Realizing dipolar spin models with arrays of superconducting qubits

We propose a platform for quantum many body simulations of dipolar spin models using current circuit QED technology. Our basic building blocks are 3D transmon qubits where we use the naturally occurring dipolar interactions to realize interacting spin systems. This opens the way toward the realization of a broad class of tunable spin models in both two- and one-dimensional geometries. We illustrate the potential offered by these systems in the context of dimerized Majumdar-Ghosh-type phases, archetypical examples of quantum magnetism, showing how such phases are robust against disorder and decoherence and could be observed within state-of-the-art experiments.

Figure 1

Circuit QED implementation of spin models. (a) Drawing of one half of a waveguide cavity with two microwave couplers. Inside the cavity is a piece of sapphire with multiple transmon qubits arranged on a triangular lattice. The rotation angles of the qubits are chosen such that only a few of them couple to the fundamental mode of the cavity with a predetermined coupling strength. (b) The cavity can be loaded with an arbitrary subset of qubits, e.g., considering the ones in the dot-dashed box in (a) realizes a triangular ladder. Here, the bond thickness denotes the interaction strength of nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) interactions. Additional longer-range contributions are represented by dashed lines (only the strongest are shown). (c) Sketch of the dimerized phase of the extended $XY$ model in Eq. (1). The phase can be understood as a solid of local triplet states, denoted by the shaded areas.

Figure 2

Microscopic finite-element simulation (see text) of the dependence of the coupling $J_{i,j}$ between two qubits. Panel (a): distance dependence at ${\theta }_i={\theta }_j={90}^{\circ }$ for three different antenna length $d_m$ (given in mm). The dashed line indicates $J_{i,j}=0$. The inset displays a typical result from HFSS simulations [27], with the dipoles surrounded by the computed electric field (intensity decreasing from red to blue). Panel (b): angular dependence illustrated for the case ${\theta }_i={\theta }_j=\theta$ at distance $r=1.5$ mm. The solid line plots Eq. (2) with $J_0,r_m$ extracted from best fits in (a).

Figure 3

Mesh used to calculate the electromagnetic fields around the qubits. The left half of the image shows the meshing needed for the cavity that contains a sapphire substrate with two qubits on it. The right half shows meshing around one of the qubits. The coordinate system used in this paper is defined in the lower left.

Figure 4

Extracted capacitance of the qubit as a function of the width of the antenna paddles. The dashed line is a linear fit to the data.

Figure 5

Qubit-cavity coupling strength $g$ as a function of its position inside the cavity. The qubit was placed in the center of the cavity in the $y$ and $z$ direction and was moved along the $x$ axis. The dashed line is a cosine fit to the data.

Figure 6

Coupling $g$ of the qubit to the cavity as a function of its antenna length. Since a change in antenna length will also change the resonance frequency, the width of the antenna has been adjusted to keep the resonance frequency constant. The dashed line is a linear fit to the data.

Figure 7

Coupling $J_1$ and $J_2$ and the ratio $J_2/J_1$ as a function of distance $S$ between qubit rows. This simulation was run with a 1 mm antenna. Each row of qubits in the ladder model can be fabricated on an individual piece of sapphire. These two pieces can then be moved relative to each other via a piezo actuated stage which allows for an in-situ change of the coupling ratio $J_2/J_1$. As the distance for all qubits in one row does not change, $J_2$ stays constant. In contrast $J_1$ will vary with the distance of the two sapphire pieces. The inset shows a side view of the two sapphire pieces with the two rows of qubits (red blocks) on top. The two rows are offset from each other on the two sapphires. This offset can be adjusted to match the desired couplings for $S$ approaching zero.

Figure 8

Comparison between the BOP $D^z$ for a large system ($L=192$ spins, dashed black line) and the bond correlation $B^z$ in the middle of the chain for samples of different sizes. The phase diagram hosts three phases: superfluid (SF), dimer (DP), and chiral phase (CP). The gray area can be realized in the setting described in Sec. 3. The dot-dashed and dot-dot-dashed vertical lines represent the DP boundaries [38]

Figure 9

From left to right: Scaling of $D^z,B^z,B^{zz}$ as a function of $J_2/J_1$ for different system sizes.

