Entanglement and squeezing of continuous-wave stationary light

In reality one has access only to a finite time interval (and correspondingly to a finite band of frequencies) of the total field a(t). These detectable intervals of the total field correspond to specific temporal modes. They can be physically defined, for example, by the temporal profile of the pumping field in pulsed experiments [8, 10, 52–54], they can also be extracted by post-processing the previously recorded time signal [55, 56], or they can be selected by the measurement apparatus as a result of the corresponding response and detection times [56–59]. In particular, in the case of experiments involving stationary fields, the detection time can be sufficiently long to select a well-defined spectral component of the total signal (as achieved, for example, with an electronic spectrum analyzer) [6, 7, 42].

In general a temporal filtered mode can be introduced in terms of a filter function ${{\phi }_{\tau }}(t)$, which defines the time profile of the mode with a duration of order τ. Correspondingly, it defines a band of spectral components, of width $1/\tau$, that are combined into the filtered signal [60, 61]. The generic form for the operators of a filtered mode can be expressed as

$$\begin{eqnarray}&&\overline{{{a}_{\tau }}}(\Omega ,t)=\int _{-\infty }^{\infty }{\rm d}s\ {{{\rm e}}^{{\rm i}\Omega s}}\ {{\phi }_{\tau }}(t-s)\ a(s)\end{eqnarray} \tag{ 4 }$$

with ${{\left\{ \overline{{{a}_{\tau }}}(\Omega ,t) \right\}}^{\dagger }}=\overline{a_{\tau }^{\dagger }}(-\Omega ,t)$, and where the symbol $\overline{\ \ }$ indicates filtered quantities. The parameter Ω defines the central frequency of the filter, and the filter function ${{\phi }_{\,\tau }}(t)$ is real and normalized according to

$$\begin{eqnarray}&&\int _{-\infty }^{\infty }{\rm d}s\ {{\phi }_{\tau }}{{(s)}^{2}}=1.\end{eqnarray} \tag{ 5 }$$

Consequently, the filtered operators are discrete bosonic operators which satisfy the standard commutation relation

$$\begin{eqnarray*}&&[\overline{{{a}_{\tau }}}(\Omega ,t),\overline{a_{\tau }^{\dagger }}(-\Omega ,t)]=1,\quad \forall \tau .\end{eqnarray*}$$

The corresponding equivalent form of the filtered operators, in terms of the spectral components of the field, is

$$\begin{eqnarray}&&\overline{{{a}_{\tau }}}(\Omega ,t)=\int _{-\infty }^{\infty }{\rm d}\omega {{{\rm e}}^{-{\rm i}(\omega -\Omega )t}}{{\tilde{\phi }}_{\tau }}(\omega -\Omega )\ \tilde{a}(\omega )\end{eqnarray} \tag{ 6 }$$

where ${{\tilde{\phi }}_{\,\tau }}(\omega )$ is the Fourier transformed filter function ${{\tilde{\phi }}_{\tau }}(\omega )=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{\infty }{\rm d}s\ {{{\rm e}}^{{\rm i}\omega s}}\ {{\phi }_{\tau }}(s),$ which peaks at $\omega =0$ and has a width on the order of $1/\tau$.

Two particular cases are worth mentioning: the exponential filter with $\phi _{\tau }^{{\rm exp} }(t)=\sqrt{2}\theta (t){{{\rm e}}^{-t/\tau }}/\sqrt{\tau }$ and $\tilde{\phi }_{\tau }^{{\rm exp} }(\omega )=\sqrt{\tau /\pi }/(1-{\rm i}\ \tau \ \omega )$, which has been used, for example, in [57] to introduce the physical spectrum of light; and the step-filter function

$$\begin{eqnarray}&&\phi _{\tau }^{step}(t)=\frac{\theta (t)\ \theta (\tau -t)}{\sqrt{\tau }},\tilde{\phi }_{\tau }^{step}(\omega )=\sqrt{\frac{2\pi }{\tau }}{{{\rm e}}^{{\rm i}\omega \frac{\tau }{2}}}\frac{{\rm sin} (\omega \ \tau /2)}{\pi \ \omega }\end{eqnarray} \tag{ 7 }$$

which we will connect to homodyne and heterodyne detection techniques in the following. In this form the time t in the filtered operator $\overline{{{a}_{\tau }}}(\Omega ,t)$ corresponds to the final time of the filtering process, that is, $\overline{{{a}_{\tau }}}(\Omega ,t)=\int _{-\infty }^{t}{\rm d}s\ {{{\rm e}}^{{\rm i}\Omega s}}\ {{\phi }_{\tau }}(t-s)\ a(s).$ Although not strictly relevant to the results presented in this article, this choice is physically motivated by the fact that in this way we define, at time t, a causal operator ${{\bar{a}}_{\tau }}(\Omega ,t)$ which depends only on the past of the continuous field a(t) [60].

In the limit of long filtering times, $\tau \to \infty$, the filter selects a single spectral component of the field. In the following we will focus on the spectral components of stationary fields for which we will use the following simplified notation:

$$\begin{eqnarray}&&\bar{a}(\Omega )\equiv \mathop{{\rm lim} }\limits_{\,\tau \to \infty }\overline{{{a}_{\tau }}}(\Omega ,t),\end{eqnarray} \tag{ 8 }$$

where we drop the label τ, the limit symbol, and the time argument t. In particular, the time t in these operators is irrelevant for stationary fields, in the sense that (as shown in the next section) the correlations of operators of this form, in the limit of large τ, are independent of the time arguments.

We finally note that we are describing the field in a reference frame rotating at the carrier frequency ${{\omega }_{L}}$; therefore $\bar{a}(\Omega )$ is, in fact, the operator for the spectral component of the field at the sideband frequency ${{\omega }_{L}}+\Omega .$

In this article we are interested in the squeezing properties of the electromagnetic field, which refers to reduced fluctuations or reduced variance of specific quadratures below the vacuum noise level, and in the corresponding entanglement features. Squeezing can be revealed from the analysis of the second-order correlations of field operators. Therefore in this section we analyze the basic properties of the correlations of filtered modes. In particular we focus on the spectral properties of stationary fields. The corresponding field operators, a(t) and $\tilde{a}(\omega )$, have diverging correlation functions as a consequence of the commutation relations in equation (3). In this case it is therefore instructive to analyze the fluctuations of the spectral modes in terms of the filtered spectral operators defined in equation (6), whose correlations, in contrast, are always finite also in the limit of long integration time $\tau \to \infty$. This approach is particularly useful for the study of the corresponding entanglement properties, and it has the advantage of providing a clear physical definition of discrete modes corresponding to the specific spectral components, hence allowing for a transparent application of the techniques developed in entanglement theory, which indeed deals with discrete modes (see appendix A).

