Deterministic preparation of superpositions of vacuum plus one photon by adaptive homodyne detection: experimental considerations

To begin, we recapitulate the protocol presented in [5] for the deterministic preparation of an arbitrary pure qubit state within the single rail photonic encoding (see figure 1). Consider for the moment the preparation of the equal superposition state, i.e.

$$\begin{eqnarray}&&\mid \psi \rangle =\frac{\mid 0\rangle +\mid 1\rangle }{\sqrt{2}}.\end{eqnarray} \tag{ 1 }$$

Assuming we can generate a single photon deterministically, we can send this through a 50:50 beam splitter, generating two out-going entangled modes in the state

$$\begin{eqnarray}&&\mid \Psi \rangle =\frac{\mid 0{{\rangle }_{1}}\mid 1{{\rangle }_{2}}-\mid 1{{\rangle }_{1}}\mid 0{{\rangle }_{2}}}{\sqrt{2}},\end{eqnarray} \tag{ 2 }$$

where the subscripts 1, 2 indicate the output ports of the beam splitter.

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Suppose that on the first mode we perform a canonical phase measurement described by the probability-operator measure (POM)

$$\begin{eqnarray}&&{{\hat{E}}_{{\rm can}}}(\theta )=\frac{\mid \theta \rangle \langle \theta \mid }{2\pi },\quad \mid \theta \rangle =\mid 0{{\rangle }_{1}}+{{{\rm e}}^{{\rm i}\theta }}\mid 1{{\rangle }_{1}},\end{eqnarray} \tag{ 3 }$$

which is normalized according to

$$\begin{eqnarray}&&\int _{0}^{2\pi }{\rm d}\theta \ {{\hat{E}}_{{\rm can}}}(\theta )=\mid 0\rangle \langle 0\mid +\mid 1\rangle \langle 1\mid =\hat{1},\end{eqnarray} \tag{ 4 }$$

where the identity $\hat{1}$ is restricted to the subspace of interest, spanned by ${{\{\mid 0\rangle }_{1}},\mid 1{{\rangle }_{1}}\}$. As a result of the measurement, we obtain the outcome θ, drawn randomly with a uniform probability distribution in the range $[0,2\pi )$. After this measurement, the quantum state on the other mode collapses to

$$\begin{eqnarray}&&\mid \psi {{\rangle }_{2}}=\frac{\mid 1{{\rangle }_{2}}-{{{\rm e}}^{-{\rm i}\theta }}\mid 0{{\rangle }_{2}}}{\sqrt{2}}=\frac{\mid 0{{\rangle }_{2}}+{{{\rm e}}^{{\rm i}(\theta +\pi )}}\mid 1{{\rangle }_{2}}}{\sqrt{2}}.\end{eqnarray} \tag{ 5 }$$

Then, by using classical feed-forward from the phase measurement, we can undo the random phase $\theta +\pi$ by passing the state through a phase modulator to create (1).

This protocol is easily generalizable to arbitrary qubit states. We can prepare non-equal superpositions by employing a beam splitter with intensity reflectivity η, to get at the output of the beam splitter the quantum state

$$\begin{eqnarray}&&\frac{\sqrt{\eta }\mid 0{{\rangle }_{1}}\mid 1{{\rangle }_{2}}-\sqrt{1-\eta }\mid 1{{\rangle }_{1}}\mid 0{{\rangle }_{2}}}{\sqrt{2}}.\end{eqnarray} \tag{ 6 }$$

Again, measuring the first mode with a canonical phase measurement, we can correct the phase on the second mode, and apply any desired additional phase φ, to obtain

$$\begin{eqnarray}&&\mid \psi \rangle =\sqrt{1-\eta }\mid 0{{\rangle }_{2}}+{{{\rm e}}^{{\rm i}\varphi }}\sqrt{\eta }\mid 1{{\rangle }_{2}}.\end{eqnarray} \tag{ 7 }$$

It is not easy in optics to implement the canonical POM (3), as we will investigate. The problem of efficiency is an obvious one, but this is simply equivalent to loss, and so can be incorporated into the final analysis by applying loss to the state. A more interesting problem is how to project onto the state $\mid \theta \rangle$, with its equal superposition of $\mid 0\rangle$ and $\mid 1\rangle$. This equal superposition is essential for ensuring that all outcomes lead to a state that is only a phase shift away from the desired qubit state. More generally, an optical measurement will involve projecting onto an unnormalized state

$$\begin{eqnarray}&&\mid R\rangle =\mid 0\rangle +R\mid 1\rangle ,\end{eqnarray} \tag{ 8 }$$

where $R\in \mathbb{C}$. The POM describing such a measurement is

$$\begin{eqnarray}&&\hat{E}(R)={{\wp }_{{\rm ost}}}(R)\mid R\rangle \langle R\mid ,\end{eqnarray} \tag{ 9 }$$

where ${{\wp }_{{\rm ost}}}(R)$ is a positive, normalized distribution satisfying

$$\begin{eqnarray}&&\int {{\wp }_{{\rm ost}}}(R){{{\rm d}}^{2}}R=\int {{\wp }_{{\rm ost}}}(R)\ \mid R{{\mid }^{2}}{{{\rm d}}^{2}}R=1.\end{eqnarray} \tag{ 10 }$$

This ensures that the POM obeys the completeness relation

$$\begin{eqnarray}&&\int \hat{E}(R){{{\rm d}}^{2}}R=\hat{1}\end{eqnarray} \tag{ 11 }$$

as required.

