From SQUID to Qubit Flux 1/f Noise: The Saga Continues Fei Yan, S. Gustavsson, A. Kamal, T. P. Orlando Massachusetts Institute of Technology, Cambridge, MA T. Gudmundsen, David Hover, A. Sears, J.L. Yoder, A. J. Kerman, W. D. Oliver MIT Lincoln Laboratory, Lexington, MA Jeffrey Birenbaum, JC University of California, Berkeley Published in Nature Communications 3 November 2016 IWSSD2016 Tsukuba, Japan 15 November 2016 Support: Intelligence Advanced Research Projects Activity (IARPA) Army Research Office (ARO)
From SQUID to Qubit The Ubiquitous 1/f Noise Three-Junction Flux Qubit Relaxation and Decoherence The C-Shunt Flux Qubit: Origin The C-Shunt Flux Qubit: Recent Experiments Concluding Remarks
The Ubiquitous 1/f Noise X(t) log S x (f) time (t) Vacuum tubes Carbon resistors Semiconductor devices Metal films Superconducting devices log f Spectral density: S x (f) 1/f α, α ~1
Random Telegraph Signals (RTS) and l/f Noise X(t) time (t) For a single characteristic time τ: S RTS (f) τ/[1 + (2πfτ) 2 ] For example, electron hops between traps in a semiconductor The superposition of Lorentzians from uncorrelated processes with a broad distribution of τ yields 1/f noise (Machlup 1954) To generate 1/f noise at frequency f 0, the particle must reside in a well for time 1/f 0 log S x (f) 1/f 2 log f 1/f For example, for f 0 = 10 4 Hz, 1/f 0 ~ 3 h
Intensity Fluctuations in Music and Speech Spectra have been offset vertically log 10 [S intensity (f)] Richard Voss and JC Nature 1975 log 10 (f)
1/f Noise in Superconducting Devices: Three basic mechanisms 1. Critical current noise: Trapping and release of electrons in tunnel barriers modify the transparency of the junction, causing its resistance and critical current to fluctuate. At low temperatures, the process may involve quantum tunneling. 2(a). Charge noise: Hopping of electrons between traps induces fluctuating charges on to nearby films and junctions 2(b). Ohmic charge noise: At high frequencies, so called ohmic charge noise arises from transitions between the quantized energy levels in the traps that produce 1/f charge noise at low frequencies. ( Ohmic because in the original Caldeira-Leggett model of relaxation and decoherence in Josephson junctions the loss mechanism was a resistor modeled by an array of quantum oscillators.)
Ohmic Charge Noise Shnirman et al. PRL 2005 Charge qubit Courtesy Gerd Schön η η η
3. Flux noise: Flux-sensitive devices (SQUIDs, flux qubits.) exhibit flux noise. This arises from the random reversal of spins at the surface of the superconductor thereby coupling magnetic flux into the loop of the SQUID or flux qubit. I Φ V V δv What do we see when we bias here? δφ 0 1 2 Φ Φ 0
DC SQUIDs: Flux 1/f α Noise Nb washer T = 90 mk Φ V α 0.8 (spectral density) First observed in 1982
From SQUID to Qubit The Ubiquitous 1/f Noise Three-Junction Flux Qubit Relaxation and Decoherence The C-Shunt Flux Qubit: Origin The C-Shunt Flux Qubit: Recent Experiments Concluding Remarks
L Three-Junction Flux Qubit DC SQUID + One Smaller Junction Φ Φ L Small junction area/ large junction area = γ I q 0 I q 1 Degeneracy point: Applied flux Φ q = Φ 0 /2 (Φ 0 h/2e) Ψ = ( 0 ± 1 )/ 2 U Superposition resolves degeneracy J.E. Mooij et al., Science 285, 1036 (1999) C.H. Van der Wal et al., Science 290, 773 (2000) Φ 0 /2 Φ q
Energies of the Flux Qubit ν = at degeneracy dε/dφ q = 2Ι q dν/dφ q = (dν/dε)(dε/dφ q ) = 2(ε/ν)Ι q ε = 2I q (Φ q Φ 0 /2) = 0 at degeneracy Josephson coupling energy: E j I 0 Φ 0 /2π, I 0 = critical current Junction charging energy: E c e 2 /2C, C is junction capacitance E j /E c is large, typically ~ 100 γ = small junction I 0 /large junction I 0 ~ 0.7 Excellent approximation to two-level system near Φ 0 /2