Let us study the correlations between the spectral components of two stationary continuous fields with annihilation operators ${{a}_{1}}(t)$ and ${{a}_{2}}(t)$ respectively, which fulfill the commutation relation $\left[ {{a}_{j}}(t),{{a}_{k}}{{(t^{\prime} )}^{\dagger }} \right]={{\delta }_{j,k}}\ \delta (t-t^{\prime} )$. (The same results that we discuss hereafter for two continuous fields can be applied with minor modifications to different spectral components belonging to a single field.) By straightforward application of the Wiener–Khintchine theorem, we find that all the information about the correlations between the spectral components is contained in the power spectrum matrix $\tilde{\mathcal{P}}(\omega )$, defined as the Fourier transform of the two-time stationary correlation matrix, which can be expressed in terms of the elements of the column vector of operators ${\bf a}(t)={{\left( {{a}_{1}}(t),{{a}_{2}}(t),a_{1}^{\dagger }(t),a_{2}^{\dagger }(t) \right)}^{T}}$ as the matrix $\mathcal{A}(t)=\left\langle {\bf a}(t){\bf a}{{(0)}^{T}} \right\rangle$, whose elements are ${{\{\mathcal{A}(t)\}}_{j,k}}=\left\langle {{\{{\bf a}(t)\}}_{j}}\;{{\{{\bf a}(0)\}}_{k}} \right\rangle$, where $j,k\in \{1,2,3,4\}$ are vector indices, not to be confused with the indices of the modes. To be specific,

$$\begin{eqnarray}&&\tilde{\mathcal{P}}(\omega )=\int _{-\infty }^{\infty }{\rm d}t\;{{{\rm e}}^{{\rm i}\,\omega \,\tau }}\mathcal{A}(t),\end{eqnarray} \tag{ 9 }$$

where we use the fact that two-time correlation functions of stationary signals depend only on the difference of the time arguments. In particular the correlations between the Fourier-transformed operators $\tilde{{\bf a}}(\omega )=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{\infty }{\rm d}t\ {{{\rm e}}^{{\rm i}\omega t}}{\bf a}(t)$ are diverging and are related to the power spectrum matrix by

$$\begin{eqnarray}&&\left\langle \tilde{{\bf a}}(\omega )\tilde{{\bf a}}{{(\omega ^{\prime} )}^{T}} \right\rangle =\delta (\omega +\omega ^{\prime} )\tilde{\mathcal{P}}(\omega ).\end{eqnarray} \tag{ 10 }$$

To gain insight into the physical meaning of these diverging quantities we employ the narrow filtered modes introduced in the preceding section. We construct the vector of filtered spectral components ${\bf \bar{a}}(\Omega )={{{\rm lim} }_{\tau \to \infty }}\int _{-\infty }^{\infty }{\rm d}s\ {{{\rm e}}^{{\rm i}\Omega s}}\ {{\phi }_{\tau }}(t-s)\ {\bf a}(t)$, which is given by ${\bf \bar{a}}(\Omega )={{\left( \overline{{{a}_{1}}}(\Omega ),\overline{{{a}_{2}}}(\Omega ),\overline{a_{1}^{\dagger }}(\Omega ),\overline{a_{2}^{\dagger }}(\Omega ) \right)}^{T}}$, and we compute the corresponding matrix of correlations:

$$\begin{eqnarray}&&\left\langle {\bf \bar{a}}(\Omega ){\bf \bar{a}}{{(\Omega ^{\prime} )}^{T}} \right\rangle =\mathop{{\rm lim} }\limits_{\tau \to \infty }\int _{-\infty }^{\infty }{\rm d}\omega ^{\prime} \ \int _{-\infty }^{\infty }{\rm d}\omega \ \left\langle \tilde{{\bf a}}(\omega )\ \tilde{{\bf a}}{{(\omega ^{\prime} )}^{T}} \right\rangle \ {{{\rm e}}^{-{\rm i}[(\omega -\Omega )t+(\omega ^{\prime} -\Omega ^{\prime} )t^{\prime} ]}}{{\tilde{\phi }}_{\tau }}(\omega -\Omega ){{\tilde{\phi }}_{\tau }}(\omega ^{\prime} -\Omega ^{\prime} ).\end{eqnarray} \tag{ 11 }$$

These quantities can be evaluated by noting that for large τ, the square modulus of the filter function approaches a delta function, ${{{\rm lim} }_{\tau \to \infty }}{{\left| {{\tilde{\phi }}_{\tau }}(\omega ) \right|}^{2}}=\delta (\omega )$, whereas its integral goes to zero, ${{{\rm lim} }_{\tau \to \infty }}\int {\rm d}\omega {{\tilde{\phi }}_{\tau }}(\omega )=0$. And, likewise, given a generic finite function $f(\omega )$, the relation

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{\tau \to \infty }\int _{-\infty }^{\infty }{\rm d}\omega {{\tilde{\phi }}_{\tau }}(\omega +\Omega )\ {{\tilde{\phi }}_{\tau }}(-\omega -\Omega ^{\prime} )\ f(\omega )={{\delta }_{\Omega ,\Omega ^{\prime} }}f(\Omega )\end{eqnarray} \tag{ 12 }$$

holds. Consequently equations (10) and (12) can be used in equation (11) to find

$$\begin{eqnarray}&&\left\langle {\bf \bar{a}}(\Omega ){\bf \bar{a}}{{(\Omega ^{\prime} )}^{T}} \right\rangle ={{\delta }_{\Omega ,-\Omega ^{\prime} }}\tilde{\mathcal{P}}(\Omega ),\end{eqnarray} \tag{ 13 }$$

which shows that the power spectrum is directly related to the correlations of narrow filtered modes. In other terms, in the limit of large integration time τ, i.e., when the bandwidth selected by the filter is sufficiently small, the correlation functions reduce to the power spectrum of the continuous field [57]. This result is valid when τ is much larger than the decay time of the signal correlations ${{\tau }_{C}}$ (the memory time of the signals), ${{\tau }_{C}}\ll \tau$. We note in particular that this relation implies the stationarity of the signal, which is reached on a time scale on the order of ${{\tau }_{C}}$.

The correlations between two modes are conveniently analyzed in terms of the corresponding correlation matrix, from which the corresponding squeezing and entanglement properties can be readily derived (see appendix A for a short review). We remark, however, that the matrix $\tilde{\mathcal{P}}(\Omega )$ is not a correlation matrix for two modes. In fact, it contains the correlations between the four spectral modes corresponding to the two pairs of sidebands at the frequency $\pm \Omega$ of the two continuous fields. On the other hand, the elements of the power spectrum matrix can be used to construct the correlation matrix for two narrow modes filtered from the two stationary continuous fields as follows. We consider two modes, at frequencies Ω and $\Omega ^{\prime}$, respectively described by the filtered operators

$$\begin{eqnarray}\begin{array}{cc} {{{\bar{a}}}_{1}}(\Omega ), & \ \ \ \overline{a_{1}^{\dagger }}(-\Omega ), \\ {{{\bar{a}}}_{2}}(\Omega ^{\prime} ), & \ \ \ \overline{a_{2}^{\dagger }}(-\Omega ^{\prime} ), \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where the narrow bandwidth limit $(\tau \to \infty )$ is implicit in their definition (see equation (8)). The corresponding correlation matrix is defined using the vector ${\bf \bar{a}}(\Omega ,\Omega ^{\prime} )={{\left( \overline{{{a}_{1}}}(\Omega ),\overline{{{a}_{2}}}(\Omega ^{\prime} ),\overline{a_{1}^{\dagger }}(-\Omega ),\overline{a_{2}^{\dagger }}(-\Omega ^{\prime} ) \right)}^{T}}$, as $\bar{\mathcal{A}}(\Omega ,\Omega ^{\prime} )=\left\langle {\bf \bar{a}}(\Omega ,\Omega ^{\prime} )\ {\bf \bar{a}}{{(\Omega ,\Omega ^{\prime} )}^{T}} \right\rangle$, and can be expressed in terms of the elements of the power spectrum matrix defined in equation (13) as

