Scalable $W$-type entanglement resource in neutral-atom arrays with Rydberg-dressed resonant dipole-dipole interaction

The system Hamiltonian, describing an itinerant dressed Rydberg excitation coupled to dispersionless bosons, can succinctly be written as

where $\mathbf{u}\equiv\{u_{n}|\>n=1,\ldots,N\}$ is a shorthand for the set of the atom displacements and $c^{\dagger}_{n}$ ($c_{n}$) creates (destroys) an excitation at site $n$, with

being its corresponding on-site energy Wüster et al. (2011). The latter depends on the boson displacements, not only on site $n$ but also on the adjacent sites $n\pm 1$. At the same time

is the excitation hopping amplitude between sites $n$ and $n+1$ Wüster et al. (2011), which depends on the difference $u_{n+1}-u_{n}$ of the respective displacements. To facilitate further analysis, it is prudent to introduce the dimensionless quantity $\zeta\equiv C_{3}/(\hbar\Delta a^{3})$, the ratio of the most relevant energy scales in the system at hand.

Before embarking on further discussion, it is useful to stress that the very existence of the dressing-induced on-site term in the Hamiltonian of Eq. 1, which has no analog in the standard resonant dipole-dipole interaction case Ate (b), is a consequence of the fact that the effective dressing-induced interaction potential for a pair of atoms has a nonzero diagonal component Wüster et al. (2011).

For small displacements ($u_{n}\ll a$) it is pertinent to expand the expressions on the right-hand-side of Eqs. (2) and (3) to linear order in the difference of displacements using the approximation $(1\pm x)^{r}\approx 1\pm rx$ ($|x|\ll 1$). The linear dependence of $\varepsilon_{n}(\mathbf{u})$ on $u_{n+1}-u_{n-1}$ captures the coupling of the excitation density at site $n$ with the boson displacements on the neighboring sites $n\pm 1$ [breathing-mode-type (B) e-b coupling]; similarly, the linear dependence of $t_{n,n+1}$ on $u_{n+1}-u_{n}$ describes how the excitation hopping between sites $n$ and $n+1$ is affected by the boson displacements [Peierls-type (P) coupling] Stojanović and Vanević (2008); Stojanović et al. (2014); Stojanović and Salom (2019). This lowest-order expansion reads

where $\xi_{\textrm{B}}$ and $\xi_{\textrm{P}}$ are given by

and the bare on-site energy and hopping amplitude by

The positive sign of $t_{e}$ [realized for $|\zeta|>1$, i.e., $C_{3}/(\hbar|\Delta|a^{3})>1$] corresponds to the conventional situation where the bare-excitation dispersion $\epsilon_{0}-2t_{e}\cos k$ has its minimum at $k=0$. However, of principal interest here is the opposite, negative sign of $t_{e}$ [for $|\zeta|<1$, i.e., $C_{3}/(\hbar|\Delta|a^{3})<1$]. This is the case with the band minimum at $k=\pi$, which corresponds to the bare-excitation Bloch state

where $|0\rangle_{\textrm{e}}$ and $|0\rangle_{\textrm{b}}$ are the excitation and boson vacuum states, respectively.

The effective system Hamiltonian has a noninteracting part that comprises free excitation- and boson terms:

Its interacting part is given by $H_{\textrm{e-b}}=H_{\textrm{B}}+H_{\textrm{P}}$, where

are the terms that correspond to the two different e-b coupling mechanisms described above [cf. Eq. (III.1)], with $g_{\textrm{B}}\equiv\xi_{\textrm{B}}/(2m\hbar\omega^{3}_{\textrm{b}})^{1/2}$ and $g_{\textrm{P}}\equiv\xi_{\textrm{P}}/(2m\hbar\omega^{3}_{\textrm{b}})^{1/2}$ being the dimensionless B- and P coupling strengths. Importantly, owing to the discrete translational symmetry of the system, the eigenstates of $H=H_{0}+H_{\textrm{e-b}}$ can be labelled by the eigenvalues of the total quasimomentum operator $K_{\mathrm{tot}}=\sum_{k}k\>c_{k}^{\dagger}c_{k}+\sum_{q}q\>b_{q}^{\dagger}b_{q}$. The permissible eigenvalues of this operator belong to the Brillouin zone that corresponds to the underlying 1D lattice.

