Putting photons to work
Interacting quantum particles can behave in peculiar ways. To understand that behavior, physicists have turned to quantum simulation, in which a tunable and clean system can be monitored as it evolves under the influence of interactions. Roushan et al. used a chain of nine superconducting qubits to create effective interactions between normally noninteracting photons and directly measured the energy levels of their system. The interplay of interactions and disorder gave rise to a transition to a localized state. With an increase in the number of qubits, the technique should be able to tackle problems that are inaccessible to classical computers.
Science, this issue p. 1175
Abstract
Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field—the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter.
Consider a system of interacting particles isolated from the environment initially prepared in a very low entropy state far from equilibrium. It is often observed that the system acts as its own thermal reservoir and approaches the equilibrium state. In this thermal phase, the system shows ergodic behavior, wherein it uniformly explores all accessible states over time. Recent works discuss the emergence of another phase in certain parameter regimes in which ergodicity breaks down and thermal equilibrium becomes unattainable (1–8). This phase is referred to as the many-body localized (MBL) phase (9–16). The conventional quantum phase transitions, e.g., from para- to ferromagnetic, are characterized by changes in the ground state of the system. However, the signature differences between the thermalized and MBL phases are in dynamical behaviors, indicating that the transition involves change in the properties of all many-body eigenstates of the system. Hence, the physics goes beyond the ground state and requires study of the entire energy spectrum, which constitutes an experimental challenge.
In quantum physics, the quantized eigenenergies and their associated wave functions provide extensive information for predicting the chemistry of molecules or physics of condensed-matter systems. Creating local perturbations and recording the subsequent response of the system as a function of time can reveal the characteristic modes of that system (17, 18). Our method for measuring the energy spectrum of a Hamiltonian is based on this principle. For fixed Hamiltonians, the state of a system evolves according to Schrödinger equation
(1)where 
is an eigenenergy of the Hamiltonian and 
is the corresponding eigenstate. Equation 1 implies that 
and 
determine the frequencies and the amplitudes of the modulations in 
, respectively. The similarity of Eq. 1 and a Fourier transform (FT) relation suggests that the frequencies observed in the FT of the evolution could in principle reveal 
. In addition, the magnitudes of FT terms provide 
; these coefficients set the relative contribution of each 
to a given dynamics.
Using nine superconducting qubits (Fig. 1A), we constructed a one-dimensional (1D) lattice of bosons and implemented a spectroscopy method based on the above-mentioned fundamental postulate of quantum mechanics. Each of our qubits can be thought of as a nonlinear photonic resonator in the microwave regime (19, 20). The Hamiltonian of the chain can be described by the Bose-Hubbard model
(2)where 
(a) denotes the bosonic creation (annihilation) operator, 
is the on-site potential, J is the hopping rate of the photons between nearest-neighbor lattice sites, and U is the on-site interaction. The qubit frequency, the nearest-neighbor coupling, and nonlinearity set 
, J, and U, respectively. In our system, we can vary the first two on nanosecond time scales, but U is fixed.
(A) Optical micrograph of the device. (B) Pulse sequence used to measure eigenvalues of a time-independent Hamiltonian, Eq. 2, with 
MHz, 
, and 
randomly chosen from [0,100] MHz. Initially, all the qubits are in the 
state. Using a microwave pulse, one of the qubits is then placed in the superposition of the 
and 
states (
depicted here). The coefficients in the Hamiltonian are set by applying square pulses on the qubits 
and couplers 
. After the evolution, a microwave 
pulse is applied to the qubit to measure 
or 
. (C) Typical data set showing 
and 
versus time. (D) The FT of 
for 
. The peaks in the FT correspond to the eigenvalues of the Hamiltonian. The probability of a Fock state on 
being in the ninth eigenstate 
is highlighted. (E) Average of the square of FT amplitudes shown in (D). Averaging is done to show all nine peaks in one curve.
