Designing frequency-dependent relaxation rates and Lamb shifts for a giant artificial atom

In traditional quantum optics, where the interaction between atoms and light at optical frequencies is studied, the atoms can be approximated as pointlike when compared to the wavelength of light. So far, this relation has also been true for artificial atoms made out of superconducting circuits or quantum dots, interacting with microwave radiation. However, recent and ongoing experiments using surface acoustic waves show that a single artificial atom can be coupled to a bosonic field at several points wavelengths apart. Here, we theoretically study this type of system. We find that the multiple coupling points give rise to a frequency dependence in the coupling strength between the atom and its environment and also in the Lamb shift of the atom. The frequency dependence is given by the discrete Fourier transform of the coupling-point coordinates and can therefore be designed. We discuss a number of possible applications for this phenomenon, including tunable coupling, single-atom lasing, and other effects that can be achieved by designing the relative coupling strengths of different transitions in a multilevel atom.

Figure 1

A sketch of the system under consideration. A multilevel atom with energy levels $|0\rangle ,|1\rangle ,|2\rangle ,...$ couples at the points $x_1,...,x_N$ to a bosonic field with right- and left-traveling modes. The distance between the coupling points can, for example, be on the order of wavelengths $\lambda =2\pi v/{\omega }_{1,0}$, where ${\omega }_{1,0}$ is the first transition frequency of the atom and $v$ is the velocity of the bosonic modes.

Figure 2

The frequency dependence of the relaxation rate $\Gamma$ and the main contribution to the Lamb shift $\Delta$ for $N=3$ [blue (dark gray) lines; solid for $\Gamma$, dashed for $\Delta$] and $N=10$ [red (light gray) lines; dash-dotted for $\Gamma$, dotted for $\Delta$] in the symmetric case. Note that ${\omega }_{1,0}(x_2-x_1)/2\pi v$ corresponds to $\varphi /2\pi$. Everything has been normalized to the maximum coupling strength for each N. We have set $J\left(\omega \right)$ constant for simplicity. It is usually a function varying slowly with $\omega$; in the Ohmic case $J\left(\omega \right)\propto \omega$.

Figure 3

Designed relaxation rate frequency dependencies. The solid black line shows two maxima of equal magnitude (parameters: $g_k=\{1,1,1,1\}$, $x_k=\{0,1,1.5,3\}{x}_2$), the dotted blue line has a wide, flat maximum (parameters: $g_k=\{1,3,3,1\}$, $x_k=\{0,1,2,3.5\}{x}_2$), and the dashed red line has two wide, shallow minima (parameters: $g_k=\{1,4,4,1\}$, $x_k=\{0,1,2,3\}{x}_2$).

Figure 4

A scheme for population inversion. The relaxation rates ${\Gamma }_{1,0}$ (solid blue line) and ${\Gamma }_{2,1}$ (dashed red line) for the first two atom transitions, plotted as a function of the first transition frequency ${\omega }_{1,0}$ for $N=10$ in the maximally symmetric case. By choosing the anharmonicity to be $-0.1\times 2\pi v/(x_2-x_1)$, we can make the global maximum of ${\Gamma }_{2,1}$ coincide with a minimum for ${\Gamma }_{1,0}$. Inset: Energy-level diagram showing the relevant driving and relaxation rates for population inversion.

Figure 5

Enhancing multiphoton relaxation rates. We plot the relaxation rate as a function of frequency for the maximally symmetric case with $N=10$ and an anharmonicity $-0.2\times 2\pi v/(x_2-x_1)$. The $|1\rangle \to |0\rangle$ and $|2\rangle \to |1\rangle$ transitions can then be placed at relaxation rate minima, while the two-photon process at ${\omega }_{2,0}/2=({\omega }_{1,0}+{\omega }_{2,1})/2$ is at a maximum. Inset: Energy-level diagram showing the transition frequencies.

Figure 6

Varying the anharmonicity. The Lamb shifts of the first (blue dotted line) and second (red dashed line) transitions of the giant atom plotted together with their difference (black solid line), the resulting change in anharmonicity, for the maximally symmetric case with $N=10$ and an anharmonicity of $-0.1\times 2\pi v/(x_2-x_1)$.

Figure 7

An example of an experimental implementation of our system, using a transmon coupled to SAWs. Adapted from [13], with thanks to M. V. Gustafsson, T. Aref, and M. K. Ekström for providing the images. (a) The lower blue part in the center of the image is the two transmon islands. SAWs propagate from left to right in the gap between the grounded yellow areas at the edges of the image. The upper blue part is an electrical gate, enabling rf excitation of the transmon. (b) Close-up of the transmon islands. The green part in the center of the image is the SQUID connecting the islands. (c) Close-up of the individual fingers of the transmon capacitance. The distance between neighboring fingers (connection points) is on the order of the SAW wavelength. The double-finger structure used here reduces mechanical reflections.

Figure 8

A sketch of a possible implementation using an xmon coupled to a meandering transmission line. The distance between coupling points can be set with great precision by choosing the transmission line length, and the capacitive coupling at each connection point can be tuned by designing the tips of the fingers of the xmon island.