1. Introduction
Optomechanical systems, in which electromagnetic radiation is coupled to the mechanical motion of vibrating elements [1, 2], are of great interest for quantum information processing. Coherent control of photonic and mechanical degrees of freedom through radiation-pressure interaction has allowed remarkable achievements in mechanical ground-state cooling [3–5], generation of squeezing [6–10], and entangled states of mechanical and/or cavity modes [11–14]. Among a variety of setups, we will focus here on the three-mode system illustrated in figure 1, where two cavities interact with a common mechanical mode. Of particular interest is that the two cavities can have very different frequencies. Significant advances in microwave-optical conversion have already been reported in such optomechanical systems [15, 16], making it a promising interface between solid-state quantum information devices and fiber optics quantum communication [17–21].
Figure 1. Schematics of the three-mode system, where two driven cavities (1 and 2) interact with a common mechanical resonator. Tripartite entanglement is generated in the output of these modes [22]. An auxiliary cavity (a) can be used to read-out the mechanical output, leading to entanglement concentration for the emitted photons. |
The same optomechanical system is also effective in generating photon entanglement, which has been studied in detail theoretically [22–26] and was realized by a recent experiment [14]. The output entanglement of the two cavities is the main focus of the present work. It can be optimized with respect to the optical drives [14, 22], but further increases are possible through more sophisticated approaches. In particular, the tripartite entanglement of the full system (i.e. considering explicitly the output of the mechanical mode) diverges when approaching the instability point [22], suggesting that the large entanglement of the three-body state may serve as a physical resource to greatly enhance the bipartite entanglement of the emitted photons.
Although successful entanglement concentration and distillation protocols have been developed for discrete-level systems [27, 28], the entanglement of continuous-variable Gaussian states cannot be distilled using Gaussian operations [29, 30]. Here, due to the quadratic nature of the linearized optomechanical interactions, the output state is a squeezed thermal state [22] and optomechanical non-linearities are generally small [31], making it difficult to implement non-Gaussian operations at the Hamiltonian level. An attractive alternative is offered by phonon-counting measurements, which have been shown to enhance the cavity output entanglement in an idealized scenario of perfect projective measurements [26]. As illustrated in figure 1, phonon-counting can be realized by coupling the mechanical mode to a strongly damped auxiliary cavity, which can then be used to indirectly measure the phonon number through standard photon counting techniques [22]. Recently, this strategy was successfully realized [32].
In the present work we analyze the robustness of phonon measurements as a simple entanglement concentration protocol, by including the (possibly large) detector inefficiency. This issue is especially important in view of the combined experimental progress of [14, 32]. We demonstrate that phonon-counting is always beneficial, even with limited detector efficiency, and determine the optimal conditions to enhance photon entanglement. We discuss both the maximum entanglement upon post-selection, as well as its typical value versus system parameters. Numerical results are well understood through an approximate analytical approach.
2. Output entanglement
We start by summarizing previous results on the output entanglement of a 3-mode optomechanical system, formed by two cavity modes with frequencies ω1 and ω2 coupled simultaneously to a mechanical mode with frequency ωm. The setup is illustrated in figure 1 and can be described by the Hamiltonian
$\begin{eqnarray}\hat{H}={\omega }_{{m}}{\hat{b}}^{\dagger }\hat{b}+\sum _{i=1,2}[{\omega }_{i}{\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}+{g}_{i}({\hat{b}}^{\dagger }+\hat{b}){\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}],\end{eqnarray}$