From SQUID to Qubit The Ubiquitous 1/f Noise Three-Junction Flux Qubit Relaxation and Decoherence The C-Shunt Flux Qubit: Origin The C-Shunt Flux Qubit: Recent Experiments Concluding Remarks
T 1 Relaxation time: Relaxation and Decoherence Time to relax from first excited state e> to ground state g> Energy loss involved The classic Caldeira-Leggett spin-boson model treats linear dissipation as a bath of harmonic oscillators and calculates the Johnson-Nyquist noise from the complex impedance of the environment. The power spectrum of this ohmic dissipation scales with frequency. T 2 Decoherence time: Time for phase difference between two eigenstates to become randomized 1/T 2 = 1/(2T 1 ) + 1/τ φ τ φ is the pure dephasing time (fluctuations in energy level splitting) Maximum value of T 2 is 2T 1
Previously Known Relaxation Processes Purcell effect (Purcell 1946). With the qubit embedded in a microwave transmission line cavity, spontaneous emission of photons from the qubit into the cavity produces relaxation. 1 High frequency ohmic charge noise arises from quantized energy levels in the two-level systems (TLSs) that at low frequencies produce 1/f charge noise. 2,3 Excess quasiparticles (above the thermal population) generate additional dissipation at microwave frequencies, reducing T 1. 4,5 [ 1 Houk et al., Phys. Rev Lett. 101, 808502 (2008). 2 stafiev et al., Phys. Rev Lett. 93, 267007 (2004). 3 Shnirman et al., Phys. Rev Lett. 94, 127002 (2005). 4 Martinis et al., Phys. Rev Lett. 103, 097002 (2009). 5 Catelani et al., Phys. Rev Lett. 106, 077002 (2011).]
Typical Values of T 1 and T 2 For many years, T 1, T 2ECHO were typically a few µs at degeneracy Famous, solitary exception: flux qubit fabricated at NEC and measured at MIT-LL 1 : T 1 = 12 µs, T 2ECHO = 23 µs More recently, six flux qubits in a 3-D cavity yielded 2 T 1 = 6-20 µs,t 2* (Ramsey) = 2 8 µs 1 Bylander et al. (2011) 2 Stern et al. (2014)
From SQUID to Qubit The Ubiquitous 1/f Noise Three-Junction Flux Qubit Relaxation and Decoherence The C-Shunt Flux Qubit: Origin The C-Shunt Flux Qubit: Recent Experiments Concluding Remarks
The C-Shunt Flux Qubit: Origin You, Hu, Ashab & Nori, Phys. Rev. B 140515 (2007) Φ q γ γ Charge noise arises from TLSs close to small junction Very small area of junction is not so easy to control, producing variations in C j and E j, and thus in qubit characteristics. Φ q γ γ Adding C s allows more flexibility in design and lower values of I q. In the presence of flux noise, since T 2 1/(dν/dΦ q ) 2 1/Ι q2, reducing I q increases T 2 while retaining high anharmonicity. Since C s >> γc j, most of the electric field energy is stored in C s, which can be designed to have a very high Q. Steffen et al. (IBM): T 1 5.7 µs, T 2E = 9.6 µs 2T 1
Importance of Anharmonicity Energy Spectrum vs. Applied Flux Frequency (GHz) Φ q /Φ 0 States 0> and 1> provide an excellent approximation to a two-level system near Φ 0 /2 Anharmonicity: A = f 12 f 01 Higher anharmonicity results in faster gates since the spread of frequencies around f 01 in a microwave pulse can be greater without exciting higher states
From SQUID to Qubit The Ubiquitous 1/f Noise Three-Junction Flux Qubit Relaxation and Decoherence The C-Shunt Flux Qubit: Origin The C-Shunt Flux Qubit: Recent Experiments Concluding Remarks
