Single-Photon Pulse Induced Transient Entanglement Force

TL;DR

This paper demonstrates that a single-photon pulse can induce a transient entanglement force between two atoms, which can be tuned and enhanced, offering new ways to manipulate and observe atom-atom entanglement.

Contribution

It reveals that shaped single-photon pulses induce a tunable, transient entanglement force between atoms, with potential enhancement via graphene plasmons.

Findings

SPP induces a transient entanglement force between atoms.

The force can be tuned from repulsive to attractive by polarization.

Graphene plasmons can enhance the entanglement force by over three orders of magnitude.

Abstract

We show that a single photon pulse (SPP) incident on two interacting two-level atoms induces a transient entanglement force between them. After absorption of a multi-mode Fock state pulse, the time-dependent atomic interaction mediated by the vacuum fluctuations changes from the van der Waals interaction to the resonant dipole-dipole interaction (RDDI). We explicitly show that the RDDI force induced by the SPP fundamentally arises from the two-body transient entanglement between the atoms. This SPP induced entanglement force can be continuously tuned from being repulsive to attractive by varying the polarization of the pulse. We further demonstrate that the entanglement force can be enhanced by more than three orders of magnitude if the atomic interactions are mediated by graphene plasmons. These results demonstrate the potential of shaped SPPs as a powerful tool to manipulate this entanglement force and also provides a new approach to witness transient atom-atom entanglement.

Tables

Table 1. Table 1: The data of the Rb 85 superscript Rb 85 \rm{{}^{85}Rb} atom used in this paper coming from Ref.  Ru85data . We note that the spontaneous decay rate can be obtained directily from Eq. ( 41 ) with ω 0 subscript 𝜔 0 \omega_{0} and d 0 subscript 𝑑 0 d_{0} .

${}_{}{}^{85}\mathrm{Rb}$	Transition frequency ${\omega }_0$	Wave length
D1 ($5^2{S}_{1/2}\to 5^2{P}_{1/2}$)	$2\pi \times 3.77\times {10}^{14}$ Hz	$794.98$ nm
Transition dipole element $d_0$	Spontaneous decay rate ${\gamma }_0$	Life time ${\tau }_0=1/{\gamma }_0$
$2.54\times {10}^{-29}C\cdot m$	$2\pi \times 5.75\times {10}^6$ Hz	$27.68\times {10}^{-9}s$

Equations

$$\hat{F}(r)\equiv -\frac{\partial }{\partial r}\hat{U}(r)=\sum _{mn}{F}_{mn}(r)|m⟩⟨n|,$$

$${\hat{F}}_{\mathrm{RDDI}}(r)=F_{eg,ge}(r)|eg⟩⟨ge|+h.c.$$

$$|{\Psi }^{\pm }⟩=\frac{1}{\sqrt{2}}\left(|eg⟩\pm |ge⟩\right),$$

$$\frac{d}{dt}\tilde{\rho }(t)=[{\widehat{\hat{\mathcal{L}}}}_{\mathrm{atom}}+{\widehat{\hat{\mathcal{L}}}}_{\mathrm{pump}}(t)]\tilde{\rho }(t),$$

$${\widehat{\hat{\mathcal{L}}}}_{\mathrm{pump}}(t)\tilde{\rho }(t)=\sum _{j=1,2}\sqrt{{\gamma }_{jj}}{\eta }_j\left\{{\xi }^{\ast }(t-t_j)\left[{\hat{\sigma }}_{j+},\tilde{\rho }(t){\hat{\tau }}_{-}\right]+h.c.\right\},$$

$$\xi (t)={\left(\frac{1}{2\pi {\tau }_f^2}\right)}^{1/4}\exp \left[-\frac{t^2}{4{\tau }_f^2}-i{\omega }_{0}t\right],$$

$${\widehat{\hat{\mathcal{L}}}}_{\mathrm{atom}}\tilde{\rho }(t)=$$

$$+\sum _{ij}\frac{1}{2}{\gamma }_{ij}\left[2{\hat{\sigma }}_i^{-}\tilde{\rho }(t){\hat{\sigma }}_j^{+}-\tilde{\rho }(t){\hat{\sigma }}_i^{+}{\hat{\sigma }}_j^{-}-{\hat{\sigma }}_i^{+}{\hat{\sigma }}_j^{-}\tilde{\rho }(t)\right],$$

$${\sigma }_{\mathrm{intra}}(\omega )$$

$${\sigma }_{\mathrm{inter}}(\omega )$$

$$H(x)$$

$$\hat{H}|n⟩=\left(\sum _l{H}_{lk}|l⟩⟨k|\right)|n⟩=\sum _l{H}_{ln}|l⟩$$

$$\left(\frac{\partial }{\partial r}\hat{H}\right)|n⟩+\hat{H}|\frac{\partial }{\partial r}n⟩=\sum _l\left[\left(\frac{\partial }{\partial r}{H}_{ln}\right)|l⟩+H_{ln}|\frac{\partial }{\partial r}l⟩\right].$$

$$⟨m|\left(\frac{\partial }{\partial r}\hat{H}\right)|n⟩=\frac{\partial }{\partial r}{H}_{mn}+\sum _l\left[H_{ln}⟨m|\frac{\partial }{\partial r}l⟩-H_{ml}⟨l|\frac{\partial }{\partial r}n⟩\right].$$

$$\hat{U}(r)=U_{mn}(r)|m⟩⟨n|,$$

$$\hat{F}(r)$$

$$\hat{H}={\hat{H}}_F+\sum _{j=1,2}{\hat{H}}_{A,j}+\sum _j{\hat{H}}_{AF,j},$$

$${\hat{H}}_F=\int d^3\mathbf{x}{\int }_0^{\infty }d\omega \hslash \omega {\hat{\mathbf{f}}}^{†}(\mathbf{x},\omega )\cdot \hat{\mathbf{f}}(\mathbf{x},\omega ),$$