$$\begin{eqnarray}&&\begin{array}{l} \bar{\mathcal{A}}(\Omega ,\Omega ^{\prime} ) \\ =\;\left( \begin{array}{ccccccccccccccc} {{\delta }_{\Omega ,0}}{{\delta }_{\Omega ^{\prime} ,0}}\ {{\left\{ \tilde{\mathcal{P}}(0) \right\}}_{1,1}} & {{\delta }_{\Omega ,-\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(\Omega ) \right\}}_{1,2}} & \ {{\left\{ \tilde{\mathcal{P}}(\Omega ) \right\}}_{1,3}} & {{\delta }_{\Omega ,\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(\Omega ) \right\}}_{1,4}} \\ {{\delta }_{\Omega ,-\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(\Omega ^{\prime} ) \right\}}_{2,1}} & {{\delta }_{\Omega ,0}}{{\delta }_{\Omega ^{\prime} ,0}}\ {{\left\{ \tilde{\mathcal{P}}(0) \right\}}_{2,2}} & {{\delta }_{\Omega ,\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(\Omega ^{\prime} ) \right\}}_{2,3}} & \ {{\left\{ \tilde{\mathcal{P}}(\Omega ^{\prime} ) \right\}}_{2,4}} \\ \ {{\left\{ \tilde{\mathcal{P}}(-\Omega ) \right\}}_{3,1}} & {{\delta }_{\Omega ,\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(-\Omega ) \right\}}_{3,2}} & {{\delta }_{\Omega ,0}}{{\delta }_{\Omega ^{\prime} ,0}}\ {{\left\{ \tilde{\mathcal{P}}(0) \right\}}_{3,3}} & {{\delta }_{\Omega ,-\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(-\Omega ) \right\}}_{3,4}} \\ {{\delta }_{\Omega ,\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(-\Omega ^{\prime} ) \right\}}_{4,1}} & \ {{\left\{ \tilde{\mathcal{P}}(-\Omega ^{\prime} ) \right\}}_{4,2}} & {{\delta }_{\Omega ,-\Omega ^{\prime} }}\ {{\left\{ \tilde{\mathcal{P}}(-\Omega ^{\prime} ) \right\}}_{4,3}} & {{\delta }_{\Omega ,0}}{{\delta }_{\Omega ^{\prime} ,0}}\ {{\left\{ \tilde{\mathcal{P}}(0) \right\}}_{4,4}} \\ \end{array} \right). \\ \end{array}\end{eqnarray} \tag{ 15 }$$

We finally note that $\bar{\mathcal{A}}(\Omega ,\Omega ^{\prime} )$ is equal to the power spectrum matrix $\tilde{\mathcal{P}}(\Omega )$ only at zero frequency, $\bar{\mathcal{A}}(0,0)=\tilde{\mathcal{P}}(0)$.

In homodyne detection the signal field is mixed on a 50:50 beam splitter with a strong monochromatic field (the local oscillator) at the same frequency as the carrier signal. The fields at the two output ports of the beam splitter are detected and the corresponding photocurrents are subtracted, resulting in a signal that contains information about a field quadrature [65] and that can be described by a photocurrent operator of the form (see appendix B)

$$\begin{eqnarray}&&{{I}^{(\theta )}}(t)={{{\rm e}}^{{\rm i}\theta }}a(t)+\ {{{\rm e}}^{-{\rm i}\theta }}{{a}^{\dagger }}(t),\end{eqnarray} \tag{ 27 }$$

where θ is the phase of the local oscillator. The power spectrum of the photocurrent contains information about the spectral components of the detected field, and in particular it quantifies the strength of the fluctuations at specific frequencies. We will refer to it as the homodyne spectrum, and it can be expressed as the autocorrelation function of the filtered photocurrent, integrated over a long time τ, of the form $J_{\tau }^{(\theta ,\varphi )}(\epsilon ,t)\propto \frac{1}{\sqrt{\tau }}$ $\int _{t-\tau }^{t}{\rm d}s$ ${\rm cos} (\epsilon s+\varphi )\ {{I}^{(\theta )}}(s)$. In detail, the homodyne spectrum can be written as

$$\begin{eqnarray}&&{{\mathcal{G}}^{(\theta )}}(\epsilon )=\mathop{{\rm lim} }\limits_{\tau \to \infty }\left\langle {{\left[ J_{\tau }^{(\theta ,\varphi )}(\epsilon ,t) \right]}^{2}} \right\rangle .\end{eqnarray} \tag{ 28 }$$

We note that, for stationary processes, this quantity is independent of the phase of the filter φ (see appendix B). However, this phase is relevant and can be useful when considering combinations of filtered photocurrents at different phases, which, as discussed hereafter, can be exploited to probe arbitrary superpositions of spectral modes. Moreover the same results for the power spectrum in equation (28) are obtained when, in the filtered photocurrent $J_{\tau }^{(\theta ,\varphi )}(\epsilon ,t)$, one uses an exponential oscillating function, as in the filters of section 2.1, in place of the sinusoidal function introduced earlier. The use of a sinusoidal function is, however, more convenient because, in this case, the filtered photocurrent is a Hermitian operator, thereby making the relation between the photocurrent and the field observables more transparent. Specifically the filtered photocurrent in the limit of long filtering time can be expressed as the sum of two filtered quadrature operators for the two spectral modes at frequencies $\pm \epsilon$ (see appendix B):

$$\begin{eqnarray}\begin{array}{rcl} {{J}^{(\theta ,\varphi )}}(\epsilon ) & = & \mathop{{\rm lim} }\limits_{\tau \to \infty }J_{\tau }^{(\theta ,\varphi )}(\epsilon ,t) \\ {} & = & \frac{1}{\sqrt{2}}\left[ \overline{{{x}^{(\theta +\varphi )}}}(\epsilon )+\overline{{{x}^{(\theta -\varphi )}}}(-\epsilon ) \right]. \\ \end{array}\end{eqnarray} \tag{ 29 }$$

Similar to equation (19), this is a symmetric superposition of two quadratures, defined as in equation (18), corresponding to the two spectral components, which in this case are filtered from the same field, and whose annihilation operators are

$$\begin{eqnarray}&&\bar{a}(\epsilon )\ \ {\rm and}\ \ \ \bar{a}(-\epsilon ),\end{eqnarray} \tag{ 30 }$$

as illustrated in figure 1. The corresponding single-mode homodyne spectrum (where, single-mode, indicates that its results form the detection of a single continuous field) is equal to equation (22) with ${{\xi }_{+}}={{\xi }_{-}}$ and ${{\theta }_{+}}+{{\theta }_{-}}=2\theta$, i.e., ${{\mathcal{G}}^{(\theta )}}(\epsilon )=1+n_{+}^{(I)}+n_{-}^{(I)}+[{{m}^{(I)}}{{{\rm e}}^{2\,{\rm i}\,\theta }}+{{m}^{(I)}}^{*}\;{{{\rm e}}^{-2\,{\rm i}\,\theta }}]$, where

$$\begin{eqnarray}&&n_{\pm }^{(I)}=\left\langle \overline{{{a}^{\dagger }}}(\mp \epsilon )\bar{a}(\pm \epsilon ) \right\rangle ,\ \ \ \ \ \ {{m}^{(I)}}=\left\langle \bar{a}(\epsilon )\bar{a}(-\epsilon ) \right\rangle .\end{eqnarray} \tag{ 31 }$$

