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Abstract
In this paper, we report on −3.5±0.2 dB vacuum squeezing (corresponding to −4.2±0.2 dB with loss correction) at 795 nm via the polarization self-rotation (PSR) effect in rubidium vapor by applying a magnetic field, whose direction is perpendicular to the propagation and polarization of the pump light. Compared with the case without the magnetic field, whose optimal squeezing degree is about −1.5 dB, this weak magnetic field can enhance the PSR effect and ultimately increase the squeezing degree. This compact squeezed light source can be potentially utilized in quantum protocols in which atomic ensembles are involved, such as in quantum memory, atomic magnetometers and quantum interferometers.
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1. Introduction
Surpassing the standard quantum limit (SQL) in a measurement is a long-term goal in quantum metrology and quantum communication. Squeezed states of light, whose quantum-field-quadrature noise is phase dependent with a minimum below SQL, can be used to achieve this goal. [1–4] . So considerable attention has been attracted to employ the squeezed states of light to enhance the sensitivity of the measurement [5–11]. A recent impressive application of this light is the gravitational wave detection [12–14].
There are many techniques for producing squeezed states of light, such as the four-wave mixing (FWM) process [15], an optical parametric oscillator (OPO) [1] and the polarization self-rotation (PSR) effect [8,9,16]. Compared with FWM and OPO, PSR can be used to generate a simple and compacted squeezed vacuum light with a continuous-wave (CW) diode laser and an atomic rubidium vapor cell. In addition, as the squeezed light is generated in an atomic ensemble, this nonclassical light can be directly applied in quantum protocols in which atomic ensembles are involved, such as in quantum memory [17], atomic magnetometers [4,18,19] and quantum interferometers [20,21]. However, until today, the best squeezing degree of such light was reported to be −3 dB [22] under readily achievable experimental conditions. In order to make this simple, compact and cost-effective squeezed light source available to a variety of experiments and applications, further improvements need to be studied.
In this paper, we report on −3.5$\pm$0.2 dB vacuum squeezing (corresponding to −4.2$\pm$0.2 dB with loss correction) at 795 nm via PSR in $\mathrm {^{87}Rb }$ atomic vapor cell. Different from previous vacuum squeezing via PSR [8–11,16,22,23] , a weak magnetic field perpendicular to the propagation and polarization directions of the input pump light is applied on the $\mathrm {^{87}Rb }$ atomic vapor cell to enhance the PSR and finally increase the squeezing degree. Furthermore, we experimentally measure the squeezing degree under different experimental conditions, including various directions and intensities of the weak magnetic field, detuning frequencies and pump powers. As a result, the optimal experimental conditions are achieved. Moreover, the optimal squeezing degree can be improved from −1.5 dB without a magnetic field to −3.5$\pm$0.2 dB just by applying a magnetic field. Our results are useful for improving the measurement sensitivity for highly sensitive practical applications.
2. Theoretical analysis
A schematic diagram of PSR is shown in Fig. 1. A y-polarized strong pump light field with amplitude $\varepsilon _{0}$ propagates along the z-axis direction. Its frequency $\omega$ is near resonant with the $^{87}$Rb D1 transition from $\left \vert 5^{2}S_{1/2},F=2\right \rangle$ to $\left \vert 5^{2}P_{1/2},F^{^{\prime }}=2\right \rangle$. When the pump field propagates through atomic vapor, its polarization ellipse rotates by an angle $\varphi =g\epsilon (0)l$ with the initial ellipticity $\epsilon (0)\ll 1$. Here, $g$ is the self-rotation (SR) parameter and $l$ is the length of the atomic vapor. This is the PSR effect. Due to this effect, a x-polarized signal is generated [16] . A polarization beam splitter (PBS) is used to separate the strong pump light and the signal. Under homodyne detection, the quantum-field-quadrature noise [16] of the signal $Q$ is given as:
where $\varepsilon _{0}^{2}$ is the standard quantum limit (SQL) associated with the vacuum states. $\chi$ is the local oscillator phase. By choosing a particular $\chi$, $Var(Q)$ can be smaller than the vacuum fluctuation $\varepsilon _{0}^{2}$. As a result, a x-polarized squeezed vacuum state of light is generated. Based on Eq. (1), the squeezing degree depends on the atomic cell length $l$ and the SR parameter $g$. For a fixed $l$ and some certain $\chi$, the squeezing degree can be enhanced by increasing $g$.