The notation ${{\wp }_{{\rm ost}}}(R)$ is because we call this an ostensible probability distribution for R [13]. The actual probability distribution for R is, for an input state $\mid \Psi \rangle$,

$$\begin{eqnarray}&&\wp (R)={\rm T}{{{\rm r}}_{{}}}[\mid \Psi \rangle \langle \Psi \mid \hat{E}(R)]={{\wp }_{{\rm ost}}}(R)\langle \Psi \mid R\rangle \langle R\mid \Psi \rangle .\end{eqnarray} \tag{ 12 }$$

Note that in a case of a vacuum input state, the ostensible statistics coincide with the actual statistics. As we will see, the properties of ${{\wp }_{{\rm ost}}}$ are critical to the performance of the protocol. Given the result R, the best estimate of the phase is always

$$\begin{eqnarray}&&\theta ={\rm arg} \;R.\end{eqnarray} \tag{ 13 }$$

Thus, the POM for this phase measurement is

$$\begin{eqnarray}\begin{array}{rcl} \hat{E}(\theta ){\rm d}\theta & = & {\rm d}\theta \int _{0}^{\infty }{\rm d}\mid R\mid \ \hat{E}(R) \\ {} & = & \frac{{\rm d}\theta }{2\pi }[\mid 0\rangle \langle 0\mid +F(\mid 1\rangle \langle 0\mid {{{\rm e}}^{{\rm i}\theta }}+\mid 0\rangle \langle 1\mid {{{\rm e}}^{-{\rm i}\theta }})+\mid 1\rangle \langle 1\mid ] \\ {} & = & F\times {{{\hat{E}}}_{{\rm can}}}(\theta )\ {\rm d}\theta +(1-F)\times \hat{1}\ \frac{{\rm d}\theta }{2\pi }, \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&F={{\langle \mid R\mid \rangle }_{{\rm ost}}}\equiv \int {{{\rm d}}^{2}}R\ \mid R\mid \ {{\wp }_{{\rm ost}}}(R).\end{eqnarray} \tag{ 15 }$$

Clearly, $F$ is a figure of merit expressing how close the measurement is to canonical. It is also directly related to the fidelity of the qubit preparation protocol above. Considering the 50:50 case for simplicity, and defining ${{\mathcal{U}}_{2}}(\alpha )\bullet ={{{\rm e}}^{{\rm i}\alpha \hat{a}_{2}^{\dagger }{{{\hat{a}}}_{2}}}}\bullet {{{\rm e}}^{-{\rm i}\alpha \hat{a}_{2}^{\dagger }{{{\hat{a}}}_{2}}}}$, the final quantum state is

$$\begin{eqnarray}&&{{\rho }_{2}}=\int {{\mathcal{U}}_{2}}(-{\rm arg} \;R-\pi )\ {\rm T}{{{\rm r}}_{1}}[\mid \Psi \rangle \langle \Psi \mid \hat{E}(R)]{{{\rm d}}^{2}}R\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&=\frac{\mid 0\rangle \langle 0{{\mid }_{2}}+F(\mid 0\rangle \langle 1{{\mid }_{2}}+\mid 1\rangle \langle 0{{\mid }_{2}})+\mid 1\rangle \langle 1{{\mid }_{2}}}{2}.\end{eqnarray} \tag{ 17 }$$

This has a purity of ${\rm T}{{{\rm r}}_{{}}}[{{({{\rho }_{2}})}^{2}}]=(1+{{F}^{2}})/2$ and a fidelity with the desired state of $\langle \psi \mid {{\rho }_{2}}\mid \psi \rangle =(1+F)/2$. Thus we can use (15) as figure of merit to evaluate the protocol.

In the homodyne detection, the LO phase $\Phi (t)$ is constant in time, i.e. $\Phi (t)={{\Phi }_{0}}$. In this case, the definition of $R$ becomes

$$\begin{eqnarray}&&R={{{\rm e}}^{{\rm i}{{\Phi }_{0}}}}X,\end{eqnarray} \tag{ 33 }$$

with

$$\begin{eqnarray}&&X=\int _{-\infty }^{\infty }\sqrt{u(s)}\xi (s){\rm d}s.\end{eqnarray} \tag{ 34 }$$

This is a real Gaussian random variable with zero mean and hence, as required by (10), unit variance. Therefore, the ostensible distribution for X is

$$\begin{eqnarray}&&{{\wp }_{{\rm ost}}}(X){\rm d}X=\frac{1}{\sqrt{2\pi }}{{{\rm e}}^{-{{X}^{2}}/2}}{\rm d}X,\end{eqnarray} \tag{ 35 }$$

and we can calculate

$$\begin{eqnarray}&&F=\int _{-\infty }^{+\infty }\mid X\mid {{\wp }_{{\rm ost}}}(X){\rm d}X=\sqrt{\frac{2}{\pi }}\approx 0.797\;.\end{eqnarray} \tag{ 36 }$$

