Abstract
Particle transport and localization phenomena in condensedmatter systems can be modeled using a tightbinding lattice Hamiltonian. The ideal experimental emulation of such a model utilizes simultaneous, highfidelity control and readout of each lattice site in a highly coherent quantum system. Here, we experimentally study quantum transport in onedimensional and twodimensional tightbinding lattices, emulated by a fully controllable 3 × 3 array of superconducting qubits. We probe the propagation of entanglement throughout the lattice and extract the degree of localization in the Anderson and WannierStark regimes in the presence of sitetunable disorder strengths and gradients. Our results are in quantitative agreement with numerical simulations and match theoretical predictions based on the tightbinding model. The demonstrated level of experimental control and accuracy in extracting the system observables of interest will enable the exploration of larger, interacting lattices where numerical simulations become intractable.
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Introduction
Singleparticle dynamics and transport in periodic solids is well described by the tightbinding model^{1}. This model is widely used to calculate the electronic band structure of condensed matter systems and to predict their transport properties, such as the conductance^{2,3}. In the presence of a periodic lattice potential, the wavefunction of a given quantum particle overlaps neighboring lattice sites, leading to extended Bloch wavefunctions^{1}. In the absence of lattice disorder, the particle propagation is ballistic and is described by a continuoustime quantum random walk^{4}. This is in contrast to classical diffusive transport, where the propagation is quadratically slower in time^{5}.
The quantum nature of transport in lattices leads to the emergence of nonlocal correlations and entanglement between lattice sites. However, lattice inhomogeneity causes scattering and leads to quantum interference that tends to inhibit particle propagation, a signature of localization^{6,7}. The wavefunction of a localized particle rapidly decays away from its initial position, effectively confining the particle to a small region of the lattice.
Here, we study Anderson localization and WannierStark localization in onedimensional (1d) and twodimensional (2d) tightbinding lattices. In the presence of random disorder of the lattice site energies, a particle wavefunction becomes spatially localized, known as Anderson localization^{6}. In the context of electrical transport, this phenomenon alters the transport properties, e.g., decreasing the conductance by increasing the degree of localization, ultimately leading to an insulating state. Alternatively, the particle can be localized in the presence of a potential gradient across the lattice, e.g., as created in the transport case by an external, static electric field, referred to as WannierStark localization^{7}. The potential gradient induces a positiondependent phase shift in the particle wavefunction and causes the particle to undergo periodic Bloch oscillations in a confined region^{8}. While the dynamics of these oscillations are challenging to observe in naturally occurring solidstate materials^{9}, they can be directly emulated and experimentally studied using engineered quantum systems.
Anderson localization has been experimentally realized in BoseEinstein condensates^{10,11}, atomic Fermi gases^{12}, and photonic lattices^{13,14}. Similarly, Bloch oscillations and WannierStark localization have been observed in optical lattices^{9,15}, semiconductor superlattices^{16}, and photonic lattices^{17,18}. However, these demonstrations were limited to varying degrees by a lack of control and simultaneous readout of each individual site. Therefore, these experiments could not fully explore the different localization regimes and spatial dimensions of the tightbinding model in a single experimental instantiation.
Superconducting quantum circuits are highlycontrollable quantum systems that enable us to probe the properties of lattice models. By engineering the Hamiltonian of the superconducting quantum circuit, we can emulate the dynamics of the lattice Hamiltonian, and we can use single and twoqubit gate operations for state preparation and tomographic state readout^{19,20,21,22,23}. Siteselective qubit control, strong qubitqubit interactions, and long coherence times relative to typical gate times combined with the capability of simultaneously applying gates and performing tomographic state readout on all sites make this a promising platform for studying models of information propagation, manybody entanglement, and quantum transport.
In this work, we use a 9qubit superconducting circuit to emulate the dynamics of singleparticle quantum transport and localization in 1d and 2d tightbinding lattices. We probe the entanglement formed in the lattice as a result of the particle propagation. We then realize Anderson localization by implementing random disorder of the onsite energies with tunable strength, and extract the dependence of the propagation mean free path on the disorder strength in a regime with no analytical solution. Additionally, we study WannierStark localization in the presence of isotropic and anisotropic potential gradients, and find close agreement between our experiments and theoretical predictions.
The close agreement between our experimental results, simulations, and theoretical predictions results from highfidelity, simultaneous qubit control and readout, an accurate calibration, and an accounting for the relevant decoherence mechanisms in this system. Although our experiments are performed on a small lattice that can still be simulated on a classical computer, they demonstrate a platform for exploring larger, interacting systems where numerical simulations become intractable.
