Time-reversal-symmetry breaking in circuit-QED-based photon lattices

Breaking time-reversal symmetry is a prerequisite for accessing certain interesting many-body states such as fractional quantum Hall states. For polaritons, charge neutrality prevents magnetic fields from providing a direct symmetry-breaking mechanism and, similar to the situation in ultracold atomic gases, an effective magnetic field has to be synthesized. We show that in the circuit-QED architecture, this can be achieved by inserting simple superconducting circuits into the resonator junctions. In the presence of such coupling elements, constant parallel magnetic and electric fields suffice to break time-reversal symmetry. We support these theoretical predictions with numerical simulations for realistic sample parameters, specify general conditions under which time reversal is broken, and discuss the application to chiral Fock-state transfer, an on-chip circulator, and tunable band structure for the Kagome lattice.

Figure 1

The Jaynes-Cummings lattice as an example of a photon lattice. Its circuit-QED realization would consist of superconducting resonators (e.g., coplanar waveguides, schematically shown as rectangular boxes), each of which would be coupled to a superconducting qubit (symbolized as dots centered in the resonators). Microwave photons would hop between nearest-neighbor resonators, with the coupling strength $\kappa$ set by the mutual capacitance between resonator ends. Interaction between the photons and the superconducting qubits with strength $g$ would induce an effective photon-photon interaction.

Figure 2

Basic scheme of a three-port coupling element, connected capacitively to three transmission-line resonators with annihilation operators $a_j$ for photons in the relevant mode of the resonators enumerated by $j=1,2,3$.

Figure 3

Illustration of the gauge-invariant phase sum around a loop, ${\sum \circlearrowleft }_{\mathcal{C}[\mathit{ij}]}{\phi }_{\mathit{ij}}={\phi }_{12}+{\phi }_{23}+{\phi }_{34}+{\phi }_{41}$, here for a particular plaquette $\mathcal{C}$ in a two-dimensional quadratic lattice.

Figure 4

Time evolution of a single-photon Fock state in the presence of a coupler with phase $\phi =\pi /6$. The quantum state at the initial time $t=0$ is a Fock state with one photon in resonator $j=1$, and both resonators 2 and 3 in the vacuum state. The photon occupation probabilities $P_j$ are plotted as a function of time and show how the photon is transferred around the loop in a direction specified by the sign in $\phi =\pm \pi /6$. The evolution is periodic with period $\tau =2\pi /\sqrt{3}\kappa$ and the initial state is transferred into a Fock state of resonators 2 and 3 at times $t=\tau /3$ and $t=2\tau /3$, respectively.

Figure 5

Circulator behavior. The plot shows the normalized outgoing power $|b_j^{\mathrm{out}}/b_1^{\mathrm{in}}|{}^2$ for the three ports $j=1,2,3$ under coherent driving of port 1 with frequency ${\omega }_d$ when the phase $\phi$ of the coupler element is adjusted to $\pi /6$. For drive frequencies close to the resonator frequency, the signal is transferred from port 1 to port 2. The bandwidth of this circulator behavior is set by the photon hopping rates ($\kappa ={\kappa }^{\prime }/2=0.1{\omega }_r$).

Figure 6

(a) Using three-resonator junctions, one obtains a photon lattice with uniform hopping, and the resonators (depicted as rectangles) form a regular honeycomb pattern. (b) The corresponding photon lattice is the Kagome lattice, a hexagonal Bravais lattice (primitive vectors ${\mathit{\Delta }}_1$, ${\mathit{\Delta }}_2$) with three atoms $A$, $B$, and $C$ in the primitive unit cell (parallelogram shaded in gray). Adding coupler circuits in the junctions breaks time-reversal symmetry and introduces a phase factor $e^{\pm i\phi }$ in the photon hopping elements, where the sign depends on whether photons are transferred with or against the sense of rotation (circular arrows).

Figure 7

Band structure of the Kagome lattice with complex hopping elements $t=\left|t\right|e^{i\phi }$ for (a) $\phi =\pi /6$, (b) $\phi =\pi /4$, and (c) $\phi =\pi /3$. In the top panels, the dispersion $({\epsilon }_s-\omega )$ of the three bands $s=1,2,3$ is plotted in units of $\left|t\right|$. The first Brillouin zone corresponds to the hexagon centered at $\mathbf{k}=0$. The bottom panels show cuts of the dispersion along axes of high symmetry (see inset). For phases $\phi \in \frac{\pi }{6}\mathbb{Z}$, the band structure exhibits flat bands. The position of the flat band can be switched from (a) middle to (c) top to bottom [for $\phi =0$, obtained from panel (c) by reflecting all bands at $(\epsilon -\omega )=0$] by varying the phase $\phi$.