In heterodyne detection, the LO is detuned, i.e. outside the bandwidth of the optical mode. That is, the phase $\Phi (t)$ is linearly increasing in time at a rate $\Delta \;\gg \;{{{\rm max} }_{t}}[u(t)]$. By definition (26) we get

$$\begin{eqnarray}&&R=\int _{-\infty }^{\infty }{{{\rm e}}^{{\rm i}{{\Phi }_{0}}+{\rm i}\Delta s}}\sqrt{u(s)}\xi (s){\rm d}s.\end{eqnarray} \tag{ 37 }$$

This is a complex Gaussian random variable with zero mean and in the limit of $\Delta \to \infty$ gives a rotationally symmetric variable A with

$$\begin{eqnarray}&&{{\wp }_{{\rm ost}}}(A){{{\rm d}}^{2}}\;A=\frac{1}{\pi }{{{\rm e}}^{-\mid A{{\mid }^{2}}}}{{{\rm d}}^{2}}A.\end{eqnarray} \tag{ 38 }$$

From this, we can evaluate

$$\begin{eqnarray}&&F=\int \mid A\mid {{\wp }_{{\rm ost}}}(A){{{\rm d}}^{2}}\;A=\frac{\sqrt{\pi }}{2}\approx 0.886,\end{eqnarray} \tag{ 39 }$$

showing that heterodyne results in a better phase measurement than homodyne.

To do even better than heterodyne detection it is necessary to implement adaptive dyne detection. In particular, [17] introduced the following adaptive algorithm:

$$\begin{eqnarray}&&\Phi (t)={\rm arg} \;{{R}_{t}}+\frac{\pi }{2}.\end{eqnarray} \tag{ 40 }$$

This gives the stochastic differential equation for ${{R}_{t}}$

$$\begin{eqnarray}&&{\rm d}{{R}_{t}}={\rm i}\frac{{{R}_{t}}}{\mid {{R}_{t}}\mid }\sqrt{u(t)}J(t){\rm d}t\end{eqnarray} \tag{ 41 }$$

and the deterministic differential equation for $\mid {{R}_{t}}{{\mid }^{2}}$

$$\begin{eqnarray}&&{\rm d}\mid {{R}_{t}}{{\mid }^{2}}=u(t){\rm d}t.\end{eqnarray} \tag{ 42 }$$

Solving this, we get $R(t)={{{\rm e}}^{{\rm i}\theta (t)}}$, with

$$\begin{eqnarray}&&\theta (t)=\int _{-\infty }^{t}\sqrt{\frac{u(s)}{U(s)}}\ J(s){\rm d}s.\end{eqnarray} \tag{ 43 }$$

The ostensible statistics for $\theta =\theta (\infty )$ are those of

$$\begin{eqnarray}&&\theta =\int _{-\infty }^{\infty }\sqrt{\frac{u(s)}{U(s)}}\ \xi (s){\rm d}s,\end{eqnarray} \tag{ 44 }$$

which has a divergent variance because $U(-\infty )=0$. This means that, as in the heterodyne case, ${{\wp }_{{\rm ost}}}(R)$ is rotationally invariant. But in this case, since $\mid R\mid =1$, we have immediately

$$\begin{eqnarray}&&F={{\langle \mid R\mid \rangle }_{{\rm ost}}}=1,\end{eqnarray} \tag{ 45 }$$

and hence this adaptive dyne protocol realizes the canonical phase measurement.

The adaptive algorithm (40) gives the optimal expression (47) to use in the feedback loop for our qubit preparation protocol. Remembering that equation (43) is equal to ${\rm arg} ({{R}_{t}})$, equation (40) can be re-expressed as integral feedback

$$\begin{eqnarray}&&\Phi (t)=\frac{\pi }{2}+\int _{-\infty }^{t}{{\lambda }_{{\rm opt}}}(s)J(s){\rm d}s,\end{eqnarray} \tag{ 46 }$$

with the time-dependent gain

$$\begin{eqnarray}&&{{\lambda }_{{\rm opt}}}(t)=\sqrt{\frac{u(t)}{U(t)}}.\end{eqnarray} \tag{ 47 }$$

Experimentally, not only it is difficult to implement a time-dependent gain in the feedback loop, but it is impossible to implement a completely divergent gain (as required for most mode-shapes when the pulse first turns on). Also, equation (40) assumes zero time delay in the feedback loop, which is unrealistic. These considerations motivates the focus of the remainder of this paper, which is to consider a feedback protocol of the form

$$\begin{eqnarray}&&\Phi (t)=\frac{\pi }{2}+\int _{-\infty }^{t-\tau }\lambda (s)J(s){\rm d}s,\end{eqnarray} \tag{ 48 }$$

that is, a LO phase which depends on the record measurement only up to $t-\tau$ with a gain function $\lambda$ subject to some physical restrictions.

In the absence of delay $\tau$, applying the feedback (40), we can attain $F=1$. In presence of the delay the optimal gain $\lambda (t)$ to apply for the feedback is not known. Indeed, it is likely that the optimal feedback in this case cannot be written in the form (48), and instead some more complex algorithm would be optimal. However, the expression (48) still allows for a great deal of flexibility, even when choosing $\lambda (s)$ from a restricted class of functions, as we will find in section 5.

Note that as we will see later, the unit of measure of the delay should be related with the characteristic duration of the mode-shape $u(t)$, since for a fair comparison of the performances for different mode shapes we have to fix the delay to a fraction of the characteristic time length.

