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Abstract
The measurement of intense E-field is a fundamental need in various research areas. An electro-optic (EO) sensor based on common path interferometer (CPI) is widely used due to its better temperature stability and controllability of optical bias. However, the small EO coefficient leads to poor sensitivity. In this paper, a quantum enhanced EO sensor is proposed by replacing the vacuum state in classical one with a squeezed-vacuum state. Theoretical analysis shows that the performance of the quantum enhanced EO sensor, including signal to noise ratio (SNR) and sensitivity, can always beat the classical one due to the noise suppression caused by the squeezed-vacuum state. Experimental results demonstrate that, there is still a 1.12dB quantum enhancement compared with the classical one when the degree of the squeezed-vacuum is 1.60dB. More importantly, except the increase of the EO coefficient or the optical power, the performance of the EO sensor can also be enhanced via quantum light source. Such a quantum enhanced EO sensor could be practically applied for the measurement of intense E-field.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The E-field is a fundamental physical parameter for various scientific research, such as high-voltage engineering [1], terahertz generation [2], high-power electromagnetic pulses detection [3,4], and other applications [5,6]. The measurement of E-field plays an important role in various fields of science and technology. In recent years, optical sensors based on the electro-optic (EO) effect have been widely studied for intense E-field detection. Normally, an external electric field induces the refractive index change of the EO materials, such as the potassium dihydrogen phosphate (KDP) crystal [7]. Such change causes the birefringence or optical path difference, leading to the phase fluctuations when the light passes through EO modulation device. Based on the working principle, interferometers can be used to achieve a high sensitive measurement for the electric field.
Up to now, three types of interferometers [8], including the Mach-Zehnder interferometer (MZI), the coupler interferometer (CI), and the common path interferometer (CPI), are applied in EO sensor. For the MZI or CI types, they both have high sensitivity while there are still some problems need to be solved. For MZI, it is complex to control the optical bias determined by the intrinsic optical path difference of the two arms [9,10]. For CI type, it has a natural quadrature optical bias but the temperature stability remains a challenge [8]. Compared to the types mentioned above, the CPI type employs two polarization components as the arms of the interferometer, which has the advantages of small dimension, notably large E-field, and high stability on both temperature [11] and optical bias [9]. However, it has poor sensitivity due to the lower EO coefficient [8]. Thus how to obtain high precision is always a central problem to be solved in CPI [12]. Normally, the sensitivity can be improved via the increase of the optical power or EO coefficient. However, the high power decreases the responsivity of the EO crystal and even damage it [13,14]. Therefore, it is essential to develop new way to further enhance the performance of CPI without increasing optical power and EO coefficient.
Fortunately, with the development of the quantum metrology technology, the squeezed state of light, as an important resource in the field of quantum optics, has emerged [15,16]. It is an optical state in which the fluctuation of one quadrature is suppressed below the shot-noise limit, which can be generated by numerous methods based on a variety of nonlinear materials, such as the atom-based sources based on the polarization self-rotation effect [17,18]. It is proved that the squeezed states of light can improve the performance of classical optical sensors limited by quantum noise [19,20]. This is because that the squeezed states of light can suppress the quantum noise to beat standard quantum limit (SQL).
In this paper, we report on a quantum enhanced EO sensor based on CPI for E-field measurement. A squeezed-vacuum state instead of the vacuum is employed in the CPI to enhance the performance of the sensor without increasing the EO coefficient and the optical power. The potential benefits of this sensor are as follows: (i) its performance is beyond SQL; (ii) its structure maintains steady-state stability; (iii) its apparatus is simple and small in size. Such an EO sensor has significantly practical value in E-field measurement.
The paper is organized as follows: in Sec. 2, the working principle of the quantum enhanced EO sensor with loss is described. In Sec. 3, the performance of the quantum enhanced EO sensor is analyzed theoretically. In Sec. 4, the performance of the quantum enhanced EO sensor is demonstrated experimentally. In Sec. 5, a summary of our results is concluded.
2. Theory and principle
Fig.1 shows a schematic diagram of the quantum enhanced EO sensor based on CPI. An electro-optic modulator (EOM) made by KDP crystal can sense the phase difference induced by external E-field. The two arms of the CPI are two polarization modes, which are y-polarization direction and x-polarization direction respectively. The operators of y-direction and x-direction are defined separately $\hat {a}_{i}$ and $\hat {S}_{i}$, where $i=1,2,3,4$.
