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Nonreciprocal microwave transmission based on Gebhard-Ruckenstein hopping

Shumpei Masuda, Shingo Kono, Keishi Suzuki, Yuuki Tokunaga, Yasunobu Nakamura, and Kazuki Koshino
Phys. Rev. A 99, 013816 – Published 10 January 2019
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Abstract
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Article Text
  • INTRODUCTION
  • GEBHARD-RUCKENSTEIN HOPPING
  • CIRCULATOR BASED ON GR HOPPING
  • RESULTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study nonreciprocal microwave transmission based on the Gebhard-Ruckenstein hopping. We consider a superconducting device that consists of microwave resonators and a coupler. The Gebhard-Ruckenstein hopping between the resonators gives rise to a linear energy dispersion which manifests chiral propagation of microwaves in the device. This device functions as an on-chip circulator with a wide bandwidth when transmission lines are attached.

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    • Received 6 April 2018
    • Revised 10 October 2018

    DOI:https://doi.org/10.1103/PhysRevA.99.013816

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Optical & microwave phenomena
    1. Physical Systems
    Superconducting devices
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

    Authors & Affiliations

    Shumpei Masuda1,*, Shingo Kono2, Keishi Suzuki2, Yuuki Tokunaga3, Yasunobu Nakamura2,4, and Kazuki Koshino1

    • 1College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Ichikawa, Chiba 272-0827, Japan
    • 2Research Center for Advanced Science and Technology, University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan
    • 3NTT Secure Platform Laboratories, NTT Corporation, Musashino 180-8585, Japan
    • 4RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan

    • *masulas@tmd.ac.jp

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    Vol. 99, Iss. 1 — January 2019

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    Images

    • Figure 1
      Figure 1

      Schematics of energy dispersions in the first Brillouin zone. (a) Spectrum with linear dispersion. There is a discontinuity at the boundary of the BZ, namely, E(π/aL)=−E(−π/aL). (b) Spectrum corresponding to a system with short-range hopping. The red circle corresponds to a positive group velocity, while the blue square corresponds to a negative group velocity.

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    • Figure 2
      Figure 2

      Schematic of a GR cluster for N=5. The blue circles represent resonators. The dashed lines represent the coupling between the resonators.

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    • Figure 3
      Figure 3

      Coupling constant in the GR model. The g̃m,n for m=1,2 are shown for (a) N=9 and (b) N=8. Note that the rightmost and leftmost values in the panels are identical. The dashed lines are guides to the eye. The arrows in (a) indicate the cyclicity of the Hamiltonian elements for N=9 and the arrows in (b) indicate the lack of the cyclicity for N=8.

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    • Figure 4
      Figure 4

      Eigenenergies of Hcluster as a functions of kν for (a) N=9 and (b) N=8. Also shown is the phase of the wave function ϕ−1(l) relative to that of ϕ−1(1) for (c) N=9 and (d) N=8. The dashed lines are guides to the eye.

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    • Figure 5
      Figure 5

      Time evolution of the population at each site for the systems with (a) N=5 and (b) N=6. Here pi denotes the population of the resonator i.

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    • Figure 6
      Figure 6

      Schematic of a circulator with N=5. The input field is applied through one of the transmission lines. The blue circles, the green circle, and gray lines represent the resonators, the coupler, and the transmission lines, respectively. The dashed lines represent the coupling between the resonators. Transmission lines are uncoupled to resonators 2 and 5 in this example.

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    • Figure 7
      Figure 7

      Circulation properties for the case of N=3 and κ1,2,3=κ. (a) Dependence of the forward transmission probabilities (|S12|2, |S23|2, and |S31|2) on κ for Δω=0. The inset shows the system configuration. (b) Dependence of the forward and backward transmission and reflection probabilities on Δω for κ=2g.

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    • Figure 8
      Figure 8

      Circulation properties for the case of N=5, κ1,3,4=κ, and κ2,5=0. (a) Dependence of the forward transmission probabilities (|S13|2, |S34|2, and |S41|2) on κ for Δω=0. The inset shows the system configuration. (b) Dependence of the forward and backward transmission and reflection probabilities on Δω for κ=4g. The thin black line represents |S12|2 for N=3, denoted by |S12(3)|2. The inset is a close-up around Δω/g=0.

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    • Figure 9
      Figure 9

      Circulation properties for the case of N=5, κ1,2,5=κ, and κ3,4=0. (a) Dependence of the forward transmission probabilities (|S12|2, |S25|2, and |S51|2) on κ for Δω=0. The inset shows the system configuration. (b) Dependence of the forward and backward transmission and reflection probabilities on Δω for κ=2.472g. The black line represents |S12(3)|2. The inset is a close-up around Δω/g=0.

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    • Figure 10
      Figure 10

      Circulation properties for the case of N=4 and κ3=0. (a) Dependence of the forward transmission probabilities (|S12|2, |S24|2, and |S41|2) on κ for Δω=0 when κ1,2,4=κ. The inset shows the system configuration. (b) Dependence of the forward and backward transmission and reflection probabilities on Δω for the case of κ1=2.14g and κ2,4=4.24g.