In Fig. 1, we show how to identify the eigenenergies of Eq. 2 when it describes the hopping of a single photon in a disordered potential. The disordered potential is realized by choosing a random number from a uniform distrbution [–Δ, Δ] for each lattice site, and detuning the qubits accordingly. Initially, there is no photon in the system and all the qubits are in 
state. Then, we place the nth qubit 
in the superposition of 
and 
(Fig. 1B). We measure the evolution of 
and 
, where 
and 
are Pauli operators (acting on the 
and 
subspace) (Fig. 1C). From the 
and 
measurements, we construct 
. Next, we vary n from 
to 
to ensure that the energy spectrum is fully resolved. By varying n, the initial states form a complete basis, and then every energy eigenstate is certain to have some overlap with one of the initial states and hence can be detected. Figure 1D shows the square of FT amplitudes of 
for each 
in which distinct peaks can be readily identified. The result of averaging the squared FT amplitudes over nine different initial states is depicted in Fig. 1E, where nine major peaks appear; their frequencies are the nine eigenenergies of the Hamiltonian. The particular choices of initial states and the observables are made to avoid appearance of undesired energy peaks in the spectrum (17, 18, 21).
Next, we demonstrate our capability to accurately set the terms in a specific Hamiltonian and resolve the corresponding eigenenergies. We simulate the problem of Bloch electrons on a 2D lattice subject to a perpendicularly applied magnetic field B. The magnetic length 
and lattice constant a characterize the electron’s motion, and their interplay sets the physics. The resulting energy spectrum was first calculated by Hofstadter and resembles a butterfly (22). For typical crystals, the magnetic field required to “squeeze” one flux quantum through the unit cell is of the order of several tens of thousands of tesla, too high to be experimentally feasible. Recently, some features associated with the Hofstadter’s butterfly were experimentally realized by using superlattices in graphene and cold-atom systems (23–28).
The Hofstadter energy spectra can be parameterized by a single dimensionless magnetic field, 
, which counts the number of magnetic flux quanta per unit cell. In the tight binding approximation, the effective Hamiltonian is the 1D Harper Hamiltonian (21)
(3)
is a special case of HBH, reached by setting 
and exciting only one photon in the system, i.e., the interaction term is zero. In this limit, whether the particles are fermionic or bosonic has no influence on the behavior of the system. In Fig. 2, we vary b from 0 to 1 and realize 100 different instances of 
. Similar to Fig. 1, for each b value, initial states with the nth qubit excited are created, the evolution of 
and 
is measured, and n is varied from 1 to 9. For each b value, Fig. 2A shows the squared magnitude summation of the FT of 
.
In Eq. 3, we set on-site potentials 
MHz and coupling 
MHz. (A) Data similar to Fig. 1D, averaged squared FT magnitude, are shown for 100 values of dimensionless magnetic field b ranging from 0 to 1. (B) For each b value, we identify nine peaks and plot their location as a colored dot. The numerically computed eigenvalues of Eq. 2 are shown as gray lines (21). The color of each dot is the absolute value of the difference between the measured eigenvalue and the numerically computed one.
For large lattices, it is known that for rational b, all energy bands split into subbands, and for irrational b, the spectra become fractal and form a Cantor set. Because there are only nine sites, what is seen in Fig. 2A are the remnants of those bands. Nevertheless, the overall measured spectrum still resembles a butterfly. We focus on this pattern of level crossings and meanderings and ask how well the measurements match simulation. In Fig. 2B, we present the numerically computed eigenenergies with solid gray lines and the measured peaks from Fig. 2A with colored dots (21). The color of the dots shows the distance in megahertz of the peaks from the simulation values. The deviations are mainly systematic calibration errors, and occasionally an erroneous peak is identified in the subroutine. The average deviation is 3.5 MHz. This implies that we can set the matrix elements of the Hamiltonian, which in this case includes 
terms, with <2% error. This capability of controlling a large quantum system is achieved through careful modeling of the qubits as nonlinear resonators.