Results
Experimental setup
Our device consists of a 3 × 3 array of capacitively coupled superconducting transmon qubits^{24} (Fig. 1b), where qubit excitations correspond to particles in the lattice model. In the singleparticle regime, the effective system Hamiltonian in the rotating frame with reference frequency ω_{ref} is described by the tightbinding Hamiltonian:
where \({\hat{\sigma }}_{i}^{{\dagger} }\) (\({\hat{\sigma }}_{i}\)) are the twolevel creation (annihilation) operators for each lattice site i. The first term in the Hamiltonian represents particle tunneling between neighboring sites with rate J_{i,j}, realized here with an average strength of J/2π = 8.1 MHz at ω_{ref} = 5.5 GHz. The second term represents the particle occupation energy of sites i with the corresponding site energies ϵ_{i} = ω_{i} − ω_{ref}, where ω_{i} is the fundamental transition frequency of the transmon at site i. By using fluxtunable transmons, the energy of each site can be individually set over the range 3.0 GHz ≲ ω_{i}/2π ≲ 5.5 GHz. This tunability enables us to realize arbitrary energy landscapes and to isolate sublattices (e.g., onedimensional chains) by detuning certain qubits from their neighbors so that they do not interact with the rest of the lattice. The transmons have an average anharmonicity of U/2π = − 244 MHz, which corresponds to the onsite interaction energy of two particles occupying the same site. The system operates in the J ≪ ∣U∣ limit such that each lattice site can be occupied by at most a single particle, generally realizing the hardcore BoseHubbard model^{25,26}.
Our experiments feature simultaneous, siteresolved, singleshot dispersive qubit readout with an average qubit state assignment fidelity of 95%. In addition, we are able to control the energy of each site with an average precision 〈Δϵ〉 < (2π) 200 kHz (≈J/40). These features are crucial to accurately emulate the 1d and 2d tightbinding model with different lattice potential configurations, and allow us to achieve close agreement between numerical simulations and experiment.
Quantum random walk
Quantum random walks (QRWs) are the quantum mechanical analog of classical random walks. The spatial propagation of the particle throughout the lattice, relative to its initial location, is quantified by the mean square displacement \(\langle {M}^{2}\rangle ={\sum }_{i}{p}_{i}{M}_{i}^{2}\), where p_{i} is the probability of finding the particle on site i, and M_{i} is the Manhattan (1norm) distance between site i and the initialization site. Quantum properties such as singleparticle superposition and interference result in a qualitative difference between classical and quantum random walks: a classical random walk propagates diffusively in time with \(\sqrt{\langle {M}^{2}\rangle }\propto \sqrt{t}\), whereas a QRW exhibits ballistic propagation with a meansquare displacement \(\sqrt{\langle {M}^{2}\rangle }\propto t\)^{27}.
In order to experimentally observe particle propagation through QRWs, we first implement a tightbinding lattice with uniform site energies (ϵ_{i} = 0). We compare the respective propagation speeds for QRWs in a sevensite 1d chain (Fig. 1d) and a 2d 3 × 3 lattice (Fig. 1e). We inject a particle at the end (corner) of the 1d (2d) lattice and observe its propagation by tracking the excitation numbers \(\langle {\hat{n}}_{i}\rangle =\langle {\hat{\sigma }}_{i}^{{\dagger} }{\hat{\sigma }}_{i}\rangle\) on each lattice site versus evolution time t.
In the 1d case, the particle traverses the lattice and eventually reflects off the opposite end of the chain^{28}. All intermediate sites are coupled to their two nearest neighbors, with the end sites being coupled to only one. Hence, the QRW propagates in both directions: an excitation is reemitted in both the forward and reverse directions (interim sites), or the direction is reversed (end sites). Quantum interference between these multiple trajectories alters the particle wavepacket as it evolves in time. The resulting QRW pattern for a sevensite chain features a nontrivial evolution that agrees well with numerical simulation (Fig. 1d).
In the 2d QRW, the particle propagates along its diagonal symmetry axis, analogous to a QRW on a binary tree^{5}. Here, the particle position is defined by the sum of the site populations at a given Manhattan distance M from the injection site. The quantum interference leading to the simple backandforth pattern as a result of reflection is exceptional and arises from the highdegree of symmetry present in a 3 × 3 lattice (Fig. 1e). Such a highdegree of symmetry is similarly observed in 1d for a fivesite chain (see Supplementary Material).
