Quantum dynamics of semiconductor quantum dot Josephson junctions
Abstract: Josephson junctions constructed from superconductor-semiconductor-superconductor heterostructures have been used to realize a variety of voltage-tunable superconducting quantum devices, including qubits and parametric amplifiers. To date theoretical descriptions of these systems have been restricted to small quantum fluctuations of the junction phase, making them inapplicable to many experiments. In this work we relax this, employing a path-integral formulation where the phase quantum dynamics is obtained self-consistently from an underlying many-body formalism. Our method recovers previously-known results for small phase fluctuations, and predicts new effects outside of that limit: (i) system capacitances undergo a gate-voltage-dependent renormalization; and (ii) an additional charge offset appears for asymmetric junctions. Our main results can be summarized in terms of a single-particle Hamiltonian, which can be directly compared to that of an ordinary Josephson junction. This more general theory could be a first step towards designing new quantum devices that go qualitatively beyond voltage-tunable variants of previously-known circuits.
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- For Γi≪Δ2−ϵg2much-less-thansubscriptΓ𝑖superscriptΔ2superscriptsubscriptitalic-ϵ𝑔2\Gamma_{i}\ll\sqrt{\Delta^{2}-\epsilon_{g}^{2}}roman_Γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≪ square-root start_ARG roman_Δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_ϵ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG, ζ=Δ2−ϵg2+Γϵg2Δ2−ϵg2+𝒪(sin2ϕ2Γi2Δ2−ϵg2)𝜁superscriptΔ2superscriptsubscriptitalic-ϵ𝑔2Γsuperscriptsubscriptitalic-ϵ𝑔2superscriptΔ2superscriptsubscriptitalic-ϵ𝑔2𝒪superscript2italic-ϕ2superscriptsubscriptΓ𝑖2superscriptΔ2superscriptsubscriptitalic-ϵ𝑔2\zeta=\sqrt{\Delta^{2}-\epsilon_{g}^{2}}+\Gamma\frac{\epsilon_{g}^{2}}{\Delta^% {2}-\epsilon_{g}^{2}}+\mathcal{O}\left(\sin^{2}\frac{\phi}{2}\frac{\Gamma_{i}^% {2}}{\Delta^{2}-\epsilon_{g}^{2}}\right)italic_ζ = square-root start_ARG roman_Δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_ϵ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + roman_Γ divide start_ARG italic_ϵ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG roman_Δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_ϵ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + caligraphic_O ( roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_ϕ end_ARG start_ARG 2 end_ARG divide start_ARG roman_Γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG roman_Δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_ϵ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) depends weakly on the phase.
- A more rigorous definition of EJ,effsubscript𝐸𝐽effE_{J,\text{eff}}italic_E start_POSTSUBSCRIPT italic_J , eff end_POSTSUBSCRIPT with exact continuum contributions does not alter this result.
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