Quantum optics of chiral spin networks

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Quantum optics of chiral spin networks

Hannes Pichler, Tomás Ramos, Andrew J. Daley, and Peter Zoller

Phys. Rev. A 91, 042116 – Published 14 April 2015

Abstract

We study the driven-dissipative dynamics of a network of spin-1/2 systems coupled to one or more chiral 1D bosonic waveguides within the framework of a Markovian master equation. We determine how the interplay between a coherent drive and collective decay processes can lead to the formation of pure multipartite entangled steady states. The key ingredient for the emergence of these many-body dark states is an asymmetric coupling of the spins to left and right propagating guided modes. Such systems are motivated by experimental possibilities with internal states of atoms coupled to optical fibers, or motional states of trapped atoms coupled to a spin-orbit coupled Bose-Einstein condensate. We discuss the characterization of the emerging multipartite entanglement in this system in terms of the Fisher information.

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Received 14 November 2014

DOI:https://doi.org/10.1103/PhysRevA.91.042116

Authors & Affiliations

Hannes Pichler^1,2,*, Tomás Ramos^1,2, Andrew J. Daley³, and Peter Zoller^1,2,4

¹Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
²Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
³Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK
⁴Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany

^*hannes.pichler@uibk.ac.at

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Vol. 91, Iss. 4 — April 2015