By placing two photons in the system, we next study the simplest interacting bosonic cases (
, with no mapping to electronic system). The rest of the data presented in this work are obtained by using the following procedure (two-photon protocol). We realize a quasi-periodic disorder potential by setting 
. In total, four different irrational values of b are randomly chosen from [0,1], and the corresponding results are averaged. The irrational choice of b ensures that the periodicity of the potential and lattice are incommensurate. In Eq. 2, we set 
MHz, which results in 
. The initial states are made by placing two qubits (
and 
) in the superposition of the 
and 
states. We measure two-point correlations and construct 
. The peaks observed in the FT of 
are the eigenenergies of HBH in the two-photon manifold (21).
Perhaps the most direct way of examining ergodic dynamics and its breakdown is by studying the distribution of the energy levels (29–31). Using the two-photon protocol, we measure the evolution of 
for various strengths of disorder 
. We identify the peaks in the FT of 
as the energy levels 
(fig. S3). Let 
be the nearest-neighbor spacings (Fig. 3A), and level separation uniformity 
. From our measured 
, we compute the associated 
and construct their probability distribution (PD) (Fig. 3B). For low disorder, the PD is mainly centered around the 
values close to half, and with an increase of disorder, the histogram’s peak shifts toward smaller 
values.
In Eq. 2, we set hopping to 
MHz, which fixes 
. To obtain a disordered potential, we set 
with four different irrational values of 
chosen and the results averaged over b. (A) The schematic of energy levels shows how 
is defined. (B) The histogram of 
measured for various values of disorder 
is presented as a color plot. (C) The measured histogram 
of 
for 
and 5. The dashed lines are plots of 
and 
according to Eq. 4, and the solid lines are numerical simulations (21). The change from the GOE toward the Poisson distribution is indicative of vanishing of level repulsion with increase in 
.
It has been postulated that in the ergodic phase, the statistics of energy levels is the same as the ensemble of real Hermitian random matrices, which follow the Gaussian orthogonal ensemble (GOE) (31). In the many-body localized phase, the energy levels become uncorrelated owing to disorder, and hence a Poisson distribution in energy landscape is expected. The probability distribution of 
for the ergodic and many-body localized phases, respectively, is
(4)In Fig. 3C, we focus on 
and 
, showing the measured histograms with dots and the numerical simulations with solid lines (21). The dashed lines are plots of Eq. 4, providing the expected behavior in the thermodynamic limit (number of sites 
), and for limiting values of 
. In contrast to these cases, the finite size of our chain results in features that can be seen in both data and simulation. When disorder is small, the energy eigenstates are extended across the chain (Fig. 4), and hence the energy levels repel each other. Consequently, there are strong correlations between the levels, and an equidistant distribution of levels would be favorable. When 
becomes larger, the eigenstates become localized in space and unaware of each other’s presence at a given energy, and level repulsion ceases. Therefore, the levels independently distribute themselves, showing a Poisson distribution in the energy landscape. The exact realization of Poisson distribution takes place only when 
; in our case, 
, which is where the peak in the histogram appears (rα = 0.2). Because the Poisson distribution is a signature of independent events, we conclude that the transition from ergodic to localized phase is associated with vanishing correlations in energy levels.
In Eq. 2, we set 
, 
MHz, which results in 
.We measure the evolution of 
for all pairs of 
as a function of time for various strengths of disorder 
. From the magnitude of the peaks seen in the FT of the data, the probabilities relating the positions of two-photon Fock states to energy eigenstates 
are extracted. See fig. S3 for details. On the basis of those data, we calculated (A) 
and (B) 
using Eq. 5 and plotted the results. The 
is the width of the energy band at a given 
.
