Research ArticlePHYSICS

Ultralong relaxation times in bistable hybrid quantum systems

See allHide authors and affiliations

Science Advances  08 Dec 2017:
Vol. 3, no. 12, e1701626
DOI: 10.1126/sciadv.1701626


  • Fig. 1 Hybrid quantum system.

    (A) Experimental setup. Schematic illustration of our experimental setup in which an ensemble of spins (described as an effective giant spin) is inductively coupled (with a coupling rate Ω) to the cavity mode. The nonlinearity stems from the anharmonicity of this coupled spin when driven beyond its linear regime, which we probe through the transmission |T|2 = Pout/Pin of the hybrid system. (B) Photograph of the system consisting of a superconducting transmission line cavity with an enhanced neutron-irradiated diamond on top of it, containing a large ensemble of NV spins (black). Two coupling capacitors provide the necessary boundary conditions for the microwave radiation.

  • Fig. 2 Rabi splitting under different drive powers.

    Evolution of the transmission spectrum for different input drive powers Pin. In the linear regime, the Rabi splitting is observable. For a drive power Pin ≈ Pref, the spin system starts to bleach and decouples from the cavity. For input drives Pin ≫ Pref, we observe the bare cavity transmission function. This behavior can be seen from the projection of the observed maximum transmission peaks on the xy plane. When driving the system resonantly (green dashed line) and with a large enough cooperativity, operating between these two regimes exhibits amplitude bistability.

  • Fig. 3 Steady-state bistability.

    We measure the steady-state bistability transmission through the cavity as a function of increasing (blue) and decreasing (red) input power Pin. In (A), the transmission measurements are plotted for the cooperativity value Ccoll ≈ 18 and κ/2π = 1.2 MHz using two subensembles in resonance with the cavity. (B) Same transmission measurement with Ccoll ≈ 49 and κ/2π = 0.44 MHz. A small bistability area is visible where the system evolves to different steady states depending on the history of the system in either upper or lower branch. (C) Same measurement as in (A), with an increased cooperativity of Ccoll ≈ 78 (by using all four NV subensembles in resonance with the cavity), again with κ/2π = 0.44 MHz. The dashed curves are numerical solutions of Eq. 3. The dashed lines in (B) show the asymptotic solutions in the limit of large and small drive amplitudes η. Two critical values of the input power, at which a phase transition between two stable branches occurs, are characterized by a saddle-node bifurcation and labeled as Pucrit and Pdcrit. For all three cases, a sketch of the corresponding potential is also depicted, which shows the occurrence of either one or two stable solutions (red and blue solid circles) and one unstable solution (B and C) (green solid circles) for a fixed value of the input power. Tunneling through the potential barrier does not occur in our case because of the large system size such that the system does not switch back and forth between the steady states in the bistable area.

  • Fig. 4 Quench dynamics measurement.

    Quench dynamics of the high cooperativity Ccoll ≈ 78 configuration and an initial state far in the strong driving branch. In (A), the intracavity intensity |T|2/|Tmax|2 is plotted over time for different drive intensities where the time to reach a steady state strongly depends on the input intensity. For drive intensities larger than a critical drive value Pdcrit (defined as the power where the system undergoes the phase transition from the upper to the lower branch, see Fig. 3), the spin system remains saturated and sets into a state on the upper branch, whereas in the opposite case, the system evolves into a steady state on the lower branch. Close to the critical drive Pdcrit, this time scale is extremely prolonged and approaches 4 × 104 s. The dashed lines correspond to predictions from our model. In (B), we show the phase diagram as d|T|2/dt over |T|2 for the evolution toward a steady state (black dotted line) for different input drives Pin. For drive powers close to the critical drive, the derivative approaches zero, and the dynamics becomes much slower compared to drive powers larger and smaller than the critical drive. In (C), the switching times between the upper and lower branch for different input drives are shown. We define the switching time tswitch as the inverse of the smallest gradient for a given curve [green circle in (A) and (B)]. Close to the critical drive, the switching time diverges, and the time to reach a steady state becomes arbitrarily long. The solid red line is a fitting function of the form tswitch ≈ |Pin − Pdcrit|−α (with α = 1.20 ± 0.04).

Stay Connected to Science Advances

Navigate This Article