To verify these experimental results, we perform Lindblad master equation simulations (see Supplementary Material) based on the Hamiltonian in Eq. (1). We find excellent agreement between experimental data and numerical simulations by using the measured coupling strengths J_{ij} between neighboring lattice sites i, j, and taking into account qubit relaxation and decoherence.
Both 1d and 2d QRWs exhibit ballistic propagation, and the propagation speed in 2d is faster than in 1d due to the constructive interference between multiple propagation paths leading to each site in 2d (Fig. 1f). The average group velocity v_{g} of transport depends on the dimensionality of the lattice and the starting position of the particle with respect to the lattice boundaries. For a 1d QRW, a particle prepared at one end of the lattice initially propagates with v_{g} = J, because it interacts with only one other lattice site. At later times, prior to reflection from the other end, it reaches a steady state value of \({v}_{g}=\sqrt{3}J\), in agreement with theory for infinitely long chains^{27}. In contrast, we observe that a particle prepared in the corner of a 3 × 3 lattice propagates with an initial group velocity \({v}_{g}=\sqrt{2}J\) (due to the coupling to its two nearest neighbors) and a steady state average velocity of \({v}_{g}=(1+\sqrt{3/2})J\) (see Supplementary Material). During the QRW, certain sites become entangled as the particle propagates through the lattice, a phenomenon that underpins its intrinsic quantum character^{28,29}. We observe the emergence of entanglement as a result of the system time evolution via nonlocal spatial correlations. We quantify the entanglement formed among the qubits at the same distance from the particle initialization site, and study how the QRW entangles this site with the rest of the lattice in a coherent manner. The amount of entanglement within a twoqubit subsystem, described by ρ_{i,j}, can be quantified using the pairwise concurrence^{30}:
where {λ_{μ}} (in decreasing order) are the eigenvalues of the Hermitian matrix \(R\equiv \sqrt{\sqrt{{\rho }_{i,j}}{\tilde{\rho }}_{i,j}\sqrt{{\rho }_{i,j}}}\). Here, \(\tilde{\rho }=({\sigma }_{y}\otimes {\sigma }_{y}){\rho }^{* }({\sigma }_{y}\otimes {\sigma }_{y})\) is the spinflipped density matrix obtained through complex conjugation and applying PauliY operators (σ_{y}) to each qubit. Concurrence is a monotone entanglement metric for twoqubit states that takes values between zero (product state) and one (maximally entangled). To reconstruct the twoqubit density matrix ρ_{i,j}(t) and calculate the pairwise concurrence \({{{{\mathcal{C}}}}}_{i,j}(t)\), we perform twoqubit state tomography for various evolution times t of the QRW.
We measure the concurrence formed during a 2d QRW between site pairs at the same Manhattan distance from the particle initialization site (Fig. 2a). Lattice sites at the same Manhattan distance become partially entangled as the particle traverses them, as indicated by an increase in the concurrence. This approach faithfully describes the quantum correlations in the subsystems comprising of two sites M = 1, 3. However, the subsystem of sites at M = 2 contains three qubits, and solely considering pairwise concurrence values does not fully capture the entanglement within this subsystem.
Using the pairwise concurrence values between the sites at M = 2, we calculate the lowerbound of the distributed concurrence \({{{{\mathcal{C}}}}}_{i(j,k)}^{\min }\) for each site in this subsystem (Fig. 2b). For a state consisting of three qubits (i, j, k), the concurrence \({{{{\mathcal{C}}}}}_{i(j,k)}\) between site i and the system comprised of the two remaining sites (j, k) can be lowerbounded using the pairwise concurrences^{31}:
Using the extracted pairwise and distributed concurrence values, we compute the average of the lowerbound concurrence \(\bar{{{{\mathcal{C}}}}}\) for subsystems consisting of qubits that are at the same distance from the initial particle position (Fig. 2c). \(\bar{{{{\mathcal{C}}}}}\) is equal to the pairwise concurrence for subsystems containing two qubits (M = 1, 3), and an average of \({{{{\mathcal{C}}}}}_{i(j,k)}^{\min }\) for the subsystem containing three qubits (M = 2). The emergence of entanglement reflects the particle trajectory during the QRW (see Supplemental Material) and is in agreement with the lightcone of the measured quantum information propagation using outoftimeordered correlators^{26} within the LiebRobinson bounds^{32} (see Supplementary Material).