where ${\hat{a}}_{i}$ and $\hat{b}$ are the annihilation operators of the two cavities (i = 1,2) and the mechanics, respectively. The radiation pressure of photons acting on the mechanical element gives rise to the standard optomechanical interaction [1], with single-photon coupling strength gi. Applying external drives detuned to the red (blue) sidebands for cavity 1 (2), a standard linearization can be performed by defining ${\hat{a}}_{i}={\bar{a}}_{i}+{\hat{d}}_{i}$, where ${\bar{a}}_{i}$ is the classical amplitude of the cavity under strong drive [1]. After applying a rotating wave approximation, the Hamiltonian in the interaction picture reads: $\begin{eqnarray}{\hat{H}}_{\mathrm{int}}=({G}_{1}{\hat{b}}^{\dagger }{\hat{d}}_{1}+{G}_{2}\hat{b}{\hat{d}}_{2})+{\rm{h}}.{\rm{c}}.,\end{eqnarray}$
with ${G}_{i}={g}_{i}{\bar{a}}_{i}$ the dressed coupling. The environment can be modeled by standard Markovian reservoirs, with κi (γ) the damping rates of the cavities (mechanics).As discussed previously, steady state entanglement of the output photons can be generated with this system. Using the Langevin equations and the input-output relation (and working in the limit of narrow bandwidth around the cavity frequencies ωi), it was found that the stationary output state takes the form of a twice-squeezed vacuum state [22]. In the Fock state basis $\left|{n}_{1},{n}_{2},{n}_{{m}}\right\rangle $ of the output modes it reads:4 ), the divergence at C2 → C1 + 1 is due to the optomechancial instability induced by a strong blue-detuned drive [22]. In the following, we characterize the entanglement through the logarithmic negativity, defined as [33]:3 ) by taking the partial trace over the mechanical output mode. The final result reads [22]:
$\begin{eqnarray}\left|{\rm{\Psi }}\right\rangle =\sum _{p,q}\sqrt{\displaystyle \frac{{C}_{p+q}^{p}{N}_{m}^{q}{N}_{1}^{p}}{{\left(1+{N}_{2}\right)}^{p+q+1}}}\left|p,p+q,q\right\rangle ,\end{eqnarray}$
where ${C}_{p+q}^{p}$ is the binomial coefficients and $\begin{eqnarray}{N}_{1}=\displaystyle \frac{4{C}_{1}{C}_{2}}{{\left(1+{C}_{1}-{C}_{2}\right)}^{2}},\quad {N}_{2}=\displaystyle \frac{4{C}_{2}({C}_{1}+1)}{{\left(1+{C}_{1}-{C}_{2}\right)}^{2}},\end{eqnarray}$
are the average number of photons in the two optical outputs, with ${C}_{i}=4{G}_{i}^{2}/\gamma {\kappa }_{i}$ the cooperativities. The average number of output phonons is Nm = N2 − N1. In equation ( $\begin{eqnarray}{{ \mathcal E }}_{N}[\hat{\rho }]\equiv \mathrm{ln}\parallel {\hat{\rho }}^{{\rm{T}}}{\parallel }_{1}=\mathrm{ln}\left[2N({\hat{\rho }}^{{\rm{T}}})+1\right],\end{eqnarray}$
where $\hat{\rho }$ is the reduced density matrix of the system, ${\hat{\rho }}^{{\rm{T}}}$ is the partial transpose with respect to one subsystem, ∥. ∥1 denotes the trace norm, and N(. ) is the negativity $N(\hat{O})={\sum }_{i}| {\lambda }_{i}| $, where λi are the negative eigenvalues of $\hat{O}$. Here we are interested in the entanglement of the two optical modes, i.e. $\hat{\rho }$ is obtained from equation ( $\begin{eqnarray}{E}_{N}=\mathrm{ln}\left(\displaystyle \frac{{\left(1+{C}_{1}-{C}_{2}\right)}^{2}}{A+B+2{C}_{2}(1+2{C}_{1})-4\sqrt{{AB}}}\right),\end{eqnarray}$
where A = C2(C1 + C2) and $B={\left(1+{C}_{1}\right)}^{2}+{C}_{1}{C}_{2}$. For given C1, the entanglement is maximized when C2 → C1 + 1 (i.e. approaching the instability point): $\begin{eqnarray}{E}_{N}^{\max }=\mathrm{ln}\left(2{C}_{1}+1\right),\end{eqnarray}$
which increases logarithmically with the cooperativity.While equation (7 ) gives an upper bound for the linear system, it is possible to obtain a further enhancement by means of non-Gaussian operations [30]. Particularly interesting is the case of phonon counting measurements, examined in [26] by assuming an ideal detector. Experimentally, the mechanical output field in optomechanical crystals could be accessed via engineered phonon waveguides [34, 35]. Alternatively, without a phonon waveguide, it is possible to access the phonon output indirectly by making use of a strongly-damped auxiliary cavity (see figure 1). Under suitable conditions, one obtains that the optical output field maps to the mechanical output as ${\hat{d}}_{{a}}^{\mathrm{out}}=-{\rm{i}}{\hat{b}}^{\mathrm{out}}$ [22]. This type of phonon counting through an auxiliary cavity was recently demonstrated [32].
