Heralded dissipative preparation of nonclassical states in a Kerr oscillator

We present a heralded state preparation scheme for driven nonlinear open quantum systems. The protocol is based on a continuous photon counting measurement of the system's decay channel. When no photons are detected for a period of time, the system has relaxed to a measurement-induced pseudosteady state. We illustrate the protocol by the creation of states with a negative Wigner function in a Kerr oscillator, a system whose unconditional steady state is strictly positive.

Figure 1

(a) A driven nonlinear open quantum system (gray box) is monitored by a photon-counting measurement of detection efficiency $\eta$. The detection signal provides a herald for the creation of a pseudosteady state in the system. (b) In a homodyne detection setup, a local oscillator (LO) signal is added before the detection, which allows one to modify the pseudosteady state.

Figure 2

(a) Steady-state dynamics of the photon-number $〈{\widehat{a}}^{†}\widehat{a}〉$ of a Kerr oscillator subjected to a semiclassical drive. An average over 500 quantum trajectories reproduces the constant steady-state result ${〈{\widehat{a}}^{†}\widehat{a}〉}_{\mathrm{ss}}$ (thin solid red line), which determines the average photon detection rate ${\mathrm{\Gamma }}_{\mathrm{jump}}$. Along a single quantum trajectory, $〈{\widehat{a}}^{†}\widehat{a}〉$ (solid green line) evolves by stochastic quantum jumps at a rate ${\mathrm{\Gamma }}_{\mathrm{jump}}$ (jump times indicated by black triangles) interchanged with a relaxation toward a pseudosteady state ${\left|\psi \right.〉}_{\mathrm{ps}}$ at a rate ${\mathrm{\Gamma }}_{\mathrm{rel}}$. The corresponding photon number ${〈{\widehat{a}}^{†}\widehat{a}〉}_{\mathrm{ps}}$ is marked by the thick dashed gray line. In the intervals highlighted in yellow (light gray), the waiting time between two adjacent quantum jumps is longer than 5 times the relaxation time. (b) The trace distance between the state $\left|\psi \left(t\right)\right.〉$ and ${\left|\psi \right.〉}_{\mathrm{ps}}$ (solid black line) decays exponentially after a quantum jump event. The decay rate is ${\mathrm{\Gamma }}_{\mathrm{rel}}$ (dashed orange line). (c) Spectrum of the non-Hermitian Hamiltonian that defines the relaxation dynamics. The relaxation rate ${\mathrm{\Gamma }}_{\mathrm{rel}}$ is the imaginary part of the smallest gap between the stable eigenstate ${\left|\psi \right.〉}_{\mathrm{ps}}$ (solid circle) and the unstable eigenstates (open circles). Parameters: $\mathrm{\Delta }/\kappa =1.5$, $K/\kappa =2.2$, ${\left|{\alpha }_1\right|}^{2}K/{\kappa }^3=1.5$, ${\alpha }_2/\kappa =0$, and $\xi =0$.

Figure 3

(a) Relaxation rate ${\mathrm{\Gamma }}_{\mathrm{rel}}$ to the pseudosteady state ${\left|\psi \right.〉}_{\mathrm{ps}}$ (solid blue), jump rate ${\mathrm{\Gamma }}_{\mathrm{jump}}$ (dotted red), and negativity of the Wigner function (solid black) for a Kerr oscillator subject to a semiclassical drive for fixed dimensionless detuning $\mathrm{\Delta }/\kappa =1.5$ and rescaled drive power ${\left|{\alpha }_1\right|}^{2}K/{\kappa }^3=1.5$. In the area highlighted in gray, the quantum trajectory is dominated by stochastic quantum jumps, ${\mathrm{\Gamma }}_{\mathrm{jump}}\ge {\mathrm{\Gamma }}_{\mathrm{rel}}$, and ${\left|\psi \right.〉}_{\mathrm{ps}}$ cannot be prepared. The open green rectangle indicates the maximum observable negativity $N_{\max }$ and the parameters of Fig. 2. (b) Adding a local oscillator signal $\sqrt{\kappa }\xi$ allows one to unravel different pseudosteady states. The ratio ${\mathrm{\Gamma }}_{\mathrm{rel}}\left(\xi \right)/{\mathrm{\Gamma }}_{\mathrm{jump}}\left(\xi \right)$ (left plot) and the negativity $N\left({\left|\psi \left(\xi \right)\right.〉}_{\mathrm{ps}}\right)$ (right plot) now depend on the complex signal strength $\xi$. All states within the black curve indicating ${\mathrm{\Gamma }}_{\mathrm{rel}}\left(\xi \right)/{\mathrm{\Gamma }}_{\mathrm{jump}}\left(\xi \right)=1$ can be prepared in a heralded way. The value of $\xi$ indicated by an open white triangle maximizes $N\left({\left|\psi \left(\xi \right)\right.〉}_{\mathrm{ps}}\right)$ under this restriction. (c) Maximum observable negativity $N_{\max }$ as a function of dimensionless detuning and rescaled drive power without (left) and with (right) an optimization of the local oscillator signal $\xi$. In the triangle enclosed by the gray lines, two semiclassical steady-state solutions $〈\widehat{a}〉$ exist.