While the initially prepared singleparticle state is separable from the rest of the lattice, the QRW, mediated by the nearestneighbor interactions, creates and distributes entanglement to varying degrees throughout the lattice. For a system containing exactly a single particle, Eq. (3) becomes an equality and can be generalized to a system with more than three sites^{31}. In our 3 × 3 lattice, the concurrence between the initialization site (site 7) and the rest of the system \({{{{\mathcal{C}}}}}_{7,{{{\rm{lattice}}}}}\) can be extracted exactly using pairwise concurrence values (see Supplementary Material):
We observe that the quantum state of the initialization site becomes fully entangled with the larger system at time t ≃ (2J)^{−1} (Fig. 2d). The coherent propagation leads to the fall and revival of the concurrence as the particle wavefunction evolves with time, with \({{{{\mathcal{C}}}}}_{7,{{{\rm{lattice}}}}}=0\) when site 7 is in a pure ground or excited state. This behavior is in agreement with the evolution of the von Neumann entropy \(S({\rho }_{7})={{{\rm{tr}}}}({\rho }_{7}{{{\rm{ln}}}}{\rho }_{7})\) (Fig. 2e), where ρ_{7} is the singlequbit density matrix of site 7. The von Neumann entropy is a measure for the entanglement of the lattice site with its environment, and takes a value of \({{{\rm{ln}}}}(2)\) for a maximally entangled state. The dynamical revival of both \({{{{\mathcal{C}}}}}_{7,{{{\rm{lattice}}}}}\) and S(ρ_{7}) indicates that site 7 becomes predominantly entangled with the rest of the lattice, rather than with the uncontrolled environmental degrees of freedom related to decoherence.
Time evolution under an interacting Hamiltonian creates entanglement among the nine sites in our system. We quantified the entanglement between different subsystems of the lattice using concurrence and von Neumann entropy, however, the overall entanglement formed in the lattice is also of interest for studying the quantum properties of the system. Without the need for full quantum state tomography, we probe the global entanglement E_{gl} in the manybody system, beyond the nonlocal correlations formed between subsystems, by extracting the average purity of all N sites^{33,34}
where ρ_{j} is the reduced density matrix describing site j. In Fig. 2f, we observe that the global entanglement reaches a maximum value when the particle is not concentrated on either corner of the lattice, but rather extends across all lattice sites. In our 2d lattice, E_{gl} is upperbounded by the global entanglement in the 9qubit Wstate \(\left{W}_{9}\right\rangle\)^{35}, where \(\left{W}_{N}\right\rangle =\frac{1}{\sqrt{N}}{\sum }_{\pi }\left\pi (0...01)\right\rangle\) is the superposition of all singleparticle state permutations. At later times, the maximum global entanglement in the system decreases as a result of system decoherence. Using this metric, we study the degree of entanglement in the lattice as a result of the particle evolution.
Anderson localization
We have so far explored quantum transport in lattices with uniform site energies. However, particle propagation is significantly impacted by the introduction of random disorder in the site energies, leading to Anderson localization^{6}. In the presence of disorder in 1d and 2d lattices, the particle wavefunction becomes spatially localized, regardless of the disorder strength in the thermodynamic limit^{36}. In our experiments, we emulate the Anderson localization regime by introducing disorder with strength δ, where site energies are randomly sampled from a uniform distribution ϵ_{i} ∈[−δ/2, δ/2]. As a result, wavefunction scattering causes the particle to lose phase coherence on the length scale of the meanfree path l. While an analytical form for the mean free path can be derived in the limit of weak (δ ≪ J) and strong (δ ≫ J) disorder^{37,38}, there are no known analytical expressions in the intermediate disorder regime^{39}.
We examine the degree of localization by measuring the increase in average population \(\overline{\langle {n}_{s}\rangle }\) of the initially prepared lattice site and the decrease in particle spread \(\sqrt{\overline{\langle {M}^{2}\rangle }}\) for increasing disorder strengths δ, each averaged over 50 random lattice realizations. As the disorder strength increases, a higher steadystate population remains on the source site as a result of the tightbinding interaction (Fig. 3a, c), while the average particle spread away from the source decreases (Fig. 3b, d) for both 1d and 2d lattices. For a given disorder strength, the particle is more confined in a 1d lattice compared to a 2d lattice; in 2d, the probability of transport is relatively greater in the presence of disorder as propagation occurs along multiple paths between two sites. As a result of localization caused by disorder, the emergence of multipartite entanglement in the tightbinding lattice is inhibited: in Fig. 3e, f, we report a decrease in the steady state value for the average global entanglement of the system, \(\overline{{E}_{{{{\rm{gl}}}}}}\), with increasing disorder strength, due to spatial localization of the particle wavefunction. With one excitation, the confinement effects for sevensite chain causes a relatively greater deviation from the maximum value (E_{gl} of \(\left{W}_{7}\right\rangle\)) compared to the 2d lattice.