A vertically polarized coherent light $\hat {a}_{0}$ propagates along the y-direction of the EOM, while a horizontally polarized squeezed-vacuum light $\hat {S}_{0}$ propagates along the x-direction. After two beams of light pass through a polarized beam splitter (PBS1) and a half-wave plate (HWP1), we have
Fig. 1. A schematic diagram of the quantum enhanced EO sensor based on CPI. $\hat {a}_{0}$ and $\hat {S}_{0}$ are coherent state and squeezed-vacuum state respectively. $\hat {a}_{i}$ and $\hat {S}_{i}$ $(i=1,2,3,4)$ denote two beams of light respectively in the different processes. PBS1 and PBS2: polarized beam splitter. HWP1 and HWP2: half-wave plate. EOM: electro-optic modulator. QWP: quarter-wave plate. A fictitious beam splitter (BS) is employed to introduce the loss in the sensor through two vacuum state $\hat {V}_{a}$ and $\hat {V}_{S}$.
To describe the influence of optical loss, a fictitious beam splitter (BS) with transmission coefficient $t$ is introduced in this sensor. We then have
Then, the lights pass through the second half-wave plate (HWP2) and the second polarized beam splitter (PBS2). We find that
Finally, a balanced detector is employed to obtain a photon-current $i$, which is
Therefore, the working principle of the quantum enhanced EO sensor is introduced. The discussion in the next section is focused on the performance analysis, such as SNR and sensitivity.
3. Performance analysis
The photon-current $i$ from the detector is measured with a spectrum analyser. We can measure the power spectrum $P$ of the photon-current $i$ to analyze the performance of the quantum enhanced EO sensor. The SNR of the sensor at arbitrary frequency $\omega$ is given by
Here $\log$ is logarithm base 10. $P_{S}$ and $P_{N}$ are the power spectrum of the signal and noise, respectively. The units used here are $W$. $i_{S}$ and $i_{N}$ are the photon-currents of the signal and noise, respectively. When $SNR=0dB$, we can obtain the sensitivity of the sensor.The coherent light $\hat {a}_{0}$ satisfies $\hat {a}_{0}|\alpha \rangle =\alpha |\alpha \rangle$ with $\alpha =\left \vert \alpha \right \vert$, while the single-mode quadrature squeezed vacuum states $\hat {S}_{0}(\xi )=\exp [\frac {1}{2}\xi ^{\ast }\hat {a}^{2}-\xi \hat {a}^{\dagger 2}]$ satisfies $|\xi \rangle =\hat {S}_{0}(\xi )|0\rangle $ with $\xi =re^{i\theta }$. Here $r$ is the degree of squeezing and $\theta$ is the orientation of the squeezing axis. The two lights go into the sensor. If $\theta =\pi$, we can obtain
Normally, an arbitrary signal can be expressed by the combination of sine and cosine waves. Thus, we take a sine wave $U=U_{s}\sin (\omega _{s}t)$ as an example to simulate the external E-field. As a result, a small phase shift $\delta =\mu U_{s}$ at frequency $\omega _{s}$ can be observed in power sperctrum. Under condition $\varphi _{0}=\frac {\pi }{2}$, $\delta \ll 1$ and $\sinh ^{2}r\ll \left \vert \alpha \right \vert ^{2}$, we have
Based on Eq. (10), Eq. (14) and Eq. (15), we can have the SNR of the EO sensor, which is(1) when $r=0$, a coherent light and a vacuum light are injected into the sensor. The SNR can be written as
In particular, when $t=1$, it represents a classical EO sensor in an ideal situation, which is (2) when $r\neq 0$, a coherent and a squeezed vacuum state are together injected into the senor. In particular, when $t=1$, it represents the quantum enhanced EO sensor in an ideal situation, which is Based on Eq. (17) and Eq. (18), it can be seen that no matter $r$ is $0$ or not, SNR is increased with $\left \vert \alpha \right \vert ^{2}$ and $U$. Comparing Eq. (19) and Eq. (20), when the sensor is in an ideal situation, it can be seen that $SNR_{22}>SNR_{11}$ if $r>0$which means that the squeezed light can enhance the SNR of the EO sensor with the same $\left \vert \alpha \right \vert ^{2}$ and $U$. For example, when $r=0.18$, a -1.60dB squeezed-vacuum state can cause 1.60dB SNR enhancement. More importantly, the higher $r$ is, the better SNR is. The SNR of the quantum enhanced EO sensor can always beat the classical one, even if the incident optical power and the amplitude of the voltage are changed.However, in practice, the optical loss can not be ignored. When $r=0.18$, the relations between SNR and transmission coefficient are shown in Fig. 2. The quantum enhanced EO sensor and the classical one are in red dot line and black square line respectively. Notice that loss can affect the performance of the sensor. When $t<1$, the enhancement is reduced. The higher the loss is, the worse quantum enhancement is achieved.
