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Spin-boson model with an engineered reservoir in circuit quantum electrodynamics

Max Haeberlein, Frank Deppe, Andreas Kurcz, Jan Goetz, Alexander Baust, Peter Eder, Kirill Fedorov, Michael Fischer, Edwin P. Menzel, Manuel J. Schwarz, Friedrich Wulschner, Edwar Xie, Ling Zhong, Enrique Solano, Achim Marx, Juan-José García-Ripoll, Rudolf Gross

arXiv:1506.09114v1cond-mat.mes-hallcond-mat.supr-conquant-ph

TL;DR

The paper characterizes a superconducting qubit coupled to an engineered broadband environment, addressing how to probe the spin-boson bath in circuit QED. It introduces partial reflectors to shape the spectral function and finds an Ohmic bare line alongside increased spontaneous emission in the engineered environment.

  • Problem

    The paper addresses how to characterize a spin-boson model coupled to a broadband engineered environment using a superconducting qubit.

    Source: p. 1

  • Method

    The authors embed a flux qubit in a transmission line with partial reflectors and use resonance fluorescence while tuning the qubit frequency to probe the spectral function.

    Source: pp. 1–2

  • Results

    The bare transmission line is Ohmic over a broad frequency range, and the engineered environment increases spontaneous emission by a factor of 8.

    Source: pp. 2, 4

  • Takeaways & Limitations

    The work demonstrates experimental characterization of a spin-boson model with a shaped reservoir using superconducting circuits.

    Source: pp. 1, 4

    The transfer-matrix model assumes frequency-independent partial reflectors and neglects losses in the reflectors and transmission line.

    Source: p. 2

Abstract

from arXiv · show

A superconducting qubit coupled to an open transmission line represents an implementation of the spin-boson model with a broadband environment. We show that this environment can be engineered by introducing partial reflectors into the transmission line, allowing to shape the spectral function, J({\omega}), of the spin-boson model. The spectral function can be accessed by measuring the resonance fluorescence of the qubit, which provides information on both the engineered environment and the coupling between qubit and transmission line. The spectral function of a transmission line without partial reflectors is found to be Ohmic over a wide frequency range, whereas a peaked spectral density is found for the shaped environment. Our work lays the ground for future quantum simulations of other, more involved, impurity models with superconducting circuits.

Supplemental Material to “Spin-boson model with an engineered reservoir in circuit

The supplemental material covers the paper’s experimental methods, data analysis, and theoretical background.

Publication information

Supplemental material

Classification

I. EXPERIMENTAL TECHNIQUES

Microwave transmission is measured through the sample in a dilution refrigerator, with flux biasing and two-tone measurements available for qubit spectroscopy.

Experimental setup

  • The sample is measured at approximately 25 mK using microwave transmission through a vector network analyzer.The setup includes attenuated input wiring, output amplification, and isolators to protect the sample from amplifier noise.Source: §I · p. 7, Figure 1 · p. 7

Flux bias

  • An external coil supplies qubit flux bias in persistent-current mode to minimize flux noise at the sample.Source: §I · p. 7

Two-tone measurements

  • Two-tone experiments add a second microwave source to the input while the network analyzer measures transmission.Source: §I · p. 7

II. TRANSMITTANCE OF AN ATOM EMBEDDED INTO A 1D TRANSMISSION LINE WITH TWO PARTIAL REFLECTORS

The section models transmission through a one-dimensional line containing a qubit between two partial reflectors, and describes how the resulting weak resonator modifies the qubit’s environment.

Model

  • The model uses transfer matrices to analyze transmittance through a one-dimensional line containing an atom between two partial reflectors.The reflectors arise from small impedance mismatches between the on-chip line and off-chip wiring.Source: §II · pp. 7–8

Pseudo-resonator

  • The two impedance mismatches form a pseudo-resonator whose resonant structure is called the weak resonance.Source: §II · p. 8

Qubit coupling

  • Coupling a qubit at the resonator center changes transmission and reflection properties as the weak resonator changes the density of states.Relative to an ideal open transmission line, the weak-resonator effect can be represented as a radiative correction.Source: §II · p. 8

A. Transmission and Reflection without Qubit

The section represents impedance mismatches as partial reflectors and combines their transfer matrices with propagation through the intervening line to describe a pseudo-resonator.

Reflector model

Transfer matrices

  • A partial reflector’s transfer matrix is expressed using its complex transmission and reflection coefficients; lossless scattering satisfies |tZr|^2 + |rZr|^2 = 1.The reflection coefficient is given as rZr = ΔZr/(Z0 + Zr).Source: A. Transmission and Reflection without… · p. 8

Reflector properties

Pseudo-resonator propagation

Resonator geometry

Transmission

Scattering relation

Resonator modes

  • The analysis constructs orthogonal scattered waves from opposite incidence directions and considers symmetric and antisymmetric superpositions inside the resonator.The intra-resonator solutions are restricted to −s/2 < x < s/2.Source: A. Transmission and Reflection without… · p. 9

B. Transmission and Reflection with Qubit

This section builds a transfer-matrix description of a qubit coupled to a resonator formed by partial reflectors, calibrates its transmission against the qubit-free system, and compares its emission response with the Purcell limit.

B. Transmission and Reflection with Qubit

  • The transfer-matrix treatment allows a less symmetric qubit matrix when probability is not conserved, accommodating possible dephasing effects.The general matrix is described for time-reversal symmetry with symmetric qubit reflectance and transmittance.Source: B. Transmission and Reflection with… · p. 10
  • The qubit transfer matrix is inserted at the resonator center between the reflector matrices, and calibration compares the hybridized system with the qubit-free transmittance.The qubit scattering description assumes low power, unit coupling efficiency, and equal emission phase in both directions; Γ1 and Γϕ denote relaxation and pure dephasing rates in open space.Source: B. Transmission and Reflection with… · p. 10
  • The coupled-system transmission is obtained by multiplying the open-space qubit transmission by a radiative correction factor.The model's real part is consistent with the calibrated density of states at the qubit, while its imaginary part marks a Lamb shift.Source: B. Transmission and Reflection with… · p. 11
  • For small κ, the model tends towards the Purcell factor for spontaneous-emission amplification.Figure 2 compares the Purcell effect (dashed line) with the model (diamonds).Source: B. Transmission and Reflection with… · p. 11, Figure 2 · p. 11

III. SPIN-BOSON MODEL OF A QUBIT IN AN OPEN TRANSMISSION LINE

This section relates inductive qubit–line coupling to the spin-boson spectral function and considers both a continuum transmission line and a single-mode resonator limit.

III. SPIN-BOSON MODEL OF A QUBIT IN AN OPEN TRANSMISSION LINE

  • The inductive interaction is expressed through the mutual inductance M and the transmission-line vacuum current Iline.The line-current expression uses l0, Z, and L for self-inductance per unit length, line impedance, and mode volume.Source: §III · p. 12
  • The single-mode resonator limit places the qubit coupling on one mode under closed boundary conditions and uses the normalization L = λ/4.Source: §III · p. 12
  • The spin-boson spectral function J(ω) characterizes a two-level system coupled to infinitely many bosonic degrees of freedom and, in the Ohmic form, is J(ω) = 2 παω.The dimensionless Kondo parameter α expresses the Ohmic spectral function.Source: §III · p. 12
  • For the continuum line, the derivation uses the wave phase velocity v and density of states D(ω); the resonator comparison assumes single-mode coupling and D(ω) = 2.Source: §III · p. 12
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