$$[{\hat{f}}_{\alpha }(\mathbf{x},\omega ),{\hat{f}}_{\beta }^{†}({\mathbf{x}}^{\prime },{\omega }^{\prime })]={\delta }_{\alpha \beta }\delta (\mathbf{x}-{\mathbf{x}}^{\prime })\delta (\omega -{\omega }^{\prime }),\alpha ,\beta =x,y,z$$

$$[{\hat{f}}_{\alpha }(\mathbf{x},\omega ),{\hat{f}}_{\beta }({\mathbf{x}}^{\prime },{\omega }^{\prime })]=[{\hat{f}}_{\alpha }^{†}(\mathbf{x},\omega ),{\hat{f}}_{\beta }^{†}({\mathbf{x}}^{\prime },{\omega }^{\prime })]=0.$$

$$H_{A,j}=\hslash {\omega }_{a,j}{\hat{\sigma }}_j^{+}{\hat{\sigma }}_j^{-},$$

$${\hat{H}}_{AF,j}=-({\mathbf{\mu }}_{j,eg}{\hat{\sigma }}_j^{+}+{\mathbf{\mu }}_{j,ge}{\hat{\sigma }}_j^{-})\cdot \hat{\mathbf{E}}({\mathbf{x}}_j),$$

$$\hat{\mathbf{E}}(\mathbf{x})={\int }_0^{\infty }d\omega \left[\hat{\mathbf{E}}(\mathbf{x},\omega )+{\hat{\mathbf{E}}}^{†}(\mathbf{x},\omega )\right],$$

$$\hat{\mathbf{E}}(\mathbf{x},\omega )$$

$${\hat{\mathbf{E}}}^{†}(\mathbf{x},\omega )$$

$${\overleftrightarrow{G}}^{†}(\mathbf{x},{\mathbf{x}}^{\prime },\omega )$$

$$\left[\vec{\nabla }\times \vec{\nabla }\times -\frac{{\omega }^2}{c^2}\epsilon (\mathbf{x},\omega )\right]\overleftrightarrow{G}(\mathbf{x},{\mathbf{x}}^{\prime },\omega )=\overleftrightarrow{I}\delta (\mathbf{x}-{\mathbf{x}}^{\prime }).$$

$${\overleftrightarrow{G}}^{\ast }(\mathbf{x},{\mathbf{x}}^{\prime },\omega )$$

$${\overleftrightarrow{G}}^T(\mathbf{x},{\mathbf{x}}^{\prime },\omega )$$

$$\int d^{3}x\frac{{\omega }^2}{c^2}{\epsilon }_I(\mathbf{x},\omega )\overleftrightarrow{G}({\mathbf{x}}_1,\mathbf{x},\omega ){\overleftrightarrow{G}}^{†}({\mathbf{x}}_2,\mathbf{x},\omega )$$

$$U_{gg,gg}(r)=-\frac{2{\mu }_0^2}{\hslash \pi }{\int }_0^{\infty }du\frac{{\omega }_{a,1}{\omega }_{a,2}{u}^4}{[{\omega }_{a,1}^2+u^2][{\omega }_{a,2}^2+u^2]}[{\mathbf{\mu }}_1\cdot \overleftrightarrow{G}({\mathbf{x}}_1,{\mathbf{x}}_2,iu)\cdot {\mathbf{\mu }}_2{]}^2.$$

$$G_{\parallel }({\mathbf{x}}_2,{\mathbf{x}}_1,\omega )$$

$$G_{\perp }({\mathbf{x}}_2,{\mathbf{x}}_1,\omega )$$

Full text

I Introduction

Single-photon-induced forces and torques correspond to the fundamental limit of optical linear momentum and angular momentum exchange with atoms tang2014prospect . Their direct detection is an open challenge since state-of-the-art quantum detectors are only sensitive to energy and arrival time of single photons eisaman2011invited . Recent advances in temporal shaping of single photon scattering from atoms has shed light on the role of the temporal waveform of Fock states Romero2012ultrfast . In light of these developments, it is an open question how single photon waveforms influence dipole-dipole interactions between atoms. Of particular interest is the exploration whether single photon shaped waveforms incident on interacting atoms can lead to experimentally observable transient effects.

During the last two decades, many techniques have been utilized to enhance the strength of the dipole-dipole interaction and the corresponding force buhmann2007dispersion , such as utilizing micro-cavity Agarwal1998microcavity ; Hopmeier1999enhanced ; Donaire2017dipole ; cortes2017fundamental , surface plasmons gonzalez2011entanglement ; Xu2011entanglement ; Haakh2014dynamical , and hyperbolic materials cortes2017super . Especially, the strong dipole-dipole interaction induced large energy shift in highly excited atoms (e.g. Rydberg atoms) has been proposed as the mechanism for “Rydberg blockade”, which provides a novel approach for quantum information processing Jaksch2000fast ; Lukin2001dipole and simulation of quantum phase transition labuhn2016tunable ; bernien2017probing . However, single-photon pulse as a tool to manipulate the transient dipole-dipole force has not been explored.

In this paper, we show the existence of a unique transient entanglement force between two neutral atoms induced by a single photon pulse. With the help of our defined force operator, we explicitly show that the resonant dipole-dipole interaction (RDDI) force fundamentally arises from two-body entanglement, which is significantly different from the van der Waals force. Our theoretical framework combines quantum theories of single-photon pulse scattering Domokos2002quantum ; wang2011efficient ; Baragiola2012n-photon ; yang2018concept and the macroscopic quantum electrodynamics approach of dipole-dipole interaction craig1998molecular ; Dung2002resonant ; Safari2006body ; shengwen2018magnetic . We thus show that the quantum statistics of the incident (Fock-state vs coherent-state) pulses lead to significant differences in the induced RDDI entanglement forces. After absorption of a single photon pulse, the inter-atomic force changes from the extremely weak van der Waals force london1937general ; Casimir1948retardation ; buhmann2007dispersion to the RDDI force mclone1964interaction ; stephen1964first with the amplitude enhanced by $\sim 10$ orders of magnitude.