With such a general expression as equation (48) there will no longer be any simple expressions for the ostensible statistics of ${{R}_{t}}$. Instead, one has to evaluate the moments of the stochastic integral

$$\begin{eqnarray}&&R=\int _{-\infty }^{\infty }{{{\rm e}}^{{\rm i}\Phi (s)}}\sqrt{u(s)}\xi (s){\rm d}s,\end{eqnarray} \tag{ 49 }$$

with $\Phi (t)$ given by

$$\begin{eqnarray}&&\Phi (t)=\frac{\pi }{2}+\int _{-\infty }^{t-\tau }\lambda (s)\xi (s){\rm d}s.\end{eqnarray} \tag{ 50 }$$

We will see in the next section how to develop an approximation for $F$ from the ostensible moments of $R$.

We first consider the case of zero delay in the feedback loop, i.e. $\tau =0$. The final expression that we obtain reads

$$\begin{eqnarray}\begin{array}{rcl} \tilde{F} & = & 1-\int _{-\infty }^{+\infty }{\rm d}t\int _{-\infty }^{{{t}_{-}}}{\rm d}s\ {{u}_{t}}{{u}_{s}}(1+{{{\rm e}}^{-2\int _{s}^{t}\lambda _{\tau }^{2}\;{\rm d}\tau }})/4 \\ {} & {} & +\int _{-\infty }^{+\infty }{\rm d}t\int _{-\infty }^{{{t}_{-}}}{\rm d}s\int _{-\infty }^{{{s}_{-}}}{\rm d}v\ {{u}_{t}}\sqrt{{{u}_{s}}{{u}_{v}}}{{\lambda }_{s}}{{\lambda }_{v}}{{{\rm e}}^{-2\int _{s}^{t}\lambda _{\tau }^{2}\;{\rm d}\tau }}{{{\rm e}}^{-\int _{v}^{s}\lambda _{\tau }^{2}\;{\rm d}\tau /2}}. \\ \end{array}\end{eqnarray} \tag{ 58 }$$

If we substitute the optimal adaptive gain ${{\lambda }_{{\rm opt}}}(t)$ of (47), the two integral terms in expression (58) cancel identically, giving $\tilde{F}=1$ as it should be. Given how complicated (58) is, this is a powerful check on its correctness.

We then consider the case with delay, $\tau \gt 0$. The expression in this case becomes even more complicated, that is

$$\begin{eqnarray}\begin{array}{rcl} \tilde{F} & = & 1-\int _{-\infty }^{+\infty }{\rm d}t\int _{0}^{+\infty }{\rm d}\Delta \ {{u}_{t}}{{u}_{t-\Delta }}(1+{{{\rm e}}^{-2\int _{t-\Delta -\tau }^{t-\tau }\lambda _{s}^{2}{\rm d}s}})/4 \\ {} & {} & +2\int _{-\infty }^{+\infty }{\rm d}t\int _{\tau }^{+\infty }{\rm d}\Delta \int _{0}^{\tau }{\rm d}\delta \ {{\lambda }_{t-\Delta }}{{\lambda }_{t-\Delta -\delta }}{{u}_{t}}\sqrt{{{u}_{t-\Delta }}{{u}_{t-\Delta -\delta }}}{{{\rm e}}^{-2\int _{t-\Delta -\tau }^{t-\tau }\lambda _{s}^{2}{\rm d}s}}{{{\rm e}}^{-\frac{1}{2}\int _{t-\Delta -\delta -\tau }^{t-\Delta -\tau }\lambda _{s}^{2}{\rm d}s}} \\ {} & {} & +\int _{-\infty }^{+\infty }{\rm d}t\int _{\tau }^{+\infty }{\rm d}\Delta \int _{\tau }^{+\infty }{\rm d}\delta \ {{\lambda }_{t-\Delta }}{{\lambda }_{t-\Delta -\delta }}{{u}_{t}}\sqrt{{{u}_{t-\Delta }}{{u}_{t-\Delta -\delta }}}{{{\rm e}}^{-2\int _{t-\Delta -\tau }^{t-\tau }\lambda _{s}^{2}{\rm d}s}}{{{\rm e}}^{-\frac{1}{2}\int _{t-\Delta -\delta -\tau }^{t-\Delta -\tau }\lambda _{s}^{2}{\rm d}s}}. \\ \end{array}\end{eqnarray} \tag{ 59 }$$

As a check, the limit for $\tau \to 0$ returns the expression (58).

In order to test the performances of the protocol against the delay, we consider four different mode-shapes:

• rectangular shape
  $$\begin{eqnarray}u(t)=\left\{ \begin{array}{ccccccccccccccc} 1, & {\rm if}\;0\lt t\lt 1, \\ 0, & {\rm otherwise}, \\ \end{array} \right.\end{eqnarray} \tag{ 60 }$$
• bilateral exponential shape
  $$\begin{eqnarray}&&u(t)=\frac{\kappa }{2}{{{\rm e}}^{-\kappa \mid t\mid }},\end{eqnarray} \tag{ 61 }$$
• falling unilateral exponential shape
  $$\begin{eqnarray}u(t)=\left\{ \begin{array}{ccccccccccccccc} k{{{\rm e}}^{-kt}}, & {\rm if}\;t\geqslant 0, \\ 0, & {\rm otherwise}, \\ \end{array} \right.\end{eqnarray} \tag{ 62 }$$
• rising unilateral exponential shape
  $$\begin{eqnarray}u(t)=\left\{ \begin{array}{ccccccccccccccc} k{{{\rm e}}^{kt}}, & {\rm if}\;t\leqslant 0, \\ 0, & {\rm otherwise}. \\ \end{array} \right.\end{eqnarray} \tag{ 63 }$$