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    • Figure 11
      Figure 11

      Circulation properties for the case of N=6, κ1,3,5=κ, and κ4,6=0. (a) Dependence of the forward transmission probabilities (|S13|2, |S35|2, and |S51|2) on κ for Δω=0. The inset shows the system configuration. (b) Dependence of the forward and backward transmission and reflection probabilities on Δω for κ=4.328g. The black line represents |S12(3)|2. The inset is a close-up around Δω/g=0.

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    • Figure 12
      Figure 12

      Circulation properties for the case of N=195, κ1,66,131=κ, and κi=0 for i≠1,66,131. (a) Dependence of the forward transmission probabilities on κ for Δω=0. The inset schematically shows the system configuration for large N. (b) Dependence of the forward and backward transmission and reflection probabilities on Δω for κ≃4g. The transmission and reflection probabilities for N=195 are indicated by arrows. The red (gray) solid and purple (gray) dash-dotted lines are for the forward transmission and the reflection probabilities for N=195, respectively. The green (light gray) dashed line, which has peaks at |Δω/g|≃π, is for the backward transmission probability for N=195. The black solid and dashed lines are for the forward transmission probabilities for the N=6 system in Fig. 11 and N=3, respectively. The right inset is a close-up around Δω/g=0. The upper black line is for N=6 and the lower red (gray) line is for N=195.

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    • Figure 13
      Figure 13

      Dependence of Δωop(N) on N for P=0.5, 0.8, 0.9, and 0.99. The results for N=3, 4, 5, 6, and 7 correspond to Figs. 7, 10, 8, 11, and 19, respectively. The circulators for N=4,5,7 have two different values of Δωop(N) corresponding to different input ports. The higher values are shown here. The other values are at most 28% lower than the higher ones (not shown).

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    • Figure 14
      Figure 14

      Dependence of the directionality parameter on Δω for (a) N=4, (b) N=5, and (c) N=6. The thin black line is for N=3. The system configurations used and the parameters for N=3, 4, 5, and 6 are the same as in Figs. 7, 10, 8, and 11, respectively.

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    • Figure 15
      Figure 15

      Dependence of Δωd(N) on N for Pd=0.5, 0.8, and 0.9. The system configurations used and the parameters are the same as in Fig. 13. The circulators for N=4,5,7 have two different values of Δωd(N) corresponding to different input ports, respectively. The higher values are shown here. The other values are at most 20% lower than the higher ones (not shown).

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    • Figure 16
      Figure 16

      Dependence of Pm(out)(Δω) in Eq. (10) on Δω for κm(int)=0.02g for (a) N=3 and (b) N=6. The solid lines correspond to the forward transmission of the noise, while the dash-dotted and dotted lines correspond to the backward transmission and reflection, respectively. The thermal equilibrium occupation numbers used are n¯1(b)=0.5 and n¯n(b)=0 for n≠1 and n¯n(c)=0. The system configurations and κm are the same as in Figs. 7 and 11, respectively.

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    • Figure 17
      Figure 17

      Dependence of the forward and backward transmission probabilities and reflection probability on Δω for κm(int)=0.02g,0.002g for (a) N=3 and (b) N=6. The system configurations and κm are the same as in Figs. 7 and 11, respectively. The data for κm(int)=0.02g and 0.002g are represented by colored (gray) and black lines, respectively. The thin line in (b) shows the forward transmission probability for N=3 for κm(int)=0.02g.

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    • Figure 18
      Figure 18

      Dependence of the forward transmission probabilities on Δω for κm(int)=0.02g for (a) N=4, (b) N=5, and (c) N=6. The system configurations and κm are the same as in Figs. 10, 8, and 11, respectively. The thin black line represents |S12(3)|2.

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    • Figure 19
      Figure 19

      Circulation properties for the case of N=7, κ1,3,6=κ, and κ2,4,5,7=0. (a) Dependence of the forward transmission probabilities on the coupling to transmission line κ for Δω=0. The inset shows the system configuration. (b) Dependence of the forward transmission probabilities on detuning Δω for κ=4.45g. The thin lines correspond to the system with N=5 in Fig. 8. (c) Close-up of (b) around Δω/g=0.

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    • Figure 20
      Figure 20

      Dependence of the directionality parameter in Eq. (9) on Δω for N=7. The parameters used and the configuration are the same as in Fig. 19. The thin black line is for the system with N=3 shown in Fig. 7.

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    • Figure 21
      Figure 21

      (a) Dependence of the forward and backward transmission probabilities on detuning Δω for κ1,2,5=2g. The thin lines correspond to the case with θ=π/10 and thick lines correspond to θ=π/8. The inset shows the system configuration. (b) Plot of |S21|2−|S12|2 as a function of Δω and θ.

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    • Figure 22
      Figure 22

      Effects of the fluctuations on the system parameters. The system with N=6 in Fig. 10 is investigated. The dependence of |S13|2 is on (a) λκ1 and λκ3, (b) λg13 and λg35, and (c) θ1 and θ2. The values next to the contour lines indicate the values of |S13|2.

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    • Figure 23
      Figure 23

      Circuit diagram of a possible physical realization of the system depicted in Fig. 6. Here Z0 is the characteristic impedance of the transmission lines, Cκ is the coupling capacitance between the resonators and the transmission lines, Cc is the coupling capacitance between the resonators and the Josephson ring, and CJ and EJ are the junction capacitance and the Josephson energy, respectively.

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