Quadrature Components
Definition
In the homodyne detection section here, we computed that the average and the variance of the signal:
\begin{aligned} & \langle \mathbf N_3 - \mathbf N_4 \rangle = \langle \boldsymbol \Psi_1 | \left(e^{-i\varphi_\lambda}\mathbf a_1 + e^{i\varphi_\lambda} \mathbf a_1^\dag \right) |\boldsymbol \Psi_1 \rangle \\ & = \Delta^2\left(\mathbf N_3 - \mathbf N_4\right) = \Delta^2 \left( e^{-i\varphi_\lambda}\mathbf a_1 + e^{i\varphi_\lambda} \mathbf a_1^\dag \right) \end {aligned}
And we noticed that are proportional to the average and the variance of the observable:
e^{-i\varphi_\lambda}\mathbf a_1 + e^{i\varphi_\lambda} \mathbf a_1^\dag = \cos\left(\varphi_\lambda\right) \left( \mathbf a_1 + \mathbf a_1^\dag \right) + \sin\left(\varphi_\lambda\right) \left( \frac{\mathbf a_1 - \mathbf a_1^\dag}{i} \right)
where it is now separated into the real and imaginary part we have:
\begin{aligned} \mathbf Q & = \sqrt{\frac{\hbar}{2}} \left( \mathbf a_1 + \mathbf a_1^\dag \right) \\ \mathbf P & = \sqrt{\frac{\hbar}{2}} \left( \frac{\mathbf a_1 - \mathbf a_1^\dag}{i} \right)\\ \end{aligned}
These quadrature component were already encountered when studying the quantization of the light in one mode here (with a factor \sqrt{\frac {\hbar}{2}}). We also computed their commutator:
[\mathbf Q, \mathbf P] = i \hbar
These can be computed separately choosing the phase of the oscillator equal to 0 or \frac{\pi}{2}.
With this convention, expressing the electric field as function of these observables:
\mathbf{E}_{\lambda}(\mathbf r,t) = i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left( \mathbf{a}_{\lambda} e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t)} - \mathbf{a}_{\lambda}^{\dagger} e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t)} \right)
inverting the relationship:
\begin{aligned} & \mathbf{a}_{\lambda} = \sqrt{\frac{2}{\hbar}} (Q_{\lambda} + iP_{\lambda}) \\ & \mathbf{a}_{\lambda}^{\dagger} = \sqrt{\frac{2}{\hbar}} (Q_{\lambda} - iP_{\lambda}) \end{aligned}
The electric field can be then express as:
\mathbf{E}(\mathbf r,t) = \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \sqrt{\frac{2}{\hbar}} \left( -Q_{\lambda} \sin(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t) - P_{\lambda} \cos(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t) \right)
The minus is for historical convention and still widely used.
The usage of these quadrature components is not new from quantum mechanics; it was already present in classical electromagnetic theory, when the classical electromagnetic field was expressed as:
\mathbf E(\mathbf r,t) = -\mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left| \alpha_\lambda \right|\sin\left(\mathbf k_\lambda \cdot \mathbf r + \varphi_\lambda -\omega_\lambda t \right)
so the state of the field is fully characterized by the complex amplitude i \mathscr{E}_{\lambda}^{(1)} \alpha_\lambda. The complex number \alpha_\lambda can be expressed as modulus and argument:
\alpha_\lambda = \left| \alpha_\lambda \right| e^{i\varphi_\lambda}
or equivalently with its real and imaginary part:
\begin{aligned} & \left| \alpha_\lambda \right| \cos\left( \varphi_\lambda \right) = \frac{\alpha_\lambda + \bar \alpha_\lambda}{2} = \frac{Q_\lambda}{\sqrt{2\hbar}} \\ & \left| \alpha_\lambda \right| \sin\left( \varphi_\lambda \right) = \frac{\alpha_\lambda - \bar \alpha_\lambda}{2i} = \frac{P_\lambda}{\sqrt{2\hbar}} \end{aligned}
When moving to the quantum optics framework, these two quantities correspond to genuine observable (Hermitian operators), while it is not possible for the module and the phase.
