Coherent controllers for optical-feedback cooling of quantum oscillators

We study the cooling performance of optical-feedback controllers for open optical and mechanical resonators in the linear quadratic Gaussian setting of stochastic control theory. We utilize analysis and numerical optimization of closed-loop models based on quantum stochastic differential equations to show that coherent control schemes, where we embed the resonator in an interferometer to achieve all-optical feedback, can outperform optimal measurement-based feedback control schemes in the quantum regime of low steady-state excitation number. These performance gains are attributed to the coherent controller's ability to simultaneously process both quadratures of an optical probe field without measurement or loss of fidelity, and may guide the design of coherent feedback schemes for more general problems of robust nonlinear and robust control.

Figure 1

Optical cavity plant system with five possible classical and coherent feedback controllers.

Figure 2

Experimental realization of an OPO with a cavity and a nonlinear crystal.

Figure 3

Output from our mathematica package, describing the plant system.

Figure 4

(b) Plant cavity photon number, as a function of noise strength $k_N$. The uncontrolled case is shown, as well as the photon number for various control schemes. (a) Photon number relative to the no-control case. Smaller is better.

Figure 5

(a) Optimal heterodyne amplification $\eta$, for a classical controller, as a function of plant noise. (b) LQR as a function of controller amplification, for five different noise values.

Figure 6

Optical parametric oscillators adiabatically eliminate into ideal squeezers.

Figure 7

(a) Optimal squeezing for the squeezer (solid) and two-mode squeezer (dotted) controllers. (b) Performance as a function of squeezing for multiple noise levels.

Figure 8

Single cavity with modes $a,a^{†}$, coupled to a mechanical oscillator with modes $b,b^{†}$.

Figure 9

Control-system setup for the mechanical-oscillator cooling problem. Four potential controller designs.

Figure 10

Equivalent view of the plant-controller setup shown in Fig. 9. See Eq. (33).

Figure 11

(b) Plot of the average phonon number $\langle N\rangle =\langle b^{†}b\rangle$ of the mechanical oscillator for three different control schemes. (a) Phonon-number reduction, relative to the no-control case. The general coherent-controller result is not shown, since it overlaps the OPO line, the optimal coherent controller being an OPO cavity.

Figure 12

Flow of $x$- and $p$-quadrature signals (blue and red, respectively) in the classical and coherent controllers.

Figure 13

Heterodyne-based measurement controllers, which measure both quadratures of the beam by splitting it, do not outperform the best homodyne controller for this system.

Figure 14

Parameters of the optimal simple cavity controller, as a function of noise strength.

Figure 15

Calculated output spectrum of light exiting the optimal simple cavity controller. Six values of $k_n$ are plotted, ${10}^5$ (darkest), ${10}^3$, ${10}^1$, ${10}^{-1}$, ${10}^{-3}$, and ${10}^{-5}$ (lightest).

Figure 16

Parameters of the optimal OPO cavity controller, as a function of noise strength.

Figure 17

Model for a non-adiabatically-eliminated cavity.

Figure 18

Model for a non-adiabatically-eliminated OPO cavity with a spring mirror.

Figure 19

Coherent control problem represented as two coupled thermodynamic systems.

Figure 20

The concatenation, series, and feedback products are the three basic operations of the Gough-James circuit algebra.