We propose an experiment to detect this single photon pulse induced force with two levitated neutral atoms (see Fig. 1), which are separated with distance $r\sim 1\rm{\mu m}$ by optical tweezers operating at the magic wavelength barredo2016atom ; endres2016atom ; liu2018building . Even with this enhancement, detection of such a weak transient RDDI force is still a difficult challenge. Therefore, we we demonstrate that the single photon pulse induced RDDI force can be significantly enhanced by placing the atoms near a graphene layer with the assistance of graphene-based surface-plasmon polaritons. By investigating the full quantum dynamics of single-photon absorption, we predict optimum entanglement generation mechanisms conducive to experimental inquiry. Finally, we argue that the proposed effect can be differentiated from previously known dipolar interactions since the single photon pulse induced entanglement force can be tuned from repulsive to attractive by tuning the polarization of the incident pulse.

II Dipole-dipole interaction force operator

With the help of the Hellmann-Feynman theorem Feynman1937force , we define a quantum operator to characterize the force generated by the coherent part of the dipole-dipole interaction in A,

[TABLE]

where $F_{mn}(r)\equiv-\partial U_{mn}(r)/\partial r$ is determined by the atom-atom interaction $\hat{U}(r)=\sum_{mn}U_{mn}(r)\left|m\right\rangle\left\langle n\right|$ induced by electromagnetic vacuum fluctuations ficek2002entangled ; Dung2002resonant and $\left|m\right\rangle\in\{|gg\rangle,|eg\rangle,\left|ge\right\rangle,\left|ee\right\rangle\}$ for a two-level-atom pair. The dipole-dipole interaction force is always along the axis joining the two atoms. Our defined force operator allows us directly to classify the dipole-dipole interaction force into two categories: (1) van der Waals force between two atoms in a direct-product state, such as the force for two ground-state atoms $\hat{F}_{\rm vdW}=F_{gg,gg}|gg\rangle\langle gg|$; (2) RDDI force for entangled atoms, e.g.,

[TABLE]

We will show how to control this force with a single photon pulse later.

We emphasize that the latter RDDI force fundamentally arises from two-body entanglement Behunin2010non . The eigenvectors of the force operator $\hat{F}_{\rm RDDI}(r)$ are the two Bell states

[TABLE]

with eigenvalues $\pm F_{eg,ge}(r)$. For a given two-atom state $\rho(t)$, the absolute value of the RDDI force is proportional to to the probability difference of the two-atom state on these two entangled states, i.e., $F_{\rm RDDI}(r,t)\propto|\langle\Psi^{+}|\rho(t)|\Psi^{+}\rangle-\langle\Psi^{-}|\rho(t)|\Psi^{-}\rangle|$. This immediately reveals that, to maximize the RDDI force, one needs to prepare the atom pair in one of these two entangled states. We also note that, the RDDI force presents a readout of two-body entanglement. This entanglement force between transition dipoles is fundamentally different from van der Waals force Casimir1948retardation and the force generated by the permanent dipole-dipole interaction craig1998molecular . We emphasize that the maximum possible RDDI force (the eigenvalue of the force operator) is determined by the atom-atom distance $r$. However, the exact time-dependent envelope of the RDDI force in a specific dynamical process is determined by the atomic two-body entanglement.

III Dynamical entanglement force

The master equation method has been broadly applied to study the dipole-dipole interaction and entanglement between neutral atoms ficek2002entangled ; Dung2002resonant ; kastel2005suppresion ; brooke2008super ; Dzsotjan2010quantum . We now incorporate the single photon pulse absorption dynamics with the traditional master equation to show the time-dependent entanglement force induced by a single photon pulse (see F),

[TABLE]

where $\tilde{\rho}(t)=\rho_{{\rm PN}}(t)\otimes\rho(t)$ is an effective density matrix. We have introduced an extra qubit degree of freedom $\rho_{{\rm PN}}(t)$ to characterize the photon number degree (see more details in Baragiola2012n-photon ). The initial value of $\tilde{\rho}(t)$ is given by $\tilde{\rho}(0)=\hat{I}_{{\rm PN}}\otimes\rho(0)$, where $\hat{I}_{{\rm PN}}$ is the two-dimensional identity matrix and $\rho(0)=\left|gg\right\rangle\left\langle gg\right|$ denotes the initial state of the atom pair.

The quantum pumping from a single photon pulse is characterized by,

[TABLE]

where $\gamma_{jj}=\gamma_{0}$ is the spontaneous decay rate of the atoms in vacuum. The coefficient $\eta_{j}$ characterizes the pumping efficiency, which is determined by the effective scattering cross section of the $j$th atom. The wave-packet amplitude of a Gaussian single photon pulse is given by

[TABLE]

with center frequency $\omega_{0}$ and pulse length $\tau_{f}$ yang2018concept . The time that the center of the pulse arrives at the $j$th atom is given by $t_{j}=\boldsymbol{k}_{0}\cdot\boldsymbol{x}_{j}/\omega_{0}$ ($|\boldsymbol{k}_{0}|=\omega_{0}/c$). The absorption of the pulse is characterized by the Pauli matrix $\hat{\tau}_{-}$ of the extra qubit degree. The interatomic RDDI are included in the regular time-independent Lindblad superoperator ficek2002entangled ; Dung2002resonant

[TABLE]

where $\omega_{0}$ is the energy splitting of the two-level atoms, and the energy shifts $\delta_{ij}=U_{eg,ge}(r)/\hbar$ and decay rates $\gamma_{ij}$ are given in  D.