The choice of the origin of time (and space, as \mathbf{k}_{\lambda} \cdot \mathbf{r} is involved) effectively sets a reference point for the phase of the wave.
The decomposition into \mathbf Q_\lambda \sin(\dots) and \mathbf P_\lambda \cos(\dots) is based on choosing sine and cosine functions as basis waves. By choosing this basis, we are implicitly defining an “origin” from which we measure phase. If we shift the origin of time, the phase of the entire wave shifts, and the values of \mathbf Q_\lambda and \mathbf P_\lambda needed to describe the same physical wave in terms of our sine and cosine basis will change.
In this respect, “in-phase” (\mathbf Q_\lambda) and “out-of-phase” (\mathbf P_\lambda) components are defined relative to the chosen sine and cosine basis, which is itself linked to an implicit choice of origin for time and space. Changing the origin will redistribute the contributions between the in-phase and out-of-phase components, although the underlying physical field remains the same.
Therefore, when considering homodyne detection, phase locking play a major role because the output of a homodyne detector depends critically on the relative phase between the signal field and the local oscillator field. If the relative phase drifts randomly, the measurement outcome will fluctuate and become meaningless. To obtain stable and reproducible measurements of a specific quadrature, the relative phase must be stable over the measurement time.
Maintaining phase lock means ensuring that the relative drift between the signal and local oscillator fields’ phases is much smaller than one optical period during the measurement. Achieving this directly by stabilizing two independent clocks is technically challenging.
The practical solution is to derive both the signal and local oscillator fields (or at least their phase references) from the same master clock. If both fields are locked to the same clock, even if that master clock drifts slightly, the relative phase between the signal and local oscillator remains constant. This common clock ensures the stability of the phase difference, which is essential for homodyne detection.
By controlling the phase of the local oscillator relative to the signal (through the common clock reference), you can select which quadrature of the signal field is measured. If the local oscillator phase is set to 0 (or a multiple of 2\pi) relative to a chosen reference, the homodyne detection will be sensitive to the in-phase quadrature component (\mathbf Q_\lambda).
If the LO phase is set to \pi/2 (or \pi/2 + 2n\pi) relative to the same reference, the detection will be sensitive to the out-of-phase quadrature component (\mathbf P_\lambda). By adjusting the LO phase to other values, you can measure linear combinations of \mathbf Q_\lambda and \mathbf P_\lambda.
It is not possible to measure both quadratures simultaneously, because they do not not commute; naturally it is possible to repeat the measurement experiment many times, and each time compute one of the components (or a combination of the two).
