Coherent imaging spectroscopy of a quantum many-body spin system

Abstract

Quantum simulators, in which well controlled quantum systems are used to reproduce the dynamics of less understood ones, have the potential to explore physics that is inaccessible to modeling with classical computers. However, checking the results of such simulations will also become classically intractable as system sizes increase. In this work, we introduce and implement a coherent imaging spectroscopic technique to validate a quantum simulation, much as magnetic resonance imaging exposes structure in condensed matter. We use this method to determine the energy levels and interaction strengths of a fully-connected quantum many-body system. Additionally, we directly measure the size of the critical energy gap near a quantum phase transition. We expect this general technique to become an important verification tool for quantum simulators once experiments advance beyond proof-of-principle demonstrations and exceed the resources of conventional computers.Certain classes of quantum many-body systems, which describe a variety of interesting problems including high-T c superconductors or spin liquids, are believed to be fundamentally inaccessible to classical modeling [1]. For example, an interacting spin system following the Ising model can map to NP-complete computational problems [2] and has been applied to understanding neural networks [3] and social behavior [4], yet quickly becomes theoretically intractable due to the exponential number of possible spin configurations [5,6].Quantum simulations [7][8][9][10], in which well-controlled quantum objects like photons [11] or ultracold atoms [12,13] are induced to emulate other quantum systems, are a promising alternative for accessing such problems. However, as these systems approach theoretically intractable physics, validating quantum simulation results will become a major challenge [1,14]. Here we introduce a technique for performing coherent imaging spectroscopy on the Hamiltonian of an interacting many-body spin system. We use spectroscopic imaging to infer spinspin interaction strengths and directly measure the critical energy gap near a quantum phase transition. This technique is a promising benchmarking tool for systems of 30+ spins, where classical computation begins to fail.Ultracold atomic systems are particularly well-suited for simulating interacting spin systems, with the ability to prepare known input states, engineer tunable interaction patterns, and measure individual particles [12,13]. Our experiment uses trapped ions to simulate chains of spin-1/2 particles subject to effective magnetic fields and long-range, inhomogenous Ising couplings generated by optical dipole forces [15][16][17][18][19][20][21][22][23]. This results in an effective N -spin Hamiltonian (with h = 1)where σ γ i (γ = x, y, z) is the Pauli matrix for spin i along direction γ; J i,j ∼ J 0 |i − j| −α is a long-range coupling strength between spins i and j with α tunable between 0 and 3 [15]; and B(t) is the strength of a time-dependent transverse magnetic field.The ability to genera...

Electromagnetically-induced-transparency ground-state cooling of long ion strings

“…EIT cooling becomes even more appealing in the context of recent experiments simulating quantum Ising models in strings of up to eighteen ions [28][29][30]. In these experiments all 2N radial modes of an N -ion string need to be cooled to close to the ground state.…”

Simulating the Haldane phase in trapped-ion spins using optical fields

“…To date, trapped ions have mostly been used to simulate spin one-half Hamiltonians, showing the phase transition from the (anti)ferromagnetic to paramagnetic phases in the Ising model [19][20][21][22][23][24][25][26][27][28][29][30] and long range correlation functions in the XX model [31,32]. By enlarging the spin's degree and moving into integer spin chains, new and subtle physics can appear [33][34][35][36][37][38][39][40][41]; for example, the local orders vanish and we are left with hidden orders only [42].…”

Bang-bang shortcut to adiabaticity in trapped-ion quantum simulators

“…It strives to ramp the field quickly when the energy gap to the lowest coupled excited state is high, and more slowly when that gap is small, but it requires detailed knowledge of the energy of the first coupled excited state as a function of magnetic field in order to determine the ramp. While this can be found experimentally utilizing different methods [5][6][7], it is a difficult procedure to carry out for large systems that have significant frustration. The bang-bang protocol is much simpler.…”

“…Original experiments focused on adiabatic state preparation [1-3] of the transverse-field Ising model by initially orienting all of the spins along the field axis (in a large initial field) and then ramping the field to zero to create the ground state of the Ising model. But when the system size was increased, and frustrated antiferromagnetic systems were examined, it became clear that these experiments would have a large amount of diabatic excitation [4], which led to the study of excited states [4][5][6][7] and to a protocol that optimizes the field ramp with a locally adiabatic criterion [8]. In addition, other experimental situations were examined, such as Lieb-Robinson bounds [9, 10] and higher-spin cases [11].…”

“…Within the first category, recent work has found an exact shortcut for adiabatic state preparation [12,13] (at least for the nearest-neighbor transverse field Ising model), but the multiple-spin interactions needed to accomplish this goal are too complicated to implement in the current generation of quantum simulators. In the second category, we already mentioned experimental [4,5,7] and theoretical [6] methods to produce or measure specific excitations. It also is possible, in some cases, for the diabatic excitations to resemble an equilibrium thermal state, especially for ferromagnetic systems [14].…”