Design Template: Interdigitated Capacitor Resonator & Qubits λ/2 CPW resonator IDC Qubit 10 μm qubit loop QB#1 small JJ shunt capacitor QB#2 CPW slide-21 CSQ 11/28/2016 2.5 mm 100 μm ground strip ground Two qubits capacitively coupled to the same coplanar waveguide (CPW) resonator via their shunt capacitors Resonator length = 8 mm, f R = 8.27 GHz
Fabricated IDC and Qubits Resonator & Qubits λ/2 CPW resonator IDC Qubit 10 μm qubit loop QB#1 small JJ Shunt capacitor QB#2 100 μm 2.5 mm slide-22 CSQ 11/28/2016
Square-Plate Capacitor for Qubits Capacitor plates: 200 µm 200 µm C s 51 ff CPW strip Small junction Small junction: 150 nm 150 nm C jsmall 1 2 ff γ 0.42 Area of the capacitor plates 10 6 times greater than the area of the small junction
Fabrication All structures except junctions deposited with: Molecular beam epitaxy (MBE) of Al or Electron beam evaporation of Al Junctions: Electron beam evaporation of Al Substrate: Sapphire annealed at 900 C for outgassing and surface reconstruction
Dilution refrigerator typically at 20 mk Dispersive cqed Readout Directional couplers Readout (resonator) frequency ω ro /2π = 8.27 GHz HEMT Qubit drive frequency ω d Magnetic field B applied to the qubit changes f 01 and thus the resonant frequency of the resonator This changes the amplitude and phase of the transmitted readout pulse, yielding f 01 Average over typically 10,000 pulses
Spectra of Qubits A and B ω q /2π (GHz) Parameters chosen to yield 5 GHz A = 4.36 GHz B = 4.70 GHz Flux Bias Current (µa)
Flux Qubits A & B: T 1 at Degeneracy Excited-state population ρ e Devices fabricated simultaneously on the same wafer with identical geometries, and measured during the same cool down τ(µs) A = 4.36 GHz B = 4.70 GHz Solid lines are exponential fits
1/T 1 = Calculation of Relaxation Processes ħ 2 g d Φ e 2 S Φ (ω q ) Flux noise S Φ (ω q ) spectral density at qubit frequency d Φ is transition magnetic dipole. + ħ 2 g d Q e 2 S Q (ω q ) S Q (ω q ) spectral density at qubit frequency Ohmic charge noise d Q is transition electric dipole. + g 2 κ g 1γ σ y a + a e 1γ 2 Energy lost by qubit due to coupling Purcell effect to the resonator with decay rate κ; g is the coupling strength. States g 1γ and e 1γ are the dressed states of the coupled qubitresonator system. The Pauli operator for the qubit is σ y.
Qubit B: T 1 vs. Frequency Qubit B = 4.70 GHz T 1 limited predominantly by ohmic charge noise at lower frequencies T 1 dominated by Purcell at higher frequencies Flux noise is relatively unimportant Ohmic charge noise: Calculated from S Q (ω) = A Q ω/(2π 1 GHz), with A Q fitted to data Purcell: Obtained from measured resonator frequency and linewidth and calculated coupling to the qubit Flux noise: Assumes S Φ (f) = [(1 µφ 0 ) 2 /Hz](2π 1 Hz/ω) 0.8 (next slide)
Flux Qubit C (Small ): T 1 vs. Frequency Qubit C = 0.88 GHz Parameters chosen to yield small : C = 0.88 GHz T 1 limited totally by flux noise below about 4 GHz ( C 0.2 B ) Flux noise observed at frequencies up to 4 GHz T 1 dominated by Purcell at higher frequencies Charge noise is relatively unimportant Flux noise: Assumes S Φ (f) = [(1 µφ 0 ) 2 /Hz](2π 1Hz/ω) 0.8 (From SQUID spectra) Purcell: Obtained from measured resonator frequency and linewidth and calculated coupling to the qubit Ohmic charge noise: Calculated from S Q (ω) = A Q ω/(2π 1 GHz), with A Q fitted to data from qubit B
Dependence of T 1 on Qubit Geometry Interdigitated Parallel bars Squares T 1 (µs) T 1 (µs) T 1 (µs) Qubit A 23 21 44 Qubit B 22 17 55 Qubit A & B geometries identical All devices fabricated simultaneously on same wafer Measured during same cool down Square geometry wins!