We quantify the localization of the wavefunction in a lattice with N sites using the participation ratio PR, defined as^{40}
If the particle wavefunction is completely delocalized, then PR = N, whereas PR = 1 for a wavefunction fully localized to a single site (Fig. 3g). In our experiments, PR for each lattice realization is calculated using the timeaveraged population on each site after the tightbinding model dynamics reach steady state^{40}. In order to reduce the impact of qubit relaxation on these measurements, we use the averaged site populations between 100 ns and 400 ns (5/J ≲ t ≲ 20/J) to calculate PR (see Supplementary Material for an example). We experimentally extract the participation ratio \(\overline{{{{\rm{PR}}}}}\) averaged over different random lattice realizations, at different disorder strengths for both a 1d and 2d lattice. As δ increases, the particle wavefunction becomes more spatially confined and \(\overline{{{{\rm{PR}}}}}\) decreases. For weak disorder strengths, we find the localization length to be larger than the lattice size (gray region in Fig. 3h, i), imposing a limitation on our experiments caused by boundary effects.
In a finite 1d lattice, the participation ratio is related to the localization length ξ_{1d} through the expression \({{{\rm{PR}}}}({\xi }_{{{{\rm{1d}}}}})=\coth (1/{\xi }_{{{{\rm{1d}}}}})\,\tanh (N/{\xi }_{{{{\rm{1d}}}}})\) (see Supplementary Material). The 1d localization length ξ_{1d} scales directly with the mean free path l as ξ_{1d} = l^{36}. We experimentally extract the dependence of the mean free path on the disorder strength, taking the form l = a(J/δ)^{γ}, in our sevensite chain by realizing 60 random lattice disorders for different disorder strengths and calculating \(\overline{{{{\rm{PR}}}}}\) (Fig. 3h). In the 1d case, we find γ = 1.0 ± 0.03 and a = 17.0 ± 1.05 (in units of the lattice spacing) through fitting.
In a 2d n × n lattice, the participation ratio is related to the localization length through the expression \({{{\rm{PR}}}}({\xi }_{2{{{\rm{d}}}}})={\coth }^{2}(1/{\xi }_{2{{{\rm{d}}}}})\,{\tanh }^{2}(n/{\xi }_{{{{\rm{2d}}}}})\) (see Supplementary Material), where the localization length takes the form \({\xi }_{{{{\rm{2d}}}}}=l\,{e}^{\frac{\pi }{2}kl}\) with a latticedependent factor k^{36}. For weak disorder, the 2d \(\overline{{{{\rm{PR}}}}}\) scales exponentially with the mean free path. Consequently, the scaling for small values of δ is difficult to observe in our small lattice, and hence the value of k cannot be obtained with high accuracy. We extract the 2d \(\overline{{{{\rm{PR}}}}}\) in our 3 × 3 lattice (Fig. 3i) over 180 random lattice realizations for different disorder strengths, and for large values of δ we find the parameters for the mean free path γ = 0.80 ± 0.02 and a = 12.98 ± 0.60 (in units of the lattice spacing). In a 2d lattice, we notice that the extracted mean free path exhibits a weaker dependence on the disorder strength, consistent with our earlier observations.
WannierStark localization
A particle in the tightbinding lattice is localized also in the presence of a static electric field, which creates a potential gradient across lattice site energies^{7}. We emulate the effect of an electric field by creating a gradient in the lattice site energies ϵ_{i}. The resulting potential gradient causes the particle to become spatially confined due to the emergence of a bandgap in the lattice band structure^{9}. This phenomenon is referred to as WannierStark localization^{7,15,41}. The effective field creates a linear gradient \(\overrightarrow{F}\) along the lattice axis, where \(\overrightarrow{F}=\nabla {\epsilon }_{i}\), and causes the particle to undergo periodic Bloch oscillations.