Figure 8

(a) Transmission-line resonator attached through capacitors $C_L$ and $C_R$ at the left and right ends to arbitrary circuits. (b) Dissection of the transmission line (capacitance and inductance per unit length denoted by $c$ and $\ell$) into a series of LC circuits. The generalized flux variables adjacent to the resonator are given by ${\varphi }_L$ and ${\varphi }_R$.

Figure 9

(a) Array consisting of transmission-line resonators and coupling circuits in the junctions between resonators. (b) The coupling circuits, attached to the resonators by capacitors $C_c$, are Josephson rings. They consist of a superconducting ring interrupted by three identical Josephson junctions with Josephson energy $E_J$ and junction capacitance $C_J$. By applying an external magnetic field perpendicular to the plane, the loops may additionally be threaded by a magnetic flux $\Phi$.

Figure 10

Dependence of the ground-state charge number $N_0$ on external magnetic flux $\Phi$ and offset charges, here for the uniform case $n_{g1}=n_{g2}=n_{g3}\equiv n_g$. As expected, $N_0$ takes on only integer values corresponding to the total number of extra Cooper pairs located on the Josephson ring. The integer-step boundaries between regions of different $N_0$ in general acquire a small finite width due to the residual coupling to the environment that allows charge relaxation. Parameters chosen for this plot are $E_J/h=10$ GHz, $C_J=0.7$ fF, and $C_c=5$ fF, yielding $E_J/E_{\Sigma }~2$.

Figure 11

Numerical results for a junction of three resonators attached to a central Josephson ring coupler. The device is tunable by varying the magnetic flux $\Phi$ [see (color) gray scale], and by changing the global offset charge $n_g$ as set by a constant electric field (see $x$ axes). (a) The lowest transition frequency ${\omega }_{01}/2\pi$ of the Josephson ring device in comparison with the resonator frequency ${\omega }_r/2\pi =7\hspace{0.5em}\mathrm{GHz}$. As one can check, the dispersive limit is maintained for the selected values of magnetic flux $\Phi$. (b) Resulting magnitude of photon hopping strengths $\left|t\right|$. The nonmonotonic behavior is explained by the crossing of the ${\omega }_{01}$ transition and the resonator frequency around $\Phi /{\Phi }_0~0.3$. (c) The corresponding results for the gauge-invariant phase sum $\sum \circlearrowleft \phi$, proving the breaking of time-reversal symmetry. As expected from general considerations, time-reversal invariance remains intact at zero offset charge and at zero magnetic flux. (Parameters are as in Fig. 10; in addition, $C_c=5$ fF, ${\omega }_r/2\pi =7$ GHz, and $\sqrt{\ell /c}=50\hspace{0.5em}\Omega$.)

Figure 12

(a) Josephson ring with attached voltage bias lines for canceling random offset charges. (b) Effective photon hopping strengths and gauge-invariant phase sums for random offset charges, with $n_{g\mu }\in [0,1]$ with uniform probability distribution. Data points are placed such that their $x$ positions correspond to the gauge-invariant phase sums $\sum \circlearrowleft \phi$ (modulo $2\pi$), and their $y$ positions display the arithmetic mean of the three photon hopping strengths $\left|t_{\mu }\right|$. For each data point, an “error” bar shows the spread from the minimum $\left|t_{\mu }\right|$ to the maximum. (Device parameters used are the same as in Fig. 11.)

Figure 13

Regions of fixed total charge $N=0,\pm 1$ in the charge regime, as a function of the three offset charges $n_{g1}$, $n_{g2}$, and $n_{g3}$. The shape of the region boundaries depends on the charging energy ratio $E_C/E_C^{\prime }$, chosen as (a) $5/4$, (b) $5/2$, and (c) $20$. Note that the coordinate axes are oriented differently in panel (c) to reveal the flatness of the boundaries for large $E_C/E_C^{\prime }$.