A SUPERCONDUCTING QUANTUM SIMULATOR BASED ON A PHOTONIC-BANDGAP METAMATERIAL
6.5 Ergodic many-body dynamics with long-range hopping
We now utilize the platform to study the effect of long-range hopping on the many- body dynamics. Specifically, the ergodicity of the 1D Bose-Hubbard model in the hardcore limit (|U/J| ≫1) depends on the range of hopping, exhibiting integrable behavior with NN hopping, and chaotic behavior with long-range hopping. We study this crossover with various hopping ranges and investigate the resulting dynamics using both conventional one- and two-site correlators, and the statistics of the global bit-strings resulting from qubit-state measurement outcomes across the lattice. This latter technique is particularly useful in identifying universal signatures of ergodicity and the effect of decoherence at long evolution times.
The crossover between integrable and ergodic dynamics can be qualitatively visual- ized by a two-particle quantum walk [258–260] with initial excitations on sites Q5
and Q6 using the sequence shown in Fig. 6.4b. The measured quantum walk at a
Site Site Site Site J i,i+1τ (rad)
0 2 4 6 8 10
0
µ2
0.1 1
a
Q1 Q10Q1 Q10Q1 Q10Q1 Q10
1 µ2e
4.50 GHz 4.55 GHz 4.72 GHz 4.80 GHz Integrable
b
Figure 6.5: Two-particle quantum walk with increasing hopping range. a, Evolution of the population ⟨nˆi⟩ on sites Q1–Q10 as a function of normalized evolution time Ji,i+1τ. The system is initialized in zinit = 0000110000 and the evolution occurs at ω01/2π = 4.50GHz, 4.55 GHz, 4.72 GHz, and 4.80 GHz with the longest evolution times of 904 ns, 781 ns, 430 ns, and 200 ns from left to right. b, The second momentµ2 as a function of normalized evolution timeJi,i+1τ. Results calculated from the data in panel a are shown in solid curves with gray scales corresponding to frames in panelaand arrows in Fig.6.3b. Result from numerical simulation of the integrable Hamiltonian is shown as the dotted curve andµe2 for a generic ergodic system is indicated by the red dashed line. This figure is adapted from [132].
few differentω01’s indicated by arrows in Fig.6.3b is shown (Fig.6.5a) as a function of normalized evolution timeJi,i+1τ, where Ji,i+1 is the average NN hopping rate (the corresponding numerical simulations are provided in App. D.7, showing that the quantum walk patterns are not visibly affected by decoherence). The excitation wave packets smear over the system whenω01is close to the band-edge frequency.
More quantitatively, this trend can be probed by computing the probability pz of measuring a certain bit-stringzin the two-excitation sector at evolution timeτ. For a generic ergodic Hamiltonian, the second momentµ2 ≡P
zp2z[254], which reflects the probability fluctuations, converges toµe2 = 2/(D+1)after initial evolution [257]
due to the chaotic nature of its quantum dynamics (D= 45is the dimension of the two-excitation Hilbert space). No such convergence is expected in an integrable Hamiltonian due to revivals associated with ballistic propagation of wave packets.
As an example, we show in Fig.6.5b the results from the spin-1/2XY model ob- tained from modifying the Hamiltonian in Eq. 6.1 by keeping only NN hopping terms in the hardcore limit. When ω01 is closer to the band-edge, the measured
second moment deviates from the simulated integrable result and converges toµe2at an earlier normalized evolution timeJi,i+1τ consistent with the breaking of integra- bility due to the extended hopping range. We note that with|U/J|>36for all the measurements illustrated in Fig.6.5, finite on-site interactions of the Bose-Hubbard model play a negligible role in the breaking of integrability (App.D.8).
pz
0.0 0.05 0.1 0.15 102
P(p z) 100 10-2
pz
0.0 0.05 0.1 0.15 pz
0.0 0.2 0.4 0.6 0.8
b
τ (μs)0.01 0.1 1
1
0.1
Experiment Theory
Integrable theory
a
µ 2
µ2e
Figure 6.6: Ergodic many-body dynamics with long-range hopping at 4.72 GHz.
a, Second moment µ2 as a function of evolution time τ in our system from the experiment (orange) and the theory with the optimized parameter set in Fig. 6.4c (blue), compared to theoretical predictions of the integrable model (green). The shading on each curve corresponds to a standard deviation of the mean second moment for 20 randomly chosen initial bit-strings zinit in the two-particle sector, and the red dashed line represents the ergodic value µe2. b, Density histogram P(pz)of the distribution of experimental bit-string probabilities {pz}with the 20 initializations zinit’s at evolution times τ = 16ns, 360 ns, and 5.4µs from left to right (indicated by the dotted lines in (A)). The solid lines show the PT distribution and the dashed line in the right plot shows the valuepz = 1/Dof a classical uniform distribution. This figure is adapted from [132].
