qubit in 3D superconducting box: one factor is reduction of electric field lines on surfaces which may host TLS. Also possibly better shielding from stray external radiation. Rigetti et al. speculate that being able to provide 4π of cryogenic coverage helped (easy to cool (normal) Cu walls down to 10 mk - more difficult to keep superconductor cool). Question: Is free-space electric dipole decay an issue? Quite generally, Γ ED d 2 ω d: transition dipole ME ω 10 : transition frequency For 2s 1s transition of H atom, ω 10 /2π Hz and Γ = sec 1. For charge qubit in cavity, let s say (optimistically) ω 10 /2π 20 GHz, then ω 3 factor reduces rate by However, d is 2l/a 0 (2 for C. pair, l qubit dimension) relative to H atom, so if d = 10µ (perhaps optimistic) d 2 factor increases by Hence would predict for charge qubit Γ ED Γ 2p 1s 10 3 i.e. T msec. since measured T 1 approaches 0.1 msec, conceivable this could be partial effect (certainly will soon need to be taken into account). Other recent improvements 8 from better materials engineering of junction itself (e.g. Al x O y TiN). As known sources of decoherence vanish, new ones appear, e.g. nonequilibrium quasiparticles... 7 Paik et al., PRL 107, (2011) Rigetti et al., Phys. Rev. B 86, (R) (2012) 8 Chang et al., APL 103, (2013) 11

12 Operation of single qubits Ideally, want to be able to adjust Hamiltonian so that Ĥ 0 = 1 2 ( ɛ 0 0 ɛ ) Ĥ(t) = Ĥ0 + δĥ(t), 1 2 ɛˆσ z( 1 2 ω 01ˆσ z ), δĥ(t) = H(t) ˆσ, H: pseudo-magnetic field, in arbitrary direction in Bloch sphere. How to do this? Depending on nature of qubit, may have one or more control parameters which automatically provide possibility of H in particular direction. If (as above) take basis so that Ĥ0 diagonal, then Charge qubit: most obvious control parameter is gate voltage V g V g0 + δv g (t). Then, H(t) = (cos θˆx + sin θẑ)2(2e)δv g (t), θ: angle of rotation of dc equilibrium state from degeneracy point (E C (n + 1) = E C (n)), so at degeneracy H(t) purely in x-direction. Can also in principle use E J as control parameter: in that case H(t) = (sin θˆx cos θẑ)δe J (t), so at degeneracy H purely in z-direction. 12

13 Flux qubit: most obvious control parameter is externally applied flux, which couples to total trapped flux Φ (quantum variable!) via term (Φ Φ ext ) 2 /2L. Hence in energy basis (so that Φ Φ ext ˆσ x at Φ ext = Φ 0 /2) Φ ext 2q 0 H(t) = (cos θˆx + sin θẑ)q 0 δφ(t)/l, so at symmetric point (Φ ext = Φ 0 /2) H(t) is purely in ˆx-direction. How to provide a y-component of pseudo-magnetic field H(t)? One possible answer: generalize what was already done for phase qubit: e.g. for flux qubit, δφ(t) = δφ dc (t) + δφ x (t) cos ω 01 t + δφ y (t) sin ω 01 t. Then when we go into rotating frame and use RW approximation, H x = q 0 L 1 δφ x (t), H y (t) = q 0 L 1 δφ y (t), H z (t) = q 0 L 1 δφ dc (t), works so long as time-dependence of δφ x,y (t) on scale ω 1 01 negligible. 13

14 Principal types of experiment with single qubits 1. Spectroscopy of avoided level crossing (Nakamura et al., van der Wal et al., Friedman et al.,) e.g. flux qubit: couple to DC SQUID apply given rf field to qubit (δφ ext (t) cos ω fr t) measure switching current of dc SQUID as f(φ ext ). Recall that for slow sweep of SQUID bias current, measures qubit in energy basis dip (or blip, depending on sign of δφ ext ) of I sw indicates increased occupation of upper state resonance condition E(Φ ext ) = ω rf. Thus plot out E(Φ ext ). ɛ Φ 0 /2 Φ ext If qubit in incoherent mixture of eigenvalues of Φ ( L and R ), expect crossing. If qubit in coherent quantum superposition a L + b R expect solid line (seen). 2. Rabi oscillations Start with qubit in GS, drive with pulse (length τ) at frequency ω 01 and amplitude H 0, wait time t, measure occupation probability of upper/lower state. Result should (a) oscillate as f(τ) with Rabi frequency H 0 τ (b) decay as exp( t/t 1 ). measurement of T 1. Now routine in all kinds of qubit. 14

15 3. Ramsey-fringe experiments Start with qubit in GS, apply resonant π/2 pulse around (say) x-axis of Bloch sphere (schematically, H x (t) = H 0 θ(t)θ(π/2h 0 t) cos ω rf t), ω rf close to ω 01, wait time t, apply second pulse around some axis, measure prob. of ES/GS. Expect ensemble-averaged result (a) oscillates as f(t) with difference frequency ω rf ω 01 (b) decays with t as exp( t/t 2 ). By spin-echo technique, can remove extrinsic contributions to T 2 and measure intrinsic dephasing rate T

16 Best current parameter values for single qubits 9 Rigetti et al., Phys. Rev. B 86, (R) (2012), (transmon in 3D cavity): T 1 = 70 µs, T 2 = 92 µs. Pop et al., Nature 508 (2014) (fluxonium in 3D cavity at sweet spot ) T 1 8 ms, (T 2 not measured) Comparison with desired parameter margins (from Devoret + Schoelkopf, Science 339, 1169 (2013)): Requirement for scalability Desired Estimated Demonstrated capability margins current capability successful performance QI operation Reset qubit 10 2 to Fidelity Rabi flop 10 2 to Fidelity 0.99 Swap to bus 10 2 to Fidelity 0.98 Readout qubit 10 2 to Fidelity 0.98 System Hamiltonian Stability 10 6 to 10 9? δf/f in 1 day < Accuracy 10 2 to to to 10% Yield > 10 4?? Complexity 10 4 to ? 1 to 10 qubits 9 Note in 1999, best value of T 2 a few nsec. 16