Thus, the single-mode phase-optimized squeezing spectrum, which is experimentally accessible by tuning the phase of the local oscillator, ${{S}^{(I)}}(\epsilon )={{{\rm min} }_{\theta }}{{\mathcal{G}}^{(\theta )}}(\epsilon )$, is equal to equation (22) evaluated for the parameters in equation (31). When it is smaller than one, it indicates that the two sideband modes $\bar{a}(\pm \epsilon )$ are entangled [37–40]. Their logarithmic negativity, which as discussed in section 3, is directly related to the squeezing spectrum in equation (23), could be measurable if one could construct a filtered photocurrent similar to equation (20), which is a non-symmetric superposition of the quadratures of the two modes. This photocurrent is, in fact, achievable by combining two filtered photocurrents detected at appropriately tuned phases of the local oscillator θ and of the filter φ. Specifically one should first detect the filtered photocurrent ${{J}^{(\theta ,\varphi )}}(\epsilon )$ for some value of θ and φ and then a second one, ${{J}^{(\theta ^{\prime} ,\varphi ^{\prime} )}}(\epsilon )$, with the phases tuned to different values $\theta ^{\prime}$ and $\varphi ^{\prime}$. The two photocurrents are then summed, resulting in the total photocurrent

$$\begin{eqnarray}&&J_{{{\xi }_{+}},{{\xi }_{-}}}^{({{\theta }_{+}},{{\theta }_{-}})}(\epsilon )=\frac{1}{\sqrt{\xi _{+}^{2}+\xi _{-}^{2}}}\left[ {{\xi }_{+}}\overline{{{x}^{({{\theta }_{+}})}}}(\epsilon )+{{\xi }_{-}}\ \overline{{{x}^{({{\theta }_{-}})}}}(-\epsilon ) \right],\end{eqnarray} \tag{ 32 }$$

which we have appropriately normalized, and where

$$\begin{eqnarray}\begin{array}{rcl} {{\theta }_{\pm }} & = & \frac{\theta +\theta ^{\prime} \pm (\varphi +\varphi ^{\prime} )}{2} \\ {{\xi }_{\pm }} & = & {\rm cos} \left[ \frac{\theta -\theta ^{\prime} \pm (\varphi -\varphi ^{\prime} )}{2} \right]. \\ \end{array}\end{eqnarray} \tag{ 33 }$$

We note that equation (32) has, indeed, the form of the composite quadrature defined in equation (20). The corresponding single-mode globally optimized squeezing spectrum $S_{{\rm min} }^{(I)}(\epsilon )\;=\;$ ${{{\rm min} }_{{{\theta }_{\pm }},{{\xi }_{\pm }}}}\left\langle {{\left[ J_{{{\xi }_{+}},{{\xi }_{-}}}^{({{\theta }_{+}},{{\theta }_{-}})}(\epsilon ) \right]}^{2}} \right\rangle$ is then equal to equation (23) evaluated for the parameters in equation (31), and it can be measured by minimizing the homodyne spectrum over the phases of both the local oscillator and the filter. In particular, although the tuning the filter phases, ${{\varphi }_{{}}}$ and ${{\varphi }^{{\prime} }}$, can be, in principle, achieved by recording the photocurrent for a sufficiently long time and then post-processing the recorded signal, the phases of the local oscillator, ${{\theta }_{{}}}$ and θ', must be adjusted during repeated homodyne measurements.

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In the preceding section we saw that the single-mode homodyne spectrum, which is measurable from a single stationary continuous field, provides information about the entanglement between two spectral components. Let us now study the two-mode squeezing spectrum obtained from the combination of two homodyne photocurrents which result from the measurement of two signal fields ${{a}_{1}}(t)$ and ${{a}_{2}}(t)$, as depicted in figure 2. This strategy detects the correlations between four spectral components [68]. Specifically, we consider the situation in which the photocurrents corresponding to the two detected fields, $I_{1}^{({{\theta }_{1}})}(t)$ and $I_{2}^{({{\theta }_{2}})}(t)$, have the form of equation (27) and are combined to construct the total photocurrent

$$\begin{eqnarray}&&I_{c}^{({{\theta }_{1}},{{\theta }_{2}})}(t)=\frac{1}{\sqrt{\mu _{1}^{2}+\mu _{2}^{2}}}\left[ {{\mu }_{1}}\;I_{1}^{({{\theta }_{1}})}(t)+{{\mu }_{2}}\;I_{2}^{({{\theta }_{2}})}(t) \right],\end{eqnarray} \tag{ 34 }$$

where we have introduced the scaling parameters ${{\mu }_{1}}$ and ${{\mu }_{2}}$, which weight the two quadratures differently and hence provide a means to select arbitrary collective modes as discussed hereafter. These parameters are controllable experimentally, including the asymmetrical amplification and/or attenuation of the two photocurrents. The total photocurrent is then analyzed by frequency. As in the preceding case, the filtered photocurrent $J_{c}^{({{\theta }_{1}}{{\theta }_{2}})}(\epsilon )={{{\rm lim} }_{\tau \to \infty }}\frac{1}{\sqrt{\tau }}\int _{t-\tau }^{t}{\rm d}s\ {\rm cos} (\epsilon s+\varphi )\ I_{c}^{({{\theta }_{1}}{{\theta }_{2}})}(s)$ can be decomposed into two spectral components at the frequencies $\pm \epsilon$:

$$\begin{eqnarray}&&J_{c}^{({{\theta }_{+}},{{\theta }_{-}})}(\epsilon )=\frac{1}{\sqrt{2}}\left[ \overline{x_{c}^{({{\theta }_{+}})}}(\epsilon )+\overline{x_{c}^{({{\theta }_{-}})}}(-\epsilon ) \right]\end{eqnarray} \tag{ 35 }$$

where ${{\theta }_{\pm }}=\frac{{{\theta }_{1}}+{{\theta }_{2}}}{2}\pm \varphi$, and where we have introduced the quadrature operators for the collective spectral modes defined as

$$\begin{eqnarray}&&\overline{x_{c}^{(\theta )}}(\epsilon )={{{\rm e}}^{{\rm i}\theta }}\bar{c}(\epsilon )+{{{\rm e}}^{-{\rm i}\theta }}\overline{{{c}^{\dagger }}}(-\epsilon )\end{eqnarray} \tag{ 36 }$$

with the collective annihilation and creation operators given by

$$\begin{eqnarray}\begin{array}{rcl} \bar{c}(\pm \epsilon ) & = & \frac{1}{\sqrt{\mu _{1}^{2}+\mu _{2}^{2}}}\left[ {{\mu }_{1}}\;{{{\rm e}}^{{\rm i}{{\theta }_{c}}}}\overline{{{a}_{1}}}(\pm \epsilon )+{{\mu }_{2}}\;{{{\rm e}}^{-{\rm i}{{\theta }_{c}}}}\overline{{{a}_{2}}}(\pm \epsilon ) \right], \\ \overline{{{c}^{\dagger }}}(\mp \epsilon ) & = & \frac{1}{\sqrt{\mu _{1}^{2}+\mu _{1}^{2}}}\left[ {{\mu }_{1}}\;{{{\rm e}}^{-{\rm i}{{\theta }_{c}}}}\overline{a_{1}^{\dagger }}(\mp \epsilon )+{{\mu }_{2}}\;{{{\rm e}}^{{\rm i}{{\theta }_{c}}}}\overline{a_{2}^{\dagger }}(\mp \epsilon ) \right], \\ \end{array}\end{eqnarray} \tag{ 37 }$$

where $\left[ \bar{c}(\pm \epsilon ),\overline{{{c}^{\dagger }}}(\mp \epsilon ) \right]=1$ and ${{\theta }_{c}}=\left( {{\theta }_{1}}-{{\theta }_{2}} \right)/2$. Also in this case, the filtered photocurrent is a composite quadrature of the from of equation (29). Thus, although it is constructed from the collective operators in equation (37), we can still apply the results of section 3 to conclude that the two-mode squeezing spectrum, ${{S}^{(II)}}(\epsilon )$, obtained as the minimum over the phases ${{\theta }_{\pm }}$ of the autocorrelation function of the filtered current, has the form of equation (22) but is evaluated with the parameters

