# Quantum control of motion

By Akash Dixit

Title: Quantum state preparation, tomography, and entanglement of mechanical oscillators

Authors: E. Alex Wollack, Agnetta Y. Cleland, Rachel G. Gruenke, Zhaoyou
Wang, Patricio Arrangoiz-Arriola, and Amir H. Safavi-Naeini

Institution: Department of Applied Physics and Ginzton Laboratory, Stanford University 348 Via Pueblo Mall, Stanford, California 94305, USA

Manuscript: Published in Nature [1], Open Access on arXiv

Introduction
The field of quantum information sciences contains a multitude of different technologies, including atoms, spins, and defect centers in diamond. This work focuses on two emerging technologies: superconducting circuits and mechanical oscillators. Each system has its advantages, but it is not obvious that any one is the best platform for building a quantum computer, developing quantum sensors, or facilitating quantum communication. To achieve these goals, it is necessary to develop hybrid quantum systems that can utilize the strengths of various quantum technologies.

In this work, the authors demonstrate the ability to couple superconducting qubits to mechanical motion. This establishes the building blocks for a hybrid quantum system that can take advantage of the the best of both systems. The qubit is customizable and easy to communicate with, making it ideal for state initialization and characterization. The mechanical modes are fabricated with small spatial footprints and have long lifetimes, making it possible to scale to larger systems and hold quantum information for long timescales. I will describe how the authors use the coupling between these two systems to both prepare and measure states of mechanical motion using the qubit. I first describe the carefully engineered device that couples one qubit to two mechanical oscillators. Then I discuss the two modes of operation, where the qubit is used to both prepare states of mechanical motion and measure the quantum state of the mechanical mode. Finally, I show how the authors use the qubit as an intermediary to prepare entangled mechanical states across two oscillators.

Device

The device used in this works consists of two mechanical oscillators and a superconducting qubit. The mechanical oscillators are fabricated in thin film lithium niobate (LiNiO3). These oscillators are formed by embedding a defect in a periodic structure of the material, called a phononic crystal. The defect is a mismatch in the periodicity of the structure and confines mechanical motion, preventing acoustic radiation and enabling long mechanical lifetimes. Like electromagnetic radiation, mechanical motion can be quantized. The individual quanta of mechanical motion are called phonons, and the mechanical oscillator can be characterized as a harmonic oscillator with equal energy level spacing. The qubit is made by fabricating an $LC$ oscillator with superconducting materials. The key element of this circuit is a Josephson junction, which is made of aluminum oxide sandwiched between layers of superconducting aluminum. The junction acts as a nonlinear inductor that modifies the energy level spacing of the $LC$ oscillator. The energy levels of the usual $LC$ oscillator (which is a harmonic oscillator) are equally spaced, meaning the transition energy between any two levels is the same. However, with the nonlinear inductor in the circuit, there are no longer equally spaced energy levels, making it possible to uniquely address the two lowest energy levels of the system, ground ($\left| g \right\rangle$) and excited ($\left| e \right\rangle$). The two levels form a quantum bit (qubit). The qubit is designed to be tunable in frequency by placing two Josephson junctions in a parallel with each other. By applying a magnetic field using a wire carrying current, a magnetic flux is threaded through the loop to change the qubit frequency.

The qubit and mechanical oscillators are fabricated on separate chips that are placed $\sim \mu m$ apart. To couple the qubit and mechanical oscillators, the authors use the piezoelectricy of the lithium niobate film. The mechanical motion of this material produces an accumulation of electric charges onto aluminum pads located on both chips, which are designed to be the capacitive element of the qubit. The qubit capacitor is charged by the motion of the mechanical oscillators, ensuring that the two systems are linked together.

