Entanglement Thresholds of Doubly Parametric Quantum Transducers

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Entanglement Thresholds of Doubly Parametric Quantum Transducers

Curtis L. Rau, Akira Kyle, Alex Kwiatkowski, Ezad Shojaee, John D. Teufel, Konrad W. Lehnert, and Tasshi Dennis

Phys. Rev. Applied 17, 044057 – Published 29 April 2022

Abstract

Doubly parametric quantum transducers, such as electro-optomechanical devices, show promise for providing the critical link between quantum information encoded in highly disparate frequencies such as in the optical and microwave domains. This technology would enable long-distance networking of superconducting quantum computers. Rapid experimental progress has resulted in impressive reductions in decoherence from mechanisms such as thermal noise, loss, and limited cooperativities. However, the fundamental requirements on transducer parameters necessary to achieve quantum operation have yet to be characterized. In this work we find simple, protocol-independent expressions for the necessary and sufficient conditions under which doubly parametric transducers in the resolved-sideband, steady-state limit are capable of entangling optical and microwave modes. Our analysis treats the transducer as a two-mode bosonic Gaussian channel capable of both beamsplitter-type and two-mode squeezing-type interactions between optical and microwave modes. For the beamsplitter-type interaction, we find parameter thresholds that distinguish regions of the channel’s separability, capacity for bound entanglement, and capacity for distillable entanglement. By contrast, the two-mode squeezing-type interaction always produces distillable entanglement with no restrictions on temperature, cooperativities, or losses. Counterintuitively, for both interactions, we find that achieving quantum operation does not require either a quantum cooperativity exceeding one, or ground-state cooling of the mediating mode. Finally, we discuss where two state-of-the-art implementations are relative to these thresholds and show that current devices operating in either mode of operation are in principle capable of entangling optical and microwave modes.

Received 29 October 2021
Revised 4 April 2022
Accepted 6 April 2022

DOI:https://doi.org/10.1103/PhysRevApplied.17.044057

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Entanglement detection Optomechanics Quantum channels Quantum information processing with continuous variables Quantum information with hybrid systems Quantum networks Quantum teleportation

Devices

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Curtis L. Rau ^1,2,†, Akira Kyle ^1,2,†, Alex Kwiatkowski ^1,2, Ezad Shojaee², John D. Teufel ², Konrad W. Lehnert^1,2,3, and Tasshi Dennis ^2,*

¹Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
²National Institute of Standards and Technology (NIST), Boulder, Colorado 80305, USA
³JILA, University of Colorado and NIST, Boulder, Colorado 80309, USA

^*tasshi.dennis@nist.gov
^†These authors contributed equally.

Article Text

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Vol. 17, Iss. 4 — April 2022

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Images

Figure 1
(a) Diagram showing the couplings in the linearized operator equations of motion in a reference frame rotating at the pump frequencies. Circles represent internal resonator modes. (b) The equivalent circuit diagram derived from input-output relations of the central frequency mode in the resolved sideband limit. In addition to the coupling losses arising from the equations of motion, this also includes the effect of transmission losses on the itinerant optical and microwave modes. Environmental modes are represented with dashed lines. Throughout this paper we use the convention of denoting optical, microwave, and mediating modes with red, blue, and green, respectively.
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Figure 2
Effective two-mode squeezed lossy state describing the state generated by a DPT under a squeezing-type interaction with vacuum inputs. The variable $r^{'}$ is the effective squeezing parameter while $τ_{a}^{'}$ and $τ_{b}^{'}$ are the effective single-mode transmissivities.
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Figure 3
(a) Single-mode down-conversion channel obtained by initializing the microwave port with vacuum and tracing out the optical output in Fig. 1. (b) Single-mode up-conversion channel obtained by initializing the optical port with vacuum and tracing out the microwave output in Fig. 1. These single-mode channels can be completely characterized by effective transmissivity $τ_{{u, d}}^{'}$ and added noise $n_{{u, d}}^{'}$ parameters.
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Figure 4
Given the channel $E$ , the Choi-Jamiołkowski isomorphism gives the dual state $\hat{ρ}$ , which is found by $E$ acting on inputs that are each maximally entangled with an ancilla. For continuous variable systems, the maximally entangled states are infinitely squeezed two-mode squeezed vacuum states ( $r \to \infty$ ). The dashed box represents the two-mode channel between optical and microwave modes that the transducer implements.
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Figure 5
Plots of entanglement thresholds for the beamsplitter-type DPT interaction with equal cooperativities ( $C = C_{a} = C_{b}$ ) and no transmission losses ( $δ_{a} = δ_{b} = ϵ_{a} = ϵ_{b} = 1$ ), in which case the thresholds can be characterized simply by the quantum cooperativity $C / n_{th}$ that is related to the average thermal occupation of the mediating mode due to radiation pressure cooling [36]. (a) Thresholds as a function of equal optical and microwave coupling transmissivities. (b) Thresholds as a function of only optical transmissivity with microwave transmissivity fixed at 0.5 to illustrate the different functional behavior between the thresholds. For both plots, below the red curve the two-mode transducer channel is separable [Eq. (6)], while between the red and green curves the channel is capable of producing bound entanglement, and above the green curve the channel is capable of producing distillable entanglement [Eq. (8)]. The entanglement breaking thresholds for the one-mode up-conversion [Eq. (11)] and down-conversion [Eq. (12)] channels are indicated with the dashed curves and are never better than the two-mode transducer channel thresholds. Finally, the logarithmic negativity [41] of the transducer’s Choi state is plotted, which shows increasing entanglement with higher coupling transmissivities and quantum cooperativities.
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