# Utilizing Sequential Control Scheme to Stabilize Squeezed Vacuum States
## 1 Introduction  
Squeezed states are one of the most principal non-classical states of light in the continuous-variable regime, which were first considered to exist in the 1920s by [PERSON], [PERSON], and [PERSON] [1; 2; 3]. In 1981, [PERSON] proposed that squeezed states can be used to surpass the quantum limit in an interferometric gravitational wave detector by injecting squeezed states into the dark port of the interferometer [4]. In addition, the squeezed state can be also used to realize quantum imaging [5; 6] beyond the diffraction limit [7], spectroscopic measurement [8], velocimetry [9], and LIDAR [10] and generate a Schrodinger cat state and an Einstein-Podolsky-Rosen state for continuous-variable quantum information [11; 12; 13; 14; 15]. All these applications require squeezed states with a high-level and stable squeezing strength. Squeezed states can be generated by using an optical parametric oscillator (OPO) based on second-order nonlinearity [16], four-wave mixing based on third-order nonlinearity [17], and other schemes [18]. Using an OPO is one of the most successful methods of squeezed state generation, which also holds the record of the squeezing level [19]. The first experimental demonstration of a squeezed state based on the OPO was performed in 1986 [16]. In ideal conditions, an infinite squeezing factor can be generated and detected at the threshold. However, the non-ideal components introduce an inevitable loss during the generation, propagation, and detection of the squeezed states, and the actual servo-loops have inherent noise and drift that result in phase fluctuation during the cavity- and phase-locking [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. A monolithic cavity with the least number of optical surfaces can effectively reduce the intracavity losses [31; 32; 33; 34]. The primary advantage of a doubly-resonant OPO cavity over a single-resonant OPO cavity is the simplicity of obtaining a cavity error signal by usingthe pump field. That means the pump field amplitude is resonantly enhanced and allows increasing the transmission of the output coupler for increasing the escape efficiency. The doubly-resonant OPO cavity has other advantages including lower pump power, perfect mode matching, and so on. However, the doubly-resonant OPO cavity needs more stringent temperature stability requirements for simultaneously realizing co-resonance and best phase matching. The photothermal effect associated with absorption of the pump field in a doubly-resonant OPO cavity may be more serious than a single-resonant OPO cavity. A semi-monolithic OPO cavity has an external piezo-driven coupling mirror that can be used to impose the feedback control signal. This is a more popular scheme [19; 24; 25; 35; 36], and it has been applied to squeezed states' generation with a high-level squeezing factor [19]. The error signal for OPO locking can be extracted from the pump field in the case of a double-resonant cavity [36]. A single-resonant cavity has a single- or double-pass configuration for the pump field, which makes it unsuitable for the locking scheme to extract the error signal from the pump field. During the generation of bright squeezed light, the seed beam injected into the OPO makes the control of the cavity length and relative phase simple [25]. In this case, a seed beam injected into the optical parametric amplifier (OPA) dramatically degenerates the squeezing factor due to noise coupling between the pump and seed fields, even if both the pump and seed fields have no classical noise [37]. As the analysis frequency decreases, the limit factor becomes significant due to the increase of laser intensity noise, until squeezing can no longer be achieved [30]. The generation of squeezed vacuum states eliminates noise coupling between the pump and seed beam, which enables the production of a high-level squeezing factor that is immune to laser noise. However, there is no coherent seed beam injected into the OPO, which serves as a locking beam.