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| Fig. 1: Experimental Rydberg Atom Array Platform. a) Individual atoms (green) are trapped using optical tweezers (vertical red beams) and arranged into 1-D chains. The lateral black arrows represent the van der Waals interactions Vij between excited atoms. b) Atoms are excited by the horizontal red and blue beams to an excited Rydberg state |r> with strength Ω and detuning Δ. Atoms are prepared in a ground state |g>. (Source: J. Georgaras, after Bernien et al. [1]) |
Computational approaches serve as the natural domain for many-body quantum experiments in the absence of a fully controlled, coherent many-body quantum system. A classical system such as a computer, however, has its limitations in its applicability and functionality when applied to a quantum system. The goal is therefore to develop a quantum simulator which can provide the same kind of insights into strongly correlated quantum systems for which we currently rely on computers. Bernien et al. suggest an experimental method for creating a deterministically prepared and reconfigurable many-body quantum arrays of individually trapped cold atoms. [1] This review is an overview of their methods and results.
Bernien et al. realize a quantum simulator by coupling electrically neutral atoms to highly excited states, known as Rydberg States. [1] This large excitation separation in space creates a strong magnetic dipole. Neighboring dipoles pairs have repulsive van der Waals interaction forces between them. [2] Fig. 1a illustrates how the cold neutral Rubidium atoms are arranged as an array in the quantum simulator. The dynamics of this constructed quantum simulator system are described by the system Hamiltonian
where Δ is the detuning of the driving lasers from the Rydberg state resonance frequency, σx is the coupling between the ground state and the Rydberg state |r> of an atom at position i in the chain, driven at a frequency Ω, n counts the number of Rydberg states, and ℏ is the reduced Planck constant. The interaction strength Vij is tuned either by varying the distance between atoms or by coupling atoms to higher-energy Rydberg states.
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| Fig. 2: Phase diagram and build-up of crystalline phase. A qualitative schematic of the ground-state phase diagram of the Hamiltonian in the above equation. Different possible broken symmetry phases are shown. (Source: J. Georgaras, after Bernien et al. [1]) |
The system Hamiltonian can be compared to that of the paradigmatic Ising model in an external magnetic field for effective spin-1/2 particles with the Rydberg and the ground states being equivalent to the spin-up and spin-down states. Additionally, the magnetic interaction between atoms in the Ising model is simulated by the Rydberg interactions.
The Ising model is the most fundamental of many-bodied systems upon which much of many-body physics can be displayed and explored. As such, this system provides a powerful foundation for exploring a rich variety of many-body physics as a quantum simulator. The ground state of the quantum simulator can be prepared into many phases depending on the interaction strength and atom spacing, and increasing the laser detuning Δ. The shaded areas illustrated in Fig. 2 indicate regions each representing a different phase and illustrates how Δ changes the systems phase from disordered to ordered. As Δ/Ω increases from negative to large positive values, the Δ term dominates and the system prefers a state with a larger number of stable Rydberg excited atoms |r> and fewer superposition states. With more Rydberg states, the final interaction term between excited atoms is no longer negligible to the overall energy. In the lowest- energy, most-stable states, clusters form and new "phases" are achieved. Interactions within a cluster cause nearby Rydberg excitations to be suppressed (known as a "Rydberg blockade"). Due to the blockade, only a single excitation exists per cluster. An alternating pattern appears of a single excited atom followed by many ground- state atoms. (Fig. 2)
With quantum simulator in hand, the many-body dynamics can be studied across the phase transition into the Z2 region (the "alternating pattern" phase shown in Fig. 2). Slowly sweeping the laser detuning results in long, ordered chains with atomic states that alternate between the Rydberg and ground states. These domains are separated by domain walls built from two neighboring atoms in the same electronic state (either |rr> or |gg>). The number of domain walls can be considered as a measure of the transition from the disordered phase to the ordered phase as a function of laser detuning Δ.
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| Fig. 3: Oscillations in many-body dynamics after sudden quench. a) Domain-wall density as a function of time after the quench. Oscillations decay. b) Model of non-interacting dimers made of a ground state atom (blue) and Rydberg atom (white). (Source: J. Georgaras, after Bernien et al. [1]) |
Beyond exploring phase transitions, the quantum simulator allows the study of many-body dynamics far from equilibrium. Bernien et al. focus on the quench dynamics of a Rydberg crystal initially prepared in the Z2-ordered phase (alternating ground / excited atoms). The quench procedure consists of preparing a Z2-ordered system (i.e. large detuning Δ/Ω as explained above) and then changing the laser detuning Δ(t) suddenly to Δ = 0. After the quench, oscillations between the initial crystal and its complimentary crystal (i.e. every atomic state in each position is inverted, |r> → |g> and |g> → |r> ) are observed. (See Fig. 3.) These oscillations are observed to be robust and persist over several periods with a frequency independent of system size.
A full explanation of these observed oscillations, or "many-body revivals", are still lacking. Many-body revivals repeatedly return to their initial state and fail to thermalize irrespective of their prepared initial state. The inconsistencies with ergodicity and thermalization of the experimental results have spurred research into so-called "many-body quantum scars" to explain the oscillations. [3] At the time of writing, Bernien et al. propose a simplified model. [1] The behaviour of the quench dynamics can be modelled by dimerized spins, as illustrated in Fig 3b. The Rydberg blockade constrains each dimer into an effective spin-1 system with three states (|rg>, |gg> and |gr>) between which the system cycles during a full oscillation.
© Johnathan Georgaras. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] H. Bernien et al., "Probing Many-Body Dynamics on a 51-Atom Quantum Simulator," Nature 551, 779 (2017).
[2] B. O. Roos, "Perspectives in Calculations on Excited State in Molecular Systems," in Computational Photochemistry, ed. by M. J. Olivucci (Elsevier, 2005).
[3] C. J. Turner et al., "Quantum Scarred Eigenstates in a Rydberg Atom Chain: Entanglement, Breakdown of Thermalization, and Stability to Perturbations," Phys. Rev. B 98 155134 (2018).