Figure 4

(a) Spectrum of even-parity (circles) and odd-parity (squares) stable (solid markers) and unstable (open markers) eigenstates of the non-Hermitian Hamiltonian that defines the relaxation dynamics for a Kerr oscillator subject to a resonant parametric drive. The imaginary part of the gap between the two stable states determines their jump-rate asymmetry ${\mathrm{\Gamma }}_{\mathrm{asy}}$. (b) Photon-number $〈{\widehat{a}}^{†}\widehat{a}〉$ in the steady-state regime. An average over 500 trajectories reproduces the steady-state result (thin red), which determines the jump rate ${\mathrm{\Gamma }}_{\mathrm{jump}}$. Each quantum trajectory (solid green line) jumps between the stable states of opposite parity (jump times indicated by black triangles). If ${\mathrm{\Gamma }}_{\mathrm{asy}}\gtrsim {\mathrm{\Gamma }}_{\mathrm{jump}}$ holds, one can prepare the stable even-parity eigenstate ${\left|\psi \right.〉}_{\mathrm{ps}}$ in a heralded way. (c) After a quantum jump event, the trace distance between $\left|\psi \left(t\right)\right.〉$ and ${\left|\psi \right.〉}_{\mathrm{ps}}$ (solid black) decays exponentially. Since parity is conserved, the relaxation happens at a rate ${\mathrm{\Gamma }}_{\mathrm{rel}}$ (dashed orange), which is the imaginary part of the second spectral gap. For comparison, the dash-dotted blue line indicates a decay at the rate ${\mathrm{\Gamma }}_{\mathrm{asy}}$ corresponding to the first spectral gap. (d) Relaxation rate ${\mathrm{\Gamma }}_{\mathrm{rel}}$ (dashed orange), jump-rate asymmetry ${\mathrm{\Gamma }}_{\mathrm{asy}}$ (dash-dotted blue), total jump rate ${\mathrm{\Gamma }}_{\mathrm{jump}}$ (dashed red), and Wigner-function negativity (solid black) as a function of the drive strength. In the gray area, the time evolution is dominated by stochastic quantum jumps, ${\mathrm{\Gamma }}_{\mathrm{jump}}\ge {\mathrm{\Gamma }}_{\mathrm{asy}}$, and ${\left|\psi \right.〉}_{\mathrm{ps}}$ cannot be prepared. (e) Maximum observable negativity as a function of the dimensionless Kerr nonlinearity $K/\kappa$. Parameters: $\mathrm{\Delta }/\kappa =0$, $K/\kappa =10$, ${\alpha }_1/\kappa =0$, ${\alpha }_2/\kappa =5.3$, and $\xi =0$.

Figure 5

(a) Impact of finite temperature or imperfect detection on the pseudosteady state of a Kerr oscillator subject to a semiclassical drive. The main plot shows the minimum of the Wigner function ${min}_{\alpha }\left[W_{{\widehat{\rho }}_{\mathrm{ps}}}\left(\alpha \right)\right]=-N\left({\widehat{\rho }}_{\mathrm{ps}}\right)$ as a function of the thermal photon number $n_{\mathrm{th}}$ and the detection efficiency $\eta$. The smaller plots show the Wigner function $W_{\widehat{\rho }}\left(\alpha \right)$ of selected states. The origin has been shifted to the steady-state expectation value ${〈\widehat{a}〉}_{\mathrm{ss}}$. (Top) Wigner function of steady-state ${\widehat{\rho }}_{\mathrm{ss}}$ and pseudosteady state ${\widehat{\rho }}_{\mathrm{ps}}$ for $n_{\mathrm{th}}=0$ and $\eta =1$. (Bottom) Wigner function of pseudosteady state ${\widehat{\rho }}_{\mathrm{ps}}$ for $n_{\mathrm{th}}=0.5$ and $\eta =1$ (left) and $n_{\mathrm{th}}=0$ and $\eta =0.25$ (right). Parameters are $\mathrm{\Delta }/\kappa =1.5$, ${\left|{\alpha }_1\right|}^{2}K/{\kappa }^3=1.5$, ${\alpha }_2/\kappa =0$, $K/\kappa =2.2$, $\xi /\sqrt{\kappa }=0.9\times exp\left(1.8i\right)$. (b) Same plots for a Kerr oscillator subject to a parametric drive. (Top) Wigner function of steady-state ${\widehat{\rho }}_{\mathrm{ss}}$ and pseudosteady state ${\widehat{\rho }}_{\mathrm{ps}}$ for $n_{\mathrm{th}}=0$ and $\eta =1$. (Bottom) Wigner function of pseudosteady state for $n_{\mathrm{th}}=0.1$ and $\eta =1$ (left) and $n_{\mathrm{th}}=0$ and $\eta =0.5$ (right). Parameters are $\mathrm{\Delta }/\kappa =0$, ${\alpha }_1/\kappa =0$, ${\alpha }_2/\kappa =5.3$, $K/\kappa =10$, $\xi =0$.