Open-system many-body dynamics through interferometric measurements and feedback

Light-matter interfaces enable the generation of entangled states of light and matter which can be exploited to steer the quantum state of matter through measurement of light and feedback. Here we consider continuous-time, interferometric homodyne measurements of light on an array of light-matter interfaces followed by local feedback acting on each material system individually. While the systems are physically noninteracting, the feedback master equation we derive describes driven-dissipative, interacting many-body quantum dynamics, and comprises pairwise Hamiltonian interactions and collective jump operators. We characterize the general class of driven-dissipative many-body systems which can be engineered in this way, and derive necessary conditions on models supporting nontrivial quantum dynamics beyond what can be generated by local operations and classical communication. We provide specific examples of models which allow for the creation of stationary many-particle entanglement, and the emulation of dissipative Ising models. Since the interaction between the systems is mediated via feedback only, there is no intrinsic limit on the range or geometry of the interaction, making the scheme quite versatile.

Figure 1

(a) Feedback setup for a single system $\mathcal{S}$, such as an atom in a cavity (schematic). A one-dimensional light field $\mathcal{A}$ couples to $\mathcal{S}$ via $H_{\mathrm{int}}$, and passes through a phase plate $U=e^{i\theta }$. Subsequently, the transformed field $\mathcal{B}=U\mathcal{A}$ is combined with a strong local oscillator (LO) on a beam splitter (BS). Two photodetectors (PD) measure the incident intensity. The difference of the photocurrents then yields the desired quadrature $\langle \overline{X}\rangle$. This classical signal is transmitted back to the system (dotted line), where it is used to generate Hermitian feedback $\langle \overline{X}\rangle F$. (b) Schematic representation of the previous setup, using the same color code. (c) Proposed generalization to $N$ systems ${\mathcal{S}}_k,k=1,\cdots ,N$, with corresponding light fields ${\mathcal{A}}_k$. After interacting with the systems, the fields traverse an $N$-port passive interferometer. Homodyne measurement of each beam yields quadratures $\langle {\overline{X}}_k\rangle$, which together allow us to steer the systems via local feedback.

Figure 2

(a) Schematic layout of the feedback scheme for $N$ systems ${\mathcal{S}}_1$ through ${\mathcal{S}}_N$. Each system interacts with a single light field ${\mathcal{A}}_k$ via $H_{\mathrm{int}}^k$. The light then traverses an interferometer given by the $N\times N$ unitary matrix $\mathbf{U}$. We perform a homodyne measurement of each resulting field ${\mathcal{B}}_k$, which yields the quadratures $\langle {\overline{X}}_k\rangle$. Each signal is transmitted back to the systems via classical channels (dotted line), and generates local feedback $\langle {\overline{X}}_k\rangle F_k^l$, where $F_k^l$ acts only on ${\mathcal{S}}_l$. The total feedback on ${\mathcal{S}}_l$ is generated by ${\sum }_{k=1}^N\langle {\overline{X}}_k\rangle F_k^l$. (b) A different schematic of the same setup. It emphasizes how each measurement $\langle {\overline{X}}_k\rangle$ is used to generate collective feedback $F_k={\sum }_{l=1}^N{F}_k^l$ on all systems. Although both schemes are identical, viewing the feedback in this way simplifies our calculations.

Figure 3

(a) Without an interferometer we can perform only local measurements. In combination with local feedback, the scheme will generate only LOCC dynamics. (b) Local phase plates before an LOCC setup, such as a real orthogonal interferometer $\mathbf{O}$, will not enable any nonlocal operations. This is an LOCC setup which does not necessarily satisfy Eq. (50). (c) Phase plates before the detectors, on the other hand, change the homodyne measurement basis. They may enable non-LOCC dynamics, even if the preceding setup is LOCC.

Figure 4

Illustration of the two-qubit protocol (2QP). Both qubits ${\mathcal{S}}_j$ interact with light fields via $S_{+}$ or $S_{-}$, respectively. The fields are superposed on a balanced beam splitter (BS), after which one beam is phase shifted by $\pi /2$. We perform a homodyne measurement (HM) of each field, and apply feedback $F_{\pm }$ proportional to the measured signal. The qubits will eventually relax into the steady state $\left|\psi \right(z)\rangle$.

Figure 5

Two-qubit protocol (2QP) is extended to three qubits. This is done by letting two light fields, ${\mathcal{A}}_2$ and ${\mathcal{B}}_2$, interact with ${\mathcal{S}}_2$, and using a second copy of the 2QP. ${\mathcal{A}}_2$ interacts via $S_{1,-}$ and then enters the first 2QP with light coming from ${\mathcal{S}}_1$. This generates feedback $F_{1,\pm }$ on ${\mathcal{S}}_1{\mathcal{S}}_2$ which gives rise to the Liouvillian ${\mathcal{L}}_{1,2}$. Analogously, ${\mathcal{B}}_2$ interacts with ${\mathcal{S}}_2$ via $S_{2,+}$ and then enters the second 2QP. This results in feedback $F_{2,\pm }$ on ${\mathcal{S}}_2{\mathcal{S}}_3$ and Liouvillian ${\mathcal{L}}_{2,3}$.

Figure 6

Purity of the steady state varies with $z$. Without feedback $\left(z=0\right)$ all systems are incoherently pumped into $|0\rangle$. As $z$ grows, however, competition between the pairwise dynamics increasingly mixes the steady state. The maximally mixed state is approached with ${\mathrm{Tr}}_{}\left[{\rho }^2\left(z\right)\right]\to 2^{-N}$ as $z\to 1$. The curves in the legend are shown from top to bottom in the figure.

Figure 7

Concurrence $\mathcal{C}({\rho }_{k,k+1}\left(z\right))$ of neighboring qubits in the steady state is nonzero for $0<z<0.4$, showing the presence of entanglement. We computed the steady state directly using the Python package QuTiP [54, 55] for small system size ($N\le 7$ qubits), and found the concurrence to be independent of $N$ (solid red line). For large $N$, we analyzed the steady state using a variational approach [56]. While the variational concurrence (dotted blue line) takes on slightly different values, it is still in good qualitative agreement with our exact results. We found the concurrence between non-neighboring qubits to vanish everywhere.

Figure 8

Logarithmic negativity $E_N(\rho \left(z\right))$ in the steady state for bipartitions into only odd or only even sites of setups comprising $N=3,\cdots ,7$ qubits. Increasing the system size causes a larger peak negativity. Notably, the entanglement persists for values of around $z\approx 0.6$, where the concurrence has long vanished. The curves in the legend are shown from bottom to top in the figure.

Figure 9

Feedback scheme to realize an open Ising model. Each two-level system ${\mathcal{S}}_k$ couples to two light fields, ${\mathcal{A}}_k$ (red) and ${\mathcal{B}}_k$ (blue). The $\mathcal{A}$ fields traverse an interferometer $\mathbf{V}$, while the $\mathcal{B}$ fields pass the complex conjugate ${\mathbf{V}}^{*}$. A homodyne measurement (HM) of each field then allows for feedback via operators $F_k$. This way one can realize the dynamics of a dissipative Ising model.

Figure 10

We plot the spin-up–density in the steady state of the dissipative Ising model against $g=B/\mathrm{\Delta }$ (continuous lines) for $N=5$ spins. We consider different values of the asymmetry parameter $\alpha =r/{\left|\mathrm{\Delta }\right|}^{1/2}$. For comparison we also plot the results of the purely dissipative steady states obtained by setting $H=0$ in the dynamics (dotted lines). A small deviation between the full and purely dissipative steady states occurs only when $\alpha$ is close to unity.