We demonstrate Bloch oscillations in a 1d chain for the field strength F/J = 1.5 (\(F= \overrightarrow{F}\)) observing a spatially periodic breathing motion with a probability up to 99% of the particle reviving at the initialization site (Fig. 4a). By repeating the experiment for different values of F, we observe that a particle initialized in the center of the lattice oscillates with a Bloch period T_{B} = 2π/F (Fig. 4b) and a maximum particle spread of \({d}_{{{{\rm{B}}}}}^{\max }=\max (\sqrt{\langle {M}^{2}\rangle })=2\sqrt{2}J/F\) (Fig. 4c) from the initialization site, in agreement with theory^{8}. The finite lattice size causes the boundaries to have a notable effect on the oscillation for small values of F (gray region of Fig. 4b, c), namely the decrease in the oscillation period and limiting the particle spread. We observe a similar periodic motion along the diagonal symmetry axis of the lattice in 2d (Fig. 4d) for an isotropic field (F = F_{x} = F_{y}). The Bloch period of a particle initialized in the corner of a 2d lattice with this symmetric gradient exhibits the same scaling with F as in 1d. We find that the maximum particle spread in a 2d lattice scales as \({d}_{{{{\rm{B}}}}}^{\max }=\max (\sqrt{\langle {M}^{2}\rangle })=3.01\pm 0.01\,J/F\), where we obtained the value 3.01 from a data fitting procedure. WannierStark localization is less pronounced in 2d compared to 1d for the same potential gradient, similar to the trend we observe in the Anderson localization regime and in agreement with theoretical predictions^{42}.
We also explore the effect of nonisotropic fields in a 2d lattice by independently controlling the potential gradients along the horizontal (F_{x}) and the vertical (F_{y}) axes, respectively. In Fig. 4g, we show the timeaveraged population of each site during the first 500 ns ( ≈ 25/J) of evolution as a result of different field ratios r = F_{y}/F_{x}. In the presence of the same field along both lattice axes (r = 1), the particle is localized along the diagonal symmetry axis of the lattice (Fig. 4g top left). As r increases by decreasing F_{x} while keeping F_{y} fixed, the particle becomes less localized along the F_{x}direction, skewing the propagation direction.
In the absence of boundary effects, the relation between Bloch oscillation periods \({T}_{{{{{\rm{B}}}}}_{x}}\) and \({T}_{{{{{\rm{B}}}}}_{y}}\) along each lattice axis depends on the net field direction, namely \({T}_{{{{{\rm{B}}}}}_{x}}=r\,{T}_{{{{{\rm{B}}}}}_{y}}\) (Fig. 4h). For rational values of r, the total 2d Bloch period T_{B} is the least common multiple of the onedimensional Bloch periods along each axis^{42}. A similar relationship holds for the maximum particle spread in each direction: \({d}_{{{{{\rm{B}}}}}_{x}}^{\max }=r\,{d}_{{{{{\rm{B}}}}}_{y}}^{\max }\) (Fig. 4i). In our experiments, we investigate these relations by measuring \({T}_{{B}_{x}}\) (see Fig. 4h) and \({d}_{{B}_{x}}^{\max }\) (see Fig. 4i) for r = 1, 2, 3 (solid circles) while varying F_{y}. The data match simulation (solid lines) of the 3 × 3 system very well. We additionally measured \({T}_{{B}_{y}}\) and \({d}_{{B}_{y}}^{\max }\), which are nominally unchanged for r = 1, 2, 3, and present the values measured for isotropic potential gradients (open circles). This enables us to obtain experimental estimates for r by taking the ratio between the experimentally measured Bloch periods \({T}_{{{{{\rm{B}}}}}_{x}}/{T}_{{{{{\rm{B}}}}}_{y}}\) (Fig. 4h, right panel) and the maximum spread \({d}_{{{{{\rm{B}}}}}_{x}}^{\max }/{d}_{{{{{\rm{B}}}}}_{y}}^{\max }\) (Fig. 4i, right panel). The ratios for each r are estimated from highlighted points in Fig. 4h, i.
Discussion
In this work, we have emulated quantum particle propagation and localization in 1d and 2d tightbinding lattices using a 3 × 3 array of superconducting qubits. We have measured the coherent dynamics of different entanglement metrics, such as concurrence and the von Neumann entropy, during quantum transport using simultaneous control and readout. We have further investigated localization in different regimes of the tightbinding model, with random disorder resulting in Anderson localization and with a potential gradient causing WannierStark localization and Bloch oscillations. We have studied Anderson localization in a disorder regime that lacks an analytical form, and used our data to extract the dependence of the propagation mean free path in finite 1d and 2d lattices on the disorder strength. We have measured the degree of WannierStark localization as a result of different potential gradients for both isotropic and anisotropic potentials, and have observed quantitative agreement in the properties of Bloch oscillations with theoretical predictions.