Repeating the experiment many times, it is then possible to compute the quantities of interest:
\begin{aligned} & \langle i_3 - i_4\rangle \Big|_{\varphi_{LO}=0} \rightarrow \langle \boldsymbol \psi_1 | \mathbf{Q}_1 | \boldsymbol \psi_1 \rangle \\ & \left( \Delta_{i_3 - i_4} \right)^2 \Big|_{\varphi_2=0} \rightarrow \left( \Delta_Q \right)^2 \\ & \langle i_3 - i_4\rangle \Big|_{\varphi_2=\pi/2} \rightarrow \langle \boldsymbol \psi_1 | \mathbf{P}_1 | \boldsymbol \psi_1 \rangle \\ & \left( \Delta_{i_3 - i_4} \right)^2 \Big|_{\varphi_2=\pi/2} \rightarrow \left( \Delta_P \right)^2 \end{aligned}
These quantities obey to the Heisenberg uncertainty principle:
\Delta_Q \cdot \Delta_P \geq \frac{\hbar}{2}
Choosing an arbitrary phase \theta, it is possible to measure the intermediate components:
\begin{aligned} & \mathbf{Q}_1(\theta)\big|_{\varphi_2 = \theta} = \sqrt{\frac{\hbar}{2}} \frac{e^{-i\theta} \mathbf{a}_1 + e^{i\theta} \mathbf{a}_1^{\dagger}}{\sqrt{2}} \\ & \mathbf{P}_1(\theta)\big|_{\varphi_2 = \theta + \frac{\pi}{2}} = \sqrt{\frac{\hbar}{2}} \frac{e^{-i\theta} \mathbf{a}_1 - e^{i\theta} \mathbf{a}_1^{\dagger}}{i\sqrt{2}} \end{aligned}
The commutator is naturally still i\hbar:
[\mathbf{Q}_1(\theta), \mathbf{P}_1(\theta)] = i\hbar
And they obey to the Heisenberg uncertainty principle:
\Delta_{\mathbf{Q}_1(\theta)} \cdot \Delta_{\mathbf{P}_1(\theta)} \geq \frac{\hbar}{2} From the above equation is is possible to derive for \mathbf{Q}_1(\theta):
\begin{aligned} \mathbf{Q}_1(\theta) &= \sqrt{\frac{\hbar}{2}} \frac{e^{-i\theta} \mathbf{a}_1 + e^{i\theta} \mathbf{a}_1^{\dagger}}{\sqrt{2}} \\ &= \sqrt{\frac{\hbar}{2}} \frac{(\cos\theta - i\sin\theta) \mathbf{a}_1 + (\cos\theta + i\sin\theta) \mathbf{a}_1^{\dagger}}{\sqrt{2}} \\ &= \sqrt{\frac{\hbar}{2}} \frac{\cos\theta (\mathbf{a}_1 + \mathbf{a}_1^{\dagger}) - i\sin\theta (\mathbf{a}_1 - \mathbf{a}_1^{\dagger})}{\sqrt{2}} \\ &= \cos\theta \left( \sqrt{\frac{\hbar}{2}} \frac{\mathbf{a}_1 + \mathbf{a}_1^{\dagger}}{\sqrt{2}} \right) + \sin\theta \left( -i \sqrt{\frac{\hbar}{2}} \frac{\mathbf{a}_1 - \mathbf{a}_1^{\dagger}}{\sqrt{2}} \right) \\ &= \cos\theta \, \mathbf{Q}_1 + \sin\theta \, \mathbf{P}_1 \end{aligned}
and for \mathbf{P}_1(\theta):
\begin{aligned} \mathbf{P}_1(\theta) &= \sqrt{\frac{\hbar}{2}} \frac{e^{-i\theta} \mathbf{a}_1 - e^{i\theta} \mathbf{a}_1^{\dagger}}{i\sqrt{2}} \\ &= \frac{1}{i} \sqrt{\frac{\hbar}{2}} \frac{(\cos\theta - i\sin\theta) \mathbf{a}_1 - (\cos\theta + i\sin\theta) \mathbf{a}_1^{\dagger}}{\sqrt{2}} \\ &= \frac{1}{i} \sqrt{\frac{\hbar}{2}} \frac{\cos\theta (\mathbf{a}_1 - \mathbf{a}_1^{\dagger}) - i\sin\theta (\mathbf{a}_1 + \mathbf{a}_1^{\dagger})}{\sqrt{2}} \\ &= \cos\theta \left( \frac{1}{i} \sqrt{\frac{\hbar}{2}} \frac{\mathbf{a}_1 - \mathbf{a}_1^{\dagger}}{\sqrt{2}} \right) - \sin\theta \left( \frac{1}{i} \sqrt{\frac{\hbar}{2}} \frac{i (\mathbf{a}_1 + \mathbf{a}_1^{\dagger})}{\sqrt{2}} \right) \\ &= \cos\theta \, \mathbf{P}_1 - \sin\theta \, \mathbf{Q}_1 \\ &= -\sin\theta \, \mathbf{Q}_1 + \cos\theta \, \mathbf{P}_1 \end{aligned}
Therefore these quantities can be expressed as:
