Entanglement witness for combined atom interferometer-mechanical oscillator setup

Entanglement is a core feature of quantum mechanics that precludes any hidden variable classical theory. Many groups have considered entanglement in macroscopic systems, particularly in the case of spin-spin systems [1, 2, 3, 4, 5], oscillator-oscillator systems [6, 7, 8, 9, 10, 11, 12], and spin-oscillator systems [13, 14, 15]. For instance, Kong et al. experimentally generate entanglement between the spins of atoms in a vapor [1], Ockeloen-Korppi et al. use a cavity to entangle two mirrors located 600 µm apart [6], and Riedinger et al. entangle two chips separated by 20 cm [7]. Thomas et al. entangle a spin ensemble with a square membrane of sides 1 mm in length [13].

Of particular interest are approaches to entangle masses and/or spins to investigate theories that incorporate quantum mechanics with gravity. For example, Kafri et al. [16], Bose et al. [2] and Marletto et al. [17] propose using two masses which interact with each other though a gravitational force; if the two masses become entangled, then we can conclude that the gravitational field is capable of generating entanglement. A different approach by Carney et al. [18] suggests gravitationally coupling an atom interferometer with a mechanical oscillator. If the interaction is capable of entanglement, the oscillator and the atoms will get entangled, which can be detected using the interferometer’s signal.

Such experiments have to use certain techniques to determine the existence of entanglement. The experiments in Ref. [1, 6, 7, 13, 2] use (or propose using) an entanglement witness (EW) that work for very low noise situations, where one prepares pure states of the two systems, and noise and decoherence are neglected. An EW is a quantity composed of observables; it has a bound satisfied by separable states, which, if violated, determines that entanglement exists between the systems. The literature on creating an entanglement witness is varied. The most well-known example of EWs might be Bell inequalities [19, 20] used to distinguish between the possibility of local hidden variable theories and entanglement. Another common EW is one which is often applied to oscillator-oscillator entanglement, whose general version was proposed by Duan et al. [21].

In this paper, we work within the context of the proposal by Carney et al. [18], introduced previously. In the progress towards such a gravity experiment, it is more reasonable to first attempt the generation of magnetic entanglement between the atom interferometer and the oscillator, since a sufficiently strong magnetic coupling is easier to obtain. Before proceeding with physically building the experiment, it is important to ensure that entanglement will exist, at least theoretically, and understand the physical parameters at which it will occur. In order to do this, we construct an entanglement witness (EW) for our setup that provides a path for violation at finite motional temperatures and with noise and dephasing. We investigate the effects of thermal noise on its violation and establish the conditions on parameters such that violation exists. We find that for appropriate values of oscillator quality factor $Q$, violation always exists for any value of magnetic coupling $\lambda$ and thermal occupancy $\bar{n}$.

General entanglement witness - A simplified setup that we can consider is shown in Figure 1. The Hamiltonian of the system is (note that $\hbar=1$ herein) [18]

$\hat{c}^{\dagger}$ and $\hat{c}$ are the oscillator creation and annihilation operators. $\hat{J}_{z}$ is a pseudo-spin operator describing the location of $N$ atoms; that is, $\hat{J}_{z}=\sum_{j=1}^{N}\hat{S}_{z,j}$, where $\hat{S}_{z,j}$ characterizes which of the two interferometer traps the $j^{th}$ atom is in. As stated earlier, the creation of entanglement between the interferometer and the oscillator can be determined by measuring an entanglement witness (EW). We find that the following quantity is an applicable EW for our setup:

Here, $\hat{J}_{\mu}=\sum_{j=1}^{N}\hat{S}_{\mu,j}$ and $\hat{q}(t)=\sqrt{1/(2m\omega)}(\hat{c}+\hat{c}^{\dagger})$. The coefficients $a_{\mu}$ and $b_{\mu}$ can be chosen as appropriate. Such spin-oscillator variances allow for the characterization of spin-oscillator correlations. Eq. 2 is bounded by $W_{b}$ for all separable states. If this bound is violated by the experimental state of concern, we can conclude that it is an entangled state.