Fig. 1. Schematic diagram of PSR. The pump field with linear y-polarization transmits the $^{87}$Rb atoms along the z-axis. The generated signal field passes through a PBS along the z-direction. PBS: polarization beam splitter. $\left \vert g\right \rangle$: $\left \vert 5^{2}S_{1/2},F=2\right \rangle$; $\left \vert e\right \rangle$: $\left \vert 5^{2}P_{1/2},F^{\prime }=2\right \rangle$. $m$: Zeeman level of $\left \vert g\right \rangle$; $m^{\prime }$: Zeeman level of $\left \vert m\right \rangle$. $\omega$: frequency of the pump field.
Normally, for linearly polarized input light, we have $\epsilon (0)\ = 0$. Thus, the coupling coefficient $g$ can be given as:
with Here, $P_{q}$ is the quadrature component of the atomic polarization $P=n\mathbf {Tr}\rho \boldsymbol {d}$ (where $n$ is the atomic density, $\mathbf {Tr}\rho$ is the trace of the density matrix of the atomic ensemble and $\boldsymbol {d}$ is the dipole operator) induced by the magnetic field and the optical field. Usually, the density matrix in the laboratory frame is given in terms of the rotating-frame density-matrix elements $\widetilde {\rho }$. As an example, when the pump field is on resonance with the $F=2\rightarrow F^{^{\prime }}=2$ transition, we have:Actually, in order to solve the Liouville equation in the presence of a magnetic field $\boldsymbol {B}$ along the x-axis direction in atomic vapor, the total Hamiltonian $\hat {H}$ of this system is employed in the Liouville equation [18,24,25] for $\widetilde { \rho }_{g,m}$, which is as follows:
Here, we assume that the energy of the lower state is zero. $\hat {H}_{0}$ is the unperturbed Hamiltonian with the reduced Planck constant $\hbar$. $\hat {H }_{l}$ is the light-atom-interaction Hamiltonian with the y-polarized optical electric field $\boldsymbol {\varepsilon }$ and the dipole operator $\boldsymbol {d}$. $\hat {H}_{B}$ is the magnetic-field-atom-interaction Hamiltonian with the total magnetic moment of the atom $\boldsymbol {\mu }$. It can be seen that the coupling coefficient $g$ is connected with magnetic field $\boldsymbol {B}$ through $\widetilde { \rho }_{g,m}$. As the Larmor frequency of the magnetic field is $\Omega _{L}^{0}=g_{F}\mu _{0}B/\hbar$ with Land$\acute {e}$ factor $g_{F}$ and magnetic permeability $\mu _{0}$, we can finally obtain the dependence of the coupling coefficient $g$ on the Larmor frequency $\Omega _{L}^{0}$, as shown in Fig. 2. It can be seen that a small magnetic field $B$ or Larmor frequency $\Omega _{L}^{0}$ can increase the SR parameter $g$. Therefore, in principle, a magnetic field in the x-axis direction can enhance squeezing. In the next section, we describe the experimental demonstration.Fig. 2. Numerical solutions for the coupling-coefficient $g$ change with the Larmor frequency of the magnetic field $\Omega ^{0}_L$, when the pump frequency is resonant with the transition $\vert 5^{2}S_{1/2},F=2\rangle$ to $\vert 5^{2}P_{1/2},F^{^{\prime }}=2\rangle$.
