Robust Macroscopic Schrödinger’s Cat on a Nucleus

A Schrödinger cat state [1, 2, 3] is an equal superposition of two distinct “classical” states of a system  [4, 5, 6], that is a probe for fundamental aspects of quantum physics, including the measurement problem [7], macro-realism [8, 9], quantum-classical transition [10], and non-locality [11]. Cat states also serve as important resources for quantum error correction [12, 13, 14, 15], quantum sensing [16, 17, 18], and quantum communication [19, 20]. Thus, practical realization of cat states is of interest for both foundational and technological applications.

In continuous-variable systems, cat states are generated using nonlinearity induced by light-matter interactions in the Jaynes-Cummings model [21, 22], e.g., a harmonic oscillator coupled to a superconducting qubit [23, 24, 25] or an atom in an optical cavity [26]. However, errors occurring in the qubit can propagate to the cavity, corrupting such nonclassical states [25]. Alternatively, Greenberger-Horne-Zeilinger (GHZ) states, which are spin cat states (SCSs) for permutationally symmetric systems, can be realized by entangling multiple qubits [27, 28, 29, 30], but such SCSs are fragile—with their decoherence rate proportional to the square of the number of qubits [31].

We propose a new platform to generate cat states deterministically, on-demand and without post selection using a nucleus, which is controlled without entanglement with an ancilla, hence, is robust against error propagation, unlike continuous-variable systems [12, 23, 24, 25, 26]. Here, we consider SCSs (implying macroscopic to differentiate them from “kitten” states [32]) on a nuclear spin, e.g., shown in Fig. 1, but, more generally, a superposition of any two maximally separated quasi-classical states, which are minimum uncertainty states that follow a classical evolution under a linear drive [5, 6]. We are motivated by recent experiments on high-spin nuclei (spin $I>1$) in a solid-state device [33, 34] accompanied by an enhanced electric field-driven quadrupole interaction, two orders of magnitude faster than the dephasing timescale, that could achieve ultrafast non-linear dynamics.

A highly coherent nuclear spin could open the possibility of robust generation and long-lived storage of SCSs. So far, quadrupole interaction has only been utilized for realizing spin squeezed states (SQSs), that too, on an ensemble of nuclear spins [35, 36]. However, individual addressing of high-spin systems is important for applications like quantum information processing and error correction [15]. We present two key results: i) a scheme to go beyond squeezed states towards generating SCSs and ii) an experimentally feasible control protocol for storing and manipulating SCSs on an individual nucleus.

We consider a high-spin donor (Group-V element with a high-spin nucleus) implanted in a nanoelectronic device based on enriched ²⁸Si (or any other spinless Group-IV atoms), as described in [33]. The chip is affixed to a dilution refrigerator maintained at a low temperature ($T\approx 0.01$ K) and placed inside a superconducting coil that produces a static magnetic field $\bm{B}_{0}\approx 1$ T. As shown in Fig. 2(a), the chip includes electrostatic gates to control the electrochemical potential of the donor, a microwave antenna to supply electromagnetic pulses, and a single electron transistor (SET) for reading out the donor spin state [37].

An implanted Group-V donor contains five electrons in its outermost shell, four of which form bonds with neighboring Si atoms and the remaining electron can be removed by ionizing the donor using electrostatic gates. The quantum state of the nuclear spin is initialized using flip-flop drive [38], where an electron and nuclear spin pair is flipped together in opposite directions. The donor atom is subject to a static potential, and a target nuclear-spin eigenstate can be achieved by applying a combination of flip-flop pulses and electron spin-resonance pulses that move population from higher and lower nuclear spin states towards a target state [34]. Then, the electrostatic potential is increased to ionize the donor atom by removing the electron and only the nuclear spin is manipulated using RF pulses. [39]. The readout process using electron spin resonance can be made dependent on the nuclear spin state, thus achieving nuclear readout via spin dependent tunneling of the electron [40, 33].

