State Preparation: Non-Gaussian Quantum State

Abstract

As we conclude from the previous chapter, in order to create non-Gaussian quantum states, a nonlinear optomechanical coupling is generally required. This is rather challenging to achieve, especially when the mass of the mechanical oscillator is large and the frequency is low. In this chapter, we propose a protocol for coherently transferring non-Gaussian quantum states from the optical field to a mechanical oscillator, which does not require a nonlinear coupling in the optomechanical system.

Appendix

10.1.1 Optomechanical Dynamics

In this section, we will briefly review the dynamics of a typical cavity-assisted optomechanical system, in order to justify some of the equations in the main text. The Hamiltonian for such an optomechanical system (shown schematically in Fig. 10.2) can be written as (cf. Refs. [8, 21, 31])

$$ \begin{aligned}[b] \hat{\mathcal H}&={\frac{\hat p^2}{2m}}+{\frac{1}{2}}m \omega_m^2\hat x^2+\hbar \omega_c \hat a^{\dag}\hat a+\hbar G_0 \hat x \hat a^{\dag}\hat a\\ &+i \hbar\sqrt{2\gamma} (\hat a_{\rm in}e^{-i \omega_0 t}\hat a^{\dag}-H.c.)+\hat H_{\gamma}+\hat H_{\gamma_m}. \end{aligned} $$

(10.19)

Here, $\hat x$ and $\hat p$ are the position and momentum operators for the oscillator; $\hat a$ is the annihilation operator for the cavity mode; $G_0\equiv \omega_m/L$ is the optomechanical coupling constant with $L$ the cavity length; $\gamma$ is the cavity bandwidth; $\hat H_{\gamma}$ and $\hat H_{\gamma_m}$ describe the dissipation mechanism of the cavity mode and the mechanical oscillator, respectively.

In the rotating frame at the laser frequency $\omega_0,$ the above Hamiltonian leads to the following standard Langevin equations for the mechanical oscillator:

$$ \dot {\hat x}(t)={\frac{\hat p(t)}{m}}, $$

(10.20)

$$ \dot {\hat p}(t)=-\gamma_m \hat p(t)-m\omega_m^2\hat x(t)+\hbar G_0 \hat a^{\dag}(t)\hat a(t)+\hat F_{\rm th}(t), $$

(10.21)

with $\hat F_{\rm th}$ the thermal force noise, and for the cavity mode:

$$ \dot {\hat a}(t)+(\gamma-i\Updelta)\hat a(t)=i G_0\hat a(t)\hat x(t)+\sqrt{2\gamma} \hat a_{\rm in}(t), $$

(10.22)

with $\Updelta\equiv \omega_0-\omega_c.$ The standard input-output relation for the cavity mode is given by (refer to Ref. [7])

$$ \hat a_{\rm in}(t)+\hat a_{\rm out}(t)=\sqrt{2\gamma} \hat a(t). $$

(10.23)

Since the cavity mode is coherently driven with a laser, the above equation can be linearized by replacing every quantity $\hat o$ with a sum of the classical steady part $\bar o$ and a perturbed part. The equations of motion for the perturbed parts of the oscillator read

$$ \dot {\hat x}(t)={\frac{\hat p(t)}{m}}, $$

(10.24)

$$ \dot {\hat p}(t)=-\gamma_m \hat p(t)-m\omega_m^2\hat x(t)+\bar{G}_0 \hat a_1(t)+\hat F_{\rm th}(t), $$

(10.25)

where the amplitude quadrature $\hat a_1(t)\equiv [\hat a(t)+\hat a^{\dag}(t)]/\sqrt{2}$ and $\bar G_0\equiv\sqrt{2}\hbar G_0\bar a.$ Similarly, for the cavity mode,

$$ \dot {\hat a}(t)+(\gamma-i\Updelta)\hat a(t)=i{\frac{\bar G_0}{\sqrt{2}\hbar}}\hat x(t)+\sqrt{2\gamma} \hat a_{\rm in}(t). $$

(10.26)

In the limit of a large cavity bandwidth considered in the order-of-magnitude estimate (refer to Sect. 10.3), the time dependence of the cavity mode can be adiabatically eliminated and we have

$$ \hat a(t)\approx i{\frac{\bar G_0}{\sqrt{2}\hbar \gamma}} \hat x(t)+\sqrt{{\frac{2}{\gamma}}} \hat a_{\rm in}(t). $$

(10.27)

Therefore, from Eq. (10.23),

$$ \hat a_{\rm out}(t)=\hat a_{\rm in}(t)+i{\frac{\alpha}{\sqrt{2}\hbar}}\hat x(t). $$

(10.28)

with $\alpha\equiv \sqrt{2} \bar G_0/\sqrt{\gamma}.$ By defining the output amplitude and phase quadratures as

$$ \hat b_1={\frac{\hat a_{\rm out}+\hat a^{\dag}_{\rm out}}{\sqrt{2}}}, \quad \hat b_2={\frac{\hat a_{\rm out}-\hat a^{\dag}_{\rm out}}{i\sqrt{2}}}, $$

(10.29)

we recover what has been shown in Eq. (10.4).

10.1.2 Causal whitening and Wiener Filter

In this section, we will briefly introduce the concepts of causal whitening and Wiener filtering techniques applied in this paper (One can refer to Ref. [27] for more details). They are implemented extensively in the classical signal filtering.