An ideal projective measurement performed on the mechanical mode of ∣$\Psi$⟩ leads to the photon states:6 ):A1 ), yielding:
$\begin{eqnarray}| {{\rm{\Psi }}}_{q}\rangle =\sum _{p=0}^{\infty }\sqrt{{f}_{p}(q)}| p,p+q\rangle ,\end{eqnarray}$
where q is the result of the measurement and $\begin{eqnarray}{f}_{p}(q)={C}_{p+q}^{p}{\zeta }^{p}{\left(1-\zeta \right)}^{1+q},\end{eqnarray}$
with $\zeta =4{C}_{1}{C}_{2}{\left(1+{C}_{1}+{C}_{2}\right)}^{-2}$. The logarithmic negativity is then obtained as: $\begin{eqnarray}{E}_{N}(q)=2\mathrm{ln}\sum _{p=0}^{\infty }\sqrt{{f}_{p}(q)}.\end{eqnarray}$
An interesting observation is that the projected state ∣$\Psi$q⟩ always has a larger photon entanglement than the original state before measurement. This is shown in figure 2, where the upper (red) dots were computed assuming an ideal phonon detector. We see that EN(q) is an increasing function of q where the q = 0 value is already larger than equation ( $\begin{eqnarray}{E}_{N}(q=0)=\mathrm{ln}\left(\displaystyle \frac{1+\sqrt{\zeta }}{1-\sqrt{\zeta }}\right)\gt {E}_{N}.\end{eqnarray}$
Therefore, projective measurements yield a conceptually simple, yet nontrivial, entanglement concentration protocol. Note that the state before measurement is a statistical mixture of all the ∣$\Psi$q⟩ states, where any ∣$\Psi$q>0⟩ has larger entanglement than ∣$\Psi$q=0⟩. Still, purification to the lowest-entanglement state of the mixture is beneficial. For an ideal detector, the growth of entanglement can be characterized through a large-q Gaussian approximation of fp(q), see [26] and equation ( $\begin{eqnarray}{E}_{N}(q)\simeq \mathrm{ln}\displaystyle \frac{\sqrt{8\pi \zeta \left(1+q\right)}}{1-\zeta }.\end{eqnarray}$
We see that not only the entanglement is larger than before the measurement, but grows logarithmically with the number of detected phonons and, in principle, can be as large as desired (by postselecting on the measured value of q). A comparison between the exact and approximate dependence on q is shown in figure 2.
Figure 2. Entanglement versus the detected phonon number q. The upper red dots refer to an ideal projective measurement, i.e. μ = 1, and are well approximated by the solid curve, given by equation ( |
3. Inefficient phonon detection
As the phonon measurements are necessarily imperfect, it is important to take into account a finite detection efficiency rather than assuming ideal projective measurements. As mentioned, the mechanical output can be mapped to the output of an auxiliary cavity. Therefore, following the standard treatment of photodetection [36, 37], we introduce phonon-counting operators:5 ):10 )], computing EN(μ, q) at finite efficiency can be very cumbersome. The numerical evaluation is helped by the natural block-diagonal structure of ${\hat{\rho }}_{q}^{{\rm{T}}}$, described in appendix A . Representative numerical results are shown in figure 2. We see that, as expected, the entanglement is reduced by a smaller μ. However, the phonon detection is still effective for entanglement concentration, since EN(μ, q) is always larger than the entanglement EN before measurement (dashed line). Another interesting remark is that the dependence on q is modified qualitatively by the detector inefficiency: EN(μ, q) becomes non-monotonic when μ < 1, and attains its maximum value at an optimal number of detected phonons.
