Fig. 1. (a) Schematic diagram of quantum gyroscope. (b) Evolution of the error δΩBA (cyan solid line) in the presence of photon loss under the Born–Markovian approximation. The blue dashed line is the local minima of δΩBA. The global minimum is marked by the red dot. (c) Numerical fitting reveals that the global minimum scales with the photon number as minδΩBA=5.4κN−0.23. The parameters are N=100, Ω=ω0, and κ=0.2ω0.

Fig. 2. (a) Energy spectrum of the total system formed by the two optical fields and their environments. Non-Markovian dynamical evolution of δΩ(t) multiplied by a magnification factor P when (ωc/ω0,P)=(2,10−1) in (b), (20, 10−2) in (c), and (25, 10−3) in (d). The blue dashed line in (d) is obtained by numerically solving Eq. (5), and the cyan solid line is obtained from the analytical form Eq. (9). We use s=1, η=0.05, Ω=10−2ω0, and N=100.

Fig. 3. Local minima of δΩ(t) as a function of (a) time and (b) N when t=2.5×104ω0−1 at different ωc. Steady-state |ul(∞)| marked by dots, which match with Zl depicted by lines, and the energy spectrum are shown in the inset of (b). We use s=1, η=7×10−4, Ω=10−2ω0, and N=100.

Fig. 4. Local minima of δΩ(t) as a function of (a) time and (b) N when t=2.5×104ω0−1 at different η. Steady-state |ul(∞)| marked by dots, which match with Zl depicted by lines, and the energy spectrum are shown in the inset of (b). We use s=1, ωc=5×103ω0, Ω=10−2ω0, and N=100.

Fig. 5. (a)–(c) Solution of Eq. (C2) determined by the intersectors of two curves of y(E)=E (red dashed lines) and y(E)=Y+(E) (blue solid lines) or y(E)=Y−(E) (magenta dotted lines). In the regime E>0, both Y±(E) have infinite intersections with E, which form a continuous energy band. As long as either Y−(0)<0 or Y+(0)<0, an isolated eigenenergy corresponding to a bound state is formed in the regime E<0. (d)–(f) Corresponding behaviors of |u+(t)| (blue dotted lines) and |u−(t)| (red dashed lines) determined by numerically solving Eq. (A3). The light blue dotted and light red dashed lines in (d) and (e) show Zl determined by Eq. (C4). Accompanying the formation of a bound state, the corresponding |ul(t)| approaches a finite value, which exactly matches with Zl. The parameters are s=1, η=0.05, Ω=10−2ω0, ωc=2ω0 in (a) and (d), 20ω0 in (b) and (e), and 25ω0 in (c) and (f).

Fig. 6. (a) Global behavior of the local minima of the steady-state δΩ(t) as a function of N. (b) Threshold Nc in (a) as a function of Z1Z2. We use the same parameter values as the ones of the blue solid line in Fig. 4(b) of the main text.