$$\begin{eqnarray}\begin{array}{rcl} n_{\pm }^{(c)} & = & \left\langle \overline{{{c}^{\dagger }}}(\mp \epsilon )\bar{c}(\pm \epsilon ) \right\rangle \\ {} & = & \frac{\mu _{1}^{2}\;v_{\pm }^{(11)}+\mu _{2}^{2}\;v_{\pm }^{(22)}+2\;{{\mu }_{1}}\;{{\mu }_{2}}\;\left| v_{\pm }^{(21)} \right|{\rm cos} \left[ 2\;{{\theta }_{-}}+{\rm arg} (v_{\pm }^{(21)}) \right]}{\mu _{1}^{2}+\mu _{2}^{2}} \\ {{m}^{(c)}} & = & \left\langle \bar{c}(\epsilon )\bar{c}(-\epsilon ) \right\rangle \\ {} & = & \frac{1}{\mu _{1}^{2}+\mu _{2}^{2}}\left[ \mu _{1}^{2}\;{{{\rm e}}^{{\rm i}({{\theta }_{1}}+\varphi )}}{{w}^{(11)}}+\mu _{2}^{2}\;{{{\rm e}}^{-{\rm i}({{\theta }_{1}}+\varphi )}}{{w}^{(22)}}+{{\mu }_{1}}\;{{\mu }_{2}}\left( {{{\rm e}}^{{\rm i}({{\theta }_{2}}+\varphi )}}{{w}^{(12)}}+{{{\rm e}}^{-{\rm i}({{\theta }_{2}}+\varphi )}}{{w}^{(21)}} \right) \right], \\ \end{array}\end{eqnarray} \tag{ 38 }$$

where

$$\begin{eqnarray}\begin{array}{rcl} v_{\pm }^{(jk)} & = & \left\langle \overline{a_{j}^{\dagger }}(\mp \epsilon )\overline{{{a}_{k}}}(\pm \epsilon ) \right\rangle \\ {{w}^{(jk)}} & = & \left\langle \overline{{{a}_{j}}}(\epsilon )\overline{{{a}_{k}}}(-\epsilon ) \right\rangle \\ \end{array}\end{eqnarray} \tag{ 39 }$$

for $j,k=1,2$. We remark that ${{S}^{(II)}}(\epsilon )$ is found by minimizing the autocorrelation function of the photocurrent in equation (35) over ${{\theta }_{\pm }}$ only, whereas ${{\theta }_{c}}$ and ${{\mu }_{j}}$ define the specific collective modes that are being probed and are fixed. Therefore, in this case the condition ${{S}^{(II)}}(\epsilon )\lt 1$ indicates the entanglement of the collective modes described by the operators in equation (37), each of which is a superpositions of the spectral components at the same sideband frequencies of the two fields. Moreover, as with the discussions of the single-mode homodyne spectrum, the corresponding logarithmic negativity can be, in turn, measured by summing two filtered photocurrents, at different phases, of the form of equation (35) and then calculating the corresponding autocorrelation function. The two-mode squeezing spectrum $S_{{\rm min} }^{(II)}(\epsilon )$ is then found by minimizing this quantity over the phases of both the local oscillator and the filter, and the result, in full similarity with the single-mode squeezing spectrum, is equal to equation (23) but now evaluated for the parameters in equation (38).

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As discussed in the previous sections, it is possible to construct arbitrary superposition of spectral modes at opposite sideband frequencies by the superposition of two homodyne photocurrents. The corresponding total photocurrent is then given by equation (32). Similarly, when the phases in equation (33) are set to some values for which one of the two parameters ${{\xi }_{\pm }}$ is equal to zero, then a single spectral mode is detected.

When applied to two distinct fields, this approach would permit the investigation of the correlations between two distinct spectral modes each belonging to a different field. Let us, for example, assume that we repeat the pair of measurements resulting in the total photocurrent in equation (32) for two different continuous fields ${{a}_{1}}(t)$ and ${{a}_{2}}(t)$. The two resultant composite filtered photocurrents are then, in general, given by $J_{{{\xi }_{+,j}},{{\xi }_{-,j}}}^{({{\theta }_{+,j}},{{\theta }_{-,j}})}(\epsilon )=\left[ {{\xi }_{+,j}}\overline{{{x}^{({{\theta }_{+,j}})}}}(\epsilon )+{{\xi }_{-,j}}\ \overline{{{x}^{({{\theta }_{-,j}})}}}(-\epsilon ) \right]/\sqrt{\xi _{+,j}^{2}+\xi _{-,j}^{2}}$ with the parameters defined as in equation (33) and where here j = 1, 2 distinguishes the parameters corresponding to the measurements of the first and second fields respectively.

If, in each pair of measurements, we tune the phases of the local oscillators and of the filtersto certain values for which ${{\theta }_{j}}-\theta _{j}^{{\prime} }-{{(-1)}^{j}}({{\varphi }_{j}}-\varphi _{j}^{{\prime} })=\pi$ and ${{\theta }_{j}}-\theta _{j}^{{\prime} }+{{(-1)}^{j}}({{\varphi }_{j}}-\varphi _{j}^{{\prime} })\ne \pi$ so that ${{\xi }_{-,1}}={{\xi }_{+,2}}=0$, then each composite photocurrent is proportional to a quadrature of a single spectral component corresponding respectively to the annihilation operators

$$\begin{eqnarray}&&{{\bar{a}}_{1}}(\epsilon )\ \ \ \ \ \ {\rm and}\ \ \ \ \ \ {{\bar{a}}_{2}}(-\epsilon ).\end{eqnarray} \tag{ 40 }$$

The two photocurrents are then summed together, after being multiplied by appropriately chosen scaling factors ${{\zeta }_{\pm }}$, so that the resulting total photocurrent is

$$\begin{eqnarray*}&&{{J}_{tot}}=\frac{1}{\sqrt{\zeta _{+}^{2}+\zeta _{-}^{2}}}\left[ {{\zeta }_{+}}\overline{x_{1}^{({{\theta }_{+,1}})}}(\epsilon )+{{\zeta }_{-}}\ \overline{x_{2}^{({{\theta }_{-,2}})}}(-\epsilon ) \right],\end{eqnarray*}$$

where the single-mode quadratures $\overline{x_{j}^{(\theta )}}(\pm \epsilon )$ are defined in equation (18). Thus, this protocol detects the combined quadrature defined in equation (20). The corresponding squeezing spectrum, ${{S}^{(III)}}(\epsilon )$, defined for ${{\zeta }_{+}}={{\zeta }_{-}}$, as the minimum of the autocorrelation function of the total photocurrent over ${{\theta }_{+,1}}$ and ${{\theta }_{-,2}}$, is equal to equation (22) and is obtained by appropriately tuning the phases of the local oscillator and the filter during repeated homodyne measurements. Similarly, $S_{{\rm min} }^{(III)}(\epsilon )$, defined as the minimum of the power spectrum of the total photocurrent over ${{\theta }_{+,1}}$, ${{\theta }_{-,2}}$ and ${{\zeta }_{\pm }}$, is equal to equation (23). In particular, also in this case we can conclude that this quantity is equivalent to the logarithmic negativity between ${{\bar{a}}_{1}}(\epsilon )$ and ${{\bar{a}}_{2}}(-\epsilon )$ when the fields are Gaussian.

An alternative strategy for probing single spectral modes and hence for measuring the squeezing spectrum, as well as the logarithmic negativity between two distinct spectral components of two distinct fields, as in section 4.3, is based on heterodyne measurements.

