From 8e717b0d31b4903fda48f2373700476a9cfa9fe3 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:06:44 -0400 Subject: [PATCH 01/12] P-matrix-element --- cubic-phase.md | 29 +++++++++++++++++++++++++++-- 1 file changed, 27 insertions(+), 2 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 054df8a..73f6f05 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -1,5 +1,4 @@ # Cubic phase gate - The cubic phase gate is defined as $``P(t) = \exp\left( i t \hat{q}^2 / 2 \hbar \right)``$ @@ -24,4 +23,30 @@ The above equations gives rise to the following decomposition in Hilbert space $``P(t) = e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right) ``$ -where $``\exp(i \varphi) = \frac{\sqrt[4]{1+\tfrac{t^2}{4}} }{\sqrt{1 - i \tfrac{t}{2}}}``$. \ No newline at end of file +where $``\exp(i \varphi) = \frac{\sqrt[4]{1+\tfrac{t^2}{4}} }{\sqrt{1 - i \tfrac{t}{2}}}``$. + +We now compute the coherent-state matrix elements of $P(t)$ +$$\braket{\alpha^{*}|P(t)|\beta} = \braket{\alpha^{*}|e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right)|\beta},$$ +where we use the following identity to express the squeezing operator as +$$\exp{\left(\frac{r}{2}(a^{\dagger 2} - a^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\exp{\left(-[a^{\dagger}a + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)},$$ +and a second identity to normal-order this expression +$$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$$ +with $:(a^{\dagger}a)^{p}: = a^{\dagger p}a^{p}$. + +Using these two identities, the decomposition of $P(t)$ takes the form +$$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([\operatorname{sech}{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ + +Evaluating the matrix element $\braket{\alpha^{*}|\dots|\beta} = e^{i\varphi}\braket{\bar{\alpha}^{*}|\dots|\bar{\beta}}$ gives +$$e^{i\varphi}\braket{\bar{\alpha}^{*}|\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([\operatorname{sech}{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}|{\beta}}$$ +where we have introduced the rotated labels $\bra{\bar{\alpha}^{*}} \equiv \bra{\alpha^{*}e^{-i\theta}} = \bra{\alpha^{*}}e^{i\theta a^{\dagger}a}$ and $\ket{\bar{\beta}} \equiv \ket{\beta e^{i(\pi/2 - \theta)}} = e^{i(\pi/2 - \theta) a^{\dagger}a}\ket{\beta}$ to simplify the expression. The resulting expression takes the form +$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ +where we used the overlap of two coherent states $\braket{\bar{\alpha}^{*}|\bar{\beta}} = \exp{(-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by +$$\begin{aligned} +f &= -\tanh{r}e^{2i\theta}, \\ +g &= -i\operatorname{sech}{r} , +\end{aligned}$$ +or, in terms of the parameter $t$ +$$\begin{aligned} +f &= \frac{t(t - 2i)}{4 + t^{2}}, \\ +g &= - \frac{2i}{\sqrt{4 + t^{2}}}. +\end{aligned}$$ \ No newline at end of file From fb0c6424303fd86f2d304d541925ea34126971dc Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:14:55 -0400 Subject: [PATCH 02/12] P-matrix-element-2 --- cubic-phase.