We can control all of the onsite energies with high precision compared to the coupling rate between neighboring sites. Furthermore, we can manipulate the lattice (qubit) energy landscape on timescales much shorter than the particle exchange (qubitqubit interaction) and system coherence times by two orders of magnitude. This ability to accurately initialize and manipulate our system Hamiltonian combined with highfidelity tomographic readout enables us to study tightbinding lattices under various configurations. For superconducting qubits, models based on calibrated qubit parameters and decoherence^{43,44} have been demonstrated to accurately predict the behavior of multiqubit processors^{45,46}. These features form the basis for agreement between our experimental results and our simulations. The demonstrated degree of control in realizing different regimes of the tightbinding model in 1d and 2d serves as a blueprint for exploring larger and strongly interacting quantum lattices in the fewbody^{47} and the manybody localization regimes^{40,48,49} or that contain a timedependent Hamiltonian leading to interesting phenomena such as quantum scars^{50} topological order^{51} and the breakdown of the Magnus expansion^{52}.
Data availability
The raw data used for generating the plots in this paper are available upon request.
References
Slater, J. C. & Koster, G. F. Simplified LCAO method for the periodic potential problem. Phys. Rev. 94, 1498–1524 (1954).
Goringe, C. M., Bowler, D. R. & Hernández, E. Tightbinding modelling of materials. Rep. Prog. Phys. 60, 1447–1512 (1997).
Cleri, F. & Rosato, V. Tightbinding potentials for transition metals and alloys. Phys. Rev. Lett. 48, 22–33 (1993).
Kempe, J. Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003).
Childs, A. M., Farhi, E. & Gutmann, S. An example of the difference between quantum and classical random walks. Quantum Inform. Process. 1, 35–43 (2002).
Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).
Emin, D. & Hart, C. F. Existence of WannierStark localization. Phys. Rev. Lett. 36, 7353–7359 (1987).
Hartmann, T., Keck, F., Korsch, H. J. & Mossmann, S. Dynamics of Bloch oscillations. N. J. Phys. 6, 2 (2004).
Ben Dahan, M., Peik, E., Reichel, J., Castin, Y. & Salomon, C. Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508–4511 (1996).
Billy, J. et al. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453, 891–894 (2008).
Roati, G. et al. Anderson localization of a noninteracting BoseEinstein condensate. Nature 453, 895–898 (2008).
Kondov, S. S., McGehee, W. R., Zirbel, J. J. & DeMarco, B. Threedimensional Anderson localization of ultracold matter. Science 334, 66–68 (2011).
Lahini, Y. et al. Anderson localization and nonlinearity in onedimensional disordered photonic lattices. Phys. Rev. Lett. 100, 013906 (2008).
Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered twodimensional photonic lattices. Nature 446, 52–55 (2007).
Preiss, P. M. et al. Strongly correlated quantum walks in optical lattices. Science 347, 1229–1233 (2015).
Feldmann, J. et al. Optical investigation of Bloch oscillations in a semiconductor superlattice. Phys. Rev. Lett. 46, 7252–7255 (1992).
Morandotti, R., Peschel, U., Aitchison, J. S., Eisenberg, H. S. & Silberberg, Y. Experimental observation of linear and nonlinear optical bloch oscillations. Phys. Rev. Lett. 83, 4756–4759 (1999).
Trompeter, H. et al. Bloch oscillations and Zener tunneling in twodimensional photonic lattices. Phys. Rev. Lett. 96, 053903 (2006).
Roushan, P. et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358, 1175–1179 (2017).
Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018).
Chiaro, B. et al. Direct measurement of nonlocal interactions in the manybody localized phase. http://arxiv.org/abs/1910.06024 (2019).
Ma, R. et al. A dissipatively stabilized Mott insulator of photons. Nature 566, 51–57 (2019).
Gong, M. et al. Quantum walks on a programmable twodimensional 62qubit superconducting processor. Science 372, 948–952 (2021).
Koch, J. et al. Chargeinsensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).
Yanay, Y., Braumüller, J., Gustavsson, S., Oliver, W. D. & Tahan, C. Twodimensional hardcore Bose–Hubbard model with superconducting qubits. npj Quantum Inf. 6, 58 (2020).
Braumüller, J. et al. Probing quantum information propagation with outoftimeordered correlators. Nat. Phys. 18, 172–178 (2022).