$\begin{eqnarray}{\hat{M}}_{\mu }\left(q\right)=\sum _{q^{\prime} =q}^{\infty }\sqrt{{C}_{q^{\prime} }^{q}{\left(1-\mu \right)}^{q^{\prime} -q}{\mu }^{q}}| q^{\prime} -q\rangle \langle q^{\prime} | ,\end{eqnarray}$
where q = 0, 1, … are possible detector results and μ is the detection efficiency. The probability of outcome q is ${P}_{M}(\mu ,q)=\langle {\rm{\Psi }}| {\hat{M}}_{\mu }^{\dagger }\left(q\right){\hat{M}}_{\mu }\left(q\right)| {\rm{\Psi }}\rangle $ and the photon state after measurement is: $\begin{eqnarray}{\hat{\rho }}_{q}=\displaystyle \frac{1}{{P}_{M}(\mu ,q)}{\mathrm{Tr}}_{m}\left\{{\hat{M}}_{\mu }\left(q\right)| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| {\hat{M}}_{\mu }^{\dagger }\left(q\right)\right\},\end{eqnarray}$
where the partial trace is performed with respect to the mechanical subsystem. This state is not Gaussian and its entanglement can be obtained directly from equation ( $\begin{eqnarray}{E}_{N}(\mu ,q)\equiv {{ \mathcal E }}_{N}[{\hat{\rho }}_{q}].\end{eqnarray}$
While the case of an ideal detector is particularly straightforward [see [26] and equation (The dependence of the maximum on system parameters can be characterized in more detail through an approximate analytical treatment. For the moment, we give a qualitative discussion of the optimal q based on the phonon probability distribution in the postselected state. After measurement, the state ${\hat{\rho }}_{q}$ reads:8 ). The interpretation of ${P}_{q}\left(q^{\prime} \right)$ is straightforward: Despite the measurement outcome q, there is a finite probability of having $q^{\prime} \gt q$ phonons in the output state. The difference $k=q^{\prime} -q$ are phonons missed by the inefficient detector. The explicit expression of Pq reads:20 ).
$\begin{eqnarray}{\hat{\rho }}_{q}=\sum _{q^{\prime} =q}^{\infty }{P}_{q}\left(q^{\prime} \right)| {{\rm{\Psi }}}_{q^{\prime} }\rangle \langle {{\rm{\Psi }}}_{q^{\prime} }| ,\end{eqnarray}$
where $| {{\rm{\Psi }}}_{q^{\prime} }\rangle $ corresponds to an ideal measurement, see equation ( $\begin{eqnarray}{P}_{q}(q+k)={C}_{q+k}^{k}{\left(1-\varepsilon \right)}^{q+1}{\varepsilon }^{k},\end{eqnarray}$
where we defined a modified inefficiency: $\begin{eqnarray}\varepsilon =\displaystyle \frac{{N}_{m}}{1+{N}_{m}}(1-\mu ).\end{eqnarray}$
For an ideal detector we recover ${P}_{q}\left(q^{\prime} \right)={\delta }_{q,q^{\prime} }$ while representative examples at μ < 1 are shown in figure 3, for various values of q. There, we can identify the two qualitatively different regimes. At small q, and assuming ϵ ≪ 1, the distribution of Pq decays quickly with k, i.e. the probability of having missed a phonon is small. Then, the output state is close to the ideal limit, ${\hat{\rho }}_{q}\simeq | {{\rm{\Psi }}}_{q}\rangle \langle {{\rm{\Psi }}}_{q}| $, and the entanglement is expected to increase with q as for an ideal detector. This changes drastically in the regime of large q, when the maximum of Pq(q + k) moves away from k = 0 and the distribution becomes progressively broader. Although the pure states $| {{\rm{\Psi }}}_{q^{\prime} \gt q}\rangle $ might have large entanglement, ${\hat{\rho }}_{q}$ is highly mixed. The decrease of entanglement at large q can be attributed to the growing number of $| {{\rm{\Psi }}}_{q^{\prime} }\rangle $ with similar statistical weight and a simple criterion to distinguish the two regimes is from the maximum of Pq(q + k). The maximum is at k = 0 when: $\begin{eqnarray}\displaystyle \frac{{P}_{q}(q+1)}{{P}_{q}(q)}=\varepsilon (q+1)\lt 1.\end{eqnarray}$
From this equation, and supposing ϵ ≪ 1, we get $\begin{eqnarray}{q}_{\max }\sim \displaystyle \frac{1}{\varepsilon }.\end{eqnarray}$
Alternatively, we can consider the variance of the distribution, Δk2 ≃ ϵ(1 + q), which becomes larger than 1 at q ≃ 1/ϵ in accordance with equation (
Figure 3. Probability distribution Pq(q + k) at different values of the detected phonon number q, as indicated by the legend. Other parameters are C1 = 10, C2 = 3, and μ = 0.6, giving ϵ ≃ 0.063. |
We see in figure 2 that equation (20 ), marked by black crosses, is a reasonable estimate of the optimal point ${q}_{\max }$. The value of equation (20 ) is generally larger than ${q}_{\max }$, but the approximate scale and its decrease with μ are captured well. Furthermore, we see in figure 2 that the dependence of EN(μ, q) around the optimal point is weak, thus the maximum entanglement is given by:31 ) by setting ${q}_{\max }\simeq 1/\varepsilon $. The lower bound only takes into account the first-order term, and the upper bound includes both the first- and second-order correction. These bounds are actually well satisfied in the numerical example of figure 4.