These mode-shapes can be realized in experimental setups with present-day technology. The rectangular mode-shapes can easily be obtained to a very good approximation with weak coherent light, which could be used to test phase measurements at a single-photon level. The bilateral mode-shape appears with a single signal photon coming from spontaneous parametric down conversion (SPDC), conditioned on a click in one of the two modes. The falling unilateral exponential shape is the mode of a photon from an initially excited atom. In the case of the rising unilateral exponential, recent papers have investigated how to realize such temporal envelope by modulating an attenuated coherent laser beam [18, 19] or by conditionally modulating one of the mode of a biphoton configuration on the herald of the other mode [20–22].

Note that the characteristic timescale of the single photon would need to be compatible with the speed of the electronics required to implement the adaptive measurement. Equally, as we shall show, the time delay in the electronics must be smaller than the characteristic timescale of the single photon. This could be a challenging task for a photon emitted by an atom or quantum dot, with lifetimes ranging from nanoseconds to hundreds of nanoseconds (see e.g. [19]). It would be easier for single photon states that are heralded from biphotons produced within an optical cavity, which can have a characteristic timescale hundreds of nanoseconds or greater [23]. Such a timescale is readily within the reach of modern electronics—both in terms of the speed and time delay of the electronics. Additionally, there is a natural match of timescales for single photons created through electronic modulation of coherent states.

When we introduce a delay $\tau$, we want to compare what effect it has on schemes using these four different mode-shapes. To make a fair comparison we need the characteristic times of these pulses to be the same. Since all but one of the above mode-shapes do not have compact support, we cannot simply use the pulse duration. Instead, we adopt the following measure of characteristic duration:

$$\begin{eqnarray}&&w={{[\int _{-\infty }^{+\infty }{{u}^{2}}(t){\rm d}t]}^{-1}}.\end{eqnarray} \tag{ 64 }$$

For the four mode-shapes above, this evaluates respectively to 1, $\frac{4}{\kappa }$, $\frac{2}{k}$, and $\frac{2}{k}$. Thus we normalize these all to have $w=1$ by setting $\kappa =4$ and $k=2$.

We first consider the case $\lambda (s)=\lambda$. We substitute the various mode-shapes into equation (59), in order to obtain analytical expressions for $\tilde{F}$ as a function of the delay $\tau$ and the gain $\lambda$. These expressions (which are lengthy and not very informative) are reported in appendix B. We then numerically optimize over $\lambda$. The results are shown in figure 3, along with the performance of heterodyne detection (the best non-adaptive scheme).

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For all the mode-shapes, adaptive measurement out-performs heterodyne detection at zero delay, but the performances decrease monotonically with delay, as expected. The rising exponential mode-shape (63) outperforms all other modes. In fact, for $\tau =0$, this mode-shape yields a perfect phase measurement for constant gain. This can be verified analytically, as the optimal solution of (47) is $\lambda =4\sqrt{2}/\kappa$ for this mode-shape. The bilateral exponential is second-best, which is heartening for experiments based upon SPDC. The order in terms of $\tilde{F}$ at $\tau =0$ is preserved for $\tau \gt 0$, and the rising exponential mode-shape is particularly robust to delay, with constant gain feedback still showing an improvement over heterodyne detection for $\tau$ as large as $1/3$ of the characteristic pulse time.

In figure 4, the values of constant gain employed in the feedback are plotted with respect to the delay, for the different mode-shapes. A common trend can be identified: as the delay increases the optimal $\lambda$ decreases. Also, the size of the optimal gain is, for the most part, inversely related to the effectiveness $\tilde{F}$ of the phase measurement.

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The results above were obtained under the assumption of constant gain. This is the simplest model to consider, having only a single parameter to optimize over. It is natural to ask how much improvement is possible with a slightly more complicated model, as this will give a guide to what might be expected with even more careful tailoring of the feedback algorithm. This motivates considering a piecewise constant gain $\lambda (t)$, with three parameters:

$$\begin{eqnarray}\lambda (t)=\left\{ \begin{array}{ccccccccccccccc} {{\lambda }_{1}} & {\rm for}\;t\leqslant {{t}_{1}}, \\ {{\lambda }_{2}} & {\rm for}\;t\gt {{t}_{1}}. \\ \end{array} \right.\end{eqnarray} \tag{ 65 }$$

Again, we find the analytical expression for $\tilde{F}$ for the four different mode-shapes by substituting the feedback (65) into equation (58). The analytical expression for $\tilde{F}$ are not reported here due to their length and complexity. Next, we optimize over the values ${{\lambda }_{1}},\ {{\lambda }_{2}}$ and ${{t}_{1}}$.

In figure 3(b) the performances of the phase measurement with the optimized feedback are plotted in continuous lines, while in dashed lines the previous performances with constant gains are reported. We can see quite substantial improvements for the bilateral, rectangular and falling unilateral exponential mode-shapes, although the improvement lessens for longer delays. The opposite is true for the rising exponential mode-shape, as expected since constant gain is optimal at $\tau =0$ for this case.