$W_{b}$ can be derived using the theorem in the paper by Hofmann et al. [23]. They showed that for operators $\hat{A}_{\mu}$ of system $A$ and $\hat{B}_{\mu}$ of system $B$, if $\sum_{\mu}\textrm{Var}(\hat{A}_{\mu})\geq U_{A}$ and $\sum_{\mu}\textrm{Var}(\hat{B}_{\mu})\geq U_{B}$, then $\sum_{\mu}\rm{Var}(\hat{A}_{\mu}+\hat{B}_{\mu})\geq U_{A}+U_{B}$ for any separable state. Starting with $\tilde{W}=\sum_{\mu}\textrm{Var}(\hat{J}_{\mu}+a_{\mu}\hat{q}+b_{\mu}\hat{p})$, where the summation $\sum_{\mu}$ is over the $x,y,$ and $z$ components, we follow Ref. [23] to arrive at the general bound $\tilde{W}\geq\left[\sum_{\mu}\textrm{Var}(\hat{J}_{\mu})\right]_{\rm{min}}+\left[\sum_{\mu}\textrm{Var}(a_{\mu}\hat{q}+b_{\mu}\hat{p})\right]_{\rm{min}}$. To obtain the witness in Eq. 2, we set $a_{x}=0$ and $b_{x}=0$ on the left and right hand sides of this inequality.

The oscillator-operator bound can be derived using the Heisenberg uncertainty principle and carrying out a minimization process; we get $\sum_{\mu\neq x}\textrm{Var}(a_{\mu}\hat{q}+b_{\mu}\hat{p})\geq|a_{y}b_{z}-a_{z}b_{y}|$. In analogy to a derivation in Ref. [23], the general bound for the spin operators is $\sum_{\mu}\textrm{Var}(\hat{J}_{\mu})\geq j$, which for a fully symmetric state will be $\sum_{\mu}\textrm{Var}(\hat{J}_{\mu})\geq N/2$. The most general case will be $\sum_{\mu}\textrm{Var}(\hat{J}_{\mu})\geq[j]_{\textrm{min,exp}}$ where $[j]_{\textrm{min,exp}}$ is the minimum value that $j$ can take at the time of measurement.

For the EW in Eq. 2, we can finally write the general bound to be

That is, for a separable state, the condition that $W\geq W_{b}$ will be satisfied. If we evaluate $W$ with respect to a possibly entangled state in order to obtain the value $W_{\rm{en}}$ and find that $W_{\rm{en}}<W_{b}$, then the bound is violated and we can conclude that the state is indeed entangled.

We explore this EW in the context of two scenarios; when the systems are noiseless and when there is thermal noise (the case of atomic dephasing can be found in the Supplementary Material). For each scenario, we determine whether the EW is violated. In an experimental setting, $W_{b}$ is a theoretical value that we have at hand, while $W_{\rm{en}}$ is the quantity that we estimate through measurement. We consider the quantity $W_{\rm{ratio}}=\left[(W_{b}-W_{\rm{en}})/{W_{b}}\right]_{\rm{max}}$; the EW is violated if $W_{\rm{ratio}}>0$. Since the imprecision is set by $W_{b}-W_{\rm{en}}$, $W_{\rm{ratio}}$ informs us of the number of measurements $n_{\rm{meas}}$ required to determine the existence of entanglement, since $n_{\rm{meas}}\sim W_{\rm{ratio}}^{-2}$.

A key step in this process is determining the coefficients $a_{\mu}$ and $b_{\mu}$, where we attempt to minimize the variance of the EW. When $|W_{b}-W_{\rm{en}}|\ll|W_{b}|$, finding $a_{\mu}$ and $b_{\mu}$ that maximizes $|W_{b}-W_{\rm{en}}|$ suffices. Practically, however, finding $|W_{b}-W_{\rm{en}}|_{\rm{max}}$ is complicated due to the absolute value term given by $|a_{y}b_{z}-a_{z}b_{y}|$. Therefore, in this paper we minimize the value $W_{\rm{en}}$.

