Quantum Zeno-based Detection and State Engineering of Ultracold Polar Molecules

TL;DR

This paper introduces a quantum Zeno-based method for detecting and engineering states of ultracold polar molecules, enabling precise control through collisional blockade and dissipative dynamics.

Contribution

It presents a novel toolbox utilizing quantum Zeno effects for manipulation and detection of ultracold molecules, with optimized parameters for high fidelity.

Findings

High-fidelity detection achievable with optimized parameters

Quantum Zeno effects enable controlled dissipation and state engineering

Residual imperfections like finite blockade strength impact performance

Abstract

We present and analyze a toolbox for the controlled manipulation of ultracold polar molecules, consisting of detection of molecules, atom-molecule entanglement, and engineering of dissipative dynamics. Our setup is based on fast chemical reactions between molecules and atoms leading to a quantum Zeno-based collisional blockade in the system. We demonstrate that the experimental parameters for achieving high fidelities can be found using a straightforward numerical optimization. We exemplify our approach for a system comprised of NaK molecules and Na atoms and we discuss the consequences of residual imperfections such as a finite strength of the quantum Zeno blockade.

Equations

$$H=\Delta (t){\sigma }_z-J{\sigma }_x-\frac{i\gamma }{2}|R⟩⟨R|,$$

$$H$$

$$+{\Delta }_a(t)|R⟩⟨R{|}_a-\frac{i\gamma }{2}|C⟩⟨C{|}_m\otimes |C⟩⟨C{|}_a$$

$$-(J_m|1,L⟩⟨1,C{|}_m+J_a|1,R⟩⟨1,C{|}_a+\text{H.c}),$$

$$U=U_{CZ}{R}_x^{(a)}(\varphi )U_H^{(m)}{U}_{CZ}{U}_H^{(m)}{R}_x^{(a)}(\varphi ),$$

$${\rho }_m^{\prime }={\rho }_m+{\varphi }^2{\sigma }_{-}^{(m)}{\rho }_m{\sigma }_{+}^{(m)}-\frac{{\varphi }^2}{2}\left\{{\sigma }_{+}^{(m)}{\sigma }_{-}^{(m)},{\rho }_m\right\}+O({\varphi }^4).$$

$$\begin{array}{rl}H=& -J(|1,L⟩⟨1,C{|}_m+|1,R⟩⟨1,C{|}_a+\text{H.c.})\\ & +\Omega ({\sigma }_x^{(m)}+{\sigma }_x^{(a)})+\delta ({\sigma }_z^{(m)}+{\sigma }_z^{(a)})\\ & -\Delta |L⟩⟨L|-\Delta |R⟩⟨R|,\end{array}$$

$$\frac{d}{dt}\rho =-\frac{i}{\hslash }[H,\rho ]+\sum _{i=\gamma ,\kappa }\left(c_i\rho c_i^{†}-\frac{1}{2}\left\{c_i^{†}{c}_i,\rho \right\}\right).$$

$$\beta =\frac{2{\hslash }^{3/2}\Gamma (1/4{)}^2}{(2{\mu }^5{C}_6{)}^{1/4}}=1.47\times 10^{-10}{\text{cm}}^3{\text{s}}^{-1},$$

Full text

Ultracold polar molecules are very promising candidates for a wide range of applications such as quantum simulation Moses2017 or searching for physics beyond the Standard Model Tarbutt2013 . However, the lack of optical control over the internal states due to the large number of vibrational levels currently presents a severe obstacle towards realizing these applications. Here, we show that strong chemical reactions between molecules and atoms enable a dissipative interaction mechanism that in the quantum Zeno regime allows to leverage the optical properties of atoms and achieve the same level of control for the molecules.

In current experiments, ultracold molecules are photo-associated from laser cooled atoms Ni2008 ; DeMarco2019 with full control over the final hyperfine state Ospelkaus2010a . The resulting molecules can be manipulated by external electric fields, microwave driving, or optical potentials, e.g., enabling to load the molecules into an optical lattice Miranda2011 ; Reichsollner2017 . Crucially, ultracold molecules can be subject to chemical reactions, both between themselves as well as with their constituent atoms Ospelkaus2010 ; Guo2018 . For some molecules such as NaK Park2015 ; Will2016 ; Seesselberg2018 , the chemical stability with respect to these atom-molecule reactions depends on the atom species involved Zuchowski2010 .

