Selective acoustic control of photon-mediated qubit-qubit interactions

Quantum technologies such as quantum sensing, quantum imaging, quantum communications, and quantum computing rely on the ability to actively manipulate the quantum state of light and matter. Quantum emitters, such as color centers trapped in solids, are a useful platform for the realization of elementary building blocks (qubits) of quantum information systems. In particular, the modular nature of such solid-state devices opens up the possibility to connect them into quantum networks and create nonclassical states of light shared among many qubits. The function of a quantum network relies on efficient and controllable interactions among individual qubits. In this context, we present a scheme where optically active qubits of differing excitation energies are mutually coupled via a dispersive interaction with a shared mode of an optical cavity. This generally off-resonant interaction is prohibitive of direct exchange of information among the qubits. However, we propose a scheme in which by acoustically modulating the qubit excitation energies it is, in fact, possible to tune to resonance a preselected pair of qubits and thus open a communication channel between them. This method potentially enables fast (approximately nanoseconds) and parallelizable on-demand control of a large number of physical qubits. We develop an analytical and a numerical theoretical model demonstrating this principle and suggest feasible experimental scenarios to test the theoretical predictions.

Figure 1

Schematically shown are (a) a modular quantum architecture and (b) quantum multiplexing. (a) A quantum node composed of a set of qubits of potentially different excitation frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ (with ${\delta }_{ij}={\omega }_i-{\omega }_j$) is embedded in a quantum network. The method described in this work can, for example, selectively trigger interactions between selected qubit pairs. (b) Using qubits of different optical frequencies enables spectral multiplexing of an optically active quantum node.

Figure 2

The principle of acoustically induced qubit-qubit interaction. (a) A set of two (or three) optically active qubits (such as diamond impurities) is placed into a photonic cavity. All of the emitters interact with an optical mode of the cavity. The excitation energies of the emitters can be modulated by acoustically driving the qubit hosting medium. (b) Schematic diagram of energy levels involved in the model of two qubits coupled to a common cavity mode. Both qubits are described as two-level systems of respective excitation frequencies ${\omega }_1$ and ${\omega }_2$ that are not equal, and both are detuned from the cavity resonance frequency ${\omega }_{\mathrm{c}}$ by $\approx \mathrm{\Delta }$. The qubit-cavity coupling is characterized by a coupling rate $g$ equal for both qubits. (c) Upon effective elimination of the cavity mode the two qubits are nonresonantly coupled via an effective rate ${\mathcal{J}}_{12}$ if the acoustic drive is turned off. (d) The acoustic drive of frequency $M$ induces phonon side bands (dressed states) around the bare qubit frequencies which can allow for sideband-mediated resonant qubit-qubit interaction.

Figure 3

Analytical results characterizing the mechanism of the acoustically induced resonant qubit-qubit interaction. (a) Relative amplitudes of phonon sidebands of a frequency-modulated qubit. (b)–(e) Effective qubit-qubit coupling $D_{12}^{\mathcal{N}}$ (normalized to ${\mathcal{J}}_{12}$) as a function of $\mathcal{D}/M$ (assuming ${\mathcal{D}}_1={\mathcal{D}}_2=\mathcal{D}$) and relative phase between the drive of qubit 1 and qubit 2 ($\mathrm{\Delta }\varphi ={\varphi }_1-{\varphi }_2$) when $\mathcal{N}\times M={\delta }_{12}$ for (b) $\mathcal{N}=1$, (c) $\mathcal{N}=2$, and (d) $\mathcal{N}=3$. In (e) we show the cuts along the white dashed lines in (a)–(c), labeled, respectively, as (i), (ii), and (iii).

Figure 4

Time evolution of populations of two distinct quantum emitters (qubits) interacting dispersively with a photonic cavity. (a) and (b) Populations of the respective emitters as a function of time and frequency of the pumping acoustic wave. We assume that the two emitters are modulated with opposite phase (${\varphi }_1={\varphi }_2+\pi$) and that the amplitude of the modulation is equal for both emitters ${\mathcal{D}}_1/M={\mathcal{D}}_2/M\approx 0.92$, which maximizes the value of $J_1(2{\mathcal{D}}_{1\left(2\right)}/M)$ and hence leads to an efficient qubit-qubit coupling. In (c)–(e) we plot cuts along the dashed lines shown in (a) and (b) and show how the populations (orange solid line for qubit 1 and blue dashed line for qubit 2) evolve if (a) $2M={\delta }_{12}$, (b) $M={\delta }_{12}$, and (c) $M=\frac{3}{2}{\delta }_{12}$. If an integer multiple of the modulation frequency $M$ is tuned to ${\delta }_{12}$, the two emitters are resonantly coupled, and an exchange of the emitters' populations occurs (the system is in the on state), which can be observed as a Rabi oscillation in (c) and (d). If $M$ is not compatible with ${\delta }_{12}$, no exchange of populations occurs (the system is in the off state), as shown in (e).