Hoyer, S., Sarovar, M. & Whaley, K. B. Limits of quantum speedup in photosynthetic light harvesting. New J. Phys. 12, 065041 (2010).
Yan, Z. et al. Strongly correlated quantum walks with a 12qubit superconducting processor. Science 364, 753–756 (2019).
Cheneau, M. et al. Lightconelike spreading of correlations in a quantum manybody system. Nature 481, 484–487 (2012).
Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998).
Coffman, V., Kundu, J. & Wootters, W. K. Distributed entanglement. Phys. Rev. A 61, 5 (2000).
Lieb, E. H. & Robinson, D. W. The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972).
Meyer, D. A. & Wallach, N. R. Global entanglement in multiparticle systems. J. Math. Phys. 43, 4273–4278 (2002).
Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in manybody systems. Rev. Mod. Phys. 80, 517–576 (2008).
Brennen, G. K. An observable measure of entanglement for pure states of multiqubit systems. Quantum Inf. Comp. 3, 619–626 (2003).
Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).
Casati, G., Guarneri, I., Izrailev, F., Fishman, S. & Molinari, L. Scaling of the information length in 1D tightbinding models. J. Phys. Cond. Mat. 4, 149–156 (1992).
Chuang, C., Lee, C. K., Moix, J. M., Knoester, J. & Cao, J. Quantum diffusion on molecular tubes: universal scaling of the 1D to 2D transition. Phys. Rev. Lett. 116, 196803 (2016).
Varga, I. & Pipek, J. Information length and localization in one dimension. J. Phys. Cond. Mater. 6, L115 (1994).
Van Nieuwenburg, E., Baum, Y. & Refael, G. From Bloch oscillations to manybody localization in clean interacting systems. Proc. Natl Acad. Sci. USA 116, 9269–9274 (2019).
Guo, X. Y. et al. Observation of Bloch oscillations and WannierStark localization on a superconducting processor. npj Quantum Inf. 7, 51 (2021).
Witthaut, D., Keck, F., Korsch, H. J. & Mossmann, S. Bloch oscillations in twodimensional lattices. N. J. Phys. 6, 41 (2004).
Yan, F. et al. The flux qubit revisited to enhance coherence and reproducibility. Nat. Comm. 7, 1–9 (2016).
Krantz, P. et al. A quantum engineer’s guide to superconducting qubits. Appl. Phys. Rev. 6, 021318 (2019).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Karamlou, A. H. et al. Analyzing the performance of variational quantum factoring on a superconducting quantum processor. npj Quantum Inf. 7, 156 (2021).
Villa, L., Thomson, S. J. & SanchezPalencia, L. Quench spectroscopy of a disordered quantum system. Phys. Rev. A 104, L021301 (2021).
Sierant, P., Delande, D. & Zakrzewski, J. Thouless time analysis of anderson and manybody localization transitions. Phys. Rev. Lett. 124, 186601 (2020).
Khemani, V., Hermele, M. & Nandkishore, R. Localization from Hilbert space shattering: from theory to physical realizations. Phys. Rev. B 101, 174204 (2020).
Mukherjee, B., Nandy, S., Sen, A., Sen, D. & Sengupta, K. Collapse and revival of quantum manybody scars via Floquet engineering. Phys. Rev. Lett. 101, 245107 (2020).
Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).
D’Alessio, L. & Rigol, M. Longtime behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).
Acknowledgements
The authors are grateful to F. Vasconcelos, and S. Lloyd for insightful discussions. AHK acknowledges support from the NSF Graduate Research Fellowship Program. This research was funded in part by the U.S. Army Research Office Grant W911NF1810411 and the Assistant Secretary of Defense for Research & Engineering under Air Force Contract No. FA872105C0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.
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A.H.K., J.B., Y.Y., C.T., and W.D.O. conceived the experiment. A.H.K. and J.B. performed the experiments. A.H.K., J.B., B.K., M.K., A.V., R.W., and S.G. developed the experiment control tools used in this work. D.K., A.M., B.N., and J.Y. fabricated the 3 × 3 qubit array. C.T., T.P.O., S.G., and W.D.O. provided experimental oversight and support. All authors contributed to the discussions of the results and to the development of the manuscript.
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Karamlou, A.H., Braumüller, J., Yanay, Y. et al. Quantum transport and localization in 1d and 2d tightbinding lattices. npj Quantum Inf 8, 35 (2022). https://doi.org/10.1038/s41534022005280
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DOI: https://doi.org/10.1038/s41534022005280
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