Figure 10

Scaling of the $B^{zz}$ order parameter as a function of the inverse length of the system. The curves are finite-size scaling fits using the functional form $a_0+a_1{L}^{a_2}$.

Figure 11

Effects of disorder on the dimer phase. Up to disorder Gaussian spreadings of order $\delta h\simeq 0.45J_1$, the bond correlation is weakly unaltered, i.e., the phase is stable against local perturbations. The blue curve represents a conservative estimate for typical experimental setups. Here, $L=10$ sites, results show averaging of 500 disorder realizations per point. The gray area can be realized in the setting described in Sec. 3.

Figure 12

Adiabatic state preparation. Panel (a): state preparation of the dimer phase for a system with $L=6, 2J_2=J_1=2\pi \times 100$ MHz, and different decay rates (in $2\pi \times$ kHz units), with $\gamma =\kappa$. The upper (lower) panel denotes the bond $x$ ($z$) correlation averaged over 2000 disorder realizations, with $\delta h/J_1=0.25$. The straight lines denote the ideal case. Up to decoherence rates of order 100 kHz, the final observables at the end of the ramp are within 2% of the expect one. The inset shows how the ramping parameters (in units of $J_1$) evolve with time. Panel (b): same as in (a) for fixed $\kappa =2\pi \times 100$ kHz and different values of $J_2$ (in units of $J_1$). Outside of the DP (red dot-dashed line), the order parameters have very different final values as the final state is not dimerized.

Figure 13

Circuit model corresponding to two qubits in a cavity. Values of the four capacitors drawn in red decay as $1/r^2$ where $r$ is the center to center distance between pads of the antenna. This models the dipole-dipole coupling of two qubits. The two purple structures on Q1 and Q2 represent the physical shapes of the qubits.

Figure 14

Example for three modes of a system with two qubits and a cavity. The minimum distance of two modes is defined as the coupling $2J_{i,j}$ if it is two qubits and as $2g$ if it is the qubit cavity coupling. Note that despite the fact that $Q_2$ has 9 nH of inductance, the crossing happens at lower inductances due to the coupling to $Q_1$. Dashed line shows how resonance frequency of a typical LC circuit would behave without coupling to the other resonators.

Figure 15

Capacitance between two coplanar paddles (solid line) as a function of the separation $s$ according to Eq. (A1). The corresponding structure is shown at the bottom. The dashed line shows the quadratic decay used in this study to estimate the coupling capacitance. The structure at top right shows which capacitors in the circuit model follow the quadratic decay.

Figure 16

Contour plot showing the coupling strength between two qubits $J_{i,j}$ as a function of the distance between them, in the $x$ and $y$ direction, in the absence of the cavity. The color scale is logarithmic to highlight the rapid change in $J_{i,j}$. In such a case, the dipole physics dominates and the coupling will be suppressed along a straight line. The white area is not accessible due to the dimensions of the qubits.

Figure 17

Contour plot showing the coupling strength between two qubits $J_{i,j}$ as a function of the distance between them, in the $x$ and $y$ direction, in the presence of the cavity. The color scale is logarithmic to highlight the rapid change in $J_{i,j}$. The cavity distorts the straight line of suppressed coupling opening possibilities for engineering of the coupling between two qubits. The white area is not accessible due to the dimensions of the qubits.

Figure 18

Coupling $J$ as a function of distance along $x$ axis. The center to center separation along the $y$ axis is kept at 1 mm, and one qubit is moved along the $x$ axis with respect to the other. The results suggest a double suppression of the coupling in agreement with the circuit model (dashed curve).

Figure 19

Lumped element circuit model for two capacitively coupled superconducting qubits.

Figure 20

Lumped element circuit for two interacting superconducting qubits in a cavity resonator.