Continuous variable quantum optical simulation for time evolution of quantum harmonic oscillators

Quantum simulation enables one to mimic the evolution of other quantum systems using a controllable quantum system. Quantum harmonic oscillator (QHO) is one of the most important model systems in quantum physics. To observe the transient dynamics of a QHO with high oscillation frequency directly is difficult. We experimentally simulate the transient behaviors of QHO in an open system during time evolution with an optical mode and a logical operation system of continuous variable quantum computation. The time evolution of an atomic ensemble in the collective spontaneous emission is analytically simulated by mapping the atomic ensemble onto a QHO. The measured fidelity, which is used for quantifying the quality of the simulation, is higher than its classical limit. The presented simulation scheme provides a new tool for studying the dynamic behaviors of QHO.

The Wigner function of a qu-mode is where x = (X, P ) T andx = ( ⟨X⟩ , ⟨P ⟩ ) T represent the vector and the mean value (displacement) of the amplitude and phase quadratures respectively, and det σ is the determinant of covariance matrix σ. The covariance matrix of the input state is where V sq , V an are the variances of the squeezing and anti-squeezing component, respectively.
The covariance matrix of the output state after the rotation operation in an open system (T < 1) is expressed by where and in which R = 99% is the reflection coefficient of the 99%R beam-splitter. When V sq = V an = 1, the covariance matrix of the output state is reduced to which corresponds to the case that a coherent state is used as input state.
The average amplitudes of the input state are given by  In the experiment, the average amplitudes of the input state are ⟨X⟩ in = 1.58 and ⟨P ⟩ in = 0, where φ 0 = 0 has been chosen in Eq. (10) for simplicity. The average amplitudes of the

II. TIME EVOLUTION OF AN ATOMIC ENSEMBLE
A spin coherent (squeezed) state of an atomic ensemble is mapped onto a coherent (squeezed) state of a QHO (qu-mode), respectively. The initial spin coherent state α √ N ⟩ a (|α| 2 ≪ N ) of an atomic ensemble can be mapped onto a coherent state |α⟩ of a qu-mode harmonic oscillator, where N 2 , n a − N 2 ⟩ a is the eigenvector ofĴ 2 andĴ z with eigenvalue N 2 ( N 2 + 1) and m = n a − N/2, respectively, and where |n⟩ is the eigenvector ofâ †â with eigenvalue n and α = |α| e iφ 0 . Substituting Eq. (2) with V sq = V an = 1 into Eq. (1), the corresponding Wigner function of the initial spin The initial spin squeezed state β of an atomic ensemble can be mapped onto a coherent squeezed stateŜ(ζ)D(β) |0⟩ of a qu-mode harmonic oscillator, and When V sq = 1 Van , it stands for a pure squeezed state. If V sq = V an = 1, Eq. (16) is reduced to Eq. (14). Substituting Eq. (2) into Eq. (1), we obtain the corresponding Wigner function of the initial spin squeezed state, which takes the same form of Eq. (16) with V sq = 0.45, V an = 7.08 for the squeezed state in the experiment (corresponding to measured squeezing and anti-squeezing noises of the quadrature-amplitude squeezed state are −3.5 dB and 8.5 dB).
According to the average amplitudes of the output state in Eq. (11) and the relationship between the collective spin operators and the position (momentum) operators of a harmonic oscillator, the mean values of the collective spin operators equal to The total upper-state population of the atomic ensemble N + equals to Obviously, in the case of the perfect EPR entangled state (r e −→ ∞), the decay rate of the total upper-state population N + is 2κ, which is proportional to the total number N of atoms in the atomic ensemble because of g k = g a k √ N .

III. THE COVARIANCE MATRIX IN THE EXPRESSION OF FIDELITY
The covariance matrix σ 2 of the output mode is given by σ 2 = 4σ out . The coefficient "4" comes from the normalization of shot noise level (SNL). Since the noise of a vacuum state is defined as 1/4, while in the fidelity formula the vacuum noise is normalized to "1", so a coefficient "4" appears in the expressions of covariance matrices. For the theoretically calculated final stateρ 1 the covariance matrix is where V X 1 , V P 1 and V XP 1 correspond to V X , V P and V XP in Eq. The covariance matrices are obtained from the noise power spectrum of the output state at 2 MHz, which is measured by HD3 (shown in Fig. 1(b) in the main text) and analyzed with a spectrum analyzer. The measured noise powers of the output states with a coherent state and an amplitude-squeezed state as the input state at different rotation angles are shown in Fig. 1 and Fig. 2, respectively. During the measurement, the modulation signal