The differential thermal expansion between the silicon substrate and the electrostatic gates induce lattice strain in the chip [33], thereby distorting the bonds between the donor atom and neighboring Si atoms, as shown in Fig. 2(b). The electric-field gradient (EFG) generated from distortion of the bonds results in a quadrupole interaction of the high-spin nucleus, which has a non-zero quadrupole moment due to its asymmetric nuclear charge distribution. For a spatially varying potential $V(x,y,z)$, the EFG tensor comprises elements $V_{\alpha\beta}:=\frac{\partial^{2}V(x,y,z)}{\partial\alpha\partial\beta},\,\alpha,\beta\in\{x,y,z\},$ which is real, traceless and symmetric. Thus, $V_{\alpha\beta}$ can be diagonalized in a principal axis system $\{x^{\prime}$, $y^{\prime}$, $z^{\prime}\}$, such that $|\mathcal{V}_{z^{\prime}z^{\prime}}|\geq|\mathcal{V}_{y^{\prime}y^{\prime}}|\geq|\mathcal{V}_{x^{\prime}x^{\prime}}|$, and the EFG is characterized by the asymmetry parameter $\eta:=\frac{\mathcal{V}_{x^{\prime}x^{\prime}}-\mathcal{V}_{y^{\prime}y^{\prime}}}{\mathcal{V}_{z^{\prime}z^{\prime}}}$ for $0\leq\eta\leq 1$. For a nucleus with electric quadrupole moment $q_{\text{n}}$, the quadrupole interaction is

where $\omega_{\text{q}}$ is the quadrupole coupling strength, e the elementary charge, $\hbar$ the reduced Planck’s constant, the nuclear spin operator vector $\hat{\bm{I}}:=[\hat{I}_{x^{\prime}},\hat{I}_{y^{\prime}},\hat{I}_{z^{\prime}}]$ and invariant $\mathbb{I}^{2}=\bm{I}\cdot\bm{I}$.

In addition to the quadrupole interaction, we manipulate the nuclear spin state using a static magnetic dipole interaction due to $\bm{B}_{0}$ and time dependent perturbation driven by magnetic field $\bm{B}_{1}$ with time-dependence $\epsilon(t)$ ($|\epsilon(t)|\leq 1$). For a nucleus with gyromagnetic ratio $\gamma$, the nuclear spin dynamics are described by

The $2I+1$ degenerate levels of a nuclear spin undergo Zeeman splitting due to $\gamma\bm{\hat{I}}\cdot\bm{B}_{0}$ which creates an equal spacing between the energy levels. The electric quadrupole interaction $\hat{H}_{\text{q}}$ causes an unequal shift of the levels such that the transition frequency $\omega_{i,i-1}$ between two consecutive levels $i$ and $i-1$, for $i\in[I,-I+1]$, is unique, as shown in Fig. 2(c). We represent spin states as $\ket{I,m_{I}}$, where $m_{I}\in[-I,I]$, and RF pulses drive transitions which follow the selection rule $\Delta m_{I}=\pm 1$. For numerical simulations [41], we use the example of a spin-$\nicefrac{{7}}{{2}}$ ¹²³Sb donor [33, 34], with Zeeman splitting $\gamma B_{0}=2\pi\times 8.25$ MHz, $\omega_{\text{q}}=2\pi\times 40$kHz and perturbation strength $\gamma B_{1}=2\pi\times 800$ Hz.

Quadrupole interaction is equivalent to one-axis twisting (OAT) due to $\hat{I}^{2}_{z^{\prime}}$ if $\eta=0$, two-axis counter twisting (TACT) due to $\hat{I}^{2}_{z^{\prime}}$ and $\hat{I}^{2}_{y^{\prime}}$ if $\eta=1$ [36, 42], and partial TACT otherwise. Generally, TACT is preferred over OAT as TACT generates faster squeezing in comparison to OAT and can help produce SQSs for metrology [43]. However, TACT cannot produce SCSs, and, in general, quadrupolar non-linear evolution with $\eta>0$ cannot go beyond SQSs on a nuclear spin [35].

