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Implementation of chiral quantum optics with Rydberg and trapped-ion setups

Benoît Vermersch, Tomás Ramos, Philipp Hauke, and Peter Zoller
Phys. Rev. A 93, 063830 – Published 17 June 2016
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  • INTRODUCTION
  • CHIRAL QUANTUM NETWORK MODEL
  • SPIN IMPLEMENTATION WITH RYDBERG ATOMS
  • PHONON IMPLEMENTATION WITH TRAPPED IONS
  • EXAMPLES USING CHIRALITY
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We propose two setups for realizing a chiral quantum network, where two-level systems representing the nodes interact via directional emission into discrete waveguides, as introduced in T. Ramos et al. [Phys. Rev. A 93, 062104 (2016)]. The first implementation realizes a spin waveguide via Rydberg states in a chain of atoms, whereas the second one realizes a phonon waveguide via the localized vibrations of a string of trapped ions. For both architectures, we show that strong chirality can be obtained by a proper design of synthetic gauge fields in the couplings from the nodes to the waveguide. In the Rydberg case, this is achieved via intrinsic spin-orbit coupling in the dipole-dipole interactions, while for the trapped ions it is obtained by engineered sideband transitions. We take long-range couplings into account that appear naturally in these implementations, discuss useful experimental parameters, and analyze potential error sources. Finally, we describe effects that can be observed in these implementations within state-of-the-art technology, such as the driven-dissipative formation of entangled dimer states.

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    • Received 30 March 2016

    DOI:https://doi.org/10.1103/PhysRevA.93.063830

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum opticsTrapped ions
    1. Physical Systems
    Rydberg atoms & molecules
    Atomic, Molecular & Optical

    Authors & Affiliations

    Benoît Vermersch*, Tomás Ramos†, Philipp Hauke, and Peter Zoller

    • Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria and Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria

    • *Corresponding author: benoit.vermersch@uibk.ac.at
    • †Corresponding author: tomas.ramos@uibk.ac.at

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    Vol. 93, Iss. 6 — June 2016

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    • Figure 1
      Figure 1

      Chiral quantum network realizations. (a) An array of two-level systems interacts via a chiral photonic waveguide, with different emission rates into the right- and left-moving modes, γR≠γL. (b),(c) Lattice analogs of (a), where the waveguide consists of (b) spin-12 particles, realizable with Rydberg atoms [cf. Sec. 3], or (c) phonons, realizable with trapped ions [cf. Sec. 4]. The chiral coupling is achieved by imprinting phases ϕm on the flip-flop interactions J̃m between the two-level systems and the localized waveguide modes. To mimic an infinite waveguide with a finite chain, we add local losses ΓnL,R at the ends, allowing the excitations to leave the network.

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    • Figure 2
      Figure 2

      Directionality in the weak-coupling regime. We plot a dipolar dispersion relation ωk (blue line) obtained for Jm=−|J1|/m3 and ΔB=(3/16)ζ(3)J1∼−0.23|J1|. When ΔS=0, the resonant waveguide modes correspond to ka=±π/2 (see shaded region). The use of the phases ϕm allows one to break the parity of the coupling function gk (red line) and therefore to emit preferentially in one direction. Here we consider the case of the ion implementation with J̃0=2J̃1, Jm≥2=0, and ϕ1=−π/2, allowing one to cancel the coupling to the left-moving resonant modes and thus to realize an unidirectional coupling to the right.

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    • Figure 3
      Figure 3

      A Rydberg implementation of a chiral spin waveguide. (a) Rydberg atoms representing system and bath spins are distributed in the (X,Y) plane. Each system spin (α=1,2,⋯; white disks) interacts via dipole-dipole interactions with its neighbors in the bath (j=L[α,m],R[α,m], m=1,2,⋯; gray disks). (Left) The internal coordinate system of the Rydberg atoms (x,y,z) with a quantization axis z that is rotated by an angle Θ with respect to the laboratory frame (X,Y,Z). (b) The dipole interaction between two Rydberg atoms α and R[α,m] is written in terms of the spherical coordinates (rm,θm,φm). (c) Level structure of the system and bath spins. The presence of an electric-field gradient ∇⃗E (cf. Appendix pp1-s1), or the use of a Förster resonance (cf. Appendix pp1-s2), combined with a homogenous magnetic field B, makes resonant the angular momentum nonconserving process d−1d−1, shown by red arrows. The sink spins placed at a distance a′ from the second-to-last bath spins are subjected to a local electric field E′ (or a local ac-Stark shift).