The bare on-site energy $\epsilon_{0}$ in Eq. (8) plays the usual role of the on-site energy in tight-binding models – that of a constant energy offset (i.e. the band-center energy) for an itinerant excitation. While $\epsilon_{0}$ is inconsequential for the physical mechanism that leads to $\pi$-twisted $W$ states in the system at hand (competition of the B- and P couplings), the fact that it depends on the dressing parameter $\alpha$ [cf. Eq. (6)] does have some bearing on the preparation of such states (see Sec. IV.2 below).

Consider now the special case of the proposed system with equal P- and B coupling strengths, i.e., $g_{\textrm{P}}=g_{\textrm{B}}\equiv g$. The latter physical situation corresponds to the “sweet-spot” (ss) value $\zeta_{\textrm{ss}}=(1+\sqrt{13})/6\approx 0.77$ of $\zeta$. Assuming that the atomic species and the principal quantum number are chosen, which fixes the interaction constant $C_{3}$, for each choice of the lattice period $a$ this last situation is realized for the detuning $\Delta^{\textrm{ss}}\equiv C_{3}/(\hbar\zeta_{\textrm{ss}}a^{3})$ and a range of values for the Rabi frequency $\Omega^{\textrm{ss}}=\Delta^{\textrm{ss}}\alpha$ [determined by the adopted range of values of the dressing parameter (see below)]. In this special case, the e-b-coupling and total Hamiltonians will be denoted by $H^{\textrm{ss}}_{\textrm{e-b}}$ and $H^{\textrm{ss}}$, respectively, in what follows.

To quantify the e-b coupling strength in the aforementioned special case of the system under consideration, one invokes the momentum-space form of $H^{\textrm{ss}}_{\textrm{e-b}}$, which reads

The explicit form of the e-b vertex function $\gamma^{\textrm{ss}}_{\textrm{e-b}}(k,q)$ in the last equation is

Consequently, the effective e-b coupling strength – generally defined as $\lambda_{\textrm{e-b}}=\langle|\gamma_{\textrm{e-b}}(k,q)|^{2}\rangle_{\textrm{BZ}}/(2|t_{\rm e}|\omega_{\textrm{b}})$ Sto (b), where $\langle\ldots\rangle_{\textrm{BZ}}$ stands for the Brillouin-zone average – in this special case evaluates to $\lambda^{\textrm{ss}}_{\textrm{e-b}}\equiv 3g^{2}\>\hbar\omega_{\textrm{b}}/|t_{\rm e}|$, i.e.,

The obtained dependence of $\lambda^{\textrm{ss}}_{\textrm{e-b}}$ on $\alpha\equiv\Omega^{\textrm{ss}}/\Delta^{\textrm{ss}}$ implies that the Rabi frequency $\Omega^{\textrm{ss}}$ is the main experimental knob in the system at hand. By varying $\Omega^{\textrm{ss}}$ different characteristic regimes of this system can be explored.

To set the stage for further analysis, it is prudent to specify at this point the realistic range of values for each of the relevant system parameters. The system at hand is mostly analyzed in what follows for $n_{\textrm{q}}=80$, with the corresponding value $C_{3}=2\pi\hbar\times 40$ GHz$\mu$m³ of the resonant dipole-dipole interaction constant for ⁸⁷Rb atoms. As usual for optical-tweezer arrays, the lattice period $a$ is in the range between about $3\>\mu$m and tens of micrometers. The corresponding values of the ss detuning can vary in an extremely wide range, depending on the choice of $a$; for example, for $a=4$ $\mu$m one obtains $\Delta^{\textrm{ss}}\approx 5.12$ GHz, for $a=10$ $\mu$m one finds $\Delta^{\textrm{ss}}\approx 327.4$ MHz, while for $a=15$ $\mu$m one has $\Delta^{\textrm{ss}}\approx 97$ MHz. At the same time, the typical values for the trapping frequency $\omega_{\textrm{b}}$ are $\omega_{\textrm{b}}/(2\pi)\sim(2-5)$ kHz. Finally, for the dressing parameter $\alpha$ it is worthwhile to consider values in the range $0.01-0.1$.