We devise a way to convert TACT to OAT by adding a large linear term to the quadrupole interaction, induced here by the Zeeman coupling. The resulting dynamics depend on the orientation of the EFG and strength of magnetic field $B_{0}$. When the Zeeman splitting is much larger than quadrupole coupling, i.e. $\gamma\bm{B}_{0}\gg\omega_{\text{q}}$, the component perpendicular to $\bm{B}_{0}$ does not affect the nuclear spin because quadrupolar transitions ($\Delta m_{I}=\pm 2$) are prevented by the large energy difference. In this regime, only the component of the quadrupole interaction parallel to $\bm{B}_{0}$ generates non-linear rotation, which effectively results in OAT, regardless of the value of $\eta$. In contrast to TACT, our scheme allows generation of SCSs for any orientation of the quadrupolar axes of symmetry $\{x^{\prime},y^{\prime},z^{\prime}\}$ with respect to linear interaction along the $z$ axis. For numerical simulations in this work, we set $\bm{B}_{0}=B_{0}\hat{z}$, $\bm{B}_{1}=B_{1}\hat{y}$, $\eta=0$, and EFG to be symmetric about $z$ axis, i.e. $\hat{z}^{\prime}=\hat{z}$, but, our results can be generalized to other configurations.

Despite the femtometer scale of a nucleus, nuclear-SCSs are macroscopic with their size, quantified by the relative quantum Fisher information [44, 45], scaling linearly with $I$. The size or degree of superposition $N_{\text{eff}}^{\text{r}}$ of a nuclear-spin state $\psi$ for measurement operator $\hat{O}$ is

i.e., proportional to the variance of the observable. For an appropriate $\hat{O}$, the variance is $I^{2}$ for bimodal SCSs and we calculate the size of a nuclear SCS to be $2I$, which denotes the maximum quantum Fisher information of the system and is equivalent to a GHZ state [46] made of $N=2I$ entangled qubits [47]. A nuclear SCS with $2I+1$ dimensions, e.g., a spin-$\nicefrac{{7}}{{2}}$ SCS with 8 dimensions is equivalent to a $2I=7$ qubit GHZ state with permutation symmetry, and not a $\log_{2}(2I+1)=3$ qubit state, and can have macroscopicity up to $2I=9$ for a ²⁰⁹Bi donor.

The degree of superposition $N_{\text{eff}}^{\text{r}}$ (Eq. (3)) increases and decreases periodically during OAT, as shown in Fig. 3. $N_{\text{eff}}^{\text{r}}=1$ for a spin coherent state on the $\hat{I}_{x}$ axis of the Bloch sphere at $t=0$, increases as the spin state squeezes, then reaches the flat-top of the “mesa” where it interferes with itself and finally increases again (“hill”) as two distinct components form resulting in a SCS at $t=\nicefrac{{\pi}}{{2\omega_{\text{q}}}}$. Under further evolution, the SCS returns to a coherent state at $t=\nicefrac{{\pi}}{{\omega_{\text{q}}}}$, which again squeezes to a SCS at $t=\nicefrac{{3\pi}}{{2\omega_{\text{q}}}}$, and so on. This periodic process of collapse-and-revival of the SCS has a time period of $\nicefrac{{\pi}}{{\omega_{\text{q}}}}$ and is a signature of coherence [47]. From the plots for different values of nuclear spin in Fig. 3, we note the macroscopicity of the SCSs through the linear scaling of their size $(N^{\text{r}}_{\text{eff}})_{\text{max}}=2I$.