In heterodyne techniques, the local oscillator is detuned from the carrier frequency of the signal field by a quantity $\Delta ={{\omega }_{LO}}-{{\omega }_{L}}$, and the corresponding operator for the photocurrent reads

$$\begin{eqnarray*}&&I_{\Delta }^{(\theta )}(t)={{{\rm e}}^{{\rm i}\Delta t}}{{{\rm e}}^{{\rm i}\theta }}a(t)+{{{\rm e}}^{-{\rm i}\Delta t}}{{{\rm e}}^{-{\rm i}\theta }}{{a}^{\dagger }}(t).\end{eqnarray*}$$

Hence, the corresponding filtered photocurrent, $J_{\Delta }^{(\theta ,\varphi )}(\epsilon )={{{\rm lim} }_{\tau \to \infty }}\frac{1}{\sqrt{\tau }}\int _{t-\tau }^{t}{\rm d}s\ {\rm cos} (\epsilon \;s+\varphi )I_{\Delta }^{(\theta )}(s)$, is the superposition of the quadratures for the spectral components at the frequencies $\Delta \pm \epsilon$, namely $J_{\Delta }^{(\theta ,\varphi )}(\epsilon )=\left[ \overline{{{x}^{(\theta +\varphi )}}}(\Delta +\epsilon )+\overline{{{x}^{(\theta -\varphi )}}}(\Delta -\epsilon ) \right]/\sqrt{2}$. Thus, whereas homodyne detection probes spectral components at opposite sideband frequencies, heterodyne techniques measure two components asymmetrically located with respect to the carrier frequency of the signal field but symmetric with respect to the local oscillator frequency, which, in our description (where all frequencies are relative to the carrier signal), is equal to Δ (see figure 3).

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In particular heterodyne techniques can be used to detect a single spectral component, as discussed in the following. Say we want to detect the component at frequency Ω; then we set the detuning Δ at a value much larger than the typical bandwidth of the signal $|\Delta |\gg {{\Delta }_{{\rm signal}}}$, which is the band of frequencies that are populated by the signal photons. We also assume that the frequency Ω is a relevant frequency for the field $|\Omega |\leqslant {{\Delta }_{{\rm signal}}}$. Then the corresponding heterodyne photocurrent is filtered at the frequency $\epsilon =\Delta -\Omega$ so that, as depicted in figure 3, $J_{\Delta }^{(\theta ,\varphi )}(\epsilon )$ is the superposition of the field quadratures at the frequencies Ω and $2\Delta -\Omega$:

$$\begin{eqnarray}&&J_{\Delta }^{(\theta ,\varphi )}(\Delta -\Omega )=\frac{1}{\sqrt{2}}\left[ \overline{{{x}^{(\theta -\varphi )}}}(\Omega )+\overline{{{x}^{(\theta +\varphi )}}}(2\Delta -\Omega ) \right].\end{eqnarray} \tag{ 41 }$$

Since the signal covers a bandwidth much smaller than Δ, the mode at $2\Delta -\Omega$ is basically in a vacuum and only the photons of the sideband Ω are detected. However, in doing this the vacuum fluctuations of the empty component at $2\Delta -\Omega$ are added to the signal, resulting in higher noise.

This approach can be exploited to measure the correlations between two spectral modes belonging to two separate fields. Specifically the two-mode heterodyne spectrum is obtained when detecting two signals with two heterodyne measurements (with $\Delta \gg {{\Delta }_{{\rm signal}}}$). The corresponding photocurrents are filtered independently at the frequencies ${{\epsilon }_{1}}$ and ${{\epsilon }_{2}}$ respectively, and then combined, with appropriately chosen scaling factors ${{\xi }_{j}}$, to construct the total filtered photocurrent ${{J}_{\Delta ,tot}}\propto {{\xi }_{1}}J_{\Delta }^{({{\theta }_{1}},{{\varphi }_{1}})}({{\epsilon }_{1}})+{{\xi }_{2}}J_{\Delta }^{({{\theta }_{2}},{{\varphi }_{2}})}({{\epsilon }_{2}})$. If, in particular, we are interested in the spectral components at frequency Ω of the first field and at frequency $-\Omega$ of the second such that $|\Omega |\leqslant {{\Delta }_{{\rm signal}}}$, we consider the filtered photocurrents at the frequencies ${{\epsilon }_{1}}=\Delta -\Omega$ and ${{\epsilon }_{2}}=\Delta +\Omega$. As in equation (41) they are equal, respectively, to the superpositions of the two quadratures at frequencies Ω and $2\;\Delta -\Omega$ of the first field and of the two quadratures at frequencies $-\Omega$ and $2\Delta +\Omega$ of the second. Correspondingly, the total photocurrent is given by

$$\begin{eqnarray}&&{{J}_{\Delta ,tot}}=\frac{{{\xi }_{1}}\overline{x_{1}^{({{\theta }_{1}}-{{\varphi }_{1}})}}(\Omega )+{{\xi }_{2}}\overline{x_{2}^{({{\theta }_{2}}-{{\varphi }_{2}})}}(-\Omega )}{\sqrt{2}\sqrt{\xi _{1}^{2}+\xi _{2}^{2}}}+\frac{{{\xi }_{1}}\overline{x_{1}^{({{\theta }_{1}}+{{\varphi }_{1}})}}(2\Delta -\Omega )+{{\xi }_{2}}\overline{x_{2}^{({{\theta }_{2}}+{{\varphi }_{2}})}}(2\Delta +\Omega )}{\sqrt{2}\sqrt{\xi _{1}^{2}+\xi _{2}^{2}}}\end{eqnarray} \tag{ 42 }$$

where the quadrature operators for a single component are defined in equation (18). Its autocorrelation function is therefore given by

$$\begin{eqnarray}&&\left\langle {{\left[ {{J}_{\Delta ,tot}} \right]}^{2}} \right\rangle =\frac{1}{2}\left\langle {{\left[ X_{{{\xi }_{1}},{{\xi }_{2}}}^{({{\theta }_{1}}-{{\varphi }_{1}},{{\theta }_{2}}-{{\varphi }_{2}})}(\Omega ) \right]}^{2}} \right\rangle +\frac{1}{2}\end{eqnarray} \tag{ 43 }$$

where the term $\frac{1}{2}$ on the right is due to the vacuum fluctuations of the spectral modes at $2\Delta \pm \Omega$, the collective quadrature $X_{{{\xi }_{1}},{{\xi }_{2}}}^{({{\theta }_{1}},{{\theta }_{2}})}(\Omega )$ has the same form of the one defined in equation (20), and its autocorrelation function, which is given in equation (22), quantifies the correlations between the modes $\overline{{{a}_{1}}}(\Omega )$ and $\overline{{{a}_{2}}}(-\Omega )$.

Also in this case we define two kinds of optimized squeezing spectra. One is obtained by minimizing the autocorrelation function in equation (43), with ${{\xi }_{1}}={{\xi }_{2}}$, only over the phases of the local oscillators $T(\Omega )={{{\rm min} }_{{{\theta }_{j}}}}\left\langle {{\left[ {{J}_{\Delta ,tot}} \right]}^{2}} \right\rangle$; the other is obtained when the minimization also runs over the scaling parameters, ${{T}_{{\rm min} }}(\Omega )={{{\rm min} }_{{{\theta }_{j}},{{\xi }_{j}}}}\left\langle {{\left[ {{J}_{\Delta ,tot}} \right]}^{2}} \right\rangle$. In both cases they can be expressed in terms of the squeezing spectra resulting from the protocol described in section 4.3 as

$$\begin{eqnarray}&&T(\Omega )=\frac{{{S}^{(III)}}(\Omega )+1}{2},\ \ \ \ \ \ \ \ {{T}_{{\rm min} }}(\Omega )=\frac{S_{{\rm min} }^{(III)}(\Omega )+1}{2}.\end{eqnarray} \tag{ 44 }$$

Thus, according to equation (26), in the Gaussian case, ${{T}_{{\rm min} }}(\Omega )$ measures the entanglement between the modes whose operators are ${{\bar{a}}_{1}}(\Omega )$ and ${{\bar{a}}_{2}}(-\Omega )$.