Noiseless entanglement witness - First, we investigate the violation for the noiseless scenario. Here the atomic states starts in the $\hat{J}_{x}$ eigenstate $\ket{++...+}$ and the oscillator in the ground state $\ket{0}$ [18]; after evolving under Eq. 1, the final entangled state is given by $\ket{\psi_{\rm{en}}}$ (explicit form is given in the Supplemental Material).

We can understand the evolution of the system described by Eq. 1 within the Heisenberg picture. The first component to consider is the Heisenberg operator $\hat{c}(t)=e^{-i\omega t}\hat{c}(0)-i\int_{0}^{t}dt^{\prime}~{}g\hat{J}_{z}e^{-i\omega(t-t^{\prime})}$. The second component is the rotation of the spin operators $\hat{J}_{x}$ and $\hat{J}_{y}$ about the z-axis by angle $\epsilon(t)=g\int_{0}^{t}dt^{\prime}~{}[\hat{c}(t^{\prime})+\hat{c}(t^{\prime})^{\dagger}]$, due to the term $g~{}\hat{q}\hat{J}_{z}$ in the Hamiltonian; this is visually shown in Figure 1. These two components show us that the different operators become correlated with each other, for instance, $\hat{q}(t)$ with $\hat{J}_{z}$. Optimizing the values of $a_{\mu}$ and $b_{\mu}$ translates to controlling the contributions of these correlations such that the variances of Eq. 2 are minimized for the entangled state. For instance, if we consider the operator combination $\hat{J}_{y}(t)+a_{y}\hat{q}(t)+b_{y}\hat{p}(t)$, to first order in $\lambda$ (where $\lambda=g/\omega$) for simplicity, we have

The first line of Eq. 4 is the Heisenberg-evolved operator $\hat{J}_{y}(t)$ expanded to $O(\lambda)$ (with the squeezing term dropped). The term $a_{y}\hat{q}(t)+b_{y}\hat{p}(t)$ added to $J_{y}(t)$ contributes the coefficients $a_{y}$ and $b_{y}$; given the explicit form of $a_{y}\hat{q}(t)+b_{y}\hat{p}(t)$ as shown in the second line of Eq. 4, it seems plausible that values for $a_{y}$ and $b_{y}$ can be chosen such that the contribution to $\hat{J}_{y}(t)$ from $\hat{J}_{x}$ can be canceled. Note that, since we are starting in the state $\ket{++...}$, the $\hat{J}_{x}$ operators here evaluate to $N/2$ and the $O(\lambda\hat{J}_{z})$ terms evaluate to 0.

If we proceed with solving for $a_{\mu}$ and $b_{\mu}$, we see that this does indeed occur. We first expand various products of the operators ($\hat{J}_{x}(t)\hat{q}(t)$, etc.) to $O(\lambda^{2})$; $O(\lambda^{2})$ should be appropriate, since the only other parameter that can greatly “cancel out” the effects of $\lambda$ is the number of atoms $N$ and the highest order of $N$ is $N^{2}$ (which arises from the squaring of the spin operators). Once we have these Heisenberg-evolved operators and their products, we can evaluate their expectation values with respect to $\ket{++...}\ket{0}$ (which is the remainder of $\ket{\psi_{\rm{en}}}$) to obtain $W_{\rm{en}}$.

Now we can minimize $W_{\rm{en}}$ with respect to $a_{\mu}$ and $b_{\mu}$. We find that

As can be seen, these coefficients and, thereby, the EW of Eq. 2, depend on the following experimental parameters: the coupling $\lambda$, the number of atoms $N$, the mass of the oscillator $m$, oscillator frequency $\omega$, and the interaction time $t$. Furthermore, these coefficients satisfy $\mathbf{a}\cdot\mathbf{b}=0$.