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 73f6f05..5eb5c71 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -34,16 +34,16 @@ $$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$$ with $:(a^{\dagger}a)^{p}: = a^{\dagger p}a^{p}$. Using these two identities, the decomposition of $P(t)$ takes the form -$$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([\operatorname{sech}{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ +$$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ Evaluating the matrix element $\braket{\alpha^{*}|\dots|\beta} = e^{i\varphi}\braket{\bar{\alpha}^{*}|\dots|\bar{\beta}}$ gives -$$e^{i\varphi}\braket{\bar{\alpha}^{*}|\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([\operatorname{sech}{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}|{\beta}}$$ +$$e^{i\varphi}\braket{\bar{\alpha}^{*}|\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}|{\beta}}$$ where we have introduced the rotated labels $\bra{\bar{\alpha}^{*}} \equiv \bra{\alpha^{*}e^{-i\theta}} = \bra{\alpha^{*}}e^{i\theta a^{\dagger}a}$ and $\ket{\bar{\beta}} \equiv \ket{\beta e^{i(\pi/2 - \theta)}} = e^{i(\pi/2 - \theta) a^{\dagger}a}\ket{\beta}$ to simplify the expression. The resulting expression takes the form $$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ -where we used the overlap of two coherent states $\braket{\bar{\alpha}^{*}|\bar{\beta}} = \exp{(-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by +where we used the overlap of two coherent states $\braket{\bar{\alpha}^{*}|\bar{\beta}} = \exp{(-\frac{1}{2}(|\alpha|^{2} + |\beta|^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by $$\begin{aligned} f &= -\tanh{r}e^{2i\theta}, \\ -g &= -i\operatorname{sech}{r} , +g &= -\frac{i}{\cosh{r}} , \end{aligned}$$ or, in terms of the parameter $t$ $$\begin{aligned} From 854c95dbe6048b9ef45146d6910439290d2adf17 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:18:12 -0400 Subject: [PATCH 03/12] P-element-matrix-3 --- cubic-phase.md | 34 ++++++++++++++++++++++++---------- 1 file changed, 24 insertions(+), 10 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 5eb5c71..4e3bb65 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -26,27 +26,41 @@ $``P(t) = e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp( where $``\exp(i \varphi) = \frac{\sqrt[4]{1+\tfrac{t^2}{4}} }{\sqrt{1 - i \tfrac{t}{2}}}``$. We now compute the coherent-state matrix elements of $P(t)$ -$$\braket{\alpha^{*}|P(t)|\beta} = \braket{\alpha^{*}|e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right)|\beta},$$ + +$\braket{\alpha^{*}|P(t)|\beta} = \braket{\alpha^{*}|e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right)|\beta},$ + where we use the following identity to express the squeezing operator as -$$\exp{\left(\frac{r}{2}(a^{\dagger 2} - a^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\exp{\left(-[a^{\dagger}a + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)},$$ + +$\exp{\left(\frac{r}{2}(a^{\dagger 2} - a^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\exp{\left(-[a^{\dagger}a + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)},$ + and a second identity to normal-order this expression -$$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$$ + +$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$ + with $:(a^{\dagger}a)^{p}: = a^{\dagger p}a^{p}$. Using these two identities, the decomposition of $P(t)$ takes the form -$$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ + +$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$ Evaluating the matrix element $\braket{\alpha^{*}|\dots|\beta} = e^{i\varphi}\braket{\bar{\alpha}^{*}|\dots|\bar{\beta}}$ gives -$$e^{i\varphi}\braket{\bar{\alpha}^{*}|\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}|{\beta}}$$ + +$e^{i\varphi}\braket{\bar{\alpha}^{*}|\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}|{\beta}}$ + where we have introduced the rotated labels $\bra{\bar{\alpha}^{*}} \equiv \bra{\alpha^{*}e^{-i\theta}} = \bra{\alpha^{*}}e^{i\theta a^{\dagger}a}$ and $\ket{\bar{\beta}} \equiv \ket{\beta e^{i(\pi/2 - \theta)}} = e^{i(\pi/2 - \theta) a^{\dagger}a}\ket{\beta}$ to simplify the expression. The resulting expression takes the form -$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ + +$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$ + where we used the overlap of two coherent states $\braket{\bar{\alpha}^{*}|\bar{\beta}} = \exp{(-\frac{1}{2}(|\alpha|^{2} + |\beta|^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by -$$\begin{aligned} + +$\begin{aligned} f &= -\tanh{r}e^{2i\theta}, \\ g &= -\frac{i}{\cosh{r}} , -\end{aligned}$$ +\end{aligned}$ + or, in terms of the parameter $t$ -$$\begin{aligned} + +$\begin{aligned} f &= \frac{t(t - 2i)}{4 + t^{2}}, \\ g &= - \frac{2i}{\sqrt{4 + t^{2}}}. -\end{aligned}$$ \ No newline at end of file +\end{aligned}$ \ No newline at end of file From 1c39bef92d9797798a3a76e3635093c627e63f21 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:23:07 -0400 Subject: [PATCH 04/12] P-matrix-element-4 --- cubic-phase.md | 30 +++++++++++++++--------------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 4e3bb65..f03a125 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -27,40 +27,40 @@ where $``\exp(i \varphi) = \frac{\sqrt[4]{1+\tfrac{t^2}{4}} }{\sqrt{1 - i \tfrac We now compute the coherent-state matrix elements of $P(t)$ -$\braket{\alpha^{*}|P(t)|\beta} = \braket{\alpha^{*}|e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right)|\beta},$ +$$\langle \alpha^{*} \vert P(t) \vert \beta \rangle = \langle \alpha^{*} \vert e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right) \vert \beta \rangle,$$ where we use the following identity to express the squeezing operator as -$\exp{\left(\frac{r}{2}(a^{\dagger 2} - a^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\exp{\left(-[a^{\dagger}a + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)},$ +$$\exp{\left(\frac{r}{2}(a^{\dagger 2} - a^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\exp{\left(-[a^{\dagger}a + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)},$$ and a second identity to normal-order this expression -$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$ +$$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$$ with $:(a^{\dagger}a)^{p}: = a^{\dagger p}a^{p}$. Using these two identities, the decomposition of $P(t)$ takes the form -$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$ +$$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ -Evaluating the matrix element $\braket{\alpha^{*}|\dots|\beta} = e^{i\varphi}\braket{\bar{\alpha}^{*}|\dots|\bar{\beta}}$ gives +Evaluating the matrix element $\langle \alpha^{*} \vert \dots \vert \beta \rangle = e^{i\varphi} \langle \bar{\alpha}^{*} \vert \dots \vert \bar{\beta} \rangle$ gives -$e^{i\varphi}\braket{\bar{\alpha}^{*}|\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}|{\beta}}$ +$$e^{i\varphi} \langle \bar{\alpha}^{*} \vert \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)} \vert \beta \rangle$$ -where we have introduced the rotated labels $\bra{\bar{\alpha}^{*}} \equiv \bra{\alpha^{*}e^{-i\theta}} = \bra{\alpha^{*}}e^{i\theta a^{\dagger}a}$ and $\ket{\bar{\beta}} \equiv \ket{\beta e^{i(\pi/2 - \theta)}} = e^{i(\pi/2 - \theta) a^{\dagger}a}\ket{\beta}$ to simplify the expression. The resulting expression takes the form +where we have introduced the rotated labels $\langle \bar{\alpha}^{*} \vert \equiv \langle \alpha^{*}e^{-i\theta} \vert = \langle \alpha^{*} \vert e^{i\theta a^{\dagger}a}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) a^{\dagger}a} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form -$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$ +$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ -where we used the overlap of two coherent states $\braket{\bar{\alpha}^{*}|\bar{\beta}} = \exp{(-\frac{1}{2}(|\alpha|^{2} + |\beta|^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by +where we used the overlap of two coherent states $\langle \bar{\alpha}^{*} \vert \bar{\beta} \rangle = \exp{(-\frac{1}{2}(\lvert\alpha\rvert^{2} + \lvert\beta\rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by -$\begin{aligned} -f &= -\tanh{r}e^{2i\theta}, \\ -g &= -\frac{i}{\cosh{r}} , -\end{aligned}$ +$$\begin{aligned} +f &= -\tanh{r}\,e^{2i\theta}, \\ +g &= -\frac{i}{\cosh{r}}, +\end{aligned}$$ or, in terms of the parameter $t$ -$\begin{aligned} +$$\begin{aligned} f &= \frac{t(t - 2i)}{4 + t^{2}}, \\ g &= - \frac{2i}{\sqrt{4 + t^{2}}}. -\end{aligned}$ \ No newline at end of file +\end{aligned}$$ \ No newline at end of file From bbbc2e8264bd3ceec5d60fb8745454103c618448 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:24:26 -0400 Subject: [PATCH 05/12] P-matrix-element-5 --- cubic-phase.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/cubic-phase.md b/cubic-phase.md index f03a125..5716512 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -27,7 +27,7 @@ where $``\exp(i \varphi) = \frac{\sqrt[4]{1+\tfrac{t^2}{4}} }{\sqrt{1 - i \tfrac We now compute the coherent-state matrix elements of $P(t)$ -$$\langle \alpha^{*} \vert P(t) \vert \beta \rangle = \langle \alpha^{*} \vert e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right) \vert \beta \rangle,$$ +$$\langle \alpha^* \vert P(t) \vert \beta \rangle = \langle \alpha^* \vert e^{i \varphi}\exp\left( i \theta \hat{a}^\dagger \hat{a} \right) \exp(\tfrac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)) \exp\left( i (\tfrac{\pi}{2} - \theta) \hat{a}^\dagger \hat{a} \right) \vert \beta \rangle,$$ where we use the following identity to express the squeezing operator as From 07533157345578ba3599a35d4f0c8496922a29b9 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:27:41 -0400 Subject: [PATCH 06/12] P-matrix-element-6 --- cubic-phase.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 5716512..6b862e7 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -43,15 +43,15 @@ Using these two identities, the decomposition of $P(t)$ takes the form $$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ -Evaluating the matrix element $\langle \alpha^{*} \vert \dots \vert \beta \rangle = e^{i\varphi} \langle \bar{\alpha}^{*} \vert \dots \vert \bar{\beta} \rangle$ gives +Evaluating the matrix element $\langle \alpha^* \vert \dots \vert \beta \rangle = e^{i\varphi} \langle \bar{\alpha}^* \vert \dots \vert \bar{\beta} \rangle$ gives -$$e^{i\varphi} \langle \bar{\alpha}^{*} \vert \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)} \vert \beta \rangle$$ +$$e^{i\varphi} \langle \bar{\alpha}^* \vert \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)} \vert \beta \rangle$$ -where we have introduced the rotated labels $\langle \bar{\alpha}^{*} \vert \equiv \langle \alpha^{*}e^{-i\theta} \vert = \langle \alpha^{*} \vert e^{i\theta a^{\dagger}a}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) a^{\dagger}a} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form +where we have introduced