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arXiv:1401.5545 · Physical Review B

Purcell effect with microwave drive: Suppression of qubit relaxation rate

Eyob Alebachew Sete, Jay Gambetta, Alexander N. Korotkov

Published Mar 21, 2014• 71 citations
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Abstract

We analyze the Purcell relaxation rate of a superconducting qubit coupled to a resonator, which is coupled to a transmission line and pumped by an external microwave drive. Considering the typical regime of the qubit measurement, we focus on the case when the qubit frequency is significantly detuned from the resonator frequency. Surprisingly, the Purcell rate decreases when the strength of the microwave drive is increased. This suppression becomes significant in the nonlinear regime. In the presence of the microwave drive, the loss of photons to the transmission line also causes excitation of the qubit; however, the excitation rate is typically much smaller than the relaxation rate. Our analysis also applies to a more general case of a two-level quantum system coupled to a cavity.

DOI
10.1103/physrevb.89.104516

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We analyze the Purcell relaxation rate of a superconducting qubit coupled to a resonator, which is coupled to a transmission line and pumped by an external microwave drive. Considering the typical regime of the qubit measurement, we focus on the case when the qubit frequency is significantly detuned from the resonator frequency. Surprisingly, the Purcell rate decreases when the strength of the microwave drive is increased. This suppression becomes significant in the nonlinear regime. In the presence of the microwave drive, the loss of photons to the transmission line also causes excitation of the qubit; however, the excitation rate is typically much smaller than the relaxation rate. Our analysis also applies to a more general case of a two-level quantum system coupled to a cavity.
The measurement fidelity of superconducting transmon and Xmon qubits is partially limited by the qubit energy relaxation through the resonator into the transmission line, which is also known as the Purcell effect. One way to suppress this energy relaxation is to employ a filter which impedes microwave propagation at the qubit frequency. We present semiclassical and quantum analyses for the bandpass Purcell filter realized by E. Jeffrey et al. [Phys. Rev. Lett. 112, 190504 (2014)]. For typical experimental parameters, the bandpass filter suppresses the qubit relaxation rate by up to two orders of magnitude while maintaining the same measurement rate. We also show that in the presence of a microwave drive the qubit relaxation rate further decreases with increasing drive strength.
We analyze a single-shot readout for superconducting qubits via the controlled catch, dispersion, and release of a microwave field. A tunable coupler is used to decouple the microwave resonator from the transmission line during the dispersive qubit-resonator interaction, thus circumventing damping from the Purcell effect. We show that, if the qubit frequency tuning is sufficiently adiabatic, a fast high-fidelity qubit readout is possible, even in the strongly nonlinear dispersive regime. Interestingly, the Jaynes-Cummings nonlinearity leads to the quadrature squeezing of the resonator field below the standard quantum limit, resulting in a significant decrease of the measurement error.
We consider a typical circuit QED setup where an artificial atom encodes a qubit and is dispersively coupled to a measurement resonator that in turn is coupled to a transmission line. We show theoretically that by placing another artificial atom in this transmission line to act as a filter, the Purcell decay of the qubit into the transmission line is suppressed. When strong control fields are applied in the transmission line, the filter is saturated and effectively switched off. Such a Purcell filtering capability permits both the control and measurement of the qubit using the single transmission line, while maintaining the long coherence time of the qubit in the absence of the control pulses. We show that high fidelity Pauli ${\ensuremath{\sigma}}_{x}$ gates on the qubit can be realized using simple pulse shapes. For devices that already use one transmission line both for control and measurement of the qubit, our paper provides a way to completely filter out the qubit frequency without removing the possibility of controlling the system. Further, combining the proposed filter with frequency multiplexing potentially enables both control and measurement of several qubits using a single Purcell-filtered transmission line. This will enhance the scalability of superconducting quantum processors by decreasing the number of the required transmission lines.
The speed and fidelity of dispersive readout of superconducting qubits should improve by increasing the amplitude of the measurement drive. Experiments show, however, that beyond some drive amplitude there is always a saturation or drop in fidelity, often associated with a decrease in qubit energy relaxation time T1T_1T1​. A simple Lindblad master equation does not capture the latter effect. More involved approaches based on effective master equations rely on strong assumptions about the spectra of the system and the bath and only partially agree with observations. Here, we perform a first-principles simulation of the full unitary dynamics of dispersive readout by considering the circuit QED Hamiltonian coupled to a microscopic model for the measurement transmission line, allowing for its arbitrary spectrum, including filters. Our access to the dynamics of the bath degrees of freedom allows us to investigate the emission spectrum of the system as a function of drive power. We show how the dependence of qubit T1T_1T1​ on readout drive amplitude is sensitive to the details of the bath spectrum. In particular, we find that T1T_1T1​ drops with increasing drive amplitude when a Purcell notch filter is placed at the qubit frequency, and that the Lindblad master equation shows general qualitative defects compared to the first-principles model.
The dynamics of a weakly anharmonic superconducting qubit in a complex electromagnetic environment is generally well described by an effective multimode Kerr Hamiltonian at sufficiently weak excitation. This Hamiltonian can be embedded in a master equation with losses determined by the details of the electromagnetic environment. Recent experiments indicate, however, that when a superconducting circuit is driven with microwave signals populating the system with sufficiently high excitations, the observed relaxation rates appear to be substantially different from expectations based on the electromagnetic environment of the qubit alone. This issue is a limiting factor in the optimization of superconducting qubit readout schemes. We claim here that an effective master equation with drive-power-dependent parameters is an efficient approach to model such quantum dynamics. In this sequence of papers, we derive effective master equations, whose parameters exhibit nonlinear dependence on the excitation level of the circuit as well as the electromagnetic environment of the qubit. We show that the number nonconserving terms in the qubit nonlinearity generally lead to a renormalization of dissipative parameters of the effective master equation, while the number conserving terms give rise to a renormalization of the system frequencies. Here, in Paper I, we consider the excitation-relaxation dynamics of a transmon qubit that is prepared in a certain initial state, but is not driven otherwise. A unitary transformation technique is introduced to study the renormalization of (i) qubit relaxation due to coupling to a generic bath and (ii) Purcell decay. Analytic expressions are provided for the dependence of the nonlinear dissipative terms on the details of the electromagnetic environment of the qubit. The perturbation technique based on unitary transformations developed here is generalized to the continuously driven case in Paper II [A. Petrescu et al., Phys. Rev. B 101, 134510 (2020)].
In the circuit quantum electrodynamics architecture, both the resonance frequency and the coupling of superconducting qubits to microwave field modes can be controlled via external electric and magnetic fields to explore qubit-photon dynamics in a wide parameter range. Here, we experimentally demonstrate and analyze a scheme for tuning the coupling between a transmon qubit and a microwave resonator using a single coherent drive tone. We treat the transmon as a three-level system, with the qubit subspace defined by the ground and the second excited states. If the drive frequency matches the difference between the resonator and the qubit frequencies, a Jaynes-Cummings-type interaction is induced, which is tunable in both amplitude and phase. We show that coupling strengths of about 10 MHz can be achieved in our setup, limited only by the anharmonicity of the transmon qubit. This scheme has been successfully used to generate microwave photons with a controlled temporal shape [M. Pechal et al., Phys. Rev. X 4, 041010 (2014)] and can be directly implemented with superconducting quantum devices featuring larger anharmonicity for higher coupling strengths.