Fig. 2. Theoretical SNR comparison between the quantum enhanced EO sensor and the classical one when $r=0.18$. $t=1$ represents the EO sensors in an ideal situation without loss, marked by the stars. The SNR is quantum enhanced 1.60dB.
Furthermore, based on Eq. (17), we can get the voltage sensitivity of the proposed sensor when $SNR=0dB$, which is
In particular, for the classical EO sensor, $r=0$, the sensitivity is Compared with the classical one, the proposed EO sensor has better sensitivity when $r>0$.So far, our theoretical calculations show that the quantum enhanced EO sensor allows better performance. Below an experimental setup is carried to demonstrate the performance of the proposed sensor.
4. Experiment and discussion
The detailed experimental layout is shown in Fig. 3. A Pyrex cell (75mm in length, 25 mm in diameter), which is enclosed in the 4-layer magnetic shields and is heated up to 62.8$^{\circ }C$, is full of $^{87}$Rb atoms. We use a continuous-wave laser (DL Pro 100) as the pumping light source, resonant on $^{87}Rb$ D1 line and coupled into a polarization maintaining single-mode fiber (PMSF) via a single-mode coupling frame (SMCF). A Glan-Laser Calcite Polarizer (GL) is then employed to make sure high quality of linear y-polarization. Then the y-polarization pumping light, broken down into left ($\sigma ^{-}$) and right ($\sigma ^{+}$) elliptically polarized light, goes through the $^{87}Rb$ atomic cell and interacts with the atoms, causing the unequal atomic populations. As a result, the vacuum fluctuations in the orthogonal polarization are modified by the interaction. Hence, the coherent pumping light (CS) with linear y-polarization transmits through the atomic vapor along z-axis to generate the squeezed vacuum light (SV), which is polarization self rotation (PSR) [18,21]. After emitting from the atomic vapor, a polarization beam splitter (PBS1) is used to separate CS and SV. As a result, only x-polarization SV going along z-axis can be used in the EO sensor. Below we measure the degree of SV and analyze the performance of the EO sensor.
Fig. 3. The experimental setup of the quantum enhanced EO sensor. DL Pro 100: the pumping light source. PMSF:Polarization maintaining single-mode fiber. SMCF: Single-mode coupling frame. GL: Glan-Laser Calcite Polarizer. M1 and M2: Mirror reflection. PBS1, PBS2, PBS3 and PBS4: Polarized beam splitter. PZT: Piezoelectric transduce. HWP1 and HWP2: Half-wave plate. FM: Flip mirror. EOM: Electro-optic modulator. QWP: Quarter-wave plate. SG: Signal generation. PD1, PD2, PD3 and PD4: Photon detector.
Firstly, the flip mirror (FM) is flipped up. A homodyne detection is built to detect SV. A piezoelectric transduce (PZT), scanning the relative phase between SV and LO, is driven by a triangular wave with 20Hz and 2Vpp. The noise spectrum versus sweeping time is plotted by the black solid line in Fig. 4. When VS is blocked, it is the homodyne detection of vacuum (V), which is the SQL shown in red dashed line in Fig. 4. We can draw the conclusion that about $-1.60\pm 0.1dB$ squeezing is measured at 1.07MHz. The video bandwidth (VBW) and resolution bandwidth (RBW) are 30Hz and 3kHz, respectively.
