Cooperative Lamb Shift in a Mesoscopic Atomic Array

According to quantum electrodynamics, the exchange of virtual photons in a system of identical quantum emitters causes a shift of its energy levels. Such shifts, known as cooperative Lamb shifts, have been studied mostly in the near-field regime. However, the resonant electromagnetic interaction persists also at large distances, providing coherent coupling between distant atoms. Here, we report a direct spectroscopic observation of the cooperative Lamb shift of an optical electric-dipole transition in an array of ${\mathrm{Sr}}^{+}$ ions suspended in a Paul trap at inter-ion separations much larger than the resonance wavelength. By controlling the precise positions of the ions, we studied the far-field resonant coupling in chains of up to eight ions, extending to a length of $40\mu \mathrm{m}$. This method provides a novel tool for experimental exploration of cooperative emission phenomena in extended mesoscopic atomic arrays.

Figure 1

(a) Energy level diagram of a two-ion system. $|g⟩$ and $|e⟩$ denote electronic ground and excited states. ($\uparrow$) and ($\downarrow$) denote the ion spin projection on the trap axis. The states $\frac{1}{2}(|eg⟩+|ge⟩)(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)$ and $\frac{1}{2}(|eg⟩-|ge⟩)(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)$, which turn into themselves upon excitation transfer from one ion to the other $|e⟩\iff |g⟩$ and simultaneous spin exchange $\uparrow \iff \downarrow$, are shifted by $\delta$ given by Eq. (2). The states which acquire a minus sign upon such exchange, $\frac{1}{2}(|eg⟩\pm |ge⟩)(|\uparrow \downarrow ⟩\mp |\downarrow \uparrow ⟩)$, are shifted by $-\delta$. (b) System geometry. Two or more ${\mathrm{Sr}}^{+}$ ions are trapped in the center of a linear Paul trap (trap electrodes not shown). The ions form a chain along the trap axis $\widehat{z}$. A linearly polarized probe beam, propagating along the $\widehat{y}$ direction, illuminates the ions uniformly. The probe beam polarization can be rotated using a $\lambda /2$ plate. In our experiments, the beam was polarized either along the trap axis (red arrows), or orthogonal to it, in the $\widehat{x}$ direction (blue arrows). The light scattered by the ions was collected by the imaging system (IS) and detected by a photomultiplier tube (PMT).

Figure 2

Cooperative line shift in a system of two ${\mathrm{Sr}}^{+}$ ions. The data points show the measured relative frequency shift as a function of distance between the ions. The blue squares and red diamonds correspond to the probe beam polarization orthogonal and parallel to the trap axis, respectively. The light blue line shows the theoretical shift of $\frac{1}{2}\delta (r)$ with $\delta (r)$ given by Eq. (2). The width of the line reflects the uncertainty in the oscillator strength value. The error bars represent one standard deviation statistical uncertainty. The observed peak to peak spectral shift is approximately $2\times 10^{-3}$ of the line width.

Figure 3

Cooperative shift in a linear chain of several ions. The panels (a)–(f) correspond to chains of three to eight ions. In each panel, the cooperative shift is measured as a function of the distance $r$ between two adjacent ions closest to the middle of the chain. The light blue lines show the theoretical shift of Eq. (4) [30]. The error bars represent one standard deviation statistical error. Both the shift scale and the distance scale are the same in all panels. The grey lines, shown for comparison, represent the theoretical shift for equidistant chains. The width of theoretical lines represents the uncertainty in the oscillator strength value.