$\begin{eqnarray}{E}_{N}(\mu ,{q}_{\max })\simeq {E}_{N}(\mu ,1/\varepsilon ).\end{eqnarray}$
Figure 4 shows that the above approximation is satisfied with high accuracy. Unfortunately, it is not immediate to find a simple analytic expression of EN(μ, q) when μ < 1. Using perturbation theory in ϵ (as described in the next section), we have found that the difference between EN(μ, 1/ϵ) and an ideal detector is of order 1. More precisely: $\begin{eqnarray}{E}_{N}\left(1,1/\varepsilon \right)-1\lesssim {E}_{N}(\mu ,{q}_{\max })\lesssim {E}_{N}\left(1,1/\varepsilon \right)-\displaystyle \frac{3}{4},\end{eqnarray}$
which is obtained from equation (
Figure 4. Maximum entanglement as a function of μ. The dashed curve is obtained numerically. The red curve, given by equation ( |
Furthermore, equation (22 ) shows that the perturbative corrections are of order one and can be neglected when the entanglement is large (EN ≫ 1). Then, we can simply estimate ${E}_{N}(\mu ,{q}_{\max })\simeq {E}_{N}(1,1/\varepsilon )$. This limit is of relevance for a nearly ideal detector, since ${q}_{\max }\simeq 1/\varepsilon $ diverges when ϵ → 0. The corresponding entanglement approaches asymptotically equation (12 ), giving:
$\begin{eqnarray}{E}_{N}(1,{q}_{\max })\simeq \mathrm{ln}\displaystyle \frac{\sqrt{8\pi \zeta /\varepsilon }}{1-\zeta }\simeq -\displaystyle \frac{1}{2}\mathrm{ln}(1-\mu ).\end{eqnarray}$
In the above estimates, we neglected the constant correction to the logarithmic divergence. The divergence of the maximum entanglement in the ideal detector limit is directly related to the logarithmic divergence of EN(1, q) in the limit of large q.4. Analytical approach
To understand the dependence of output entanglement on q, as well as the influence on C1, C2, and μ, it is desirable to develop an analytic treatment. A relatively straightforward approach is based on a perturbative expansion around the ideal detector limit. As it turns out, this approximation is successful in describing several important features of EN(μ, q).
We start by writing the partial transpose of the density matrix as ${\hat{\rho }}_{q}^{{\rm{T}}}={\sum }_{k=0}^{\infty }{\hat{\rho }}_{q,k}^{{\rm{T}}}$, where the kth term reads:16 ) and (17 ), thus ${\hat{\rho }}_{q,0}^{{\rm{T}}}$ can be taken as unperturbed operator. The eigenstates of ${\hat{\rho }}_{q,0}^{{\rm{T}}}$ are ∣p1, p1 + q⟩, with eigenvalues ${P}_{q}(q){f}_{{p}_{1}}(q)$, and (if p1 ≠ p2):10 ) for an ideal detector, using that Pq(q) = 1 at μ = 1.
