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Flux Qubit Noise Spectroscopy Using Rabi Oscillations Under Strong Driving Conditions

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Flux Qubit Noise Spectroscopy Using Rabi Oscillations Under Strong Driving Conditions Flux qubit noise spectroscopy using Rabi oscillations under strong driving conditions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Yoshihara, Fumiki, Yasunobu Nakamura, Fei Yan, Simon Gustavsson, Jonas Bylander, William D. Oliver, and Jaw-Shen Tsai. “Flux Qubit Noise Spectroscopy Using Rabi Oscillations Under Strong Driving Conditions.” Phys. Rev. B 89, no. 2 (January 2014). © 2014 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevB.89.020503 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/86074 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. RAPID COMMUNICATIONS PHYSICAL REVIEW B 89, 020503(R) (2014) Flux qubit noise spectroscopy using Rabi oscillations under strong driving conditions Fumiki Yoshihara,1,* Yasunobu Nakamura,1,2 Fei Yan,3 Simon Gustavsson,4 Jonas Bylander,4,5 William D. Oliver,4,6 and Jaw-Shen Tsai1,7 1Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan 2Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8904, Japan 3Department of Nuclear Science and Engineering, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts 02139, USA 4Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA 5Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden 6MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA 7NEC Smart Energy Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan (Received 9 October 2013; published 9 January 2014) We infer the high-frequency flux noise spectrum in a superconducting flux qubit by studying the decay of Rabi oscillations under strong driving conditions. The large anharmonicity of the qubit and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi frequencies up to 1.7 GHz were achieved, approaching the qubit’s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide frequency range decreases with increasing frequency up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins. DOI: 10.1103/PhysRevB.89.020503 PACS number(s): 03.67.Lx, 74.50.+r, 85.25.Cp Flux noise has been investigated for decades to improve |(ω12 − ω01)/ω01|, to avoid unwanted excitations to the higher stability and sensitivity in superconducting flux-based de- energy levels, where ωij is the transition frequency between vices. Its power spectral density (PSD) has been studied the |i and |j states. We measured Rabi oscillations in a wide in superconducting quantum interference devices (SQUIDs) range of R/2π from 2.7MHzto1.7 GHz and evaluated [1,2] and in various types of superconducting qubits, such as the PSD of flux fluctuations at each R. The observed PSD charge [3], flux [4–10], and phase qubits [11–14]. The spectra decreases up to 300 MHz, where the spectral density is −20 2 −1 typically follow 1/f frequency√ dependence with a spectral approximately 10 (0) rad s. Above 300 MHz, the PSD density of 1–10 μ0/ Hz at 1 Hz, where 0 = h/2e is scatters and slightly increases. We discuss a possible origin of the superconducting flux quantum. The accessible frequency the flux fluctuations due to surface electron spins. range of the PSD was limited to approximately 10 MHz in The Hamiltonian of a flux qubit with a flux drive and in the spin-echo measurements [4,5,9,15] and was extended to a presence of fluctuations can be written in the persistent current few tens of megahertz using Carr-Purcell-Meiboom-Gill pulse basis as sequences [9]. Recently, spin-locking measurements provided H =− [(σ + εσ ) + ε cos(ω t) σ the PSD up to approximately 100 MHz [16], and a study of pc 2 x z mw mw z qubit relaxation due to dressed dephasing in a driven resonator + δ(t)σ + δε(t)σ ], (1) revealed the PSD at approximately 1 GHz [17]. The spectrum x z in a higher-frequency range would give further information where σx and σz are Pauli matrices, is the tunnel splitting for better understanding of the microscopic origin of the flux between two states with opposite persistent current direction fluctuations. along the qubit loop Ip, and ε = 2Ip0nφ is the energy