Lanczos-type exact diagonalization Sto (b) of $H^{\textrm{ss}}=H_{0}+H^{\textrm{ss}}_{\textrm{e-b}}$ is carried out here for a system with $N=10$ sites (i.e., atoms) and the maximal number $M=8$ of bosons in the truncated boson Hilbert space. This is done using a well-established procedure for a controlled truncation of bosonic Hilbert spaces. This procedure entails a gradual increase of $N$, with the concomitant increase of $M$, until a numerical convergence of the obtained results for the ground-state energy and other relevant quantities is achieved Sto (b).

The performed numerical calculation shows that the ground state of $H^{\textrm{ss}}$ undergoes a sharp level-crossing transition Sto (b) at a certain critical value $(\lambda^{\textrm{ss}}_{\textrm{e-b}})_{\textrm{c}}$ of $\lambda^{\textrm{ss}}_{\textrm{e-b}}$. For $\lambda^{\textrm{ss}}_{\textrm{e-b}}<(\lambda^{\textrm{ss}}_{\textrm{e-b}})_{\textrm{c}}$ the ground state corresponds to the eigenvalue $\pi$ of $K_{\mathrm{tot}}$ and has a peculiar character. Namely, despite being the ground state of an interacting e-b Hamiltonian, it has the form of the bare-excitation Bloch state $|\Psi_{k=\pi}\rangle$ [cf. Eq. (7)] and its energy $\epsilon_{0}-2|t_{e}|$ corresponds to a minimum of a 1D tight-binding dispersion. By contrast, the strongly boson-dressed ground state for $\lambda^{\textrm{ss}}_{\textrm{e-b}}\geq(\lambda^{\textrm{ss}}_{\textrm{e-b}})_{\textrm{c}}$ is twofold-degenerate and corresponds to $K=\pm K_{\textrm{gs}}$, where $0<K_{\textrm{gs}}<\pi$. The dependence of the ground-state energy $E_{\textrm{gs}}$ (without the constant contribution $\epsilon_{0}$), expressed in units of $|t_{e}|$, on $\lambda^{\textrm{ss}}_{\textrm{e-b}}$ is depicted in Fig. 2.

Figure 3 illustrates how the ground-state total quasimomentum $K_{\textrm{gs}}$ depends on $\lambda^{\textrm{ss}}_{\textrm{e-b}}$. In particular, $K_{\textrm{gs}}=\pi$ in the bare-excitation ground state $|\Psi_{k=\pi}\rangle$ has a vanishing bosonic contribution as $\langle\Psi_{k=\pi}|\sum_{n}b^{\dagger}_{n}b_{n}|\Psi_{k=\pi}\rangle=0$. The fact that this bare-excitation Bloch state is a ground-state of an interacting e-b system is a direct implication of the assumption that $g_{\textrm{P}}=g_{\textrm{B}}\equiv g$ (recall Sec. III.2), i.e. it is a consequence of an effective mutual cancellation of P- and B couplings for a bare excitation with this particular quasimomentum ($k=\pi$).

It is pertinent at this point to establish the connection between the ground states of the system at hand and the desired $N$-qubit $W$ states. The bare-excitation Bloch state $|\Psi_{k}\rangle$, recast in terms of the pseudospin-$1/2$ (qubit) degrees of freedom, coincides with the twisted $W$ state

parameterized by the quasimomentum $k$ from the Brillouin zone (i.e. $-\pi<k\leq\pi$). In particular, $|\Psi_{k=\pi}\rangle$ – the ground state of the system at hand for $\lambda^{\textrm{ss}}_{\textrm{e-b}}<(\lambda^{\textrm{ss}}_{\textrm{e-b}})_{\textrm{c}}$ – corresponds to the $\pi$-twisted $W$ state $|W_{N}(k=\pi)\rangle$ of Rydberg-dressed qubits.