In section 4 we described three different strategies for the experimental investigation of the spectral properties of stationary continuous fields, which are based on homodyne and heterodyne techinques. They probe different pairs of spectral modes given, respectively, by equations (30), (37), and (40). When applied to the investigation of the optomechanical system, these techniques allow the detection of the corresponding spectral components of the two output fields $a_{j}^{{\rm out}}(t)$, as depicted in figure 4. In particular, using these techniques it is possible to study composite quadratures of these pairs of modes and their squeezing and entanglement properties. We have identified two different kinds of optimized squeezing spectra, corresponding to different experimental approaches for the measurement and the minimization of the homodyne photocurrent fluctuations. Specifically, to probe symmetric superpositions of quadratures of the two modes it is sufficient to apply standard homodyne techniques; thus minimization is achieved by tuning the relative phase of the two quadratures, which is controlled experimentally by the phase of the local oscillator. We indicate this phase-optimized spectrum with the symbol ${{S}^{(\ell )}}(\omega )$, where the label $\ell =I,II,III$ is used to distinguish the three detection strategies (see figure 4). At the same time, non-symmetric superpositions, with different weights of the two quadratures, can be probed by combining different filtered photocurrents detected with appropriately selected phases of both the filter and the local oscillator. The globally optimized squeezing spectrum $S_{{\rm min} }^{(\ell )}(\omega )$ is then obtained, minimizing the corresponding fluctuations over the phases of both the local oscillator and the filter.

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In all cases the squeezing spectra ${{S}^{(\ell )}}(\omega )$ and $S_{{\rm min} }^{(\ell )}(\omega )$ are equal to equation (22) and (23), evaluated in each case for the specific parameters ${{n}_{\pm }}$ and m, which correspond to the detected spectral modes. Specifically, ${{S}^{(I)}}(\omega )$ and $S_{{\rm min} }^{(I)}(\omega )$ are obtained by the single-mode homodyne detection of a single field (either one of the two output fields), as discussed in section 4.1, and if applied to the output from the first mirror, then $n_{\pm }^{(I)}=\left\langle \overline{a{{_{1}^{{\rm out}}}^{\dagger }}}(\mp \omega )\ \overline{a_{1}^{{\rm out}}}(\pm \omega ) \right\rangle$ and ${{m}^{(I)}}=\left\langle \overline{a_{1}^{{\rm out}}}(\omega )\ \overline{a_{1}^{{\rm out}}}(-\omega ) \right\rangle$. The second strategy is based on the two-mode homodyne detection of the two output fields (see section 4.2), and the corresponding spectra, ${{S}^{(II)}}(\omega )$ and $S_{{\rm min} }^{(II)}(\omega )$, are evaluated for $n_{\pm }^{(II)}=\left\langle \overline{c{{_{j}^{{\rm out}}}^{\dagger }}}(\mp \omega )\ \overline{c_{j}^{{\rm out}}}(\pm \omega ) \right\rangle$ and ${{m}^{(II)}}=\left\langle \overline{c_{j}^{{\rm out}}}(\omega )\ \overline{c_{j}^{{\rm out}}}(-\omega ) \right\rangle$, where the operators $\overline{{{c}^{(out)}}}(\pm \omega )$ have the same form of equation (37), but in this case, they are constructed as the superpositions of the two filtered output fields $\overline{a_{j}^{{\rm out}}}(\pm \omega )$. Finally, ${{S}^{(III)}}(\omega )$ and $S_{{\rm min} }^{(III)}(\omega )$ correspond to the two-mode heterodyne detection of the two output fields as discussed in section 4.4. If it is applied to the spectral component at frequency ω of the first output and at $-\omega$ of the second then ${{S}^{(III)}}(\omega )$ and $S_{{\rm min} }^{(III)}(\omega )$ are evaluated for $n_{\pm }^{(III)}=\left\langle \overline{a{{_{1}^{{\rm out}}}^{\dagger }}}(\mp \omega )\ \overline{a_{2}^{{\rm out}}}(\pm \omega ) \right\rangle$ and ${{m}^{(III)}}=\left\langle \overline{a_{1}^{{\rm out}}}(\omega )\ \overline{a_{2}^{{\rm out}}}(-\omega ) \right\rangle$. We note that these last two spectra can also be retrieved by combining various homodyne photocurrents as discussed in section 4.3.

The power spectrum matrix in equation (45) can be used to find

$$\begin{eqnarray*}\begin{array}{rcl} {{S}^{(\ell )}}(\omega ) & = & 1+\frac{4\;{{g}^{2}}}{|f(\omega ){{|}^{2}}}\left[ {{\left| q_{+}^{(\ell )} \right|}^{2}}\;{{\beta }_{\omega }}+{{\left| q_{-}^{(\ell )} \right|}^{2}}\;{{\beta }_{-\omega }}-2\left| q_{+}^{(\ell )}q_{-}^{(\ell )}\;\alpha \right| \right], \\ S_{{\rm min} }^{(\ell )}(\omega ) & = & 1+\frac{4\;{{g}^{2}}}{|f(\omega ){{|}^{2}}}\left[ {{\left| q_{+}^{(\ell )} \right|}^{2}}\;{{\beta }_{\omega }}+{{\left| q_{-}^{(\ell )} \right|}^{2}}\;{{\beta }_{-\omega }}-\sqrt{4{{\left| q_{+}^{(\ell )}q_{-}^{(\ell )}\;\alpha \right|}^{2}}+{{\left( {{\left| q_{+}^{(\ell )} \right|}^{2}}\;{{\beta }_{\omega }}-{{\left| q_{-}^{(\ell )} \right|}^{2}}\;{{\beta }_{-\omega }} \right)}^{2}}} \right] \\ \end{array}\end{eqnarray*}$$

where

$$\begin{eqnarray}\begin{array}{rcl} q_{+}^{(I)} & = & q_{-}^{(I)}=\sqrt{{{\kappa }_{1}}} \\ q_{+}^{(II)} & = & q_{-}^{(II)}=\frac{{{\mu }_{1}}\;{{{\rm e}}^{{\rm i}{{\theta }_{c}}}}\sqrt{{{\kappa }_{1}}}+{{\mu }_{2}}\;{{{\rm e}}^{-{\rm i}{{\theta }_{c}}}}\sqrt{{{\kappa }_{2}}}}{\sqrt{\mu _{1}^{2}+\mu _{2}^{2}}} \\ q_{+}^{(III)} & = & \sqrt{{{\kappa }_{1}}},\ \ \ \ \ \ \ \ \ q_{-}^{(III)}=\sqrt{{{\kappa }_{2}}}, \\ \end{array}\end{eqnarray} \tag{ 49 }$$

In the case of strategy $(II)$, the parameters ${{\mu }_{j}}$ and ${{\theta }_{c}}$, in the expression for $q_{\pm }^{(II)}$, determine the specific detected composite modes, which are defined as in equation (37). We observe that when

$$\begin{eqnarray}&&{{\theta }_{c}}=0\ \ \ \ \ \ {\rm and}\ \ \ \ \ \ \ {{\mu }_{1}}/{{\mu }_{2}}=\sqrt{{{\kappa }_{1}}/{{\kappa }_{2}}}\end{eqnarray} \tag{ 50 }$$

then $q_{\pm }^{(II)}={{\kappa }_{1}}+{{\kappa }_{2}}$; hence we can conclude that $S_{{\rm min} }^{(II)}(\omega )$ evaluated for a two-sided configuration with decay rates ${{\kappa }_{1}}$ and ${{\kappa }_{2}}$ is equal to $S_{{\rm min} }^{(I)}(\omega )$ when it is evaluated for a single-sided cavity with the decay rate equal to the sum of the decay rates ${{\kappa }_{1}}+{{\kappa }_{2}}$ of the two-sided configuration. It indicates that the entanglement between $\overline{a_{1}^{{\rm out}}}(\omega )$ and $\overline{a_{1}^{{\rm out}}}(-\omega )$, in a single-sided cavity, is redistributed among the four spectral components $\overline{a_{1}^{{\rm out}}}(\pm \omega )$ and $\overline{a_{2}^{{\rm out}}}(\pm \omega )$ in the case of a two-sided cavity. For this reason, the same amount of squeezing found in the case of a single-sided cavity, can be recovered when the information from the two decay channels of a two-sided cavity are properly combined. In particular $q_{\pm }^{(II)}$ is maximum for the parameters of equation (50) and consequently the corresponding modes are the collective modes that are maximally squeezed (and entangled).