bias Decay of Rabi oscillations has also been used as a tool between the two states. Here the flux bias through the loop to characterize the decoherence in superconducting qubits. ex is normalized by 0 as nφ = ex/0 − 0.5. The first PSDs of fluctuating parameters, such as charge, flux, or term on the right-hand side of Eq. (1) represents the flux qubit coupling strength to an external two-level system, at the Rabi with a static√ flux bias. The transition frequency can be written frequency can be detected [3,9,18,19]. The Rabi frequency 2 2 R as ω01 = + ε . We find ω01 = and ∂ω01/∂nφ = 0at is proportional to the amplitude of the driving field for weak nφ = 0; this is the optimal flux bias condition where dephasing to moderate driving at the qubit transition frequency, and Rabi due to fluctuations of nφ is minimal. The second term is an ac frequencies in the gigahertz range have been achieved under drive at frequency ωmw with the amplitude εmw to induce Rabi a strong driving field [20–22]. However, the decay was not oscillations. The third and fourth terms represent fluctuations systematically studied because of the presence of extrinsic of and ε, respectively. In the present sample, ε is tunable decoherence mechanisms under the strong driving conditions. via nφ while is fixed. To induce fast Rabi oscillations without significant extra There exist a few dominant contributors to the decay of decoherence, we choose a flux qubit having strong induc- Rabi oscillations: the quasistatic noise; the noise at ω01, which tive coupling to a microwave line and large anharmonicity, causes the qubit energy relaxation; and the noise at R [3,9,23]. The resulting decay envelope Aenv(t) is described as * = − exp [email protected] Aenv(t) Ast(t)exp R t , (2) 1098-0121/2014/89(2)/020503(5) 020503-1 ©2014 American Physical Society RAPID COMMUNICATIONS FUMIKI YOSHIHARA et al. PHYSICAL REVIEW B 89, 020503(R) (2014) where A (t) is the contribution from the quasistatic noise, 2.5 st ) 1.60 exp 1.2 (a) -1 (b) R s 2.0 /2 which is usually nonexponential, and is the decay rate 7 R Bloch-Siegert 1.58 Hz) 8 1.5 R (10 0.8 numerical calc. (10 of the exponentially decaying term. As we are interested in 1/e 1.56 (10 1.0 R 9 Hz) 1/e R st the flux fluctuations at the Rabi frequency, contributions from 0.4 1.54 /2 R , 0.5 other sources are to be separated out. st R 0.0 0.0 1.52 0.8 1.2 1.6 -400 -200 0 The quasistatic noise, which results in Ast(t)inEq.(2), 9 /2 (10 Hz) /2 (MHz) is attributed to the fluctuations of the time-averaged values R0 mw 9 /2 (10 Hz) of δε(t) and δ(t) during a single decoherence measurement 01 2 5.8 6.0 6.2 6.4 6.6 trial. The variances of the quasistatic flux noise, σδε, and the 3.0 2 1.75 (c) (d) ) 3 /4 σ 1 noise, δ, are determined from the result of free-induction- -1 Hz) 1.70 flux s 9 2 2 6 2.0 decay (FID) measurements [24], where we find σ σ . 1.65 insensitive δε δ (10 (10 To evaluate the decay envelope A (t) due to the quasistatic 1.60 1.0 st /2 on resonance 1/e R R 1.55 flux noise, we numerically calculate the time evolution of the constant mw 0.0 density matrix of the qubit ρ (t) under H . 3.2 3.6 4.0 4.4 7 8 9 qubit pc 9 10 10 10 /2 (10 Hz) /2 (Hz) The exponentially decaying component of the envelope is R0 caused by the fluctuations at ω and , and the rate is written 01 R FIG. 1. (Color online) (a) Numerically calculated shift of the as [3] resonant frequency δω (black open circles) and the Bloch–Siegert 2 st (3 − cos ζ ) shift δωBS (blue line). (b) Numerically calculated decay rate R (black exp = 1 + , (3) R 4 R open circles) and Rabi frequency R (red solid triangles) as functions of the detuning δωmw from ω01. The purple solid line is a fit based where on Eq. (6). The measured 1/e decay rates 1/e at ε/2π = 4.16 GHz R H 2 for the range of Rabi frequencies R/2π between 1.5 and 1.6 GHz 2π ∂ pc 1 = Sλ(ω01) 1 0 (4) (blue solid circles) are also plotted. (c) Calculated Rabi frequency R 2 ∂λ = + λ based on Eq. (6), as a function of ε for the cases (i) ωmw ω01 δω (black solid line) and (ii) ω /2π = 6.1 GHz (red dashed line). The and mw upper axis indicates ω01, corresponding to ε in the bottom axis.
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