External control of a high-spin nucleus is required for initialization, manipulation and detection of nuclear-SCSs. For initialization, the eigenstate $\ket{I,I}$, which is also a coherent state, should first be rotated to reside on the $\hat{I}_{x}$ axis of the Bloch sphere. Then, OAT results in a SCS on the $\hat{O}=\hat{I}_{y}$ axis for a half-integer nuclear spin, but, for donors in solid-state devices, nuclear-spin initialization and measurement is limited to the eigenstates of $\hat{I}_{z}$. Observing nuclear SCSs could be possible with angular momentum operations, which are trivial when $\gamma B_{1}\gg\omega_{\text{q}}$ [42, 48] or with a tunable non-linearity [49]. However, quantum control is needed as, experimentally, $\gamma B_{1}=2\pi\times 800$ Hz $\ll\omega_{\text{q}}=2\pi\times 40$ kHz such that a RF pulse can only selectively drive a particular nuclear spin transition leading to Givens rotations, and the quadrupole coupling cannot be turned on-and-off.

For applying global SU(2) rotations with an always-on non-linearity and low drive strength, we use a multi-tone (MT) pulse that is a superposition of $2I$ tones of equal amplitudes, corresponding to the $2I$ nearest neighbor sub-spaces with transition frequencies represented by frequency vector $\bm{\omega}:=\{\omega_{i,i-1}\}_{i=I}^{-I-1}$. Each tone has a global phase $\phi$ and a pulse envelope switched on at a time $t_{0}$ is described by the amplitude $\epsilon^{\text{MT}}(t;\bm{\omega},\phi,t_{0})=\frac{1}{2I}\sum_{\omega_{j}\in\bm{\omega}}\cos(\omega_{j}(t-t_{0})+\phi)$. Here, $\phi$ tunes the axis of rotation on the Bloch sphere while preserving the relative phase between different energy levels. The frequency $\Omega$ of SU(2) rotations is generalized from the Rabi frequency $\nicefrac{{\gamma B_{1}}}{{2}}$ for a two-level nuclear spin, to a high-spin nucleus by dividing the Rabi frequency by the number of tones $2I$, i.e. $\Omega=\nicefrac{{\gamma B_{1}}}{{4I}}$. Applying this pulse for time $\Delta t$ rotates the nuclear spin by angle $\Theta$ given by $\Theta=\Omega\Delta t$. In contrast to other schemes that employ numerical techniques with multi-tone control [50, 51], we derive exact expressions for phase-locked pulses, which can enable novel experimental applications such as full-state tomography and interferometry with a high-spin nucleus.

Figure 4(a) shows our Ramsey-like control sequence to realize SCSs, where a square-shaped MT RF pulse is applied to the initial state $\ket{I,I}$ between time $t_{0}=0$ and $t_{1}=t_{\nicefrac{{\pi}}{{2}}}$. Then the pulse is switched off for time $T$, and again applied between $t_{2}=T+t_{\nicefrac{{\pi}}{{2}}}$ and $t_{3}=T+2t_{\nicefrac{{\pi}}{{2}}}$, with the amplitude

for $\sqcap$ the top-hat function. Here, the global rotation of $\Theta=\nicefrac{{\pi}}{{2}}$ would take time $\Delta t_{\nicefrac{{\pi}}{{2}}}=\frac{\pi}{2}\cdot\frac{4I}{\gamma B_{1}}$ and setting the phase difference $\Delta\phi=\nicefrac{{\pi}}{{2}}$ between the first and the second global rotations accounts for the angle between a coherent state and the resulting SCS on the equator. The resulting dynamics are shown in Fig. 4(d), where we note that SCSs along any axis can be realized using global rotations (Fig. 4(d)(vi)). The quadrupole coupling cannot be turned off and these states squeeze during rotations. A SCS can be stored for long times by aligning it to the fixed points on the $z$ axis (Fig. 4(d)(vii)).