$\begin{eqnarray}\begin{array}{rcl}{\hat{\rho }}_{q,k}^{{\rm{T}}} & = & {P}_{q}(q+k)\displaystyle \sum _{{p}_{1},{p}_{2}=0}^{\infty }\sqrt{{f}_{{p}_{1}}(q+k){f}_{{p}_{2}}(q+k)}\\ & & \times | {p}_{1},{p}_{2}+q+k\rangle \langle {p}_{2},{p}_{1}+q+k| .\end{array}\end{eqnarray}$
Following the discussion of previous section, in the good detector limit Pq(q + k) is a fast-decaying probability distribution, see equations ( $\begin{eqnarray}\left|{p}_{1},{p}_{2};\pm \right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(| {p}_{1},{p}_{2}+q\rangle \pm | {p}_{2},{p}_{1}+q\rangle \right),\end{eqnarray}$
with eigenvalues: $\begin{eqnarray}{\epsilon }_{\pm }^{(0)}({p}_{1},{p}_{2})=\pm {P}_{q}(q)\sqrt{{f}_{{p}_{1}}(q){f}_{{p}_{2}}(q)}.\end{eqnarray}$
The negativity is obtained from ${\epsilon }_{-}^{(0)}({p}_{1},{p}_{2})$ as $N[{\hat{\rho }}_{q,0}^{{\rm{T}}}]={\sum }_{{p}_{1}\gt {p}_{2}}{P}_{q}(q)\sqrt{{f}_{{p}_{1}}(q){f}_{{p}_{2}}(q)}$. By using ∑pfp(q) = 1 we can also write the negativity as: $\begin{eqnarray}N[{\hat{\rho }}_{q}^{{\rm{T}}}]\simeq -\displaystyle \frac{{P}_{q}(q)}{2}+\displaystyle \frac{{P}_{q}(q)}{2}{\left(\sum _{p=0}^{\infty }\sqrt{{f}_{p}(q)}\right)}^{2}+\ldots ,\end{eqnarray}$
which immediately leads to equation (The higher-order terms can be obtained using standard second-order perturbation theory on the hermitian operator ${\hat{\rho }}_{q}^{{\rm{T}}}={\hat{\rho }}_{q,0}^{{\rm{T}}}+\delta {\hat{\rho }}_{q}^{{\rm{T}}}$. The calculations are rather straightforward but tedious, and we describe them in appendix B . In performing the perturbative treatment, only corrections to ${\epsilon }_{-}^{(0)}({p}_{1},{p}_{2})$ are of relevance, as the entanglement EN(μ, q) can be obtained from the negativity. The final result reads:9 ) and28 ) we see that the terms in the expansion decrease quickly when:19 ).
$\begin{eqnarray}\begin{array}{l}{E}_{N}(\mu ,q)={E}_{N}\left(1,q\right)-(q+1)\varepsilon -\displaystyle \frac{1}{2}(q+1){\varepsilon }^{2}\\ \quad +\displaystyle \frac{1}{2}\displaystyle \frac{\sum _{{p}_{1},{p}_{2}}{g}_{q}({p}_{1},{p}_{2})}{{\left(\sum _{p}\sqrt{{f}_{p}(q)}\right)}^{2}}{\left(q+1\right)}^{2}{\varepsilon }^{2}+\ldots \,,\end{array}\end{eqnarray}$
where fp(q) is defined in equation ( $\begin{eqnarray}{g}_{q}({p}_{1},{p}_{2})=\displaystyle \frac{{f}_{{p}_{1}}(q+1){f}_{{p}_{2}}(q+1)}{\sqrt{{f}_{{p}_{1}+1}(q){f}_{{p}_{2}}(q)}+\sqrt{{f}_{{p}_{1}}(q){f}_{{p}_{2}+1}(q)}}.\end{eqnarray}$
In equation ( $\begin{eqnarray}(q+1)\varepsilon \ll 1,\end{eqnarray}$
which is consistent with equation (A comparison between equation (28 ) and the numerical results is shown in figure 5, for a specific choice of C1, C2 and relatively small values of q. We see that the leading term (i.e. the simple linear correction) describes the initial decrease of entanglement away from μ = 1 well. However, deviations appear at smaller μ, when nonlinear corrections start to play an important role. This effect is captured by the quadratic terms of equation (28 ), which give much better agreement over the whole range of μ. We note that, although the expansion is performed for ‘small' deviations from the ideal detector limit, the appropriate expansion parameter is (q + 1)ϵ and not 1 − μ. Therefore, the expansion is still valid for a very poor detector with 1 − μ ∼ 1, if ϵ ≪ 1 and q is sufficiently small.