To further probe this ergodic nature of Hamiltonian with long-range hopping, we use the experimental evolution at ω01/2π = 4.72 GHz as an example. At a short time (τ = 16 ns), the excitations remain in their initial sites. This is visualized for a quantum walk with initial excitations on sites Q5 and Q6 in the left panel of Fig. 6.7a (evolution of population ⟨nˆi⟩) and in the bottom left panel of Fig. 6.7b (two-site correlator ⟨nˆinˆj⟩). The histogram P(pz) of experimentally measured bit-string probabilities {pz} at this early evolution stage (Fig. 6.6b, left) shows a
distribution with a tail of large pz values, giving a largeµ2 (Experiment curve in Fig.6.6a). This is associated with an insufficient scrambling of the initially localized quantum information. At an intermediate time (τ = 360 ns), the excitations are more spread out over the entire 1D lattice (middle left panel of Fig.6.7a), forming a
“speckle” pattern with site-to-site fluctuation associated with quantum interference.
The quantitative signatures of this speckle pattern manifest in the histogramP(pz) following the Porter-Thomas (PT) distribution [261] (Fig.6.6b, middle) and in the second moment µ2 settling to the ergodic value µe2. The PT distribution results from the randomness in the distribution of wavefunction magnitudes, which is predicted by Berry’s conjecture [262] stating that the single-particle eigenstates of a chaotic system behave like random superpositions of plane waves. Similarly, in the many-body settings, the distribution of wavefunction magnitudes across basis states also follow the PT distribution. Our observation is the first experimental verification of this many-body version of Berry’s conjecture in a Bose-Hubbard system, whose extension in the thermodynamic limit provides the modern theory of quantum thermalization such as eigenstate thermalization hypothesis [263,264].
This draws connection between quantum many-body chaos and random matrix theory, leading to a deeper understanding of the randomness in many-body dynamics [256]. Note that the randomness in our case originates from the ergodicity of the time-independent Hamiltonian instead of the randomness inherent in random circuits [49,254]. In contrast to the experimental results, theoretical calculations at the same evolution time using the integrable Hamiltonian shows aggregated excitations on a few sites (Fig.6.7c, middle) and the resulting larger value ofµ2 (Integrable theory curve in Fig.6.6a). This comparison highlights the effect of long-range hopping in probing the ergodic regime.
Finally, we study the impact of decoherence by juxtaposing the measurement results and the decoherence-free theoretical calculation using the optimal learned Hamilto- nian with long-range hopping. Before the evolution time ofτ ≈1µs, the two cases agree in the second moment µ2 (Experiment and Theory curves in Fig.6.6a), the quantum walk population (Fig. 6.7a, left and middle), and the two-site correlator (middle panels of Fig. 6.7b), suggesting these results are not affected by decoher- ence. After a long evolution time (τ = 5.4µs, larger than the averaged Ramsey coherence timeT2,i∗ = 1.16µs), the second moment of the two cases deviates from one another, and the experimental speckle pattern begins to wash out compared to the theoretical modeling (top panels of Fig.6.7b). Another probe of the decoherence is the histogram P(pz) of the measured bit-string probabilities (Fig. 6.6b, right).
a
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
τ (μs)
Site Site Site
Q1 Q10Q1 Q10 Q1 Q10
0.15 0.10 0.05 0.15 0.10 0.05
0.6 0.4 0.2 0.0
c
Site Qi Q1 Q10 Site Q j Q 1Q 10Site Q j Q 1Q 10Site Q j Q 1Q 10
0.06 0.04 0.02
b
Site Qi Q1 Q10
Site Qi Q1 Q10 Site Q j Q 1Q 10Site Q j Q 1Q 10Site Q j Q 1Q 10
0.10 0.05 0.000.75 0.50
0.00 0.25 0.00
Figure 6.7: Quantum walk under ergodic many-body dynamics. a, Evolution of the population⟨nˆi⟩on sites Q1–Q10as a function of timeτ withzinit = 0000110000 in the cases of experiment, theory, and integrable theory from left to right. The white dashed lines at the bottom (in the middle) indicatesτ = 16ns (360 ns). b–c, Two-site correlator ⟨nˆinˆj⟩ withzinit = 0000110000 at evolution times τ = 16ns, 360 ns, and 5.4µs from bottom to top in the cases of experiment (left column of panelb), theory (right column of panel b), and integrable theory in panel c. This figure is adapted from [132].
Here, the histogram deviates from the PT distribution, narrows substantially, and approaches a uniform distribution corresponding to a completely decohered, max- imally mixed state. Additional numerical simulations of µ2 andP(pz)for ergodic and integrable systems can be found in App.D.8.