Figure 5

Selective qubit-qubit interaction demonstrated on a system of three qubits interacting with a single cavity mode. Populations of (a) qubit 1, (b) qubit 2, and (c) qubit 3 as a function of time and the drive frequency $M$. The drive amplitude is adjusted such that ${\mathcal{D}}_i/M=\mathcal{D}/M\approx 0.92$, and the phases of the drive are chosen to be ${\varphi }_2={\varphi }_3={\varphi }_1+\pi$ (i.e., qubit 1 interacts efficiently with qubit 2 and qubit 3, but qubits 2 and 3 do not mutually interact). Qubit 1 is initially in its excited state, and the remaining parts of the system are in their ground state. If the integer multiple of the drive frequency matches the difference of the excitation frequencies between qubit $i$ and qubit $j$, resonant interaction between these qubits is activated. This interaction selectively affects only the pair of qubits $i$ and $j$ if the difference ${\delta }_{ij}$ of the excitation frequencies is not equal to an integer multiple of ${\delta }_{ik}$ or ${\delta }_{jk}$. In (d) and (e) we show cuts along the lines drawn in (a) and (b) using the solid orange line for the population of qubit 1, the yellow dashed line for qubit 2, and the blue dotted line for qubit 3. (d) Rabi oscillations appear between qubits 1 and 2 for $M={\delta }_{12}$, while qubit 3 is effectively decoupled from the dynamics. The reverse situation is shown in (e) for $M={\delta }_{13}$.

Figure 6

Breakdown of the selectivity of the acoustically driven qubit-qubit interaction. Populations of qubits (a) and (d) 1, (b) and (e) 2, and (c) and (f) 3 as a function of time and frequency $M$ of the acoustic drive. We consider three coupled qubits of frequencies ${\delta }_1=2\pi \times 0$ Hz, ${\delta }_2=2\pi \times 6$ GHz, and (a)–(c) ${\delta }_3=2\pi \times 6.4$ GHz and (d)–(f) ${\delta }_3=2\pi \times 6.1$ GHz. The qubits are driven with an equal amplitude ${\mathcal{D}}_i/M=\mathcal{D}/M=0.92$ but a different phase ${\varphi }_2={\varphi }_3={\varphi }_1+\pi$. The frequency differences ${\delta }_{21}$ and ${\delta }_{31}$ are marked by the white dashed lines. We observe that the coherent exchange of population between qubit 1 and qubit 2 (qubit 3) is selective even for small differences between the target qubit frequencies as small as ${\delta }_{32}=2\pi \times 400$ MHz in (a)–(c); however, for ${\delta }_{32}=2\pi \times 100$ MHz we see that despite the dominantly selective population transfer a considerable population is transferred to the off-resonant qubit in (d)–(f). All the remaining parameters are given in Sec. 3.

Figure 7

Evaluation of cross talk. We consider three qubits (as described in Sec. 3) and apply a resonant acoustic drive selectively enabling the coherent interaction between qubit 1 and qubit 2 (${\delta }_2=M=2\pi \times 8$ GHz). To evaluate cross talk we calculate the population of qubit 3 as a function of its detuning from qubit 2. The results are shown using (a) linear and (b) logarithmic scales.

Figure 8

Influence of qubit intrinsic damping and pure dephasing on the acoustically induced resonant interaction. The populations of qubit 1 and the corresponding populations of qubit 2 are plotted in (a), (c), and (e) and (b), (d), and (f), respectively, as a function of time and the frequency $M$ of the acoustic drive. The qubits are driven with an amplitude of ${\mathcal{D}}_i/M=\mathcal{D}/M=0.92$ and opposite relative phase ${\varphi }_1={\varphi }_2+\pi$. The respective values of damping and dephasing are given in the inset and are identical for (a) and (b), (c) and (d), and (e) and (f), respectively. All the remaining parameters are given in Sec. 3.