Figure 5. Entanglement versus detector efficiency μ. The numerical results (red solid curves) are compared to equation ( |
Further comparison of the analytic and numerical results as a function of q is shown in figure 6. Besides confirming that the second-order terms generally improve the agreement, we see that the series becomes unreliable at q ∼ 1/ϵ, consistently with previous remarks. Interestingly, the exact numerical results always lie between the first- and second-order approximation, thus the expansion appears to be formed by terms with alternating signs or, at least, the third-order term is negative.
Figure 6. Further comparisons of the perturbation theory of EN with exact numerical values (red middle curve). The upper (lower) curve is a plot of equation ( |
The latter observation is useful to estimate the entanglement around q ∼ 1/ϵ ≫ 1, which is at the boundary of validity of the perturbative expression. This value of q is interesting because it approximates the optimal point for the output entanglement [see the discussions of the previous section and, in particular, equations (20 ) and (21 )]. In appendix B we analyze the coefficient of the second-order term at q ≫ 1 and obtain [see equations (B4 )]:20 ) we obtain the upper and lower bounds of equation (22 ). See also figure 4 for a quantitative check. While the lower bound only takes into account the first-order term, the upper bound includes both the first- and second-order corrections and is a more accurate estimate of the true value.
$\begin{eqnarray}{E}_{N}\left(\mu ,q\gg 1\right)={E}_{N}\left(1,q\right)-q\varepsilon +\displaystyle \frac{1}{4}{\left(q\varepsilon \right)}^{2}+\ldots \end{eqnarray}$
Combining this simplified expression with equations (5. Typical entanglement
So far, we have mainly analyzed the maximum entanglement which can be obtained with an imperfect detector. However, this quantity is not always a practical characterization of EN(μ, q). As the probability of detecting a large number of phonons decreases quickly with q, it might be very difficult to generate a maximally entangled state. More precisely, the probability distribution of phonon detection outcome q is:
$\begin{eqnarray}{P}_{M}(\mu ,q)=\displaystyle \frac{{\left(\mu {N}_{m}\right)}^{q}}{{\left(1+\mu {N}_{m}\right)}^{q+1}},\end{eqnarray}$
which gives an average number μNm of detected phonons. In the specific case of figure 2, we have Nm ≃ 0.2, thus it is very difficult to get a value of entanglement much larger than EN(μ, q = 0) ∼ 2. This motivates us to introduce the ‘typical' entanglement, defined as follows: $\begin{eqnarray}{\overline{E}}_{N}(\mu )={E}_{N}(\mu ,\overline{q}),\end{eqnarray}$
where $\overline{q}=\lfloor \mu {N}_{m}\rceil $ is the nearest integer to μNm. Representative data for the typical entanglement, as function of C2 at different detector efficiencies, are shown in figure 7.
Figure 7. Dependence of the typical entanglement ${\overline{E}}_{N}(\mu )$ on C2 for different detector efficiencies μ. The baseline data (bottom blue dots) are the entanglement before measurement. The other sets of data are for μ = 1, 0.99, 0.95, 0.9, 0.7 and 0.5 (from top to bottom). The green dashed curve is equation ( |
For an ideal detector, we recall that EN is a monotonic function of q. Therefore, to increase the typical entanglement one has to simply attain a large value of Nm, which is possible by approaching the instability point C2 → C1 + 1. In this limit, equation (12 ) immediately gives:
$\begin{eqnarray}{\overline{E}}_{N}(1)\simeq \mathrm{ln}\displaystyle \frac{\sqrt{8\pi \zeta \left(1+{N}_{m}\right)}}{1-\zeta }\simeq \mathrm{ln}\displaystyle \frac{4(1+{C}_{1})\sqrt{2\pi {C}_{1}}}{1+{C}_{1}-{C}_{2}},\end{eqnarray}$
which is in good agreement with the numerical results, as shown in figure 7. We see that the divergence of $\overline{q}$ at the instability point is translated into a logarithmic divergence of the typical entanglement. On the other hand, figure 7 shows that the behavior at μ < 1 is quite different: For a non-ideal detector, the dependence of ${\overline{E}}_{N}(\mu \lt 1)$ is non-monotonic when approaching the instability point.To understand this behavior we note that, by increasing C2 from zero towards the unstable condition, both Nm and $\overline{q}\sim \mu {N}_{m}$ grow from 0 to ∞ . At the same time, the value ${q}_{\max }=1/\varepsilon $ which maximizes the entanglement decreases toward a finite value ${q}_{\max }\to {\left(1-\mu \right)}^{-1}$, see equation (18 ). As exemplified in figure 8(a), there is a value of C2 when $\overline{q}={q}_{\max }$ and we associate this point to the maximum typical entanglement. That this condition corresponds to the optimal point is confirmed explicitly by figure 8(b). We can also note that, close to the instability condition, a relatively small increase of C2 results in a great change of $\overline{q}$. Therefore, the non-monotonic behavior of entanglement we have discussed previously (as function of q, see figure 2) gets reflected in the C2 dependence of the typical entanglement.