In figure 5 the family of the optimized piecewise constant feedback $\lambda (t)$ are plotted, as a function of the delay $\tau$. In each figure, black stripes has been plotted parallel to the time (t) axis, to indicate the shape of the function $\lambda (t)$ corresponding to a specific parameter $\tau$. In all cases except the rising unilateral exponential, $\lambda (t)$ decreases with time, that is, ${{\lambda }_{1}}\geqslant {{\lambda }_{2}}$. In the case of the rectangular and bilateral exponential shapes (figures 5(a) and 5(b) respectively), ${{\lambda }_{2}}$ and ${{t}_{1}}$ shows almost no dependence, while ${{\lambda }_{1}}$ decreases with the delay. A similar behavior is exhibited in figure 5(c) for the falling unilateral exponential, with a little dependance of ${{t}_{1}}$ on the delay. By contrast, in the case of the rising unilateral exponential shape, figure 5(d), $\lambda (t)$ increases with time, that is, ${{\lambda }_{1}}\leqslant {{\lambda }_{2}}$. As the delay increases ${{\lambda }_{1}}$ and ${{t}_{1}}$ decrease, while ${{\lambda }_{2}}$ initially decreases but then increases. Figure 6 shows the same data as figure 5(d), plotted as ${{\lambda }_{1}}$, ${{\lambda }_{2}}$ and ${{t}_{1}}$ versus $\tau$ for clarity. In the lower right plot, the optimal parameter for $\tau =0.1$ are collected to picture the optimal feedback gain $\lambda (t)$.

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Consider for simplicity a rectangular pulse shape, which is zero everywhere except

$$\begin{eqnarray}&&u(t)=1,\quad 0\leqslant t\leqslant 1,\end{eqnarray} \tag{ A.1 }$$

so that we can restrict the integrals to the domain $[0,1]$. By the definition of (49), we have

$$\begin{eqnarray}&&\mid R{{\mid }^{2}}=\int _{0}^{1}{\rm d}t\ \int _{0}^{1}{\rm d}s\ {{{\rm e}}^{{\rm i}{{\Phi }_{t}}-{\rm i}{{\Phi }_{s}}}}\ {{\xi }_{t}}{{\xi }_{s}}=\int _{0}^{1}{\rm d}t\ \int _{0}^{1}{\rm d}s\ {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}})\ {{\xi }_{t}}{{\xi }_{s}}=1+2x,\end{eqnarray} \tag{ A.2 }$$

where

$$\begin{eqnarray}&&x\triangleq \int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\ {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}})\ {{\xi }_{t}}{{\xi }_{s}}.\end{eqnarray} \tag{ A.3 }$$

Note that $\langle x\rangle$ must vanish since $\langle \mid R{{\mid }^{2}}\rangle =1$. In terms of $x$, our figure of merit (54) is thus

$$\begin{eqnarray}&&\tilde{F}=1-\langle {{x}^{2}}\rangle /2.\end{eqnarray} \tag{ A.4 }$$

The integrand function ${\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}})$, as well as others that follow, depends on the stochastic process

$$\begin{eqnarray}&&{{\Phi }_{t}}-{{\Phi }_{s}}=\int _{s}^{{{t}_{-}}}\xi (v)\ {\rm d}v=\int _{s}^{{{t}_{-}}}{\rm d}W(v),\end{eqnarray} \tag{ A.5 }$$

that is non-anticipating in t, but correlated with ${\rm d}W(v),\ v\in [s,t)$. The increment ${{\xi }_{t}}$ is hence uncorrelated from the other integrand function, allowing one to factor the expectation

$$\begin{eqnarray}&&\langle x\rangle =\langle \int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\ {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}})\ {{\xi }_{t}}{{\xi }_{s}}\rangle =\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\ \langle {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}})\ {{\xi }_{s}}\rangle \langle {{\xi }_{t}}\rangle =0\end{eqnarray} \tag{ A.6 }$$

by the stochastic Itō calculus. As noted above, this result is necessary, so we have the first check on the consistency of our approach.

The term $\langle {{x}^{2}}\rangle$ is

$$\begin{eqnarray}&&\langle {{x}^{2}}\rangle =\int _{0}^{1}{\rm d}t\int _{0}^{t}{\rm d}s\int _{0}^{1}{\rm d}v\int _{0}^{v}{\rm d}z\ \langle {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}}){\rm cos} ({{\Phi }_{v}}-{{\Phi }_{z}})\ {{\xi }_{t}}{{\xi }_{s}}{{\xi }_{v}}{{\xi }_{z}}\rangle .\end{eqnarray} \tag{ A.7 }$$

On the range of integration where $t\ne v$, ${{\xi }_{t}}$ and ${{\xi }_{v}}$ are uncorrelated, the expectation factorizes, and hence these regions do not contribute to the integral. Equation (A.7) thus reduces to

$$\begin{eqnarray*}&&\langle {{x}^{2}}\rangle =\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\int _{0}^{{{t}_{-}}}{\rm d}z\ \langle {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}}){\rm cos} ({{\Phi }_{t}}-{{\Phi }_{z}})\ {{\xi }_{s}}{{\xi }_{z}}\rangle \langle \xi _{t}^{2}\rangle .\end{eqnarray*}$$