3. Experimental setup
The experimental setup is presented in Fig. 3. The $^{87}$Rb atomic vapor cell without buffer gas and paraffin (75 mm in length; 25 mm in diameter) and the transmittance of the atomic cell is 90% (double-side), which is placed in a 4-layer magnetic shielding and heated to $\mathrm {74 ^{o}}$C (the atomic density is$\sim$ $10^{10}$). The output of an external cavity diode laser (Toptica DL pro), whose total optical power is 100mW and the frequency is near resonant with the atomic transition from $\left \vert 5^{2}S_{1/2},F=2\right \rangle$ to $\left \vert 5^{2}P_{1/2},F^{^{\prime }}=2\right \rangle$, is coupled into a polarization maintaining single-mode optical fiber (SMF), and then passes through a Glan polarizer (GL) to ensure a high-quality vertical polarization (y-polarization in atomic vapor cell) and single-mode spatial distribution for the pump beam. This collimated pump beam with a waist of 2 mm enters into the atom vapor cell to generate squeezed vacuum light via the PSR effect. The half-wave plate (HWP1) and the quarter-wave plate (QWP1) are used to eliminate the Faraday effect on the pump and signal lights caused by the atoms located in the state $\left \vert 5^{2}S_{1/2},F=1\right \rangle$. Finally, a x-polarized squeezed vacuum light is separated from the strong pump field after the polarization beam splitter (PBS1).
Subsequently, a balanced homodyne detection technique is built after PBS1 to measure the quantum-field-quadrature noise. The original pump light serves as a local oscillator, whose polarization is rotated by HWP2 and then spatially overlaps with the squeezed light at PBS2. The resulting Mach-Zehnder interferometer is carefully aligned for a good visibility. In our experiment, the visibility is 99%. A piezoelectric transducer (PZT) is used to scan the local oscillator phase and thus the fluctuation of the field quadrature can be measured. The outputs of PBS3 are measured by a balanced photodetector (Thorlabs, PDB450A) with a gain of $10^4$ and a bandwidth of 50 MHz. Here the photodiodes are Hamamatsu S3883 with the quantum efficiency are 95%. The phase-dependent quadrature fluctuation is finally measured with a spectrum analyzer operated in the zero-span mode at a set of the video bandwidth (VBW) and the resolution bandwidth (RBW) are 30Hz and 3KHz, respectively. The electronic noise is 8 dB below the SQL.
Fig. 3. Experiment setup. OI: optical isolator; SMF: single mode fiber; PBS: polarization beam splitter; GL: Glan polarizer; HWP: half-wave plate; QWP: quarter-wave plate; PD: balanced photodetector; PZT: piezoelectric transducer. B: The additional magnetic field is along the x-axis direction.
In our experiment, different from other experimental setup to shield almost all the magnetic fields around the atomic cell, we apply the weak DC magnetic fields with different directions inside the magnetic shield to analyze the performance of the squeezed light. Here, we use a solenoid to generate a magnetic field along the z-axis direction and use a Helmholtz coil to generate a magnetic field along the x-axis direction. The variance of our magnetic field is smaller than one percent.
4. Results and discussion
Based on the experimental setup shown in Fig. 3, the phase-dependent noise spectra of the x-polarized signal are shown in Fig. 4. It is compared to the SQL measured by blocking the x-polarized signal. In Fig. 4(a), there is no external magnetic field. It can be seen that the minima of noise spectra fall below the SQL to exhibit a squeezing of 0.8 dB. In contrast, when a 100 mG magnetic field is applied along the x-axis direction on the atomic vapor cell, the squeezing is about 3 dB. Thus, a magnetic field in the x-axis direction can enhance the squeezing and the experimental results are consistent with our previous theoretical expectations. Below, we pay more attention to obtaining the optimum squeezing. The experimental conditions, including the direction and intensity of the weak magnetic field, the detuning frequency and the pump power, are analyzed.
Fig. 4. Noise spectra without a magnetic field (a) and with a 100 mG magnetic field along the x-axis (b). SQL is set to zero. The pump power is 6 mW, and the pump frequency is resonant with the transition $\vert 5^{2}S_{1/2},F=2\rangle$ to $\vert 5^{2}P_{1/2},F^{^{\prime }}=2\rangle$.