In this Letter, we present a complete toolbox for transferring the optical control over atoms to the realm of polar molecules. Our approach crucially relies on the presence of strong chemical reactions between atoms and molecules. In the quantum Zeno regime Zhu2014 , this dissipative atom-molecule interaction implements a collisional blockade that allows for the efficient detection of molecules, overcoming a longstanding challenge. In contrast to previous proposals Covey2018 , our approach does not require high-fidelity association and deassociation of the molecules. Furthermore, we demonstrate how to extend our detection method to the generation of atom-molecule entanglement and to the controlled dissipation of rotational excitations of the molecules. We dicuss the expected fidelities for experimentally realistic parameters, as well as the role of residual imperfections such as an imperfect quantum Zeno blockade.

In the following, we will first discuss the basic mechanism to implement a detection scheme for the molecules. Later on, we will show how this can be generalized towards the generation of entanglement between the internal degrees of freedom of a molecule and an atom. Finally, we will show how the entangling operation can also be employed towards the controlled dissipation of rotational excitations of the molecule, which can be readily used within laser cooling protocols.

Detection of molecules.— For the detection of the molecules, we consider a bilayer setup of two two-dimensional optical lattice potentials, see Fig. 1. Different layers for the atoms and molecules can be selected by magnetic and electric field gradients, respectively, in combination with microwave fields. Hence, it is possible to initially load all the Na atoms in one layer, while the NaK molecules are located in the other layer. We assume that the tunneling within the two lattice planes is strongly suppressed by the lattice potentials, hence it is sufficient to consider a single double-well setup. Additionally, we consider the atomic layer prepared in a Mott insulator state with one atom per site. Owing to the differences in the AC polarizability Holmgren2010 ; Neyenhuis2012 , as well as the fact that the atoms are much lighter than the molecules, we limit the following analysis to the case where only the atom moves with a hopping matrix element $J$, while the molecule remains stationary throughout the process. The crucial dissipative element is provided by a loss with a rate $\gamma$ induced by the chemical reaction between Na atoms and NaK molecules Idziaszek2010 ; Ospelkaus2010 .

The central idea behind the detection mechanism is to attempt a Landau-Zener transfer of the atom to the site potentially occupied by a molecule, e.g., by adiabatically reversing the magnetic field gradient. If there is no molecule on the second site, the transfer will succeed and the atom can be detected using standard quantum gas microscope techniques Bakr2009 ; Sherson2010 ; Haller2015 . However, in the presence of molecule in the second site, the strong chemical reaction creates a quantum Zeno blockade Zhu2014 of the Landau-Zener process. Hence, detecting the position of the atom after completion of the Landau-Zener sweep allows to deduce the positions of the molecules.

To be specific, we model the detection in terms of two independent Landau-Zener processes, with and without the presence of a molecule on the second site. For simplicity, we assume half filling of the molecular layer, i.e., equal probabilities for both processes, however the subsequent optimization can also be tailored towards other molecular densities. Then, the probability for a detection failure (either false positive or false negative) is simply given by the averaged error probability of the two processes. We model the Landau-Zener processes in terms of a non-Hermitian Hamiltonian

[TABLE]

where we have used Pauli matrices $\sigma_{\alpha}$ in the basis of the two atom positons $\ket{L}$ and $\ket{R}$, while $\Delta(t)$ describes the influence of the time-dependent magnetic field gradient. The dissipation $\gamma$ vanishes in the absence of a molecule on the right site. The time-dependent term $\Delta(t)$ is taken to be an odd polynomial in $t$ up to fifth order, i.e., satisfying the constraints $\Delta(\pm\infty)=\pm\infty$ and $\Delta(0)=0$. Then, we can minimize the total detection error $\varepsilon$ by optimizing the coefficients of the polynomial as well as the initial time $t_{i}$, while the final time $t_{f}$ is given by $t_{f}=-t_{i}$. As shown in Fig. 2, both Landau-Zener processes can be implemented with high success rate, i.e., the total error $\varepsilon=0.03$ is low. We note that this error can be even further reduced when starting from a high fidelity Mott insulator for the atoms Bakr2011 , as observing no atom in either site clearly signals the presence of a molecule.

Atom-molecule entanglement.— As the next step, we extend the Zeno-based detection scheme towards entanglement of the internal states of atoms and molecules. The internal states $\ket{0}_{a/m}$ and $\ket{1}_{a/m}$ refer to the rovibrational ground state and a rotationally excited state for the molecule ($m$) and two hyperfine states of the atom ($a$), respectively. To realize an entangling operation, we consider a setup consisting of three two-dimensional planes, of which we again consider a single three well unit, see Fig. 3a. We constrain the motion of the molecule (atom) to the left (right) site and the center site, respectively. Additionally, we assume that the motion takes only place within the $\ket{1}$ state of both atom and molecule.