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    • Figure 4
      Figure 4

      (a) In the Rydberg implementation, the chirality γL/γR is highly tunable via Θ and ℓ/a, with large plateaus where unidirectionality is achieved. (b),(c) Decay of an excitation from a system spin to the bath with unidirectional emission. (b) The system spin population decays exponentially as expected in the Markovian regime (cf. Sec. 2b). (c) The bath occupation shows that the emitted wave packet propagates towards the right before being absorbed perfectly by the sink at the boundary. The dashed line indicates the position of the system spin. Parameters are given in Sec. 3d.

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    • Figure 5
      Figure 5

      Chiral system-bath coupling in a trapped-ion chain using global lasers and single-site initial-state preparation. (a) Schematic representation of the setup, where localized radial vibrations bj of j=1,⋯,NB ions realize a discrete waveguide of phononic excitations and which interact via long-range Coulomb-mediated hoppings Jlj. The internal states of selected NS (<NB) ions realize the system spins that couple chirally to the phonon waveguide. Sitting adjacently to the right and left of each system spin we prepare “auxiliary ions” in another long-lived internal state, and the rest of the ions are shelved in a third long-lived state such that only their vibrations participate in the dynamics. At the ends of the ion chain, we apply localized laser cooling to engineer losses or “sinks” of phononic excitations and thus mimic the output ports of an infinite waveguide (b) Each system spin at site j=c[α] couples to its own phonon vibrations with strength J̃0 and with (possibly inhomogeneous) strengths and phases, J̃1(α,ν) and ϕ1(α,ν), to the vibrations of the auxiliary ions at sites j=ν[α,1], with ν=L,R. Off-resonant transitions to the excited state |E〉α,ν of the auxiliary ions mediate the nonlocal coupling in a third-order process (cf. Fig. 6 for more details). The combination of these local and nonlocal couplings with phases allows one to achieve a chiral coupling [cf. Sec. 4c4].

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    • Figure 6
      Figure 6

      Level scheme and laser-mediated couplings for realizing a chiral network in trapped ions. A laser with Rabi frequency Ωd drives the carrier transition of system spin α, |g〉α↔|e〉α, with a small detuning ΔS (cyan line). A local system-bath interaction is induced by a first sideband transition coupling the upper state of the system spin resonantly to the vibrational modes |1〉n with frequency ω¯ and Rabi frequency Ω0 (blue line). The nonlocal coupling between a system spin at site j=c[α] and the local vibrations of its auxiliary ions at sites j=ν[α,1], with ν=L,R, is obtained in a third-order process from lasers p={1,2,3} (red lines). Here p=1 couples off-resonantly to the red sideband with detuning δ1, then p=2 couples from there off-resonantly to the excited state |E〉α,ν of the auxiliary ion, and finally p=3 couples |E〉α,ν off-resonantly with detuning δ2 back to the red sideband around the reference frequency ω¯. The ηp denote the Lamb-Dicke parameters determining the effective coupling strength.