Both the imaginary part (the cooperative decay rates $\gamma_{12}=\gamma_{21}$) and the real part (the energy shift $\delta_{12}=\delta_{21}$) of the RDDI are dependent on the polarization of the atomic dipoles $\boldsymbol{\mu}_{j}$ with respect to the relative displacement vector $\boldsymbol{r}$. As shown in Fig. 2(a), the cooperative decay rates decrease monotonously with atom-atom distance $r$ in the near region, begins to oscillate in the medium region, and vanishes in the far region. Note that, the sub-indices ∥ and ⟂ denote the cases when $\boldsymbol{\mu}_{j}$ is parallel and perpendicular to $\boldsymbol{r}$, respectively. Although $\gamma_{12,\parallel}$ and $\gamma_{12,\perp}$ behave differently, both of them converges to the spontaneous decay rate $\gamma_{0}$ in the near region and decrease to zero in the far region [see the subplot in Fig. 2(a)]. Rewriting the master equation (7) in the bright and dark states basis, this will automatically give the superradiance and subradiance DeVoe1996observation . The coherent part of the RDDI diverges in the near region. More importantly, $\delta_{12,\parallel}$ and $\delta_{12,\perp}$ usually have opposite signs, especially in the near region. This lays the foundation to tune the RDDI force by tuning the polarization of the pulse as explained in the following.

The time-dependent RDDI entanglement force, $F_{{\rm RDDI}}(r,t)=Tr[\rho(t)\hat{F}_{\rm RDDI}(r)]$, induced by a single photon pulse for different atom distance is displayed in Fig. 3. For a fixed inter-atomic distance, the RDDI force increases after the pulse excites the atoms and decreases with time when atoms re-emit the photon. We can also see the amplitude of the RDDI force oscillates with atom distance $r$, due to the oscillation in the matrix elements $F_{ge,eg}(r)$ of the RDDI force operator. The van der Waals force has been neglected here as it is negligibly small as shown in C. The impulse force from the incident pulse is estimated to be $F_{\rm imp}\approx\hbar\omega_{0}/c\tau_{f}\sim 10^{-20}$N with center frequency $\omega_{0}\approx 2\pi\times 3.77\times 10^{14}\>{\rm Hz}$ and pulse length $\tau_{f}\sim 30\,{\rm ns}$. But this force is along $y$-axis, which is perpendicular to the inter-atomic force in $x$-direction and can be relieved by the trapping force in $y$-axis. Thus, the only relevant force along the axis joining the two atoms is the RDDI entanglement force.

Quantum entanglement fundamentally determines the time-dependent RDDI force induced by a single photon pulse. Here, we use the concurrence to quantitatively characterize the two-qubit entanglement Wootters1998entanglement . As shown in Fig. 4 (a), for fixed atom-atom distance $r=1.2\,{\rm\mu m}$, the concurrence $\mathcal{C}(t)$(the dashed-pink line) and the RDDI force $F_{{\rm RDDI}}(t)$ (the solid-blue line), as well as the excitation probability of the first atom $P_{1e}(t)$ (the dotted-red line), reach their maxima simultaneously for homogeneous pumping case ($\eta_{1}=\eta_{2}$). But for the local pumping of the first atom case with $\eta_{1}=1$ and $\eta_{2}=0$ [see Fig. 4 (b)], $\mathcal{C}(t)$ and $F_{{\rm RDDI}}(t)$ reach their peaks at the time, which is later than the time when $P_{1e}(t)$ reaches its maximum. Thus, it is the entanglement instead of the total excitation probability that maximizes the RDDI force. We also see that there are two ways to generate the quantum entanglement between the atoms: (1) homogeneous pumping to the symmetric state $\left|\Psi^{+}\right\rangle$ directly by the single photon pulse; (2) local pumping of single atom to state $\left|eg\right\rangle$ and then the RDDI evolves the atoms to entangled states. Here, we show that the first one is more efficient for entanglement generation. The total photon absorption probability $P_{e,tot}(t)$ for both homogeneous [$P_{e,tot}(t)=2P_{1e}(t)$ in Fig. 4(a)] and local pumping cases [$P_{e,tot}(t)=P_{1e}(t)$ in Fig. 4(b)] are almost the same. But the entanglement and the RDDI entanglement force under homogeneous pumping are much larger than that of local pumping case. This is because the projection of the atomic state $\rho(t)$ on the entangled state $\left|\Psi^{+}\right\rangle$ under homogeneous pumping is much larger.

The existing theory craig1998molecular ; buhmann2007dispersion ; Dung2002resonant ; ficek2002entangled can not describe the quantum pulse induced dipole-dipole interaction force. Now, we show that the force induced by a Fock-state pulse is significantly different from the one induced by a coherent-state pulse. As explained in Ref. yang2018concept , the absorption probability of Fock-state single photon pulse by a two-level atom is much higher than that of coherent-state pulse. Thus, the corresponding force is larger as shown by the blue lines in Fig. 5. However, there exists an optimal pulse length $\tau_{f,{\rm opt}}$ to reach the largest excitation probability of the atoms for Fock-state pulses Baragiola2012n-photon . For fixed pulse length $\tau_{f}\gamma_{0}=0.3$, the maximum entanglement force decreases with photon number in Fig. 5 (a), as the total excitation probability decreases Baragiola2012n-photon . But the force induced by coherent pulse always increases with the mean photon number [see Fig. 5 (b)]. In an experiment, larger entanglement force can be obtained by optimizing the pulse length to increase the atomic excitation probability for given atomic transition frequency and dipole-dipole interaction strength as shown in F.

IV Near-field enhancement of the entanglement force

The entanglement force can be enhanced significantly by engineering the nanophotonic environment near the atoms. As a practical illustration, we demonstrate this enhancement by placing the atoms near a graphene layer as depicted in Fig. 1 (b). The surface plasmon polaritons of graphene have been previously shown to allow conventionally forbidden atomic transitions rivera2016shrinking in addition to enhancing other well-known physical effects such as decay rate of emitters koppens2011graphene and Förster energy transfer rate biehs2013large . This enhancement fundamentally originates from the strong light-matter interaction due to the large density of states of the surface plasmon modes, i.e. the polaritons generated by the strong coupling between the electromagnetic field and the charge excitations at a conductor surface koppens2011graphene . Since the field is strongly confined at the surface, thus the corresponding enhancement only occurs when the emitters are placed close to the surface.

