• Chinese Optics Letters
  • Vol. 19, Issue 7, 072701 (2021)
Boya Xie and Sheng Feng*
Author Affiliations
  • Hubei Key Laboratory of Modern Manufacturing Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China
  • show less
    DOI: 10.3788/COL202119.072701 Cite this Article Set citation alerts
    Boya Xie, Sheng Feng. Heterodyne detection enhanced by quantum correlation[J]. Chinese Optics Letters, 2021, 19(7): 072701 Copy Citation Text show less

    Abstract

    Heterodyne detectors as phase-insensitive (PI) devices have found important applications in precision measurements such as space-based gravitational-wave (GW) observation. However, the output signal of a PI heterodyne detector is supposed to suffer from signal-to-noise ratio (SNR) degradation due to image band vacuum and imperfect quantum efficiency. Here, we show that the SNR degradation can be overcome when the image band vacuum is quantum correlated with the input signal. We calculate the noise figure of the detector and prove the feasibility of heterodyne detection with enhanced noise performance through quantum correlation. This work should be of great interest to ongoing space-borne GW signal searching experiments.

    1. Introduction

    Heterodyne detection is a powerful tool to capture low-frequency weak signals (1kHz) carried by optical beams, e.g.,  interferometric signals generated by gravitational-wave (GW) disturbances at space-borne observatories[13]. Nevertheless, traditional heterodyne detectors as phase-insensitive (PI) devices inevitably suffer from signal-to-noise ratio (SNR) degradation, caused by the image band vacuum at its input[47] and imperfect-quantum-efficiency-induced vacuum at its output[8]. As space-based GW detection systems are approaching their quantum noise limits[9], the quantum noise property of heterodyne detectors will become an important limiting factor for further system sensitivity improvement in the near future. Therefore, conquering the SNR degradation in heterodyne detection, if possible, may considerably benefit GW signal searching experiments, because the volume of space that is probed for potential GW sources increases as the cube of the strain sensitivity.

    According to the current heterodyne detection theory[4,5], 3 dB noise penalty occurs in heterodyne detection due to the image band vacuum at the detector’s input. It was suggested that the 3 dB heterodyne noise might be suppressed by injection of light in two-photon coherent states at the degenerate frequency of the image band vacuum into the detector[10], which unfortunately has never been implemented so far. Using an amplitude-squeezed local oscillator (LO), the quantum noise of a one-port heterodyne detector may be reduced[11], yet the problem of the 3 dB extra heterodyne noise due to the image band vacuum was not addressed. A phase-sensitive (PS) heterodyne detector with a bichromatic LO has proven to be noiseless[1216], but its phase sensitivity[14] requires phase control for the input signal that is intractable in the detection scheme of ongoing space-based GW experiments, where violent disturbance to the phase of the input signal is inevitable[13].

    Inspired by the work on quantum noise cancellation of a parametric amplifier by correlating the amplifier’s internal degree with the input signal through quantum entanglement[17], we study heterodyne detection enhanced by quantum correlation between the image band vacuum and the signal, given that the image band vacuum may be thought of as the internal degree of a detector according to the theory of the linear amplifier[18]. In this study, we focus on the following detection scenario: prior to being sent to a heterodyne detector for detection, the signal light is firstly fed into a noiseless parametric amplifier[19], whose pump light is at twice the frequency of the heterodyne LO, and, hence, quantum correlation is established between the signal mode and the image band (idler) mode in a vacuum state (Fig. 1). Here, we show that, if the amplified signal light is received by the heterodyne detector, the aforementioned 3 dB heterodyne noise can be completely eliminated due to quantum correlation. Moreover, our theoretical results show that optical amplification prior to signal detection may also serve to defeat the noise performance degradation in heterodyne detection due to imperfect quantum efficiency, akin to the case of direct detection[8].

    Schematics for heterodyne detection of light. (a) The signal mode enters into the detector together with an unexcited mode (image band vacuum) that gives rise to 3 dB heterodyne noise[4,5,7]. (b) Before being received by the detector, the signal light passes through a parametric amplifier where quantum correlation[17] is generated between the signal mode and the image band vacuum for reduction of the 3 dB heterodyne noise. ωs,i,l, angular frequency of the signal/image band/local oscillator mode; E^s,a(+)(t), quantum field of signal/amplified light beam; εl(+)(t), classical field of local oscillator light beam; 50-50, balanced beamsplitter; LA, linear amplifier; J−(t) ≡ J2(t) − J1(t), average differential photocurrent signal from the detector. Inset: a typical parametric amplifier consisting of a type-I periodically-poled KTiOPO4 (PPKTP) crystal inside an optical cavity and a laser pump[20] may be used to realize the proposed heterodyne detection with ωp = ωs + ωi, where ωp is the pump angular frequency. M1 & M2, cavity mirrors; DM, dichromatic mirror.