We can substitute these coefficients into Eq. 4 to see how they can be used to cancel out correlations between operators. If we were to set $\bra{++...}\hat{J}_{x}\ket{++...}=N/2$, we find that $\bra{\psi_{\rm{en}}}\hat{J}_{y}+a_{y}\hat{q}+b_{y}\hat{p}\ket{\psi_{\rm{en}}}=\bra{0}\bra{++...}\hat{J}_{y}\ket{++...}\ket{0}$.

We can now finally obtain the bound

and the entangled-state value

We see that $\ket{\psi}_{\rm{en}}$ violates the EW bound by a value of $(3/2)\lambda^{2}N^{2}[1-\cos(\omega t)]$. It must be emphasized again that these calculations hold for $\lambda\ll 1$. The exact limit on $\lambda$ is such that Eq. 10 is always positive; that is, the result is valid for $\lambda<1/\sqrt{2N}$, which for $N=10^{6}$ would be $\lambda<7\times 10^{-4}$. Fig. 2 shows a plot of the violation (black line); it is highest at $t=\pi/\omega$, which is consistent with the results in Carney et al. [18] where the oscillator is completely entangled with the atoms at $t=\pi/\omega$.

Entanglement witness with a thermal state and white noise bath - We can now consider the case of thermal noise. There are three scenarios that can be considered; the first when the oscillator starts in a thermal state, the second when the oscillator starts in a thermal state and remains in contact with a white noise bath, and the third when the oscillator starts in the ground state and remains in contact with a bath.

For a thermal state $\ \hat{\rho}=\sum_{n=0}^{\infty}\frac{e^{-n\beta\omega}}{1-e^{-\beta\omega}}\ket{n}\bra{n}$ [24], we find that, to $O(\lambda)$, the new coefficients are

The next order of these coefficients is $O(\lambda^{3})$. Comparing with Equations 5-8, it can be seen that $a_{\rm{z,\bar{n}}}$ and $b_{\rm{z,\bar{n}}}$ have been modified by a factor of $1/(2\bar{n}+1)$. However, they no longer satisfy $\mathbf{a}\cdot\mathbf{b}=0$. As explained previously, and as can be seen by their similar forms, these coefficients work together with the covariances and correlations to minimize the value of $W_{\rm{en}}$ when starting in a thermal state.

The bound to $O(\lambda^{2})$ is

and the entangled state value of the EW to $O(\lambda^{2})$ is

The expansion to $O(\lambda^{2})$ is only valid if $\bar{n}\lambda^{2}\ll 1$; larger $\bar{n}\lambda^{2}$ leads to key corrections associated with atomic phase evolution of order unity. Plots for different values of $\bar{n}$ are shown in Fig. 2; the higher the value of $\bar{n}$, the weaker the violation.

Next, we consider a scenario where the oscillator is in continuous contact with white noise, which approximates a thermal bath at high temperature. The Hamiltonian is now given by

