Optimizing for an arbitrary Schrödinger cat state

We derive a set of functionals for optimization towards an arbitrary cat state and demonstrate their application by optimizing the dynamics of a Kerr-nonlinear Hamiltonian with two-photon driving. The versatility of our framework allows us to adapt our functional towards optimization of maximally entangled cat states, applying it to a Jaynes-Cummings model. We identify the strategy of the obtained control fields and determine the quantum speed limit as a function of the cat state's excitation. Finally, we extend our optimization functionals to open quantum system dynamics and apply it to the Jaynes-Cummings model with decay on the oscillator. For strong dissipation and large cat radii, we find a change in the control strategy compared to the case without dissipation. Our results highlight the power of optimal control with functionals specifically crafted for complex physical tasks and the versatility of the quantum optimal control toolbox for practical applications in the quantum technologies.

Figure 1

Comparison of the optimization results for two different functionals. The plot shows the Wigner distribution of the final states obtained with the cat-state functional described in Sec. 2 in (a) and with the state-to-state functional [Eq. (35)] in (b).

Figure 2

Spectra of the optimized pulses for optimization towards maximally entangled cat states with $|{\alpha }_{\mathrm{tgt}}|=1$ (top) and $|{\alpha }_{\mathrm{tgt}}|=2$ (bottom). The dashed lines indicate the transition frequencies between the eigenstates $|n,\pm \rangle$ of the drift Hamiltonian, defined in Eq. (39).

Figure 3

Detailed analysis of the optimized pulse towards $|{\alpha }_{\mathrm{tgt}}|=2.0$, showing the pulse in time domain (c), in frequency domain (a), and a time-frequency distribution (b) calculated via the Gabor transform. All quantities are expressed in units of the coupling strength $g$.

Figure 4

The final value of $\left|\alpha \right|$ plotted against the pulse duration $T$ for different optimizations. The different marker styles indicate optimizations performed with different target values $|{\alpha }_{\mathrm{tgt}}|$. The dashed auxiliary lines on the right, colored with the same color as the marker styles, are added to guide the eye.

Figure 5

Dependence of the final state errors on the dissipation strength $\kappa$ for different target values $|{\alpha }_{\mathrm{tgt}}|$: (a) purity error, (b) cat infidelity, and (c) cat radius error. The solid lines correspond to pulses optimized with dissipation taken into account whereas the dashed lines display the results obtained using the coherently optimized pulses, but propagated with the corresponding dissipation strength.

Figure 6

The same data as in Fig. 5 but with linear scale for the purity to illustrate the different improvements under reoptimization with dissipation (solid) compared to coherently optimized results (dashed). Same line styles and color code as in Fig. 5.

Figure 7

Comparison of different strategies: The solid lines depict the dynamics induced by the pulse optimized with strong dissipation and the dashed lines depict the dynamics with the coherently optimized pulses. The left and right columns correspond to $|{\alpha }_{\mathrm{tgt}}|=1.5$ and $|{\alpha }_{\mathrm{tgt}}|=2.0$, respectively. (a), (b), (c), (d) The average excitation of the harmonic oscillator, Eq. (47), and of the qubit, Eq. (48). (e), (f) The mutual information between the harmonic oscillator and the qubit, Eq. (31).

Figure 8

Visualization of the strategy change, storing excitation in the qubit instead of the harmonic oscillator, when optimizing with strong dissipation. (a) The same plot as in Fig. 7. (b), (c) The dynamics of the qubit in Bloch sphere representation. For the sake of clarity, we only show the final part of the dynamics in (b) and (c), as indicated by the gray dashed lines in (a).