    Figure 1.Schematics for heterodyne detection of light. (a) The signal mode enters into the detector together with an unexcited mode (image band vacuum) that gives rise to 3 dB heterodyne noise[4,5,7]. (b) Before being received by the detector, the signal light passes through a parametric amplifier where quantum correlation[17] is generated between the signal mode and the image band vacuum for reduction of the 3 dB heterodyne noise. ωs,i,l, angular frequency of the signal/image band/local oscillator mode; E^s,a(+)(t), quantum field of signal/amplified light beam; εl(+)(t), classical field of local oscillator light beam; 50-50, balanced beamsplitter; LA, linear amplifier; J(t) ≡ J2(t) − J1(t), average differential photocurrent signal from the detector. Inset: a typical parametric amplifier consisting of a type-I periodically-poled KTiOPO4 (PPKTP) crystal inside an optical cavity and a laser pump[20] may be used to realize the proposed heterodyne detection with ωp = ωs + ωi, where ωp is the pump angular frequency. M1 & M2, cavity mirrors; DM, dichromatic mirror.

    2. Heterodyne Detector

    Let us consider a quantum field of signal light that has a continuum of frequency modes[14,2123]: E^s(+)(r,t)=iε0Vk(12ωk)12a^kei(k·rωkt),wherein V stands for the quantization volume, ε0 is the dielectric permittivity of vacuum, k is the set of plane-wave modes with ωk the corresponding angular frequency of each mode, and h/2π, in which h represents the Planck constant. The amplitude operator a^k is the photon annihilation operator for mode k and stays constant if there is no free electrical charge in the space[21]. The two mutually adjoint operators a^k and a^k obey the following commutation relations: [a^k,a^k]=[a^k,a^k]=0,[a^k,a^k]=δk,k.

    For simplicity, let us further assume that the detected light is a single-frequency coherent field with an excited mode at the angular frequency of ωs. If directly received by a detector with unity active detection area, the light field has an intrinsic SNR, SNRin=cε0ωsBE^s()(t)E^s(+)(t)=cε02ωsB|αs|2.

    Here, c stands for the speed of light in vacuum, B represents the measurement bandwidth inversely proportional to the measurement time, αsa^s×ωs/ε0V, and E^s()(t)=[E^s(+)(t)].

    To quantitatively evaluate the noise performance of a heterodyne detector, we make use of the quantity of noise figure (NF), NF=10log10SNRinSNRout,where SNRout is the signal SNR at the detector’s output, SNRoutPoutχ(Ω)·B.

    Here, Pout is the average power of the output photoelectric signal produced by the detector, and χ(Ω) represents the average noise power density of the photoelectric signal at Ω=(ωsωi)/2 (ωi stands for the angular frequency of the image band mode, and ωs>ωi is assumed; moreover, the frequency of the signal carried by the optical beam is much below the heterodyne frequency). A lower NF value indicates better noise performance for the detector. From Eqs. (3)–(5), it follows that one needs the values of Pout and χ(Ω) to calculate the NF of the detector. The average signal power Pout may be figured out with Pout=1T0TdtJ2(t),wherein J(t) is the average differential photocurrent signal at the detector’s output[14,22], J(t)=η0dtj(t)I^2(tt)I^1(tt),to which a similar result may be obtained in semiclassical treatment[23]. Here, η is the quantum efficiency of the detector in units of (ωs)1 (the average number of photoelectrons per photon energy), j(t) is the output current pulse produced by a photoemission, and j(t)=0 for t<0. I^1,2(t) are the light intensities at the two output ports of the detector, I^1,2(t)=(cε0/2){εl()(t)εl(+)(t)+E^a()(t)E^a(+)(t)±i[εl(+)(t)E^a()(t)εl()(t)E^a(+)(t)]},where εl(+)(t)=εleiωlt+ikl·r+iθl is the single-frequency classical field of the LO, with both the amplitude εl and phase θl being real numbers. εl()(t)=[εl(+)(t)]*, and E^a(+)(t) is the field of the amplified signal produced in the amplifier [Fig. 1(b)], E^a(+)(r,t)=iε0Vk(12ωk)12b^kei(k·rωkt),which is related to the input light field E^s(+)(t) through the linear evolution equation[18], b^s=a^scoshr+a^isinhr,b^i=a^ssinhr+a^icoshr.