where $F_{\rm{in}}(t)$ is a classical white noise force; this Hamiltonian is simply that of Eq. 1 to which the effect from $F_{\rm{in}}(t)$ has been added. The auto-correlation function is a delta function with an amplitude set by the fluctuation-dissipation theorem $\langle\langle F_{\rm{in}}(t)F_{\rm{in}}(t^{\prime})\rangle\rangle=\bar{n}\gamma\delta(t-t^{\prime})$. Proceeding to understand within the Heisenberg picture as previously, we can solve for $\hat{c}(t)$ to obtain $\hat{c}(t)=e^{-i\omega t}\hat{c}(0)-i\int_{0}^{t}dt^{\prime}~{}(g\hat{J}_{z}+F_{in}(t^{\prime}))e^{-i\omega(t-t^{\prime})}$. With $\hat{q}(t)=q_{\rm{zpf}}[\hat{c}(t)+\hat{c}^{\dagger}(t)]$ and $\hat{p}(t)=ip_{\rm{zpf}}[-\hat{c}(t)+\hat{c}^{\dagger}(t)]$, the noise contributions to $\hat{q}(t)$ and $\hat{p}(t)$ can be expressed as $\mathcal{Q}(t)=-2q_{\rm{zpf}}\int_{0}^{t}dt^{\prime}~{}F_{\rm{in}}(t^{\prime})\sin[\omega(t-t^{\prime})]$ and $\mathcal{P}(t)=-2p_{\rm{zpf}}\int_{0}^{t}dt^{\prime}~{}F_{\rm{in}}(t^{\prime})\cos[\omega(t-t^{\prime})]$. Further, $\hat{J}_{x}$ and $\hat{J}_{y}$ will rotate about the z-axis by an angle $\xi(t)=g\int_{0}^{t}dt^{\prime}~{}[\hat{c}(t^{\prime})+\hat{c}(t^{\prime})^{\dagger}]$ as follows: $\hat{J}_{y}(t)=\hat{J}_{y}\cos[\xi(t)]+\hat{J}_{x}\sin[\xi(t)]$ and $\hat{J}_{x}(t)=\hat{J}_{x}\cos[\xi(t)]-\hat{J}_{y}\sin[\xi(t)]$. The white noise contribution to this angle is given by $\Xi(t)=-g\int_{0}^{t}dt^{\prime}~{}\mathcal{Q}(t^{\prime})/q_{\rm{zpf}}=2g\int_{0}^{t}\int_{0}^{t^{\prime}}dt^{\prime\prime}dt^{\prime}~{}F_{\rm{in}}(t^{\prime\prime})\sin[\omega(t^{\prime}-t^{\prime\prime})]$. Note that here, $q_{\rm{zpf}}=\sqrt{1/(2m\omega)}$ and $p_{\rm{zpf}}=\sqrt{m\omega/2}$.

Note that $\mathcal{Q}(t)$, $\mathcal{P}(t)$, and $\Xi(t)$ are not operators since they arise from the classical noise $F_{\rm{in}}(t)$; therefore, their covariances consist of averages over classical noise and are hence represented with the notation $\langle\langle...\rangle\rangle$. For instance, $\langle\langle\Xi(t)\Xi(t)\rangle\rangle$ contributes to the noise in the EW term $\Delta\langle\langle\hat{J}_{\rm{x}}\rangle\rangle^{2}$ and $\langle\langle\mathcal{Q}(t)\mathcal{Q}(t)\rangle\rangle$ contributes to the noise in $\langle\langle\hat{q}^{2}\rangle\rangle$. Therefore, the expectation values that compose the EW must be modified by adding the noise contributions $\Delta\langle\langle\hat{J}_{\rm{x}}\rangle\rangle^{2}$, $\Delta\langle\langle q^{2}\rangle\rangle$, $\Delta\langle\langle p^{2}\rangle\rangle$, $\Delta\langle\langle\hat{p}\hat{J}_{y}+\hat{J}_{y}\hat{p}\rangle\rangle$, $\Delta\langle\langle\hat{q}\hat{J}_{y}+\hat{J}_{y}\hat{q}\rangle\rangle$ and $\Delta\langle\langle\hat{q}\hat{p}+\hat{q}\hat{p}\rangle\rangle$; an appropriate linear combination of these gives $\Delta W_{\rm{\bar{n},noise}}$. Thus, we now have, $W_{\rm{en,\bar{n}},noise}=W_{\rm{en,\bar{n}}}+\Delta W_{\rm{\bar{n},noise}}$, where

with the coefficients being those in Equations 11-14. Note that the terms where $\lambda$ appears have been truncated to $O(\lambda^{2})$. The term $\langle\langle\hat{J}_{x}^{2}+J_{y}^{2}\rangle\rangle$ is unaffected by noise as it commutes through the Hamiltonian. Therefore, for the spin variances, only the term $\Delta\langle\langle\hat{J}_{\rm{x}}\rangle\rangle^{2}$ comes into play; starting in $\hat{J}_{x}$ eigenstate means that, to $O(\lambda^{2})$, there is no noise contribution from $\langle\langle\hat{J}_{\rm{y}}\rangle\rangle$ either.