\begin{aligned} & \mathbf{Q}_1(\theta)\big|_{\varphi_2 = \theta} = \cos\theta \, \mathbf{Q}_1 + \sin\theta \, \mathbf{P}_1 \\ & \mathbf{P}_1(\theta)\big|_{\varphi_2 = \theta + \frac{\pi}{2}} = -\sin\theta \, \mathbf{Q}_1 + \cos\theta \, \mathbf{P}_1 \end{aligned}
Complex plane representation
Let’s consider a quasi classical state:
| \boldsymbol \psi_1 \rangle = | \boldsymbol \alpha_1 \rangle with the complex number as:
\alpha_1= \left| \alpha_1 \right| e^{i \varphi_1}
The average of the quadrature components are:
\begin{aligned} \langle \boldsymbol \alpha_1 | \mathbf Q_1 | \boldsymbol \alpha_1 \rangle & = \sqrt{\frac{\hbar}{2}}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 + \mathbf a_1^\dag \right) | \boldsymbol \alpha_1 \rangle = \sqrt{\frac{\hbar}{2}}\left[\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 \boldsymbol \alpha_1 \rangle \right) + \left( \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \right) | \boldsymbol \alpha_1 \rangle \right]\\ & = \sqrt{\frac{\hbar}{2}}\left(\langle \boldsymbol \alpha_1 \alpha_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 \bar \alpha_1 | \boldsymbol \alpha_1 \rangle \right) = \sqrt{\frac{\hbar}{2}}\left( \alpha_1 + \bar \alpha_1 \right) \\ & = \sqrt{\frac{\hbar}{2}}\left( \left| \alpha_1 \right| e^{i \varphi_1} + \left| \alpha_1 \right| e^{-i \varphi_1} \right) = \sqrt{\frac{\hbar}{2}} \left| \alpha_1 \right|\left( e^{i \varphi_1} + e^{-i \varphi_1} \right) \\ & = \sqrt{2\hbar} \left| \alpha_1 \right| \cos (\varphi_1) \\ \langle \boldsymbol \alpha_1 | \mathbf P_1 | \boldsymbol \alpha_1 \rangle & = \sqrt{\frac{\hbar}{2}} \langle \boldsymbol \alpha_1 | \left( \frac{\mathbf a_1 - \mathbf a_1^\dag}{i} \right) | \boldsymbol \alpha_1 \rangle = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left[\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 \boldsymbol \alpha_1 \rangle \right) - \left( \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \right)| \boldsymbol \alpha_1 \rangle \right] \\ & = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left(\langle \boldsymbol \alpha_1 \alpha_1 | \boldsymbol \alpha_1 \rangle - \langle \boldsymbol \alpha_1 \bar \alpha_1 | \boldsymbol \alpha_1 \rangle \right) = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left( \alpha_1 - \bar \alpha_1 \right) \\ & = \sqrt{\frac{\hbar}{2}} \frac{1}{i}\left( \left| \alpha_1 \right| e^{i \varphi_1} - \left| \alpha_1 \right| e^{-i \varphi_1} \right) = \sqrt{\frac{\hbar}{2}} \frac{1}{i}\left| \alpha_1 \right|\left( e^{i \varphi_1} - e^{-i \varphi_1} \right) \\ & = \sqrt{2\hbar}\left| \alpha_1 \right| \sin (\varphi_1) \end{aligned}
A similar calculation can be done for \mathbf Q_1(\theta) and \mathbf P_1(\theta):