Initializing a mechanical state
The authors design the qubit to interact in two different ways with the mechanical oscillators. In the first mode, the qubit is tuned to be on resonance with a particular mechanical oscillator ($\omega_q = \omega_1, \omega_2$). Note that the mechanical frequencies of the two oscillators are different, so the qubit can only be in resonance with one at a time. This allows for the direct exchange of energy between qubit and either oscillator at a rate related to the capacitive coupling between the two, $g_1 = 2 \pi \times$ 9.5 MHz, $g_2 = 2 \pi \times$ 10.5 MHz. The Hamiltonian that describes the interaction between a qubit and mechanical oscillator on resonance the Jaynes-Cummings interaction:

$\mathcal{H}_{\mathrm{on}} = g(a^{\dagger} \sigma^{-} + a \sigma^{+})$
[Equation 1].

$a^{\dagger}, a$ and $\sigma^{+}, \sigma^{-}$ are the creation, annihilation operators for the mechanical oscillator and qubit respectively. When on resonance, the qubit and mechanical oscillator swap their respective states in time $\pi/g \sim$ 24-26 $ns$ depending on the particular oscillator.

This swap can be used as a method of mechanical state preparation. The authors first tune the qubit so that it is off resonant from either mechanical oscillator. Then with the mechanical mode containing no quanta, the qubit is initialized so the joint states are $\left| 0,g \right\rangle, \left| 0,e \right\rangle$, or $\left| 0,g \right\rangle + \left| 0,e \right\rangle$ state. The joint state $\left| m, q \right\rangle$, describe the phonon number of a particular mechanical oscillator, $m = 0, 1, 2$…, and whether the qubit is in the ground or excited state, $q = g, e$. The qubit frequency is tuned to be on resonance with either mechanical mode for a time corresponding to a full swap. When the swap operation is applied to the joint state $\left| 0,g \right\rangle$, the system remains unchanged since both subsystems are in their ground state and there is no energy to exchange. Under the swap, the state $\left| 0,e \right\rangle$ becomes $\left| 1,g \right\rangle$ as shown in Figure 1. When the qubit is initialized in a superposition state, the joint state is $\left| 0,g \right\rangle + \left| 0,e \right\rangle$. The swap operation acts on both parts of this superposition leading to the final state $\left| 0,g \right\rangle + \left| 1,g \right\rangle$. The mechanical oscillator is now in a superposition state, but the state of the mechanical oscillator is not entangled with the qubit state.

Measuring a mechanical state
In the second mode of operation, the qubit is off resonance from either mechanical oscillator, usually called a dispersive interaction. The dispersive interaction rate between qubit and mechanical oscillator, $\chi$, is now set by the direct capactive coupling, $g$, the detuning between qubit and mechanics, $\Delta$, and other qubit parameters. In the limit that the detuning between qubit and mechanics is larger than the the capacitive interaction rate ($\Delta \gg g$), the interaction shown in Equation 1 is approximated by the off resonant Hamilitonian:

$\mathcal{H}_{\mathrm{off}} = \chi a^{\dagger} a \sigma_z$
[Equation 2].

The combination $a^{\dagger}a$ is the operator version of the number of phonons, $m$, in the mechanical oscillator. $\sigma_z$ is the operator version of the qubit state, either $\left| g \right\rangle$ or $\left| e \right\rangle$.

Without the interaction between the qubit and mechanics, the Hamiltonian of the just the qubit would look like $\mathcal{H}_{q} = \omega_q \sigma_z$, where $\omega_q$ is the transition frequency of the qubit. When we add in the off resonant interaction, the Hamiltonian can be expressed as $\mathcal{H}_{q} + \mathcal{H}_{\mathrm{off}} = (\omega_q - \chi a^{\dagger}a )\sigma_z$. By comparing the combined Hamiltonian with the one of just the qubit, we see that the effect of the interaction is to modify the transition frequency of the qubit (represented by everything before the $\sigma_z$). So now, the qubit transition frequency is dependent on the number of phonons in the mechanical oscillator ($m = a^{\dagger}a$). For every additional phonon in the mechanical oscillator, the qubit transition frequency shifts by $\chi$.