Here, we show that the RDDI strength and the time-dependent entanglement force can be enhanced significantly by placing the atoms near a graphene layer. As presented in D, the RDDI strength can be directly evaluated via the classical Green’s tensor $\overleftrightarrow{G}(\mathbf{x}_{1},\mathbf{x}_{2},\omega)$. In the presence of a planar surface, the Green tensor in the upper half-space can always be split into two parts novotny2012principles : $\overleftrightarrow{G}(\mathbf{x}_{1},\mathbf{x}_{2},\omega)=\overleftrightarrow{G}_{0}(\mathbf{x}_{1},\mathbf{x}_{2},\omega)+\overleftrightarrow{G}_{R}(\mathbf{x}_{1},\mathbf{x}_{2},\omega)$ corresponding to the contributions from the free space and the reflection by graphene, respectively. The free space Green tensor has been analytically given in Refs. knoll2000qed ; Safari2015body ; Morice1995refractive . The reflection Green tensor can be obtained from the optical conductivity of a graphene layer (see more details in E). The in-plane optical conductivity of graphene includes intra-band and inter-band contributions koppens2011graphene ; biehs2013large ; nikitin2011fields ; falkovsky2007space ; stauber2008optical $\sigma(\omega)=\sigma_{{\rm intra}}(\omega)+\sigma_{{\rm inter}}(\omega)$ with

[TABLE]

and

[TABLE]

where $\tau_{D}$ is the relaxation time in the Drude model, $E_{F}$ is the graphene’s Fermi energy, $T$ is the temperature, and the function

[TABLE]

Figure 6 (a) demonstrates the distance dependence of the entanglement force. For atomic transition frequency close to graphene surface plasmon polaritons (exact data provided in E), the enhancement factor is larger than $1000$ at atom-surface distance $z_{0}=10$ nm (red curve). When the two atoms are very close to the graphene layer, the RDDI is primarily mediated by the surface plasmon polaritons in the graphene layer instead of the vacuum fluctuations. The large density of states of surface polaritons enhances the strength of RDDI by orders of magnitude. While the graphene-based surface plasmon polaritons occur in the terahertz to near-infrared band koppens2011graphene ; nikitin2011fields , similar enhancement at optical frequencies are feasible with other plasmonic materials such as gold and silver west2010searching .

V Precise control of the entanglement force

Now, we show single photon pulse as a novel tool to precisely control the atomic force: (1) a more than ten orders of dipole-dipole interaction force amplitude change can be induced by a single photon pulse; (2) the induced entanglement force can be continuously tuned from being repulsive to attractive by varying the polarization of the pulse. For relevant inter-atomic separations ($r\sim 1\,{\rm\mu m}$), the van der Waals force is around $\sim 5\times 10^{-35}$ N (see Fig. 7), which is far beyond the state-of-art force sensitivity. As the van der Waals force arises from higher-order process, thus it is much weaker than the RDDI force. After absorption of a single photon pulse, the RDDI force dominates with a greatly enhanced amplitude $\sim 10^{-22}$ N. This force can be further enhanced upto $10^{-19}$ N with surface plasmons-plaritons. Using phase-coherent Doppler velocimetry, force sensitivity of $\sim 10^{-24}{\rm N/\sqrt{Hz}}$ can be approached in trapped ion systems biercuk2010ultrasensitive . In a Mach-Zehnder-type interferometer with a free fall cesium atom from an optical tweezer, a force of magnitude $3.2\times 10^{-27}$ N has been measured in an experiment Parazzoli2012observation . Therefore, we are confident that that the transient entanglement force induced by a single-photon pulse can also be detected in the near future.

For atomic transition between states connected by linearly polarized light, the direction of the corresponding transition dipole is determined by the polarization of the incident pulse. As shown in Fig. 6 (b), both the forces induced by parallelly ($\parallel$) and perpendicularly ($\perp$) polarized pulses oscillate with the atomic distance around $r\sim 1\,{\rm\mu m}$. But these two forces have a phase shift and usually have opposite signs (especially in the near region $r<0.5\,{\rm\mu m}$). Thus, we can control the force to be repulsive or attractive by changing only the polarization of the pulse. More importantly, we can continuously tune the value of the RDDI force via tuning the pulse polarization angle $\theta$ in $xz$-plane with fixed atom-atom distance ($r$) (see the subplot).

Conclusion and discussion

We reveals the essential role of the two-body entanglement in the RDDI force. We utilize a time-dependent theoretical framework to study the transient entanglement force between two neutral atoms induced by a quantum pulse. We also show that this entanglement force can be significantly enhanced by engineering their nano-photonic environment and precisely controlled by tuning the polarization of the incident pulse.

Looking ahead, our work provides a natural platform to investigate photoassociation in chemical reactions and bioprocesses  liu2018building . By generalizing the force operator to multi-atom case, we can also study the role of the many-body entanglement in the collective force of neutral atom ensembleSinha2018collective ; yang2018concept . The photon absorption probability and atom-atom entanglement can be enhanced by tailoring the shape and the time-frequency correlation of photon pulses yang2018concept .

Acknowledgments

This work is supported by the DARPA DETECT ARO award (W911NF-18-1-0074).

Appendix A Dipole-dipole interaction force operator

According to the Hellmann-Feynman theorem Feynman1937force , we perform the derivation to the secular equation with respect to the atom-atom separation $r$,

[TABLE]

to obtain

[TABLE]

Multiply both side with $\left\langle m\right|$, we have

[TABLE]

In most case, due to the non-adiabatic transition terms in the square brackets, there does not exist a well defined force operator for a microscopic system, such as the exchanging interaction in a condensed-matter lattice. But in our case, the distance between the two atoms is much larger than the size the the atoms. Thus, the atomic wave function is not dependent on the relative distance $r$ and the second term at the right-hand-side disappears (i.e., $\langle l|\partial n/\partial r\rangle=0$).