By again splitting the range of integration $[0,t)\times [0,t)$ in the variables ${\rm d}s,{\rm d}z$ into three regions, i.e. s = z, $s\lt z$ and $s\gt z$, and then using the symmetry of the integrand on the regions $s\lt z$ and $s\gt z$, we obtain two terms:

$$\begin{eqnarray}&&\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\int _{0}^{{{t}_{-}}}{\rm d}z\ \langle {{{\rm cos} }^{2}}({{\Phi }_{t}}-{{\Phi }_{s}})\delta (s-z){{\xi }_{s}}{{\xi }_{z}}\rangle \end{eqnarray} \tag{ A.8 }$$

and

$$\begin{eqnarray}&&2\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\int _{0}^{{{s}_{-}}}{\rm d}z\langle {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}}){\rm cos} ({{\Phi }_{t}}-{{\Phi }_{z}})\ {{\xi }_{s}}{{\xi }_{z}}\rangle .\end{eqnarray} \tag{ A.9 }$$

The functions ${{{\rm cos} }^{2}}({{\Phi }_{t}}-{{\Phi }_{s}})$, ${\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}})$ and ${\rm cos} ({{\Phi }_{t}}-{{\Phi }_{z}})$ are not uncorrelated with ${{\xi }_{s}}$ and ${{\xi }_{z}}$. However, we can apply Novikov's formula [24], which in our case reads

$$\begin{eqnarray}&&\langle {{\xi }_{t}}\mathcal{F}\rangle =\langle \frac{\delta \mathcal{F}}{\delta {{\xi }_{t}}}\rangle ,\end{eqnarray} \tag{ A.10 }$$

where $\mathcal{F}$ is an arbitrary differentiable functional of the stochastic process ${\boldsymbol{ \xi }}$. Applying the formula, we get for the term (A.8)

$$\begin{eqnarray}&&\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\ \langle {{{\rm cos} }^{2}}({{\Phi }_{t}}-{{\Phi }_{s}})\rangle =\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\frac{1+\langle {\rm cos} (2{{\Phi }_{t}}-2{{\Phi }_{s}})\rangle }{2}\end{eqnarray} \tag{ A.11 }$$

while for term (A.9), splitting the interval $[u,t)$ in $[u,s)\cup [s,t)$ in order to consider uncorrelated stochastic process ${{\Phi }_{t}}-{{\Phi }_{s}}$ and ${{\Phi }_{s}}-{{\Phi }_{z}}$, we get

$$\begin{eqnarray}\begin{array}{rcl} \langle {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{s}}){\rm cos} ({{\Phi }_{t}}-{{\Phi }_{z}}){{\xi }_{s}}{{\xi }_{z}}\rangle & = & \langle \lambda (z)\lambda (s){\rm sin} (2{{\Phi }_{t}}-2{{\Phi }_{s}}){\rm sin} ({{\Phi }_{s}}-{{\Phi }_{z}}) \\ {} & {} & -\lambda (z)\lambda (s){\rm cos} (2{{\Phi }_{t}}-2{{\Phi }_{s}}){\rm cos} ({{\Phi }_{s}}-{{\Phi }_{z}})\rangle \\ \end{array}\end{eqnarray} \tag{ A.12 }$$

$$\begin{eqnarray}\begin{array}{cll} {} & = & \lambda (z)\lambda (s)\langle {\rm sin} (2{{\Phi }_{t}}-2{{\Phi }_{s}})\rangle \langle {\rm sin} ({{\Phi }_{s}}-{{\Phi }_{z}})\rangle \\ {} & {} & -\lambda (z)\lambda (s)\langle {\rm cos} (2{{\Phi }_{t}}-2{{\Phi }_{s}})\rangle \langle {\rm cos} ({{\Phi }_{s}}-{{\Phi }_{z}})\rangle . \\ \end{array}\end{eqnarray} \tag{ A.13 }$$

Finally, we can calculate the integrand function using their Taylor series and the moments for a Gaussian stochastic process of mean 0 and standard deviation σ,

$$\begin{eqnarray}\langle {{G}^{p}}\rangle =\left\{ \begin{array}{ccccccccccccccc} 0 & {\rm if}\;p\;{\rm is}\;{\rm odd}, \\ {{\sigma }^{p}}(p-1)!! & {\rm if}\;p\;{\rm is}\;{\rm even}, \\ \end{array} \right.\end{eqnarray} \tag{ A.14 }$$

to obtain

$$\begin{eqnarray}\begin{array}{rcl} \langle {\rm sin} ({{\Phi }_{i}}-{{\Phi }_{j}})\rangle & = & 0, \\ \langle {\rm cos} ({{\Phi }_{i}}-{{\Phi }_{j}})\rangle & = & {{{\rm e}}^{-{\rm Var}[{{\Phi }_{i}}-{{\Phi }_{j}}]/2}}, \\ \langle {\rm cos} (2{{\Phi }_{i}}-2{{\Phi }_{j}})\rangle & = & {{{\rm e}}^{-2{\rm Var}[{{\Phi }_{i}}-{{\Phi }_{j}}]}}, \\ \end{array}\end{eqnarray} \tag{ A.15 }$$

where

$$\begin{eqnarray}&&{\rm Var}[{{\Phi }_{i}}-{{\Phi }_{j}}]=\int _{j}^{i}{{\lambda }^{2}}(\tau )\ {\rm d}\tau .\end{eqnarray} \tag{ A.16 }$$