Dependence of A Q on Qubit Geometry Interdigitated A Q = (15.4 10 9 e) 2 /Hz Squares A Q = (7.4 10 9 e) 2 /Hz Large area capacitor (C s >> C j ) stores most of electric field energy with greatly reduced electric field strength compared with that of the junctions. Reduced electric field participation of surface and interface losses. Large shunt capacitor improves qubit reproducibility: effect of variation in junction capacitance is reduced, and unwanted stray capacitances are overwhelmed.
Parameters of 22 Qubits
Parameters of 22 Qubits Anharmonicity A 800 910 MHz Longest T 1 s
Measured & Predicted Values of T 1 at Degeneracy: 22 Qubits with Different Parameters & C s C s : 18 51 ff E c : 0.38 1.1 GHz E j : 36 140 GHz Frequency at deg: 0.5 5 GHz Anharmonicity for longest T 1 s 800-910 MHz Values of T 1 at degeneracy calculated from known qubit parameters using fixed values of flux and charge noise. Good agreement given wide variation in design and critical current density over five fabrication runs
Flux Noise: α = 0.8 It appears that α = 0.8 is maintained rather accurately over the entire frequency range of flux noise below temperatures of, say, 100 mk. A minor deviation from α = 0.8 would have a major impact. As an example, consider two flux noise spectral densities: S 1 (f) 1/f, S 2 (f) 1/f 0.8 And S 1 (1 Hz) = S 2 (1 Hz) At 5 GHz, S 2 (5 GHz)/S 1 (5 GHz) = 87! Flux 1/f noise persists from 10 4 Hz (SQUID) to 4 GHz (flux qubit) with constant slope.
Qubit A T 2 : Spin-Echo at Degeneracy T 1 = 44 µs Maximum T 2Echo = 2T 1 = 88 µs Using Carr-Purcell-Meiboom-Gill (CPMG) Sequence: T 2CPMG 85 µs 2T 1 slide-37 CSQ 11/28/2016
From SQUID to Qubit The Ubiquitous 1/f Noise Three-Junction Flux Qubit Relaxation and Decoherence The C-Shunt Flux Qubit: Origin The C-Shunt Flux Qubit: Recent Experiments Concluding Remarks
C-Shunt Flux Qubit: Concluding Remarks Planar device. Combination of C-shunt, which increases design flexibility, and low loss materials yields: T 1 55 µs. Qubits with highest values of T 1 and T 2 have an anharmonicity as high as 800-910 MHz. C-shunt which stores virtually all of the electrical energy increases reproducibility and largely eliminates the effects of stray capacitances near the small junction. Large capacitor (C s >> C j ) reduces sensitivity to charge noise. Large area increases T 1 & T 2 by reducing the electric field at its edges, and hence the coupling of the qubit to two-level systems. Flux 1/f noise extends over 14 decades of frequency, from 10 4 Hz (SQUIDs) to 4 10 9 Hz (flux qubits). Future Simulations of the electric field losses around different capacitor designs are in progress. The C-shunt flux qubits made to date are not necessarily optimized. The trade offs between ohmic charge noise, flux noise and the Purcell effect are subtle, and detailed simulations are required to optimize the performance.