Scalable Dissipative Preparation of Many-Body Entanglement

We present a technique for the dissipative preparation of highly entangled multiparticle states of atoms coupled to common oscillator modes. By combining local spontaneous emission with coherent couplings, we engineer many-body dissipation that drives the system from an arbitrary initial state into a Greenberger-Horne-Zeilinger state. We demonstrate that using our technique highly entangled steady states can be prepared efficiently in a time that scales polynomially with the system size. Our protocol assumes generic couplings and will thus enable the dissipative production of multiparticle entanglement in a wide range of physical systems. As an example, we demonstrate the feasibility of our scheme in state-of-the-art trapped-ion systems.

Figure 1

Protocol. Preparation of a GHZ state of $N$ qubits is realized by two operations: (i) “$Z$ pumping” of states with other than 0 or $N$ atoms in state $|1⟩$ to $|0{⟩}^{\otimes N}$ and (ii) $|0{⟩}^{\otimes N}$ is a superposition of $|\mathrm{GHZ}⟩$ and $|{\mathrm{GHZ}}_{-}⟩=(|0{⟩}^{\otimes N}-|1{⟩}^{\otimes N})/\sqrt{2}$, so that $|{\mathrm{GHZ}}_{-}⟩$ needs to be removed by the parity-selective “$X$ pumping.”

Figure 2

Setup (a)–(c) and dissipative mechanism (d) for GHZ preparation, shown for $N=3$ qubits. Setup (a): We consider a chain of $N$ subsystems (“atoms”) with four levels, coupled to two common harmonic oscillators. As dissipative processes, we assume spontaneous emission from the excited levels ${\gamma }_{e/f}$. We apply coupling configurations $Z$ (b) and $X$ (c) consisting of $2(N-1)$ driving tones ${\mathrm{\Omega }}_Z^{(F)}$ with detunings ${\mathrm{\Delta }}_Z^{(F)}$ in (b), $2\lfloor (N+1)/2\rfloor$ tones with ${\mathrm{\Omega }}_X^{(F)}$ and ${\mathrm{\Delta }}_X^{(F)}$ in (c), and couplings of the atoms to the oscillator modes $g$. (d) $Z$ pumping towards $|0{⟩}^{\otimes N}$ using the couplings in (b). Ground states are coupled to atom-excited states by weak driving. Depending on the number $n_1$ of atoms in $|1⟩$, the excited states form dressed states with oscillator-excited states at energies $\pm \sqrt{n_1}g$. By applying fields with detunings $|{\mathrm{\Delta }}_Z^{(F)}|=\sqrt{F}g$ for $1\le F\le N-1$ (only the red-detuned fields are shown), all states except $|\mathrm{GHZ}⟩$ and $|{\mathrm{GHZ}}_{-}⟩$ are pumped to $|0{⟩}^{\otimes N}=(|\mathrm{GHZ}⟩+|{\mathrm{GHZ}}_{-}⟩)/\sqrt{2}$. $|{\mathrm{GHZ}}_{-}⟩$ is emptied by the parity-selective $X$ pumping as described in the text.

Figure 3

Evolution towards a steady GHZ state. Starting from a fully mixed state, we numerically solve an effective master equation. The curves in (a) show the evolution for 2–8 qubits (different colors from blue to black) and are obtained by numerically optimizing all available parameters to reach a fidelity of $F_{\mathrm{GHZ}}=0.9$ (black dashed line) within as short a time as possible. In (b) we show the scaling of the preparation time with the number of qubits. Both (a) and (b) show different degrees of truncation of the Hilbert space (dash-dotted lines and blue squares, effective ground state dynamics after adiabatic elimination; solid lines and green circles, one excitation; dashed lines and red triangles, two excitations). In (b), small symbols represent analytically optimized parameters and large symbols numerically optimized parameters. We find a polynomial scaling of the preparation time that is within our analytical bound [black dashed line in (b)].