High-Contrast Qubit Interactions Using Multimode Cavity QED

David C. McKay, Ravi Naik, Philip Reinhold, Lev S. Bishop, and David I. Schuster

Phys. Rev. Lett. 114, 080501 – Published 27 February 2015

Abstract

We introduce a new multimode cavity QED architecture for superconducting circuits that can be used to implement photonic memories, more efficient Purcell filters, and quantum simulations of photonic materials. We show that qubit interactions mediated by multimode cavities can have exponentially improved contrast for two qubit gates without sacrificing gate speed. Using two qubits coupled via a three-mode cavity system we spectroscopically observe multimode strong couplings up to 102 MHz and demonstrate suppressed interactions off resonance of 10 kHz when the qubits are $\approx 600 MHz$ detuned from the cavity resonance. We study Landau-Zener transitions in our multimode systems and demonstrate quasiadiabatic loading of single photons into the multimode cavity in 25 ns. We introduce an adiabatic gate protocol to realize a controlled- $Z$ gate between the qubits in 95 ns and create a Bell state with 94.7% fidelity. This corresponds to an on/off ratio (gate contrast) of 1000.

Received 7 May 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.080501

Authors & Affiliations

David C. McKay^1,*, Ravi Naik¹, Philip Reinhold^1,†, Lev S. Bishop^2,3, and David I. Schuster¹

¹James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
²Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
³IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA

^*dcmckay@uchicago.edu
^†Present address: Yale Department of Applied Physics, New Haven, Connecticut 06511, USA.

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Vol. 114, Iss. 8 — 27 February 2015

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Figure 1
Multimode device schematic and spectroscopy. (a) Schematic (top) and optical image (bottom) of our cQED device consisting of three lumped LC resonators (blue dotted line), which couple two transmon-type qubits (red dotted line). The qubits are coupled to readout resonators at $ν_{1 (2)} = 4.20 (4.65) GHz$ . The qubit 1(qubit 2) lifetime $T_{1} = 2.36 (2.14) μ s$ and the decay of Ramsey coherence (fit to Gaussian decay $e^{- t^{2} / 2 σ^{2}}$ ) is $σ = 312 (492) ns$ . Full fabrication details, qubit properties, instrumentation, and cryogenic setup are given in Ref. [23]. (b) Single qubit spectroscopy as the qubit frequency $ν_{Q}$ is tuned using the flux line. The dashed line is a fit obtained by diagonalizing the energy levels of the transmon in the charge basis. (c) Spectroscopy of the region where the qubit frequency crosses through the filter modes [dashed box in (b)]. The frequency of the other qubit is fixed and below the filter. The dashed lines are the eigenvalues of the Hamiltonian given by Eq. (1) using the qubit-filter parameters listed in the main text. The inset is a cross section of similar spectroscopy data demonstrating multimode strong coupling. (d) Spectroscopy of the qubit-qubit avoided crossing [dashed box in (c)]. In (b), (c), and (d) flux (in $Φ_{0}$ ) is obtained from experimental units as a fit parameter.
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Figure 2
Off-resonant coupling. Qubit-qubit exchange rate $J$ as a function of the qubit frequency $ν_{Q}$ (top axis, detuning $Δ$ from the bare cavity frequency) for two different scaling laws (dashed lines), by numerical diagonalizing Eq. (1) (green line), and by measuring the exchange splitting (data points). We measure the exchange splitting from qubit spectroscopy, sample data is shown in Fig. 1. We also plot a numerical calculation of the c-phase rate [from Eq. (1)] versus the qubit 1 frequency using the filter parameters determined by the fit in Fig. 1 where qubit 2 is detuned below qubit 1 by 50 MHz (red line).
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Figure 3
Single photon loading and Stark shift. (a) To study the dynamics of loading single photons into the filter we traverse the qubit-filter avoided crossing in variable time $t$ (protocol illustrated in inset and described in the main text). We plot the qubit excited state population versus the ramp time $t$ : the solid black line is a guide to the eye, the dashed gray line is the expected maximum state population given $T_{1}$ decay, and the green solid line is a numerical solution of the Schrödinger equation using the Hamiltonian given by Eq. (1) and scaled by $T_{1}$ decay. (b) To measure the Stark shift between a single photon in the lowest mode of the filter and a qubit at bare frequency $ν_{Q 2, f}$ we perform a Ramsey experiment (protocol illustrated in inset and described in the main text). We plot the Stark shift as a function of $ν_{Q 2, f}$ and compare against a theory curve (blue solid line) with no free parameters. We use $ν_{Q 2, f} = 5.3 GHz$ as the reference height (i.e., set the Stark shift at that point to zero).
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Figure 4
Bell state and CZ gate. (a) Timing diagram for our Bell state experiment illustrating the flux pulses (solid lines) and microwave pulses (Gaussian). (b) The energy levels calculated from Eq. (1) in the orange region highlighted in (a). When qubit 1 crosses the filter, the qubit excitation is converted into a photon in the lowest filter mode. When qubit 2 is raised, the energy of the filter photon depends on the state of qubit 2, which generates a $c$ phase; the total phase is the yellow area. (c) Absolute value of the density matrix elements after state tomography of the Bell state produced by the gate [23].
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