A possible scheme to engineer these constrains may be as follows. In addition to the time-dependent magnetic or electric field gradient, the central site $\ket{C}$ may be shifted by an energy difference $\delta$ from that $\ket{L}$ and $\ket{R}$, such that direct hopping is prevented. Sites $\ket{L}$ and $\ket{C}$ ($\ket{R}$ and $\ket{C}$) are connected along the $x$ ($y$) direction. A first Raman pair (or rf frequency) with frequency transfer $\Delta\omega_{1}=\Delta$, Rabi frequency $\Omega_{1}$, and momentum transfer $\vec{\Delta}k_{1}=\Delta k_{1}\vec{e}_{x}$, induces a molecular hopping rate $J_{m}=\hbar\Omega_{1}\langle 1,L|_{m}e^{i\Delta k_{1}x}|1,C\rangle_{m}$. Crucially, $\langle 1,R|_{m}e^{i\Delta k_{1}x}|1,C\rangle_{m}=0$ due to orthogonality of the Wannier-like functions characterizing each lattice site. As a result, molecules in the $|1\rangle_{m}$ state only hop from $\ket{L}$ to $\ket{C}$. Similarly, using a second Raman pair with $\Delta\omega_{2}=\Delta$, $\Omega_{2}$, and $\vec{\Delta}k_{2}=\Delta k_{2}\vec{e}_{y}$, we may induce a hopping rate $J_{a}=\hbar\Omega_{2}\langle 1,R|_{a}e^{i\Delta k_{2}y}|1,C\rangle_{a}$, for atoms between $\ket{R}$ and $\ket{C}$ without any atomic hopping from $\ket{C}$ to $\ket{L}$.

The Hamiltonian of the three-well setup is of the form

[TABLE]

where $B$ is the rotational constant of the molecule, $E_{HFS}$ is the hyperfine splitting of the atom and $\Delta_{m/a}$ refers to the electric and magnetic field gradients, respectively. For simplicity, we assume the condition $\Delta_{m}(t)=-\Delta_{a}(t)\equiv\Delta(t)$, although this choice is not essential for achieving high fidelity entanglement.

The entangling operation we are realizing corresponds to a Controlled-Z (CZ) quantum gate for the internal states. We restrict the computational states to the molecule being in the motional state $\ket{L}$ and the atom being in $\ket{R}$, i.e., any residual occupation of the central site constitutes a gate error. Without loss of generality, we set the tunnelling matrix elements to be identical, i.e., $J_{a}=J_{m}\equiv J$. The CZ gate is implemented using a similar Landau-Zener protocol as our detection scheme, however the constraints on $\Delta(t)$ are different. Since the motional degrees of freedom need to return to the initial value at the end of the gate, this requires $\Delta(t)=\Delta(-t)$, therefore we take $\Delta(t)$ to be an even polynomial in $t$ up to fourth order. Then, optimizing the coefficients of the polynomial as well as the gate time result in a close approximation to an exact CZ gate, see Fig. 3b. To capture the performance of the gate in a single number, we use the fidelity $F=|\braket{\psi_{+}}{\psi(t_{f})}|^{2}$ to create the maximally entangled state $\ket{\psi_{+}}=\frac{1}{\sqrt{2}}(\ket{0L_{m}0R_{a}}+\ket{1L_{m}1R_{a}})$, as this constitutes a worst case scenario for a large class of error mechanisms Weimer2012 . For a dissipation strength of $\gamma=50\,J$, we can achieve a fidelity of $F=0.913$, which approaches a regime that is sufficient for the digital quantum simulation of important many-body spin models Weimer2013a . Finally, we would like to note that the presented atom-molecule entanglement can be readily converted into molecule-molecule entanglement using standard entanglement transfer protocols Barenco1995 .

In the previous analyses, we have assumed that the loss rate $\gamma=50\,J$ is fixed with respect to the tunnelling matrix element. However, by tuning the laser parameters for generating the potential wells, it is possible to make this ratio highly tunable. Therefore, it is interesting to investigate how our protocols responds to such a change, it particular there might be practical limitations on how low $J$ can become before other error mechanisms such as thermal losses, buffer gas collisions, etc. kick in. Figure 4 shows that both the detection and the entanglement error $\varepsilon=1-F$ both follow the expected $1/\gamma$ quantum Zeno profile, with the detection error being slightly below the entanglement error.

Dissipative quantum state engineering.— Let us now extend the mechanism behind the CZ gate towards the controlled dissipation of rotational excitations. For polar molecules, such a mechanism cannot reliably be realized using optical pumping due to the large number of vibrational levels an optical excitation can decay into. Nevertheless, such a controlled dissipation channel has immediate application for the laser cooling of molecules Zhao2012 , as well as for the dissipative engineering of tailored quantum many-body states Diehl2008 ; Verstraete2009 ; Weimer2010 .