\begin{aligned} \langle \boldsymbol \alpha_1 | \mathbf P_1 | \boldsymbol \alpha_1 \rangle & = \sqrt{\frac{\hbar}{2}} \langle \boldsymbol \alpha_1 | \left( \frac{e^{-i\theta} \mathbf a_1 + e^{i\theta} \mathbf a_1^\dag}{i} \right) | \boldsymbol \alpha_1 \rangle = \sqrt{\frac{\hbar}{2}}\left[e^{-i\theta}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 \boldsymbol \alpha_1 \rangle \right) + e^{i\theta}\left( \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \right) | \boldsymbol \alpha_1 \rangle \right] \\ & = \sqrt{\frac{\hbar}{2}}\left(e^{-i\theta}\langle \boldsymbol \alpha_1 \alpha_1 | \boldsymbol \alpha_1 \rangle + e^{i\theta} \langle \boldsymbol \alpha_1 \bar \alpha_1 | \boldsymbol \alpha_1 \rangle \right) = \sqrt{\frac{\hbar}{2}}\left( e^{-i\theta}\alpha_1 + e^{i\theta}\bar \alpha_1 \right) \\ & = \sqrt{\frac{\hbar}{2}} \left( \left| \alpha_1 \right| e^{-i\theta} e^{i \varphi_1} + \left| \alpha_1 \right| e^{i\theta}e^{-i \varphi_1} \right) = \sqrt{\frac{\hbar}{2}}\left| \alpha_1 \right|\left( e^{i (\varphi_1 -\theta)} + e^{-i (\varphi_1 -\theta)} \right) \\ & = \sqrt{2\hbar} \left| \alpha_1 \right| \cos (\varphi_1 - \theta) \\ \langle \boldsymbol \alpha_1 | \mathbf P_1 | \boldsymbol \alpha_1 \rangle & = \sqrt{\frac{\hbar}{2}} \langle \boldsymbol \alpha_1 | \left( \frac{e^{-i\theta} \mathbf a_1 - e^{i\theta} \mathbf a_1^\dag}{i} \right) | \boldsymbol \alpha_1 \rangle = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left[e^{-i\theta}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 \boldsymbol \alpha_1 \rangle \right) - e^{i\theta}\left( \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \right) | \boldsymbol \alpha_1 \rangle \right] \\ & = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left(e^{-i\theta}\langle \boldsymbol \alpha_1 \alpha_1 | \boldsymbol \alpha_1 \rangle - e^{i\theta} \langle \boldsymbol \alpha_1 \bar \alpha_1 | \boldsymbol \alpha_1 \rangle \right) = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left( e^{-i\theta}\alpha_1 - e^{i\theta}\bar \alpha_1 \right) \\ & = \sqrt{\frac{\hbar}{2}} \frac{1}{i}\left( \left| \alpha_1 \right| e^{-i\theta} e^{i \varphi_1} - \left| \alpha_1 \right| e^{i\theta}e^{-i \varphi_1} \right) = \sqrt{\frac{\hbar}{2}} \frac{1}{i} \left| \alpha_1 \right|\left( e^{i (\varphi_1 -\theta)} - e^{-i (\varphi_1 -\theta)} \right) \\ & = \sqrt{2\hbar} \left| \alpha_1 \right| \sin (\varphi_1 - \theta) \end{aligned}
It is possible to represent the results in a complex plane with the horizontal axis corresponding to \mathbf Q and the vertical axis to \mathbf P, in units of \sqrt{2}\hbar.
The averages of \mathbf Q and \mathbf P are then plotted on these axes, and the quadratures are simply the real and imaginary parts of the complex number \alpha, once it is mapped onto the complex plane.
If one considers the average of the quadrature \mathbf Q_\theta, associated with the phase \theta of the local oscillator, it is equivalent to projecting \alpha onto an axis rotated by \theta from the \mathbf Q axis.
This representation directly corresponds to performing many measurements on identical radiation states or taking successive measurements. For each observable, multiple measurements are needed to determine the average value since the quadratures do not commute and therefore simultaneous measurement is impossible, requiring different sets of measurements for different quadrature.
Let’s compute now the dispersion of the quadrature using that the variance of a random variable can be expressed as the average of the square minus the square of the average.
