Quantum chaos in a crystal of one hundred ions
Context
The Dicke model, proposed in 1954, describes a collective spin interacting with a single bosonic mode and serves as a minimal arena for studying the transition from integrable behavior to chaos. In neighboring works on our map, quantum simulators have already reproduced neutrino entanglement, time crystals and the synchronized dance of quantum tops, but all of them stayed in the regime of few particles or mean field. Here, for the first time, fully coherent Dicke dynamics is realized in a closed system of one hundred ions, where phonons actively participate in the evolution rather than merely acting as mediators. This makes it possible to see directly how quantum fluctuations destroy classical chaos and give rise to nonclassical correlations.
Methods
A crystal of roughly one hundred 9Be+ ions is held in a Penning trap at a magnetic field of 4.46 T. The internal spin states of the ions encode the collective spin, while the axial center-of-mass motion at a frequency of 1.59 MHz plays the role of the bosonic mode. A spin-dependent optical dipole force couples spin and motion with a coupling frequency on the order of 2π×1 kHz, and a microwave field with a Rabi frequency up to 5.7 kHz controls the transverse field. Cooling to sub-Doppler temperatures via electromagnetically induced transparency reduces the thermal phonon occupancy to 0.5. The dynamics is measured by fluorescence readout of spin populations after a variable evolution time of up to 3 ms. The theoretical description relies on the truncated Wigner approximation, which accounts for quantum and thermal fluctuations of the initial state as well as technical noise.
Results
In the integrable regime, where phonons are adiabatically eliminated, a dynamical phase transition is observed at a critical transverse field equal to half the equilibrium critical value. The time-averaged magnetization jumps to zero when the spin leaves the trap below the equator of the Bloch sphere. As the detuning decreases, phonons become active and the trajectories in phase space turn erratic, consistent with a positive Lyapunov exponent and the nonintegrability of the model. Quantum fluctuations cause rapid damping of the oscillations that is not reproduced by mean field. In the resonant regime, where the initial state is an unstable fixed point of the classical dynamics, quantum vacuum noise triggers exponential growth of paired spin-phonon excitations. The Renyi entropy of a single spin reaches its maximum value, indicating thermalization and entanglement. Numerical modeling predicts two-mode squeezing with a reduction of the hybrid quadrature variance 2.6 dB below the standard quantum limit, and 8.2 dB in the ideal case. At longer times, collapses and revivals of vacuum Rabi oscillations are observed, confirming the coherence of the many-body dynamics.
Implications
The work stands out from neighboring studies in that it is the first to bring a quantum simulator of the Dicke model into the strong-coupling regime with an active bosonic mode and one hundred particles. Neighboring works were limited either to small systems or to adiabatic elimination of phonons, where quantum effects are suppressed. Here quantum noise is not merely a correction but the driving force of the dynamics, which opens a path to studying information scrambling and entanglement in closed many-body systems. The resulting platform allows quantitative tests of theories of quantum chaos and thermalization, and also enables the preparation of squeezed states for quantum metrology.
Future development
Further increasing the number of ions and improving coherence by suppressing magnetic noise will make it possible to keep the system in the quantum regime longer and observe subtler effects, such as many-body localization or prethermalization. Direct measurement of the squeezing of hybrid quadratures, instead of inferring it numerically, may become feasible, which would confirm the generation of nonclassical correlations. A connection to thermofield double states opens the prospect of testing holographic ideas under laboratory conditions.
Impact
The results will influence quantum metrology by providing squeezed states to boost sensor sensitivity, and quantum information by offering a controlled environment for studying information scrambling. They will also stimulate the development of analog quantum simulators for problems in condensed matter physics and high energy physics.
Next steps
The immediate steps include direct measurement of the hybrid quadratures to confirm squeezing and extending the coherence time by shielding magnetic noise. There are also plans to study the dynamics under different initial phonon states in order to separate the contributions of quantum and thermal fluctuations.
Key open problems
For quantum many-body physics, the key unsolved problem remains the connection between classical chaos and quantum thermalization. This work provides an experimental platform where that connection can be tested quantitatively by measuring the growth of the Renyi entropy and the damping of oscillations as functions of the model parameters. The role of long-range interactions in ion crystals for information scrambling also remains an open question.
A crystal of one hundred ions at a temperature of 0.4 mK behaves as a single quantum object, even though each ion sits about 10 micrometers from its neighbors. The frequency of the axial center-of-mass motion matches the vibration frequency of a violin string, only a million times higher.
The idea of quantum chaos and entanglement in many-body systems echoes Greg Egan's novel Permutation City, where consciousness is modeled on a quantum computer and the boundary between order and chaos blurs.
Key numbers
- number of ions: about 100
- COM mode frequency: 1.59 MHz
- squeezing below the standard quantum limit: 2.6 dB
- evolution time: up to 3 ms
- temperature after EIT cooling: 0.4 mK (n≈0.5)