the rotated labels $\langle \bar{\alpha}^* \vert \equiv \langle \alpha^* e^{-i\theta} \vert = \langle \alpha^* \vert e^{i\theta a^{\dagger}a}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) a^{\dagger}a} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form -$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^{*} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ +$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^* \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ -where we used the overlap of two coherent states $\langle \bar{\alpha}^{*} \vert \bar{\beta} \rangle = \exp{(-\frac{1}{2}(\lvert\alpha\rvert^{2} + \lvert\beta\rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by +where we used the overlap of two coherent states $\langle \bar{\alpha}^* \vert \bar{\beta} \rangle = \exp{(-\frac{1}{2}(\lvert\alpha\rvert^{2} + \lvert\beta\rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by $$\begin{aligned} f &= -\tanh{r}\,e^{2i\theta}, \\ From 8bd463bbd4d8abc497bd021ec1e1064ee330c109 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:31:38 -0400 Subject: [PATCH 07/12] P-matrix-element-7 --- cubic-phase.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/cubic-phase.md b/cubic-phase.md index 6b862e7..8955c86 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -49,7 +49,9 @@ $$e^{i\varphi} \langle \bar{\alpha}^* \vert \exp{\left(\frac{\tanh{r}}{2}a^{\dag where we have introduced the rotated labels $\langle \bar{\alpha}^* \vert \equiv \langle \alpha^* e^{-i\theta} \vert = \langle \alpha^* \vert e^{i\theta a^{\dagger}a}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) a^{\dagger}a} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form -$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^* \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)}$$ +$$ +\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^* \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)} +$$ where we used the overlap of two coherent states $\langle \bar{\alpha}^* \vert \bar{\beta} \rangle = \exp{(-\frac{1}{2}(\lvert\alpha\rvert^{2} + \lvert\beta\rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by From d2e9d1378eceba65efe8a41b5ae7ce434db58eb7 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Tue, 7 Jul 2026 16:32:23 -0400 Subject: [PATCH 08/12] P-matrix-element-8 --- cubic-phase.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 8955c86..1ba410d 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -49,9 +49,7 @@ $$e^{i\varphi} \langle \bar{\alpha}^* \vert \exp{\left(\frac{\tanh{r}}{2}a^{\dag where we have introduced the rotated labels $\langle \bar{\alpha}^* \vert \equiv \langle \alpha^* e^{-i\theta} \vert = \langle \alpha^* \vert e^{i\theta a^{\dagger}a}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) a^{\dagger}a} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form -$$ -\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\ g & f^* \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \right)} -$$ +$$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\\\ g & f^* \end{bmatrix} \begin{bmatrix} \alpha \\\\ \beta \end{bmatrix} \right)}$$ where we used the overlap of two coherent states $\langle \bar{\alpha}^* \vert \bar{\beta} \rangle = \exp{(-\frac{1}{2}(\lvert\alpha\rvert^{2} + \lvert\beta\rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by From 3763b469f51f1cc39f16dfc5e0d9a4403aef9fa3 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Wed, 8 Jul 2026 12:36:57 -0400 Subject: [PATCH 09/12] P-matrix-element-9 --- cubic-phase.