# How can I write a Gaussian state as a squeezed, displaced thermal state?

I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\dagger(\alpha) \hat{S}^\dagger(\zeta) . \end{gather} Here, \begin{gather} \rho_{\bar{n}} = \int_{\mathbb{C}} P_{\bar{n}}(\alpha) |\alpha\rangle\langle\alpha|d\alpha \text{ with } P_{\bar{n}}(\alpha) = \frac{1}{\pi \bar{n}} e^{- |\alpha|^2 / \bar{n}} \end{gather} is a thermal state with occupation $\bar{n}$, \begin{gather} \hat{S}(\zeta) = e^{(\zeta^* \hat{a}^2 + \zeta \hat{a}^{\dagger 2}) / 2} \end{gather} is the squeezing operator, and \begin{gather} \hat{D}(\alpha) = e^{\alpha^* \hat{a} - \alpha \hat{a}^\dagger} . \end{gather} is the displacement operator. I prefer to use the convention $\hat{x} = (\hat{a} + \hat{a}^\dagger) / \sqrt{2}$ and $\hat{p} = (\hat{a} - \hat{a}^\dagger) / \sqrt{2} i$.

I assume that the way to accomplish this is to derive the mean and variance of our Gaussian state $\rho$ and thereby determine $\zeta$ and $\alpha$. However, I have been unsuccessful in doing so. That is, given the mean and variance of our Gaussian state $\rho$, what are $\zeta$ and $\alpha$?

On a side note, I was also wondering if there is a standard result for the commutator of $\hat{S}(\zeta)$ and $\hat{D}(\alpha)$?

• There is a result (without demonstration) in formula $(11)$ (and following lines) of this paper. A pseudo-commutation relation for $D$ and $S$ is given in formula $(15)$ of this paper – Trimok Sep 11 '14 at 10:36
• Probably you could demonstrate the result thanks to the action of $D$ and $S$ on $a, a^+$ (see pages $15$ and $28$ of this presentation), and the expression of the thermal density matrix in the Fock basis (see $(3.87)$ in this ref) – Trimok Sep 11 '14 at 10:57
• @Trimok Without your reference (which does not really answer the problem) I was wondering why this result should even be true ? Is there a way to map the Gaussian to any other states or what ? – FraSchelle Sep 17 '14 at 5:29
• @FraSchelle : If you inverse the order of $S$ and $D$ (relatively to the OP), the result in my first ref is correct (I have checked the mean, but the variance should be correct too), and it gives a formula between mean, variance, $\zeta$ , $\alpha$, and $\bar n$ (which was the OP question). – Trimok Sep 17 '14 at 9:24
• @FraSchelle : Now, in my last ref, a "gaussian" state is diagonal in the coherent basis ($3.86$), and is also diagonal in the Fock basis ($3.87$), but it does not appear "gaussian" in the Fock basis. – Trimok Sep 17 '14 at 9:24

## 1 Answer

I will follow the notes by A. Ferraro et al.

A state $$\rho$$ of a system with $$n$$ degrees of freedom is said to be Gaussian if its Wigner function can be written as $$W[\rho](\boldsymbol{\alpha}) = \frac{\exp\left( -\frac{1}{2}(\boldsymbol{\alpha} - \bar{\boldsymbol{\alpha}})^T \boldsymbol{\sigma}^{-1}_\alpha (\boldsymbol{\alpha} - \bar{\boldsymbol{\alpha}}) \right) }{(2\pi)^n\sqrt{\text{Det}[\boldsymbol{\sigma}_\alpha ]}},$$ where $$\boldsymbol{\alpha}$$ and $$\bar{\boldsymbol{\alpha}}$$ are vectors containing all the $$2n$$ quadratures of the system and their average values, respectively, and $$\boldsymbol{\alpha}$$ is the covariance matrix, whose elements are defined as

$$[\boldsymbol{\sigma}]_{kl} := \frac{1}{2} \langle \{R_k,R_l \} \rangle - \langle R_k\rangle \langle R_l\rangle,$$ where $$\{\cdot,\cdot\}$$ is the anticommutator, and $$R_k$$ is the $$k-th$$ element of the vector $$\boldsymbol{R}= (q_1,p_1,\ldots,q_n,p_n)^T$$ with the $$q$$s and $$p$$s being the position and momentum-like operators.

A very important result is that it turns out that Gaussian states can be fully characterised by their covariace matrix plus the vector of expectation values of the quadratures, $$\bar{\boldsymbol{\alpha}}$$. If your system only has one mode (one boson), then you only need a symmetric $$2\times2$$ matrix and two real numbers ($$q$$ and $$p$$) to describe it! This means a total of five parameters.

As you point out, we can write any Gaussian state as

$$\rho = D(\bar{\alpha})S(\xi)\rho_{th}S^\dagger(\xi)D^\dagger(\bar{\alpha})$$ where here $$\bar{\alpha}\equiv\frac{1}{\sqrt{2}}(\bar{x}+i\bar{p})$$ and $$\xi = r e^{i\varphi}$$. If your thermal state has mean photon number $$N$$, then it suffices to know $$N$$, $$r$$ and $$\varphi$$ to compute the covariance matrix. Its elements are given by

$$\sigma_{11} =\frac{2N+1}{2} \left(\cosh (2r)+\sinh (2r)\cos (\varphi)\right)$$ $$\sigma_{22} =\frac{2N+1}{2}\left( \cosh(2r)-\sinh(2r)\cos(\varphi) \right)$$ $$\sigma_{12} =\sigma_{21}=-\frac{2N+1}{2}\sinh(2r)\sin(\varphi).$$

You can see that the main properties of the state are captured by the covariance matrix, because the displacement $$\bar{\alpha}$$ can always be disregarded by local operations (it is a phase-space translation). In other words, you can always put it to zero.

Answering your question, note that a Gaussian state is not simply a Gaussian distribution. You need more parameters than simply the variance and the mean (as you would to define a classical Gaussian probability distribution). These are in general five real values, but the essential ones are the ones entering the covariance matrix, as explained before.

As for the commutator, I am not aware of any closed formula. But I do know that displacing and then squeezing produces a state which has the same squeezing as the zqueezed-displaced state, but a different displacement.