In the atomic Hamiltonian, only the dipole-dipole interaction part

[TABLE]

depends on the inter-atomic distance $r$. As the corresponding force is always along the co-axis line, we can define a scalar operator for this force as,

[TABLE]

We note that this force operator only works for weak atom-field coupling case. If the two atoms strongly coupled to a resonant cavity field, one can not eliminate the degree of the cavity mode to obtain an effective interaction Hamiltonian as shown in Eq. (14). In this case, the inter-atomic force is not only dependent on atom-atom separation, but also the position of each atom esfandiarpour2018cavity . More important, the magnitude of the forces experienced by the two atoms can be different, which violates Newton’s third law for a macroscopic body. We do not consider this case in this paper.

Different elements in the operator $\hat{F}(r)$ correspond to different virtual processes generated forces. We emphasize that only the anti-diagonal elements of the two-body interaction in (14) can be mediated by second-order processes craig1998molecular and all the other terms result mainly from fourth order processes. Thus, the corresponding forces are weak. In this paper, we only focus on two forces. The first one is the van der Waals (vdW) force between two ground-state atoms $F_{{\rm vdW}}\propto F_{gg,gg}(r)$, which mainly arises from fourth-order process craig1998molecular ; Safari2006body and usually is extremely small. An incident single-photon pulse can pump the atom pair to an entangled state. In this case, the interaction changes to the RDDI, which plays the key role in energy transfer between different molecules in chemical and biological processes. As the RDDI is mediated by second-order processes, the corresponding force $F_{{\rm RDDI}}\sim F_{ge,eg}(r)$ between the two atoms will be greatly enhanced. In the following, we present the approach to calculate the elements of $\hat{U}(r)$ and $\hat{F}(r)$.

Appendix B Model Hamiltonian for Atom-Field Interaction

The Hamiltonian of the total system is given by

[TABLE]

where the Hamiltonian of the field modes in an arbitrary linear (non-magnetic) media is given by Dung1998three ; knoll2000qed

[TABLE]

and the ladder operators of the eigen modes satisfy the commutation relations

[TABLE]

and

[TABLE]

The Hamiltonian of the two atoms is

[TABLE]

where $\omega_{a,j}$ is the energy splitting of the $j$th atom and $\hat{\sigma}_{j}^{+}=(\hat{\sigma}_{j}^{-})^{\dagger}=\left|e_{j}\right\rangle\left\langle g_{j}\right|$ is the Puali matrix. There are two forms of Hamiltonian to describe the interaction between the atoms and the electromagnetic field. One is the minimum coupling and the other one is the multipolar coupling craig1998molecular . The difference and relation between these two forms of interaction can be found in craig1998molecular ; Buhmann2004Casimir . Here, we use the multiploar interaction Hamiltonian

[TABLE]

where $\boldsymbol{\mu}_{j,eg}$ is the electric dipole transition element of the $j$th atom. In the following, for simplicity, we consider two identical atom case $\boldsymbol{\mu}_{j,eg}=\boldsymbol{\mu}_{j,eg}=\boldsymbol{\mu}_{j}=d_{0}\boldsymbol{e}_{j}$.

The electric field operator can be expanded with the eigen modes of the field as

[TABLE]

where

[TABLE]

with $\sqrt{\varepsilon_{I}(\boldsymbol{x}^{\prime},\omega)}$ the imaginary part of the complex permittivity $\varepsilon(\boldsymbol{x},\omega)$, the vacuum permittivity $\varepsilon_{0}$, and the speed of light $c$ in vacuum. The function $\overleftrightarrow{G}(\mathbf{x},\mathbf{x}^{\prime},\omega)$ is the classical Green tensor obeying the equation

[TABLE]

Here, we assume that the media is a non-magnetic material with constant permeability $\mu_{0}=1$ and the frequency-dependent complex permittivity $\varepsilon(\mathbf{x},\omega)$. The Green tensor has the properties

[TABLE]

We will show that both the van der Waals interaction and the resonant dipole-dipole interaction can be easily obtained with the Green tensor.

Appendix C van der Waals Interaction

The van der Waals interaction between two atoms has between well studied. A detailed calculation of the coherent van der Waals interaction in free space is presented in Ref. craig1998molecular . Here, we only present the more general form of van der Waals interaction between two identical atoms obtained by Safari and his collaborators Safari2006body ,

[TABLE]

The incoherent part of van der Waals interaction has been neglected, as it is usually negligible small compared to the spontaneous decay rate of the atoms.

C.1 Free-space case

In this subsection, we recover the well known van der Waals force in free space. It is easy to find that if we let $\mathbf{r}=\mathbf{x}_{2}-\mathbf{x}_{1}=(r,0,0)$, only the diagonal elements of the free space Green tensor are nonzero knoll2000qed ; Safari2015body ,

[TABLE]

Here, the sub-indices $\parallel$ and $\perp$ denote parallel and perpendicular to $\boldsymbol{r}$, respectively.

As the ground-state atoms can be excited by arbitrarily polarized virtual photons. Thus, to calculate the van der Waals interaction, we need average out the polarization angle by taking the spherically symmetric polarizability tensor [see Eq. (49) in Ref. Safari2006body ]. Finally, the van der Waals interaction between two ground-state atom is given by

[TABLE]

Using the method presented in craig1998molecular (see Chaps. 7.5 and 7.6), we can verify that:

[TABLE]

Thus, the corresponding force $F_{{\rm vdW}}(r)$ will be of scale$\sim 1/r^{7}$ in the near region and $\sim 1/r^{8}$ in the far region. As shown by the pink curve in Fig. 7, the van der Waals force $F_{{\rm vdW}}(r)$ deviate from the line $1/r^{7}$ (the thin black line) slightly in the far region.