We start calculating the average of the square of \mathbf Q_1:
\begin{aligned} \langle \boldsymbol \alpha_1 | \mathbf Q_1^2 | \boldsymbol \alpha_1 \rangle & = \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 + \mathbf a_1^\dag \right)^2 | \boldsymbol \alpha_1 \rangle = \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 + \mathbf a_1 \mathbf a_1^\dag + \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle = \\ & = \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 + (1 + \mathbf a_1^\dag \mathbf a_1 ) + \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle \\ & = \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 + 1 + 2 \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle \\ & = \frac{\hbar}{2}\left[ 1 + \langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 + 2 \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \langle \boldsymbol \alpha_1 | \mathbf a_1^2 | \boldsymbol \alpha_1 \rangle + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 | (\mathbf a_1^\dag)^2 | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \langle \boldsymbol \alpha_1 | \mathbf a_1 (\mathbf a_1 | \boldsymbol \alpha_1 \rangle) | \boldsymbol \alpha_1 \rangle + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\mathbf a_1^\dag | \boldsymbol \alpha_1 \rangle) | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \langle \boldsymbol \alpha_1 | \mathbf a_1 (\alpha_1 | \boldsymbol \alpha_1 \rangle) | \boldsymbol \alpha_1 \rangle + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\bar \alpha_1 \langle \boldsymbol \alpha_1 |) | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1 \langle \boldsymbol \alpha_1 | \mathbf a_1 | \boldsymbol \alpha_1 \rangle + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1 \langle \boldsymbol \alpha_1 | (\alpha_1 | \boldsymbol \alpha_1 \rangle) + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1 \langle \boldsymbol \alpha_1 | (\bar \alpha_1 \langle \boldsymbol \alpha_1 |) | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 \langle \boldsymbol \alpha_1 | \boldsymbol \alpha_1 \rangle + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \langle \boldsymbol \alpha_1 | \boldsymbol \alpha_1 \rangle \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\mathbf a_1 | \boldsymbol \alpha_1 \rangle) + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\alpha_1 | \boldsymbol \alpha_1 \rangle) + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \alpha_1 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \alpha_1 (\bar \alpha_1 \langle \boldsymbol \alpha_1 |) | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \alpha_1 \bar \alpha_1 \langle \boldsymbol \alpha_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \alpha_1^2 + 2 \alpha_1 \bar \alpha_1 + \bar \alpha_1^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 + \left(\alpha_1 + \bar \alpha_1\right)^2 \right] \end{aligned}
We previously computed:
\langle \boldsymbol \alpha_1 | \mathbf Q_1 | \boldsymbol \alpha_1 \rangle = \sqrt{\frac{\hbar}{2}}\left( \alpha_1 + \bar \alpha_1 \right) Taking the square we can now compute the variance \Delta Q_1^2 as:
\begin{aligned} \Delta Q_1^2 & = \langle \boldsymbol \alpha_1 | \mathbf Q_1^2 | \boldsymbol \alpha_1 \rangle - \left(\langle \boldsymbol \alpha_1 | \mathbf Q_1^2 | \boldsymbol \alpha_1 \rangle\right)^2 \\ & = \frac{\hbar}{2}\left[ 1 + \left(\alpha_1 + \bar \alpha_1\right)^2 \right] - \frac{\hbar}{2}\left(\alpha_1 + \bar \alpha_1\right)^2= \frac{\hbar}{2} \end{aligned}
It is possible to perform a similar calculation for \mathbf P_1:
\begin{aligned} \langle \boldsymbol \alpha_1 | \mathbf P_1^2 | \boldsymbol \alpha_1 \rangle & = - \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1 - \mathbf a_1^\dag \right)^2 | \boldsymbol \alpha_1 \rangle = - \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 - \mathbf a_1 \mathbf a_1^\dag - \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle = \\ & = - \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 - (1 + \mathbf a_1^\dag \mathbf a_1 ) - \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle \\ & = - \frac{\hbar}{2}\langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 - 1 - 2 \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle \\ & = - \frac{\hbar}{2}\left[ -1 + \langle \boldsymbol \alpha_1 | \left( \mathbf a_1^2 - 2 \mathbf a_1^\dag \mathbf a_1 + \left(\mathbf a_1^\dag \right)^2 \right) | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \langle \boldsymbol \alpha_1 | \mathbf a_1^2 | \boldsymbol \alpha_1 \rangle - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 | (\mathbf a_1^\dag)^2 | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \langle \boldsymbol \alpha_1 | \mathbf a_1 (\mathbf a_1 | \boldsymbol \alpha_1 \rangle) | \boldsymbol \alpha_1 \rangle - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\mathbf a_1^\dag | \boldsymbol \alpha_1 \rangle) | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \langle \boldsymbol \alpha_1 | \mathbf a_1 (\alpha_1 | \boldsymbol \alpha_1 \rangle) | \boldsymbol \alpha_1 \rangle - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\bar \alpha_1 \langle \boldsymbol \alpha_1 |) | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1 \langle \boldsymbol \alpha_1 | \mathbf a_1 | \boldsymbol \alpha_1 \rangle - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1 \langle \boldsymbol \alpha_1 | (\alpha_1 | \boldsymbol \alpha_1 \rangle) - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1 \langle \boldsymbol \alpha_1 | (\bar \alpha_1 \langle \boldsymbol \alpha_1 |) | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 \langle \boldsymbol \alpha_1 | \boldsymbol \alpha_1 \rangle - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \langle \boldsymbol \alpha_1 | \boldsymbol \alpha_1 \rangle \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag \mathbf a_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\mathbf a_1 | \boldsymbol \alpha_1 \rangle) + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag (\alpha_1 | \boldsymbol \alpha_1 \rangle) + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \alpha_1 \langle \boldsymbol \alpha_1 | \mathbf a_1^\dag | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \alpha_1 (\bar \alpha_1 \langle \boldsymbol \alpha_1 |) | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \alpha_1 \bar \alpha_1 \langle \boldsymbol \alpha_1 | \boldsymbol \alpha_1 \rangle + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \alpha_1^2 - 2 \alpha_1 \bar \alpha_1 + \bar \alpha_1^2 \right] \\ & = - \frac{\hbar}{2}\left[ -1 + \left(\alpha_1 - \bar \alpha_1\right)^2 \right] \\ & = \frac{\hbar}{2}\left[ 1 - \left(\alpha_1 - \bar \alpha_1\right)^2 \right] \\ \end{aligned}
We previously computed:
\langle \boldsymbol \alpha_1 | \mathbf P_1 | \boldsymbol \alpha_1 \rangle = \sqrt{\frac{\hbar}{2}}\frac{1}{i}\left( \alpha_1 - \bar \alpha_1 \right)
Taking the square we can now compute the variance \Delta P_1^2 as:
\begin{aligned} \Delta P_1^2 & = \langle \boldsymbol \alpha_1 | \mathbf P_1^2 | \boldsymbol \alpha_1 \rangle - \left(\langle \boldsymbol \alpha_1 | \mathbf P_1^2 | \boldsymbol \alpha_1 \rangle\right)^2 \\ & = \frac{\hbar}{2}\left[ 1 - \left(\alpha_1 - \bar \alpha_1\right)^2 \right] + \frac{\hbar}{2}\left(\alpha_1 - \bar \alpha_1\right)^2= \frac{\hbar}{2} \end{aligned}
We arrive at the results that the standard deviation for the two canonical conjugated variable is the square root of the variance:
\Delta Q_1 = \Delta P_1 = \sqrt{\frac{\hbar}{2}}
Performing the same type of computation for each angle theta would lead to the same result:
\Delta Q_1(\theta) = \Delta P_1(\theta) = \sqrt{\frac{\hbar}{2}}
Therefore the Heisenberg uncertainty principle gives:
\Delta Q_1 \Delta P_1 = \sqrt{\left(\Delta Q_1\right)^2}\sqrt{\left(\Delta P_1\right)^2} = \frac{\hbar}{2}
It shows that quasi-classical states are minimum uncertainty states, where the product of the uncertainties reaches the theoretical lower bound established by quantum mechanics.
For each observable, the spread is shown as a segment of length \sqrt{2\hbar} standard deviation units.
Projecting a disc of radius one-half around \mathbf \alpha onto the corresponding axis represents these segments. Measuring a quadrature \mathbf Q_\theta yields a distribution of results, which for a quasi-classical state is a Gaussian centered on the average with the mentioned standard deviation:
\mathcal P\left(\mathbf Q_1\right) = \frac{1}{\sqrt{2\pi}\Delta Q_1}e^{-\frac{\left(\mathbf Q_1 - \langle \mathbf Q_1 \rangle\right)^2}{2\Delta Q_1^2}}
This is representable as a two-dimensional Gaussian centered on \alpha:
\mathcal P\left(\mathbf Q_1, \mathbf P_1\right) = \frac{1}{2\pi\Delta Q_1^2}e^{-\frac{\left(\mathbf Q_1 - \langle \mathbf Q_1 \rangle\right)^2 + \left(\mathbf P_1 - \langle \mathbf P_1 \rangle\right)^2}{2\Delta Q_1^2}}
Integrating this 2D Gaussian along the direction perpendicular to the Q_\theta axis results in a Gaussian probability distribution for Q_\theta. Projecting a rotationally symmetric 2D Gaussian onto any axis yields a 1D Gaussian with the same standard deviation.