To realize this dissipative state engineering, we consider two different strategies. The first strategy is to apply a pulsed scheme, in which the CZ gate $U_{CZ}$ is combined with single qubit gates in both the atom and the molecule, in close analogy to a digital quantum simulator. We consider the gate sequence

[TABLE]

where $R_{x}(\phi)=\exp(i\phi\sigma_{x})$ is the single qubit rotation about the $x$ axis and $U_{H}$ is the Hadamard gate. After the gate sequence, the molecule and the atom will be entangled. However, a subsequent optical pumping of the atom according to $\ket{1}\to\ket{0}$ realizes a discrete quantum master equation in Lindblad form according to

[TABLE]

This dissipative dynamics can then be interspersed with coherent driving of the rotational states of the molecules to achieve the desired quantum state engineering.

The second strategy for this dissipative state engineering uses a continuous driving of both the molecular and the atomic transitions rather than a pulsed quantum gate sequence. This has two advantages. On the one hand, the mechanism behind the CZ gate can be understood to implement an effective $\sigma_{z}^{(m)}\sigma_{z}^{(a)}$ interaction, which cannot transfer energy from the molecule to the atom. By going into the rotating frame of the driving, this is converted into a $\sigma_{x}^{(m)}\sigma_{x}^{(a)}$ interaction, which contains terms that lead to an energy transfer. The second advantage is that the energy difference between the rotational constant $B$ and the hyperfine splitting $E_{HFS}$ is too large to observe direct energy transfer from the molecule to the atom. In the rotating frame, however, this large energy difference is converted into the energy difference involving the respective detunings and Rabi frequencies of the driving. When the differences in the detunings and the Rabi frequencies vanish, efficient energy transfer becomes possible under a condition very similar to the Hartmann-Hahn driving in nuclear magnetic resonance Hartmann1962 . In this case, the Hamiltonian is given by

[TABLE]

where $\Omega$ is the Rabi frequency and $\delta$ is the detuning used both for the driving of the molecule and the atom. Addionally, we consider optical pumping of the atom by including a quantum jump operator $c_{\kappa}=\sqrt{\kappa}\sigma_{-}^{(a)}$ in the quantum master equation describing the dynamics,

[TABLE]

Here, the jump operator $c_{\gamma}$ describes the occurence of a chemical reation, which removes both the atom and the molecule from the system. We then optimize the set of parameters $(\Omega,\delta,\Delta,\kappa)$ such that an initial state of $\ket{1}_{m}\ket{0}_{a}$ goes to $\ket{0}_{m}\ket{0}_{a}$. Figure 5 shows that this optimization leads to the initial rotational excitation being dissipated with high probability, while the loss resulting from chemical reaction remains low. We would like to note that this continuous dissipative process might be experimentally easier to realize than a pulsed sequence, however coming at the expense of requiring longer preparation times. We checked the robustness of our protocol with respect to an imperfect experimental implementation of the optimized parameters. Upon decreasing the atomic Rabi frequency by $10\%$ compared to its optimal value, we observe that the final probability to reach the $\ket{0}_{m}\ket{0}a$ state decreses by merely $2\%$. This underlines the robustness of our optimization protocol.

Experimental parameters.— In the following, we will calculate the loss coefficient $\gamma$ for the exothermic reaction $\text{NaK}+\text{Na}\to\text{Na}_{2}+\text{K}$. The $\text{NaK}+\text{Na}$ system has a van der Waals coefficient of $C_{6}=3592\,\text{a.u.}$ Zuchowski2013 , from which we can calculate the universal rate coefficient $\beta$ of the chemical reaction Idziaszek2010 , which is given by

[TABLE]

where $\mu$ is the reduced mass. In a potential well corresponding to an oscillator length of $a_{0}=100\,\text{nm}$, this will result in a reaction rate of $\gamma=50\,\text{kHz}$. Compared to the typical tunnelling matrix element of $J\sim 100\text{--}1000\,\text{Hz}$ Bloch2008 , dissipation rates $\gamma=50\text{--}500\,J$ deep in the quantum Zeno regime can be achieved.

In summary, we have presented a toolbox for the controlled manipulation of ultracold polar molecules via dissipative interactions with atoms. Our results pave the way for important experimental advances in the detection, entanglement, and dissipative quantum state preparation of molecular quantum many-body systems. While we have exemplified our approach for NaK, it can also be readily applied to other molecules, provided there is at least one exothermic reaction with one of its constituent atoms. An interesting question for future work is whether the dynamics leading to the formation of ultracold collision complexes Mayle2012 can be harnessed for coherent manipulation of molecules in a similar way.