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Figure 1
Spin networks with chiral coupling to 1D bosonic reservoirs. (a) Driven spins can emit photons to the left and right propagating reservoir modes, where the chirality of the system-reservoir interaction is reflected in the asymmetry of the corresponding decay rates $γ_{L} \neq γ_{R}$ . (b) Spin network coupled via three different chiral waveguides $m = 1, 2, 3$ . Waveguide $m = 1$ couples the spins in the order $(1, 2, 3, 4)$ , whereas $m = 2$ couples them in order $(1, 3, 2, 4)$ and $m = 3$ in order $(2, 1, 4, 3)$ . Note that only waveguides without closed loops are considered in this work.
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Figure 2
Dynamical purification of a chiral spin network of $N = 8$ spins into different entangled multimer steady states. As a function of time, we plot the purity of the total state $P$ and the purities of reduced density matrices of different spin subsets $P_{j_{1}, j_{2}, \dots} \equiv Tr {{(ρ_{j_{1}, j_{2}, \dots})}^{2}}$ to probe the entanglement structure of the steady states. (a) Dimers are formed when $δ_{j} = 0$ and $γ_{L} = 0.1 γ_{R}$ . (b) Tetramers are formed for the indicated detuning pattern. Here $δ_{a} = 0, δ_{b} = 0.3 γ_{R}$ , and $γ_{L} = 0.1 γ_{R}$ . (c) Genuine 8-partite entangled octamer formed as the result of coupling the spins to two chiral channels, when driven on resonance $δ_{j} = 0$ . For the second chiral channel we assume $γ_{R}^{(2)} = γ_{R}, γ_{L}^{(2)} = 0.5 γ_{R}$ and the order of coupling the spins is indicated above. Additionally, $γ_{L}^{(1)} = 0.1 γ_{R}$ and $γ_{R}^{(1)} = γ_{R}$ . (d) Nonlocal dimers in a single bidirectional channel, $γ_{L} = γ_{R}$ . The detuning pattern is as indicated, with $δ_{a} = 0.6 γ_{R}, δ_{b} = 0.4 γ_{R}, δ_{c} = 0.2 γ_{R}$ , and $δ_{d} = 0.1 γ_{R}$ . All calculations assume $Ω = 0.5 γ_{R}$ .
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Figure 3
Photonic and phononic realizations of spin chains with chiral coupling to a 1D reservoir. (a) Atoms coupled to the guided modes of a tapered nanofiber. The directionality of the photon emission $γ_{L} \neq γ_{R}$ stems from coupling between the transverse spin density of light and its propagation direction. A current experimental challenge is the control of photon emission into nonguided modes, indicated by $γ^{'}$ . (b) Cold atoms in an optical lattice immersed in a 1D quasi-BEC of a second species of atoms, where the latter represents the reservoir [3, 4]. Including synthetic spin-orbit coupling (SOC) of the quasi-BEC [44] allows the breaking of the symmetry of decay into left and right moving Bogoliubov excitations [27].
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Figure 4
A chiral waveguide couples two spins that are separated by a distance commensurate with the photon wavelength. They are additionally driven homogeneously $(Ω_{1} = Ω_{2} = Ω)$ and with opposite detunings $(δ_{1} = - δ_{2} = δ)$ . (a) The superradiant collective decay couples dissipatively only the spin triplet states. The spin singlet $| S 〉$ does not emit into the waveguide (subradiance) and couples coherently only to $| T 〉$ . (b) The level diagram of states $| g g 〉, | S 〉$ , and $| T 〉$ resembles a $Λ$ system and thus there is a dark state $| D 〉$ in the subspace spanned by $| S 〉$ and $| g g 〉$ .
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Figure 5
Deviations from the dark state conditions for $N = 2$ and their effect on the dark states. (a) A nonhomogeneous drive $\tilde{Ω} = Ω_{1} - Ω_{2} \neq 0$ couples $| S 〉$ coherently to $| e e 〉$ and $| g g 〉$ , inhibiting the formation of a dark state. (b) A homogeneous offset in the detunings $Δ = (δ_{1} + δ_{2}) / 2 \neq 0$ destroys the Raman resonance between the states $| S 〉$ and $| g g 〉$ . (c) Decay processes for spins at arbitrary distance, i.e., noncommensurate with the wavelength. The state $| S 〉$ is in general not perfectly subradiant and thus decays to the state $| g g 〉$ . (d) Additional decay channels such as emission of the spins into independent reservoirs different from the chiral waveguide also lead to a decay of the singlet state.
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Figure 6
Purity of the steady state for $N = 2$ if the dark-state conditions are not met (cf. Fig. 5). (a) Shown as a function of a homogeneous offset in the detuning $Δ$ and a staggered component of the coherent drive $\tilde{Ω}$ . (b) As a function of the distance between the spins relative to the wavelength (modulo integers) and an on-site decay $γ^{'}$ . Parameters are $Ω / γ_{R} = 0.5, δ / γ_{R} = 0.3$ , and $γ_{L} / γ_{R} = 0.5$ .