First of all, we demonstrate the effect of the magnetic field direction on squeezing. When a magnetic field in the z-axis direction is applied on the atomic vapor cell, the squeezing degree decreases with the strength of the magnetic field as shown in Fig. 5(a), which agrees with the observation in Ref. [18]. In contrast, when the magnetic field in the x-axis direction is applied, the squeezing degree increases with the strength of magnetic field, and then reaches and maintains a maximum value. This is because that the magnetic field is strong enough to quantize most atomic spins along the x-axis direction. And thus, the squeezing degree can be improved by applying a magnetic field along the x-axis direction, which matches the theoretical analysis.
Fig. 5. Minimum noise of light quadrature normalized to SQL as a function of the strength of the magnetic field applied in the z-(a) or x- (b) direction. The power of the pump light is 6 mW, and the frequency is resonant with the transition from $\vert 5^{2}S_{1/2},F=2\rangle$ to $\vert 5^{2}P_{1/2},F^{^{\prime }}=2\rangle$.
Secondly, we study the dependence of the squeezing degree on the frequency and the power of the pump laser when a 100 mG magnetic field along the x-axis direction is applied. The results are shown in Figs. 6(a) and (b). The largest squeezing occurs when the frequency of the pump laser is around the $F$ = 2$\to F^{^{\prime }}$=2 transition. Meanwhile, the squeezing degree increases with the power of the pump light until reaching a maximum level as a balance between the optical pumping and the AC Stark shift [19,23].
Fig. 6. Minimum noise of light quadrature normalized to the shot-noise level as a function of the detuning frequency (a) and the pump power (b). In (a), the pump power is 6 mW, and a 100 mG magnetic field along the x-axis is applied. The black line is the saturated absorption spectra of the $^{87}$Rb atom. In (b), the pump frequency is resonant with the $^{87}$ Rb D1 transition from $\vert 5^{2}S_{1/2},F=2\rangle$ to $\vert 5^{2}P_{1/2},F^{^{\prime }}=2\rangle$. A 100 mG magnetic field is applied along x-axis.
Finally, the maximum squeezing degree in our experiment is −3.5 dB at 2 MHz by applying a 150 mG the magnetic field along the x-axis direction on the atomic vapor and a 6 mW pump field resonant with the $F$ = 2$\to F^{^{\prime }}$=2 transition. The loss of the signal light between the cell and the detector is 20$\%$. The quantum efficiency of the detector is 95$\%$ and the visibility between the squeezing vacuum light and the local oscillator is 99$\%$. The squeezing degree reaches −4.2$\pm$0.2dB after considering these effects.
5. Conclusion
In conclusion, a compact squeezed light source at 795 nm based on PSR in a hot $\mathrm {^{87}Rb }$ vapor cell is reported. Compared with other reported experimental implementations, we applied a weak magnetic field on the atomic vapor. Our experimental results show that the squeezing degree of the signal field can be improved when the direction of the applied magnetic field is both perpendicular to the propagation and polarization directions of the pump laser. By optimizing the frequency and the power of the pump field, −3.5$\pm$0.2dB vacuum squeezing (corresponding to −4.2$\pm$0.2 dB with loss correction) below the SQL is observed. Such a squeezed light source can be directly applied in quantum protocols in which atomic ensembles are involved, such as in a quantum repeater, an atomic magnetometers and a quantum interferometers.
Funding
National Natural Science Foundation of China (11874152, 11904227, 12104161, 91536114); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01); Shanghai Municipal Education Commission (202101070008E00099); Fundamental Research Funds for the Central Universities; Science and Technology Commission of Shanghai Municipality (19YF1414300, 19YF1421800); China Postdoctoral Science Foundation (2020TQ0193).
Acknowledgment
We thank Xiaotian Feng from East China Normal university, Heng Shen from Shanxi University, Weiping Zhang from Shanghai Jiao Tong University for the helpful discussion.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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