We also note that according to equation (24) we find that, in all cases, the pairs of spectral components are entangled (and squeezed), although possibly with different degrees of entanglement (and squeezing), when

$$\begin{eqnarray}&&{{\beta }_{\omega }}\;{{\beta }_{-\omega }}\lt |\alpha {{|}^{2}}.\end{eqnarray} \tag{ 51 }$$

Moreover we note that the logarithmic negativity between the pair of spectral components detected with each strategy is obtained by applying the definition in equation (25) to the parameter

$$\begin{eqnarray*}&&{{\nu }^{(\ell )}}(\omega )=S_{{\rm min} }^{(\ell )}(\omega ),\end{eqnarray*}$$

which is equal to the minimum symplectic eigenvalue of the corresponding partially transposed covariance matrix.

In figure 5 we compare the results for the spectra evaluated for realistic parameters and corresponding to the three detection strategies, and hence to different pairs of spectral components. Each plot in figure 5 is evaluated for different values of the relative decay rate ${{\kappa }_{2}}/{{\kappa }_{1}}$ of the two mirrors, whereas the total decay rate ${{\kappa }_{1}}+{{\kappa }_{2}}$ and all the other parameters are kept fixed. In plot (a) we study a single-sided cavity with ${{\kappa }_{2}}=0$. In plot (b) the mirrors are lossy and non-symmetric, and finally in (c) the two mirrors are symmetric: ${{\kappa }_{1}}={{\kappa }_{2}}$. When ${{\kappa }_{2}}=0$, in plot (a), the curves for ${{S}^{(II)}}$, ${{\nu }^{(II)}}$, ${{S}^{(III)}}$, and ${{\nu }^{(III)}}$ correspond to the situation in which the detector on the second mirror detects only vacuum fluctuations. It is therefore clear that the curves for ${{S}^{(III)}}$ and ${{\nu }^{(III)}}$, which measure the correlations between the single spectral component $\overline{a_{1}^{{\rm out}}}(\omega )$ of the field lost from the first mirror and the single spectral component $\overline{a_{2}^{{\rm out}}}(-\omega )$ of the field lost from the second show no squeezing. On the other hand, maximum two-mode squeezing and entanglement are observed for the single-mode homodyne spectra corresponding to the curves ${{S}^{(I)}}$ and ${{\nu }^{(I)}}$, which measure the correlations between $\overline{a_{1}^{{\rm out}}}(\omega )$ and $\overline{a_{1}^{{\rm out}}}(-\omega )$. The curves ${{S}^{(II)}}$, ${{\nu }^{(II)}}$, that measure the correlations between the composite modes in equation (37), are at an intermediate value as a result of the vacuum fluctuations of the modes $\overline{a_{2}^{{\rm out}}}(\pm \omega )$, which reduces the visibility of the two-mode squeezing between $\overline{a_{1}^{{\rm out}}}(\omega )$ and $\overline{a_{1}^{{\rm out}}}(-\omega )$.

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Plot (c) corresponds to a symmetric two-sided cavity with ${{\kappa }_{1}}={{\kappa }_{2}}$. In this case the maximum squeezing and entanglement are obtained for the curves ${{S}^{(II)}}$ and ${{\nu }^{(II)}}$, indicating that in this configuration the maximally entangled spectral components correspond to the composite modes in equation (37). In particular ${{S}^{(II)}}$ and ${{\nu }^{(II)}}$ are equal to the curves for ${{S}^{(I)}}$ and ${{\nu }^{(I)}}$ in the single-sided cavity reported in plot (a). In the two cases, the correlations of the cavity field built by the optomechanical dynamics are equal as a result of the equal total decay rate. The only difference is that, in one case all the photons are lost through a single mirror, whereas in the other they are split in the two decay channels and only their combined detection can reveal the corresponding total degree of squeezing and entanglement. Moreover in plot (c) we observe that ${{S}^{(III)}}$ and ${{\nu }^{(III)}}$ are equal to ${{S}^{(I)}}$ and ${{\nu }^{(I)}}$. In this case, the two output fields are symmetric, hence the correlations between $\overline{a_{1}^{({\rm out})}}(\omega )$ and $\overline{a_{1}^{({\rm out})}}(-\omega )$ are equal to those between $\overline{a_{1}^{({\rm out})}}(\omega )$ and $\overline{a_{2}^{({\rm out})}}(-\omega )$. Finally, plot (b) corresponds to an intermediate situation between the two described in (a) and (c), and it shows that the three detection strategies can display different degrees of squeezing. In general, the values of all the squeezing spectra can lie at any value between the extremes set by the corresponding curves in plot (a) and (c), depending on the actual value of the ratio ${{\kappa }_{2}}/{{\kappa }_{1}}\in [0,1]$.

We also emphasize that the values of ${{S}^{(II)}}$ and ${{\nu }^{(II)}}$ are reported, in the three plots, for the same values of the parameters ${{\mu }_{j}}$ and ${{\theta }_{c}}$ which define the specific superposition of spectral components that are being probed as defined in equation (37). However, for each value of ${{\kappa }_{2}}/{{\kappa }_{1}}$ the values of ${{\mu }_{j}}$ and ${{\theta }_{c}}$ can be appropriately tuned to find the composite modes that are characterized by the same maximum amount of squeezing as in plot (c). This is shown for the parameters of plot (b) and at $\omega =0$ in figure 6, where maximum squeezing and entanglement (recovering the maximum value found in figure 5(c)) is found when ${{\mu }_{2}}/{{\mu }_{1}}=\sqrt{{{\kappa }_{2}}/{{\kappa }_{1}}}$. We further observe that, in figure 5, the results for ${{S}^{(I)}}$ and ${{S}^{(II)}}$ are very close to ${{\nu }^{(I)}}$ and ${{\nu }^{(II)}}$ respectively. They are substantially different only close to the mechanical resonances ($\omega =\pm {{\omega }_{m}}$), where, although the two spectral modes are entangled (${{\nu }^{(\ell )}}\lt 1$), this feature is not reflected in the corresponding squeezing spectrum ${{S}^{(\ell )}}$, which is significantly larger and very close to 1. This happens when the two corresponding spectral modes are significantly asymmetric so that $n_{+}^{(\ell )}\ne n_{-}^{(\ell )}$. This situation is realized very close to the condition $\omega =\pm {{\omega }_{m}}$ and for a bandwidth on the order of the mechanical dissipation rate γ, which, in a typical optomechanical system, can be very small, as shown in the insets of figure 5. On the other hand, the discrepancy between ${{S}^{(III)}}$ and ${{\nu }^{(III)}}$ can be considerably larger, covering, for example, the full spectrum in plots (a) and (b). This is because in this case the difference between the corresponding $n_{+}^{(III)}$ and $n_{-}^{(III)}$ is proportional to the difference between ${{\kappa }_{1}}$ and ${{\kappa }_{2}}$, which is relatively large in plots (a) and (b). In plot (c), on the other hand, the two mirrors are symmetric and likewise ${{S}^{(III)}}$ and ${{\nu }^{(III)}}$ are very close, reproducing the results for ${{S}^{(I)}}$ and ${{\nu }^{(I)}}$.

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