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Figure 7
Level diagram of the $N = 4$ spin system in a total angular momentum basis, obtained by first adding the subspaces of spin 1 with 2 and spin 3 with 4, separately. The resulting 16 states are grouped into 6 angular momentum manifolds of given total angular momentum, which ranges from 0 to 2 (see text). In each manifold, states are ordered by increasing number of excited spins, i.e., eigenvalue of $J^{z}$ (see text). The coherent driving $\sim Ω$ and dissipative terms $\sim (γ_{L} + γ_{R})$ couple them vertically. The interactions $Δ γ$ and different detunings $\sim δ_{j}$ couple states of different manifolds, but conserve the number of excitations. The null space of the collective jump operator $c = \sum_{j} σ_{j}$ is spanned by the 6 states marked in red. All these are superpositions of products of singlet ${| S 〉}_{j, l} \equiv (| g e 〉 - | e g 〉) / \sqrt{2}$ and ${| g 〉}_{j} {| g 〉}_{l}$ states between the different spins $j, l$ , as indicated in the figure.
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Figure 8
Dynamical purification in the cascaded setup $(γ_{L} = 0$ , first row), chiral setup $(γ_{L} = 0.5 γ_{R}$ , second row), and bidirectional setup $(γ_{L} = γ_{R}$ , third row). We show the entropies of reduced density matrices $S_{j_{1}, j_{2}, \dots}$ of spins ${j_{1}, j_{2}, \dots}$ (colored solid lines) and the purity of the total state $P$ (dashed black lines) as a function of time. In the first column the detuning pattern is chosen such that the steady state dimerizes, which is signaled by a vanishing entropy of the reduced density matrix of the corresponding spin pairs (see text). While in the cascaded setup the system purifies from left to right, in the chiral case the system purifies as a whole. In the second column the detuning pattern is chosen such that the steady state breaks up into a tetramer and two dimers. The last two columns show analogous situations for detuning pattern corresponding to two tetramers and an octamer, respectively. Note that in the bidirectional case $(γ_{L} = γ_{R}$ , last row), the steady state is always dimerized, but the dimers can be nonlocal, depending on the detuning pattern. Other parameters are $Ω = 0.5 γ_{R}, δ_{a} = 0.6 γ_{R}, δ_{b} = 0.4 γ_{R}, δ_{c} = 0.2 γ_{R}$ , and $δ_{d} = 0.1 γ_{R}$ .
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Figure 9
Typical behavior of a system with an odd number of spins in the case of $N = 7$ . We show the entropies of reduced density matrices $S_{j_{1}, j_{2}, \dots}$ of spins ${j_{1}, j_{2}, \dots}$ (colored solid lines) and the purity of the total state $P$ (dashed black lines) as a function of time. (a) In the strict cascaded limit $(γ_{L} = 0)$ , dimers are formed, but the last spin stays mixed and renders the steady-state nondark (cf. red and black dashed curves). (b) If $γ_{L} \neq 0$ the steady state is mixed, and no substructure is formed. Other parameters are the same as in Figs. 8 and 8.
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Figure 10
(a) Dimerized state as the steady state of the chiral spin chain, when driven on resonance with a staggered detuning pattern. (b) Tetramerized steady state, when the spin chain is driven with a permuted detuning pattern with respect to (a). (c) Simple 2-waveguide chiral network with a tetramerized pure steady state. The connection between the states generated in these 3 different situations is outlined in Secs. 4b–4d.
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Figure 11
Steady states in multiple-waveguide chiral networks. We show the entropies of reduced density matrices $S_{j_{1}, j_{2}, \dots}$ of spins ${j_{1}, j_{2}, \dots}$ (colored solid lines) and the purity of the total state $P$ (dashed black lines) as a function of time. (a) Pure tetramerized state as dark state of a 2-waveguide network. (b) 2-waveguide network with a wiring such that the steady state is mixed and without internal structure. Parameters as in Fig. 2.
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Figure 12
Imperfections for different system sizes. We consider a homogeneous offset in the detuning $Δ$ on top of the ideal detuning pattern in (a) and additional on-site decay channel with decay rate $γ^{'}$ in (b). For small $Δ$ the purity of the steady states behaves like $P = 1 - (1 / 2) {(Δ / Δ_{0})}^{2}$ , whereas for on-site decay, it scales linearly as $P = 1 - γ^{'} / γ_{0}^{'}$ (see text). The figure shows the corresponding error susceptibilities $Δ_{0}$ in (a) and $γ_{0}^{'}$ in (b) for systems with $N = 2, 4, 6, 8$ spins. Parameters: (1) and (2) show the fully $N$ -partite entangled situation with an ideal detuning pattern satisfying $δ_{1} = δ_{N} = 0$ and $δ_{2 j} = - δ_{2 j + 1} = 0.3 γ_{R}$ , else. (3) and (4) show the dimerized situation $δ_{j} = 0$ . For the decay asymmetries we choose $γ_{L} / γ_{R} = 0$ in (1) and (3), $γ_{L} / γ_{R} = 0.3$ in (2) and (4). We further fix $Ω / γ_{R} = 0.5$ .
