Coherent quantum feedback

In coherent quantum feedback, a quantum controller interacts with the controlled system without first converting the control signal into a classical measurement record. Optical feedback through a beam splitter is one example. The input-output interconnection must preserve the physical quantum dynamics, including noise and commutation relations; a classical state-feedback formula does not automatically define a valid quantum controller.

The rotating-wave approximation to a weak quadrature coupling of a cavity mode to a broadband field retains photon-exchange terms and removes counter-rotating pair creation and annihilation. Flat coupling around resonance produces the displayed interaction in units $\hslash =1$. Its short-time propagator is generated by $\sqrt{\gamma }(ad{B}^{†}-a^{†}dB)$; the Gaussian quantum noise Ito table supplies the quadratic drift correction. The resulting passive cavity has the passive cavity input-output transfer function.

For a broadband bosonic input with $[b(t),b^{†}(s)]=\delta (t-s)$, disjoint field increments commute and $[d{B}_t,d{B}_t^{†}]=dt$. An adapted system operator commutes with the next incoming increment. Noise increments scale as $\sqrt{dt}$, so quadratic products contribute to first-order stochastic evolution.

For a zero-mean Gaussian bosonic input, $d{B}^{†}d{B}^{†}=M^{\ast }dt$ completes the displayed table, with $N\ge 0$ and $|M{|}^2\le N(N+1)$. The vacuum has $N=M=0$; an unsqueezed thermal input has $M=0$. Products with $dt$ and higher differentials vanish in the first-order Ito equation. The difference between the first two products preserves the bosonic commutator.

For $\dot{a}=-\gamma a/2-\sqrt{\gamma }{b}_0$ and $b_1=\sqrt{\gamma }a+b_0$, zero-initial-data Laplace transforms give the displayed transfer function. For $\gamma >0$ its pole is in the open left half-plane, while $|G(i\omega )|=1$ and its zero lies in the right half-plane. Passing through $-1$ in this plant's frequency plot is not evidence of an unstable open cavity: a Nyquist stability criterion needs an explicitly specified feedback return ratio.

With $b_0=\beta b_{\mathrm{in}}-\alpha b_1$ and $b_2=\alpha b_{\mathrm{in}}+\beta b_1$, real ${\alpha }^2+{\beta }^2=1$ and $|\alpha |<1$, eliminate the internal fields to obtain $\dot{a}=-{\gamma }_{\mathrm{eff}}a/2-c{b}_{\mathrm{in}}$, where $c=\beta \sqrt{\gamma }/(1+\alpha )$ and $c^2={\gamma }_{\mathrm{eff}}$. The internal output transfer is $b_1/b_{\mathrm{in}}=\beta G/(1+\alpha G)$; the external output transfer is $b_2/b_{\mathrm{in}}=(\alpha +G)/(1+\alpha G)=(s-{\gamma }_{\mathrm{eff}}/2)/(s+{\gamma }_{\mathrm{eff}}/2)$. The loop is asymptotically stable in this nondegenerate range. At $\alpha =1$ the cavity mode is undamped and decoupled; at $\alpha =-1$ the instantaneous algebraic feedback connection is singular.