    Here, b^s,i is the photon annihilation operator of the signal (idler or image band) mode of E^a(+)(r,t), and r is a real constant determined by the strength and duration of the parametric amplification. From Eqs. (1), (9), and (10), it is not difficult to show E^a(+)(r,t)=E^s(+)(r,t)coshrE^i()(r,t)e2i(kl·rωlt)sinhr,in which E^i(+)(r,t)iε0Vk(12|2ωlωk|)12a^kei(k·rωkt),provided that ωs+ωi=2ωl and ks+ki=2kl.

    3. Detector’s Noise Figure

    Plugging Eqs. (8), (11), and (12) into Eq. (7), one arrives at J(t)=2ceε0ηεl|αs|×[ercosθlcos(ΩtΔθ)ersinθlsin(ΩtΔθ)],where Δθ=Δk·rθs, θs is the phase of the signal mode, Δkkskl=klki, and we make use of ωsωi since |ωsωi|<ωs,i for heterodyne detection. In addition, the detector assumes a sufficient response speed in photoemission, and, hence, j(t)=eδ(t), where δ(t) is the Dirac function with e being the charge of the electron.

    From Eq. (13), it follows that the photoelectric signal from the detector consists of a quadrature component cos(ΩtΔθ) that is amplified by a factor of er and a conjugate quadrature component sin(ΩtΔθ) that is reduced by the same factor, which holds true no matter what the input signal phase θs is. When the LO phase θl is controlled such that θl=mπ (m is any integer), the heterodyne detector produces an amplified signal J(t)=±2ceε0ηεl|αs|ercos(ΩtΔθ) whose average power is, according to Eq. (6), Pout=(ceε0ηεl|αs|)2e2r,with an amplification gain of Ge2r. On the other hand, if the LO phase θl=(2m+1)π/2, the heterodyne detector produces a reduced signal J_(t)=±2ceε0ηεl|αs|ersin(ΩtΔθ). What is interesting is that the phase of the input signal, θs, does not need to be controlled, which is of technical essence for the studied detection scheme to be adapted to space-based GW experiments, where violent signal phase disturbances are expected.

    Next, we proceed to calculate the noise power density χ(Ω) of the heterodyne signal with the Fourier transform[14], χ(ω)=1T0Tdt+dτeiωτΔJ(t)ΔJ(t+τ),wherein the auto-correlation function of the differential photocurrent fluctuations is[22]ΔJ(t)ΔJ(t+τ)=i=12η0dtI^i(tt)ji(t)ji(t+τ)+i,j=12η2(1)i+j×0dtdtji(t)jj(t)λij(tt,τ+tt),which may be derived also under semiclassical approximations[23]. Here, ji,j(t) is a photoemission-induced current pulse at the output ports of the detector, λij(t,i)T:ΔI^i(t)ΔI^j(t+i): is the correlation function of light-intensity fluctuations, and the symbol T:: means time and normal ordering of the field operators E^a(±)(t). Photodiode noise is not included in Eq. (15) since we consider only the situation in which the system sensitivity in the frequency band of interest is limited by the quantum noise of light[9].

    Under the approximations of a strong oscillator and fast response speed for the detector, the auto-correlation function Eq. (16) may be readily reduced to ΔJ(t)ΔJ(t+τ)=ηcε0e2εl2δ(τ)+η2e2i,j=12(1)i+jλij(t,τ).

    Plugging Eq. (17) into Eq. (15) leads to χ(ω)=ηcε0e2εl2+η2e2i,j=12(1)i+j×1T0Tdt+dτeiωτλij(t,τ),wherein the first term on the right hand side represents the detection shot noise.