For a spin-$\nicefrac{{7}}{{2}}$ ¹²³Sb nucleus, Fig. 4(b) shows the change in $N_{\text{eff}}^{\text{r}}$, for $\hat{O}=\hat{I}_{z}$, as $T$ is varied, similar to Fig. 3. When $\Delta\phi=\nicefrac{{\pi}}{{2}}$, we observe a rapid variation of $N_{\text{eff}}^{\text{r}}$ with a frequency of $\nicefrac{{\gamma B_{0}}}{{\pi}}$, because nuclear-spin states generated by OAT undergo Larmor precession and when the axis of rotation corresponds with the axis of the SCS, the second global rotation is unable to move the state. To remedy this immobility, we assign a $T$-dependent shift to the phase difference: $\Delta\phi=\nicefrac{{\pi}}{{2}}+\omega T$ in (IV), which is equivalent to a rotating-frame transformation, and results in the periodic collapses-and-revivals of a SCS solely due to OAT, which would enable efficient experimental detection.

In this section, we show that cat-state generation is possible with just phase modulation of multi-tone pulses in the generalized rotating frame of reference. First, we note that, in the limit of large Zeeman interaction $\gamma B_{0}\gg\omega_{\text{q}}$, we can neglect non-commuting terms, such as $\hat{I}_{x^{\prime}}^{2}$ of the quadrupole interaction (1) so that the non-linear evolution is only described in terms of the strength of quadrupole tensor, $\omega_{\text{q}}^{\text{eff}}$, parallel to the magnetic field $\bm{B}_{0}$. Under this approximation and using the notation $[\ell,\ell^{\prime}]:=\{\ell,\ell+1,\dots,\ell^{\prime}\}$, the effective Hamiltonian is

where $\omega_{j}$, $\epsilon_{j}$, and $\phi_{j}$ are respectively the frequency, amplitude and phase of the $j^{\text{th}}$ component of the multi-tone control. The eigenstates $\ket{I,k}$ of the nuclear spin have energy

for $k\in[-I,I]$.

Now we transform the Hamiltonian to the generalized rotating frame, defined by the diagonal matrix [50]

and obtain

where

We note that global rotations are applied by setting $\epsilon_{j}=\nicefrac{{1}}{{2I}}$ and $\phi_{j}=\phi$; for example, using $\phi=0$ is equivalent to

and $\phi=\nicefrac{{\pi}}{{2}}$ is equivalent to $\hat{H}_{\text{rot}}=\nicefrac{{\gamma B_{1}\hat{I}_{y}}}{{2I}}$.

One-axis twisting can be realized by virtually updating the phases of a multi-tone pulse. To see this, we note that that the effective quadrupolar operator applied for duration $T$ generates the operation $e^{-iT\omega_{\text{q}}^{\text{eff}}\hat{I}_{z}^{2}}$, which results in a non-linear phase shift between the different eigenstates of a nuclear-spin Hamiltonian. Using the dummy variable $k$ for integers in the interval $[-I,I]$, a state

is transformed as

where $-T\omega_{\text{q}}^{\text{eff}}k^{2}$ is the resulting phase shift on the energy eigenstate $\ket{I,k}$.

Now, we can factor out the global phase $-T\omega_{\text{q}}^{\text{eff}}I^{2}$ and instead of freely evolving a state under one-axis twisting, apply equivalent phase updates,

to the subsequent multi-tone pulse. Such virtual phase updates would have same effect as free non-linear evolution and would result in cat state formation when $T=\nicefrac{{\pi}}{{2\omega_{\text{q}^{\text{eff}}}}}$. Thus, in the generalized rotating frame, we can start with the initial state $\ket{I,I}$, apply a global $\nicefrac{{\pi}}{{2}}$ rotation with $\phi_{j}=\phi$, followed by a second by a global $\nicefrac{{\pi}}{{2}}$ rotation with

realizing cat states by only phase modulation of multi-tone pulses.