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 1ba410d..c5d97f2 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -31,30 +31,30 @@ $$\langle \alpha^* \vert P(t) \vert \beta \rangle = \langle \alpha^* \vert e^{i where we use the following identity to express the squeezing operator as -$$\exp{\left(\frac{r}{2}(a^{\dagger 2} - a^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\exp{\left(-[a^{\dagger}a + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)},$$ +$$\exp{\left(\frac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^{2})\right)} = \exp{\left(\frac{\tanh{r}}{2}\hat{a}^{\dagger 2}\right)}\exp{\left(-[\hat{a}^{\dagger}\hat{a} + 1/2]\ln{\cosh{r}}\right)}\exp{\left(-\frac{\tanh{r}}{2}\hat{a}^{2}\right)},$$ and a second identity to normal-order this expression -$$\exp{(\theta a^{\dagger}a)} = :\exp{([\exp{(\theta)} - 1]a^{\dagger}a)}:,$$ +$$\exp{(\theta \hat{a}^{\dagger}\hat{a})} = :\exp{([\exp{(\theta)} - 1]\hat{a}^{\dagger}\hat{a})}:,$$ -with $:(a^{\dagger}a)^{p}: = a^{\dagger p}a^{p}$. +with $:(\hat{a}^{\dagger}\hat{a})^{p}: = \hat{a}^{\dagger p}\hat{a}^{p}$. Using these two identities, the decomposition of $P(t)$ takes the form -$$P(t) = e^{i\varphi} \exp(i\theta a^{\dagger}a)\exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)}\exp{(i(\pi/2-\theta)a^{\dagger}a)}.$$ +$$P(t) = e^{i\varphi} \exp(i\theta \hat{a}^{\dagger}\hat{a})\exp{\left(\frac{\tanh{r}}{2}\hat{a}^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]\hat{a}^{\dagger}\hat{a})}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}\hat{a}^{2}\right)}\exp{(i(\pi/2-\theta)\hat{a}^{\dagger}\hat{a})}.$$ Evaluating the matrix element $\langle \alpha^* \vert \dots \vert \beta \rangle = e^{i\varphi} \langle \bar{\alpha}^* \vert \dots \vert \bar{\beta} \rangle$ gives -$$e^{i\varphi} \langle \bar{\alpha}^* \vert \exp{\left(\frac{\tanh{r}}{2}a^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]a^{\dagger}a)}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}a^{2}\right)} \vert \beta \rangle$$ +$$e^{i\varphi} \langle \bar{\alpha}^* \vert \exp{\left(\frac{\tanh{r}}{2}\hat{a}^{\dagger 2}\right)}\times:\frac{\exp{([1/\cosh{r}-1]\hat{a}^{\dagger}\hat{a})}}{\sqrt{\cosh{r}}}:\times\exp{\left(-\frac{\tanh{r}}{2}\hat{a}^{2}\right)} \vert \bar{\beta} \rangle$$ -where we have introduced the rotated labels $\langle \bar{\alpha}^* \vert \equiv \langle \alpha^* e^{-i\theta} \vert = \langle \alpha^* \vert e^{i\theta a^{\dagger}a}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) a^{\dagger}a} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form +where we have introduced the rotated labels $\langle \bar{\alpha}^* \vert \equiv \langle \alpha^* e^{-i\theta} \vert = \langle \alpha^* \vert e^{i\theta \hat{a}^{\dagger}\hat{a}}$ and $\vert \bar{\beta} \rangle \equiv \vert \beta e^{i(\pi/2 - \theta)} \rangle = e^{i(\pi/2 - \theta) \hat{a}^{\dagger}\hat{a}} \vert \beta \rangle$ to simplify the expression. The resulting expression takes the form $$\sqrt{2}e^{i\varphi} \frac{\exp{[-\frac{1}{2}(\lvert \alpha \rvert^{2} + \lvert \beta \rvert^{2})]}}{\sqrt[4]{4+t^{2}}} \exp{\left(-\frac{1}{2} \begin{bmatrix} \alpha & \beta \end{bmatrix} \begin{bmatrix} f & g \\\\ g & f^* \end{bmatrix} \begin{bmatrix} \alpha \\\\ \beta \end{bmatrix} \right)}$$ where we used the overlap of two coherent states $\langle \bar{\alpha}^* \vert \bar{\beta} \rangle = \exp{(-\frac{1}{2}(\lvert\alpha\rvert^{2} + \lvert\beta\rvert^{2} - 2\bar{\alpha}\bar{\beta}))}$, and