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Figure 13
(a) Illustration of the time-dependent process of formation of dimers, through a single random trajectory in a quantum trajectories calculation, as described in the text, with $γ_{L} = 0$ . We plot the purity of each spin pair $j$ , determined by $P_{j, j + 1} = Tr {ρ_{j, j + 1}^{2}}$ , where $ρ_{j, j + 1}$ is the reduced density operator for spins $j$ and $j + 1$ , shown as a function of the spin pair index $j$ and time $t$ . Here, we take a chain of $N = 18$ spins, and choose $Ω = 1.8 γ_{R}$ . The shading is interpolated across the plot, so that we clearly see the formation of pure spin pairs between all pairs $(j, j + 1)$ with $j$ odd, while the reduced density operators for all pairs $(j, j + 1)$ with $j$ even remain in a mixed state. (b) Same as in (a) but with $γ_{L} = 0.05 γ_{R}$ . We can clearly see that quantum jumps with $γ_{L} \neq 0$ can lead to breakup of already formed dimers, that tends to lengthen the process of reaching the steady state.
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Figure 14
(a) Schematic illustration of different ways to “cool” to the many-body dark steady state (see text). The thick black arrow corresponds to an adiabatic path, while the red arrows indicate a nonadiabatic one. (b) Dynamical purification of a chain of $N = 8$ spins into two tetramers initially in the state ${| g 〉}^{\otimes N}$ for a sudden switch on of the constant coherent driving field (dashed lines), and for an “adiabatic” switching on of the driving field (solid lines). The black lines correspond to the purities $P$ of the total system in the two cases. The total number of photons leaving the system in both cases is plotted in red; residual photons in the “adiabatic case” are due to nonadiabatic effects stemming from a finite $T_{\max} γ_{R} = 300$ . Parameters are $γ_{L} / γ_{R} = 0.5$ , and the detuning pattern chosen is $δ_{j} / γ_{R} = {0, 0.4, - 0.4, 0, 0, 0.4, - 0.4, 0}$ . (c) Total number of photons scattered for $N = 6$ (solid line) and $N = 4$ (dashed line) as a function of the ramp time $T_{\max}$ . Parameters are $Ω_{\max} = 0.5 γ_{R}, γ_{L} / γ_{R} = 0.5, δ_{j} / γ_{R} = {0, 0.4, - 0.4, 0, 0, 0}$ (solid line), and $δ_{j} / γ_{R} = {0, 0.4, - 0.4, 0}$ (dashed line).
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Figure 15
(a) Fisher information calculated via Eq. (33) for a generator $G = (1 / 2) \sum_{j} {(- 1)}^{j} σ_{j}^{x}$ and a measurement of $J^{z} = \sum_{j} σ_{j}^{z}$ , in the case of $N = 6$ spins driven on resonance. The solid line shows the standard quantum limit $F = N$ . (b) Quantum Fisher information calculated via Eq. (34), for a generator $G = (1 / 2) (σ_{1}^{x} - σ_{2}^{x} - σ_{3}^{x} + σ_{4}^{x})$ in the case of $N = 4$ spins driven with a strength $Ω / γ_{R} = 5$ and a detuning pattern $δ_{j} = {δ_{a}, δ_{b}, - δ_{b}, - δ_{a}}$ . The solid lines distinguish regions where $F_{Q}$ detects (at least) $n$ -partite entanglement (for $n = 2, 3, 4)$ . There is a parameter region where full four-partite entanglement is detected. All calculations are shown for a chirality of $γ_{L} = 0.5 γ_{R}$ .
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Figure 16
Optimal directions for local rotations to detect entanglement via the quantum Fisher information. In each panel (a)–(c) we indicate the directions ${\vec{n}}_{i}$ that give the maximum $F_{Q}$ for different steady states of a chiral spin network. We show examples of detuning patterns that give rise to (a) a dimerized state, (b) a tetramerized state, and (c) a fully eight-partite entangled octamer. The absolute values of the detunings are chosen (numerically) such that the steady state maximizes $F_{Q}^{\max}$ . The color map on each sphere corresponds to $F_{Q}$ as a function of the local rotation direction ${\vec{n}}_{i}$ , while keeping all other ${\vec{n}}_{j}$ $(j \neq i)$ at their optimal value. One finds that $F_{Q}$ is able to detect the bipartite entanglement in the dimerized state, three-partite entanglement in the tetramerized state, and four-partite entanglement in the octamer. Other parameters are $Ω / γ_{R} = 5$ and $γ_{L} / γ_{R} = 0.2$ .
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