A key signature of the transition from ergodic to MBL phase is the change in the localization length of the system from being extended over the entire system to localized over a few lattice sites. This physics can be studied by measuring the probability of each energy eigenstate being present at each lattice site 
(21). In our method, the frequencies of the FT signal give the eigenenergies, and from the magnitude of the FT terms, 
can be measured; for instance, 
is highlighted in Fig. 1C. In the study of metal–insulator transitions (32, 33), a common way to quantify the extension in real space or energy landscape is via the second moment of the probabilities, defined by participation ratio (PR)
(5)
indicates the number of sites over which an energy eigenstate 
has an appreciable magnitude. Similarly, 
measures how many energy eigenstates have a discernable presence on lattice site n. Note that the first moments of the probability distributions are normalization conditions 
and 
.
Having demonstrated that we can fully resolve the energy spectrum of the two-photon energy manifold, we now extract 
. In Fig. 4A, we plot 
for various disorder strengths in the order of increasing energy. In this energy manifold, there are 36 single- 
and 9 double-occupancy states 
, which gives 
energy levels. For low disorder 
, 
is about 8, indicating that almost all energy eigenstates are extended over the entire chain of nine qubit lattice sites. As the strength of disorder increases, the eigenstates with their energies close to the edge of the energy band start to shrink, whereas the eigenstates with energies in the middle of the band remain extended at larger disorders. This is similar to the Anderson localization picture, in which localization begins at the edges of the band, and a mobility edge forms (the yellow hue) and approaches the center of the band as disorder becomes stronger (32). The existence of the mobility edge in MBL has been theoretically questioned, and proper investigation of it requires going to larger systems and finite size scaling (34). Given that numerical exact diagonalization is limited to small systems, scaling up the experiment could shed light on this matter and the general understanding of MBL (33, 35).
In Fig. 4B, we plot the 
, which shows that as the disorder becomes stronger, the number of eigenstates present at a given lattice site is reduced, indicating that eigenstates are becoming localized on lattice sites. Furthermore, with increasing disorder, the eigenstates are avoiding the edges of the chain, and more eigenstates are present toward the center of the chain. The changes in 
and 
are the fastest near 
, suggestive of a phase transition that has been smeared out owing to finite-size effects. Nevertheless, we emphasize that the quantum phase transition to the MBL phase is only defined in the thermodynamic limit (
) (15). Given the finite size of our system and the presence of only two interacting particles, it is interesting that we see finite-size precursors associated with the MBL phase transition.
It is worth noting that the signatures of localization presented in this work are common between single-particle localization (Anderson) and MBL. We refer to our observations as MBL because of the existence of nonzero interaction U whose value has been established through independent experiments. However, a decisive distinction could result from carefully engineered pulse sequences, as suggested in (11).
Our work demonstrates that properties of various phases can be extracted by directly measuring the energy levels of a system. It is interesting to consider the application of this method to a device with a few tens of qubits, where classical simulations will begin to fail. The technique presented here is scalable to more qubits but is ultimately limited by the frequency broadening that results from decoherence. For large systems, the level spacing becomes exponentially denser and the current approach needs to be revised; promising methods are suggested in (35, 36). Nevertheless, the valuable computational resource that our platform offers resides in measuring the dynamics of observables; those dynamics quickly become intractable for classical computers as the size of the system grows.
Supplementary Materials
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References and Notes
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- Acknowledgments: We acknowledge discussions with Y. Bahri, E. Kapit, O. Kyriienko, M. F. Maghrebi, V. Oganesyan, V. N. Smelyanskiy, and G. Zhu. P.R. and C.N. performed the experiment. J.T., V.M.B., and D.G.A. provided theoretical assistance. P.R., C.N., J.T., V.M.B., and D.G.A. analyzed the data and cowrote the manuscript. All of the UCSB and Google team members contributed to the experimental setup. At CQT, this research was supported by Singapore Ministry of Education Academic Research Fund Tier 3 (grant no. MOE2012-T3-1-009); National Research Foundation (NRF) Singapore; and the Ministry of Education, Singapore under the Research Centres of Excellence Program. The data that support the plots presented in this paper and other findings of this study are available from the corresponding author upon reasonable request. The authors declare no competing financial interests.