Here, the reason why these classical techniques can be applied lies in following fact: In a linear continuous measurement, the degrees of freedom of the measurement output $\hat y(t)$ at different times commute with each other [2], i.e.,

$$ [\hat y(t), \hat y(t^{\prime})]=0. $$

(10.30)

This basically means that in principle, they can be simultaneously measured with arbitrarily high accuracy without imposing any limit. Therefore, they can be treated just as classical entities, and classical filtering techniques apply.

Causal whitening—Causal whitening is a powerful tool for simplifying the statistic of a random variable (the measurement output in this context), while maintaining its complete information. Mathematically, given the spectrum $S_{yy}(\Upomega)$ of the output $\hat y(t),$ we can factorize it as:

$$ S_{yy}(\Upomega)=\phi_+(\Upomega)\phi_-(\Upomega), $$

(10.31)

such that $\phi_+ (\phi_-)$ and its inverse are analytical functions in the upper- (lower-) half complex plane, and $\phi^*_+=\phi_-.$ The causally-whitened output in the frequency domain is defined as

$$ \hat z(\Upomega)\equiv{\frac{\hat y(\Upomega)}{\phi_+(\Upomega)}}. $$

(10.32)

Since

$$ \langle \hat z(\Upomega)\hat z(\Upomega^{\prime})\rangle=2\pi{\frac{S_{yy}} {\phi_+\phi_-}}\delta(\Upomega-\Upomega^{\prime}) =2\pi\delta(\Upomega-\Upomega^{\prime}), $$

(10.33)

the corresponding correlation function is:

$$ \langle\hat z(t)\hat z(t') \rangle=\delta(t-t^{\prime}), $$

(10.34)

which corresponds to a white noise with no correlations at different times. This not only simplifies the statistics, because $\hat z$ is uniquely defined from $\hat y,$ it also possesses the same amount of information concerning the motion of the mechanical oscillator.

Wiener filter—A Wiener filter is the optimal filter satisfying the least mean-square error criterion. Given a random variable $\hat x$ (here the oscillator position), we can extract a maximal amount of information about $\hat x(0)$ from the measurement data $\hat y(t)$ (from $-\infty$ to $0)$ with the Wiener filter $K_x(t).$ The conditional mean of $\hat x(0)$ is

$$ \hat x^{\rm cond}(0)=\int\nolimits_{-\infty}^0 dt K_x(-t)\hat y(t). $$

(10.35)

The corresponding error $\hat R_x(t)=\hat x(0)-\hat x^{\rm cond}(0)$ defines the remaining uncertainty that we cannot learn from $\hat y(t).$ Mathematically, this dictates that such an error is not correlated with $\hat y$ and is orthogonal to the space defined by the measurement results, namely,

$$ \langle \hat R_x(0)\hat y(t)\rangle=0. $$

(10.36)

Therefore, the decomposition that we applied in this paper and also Ref. [27]:

$$ \hat x(0)=\int\nolimits_{-\infty}^0 dt K_x(-t)\hat y(t)+\hat R_x(0) $$

(10.37)

is very useful in separating the statistical dependence of $\hat x$ on the measurement $\hat y$ and facilitates the analysis of the conditional dynamics.

The Wiener filter can be obtained using the standard Wiener-Hopf method, and its frequency domain representation is:

$$ K_x(\Upomega)={\frac{1}{\phi_+(\Upomega)}} \left[{\frac{S_{xy}(\Upomega)}{\phi_-(\Upomega)}}\right]_+, $$

(10.38)

where $S_{xy}$ is the cross-correlation between $\hat x$ and $\hat y,$ and $[f(\Upomega)]_+$ means taking the component of $f(\Upomega)$ that is analytical in the upper-half complex plane.

10.1.3 State Transfer Fidelity

To quantify the state transfer, we follow Refs. [12, 19] by defining the fidelity from the overlap between the two Wigner functions of the prepared oscillator state $W_{m}(X, P)$ and the target state $W_{\rm tag}(X, P):$

$$ \digamma \equiv 2\pi \int\nolimits_{-\infty}^{\infty}dX \int\nolimits_{-\infty}^{\infty}dP W_{m}(X, P)W_{\rm tag}(X, P). $$

(10.39)

Depending on the situation, X and P can be normalized with respect to either the zero-point uncertainty $x_q$ and $p_q$ or $x_q\sqrt{\omega_m/\Upomega_q},$ and $p_q\sqrt{\Upomega_q/\omega_m}.$

Since the center of the prepared state is given by $x_c$ and $p_c,$ we need to shift it to the center to compare with the target state, and this will not introduce any statistical difference. In addition, the prepared state is a squeezed state defined by ${{\mathbb{V}}}_c.$ To properly evaluate the overlap, we will apply the well-known Bogoliubov transformation to the coordinates of the prepared state:

$$ \hat X^{\prime}=\hat X(\sinh\beta+\cosh\beta\cos2\phi)-\hat P\cosh\beta\sin2\phi, $$

(10.40)

$$ \hat P^{\prime}=\hat P(\sinh\beta+\cosh\beta\cos2\phi)+\hat X\cosh\beta\sin2\phi. $$

(10.41)

By choosing an appropriate set of squeezing factor $\beta$ and rotation angle $\phi,$ the overlap with the target state can be maximized. Therefore, a properly modified definition for the fidelity should be:

$$ \digamma^{\prime} \equiv \max[\digamma, \{\beta, \phi\}]. $$

(10.42)

In the case of the single-photon injection, the Wigner function of the target mechanical state is simply:

$$ W_{\rm tag}(X, P)={\frac{X^2+P^2-1}{2\pi}}\exp\left[-{\frac{X^2+P^2}{2}}\right]. $$

(10.43)