the functions $f$ and $g$ are given by $$\begin{aligned} -f &= -\tanh{r}\,e^{2i\theta}, \\ +f &= -\tanh{r}e^{2i\theta}, \\ g &= -\frac{i}{\cosh{r}}, \end{aligned}$$ From afea7384c66b4caaef12904aea466cce20bb831b Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Wed, 8 Jul 2026 16:57:33 -0400 Subject: [PATCH 10/12] recurrence relation of P --- cubic-phase.md | 16 +++++++++++++++- 1 file changed, 15 insertions(+), 1 deletion(-) diff --git a/cubic-phase.md b/cubic-phase.md index c5d97f2..318bf3b 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -63,4 +63,18 @@ or, in terms of the parameter $t$ $$\begin{aligned} f &= \frac{t(t - 2i)}{4 + t^{2}}, \\ g &= - \frac{2i}{\sqrt{4 + t^{2}}}. -\end{aligned}$$ \ No newline at end of file +\end{aligned}$$ + +An important result is + +$$ +[\hat{a},P(t)] = P\left\{\frac{it}{2}(\hat{a}^{\dagger} + \hat{a})\right\} +$$ + +We calculate the Fock-states matrix elements of the commutator above + +$$ +\langle m|[\hat{a},P(t)]|n \rangle = \sqrt{m+1}P_{m+1,n} - \sqrt{n}P_{m,n-1} = \frac{it}{2}\left\{\sqrt{n+1}P_{m,n+1} + \sqrt{n}V_{m,n-1} \right\}, +$$ + +with the initial condition $P_{0,0} = \frac{e^{i\varphi}}{\cosh{r}} = \frac{1}{2 - it}$. \ No newline at end of file From dd2cf30b431f64f143b29618e92058ee74f6d8dd Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Thu, 9 Jul 2026 10:48:26 -0400 Subject: [PATCH 11/12] Recurrence relation P solved --- cubic-phase.md | 18 ++++++++++++++---- 1 file changed, 14 insertions(+), 4 deletions(-) diff --git a/cubic-phase.md b/cubic-phase.md index 318bf3b..d371a09 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -68,13 +68,23 @@ g &= - \frac{2i}{\sqrt{4 + t^{2}}}. An important result is $$ -[\hat{a},P(t)] = P\left\{\frac{it}{2}(\hat{a}^{\dagger} + \hat{a})\right\} +[\hat{a},P(t)] = P\left\lbrace \frac{it}{2}(\hat{a}^{\dagger} + \hat{a}) \right\rbrace $$ -We calculate the Fock-states matrix elements of the commutator above +We calculate the Fock-state matrix elements of the commutator above $$ -\langle m|[\hat{a},P(t)]|n \rangle = \sqrt{m+1}P_{m+1,n} - \sqrt{n}P_{m,n-1} = \frac{it}{2}\left\{\sqrt{n+1}P_{m,n+1} + \sqrt{n}V_{m,n-1} \right\}, +\langle m|[\hat{a},P(t)]|n \rangle = \sqrt{m+1}\,P_{m+1,n} - \sqrt{n}\,P_{m,n-1} = \frac{it}{2}\left\lbrace \sqrt{n+1}\,P_{m,n+1} + \sqrt{n}\,P_{m,n-1} \right\rbrace, $$ -with the initial condition $P_{0,0} = \frac{e^{i\varphi}}{\cosh{r}} = \frac{1}{2 - it}$. \ No newline at end of file +with the initial condition $P_{0,0} = \frac{e^{i\varphi}}{\sqrt{\cosh{r}}} = \frac{1}{\sqrt{1 - it/2}}$. + +Solving for the term with highest index + +$$ +P_{m+1,n} = \frac{1}{\sqrt{m+1}} \left\lbrace \sqrt{n}\left(1 + \frac{it}{2}\right)P_{m,n-1} + \frac{it}{2}\sqrt{n+1}P_{m,n+1} \right\rbrace, +$$ + +$$ +P_{m+1,0} = \frac{it}{2\sqrt{m+1}} P_{m,1}. +$$ \ No newline at end of file From 6e4952686570a0853113bc1ea8d892c17e2d66c2 Mon Sep 17 00:00:00 2001 From: Kevin Giraldo Date: Thu, 9 Jul 2026 10:49:07 -0400 Subject: [PATCH 12/12] dot --- cubic-phase.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/cubic-phase.md b/cubic-phase.md index d371a09..e3146b0 100644 --- a/cubic-phase.md +++ b/cubic-phase.md @@ -68,7 +68,7 @@ g &= - \frac{2i}{\sqrt{4 + t^{2}}}. An important result is $$ -[\hat{a},P(t)] = P\left\lbrace \frac{it}{2}(\hat{a}^{\dagger} + \hat{a}) \right\rbrace +[\hat{a},P(t)] = P\left\lbrace \frac{it}{2}(\hat{a}^{\dagger} + \hat{a}) \right\rbrace. $$ We calculate the Fock-state matrix elements of the commutator above