Appendix D Master-Equation Method to Calculate The Resonant Dipole-Dipole Interaction

In this section, we calculate the RDDI strength via the Lindblad form master equation for a two-level-atom pair

[TABLE]

where the decay rates are given by

[TABLE]

and the RDDI energy

[TABLE]

can also be obtained with Heisenberg equations shengwen2018magnetic ; supplementary .

This master equation can also be found in Refs. ficek2002entangled ; Dung2002resonant ; Dzsotjan2010quantum . For atomic states connected by linearly polarized light, the direction of the transition dipoles $\boldsymbol{e}_{j}$ are determined by the polarization of the incident pulse. This makes it possible to precisely control the RDDI force by tuning the polarization of the pulse as shown in the main text.

D.1 Resonant dipole-dipole interaction force in free space

In this subsection, we calculate the RDDI force in free space. It is straightforward to verify that, for the free space single point Green’s function, the real part diverges, but the imaginary part does not,

[TABLE]

[TABLE]

Then, we can obtain the well known spontaneous decay rate of an atoms in free space,

[TABLE]

We will take $\gamma_{0}=1$ as the unit of frequency and $1/\gamma_{0}$ as the unit of time in this paper. As shown in the next section, both the coherent and incoherent dipole-dipole interaction can be greatly enhanced by engineering the electromagnetic environment to change the Green tensor.

Substitute the free space Green’s tensor (31) and (32) back to the incoherent part (36) and coherent part (38) of the RDDI, we can obtain the corresponding cooperative decay rates and the energy shifts of the atoms in free space,

[TABLE]

[TABLE]

and

[TABLE]

where $k_{0}=\omega_{0}/c$.

The matrix element of the force operator $\hat{F}_{\rm RDDI}$ are given by

[TABLE]

and

[TABLE]

The numerical simulation of the forces are displayed in Fig. 7. In the near region, the RDDI force decreases with $1/r^{4}$. In the far region, $F_{\rm RDDI,\parallel}$ decreases with $1/r^{2}$ (green solid line) and $F_{\rm RDDI,\perp}$ vanishes with scaling $1/r$ (blue dotted line).

Appendix E Dipole-dipole Force near Planar Interface

As shown in previous sections, the Green tensor plays the key role in evaluation of the dipole-dipole interaction as well as the corresponding force. In this section, we explain how to calculate the RDDI force near a planar interface via the Green tensor.

The Green tensor near a planar interface is given by novotny2012principles

[TABLE]

where $\overleftrightarrow{G}_{0}$ is the Green tensor in the free space, and $\overleftrightarrow{G}_{R}$ and $\overleftrightarrow{G}_{T}$ are the contribution due to the reflection and transmission, respectively. The interface is at the plane $z=0$ and the dipole source (the atoms) are placed above the interface. Thus, all the reflected field has $z>0$ and all the transmitted field has $z<0$.

The free-space dyadic Green Tensor in real space can be written as the sum of the following terms arnoldus2003transverse

[TABLE]

where the far-, intermediate-, and near-field terms are given by,

[TABLE]

[TABLE]

and

[TABLE]

with $\mathbf{e}_{r}=\mathbf{r}/r$. The Green tensor $\overleftrightarrow{G}_{0}$ in (49) is the exact same as the one given in Eqs. (31) and (32).

Usually, the reflection Green tenor $\overleftrightarrow{G}_{R}=\overleftrightarrow{G}_{R}^{s}+\overleftrightarrow{G}_{R}^{p}$ and the transmission Green tensor $\overleftrightarrow{G}_{T}=\overleftrightarrow{G}_{T}^{s}+\overleftrightarrow{G}_{T}^{p}$ (the index $s$ and $p$ denote the s-polarized part and the p-polarized part, respectively) can only be obtained numerically via nikitin2011fields ,

[TABLE]

and

[TABLE]

where $k_{\omega}=\omega/c$ is the modular of the wave vector in free space, $q_{\alpha}=k_{\alpha}/k_{\omega},\alpha=x,y,z$ is the normalized dimensionless wave vector, $q=\sqrt{q_{x}^{2}+q_{y}^{2}}$ the projection of $\vec{q}$ on the $xy$-plane, and $q^{\prime}_{z}=\sqrt{\varepsilon(\omega)-q^{2}}$ with the relative permittivity of the outgoing media $\varepsilon(\omega)$. The kernals in the integrals are given by,

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Here, we have carried out the azimuth angle integral of $\vec{q}$ on the $xy$-plane and re-expressed the displacement $\mathbf{r}$ in the cylinder coordinate as $\mathbf{r}=r_{\perp}\mathbf{e}_{\rho}+z\mathbf{e}_{z}$ with $x=r_{\perp}\cos\phi_{0}$ and $\ y=r_{\perp}\sin\phi_{0}$. In these $M$-matrices, $J_{n}$ denotes Bessel function of $n$th order $J[n,qk_{\omega}r_{\perp}]$.

The Fresnel reflection and transmission coefficients of graphene-layer interface are given by koppens2011graphene ; nikitin2011fields

[TABLE]

[TABLE]

[TABLE]

where $\alpha(\omega)=2\pi\sigma(\omega)/\varepsilon_{0}c$ is the dimensionless in-plane conductivity of the graphene. The optical conductivity of a graphene layer can be split into intra-band and inter-band contributions $\sigma(\omega)=\sigma_{{\rm intra}}(\omega)+\sigma_{{\rm inter}}(\omega)$ with koppens2011graphene ; biehs2013large

[TABLE]

and

[TABLE]

where $\tau_{D}$ is the relaxation time in the Drude model, $E_{F}$ the graphene’s Fermi energy, and the function

[TABLE]

The RDDI strength for two atoms on top of a graphene layer is given by

[TABLE]

Then, the eigen value of the RDDI force operator on the state $|\Psi^{+}\rangle$ is obtained as $F(r)=-\partial U_{eg,ge}(r)/\partial r$. In Fig. 8, to show the enhancement in the RDDI force due to the graphene layer, we re-scale $F(r)$ with the eigen value $F_{\rm vacuum}(r_{0})$ of the corresponding RDDI force operator in vacuum at $r_{0}=1.05{\rm\mu m}$ (denoted by the vertical black line). Comparing with the subplot, we see that more than three order enhancement in the force can be obtained if the atoms are very close to the graphene layer ($z_{0}=10$ nm). We also see that this enhancement decreases fast with the hight of the atoms $z_{0}$ and vanishes for $z_{0}>500$ nm.