This interaction is crucial to being able to characterize the state of the mechanical oscillator. Since the different phonon numbers impart a different frequency shift on the qubit, the mechanical state is imprinted on the frequency of the qubit. To resolve the probabilities of different phonon numbers in the mechanical oscillator, a qubit interferometry measurement is performed. The mechanical oscillator is prepared in a Fock state with 0 or 1 phonons or in a superposition of many phonon 0, 1, 2,… Then the qubit is placed in a superposition state $\left| g \right\rangle + \left| e \right\rangle$ and allowed to precess for a variable time, $t$. During this time, the superposition state accumulates a phase at rate $\chi$ if there is one phonon, $2\chi$ for two phonons, and so on. The phase accumulated then reflects the probability ($A_n$) that the mechanical oscillator contained zero, one, two, etc… phonons. The qubit state evolves to $\left| g \right\rangle + e^{i\phi} \left| e \right\rangle$, where the phase accumulated is $\phi = \sum_n A_n n \chi t$. The authors rotate the qubit back into its measurement basis and monitor the final population of the qubit excited state as a function of the interaction time, $t$, and fit the trajectory to the functional form

$S(t) = \sum_n A_n e^{-\kappa t/2} \cos [(2 n \chi t) + \phi_n]$
[Equation 3]

This form includes the phonon number probabilities, $A_n$, as well as the number dependent precession rate, $n \chi$. It also includes a number dependent phase offset, $\phi_n$, and the phonon decay constant, $\kappa$. This captures the dynamics of the qubit trajectory even when the phonon probabilities are changing due to energy decay. The figure below shows an interferometry trace and the fit used to extract the phonon population in the mechanical oscillator. The trace contains a combination of various frequency oscillations each corresponding to a different phonon number. The weight of a particular frequency in the combination represents the probability of the corresponding phonon number to be present in the mechanical state being measured.

Entangling two mechanical oscillators
With the ability to control and measure the state of each mechanical oscillator, the next step is to prepare a joint state where the motion of the two oscillators is entangled together. We write the joint state of the qubit and two mechanical oscillators as $\left| m_1, q, m_2 \right\rangle$, where the mechanical oscillators can contain $m_1, m_2=0,1,2,..$ phonons, and the qubit can be in either the ground ($g$) or excited ($e$) state. First the qubit is prepared in its excited state with $\left| 0,e,0 \right\rangle$. A half swap between the qubit and the first mechanical oscillator entangles the two, $\left| 1, g, 0 \right\rangle + \left| 0, e, 0 \right\rangle$. This is accomplished by bringing the qubit into resonance with the mechanical oscillator for only half the time required the perform a full swap as seen in Figure 2. Finally, the qubit state is fully swapped with the second mechanical state resulting in the state $\left| 1, g, 0 \right\rangle + \left| 0, g, 1 \right\rangle$. This leaves the qubit in the ground state with the two mechanical state fully entangled together $(\left| 1,0 \right\rangle + \left| 0,1 \right\rangle) \bigotimes \left| g \right\rangle$.

Future outlook
The authors construct a device that couples mechanical motion to a superconducting qubit. The qubit is used to prepare and measure the modes of individual mechanical modes. The authors present a protocol that prepares two mechanical modes, both coupled to the same qubit, in an entangled state. This work demonstrates the building blocks needed to construct a hybrid quantum system by combining two disparate quantum systems. The authors match the precise control of a superconducting qubit with the long lifetimes of the mechanical modes to construct a devices that engages the strengths of both systems. This kind of design will enable future advances in quantum computing, sensing, and communication by drawing from many different technologies.

References

[1] Wollack, E.A., Cleland, A.Y., Gruenke, R.G. et al. Quantum state preparation and tomography of entangled mechanical resonators. Nature 604, 463–467 (2022).

Akash Dixit builds superconducting qubits and couples them to 3D cavities to develop novel quantum architectures and search for dark matter.

Thanks to Joe Kitzman for great discussions and feedback in editing this article.