Fig. 4. Noise spectrum comparison between the squeezed-vacuum state and vacuum state. All traces are recorded in zero span mode at a detection frequency of 1.07MHz, RBW=3 kHz, and VBW=30Hz when the FM is flipped up. The PZT is driven by the triangular wave with 20Hz and 2Vpp.
Afterwards, FM is flipped down. The CS and SV go into the EOM (LINOS LM0202 P) and are detected by a balanced detector, which is the quantum enhanced EO sensor. An external sinusoidal E-field simulated by a signal generator is applied to EOM. The power spectrum is shown by red solid line in Fig. 5. When SV is blocked, the CS and V are injected into the senor, which is a classical sensor. The corresponding power spectrum is shown by the black dashed line in Fig. 5. It is clear that the noise of the quantum enhanced EO sensor can be suppressed compared with the classical one, while there is no change in signal. The noise floor is enlarged and shown in the inset of Fig. 5, which clearly demonstrates about $1.12\pm 0.1dB$ noise suppression. Thus, the SNR is enhanced by about $1.12\pm 0.1dB$. Notice that there is a 1.60dB squeezed light injection, but only 1.12dB SNR improvement in the sensor. This is due to the loss in the sensor, including 87.01% the transmission rate of the optical path and 82.61% match of the two polarization modes. The experimental loss can only achieve about $1.15dB$ according to the theoretical simulation.
Fig. 5. Power spectrum comparison between the quantum enhance EO sensor and the classical one. All traces are recorded when optical power is 1.05mW and the modulation signal is a sine wave with 16mVpp at 1.07MHz.
By changing the amplitude of the modulation voltage $U_{s}$, we can obtain the relation between SNR and amplitude as shown in Fig. 6. The black-square line represents the classical EO sensor while the red-circle line shows the quantum enhanced one. It can been seen that the relation between SNR and the amplitude of the modulation voltage $U_{s}$ is consistent with Eq. (17) . Noticeably, when the optical power is fixed at $1.05mW$, the quantum enhanced EO sensor is always $1.12\pm 0.1dB$ better than the classical one under the same voltage at $\omega _{s}=1.07MHz$, which is also consistent with the theory. Meanwhile, when $SNR=0dB$, we can obtain the sensitivity of the sensor which is enlarged and shown in the inset of Fig. 6. It can be seen that the value of the sensitivity is also decreased by $1.12\pm 0.1dB$.
Fig. 6. SNR comparison between the quantum enhance EO sensor and the classical one with different amplitude of voltage when the optical power is 1.05mW and $\omega _{s}=1.07MHz$.
To further decrease the value of sensitivity, according to Eq. (21), we can increase the optical power of CS. The corresponding results are shown in Fig. 7 when $U_{s}=16mVpp$ and $\omega _{s}=1.07MHz$. It can be seen that the larger of the optical power, the smaller the sensitivity is. Meanwhile, the quantum enhanced EO sensor, shown in red-circle line, is always decreased by $1.12\pm 0.1dB$ compared with the classical one, shown in the black-square line. However, no matter how large the CS power is, the quantum enhanced EO sensor can always beat classical one with the help of the quantum squeezed light. Moreover, the quantum enhanced effect provides another way to improve the performance of the EO sensor.
Fig. 7. Sensitivity comparison between the quantum enhance EO sensor and the classical one with different optical power of the CS when $U_{s}=16mVpp$ and $\omega _{s}=1.07MHz$.
5. Conclusion
A quantum enhanced EO sensor, which is a combination of CPI and a squeezed vacuum state, is proposed to achieve high sensitivity beyond SQL. By analyzing the performance of the sensor, including the SNR and sensitivity, the quantum enhanced EO sensor can always beat the classical one even with loss.
In the future, we can further enhance the performance of quantum enhanced EO sensor by improving the degree of the squeeze as well as the optical power and the EO coefficient. The proposal provides another way to enhance the performance of the EO sensor via the quantum light source. This quantum enhanced EO sensor could find practical application in measuring the intense of E-field.
Funding
The National Natural Science Foundation of China (11904227, 11874152); the Sailing Program of the Science and Technology Commission of Shanghai Municipality (19YF1414300, 19YF1421800); the Innovation Program of Shanghai Municipal Education Commission (2021-01-07-00-08-E00099); Fundamental Research Funds for the Central Universities.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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