Cat states with a lifetime of several milliseconds can be achieved using silicon-based chips with an implanted spin-$\nicefrac{{7}}{{2}}$ ¹²³Sb donor atom, that have a nuclear spin coherence time $T_{2}^{*}\approx 100$ ms for two-level transitions inferred from experiments on nuclear electric resonance [33]. SCSs along the $z$-axis would have a reduced coherence time, lower by a factor of $2I$, due to the large separation between the two components, resulting in a coherence time of $\nicefrac{{100}}{{7}}\approx 14$ ms for ¹²³Sb. Even for a modest driving strength $\gamma B_{1}=2\pi\times 800$ Hz, a $\nicefrac{{\pi}}{{2}}$-global rotation would take $\Delta t_{\nicefrac{{\pi}}{{2}}}=\frac{\pi}{2}\cdot\frac{4I}{\gamma B_{1}}=4.375$ ms. Thus, our approach, using two MT pulses and OAT, can yield highly coherent SCSs in less than 9 ms—within the dephasing time on current hardware. Faster Rabi frequencies are easily achieved for nuclear magnetic resonance (NMR) methods [40, 34], that could reduce the time needed for MT pulses and lead to faster generation and detection of nuclear SCSs.

We compare the performance of OAT to an intuitively appealing gate sequence for creating SCSs, achieved by sequential Givens rotations [52]. Starting with $\ket{I,I}$, we apply a $\nicefrac{{\pi}}{{2}}$-pulse of frequency $\omega_{I,I-1}$ to form the superposition $\ket{I,I}+\ket{I,I-1}$, with unit norm implied. Then, we apply a sequence of $2I-1$ $\pi$-pulses to create the SCS $\ket{I,I}+\ket{I,-I}$. For observing a collapse of the SCS, repeating the same sequence evolves the SCS into the coherent state $\ket{I,-I}$, opposite to the initial state $\ket{I,I}$. In total, $4I$ pulses would be needed for each collapse-and-revival using Givens rotations, that would take $\sim 9$ ms for a $I=\nicefrac{{7}}{{2}}$ with Rabi frequency $2\pi\times 800$ Hz, in contrast to OAT which takes just $\nicefrac{{\pi}}{{\omega_{\text{q}}}}=12.5\mu$s when $\omega_{\text{q}}=2\pi\times 40$ kHz.

We simulate our scheme under decoherence from a magnetic field fluctuating at rate $\Gamma_{\text{m}}$, modeled using Lindblad operator $\hat{L}_{\text{m}}=\hat{I}_{z}$, and an electric field fluctuating at rate $\Gamma_{\text{e}}$, modeled using $\hat{L}_{\text{e}}=\hat{I}_{z}^{2}$. Figure 4(c) shows $N_{\text{eff}}^{r}$ vs evolution time beyond 9 ms (approximate time to form the first SCS). We note that OAT is two orders of magnitude faster than both Givens rotations and the dephasing timescale, opening the possibility of realizing hundreds of SCS collapses-and-revivals by using a strong non-linearity relative to decoherence, which makes our SCS robust, in contrast to alternative platforms [53]. Additionally, SCSs generated by Givens rotations lie on the $\hat{I}_{z}$ axis, where the dephasing rate of the cat state increases with the nuclear spin increases; in comparison, cat states produced by one-axis twisting lie on the equator, perpendicular to the dephasing axis, and are highly protected against decoherence.

In summary, Schrödinger cat states are important for investigating foundations of quantum mechanics as well as for applications such as quantum error correction but are fragile and hard to realize. We propose a new platform for realizing macroscopic spin cat states by converting quadrupolar two-axis counter twisting to one-axis twisting on a high-spin donor in a solid-state device. Our scheme can deterministically generate spin cat states on-demand without post-selection and is two orders of magnitude faster than the dephasing timescale, resulting in highly coherent states. We show that, in the generalized rotating frame, a spin coherent state can be converted to a cat states by only phase modulation of multi-tone pulses. Unlike continuous-variable systems, we generate cat states without entanglement with an ancilla, which renders robustness against error propagation [25] and opens the possibility of storing long-lived nuclear-spin cat states. Furthermore, our cat states can be experimentally realized on a single nucleus, which could lead to novel applications in quantum information processing.