Figure 8. Comparison between typical entanglement and maximum entanglement, as function of C2 at fixed C1 = 10 and μ = 0.9. Panel (a) shows the dependence of $\bar{q}$ and ${q}_{\max }$ on C2. Panel (b) compares the values of ${E}_{N}(\mu ,{q}_{\max })$ and ${\overline{E}}_{N}(\mu )$. The curves in panel (a) meet when $\bar{q}={q}_{\max }$, which corresponds well to the maximum of ${\overline{E}}_{N}(\mu )$ shown in panel (b). |
The condition $\overline{q}={q}_{\max }$ gives, using equation (20 ):35 ) by using ${N}_{m}=4{C}_{2}/{\left(1+{C}_{1}-{C}_{2}\right)}^{2}$. This gives:31 ):36 ) and (37 ). As seen, they capture well the optimal operating point at different values of μ.
$\begin{eqnarray}\mu {N}_{m}=\displaystyle \frac{1+{N}_{m}}{{N}_{m}(1-\mu )}\simeq \displaystyle \frac{1}{1-\mu },\end{eqnarray}$
where in the last step we suppose Nm ≫ 1. To find the corresponding value of C2, we can rewrite equation ( $\begin{eqnarray}{C}_{2,\max }\simeq 1+{C}_{1}+2\delta \mu -2\sqrt{\delta \mu (1+{C}_{1}+\delta \mu )},\end{eqnarray}$
where we defined δμ = μ(1 − μ). Finally, we estimate the value of the typical entanglement by using equation ( $\begin{eqnarray}{\overline{E}}_{N,\max }\simeq {E}_{N}(1,\displaystyle \frac{1}{1-\mu })-\displaystyle \frac{3}{4}.\end{eqnarray}$
We show in figure 7 the points $({C}_{2,\max },{\overline{E}}_{N,\max })$, obtained from equations (6. Discussions and conclusion
In this work, we have investigated an entanglement concentration scheme applicable to a 3-mode optomechanical system, where the entanglement between emitted photons can be enhanced through phonon counting measurements. Motivated by recent experimental progress [14, 32], we have analyzed in detail the role of detector inefficiency. We find that the entanglement concentration scheme is generally advantageous, also considering a poor efficiency. Furthermore, we have obtained the maximum value of entanglement as a function of the measurement outcome q. The optimal q might be quite different from the most likely measurement outcome, which motivates us to discuss how the ‘typical' entanglement depends on system parameters.
This scheme takes advantage of the strong tripartite entanglement before measurement. In the absence of phonon-counting, the bipartite entanglement is maximized when C2 → C1 + 1 but remains finite, even if the system is approaching the unstable regime. This is possible as the two optical modes are strongly entangled with the mechanics, and a divergence can only be found by considering the genuine tripartite entanglement [22]. Phonon-counting measurements are able to recover the hidden entanglement, i.e. they transform effectively the tripartite correlations into a bipartite entanglement for the outgoing photons. For an ideal projective measurement, the concentrated bipartite entanglement leads to a diverging behavior when approaching the instability (both in the maximum and typical entanglement). Such divergence is cut-off by a finite detector efficiency, which makes the optimal operation point nontrivial.