The conditions for realizing the desired states $|W_{N}(k=\pi)\rangle$ in the system under consideration are easily reached with realistic values of the relevant experimental parameters ($a,\omega_{\textrm{b}},\alpha$). To justify that, it is worthwhile to immediately note that Figs. 2 and 3 correspond to $a=4\>\mu$m, a relatively small lattice period which favors larger coupling strengths [cf. Eq. (12)] and allows the onset of a sharp transition. Yet, already for this choice of $a$, with a sufficiently large trapping frequency ($\omega_{\textrm{b}}\gtrsim 2\pi\times 3.5$ KHz) the effective coupling strength $\lambda^{\textrm{ss}}_{\textrm{e-b}}$ is always below the critical one, i.e., $\pi$-twisted $W$ states are accessible in the entire adopted range of values ($0.01-0.1$) for the dressing parameter $\alpha$. The fast decay of $\lambda^{\textrm{ss}}_{\textrm{e-b}}$ with $a$ ensures that for $a\gtrsim 5\>\mu$m the sought-after $W$ states are the ground states of the system for any realistic choice of $\omega_{\textrm{b}}$ and $\alpha$. For the sake of completeness, it is worthwhile mentioning that by choosing a smaller principal quantum number these conditions are even easier to satisfy because of the smaller value of $C_{3}$; for instance, for $n_{\textrm{q}}=50$ the corresponding value of this interaction constant for ⁸⁷Rb is an order of magnitude smaller than for $n_{\textrm{q}}=80$.

It is interesting to observe that – in addition to being the ground state of $H^{\textrm{ss}}$ for coupling strengths below the critical one – the state $|\Psi_{k=\pi}\rangle$ is an exact eigenstate of this Hamiltonian for an arbitrary $\lambda^{\textrm{ss}}_{\textrm{e-b}}$. Namely, given that $\gamma^{\textrm{ss}}_{\textrm{e-b}}(k=\pi,q)=0$ for an arbitrary $q$ [cf. Eq. 11], it is straightforward to show that $H^{\textrm{ss}}_{\textrm{e-b}}|\Psi_{k=\pi}\rangle=0$. Thus, $|\Psi_{k=\pi}\rangle$ is an eigenstate of $H^{\textrm{ss}}_{\textrm{e-b}}$. Because this last state is an eigenstate of the free Hamiltonian $H_{0}$ as well, it follows immediately that it is also an eigenstate of the total Hamiltonian $H^{\textrm{ss}}=H_{0}+H^{\textrm{ss}}_{\textrm{e-b}}$. To conclude, even for those parameters (i.e., values of the Rabi frequency that lead to coupling strengths above the critical one) for which $|\Psi_{k=\pi}\rangle$ does not coincide with the lowest-energy $K=\pi$ eigenstate of the system (for an illustration, see Fig. 4) this state still remains an eigenstate in the discrete (bound-state) part of the spectrum of $H^{\textrm{ss}}$.

A deterministic Rabi-type driving protocol for the preparation of $\pi$-twisted $W$ state is discussed in what follows, assuming that the initial state of the system is $|0\rangle_{\textrm{e-b}}\equiv|0\rangle_{\textrm{e}}\otimes|0\rangle_{\textrm{b}}$, where $|0\rangle_{\textrm{e}}$ is the shorthand for an $N$-atom state with zero Rydberg-dressed excitations (i.e., all atoms occupying the dressed state $|0\rangle$), while $|0\rangle_{\textrm{b}}$ is the collective boson vacuum, i.e. the state with all atoms being in their motional ground state. For the system to be prepared in the state $|0\rangle_{\textrm{e-b}}$, two preliminary steps ought to be carried out. The first one is to prepare the state $|0\rangle_{\textrm{e}}$ via Rydberg-dressing starting from the state $|g\rangle_{\textrm{e}}$ with all atoms in their absolute ground state $|g\rangle$. The other one is to prepare all atoms in their motional ground states using the established methodology for this purpose Kau .