Using the previous bound in Eq. 15, we plot the EW violation in Figure 2 (dotted lines). For violation to take place, the condition that

is required when $t=\pi/\omega$. The bound is approximately $\pi\bar{n}$ for $\bar{n}\gg 1$.

We can further consider the scenario where the oscillator is cooled to its ground state at the start, but remains in contact with a bath for the rest of time; this is given by $W_{\rm{en,0},noise}=W_{\rm{en,0}}+\Delta W_{\rm{\bar{n},noise}}$, with the coefficients of Eqs. 5-8 being used in Eq. 18 (see Supplemental Material for “re-optimized” coefficients). The dot-dashed lines in Fig. 2 shows the violation for different values of $\bar{n}/Q$. The condition for violation to take place is now

which for $t=\pi/\omega$ is $\frac{Q}{\bar{n}}>(7\pi/6)$. This is $O(\bar{n})$ better than for an initial thermal state.

We now investigate the value of the magnetic coupling in Figure 1. A large diamagnetic coupling can be attained by using superconducting masses and placing the atoms in different magnetically sensitive states. For instance, we can place one atom in the $m_{F}=1$ state and the other in $m_{F}=-1$. In such a scenario, the coupling would primarily be due to the first order Zeeman shift, which is linear in $B$ [25]. For a cylinder with $2R=L=5$ mm, $\chi=-1$, density$=11.34$ g/cm³ [26], $\Delta z=5~{}\mu$m, $s=8$ mm, $R_{0}=4.5$ cm, $B_{\rm{ext}}=$ 50 mT, $\omega=2\pi/20$, and $N=10^{6}$, we get $\lambda=6.3\times 10^{-4}$. A coupling approximately $\times 10^{2}$ weaker is available for atomic clock ($m_{F}=0$) states. For details on this calculation, see the Supplemental Material; this includes the cylindrical field expressions [27], and calculations for regular metals like tungsten [28, 29].

Conclusion - Within the context of a combined atom interferometer and oscillator system, we can use an entanglement witness to show that entanglement exists even in the case where the oscillator starts in a thermal state and remains in contact with a bath. In fact, we find that for the appropriate oscillator quality factor $Q$ and thermal occupancy $\bar{n}$, violation always exists for any value of magnetic coupling $\lambda$. Additionally, an $O(\bar{n})$ improvement in the EW violation if the oscillator is cooled to its ground state before running the experiment. However, to observe this violation, the number of measurements required would be at least on the order of 10⁵. Hence, experimentally using the proposed EW to measure entanglement may not be ideal; rather our results tell us that with the right physical parameters we can indeed experimentally generate entanglement between an atom interferometer and a mechanical oscillator, even with thermal noise. For practical detection of entanglement, other methods will have to be employed.

Acknowledgments - We would like to acknowledge Yuxin Wang and Jon Kunjummen for conceptual and mathematical discussions. We would also like to thank our collaborators for helping with experimental details and overall discussions, in particular, Jon Pratt, Garrett Louie, Cristian Panda, Prabudhya Bhattacharyya, Matt Tao, Lorenz Keck, James Egelhoff, John Manley, Stephan Schlamminger, Holger Müller, and Daniel Carney. We thank Daniel Barker and John Manley for comments on our paper. The work of G. P. and D. B. was made possible by the Heising-Simons Foundation grant 2023-4467 “Testing the Quantum Coherence of Gravity” and through the support of Grant 63121 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. J. T. is solely funded by the National Institute of Standards and Technology.