The Vacuum (|\boldsymbol \Psi_1\rangle = |\mathbf 0\rangle) is a specific case of a quasi-classical state where the complex amplitude \alpha is zero. In this vacuum state, the dispersion of any quadrature component is identical and equal to \sqrt{\frac{\hbar}{2}}:
\Delta Q_1 = \Delta P_1 = \Delta Q_1(0) = \sqrt{\frac{\hbar}{2}}
This uniform dispersion allows for a disk representation centered at zero in phase space.
A quasi-classical state with amplitude \alpha can be understood as a vacuum state displaced by the complex number \alpha:
|\boldsymbol \alpha_1\rangle = e^{ - \frac{\left|\alpha_1\right|^2}{2}} e^ { \alpha_1 \mathbf{a}_1^\dagger} |0\rangle
A coherent state |\alpha\rangle can be generated by applying the more general displacement operator:
\mathbf{D}(\alpha_1) = e^{ - \frac{|\alpha_1|^2}{2}} e^ {\left(\alpha_1 \mathbf{a}_1^\dagger - \bar \alpha_1 \mathbf{a}_1\right)}
to the vacuum state |\mathbf 0\rangle. This operator is Hermitian and leads to the above results for quasi-classical state since \mathbf{a}_1 | \mathbf 0\rangle = 0.
We can utilize the quasi-classical state representation to depict the time evolution of the electric field average and its dispersion.
The electric field average for a quasi-classical state with complex amplitude \alpha_1 is:
\langle\boldsymbol \alpha_1|\mathbf e_1\cdot\mathbf{E}_1(\mathbf{r},t)|\boldsymbol \alpha_1\rangle = i\mathscr E^{(1)}_1(\alpha_1 e^{(i\mathbf{k}_1\cdot\mathbf{r} - \omega_1t)} - \bar \alpha_1 e^{-i(\mathbf{k}_1\cdot\mathbf{r} - \omega_1t)}) Without loss of generality, to have simpler expressions and to concentrate of the time evolution, is is possible to take at position \mathbf r = 0,
\begin{aligned} \langle\boldsymbol \alpha_1|\mathbf e_1\cdot\mathbf{E}_1(\mathbf 0, t)|\boldsymbol \alpha_1\rangle & = i\mathscr E^{(1)}_1(\alpha_1 e^{-i\omega_1t} - \bar \alpha_1 e^{i\omega_1t}) \\ & = -2i\mathscr E^{(1)}_1 \left|\alpha_1\right| \sin(-\omega_1t + \varphi_1) \end{aligned}
evolves in time such that, within a factor of -2\mathscr E^{(1)}_1, it can be visualized by the rotation of \alpha_1 in the complex plane at angular frequency -\omega t.
Its projection onto the imaginary axis yielding the sinusoidal time dependence with a factor:
-\frac{\langle\boldsymbol \alpha_1|\mathbf e_1\cdot\mathbf{E}_1(\mathbf 0, t)|\boldsymbol \alpha_1\rangle}{2i\mathscr E^{(1)}_1}
This representation is advantageous as it allows us to visualize not only the average field but also its dispersion.
A disk, centered on the rotating complex number \alpha_1, represents this dispersion. Projecting this disk onto the imaginary axis generates a band around the evolving average, with a half-width equivalent to the standard deviation, which is constant and equal to one-half in units of 2\mathscr E^{(1)}_1.
I created a simulation were on the left the field is represented in the complex plane, rotating at a constant angular speed. On the right, the field progresses over time, maintaining a coherent quantum dispersion, depicted as a stable circular uncertainty region.
It is essential to distinguish between this time-dependent field evolution representation, which is not directly observable with current technology, and the static quadrature representation.
Quadratures are time-independent quantities, measurable through balanced homodyne detection, allowing for the determination of their averages and dispersions.
By varying the local oscillator phase, we can measure quadratures at different angles in the Q-plane. The ensemble of these measurements constructs the red disk, which is effectively the field evolution diagram at time t=0.
The graphs were produced for a quasi-classical state, but this tool is general and can be used to represent other type of states for single mode.