The envisioned state-preparation protocol is based on an external driving given by

where $\beta(t)$ accounts for its time dependence and the factors $e^{\pm iq_{d}n}$ allow for the possibility of applying external driving to different Rydberg-atom qubits with a nontrivial phase difference. The transition matrix element of $F_{q_{d}}(t)$ between $|0\rangle_{\textrm{e-b}}$ and $|\Psi_{k=\pi}\rangle\equiv|W_{N}(k=\pi)\rangle\otimes|0\rangle_{\textrm{b}}$ is equal to $\hbar\beta(t)\>\delta_{q_{d},k=\pi}$, indicating that the required phase difference between adjacent qubits is $q_{d}=\pi$. Furthermore, by assuming that $\beta(t)=2\beta_{p}\cos(\omega_{d}t)$, where $\hbar\omega_{d}\equiv\epsilon_{0}-2|t_{e}|$ is the energy difference between the two relevant states, in the rotating-wave approximation these states are Rabi-coupled with the Rabi frequency $\beta_{p}$ Shore (1990). Therefore, $\pi$-twisted $W$ states are prepared within time $\tau_{\textrm{prep}}=\pi\hbar/(2\beta_{p})$, which does not depend on the system size ($N$) at all. For example, taking $\beta_{p}/(2\pi\hbar)$ to be in the range $(10-100)$ MHz, one finds $\tau_{\textrm{prep}}\approx 3-25$ ns, which is $4-5$ orders of magnitude shorter than typical lifetimes of ordinary Rydberg states and $7-8$ orders than those of Rydberg-dressed states, as they are scaled by an additional factor of $\alpha^{-2}$.

Apart from being the ground state of the system in a broad parameter window and remaining its eigenstate even outside of that window, the desired $W$ state has another property that facilitates its preparation. Namely, it is separated from the other eigenstates of $H^{\textrm{ss}}$ by an energy gap of $\hbar\omega_{b}$. Systems in which dispersionless bosons interact with a single itinerant particle Stojanović et al. (2014); Stojanović and Salom (2019) generically posses such a gap, equal to the single-boson energy, which separates their ground state from the one-boson continuum (inelastic scattering threshold). In the weak-coupling regime ground states of such systems are typically the only bound states they have Stojanović et al. (2014). Importantly, apart from increasing the parameter window where $W$ states can be engineered, another advantage of increasing $a$ is a better separation of those states from other states. Namely, an increase of $a$ leads to a decrease of $\omega_{d}\propto\alpha^{4}C_{3}/a^{3}$, so that $\hbar\omega_{b}$ becomes a progressively larger fraction of the energy difference $\hbar\omega_{d}$. For instance, with $\alpha=0.05$ and $a=15\>\mu$m, for the chosen range of trapping frequencies $\omega_{b}$ the gap energy amounts to $15-40\>\%$ of this energy difference, which ensures that the above Rabi-type state-preparation protocol will not be hampered by an inadvertent population of undesired states.

The analysis of the ground-state properties of the system under consideration in Sec. IV.1 mostly featured the results that correspond to the relatively small latice period $a=4$ $\mu$m. It is important to stress that this choice, which favors large effective e-b coupling strengths and a possible onset of a sharp transition [cf. Sec. IV.1], was intentionally made in order to highlight the worst-case scenario as far as the realization of the desired $W$-type entanglement resurce in the system at hand is concerned. However, from the point of view of an actual $W$-state preparation, for the reasons stated above it is more favorable to choose an intermediate or large lattice period, say $a\gtrsim 12\>\mu$m. Not only that this precludes an inadvertent population of undesired states in the continuum part of the spectrum of the relevant coupled e-b system, but it also leads to smaller values of the ss detuning (note that $\Delta^{\textrm{ss}}\propto a^{-3}$), which makes this last detuning far smaller than the typical energy spacings of Rydberg levels. [Recall that the distribution of Rydberg energy levels becomes denser as the principal quantum number $n_{q}$ increases, such that the energy difference $\Delta E$ of adjacent levels scales as $n_{q}^{-3}$; note also that $\Delta E\sim 1$ GHz for $n_{q}\sim 100$. Gal ] This, in turn, prevents the possibility of inadvertently populating higher Rydberg levels in the initial, Rydberg-dressing step (i.e. in the preliminary preparation of the state $|0\rangle_{\textrm{e}}$) of the proposed $W$-state preparation protocol.