These expressions are used to evaluate (A.13), (A.8) and (A.9). Altogether, the final expression for (A.7) is

$$\begin{eqnarray}\begin{array}{rcl} \langle {{x}^{2}}\rangle & = & \int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\ (1+{{{\rm e}}^{-2\int _{s}^{t}\lambda _{\tau }^{2}\;{\rm d}\tau }})/2 \\ {} & {} & -2\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\int _{0}^{{{s}_{-}}}{\rm d}z\ {{\lambda }_{s}}{{\lambda }_{z}}{{{\rm e}}^{-2\int _{s}^{t}\lambda _{\tau }^{2}\;{\rm d}\tau }}{{{\rm e}}^{-\int _{v}^{s}\lambda _{\tau }^{2}\;{\rm d}\tau /2}}, \\ \end{array}\end{eqnarray} \tag{ A.17 }$$

and by the definition (A.4) we obtain

$$\begin{eqnarray}\begin{array}{rcl} \tilde{F} & = & 1-\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\ (1+{{{\rm e}}^{-2\int _{s}^{t}\lambda _{\tau }^{2}\;{\rm d}\tau }})/4 \\ {} & {} & +\int _{0}^{1}{\rm d}t\int _{0}^{{{t}_{-}}}{\rm d}s\int _{0}^{{{s}_{-}}}{\rm d}z\ {{\lambda }_{s}}{{\lambda }_{z}}{{{\rm e}}^{-2\int _{s}^{t}\lambda _{\tau }^{2}\;{\rm d}\tau }}{{{\rm e}}^{-\int _{v}^{s}\lambda _{\tau }^{2}\;{\rm d}\tau /2}}. \\ \end{array}\end{eqnarray} \tag{ A.18 }$$

The generalization of equation (A.18) for an arbitrary mode-shape $u(t)$ leads to the expression (58).

In this section we generalize (58) for a delay $\tau$ in the feedback scheme, by defining

$$\begin{eqnarray}&&\Phi (t)=\frac{\pi }{2}+\int _{-\infty }^{t-\tau }\lambda (s)\ {\rm d}W(s).\end{eqnarray} \tag{ A.19 }$$

In this case, $x$ can be written as

$$\begin{eqnarray}&&x=\int _{-\infty }^{+\infty }{\rm d}t\int _{0}^{+\infty }{\rm d}\Delta {\rm cos} ({{\Phi }_{t}}-{{\Phi }_{t-\Delta }})\sqrt{{{u}_{t}}{{u}_{t-\Delta }}}{{\xi }_{t}}{{\xi }_{t-\Delta }}.\end{eqnarray} \tag{ A.20 }$$

As before, the term $\langle x\rangle$ vanishes, while after some algebra the term $\langle {{x}^{2}}\rangle$ becomes

$$\begin{eqnarray*}\begin{array}{rcl} \langle {{x}^{2}}\rangle & = & \int _{-\infty }^{+\infty }{\rm d}t\int _{0}^{+\infty }{\rm d}\Delta \ \langle {{{\rm cos} }^{2}}({{\Phi }_{t}}-{{\Phi }_{t-\Delta }}){{u}_{t}}{{u}_{t-\Delta }}\rangle \\ {} & {} & -4\int _{-\infty }^{+\infty }{\rm d}t\int _{\tau }^{+\infty }{\rm d}\Delta \int _{0}^{\tau }{\rm d}\delta \ {{\lambda }_{t-\Delta }}{{\lambda }_{t-\Delta -\delta }}{{u}_{t}}\sqrt{{{u}_{t-\Delta }}{{u}_{t-\Delta -\delta }}} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{lll} {} & {} & \times \langle {\rm cos} (2{{\Phi }_{t}}-2{{\Phi }_{t-\Delta }}){\rm cos} ({{\Phi }_{t-\Delta }}-{{\Phi }_{t-\Delta -\delta }})\rangle \\ {} & {} & -2\int _{-\infty }^{+\infty }{\rm d}t\int _{\tau }^{+\infty }{\rm d}\Delta \int _{\tau }^{+\infty }{\rm d}\delta \ {{\lambda }_{t-\Delta }}{{\lambda }_{t-\Delta -\delta }}{{u}_{t}}\sqrt{{{u}_{t-\Delta }}{{u}_{t-\Delta -\delta }}} \\ {} & {} & \times \langle {\rm cos} (2{{\Phi }_{t}}-2{{\Phi }_{t-\Delta }}){\rm cos} ({{\Phi }_{t-\Delta }}-{{\Phi }_{t-\Delta -\delta }})\rangle . \\ \end{array}\end{eqnarray} \tag{ A.21 }$$

The three terms in the integral comes from splitting the range of integration opportunely, in order to identify regions where only a few of the stochastic process ${{\Phi }_{t}}-{{\Phi }_{t-\Delta }},\ {{\Phi }_{t}}-{{\Phi }_{t-\Delta -\delta }},\ {{\xi }_{t-\Delta }},\ {{\xi }_{t-\delta }}$ are correlated. Then, we use the rule (A.10) to get rid of the ${{\xi }_{t-\Delta }},\ {{\xi }_{t-\delta }}$. Finally, we resolve the expectation employing (A.15), to obtain (59).