In the main text, the corresponding time-dependent entanglement force induced by a single photon pulse has been displayed. The inter-atomic distance is set as $r=1.05{\rm\mu m}$ as marked by the dark vertical line in Fig. 8 and the atom-interface distance is set as $z_{0}=10,20,50$ nm. The pulse length $\tau_{f}$ has been optimized to get the maximum entanglement force as both the local spontaneous decay rate $\gamma_{ii}$ and the cooperative decay rates $\gamma_{ij}$ defined in Eq. 36 have also been greatly enhanced by the graphene layer.

Appendix F Time-dependent Master Equation for Quantum Pulse Scattering Processes

In this section, we study the dynamics of a two-level-atom pair. Different from the previous literatures, we prepare the atom pair in the ground state $\left|gg\right\rangle$ instead of a single-excited state (e.g., $\left|eg\right\rangle$). In 2012, Ben et al. derived a powerful time-dependent master equation for $n$-photon broadband pulse interacting with an arbitrary quantum system. Here, we generalize this method to calculate the dynamical RDDI force.

The total master equation including the single-photon pumping process is given by,

[TABLE]

where $\tilde{\rho}(t)=\rho_{{\rm PN}}(t)\otimes\rho(t)$ is an effective density matrix and we have introduced an extra qubit degree of freedom $\rho_{{\rm PN}}(t)$ to characterize the photon number degree (see more details in Ref. Baragiola2012n-photon ). The initial value of $\tilde{\rho}(t)$ is given by $\tilde{\rho}(0)=\hat{I}_{{\rm PN}}\otimes\rho(0)$, where $\hat{I}_{{\rm PN}}$ is the two-dimensional identity matrix and $\rho(0)=\left|gg\right\rangle\left\langle gg\right|$ is the initial state of the atom pair.

The the first term at right hand side (r.h.s.) of Eq. (81) characterizes the free evolution of the atom pair without the pumping

[TABLE]

The second term characterizes the pumping of the single-photon pulse,

[TABLE]

with Pauli matrices $\hat{\tau}_{-}$ characterizing the absorption of the single photon pulse. The parameter $\eta_{j}$ characterizes the pumping efficiency of the $j$th atom determined by its effective scattering cross section, $t_{j}=(\mathbf{x}_{j}\cdot\vec{e}_{p})/c$ is the time of the center of the pulse arriving the $j$th atom, and

[TABLE]

is the Fourier transform of the pulse spectrum function. For a Gaussian single photon pulse,

[TABLE]

its wave packet amplitude in the time-space domain is given by,

[TABLE]

In the main text, we assume the pulse propagates along the $x$-axis and arrives at the two atoms at the same time $t_{1}=t_{2}=0$. The pumping efficiency $\eta_{j}$ in practice shoule be much smaller than $1$ wang2011efficient ; yang2018concept , but its can be enhanced by adding a mode converter sondermann2007design . In our simulation, we take $\eta_{1}=\eta_{2}=1/\sqrt{2}$ for the homogeneous pumping case and $\eta_{1}=1,\,\eta_{2}=0$ for the locally pumping case.

This effective master equation method can be straightforwardly generalized to $n$-photon Fock-State pulse case by replacing the Pauli matrix $\hat{\tau}_{\pm}$ in Eqs. (81-83) with

[TABLE]

and replacing the $2\times 2$ identity matrix $\hat{I}_{{\rm PN}}$ with the $(n+1)\times(n+1)$ identity matrix.

Actually, $\tilde{\rho}(t)$ is not a real density matrix of a physical system, as ${\rm Tr}\tilde{\rho}(0)=n$ for $n$-photon Fock-state pulse. Thus, only its projection on the specific subspace has physical meaning. The expected value of any atomic operator $\hat{O}$ is given by

[TABLE]

where $\hat{P}$ is the projection operator of the extra qubit degree with the only non-zero element $P_{11}=1$. We also note that, to handle the coherent-state pulse case, we only need to replace all the photon related operators (i.e., $\hat{\tau}_{\pm}$, $\hat{I}_{\rm PN}$, and $\hat{P}$) with the constant $1$. This powerful time-dependent master equation (81) can be used to uniformly study the quantum photon pulse scattering process.

We can also enhance the RDDI force by changing the pulse length $\tau_{f}$ to optimize the two-body entanglement (see Fig. 9). Here, we see that, for homogeneous pumping case with $\eta_{1}=\eta_{2}=1/\sqrt{2}$, the optimal pulse length maximizes the local excitation probability of the first atom $P_{1e}$, the inter-atomic force $F_{\rm RDDI}$, and the concurrence $\mathcal{C}$ simultaneously [see Fig. 9(a)]. But, for local pumping case with $\eta_{1}=1$ and $\eta_{2}=0$, only the pulse length optimizing $\mathcal{C}$ maximizes the RDDI force [see Fig. 9(b)]. A shorter pulse optimizes the photon absorption probability $P_{1e}$, but the entanglement and the force are suppressed due to the low entanglement generation rate via the weak RDDI coupling and the fast spontaneous decay rates of the atoms. Thus, the homogeneous pumping is a more efficient way to generate the entanglement force.