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    • Figure 7
      Figure 7

      Chiral emission into the phonon modes of an inhomogeneous ion chain. (a) Control of the directionality of emission γL/γR by tuning the ratio J̃0/(2J̃1). For J̃0=2J̃1, one achieves nearly perfect chiral emission. (b) Decay with rate γ=2π×218Hz of an initially excited system spin on a time scale t∼1/γ∼1ms. (c) Real-space occupations 〈bj†bj〉 of a 16-ion chain as a function of time, showing a unidirectional emission into the ion vibrations to the right of the system spin, which sits at j=c[1]=6. From site j=10 to j=16 we have included seven local losses with rates ΓnR increasing quadratically towards the boundaries, and with a maximum of Γ1R=0.27ωx. On the left side of the chain there is one local loss at j=1 with Γ1L=0.1Γ1R. This allows us to realize nearly perfect absorbing boundary conditions, simulating the physics of an infinite waveguide. (d) Occupation of the discrete momentum modes 〈bκ†bκ〉 as a function of time, showing that (even in the presence of inhomogeneities) the phonons emitted by the system spin are mainly around the mode κ=−π/2, as expected. Other parameters are Ωα=ΔS=0, J̃1(1,L)=0.7J̃1(1,R)=3.85γ, J̃0=∑νJ̃1(1,ν), ϕ1(1,L)=−1.01(π/2), ϕ1(1,R)=−0.99(π/2), ω¯=0.964ωx, ωz=0.05ωx, and ωx=2π×3MHz.

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    • Figure 8
      Figure 8

      Dissipative preparation of a pure dimer steady state in the Rydberg (a)–(c) and the trapped-ion implementation (d)–(f). (a),(d) Time evolution of the system spin purity P (black line), the singlet occupation 〈S〉 (red line), and the ground-state occupation 〈gg〉 (red dashed line), showing that the steady-state properties of the dimer are well reached in a time scale t≳25/γ. (b),(e) Bath occupation probabilities, where the dashed lines indicate the positions of the two system spins on the waveguide (separated by eight sites in the Rydberg and by four sites in the ion implementation). The steady-state occupation, with finite excitation flux between the two system spins, but nearly zero outside, evidences the formation of the dimer. (c),(f) Snapshots of the bath occupations at γt=5,30 (increasing for darker color). Parameters for the Rydberg simulation (a)–(c) are given in Sec. 3d, considering Ω=γR and γL/γR≈1/400. In the ion simulation (d)–(f), the system spins are driven with Ω=γ/2, and the chirality is γL/γR≈0.1. Other parameters are listed in [85].

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    • Figure 9
      Figure 9

      Comparison of the dynamics of system spins emitting in opposite directions into (a) a spin and (b) a bosonic waveguide. (a) In a spin waveguide, the counterpropagating wave packets can collide, resulting in a π phase shift. (b) This shift is absent in a bosonic waveguide. (a),(b) In both cases, the right-moving wave packet leaves the network, but the left-moving one is reflected at the left boundary, after which it can be reabsorbed by the system spin α=1 at time t=τ. (c) For a bosonic waveguide, the phase accumulated at the moment of the reabsorption induces a constructive interference and 〈σ1+σ1−〉 increases. If the waveguide consists of spins, its population continues to decrease as the extra π phase reduces the constructive interference. In both cases, the population dynamics clearly deviates from a Markovian exponential decay for t>τ, which does not include the reabsorption (black dashed line). (d) The waveguide occupation shows the emission from the two system spins of two counterpropagating wave packets which collide in the central region of the waveguide. We considered the parameters for the Rydberg implementation given in Fig. 8 with ℓ=1.8a, removing the left Rydberg-excitation sink and including four bath spins on the left side of system spin α=1.

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    • Figure 10
      Figure 10

      Förster defect ωB,0−ωS,0 as a function of n. The resonance at around n=81 (dashed line) makes it possible to implement a resonant complex hopping between system and bath.

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    • Figure 11
      Figure 11

      Excitation sinks are obtained by coupling the Rydberg state |↑〉 to a short-lived state and encoding |↓〉′ in a hyperfine ground state. The loss is engineered via a laser with Rabi frequency Ωd, which couples |↑〉 to a short-lived hyperfine state of the 5P1/2 manifold, which decays to the ground-state level |↓〉′ with rate Γ′. A dressing laser admixes |↓〉′ to an additional Rydberg level, leading to flip-flop interactions with the neighboring bath spins. The process is made resonant by applying a local electric field E′ (or a local ac-Stark shift).

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