Owing to the rich energy-level structure of Rydberg atoms and the existing wealth of techniques for the coherent manipulation of atomic internal states, there are several possibilities for storing and manipulating quantum information, i.e., QIP with Rydberg qubits. Depending on the number of ground- or Rydberg states that make up the qubit (i.e., serve as its logical $|0\rangle$ and $|1\rangle$ states), there are three main types of Rydberg qubits: ($\imath$) those based on one weakly-interacting state $|g\rangle\equiv|0\rangle$ and one strongly-interacting Rydberg state $|r\rangle\equiv|1\rangle$ ($gr$-qubits), ($\imath\imath$) those encoded using two different Rydberg states ($rr$-qubits, where $|r\rangle\equiv|0\rangle$ and $|r^{\prime}\rangle\equiv|1\rangle$), and, finally, ($\imath\imath\imath$) those encoded in two long-lived low-lying atomic states $|g\rangle\equiv|0\rangle$ and $|h\rangle\equiv|1\rangle$ ($gg$-qubits).

In particular, $gg$-qubits typically involve two (usually magnetically insensitive) hyperfine sublevels of the electronic ground state – like the states $|g\rangle$ and $|h\rangle$ of the system under consideration [cf. Fig. 1(b)]. Such qubits offer the best trade-off between coherence times and switchable interactions, which makes them promising candidates for universal quantum computing. On the other hand, compared to their $gr$- and $rr$ counterparts, $gg$-qubits are weakly interacting. One possible approach for inducing stronger interactions between such qubits relies on momentarily exciting and de-exciting them via Rydberg states using precisely timed or shaped optical fields. An alternative approach for making $gg$-qubits interact more strongly – of relevance for the present work – is to weakly admix some Rydberg-state character to the ground states using an off-resonant laser coupling, thereby effectively transforming them into Rydberg-dressed qubits Pet ; Jau ; Ari ; Mit .

Generally speaking, creating quantum entanglement in large systems on timescales much shorter than the relevant coherence times is key to efficient QIP. In particular, it is shown here that in neutral-atom arrays $\pi$-twisted $W$ states of Rydberg-dressed qubits can be engineered with the corresponding preparation times being independent of the system size. Importantly, those preparation times are several orders of magnitude shorter than the typical lifetimes of the relevant Rydberg states, being at the same time an even much smaller fraction of the effective lifetimes of their Rydberg-dressed counterparts.

Another favorable feature of the system at hand – and a prerequisite for universal quantum computation – stems from the $XY$ character Sch (b) of the effective qubit-qubit interaction in this system, which is given by the free-excitation hopping term in Eq. (8). and the Peierls-coupling term in Eq. (9). When recast in terms of the pseudospin-$1/2$ degrees of freedom of Rydberg-dressed qubits, the coupling between qubits $n$ and $n+1$ is given by $J_{n,n+1}(\sigma^{x}_{n}\sigma^{x}_{n+1}+\sigma^{y}_{n}\sigma^{y}_{n+1})$, where $J_{n,n+1}\equiv 2[-t_{e}+g_{\textrm{P}}\hbar\omega_{\textrm{b}}(b^{\dagger}_{n+1}+b_{n+1}-b^{\dagger}_{n}-b_{n})]$ is the effective $XY$-coupling strength that depends dynamically on the boson degrees of freedom. Such boson-dependent coupling strengths are characteristic, for instance, of certain trapped-ion systems where collective motional modes (phonons) Tra play the role of bosons Wal . Unlike such trapped-ion systems, whose phonon spectra have a quasicontinuous character Tra , the system at hand merely involves dispersionless bosons of one single frequency. This circumvents the spectral-crowding problem that poses an obstacle for QIP in large trapped-ion chains Lan .