    From the definition of the correlation function λij(t,τ) and Eq. (8), it follows that λij(t,τ)=41c2ε02(1)i+j×[ΔE^a()(t)ΔE^a(+)(t+τ)εl(+)(t)εl()(t+τ)+ΔE^a()(t+τ)ΔE^a(+)(t)εl()(t)εl(+)(t+τ)ΔE^a()(t)ΔE^a()(t+τ)εl(+)(t)εl(+)(t+τ)ΔE^a(+)(t+τ)ΔE^a(+)(t)εl()(t)εl()(t+τ)],from which all the low-order terms in εl are dropped. With the help of the definitions of Γx(1,1)(t,τ)ΔE^x()(t)ΔE^x(+)(t+τ)eiωlτ,Γx(2,0)(t,τ)ΔE^x()(t)ΔE^x()(t+τ)eiωl(2t+τ),wherein x=s,i,a, Eq. (19) may be rewritten as λij(t,τ)=41c2ε02ε12(1)i+j×[Γa(1,1)(t,τ)Γa(2,0)(t,τ)e2iθ1+c.c.],in which θ1kl·r+θl. From Eqs. (18) and (21), it follows that χ(ω)=ηcε0e2εl2+η2c2ε02e2εl21T0Tdt+dτeiωτ×[Γa(1,1)(t,τ)Γa(2,0)(t,τ)e2iθ1+c.c.].

    In the following, we are going to evaluate Γa(1,1)(t,τ) and Γa(2,0)(t,τ) in Eq. (22) using Eqs. (11), (12), and (20). One may show without much difficulty that Γa(1,1)(t,τ)=cosh2rΓs(1,1)(t,τ)+sinh2reiωsτΔE^i(+)(t)ΔE^i()(t+τ)sinhrcoshr[Γ(2,0)(t,τ)e2ikl·r+c.c.]=sinh2reiωlτ[E^i(+)(t),E^i()(t+τ)],Γa(2,0)(t,τ)=cosh2rΓs(2,0)(t,τ)+sinh2re4ikl·r[Γi(2,0)(t,τ)]*sinhrcoshr×ei(ωlτ+2kl·r)ΔE^i(+)(t)ΔE^s()(t+τ)sinhrcoshr×ei(ωlτ2kl·r)ΔEs()(t)ΔEi(+)(t+τ)=sinhrcoshr×ei(ωlτ+2kl·r)[E^i(+)(t),E^s()(t+τ)].

    Here, Γ(2,0)(t,τ)ΔE^s()(t)ΔE^i()(t+τ)eiωl(2t+τ). In the last steps, Γs(1,1)(t,τ)=0, Γ(2,0)(t,τ)=0, Γs,i(2,0)(t,τ)=0, ΔE^s()(t+τ)ΔE^i(+)(t)=0, and ΔE^s()(t)ΔE^i(+)(t+τ)=0, given that the fields E^s,i(+)(t) are initially in coherent states[24].

    Although E^s,i(+)(t) in Eqs. (1) and (12) are expressed in three-dimensional (3D) expansions, all of the above calculations hold valid for their one-dimensional (1D) expansions as well. For optical fields in the form of collimated beams, one may substitute the 1D versions of Eqs. (1) and (12) into Eqs. (23) and (24), leading to Γa(1,1)(t,τ)Γa(2,0)(t,τ)e2iθ1=sinhr2πcε00+dωei(ωωl)τ×(sinhr|2ωlω|+e2iθlcoshrω|2ωlω|),after the summation over k is replaced by an integration: (1/V)k(1/2π)dk (k=±ω/c is the wave number of light). Plugging Eq. (25) into Eq. (22) and after some mathematical manipulations, one arrives at χ(ω)=ηcε0e2εl2[1+ηsinhrF(ω)],in which F(ω)sinhr|ωl+ω|+sinhr|ωlω|+(e2iθlcoshrωl2ω2+c.c.).

    With higher amplification gains, stronger quantum correlations between the signal and image band (idler) modes are expected for better suppression of the 3 dB heterodyne noise. The gain may be limited by practically available LO power levels for the heterodyne detection, but a high gain of up to 45 dB is still allowed if a 20 mW LO is used for space-based GW searching[2]. Therefore, we will consider only the high-gain cases for NF calculations, i.e., sinhrcoshrer/21, and suppose that the LO phase is controlled for detection of the amplified (anti-squeezed) quadrature of the signal. In addition, the LO optical frequency is much higher than the heterodyne frequency, i.e., ωlω. Under these approximations, Eq. (26) becomes χ(ω)=2ηcε0e2εl2[1+(ηωl)e2rcos2θl].

    The factor of two here accounts for the contribution of negative-frequency components when the calculation is compared with practical measurement[14]. From Eq. (28), it follows that the heterodyne detector produces a maximal amplified signal at its output when θl=0, and the corresponding noise power level of the output signal is χ(ω)=2ηcε0e2εl2(ηωl)e2r.

    From Eqs. (5), (14), and (29), it follows that the SNR of the amplified signal at the detector’s output is SNRout=Poutχ(Ω)·B=(ceε0ηεl|αs|)2e2r2ηcε0e2εl2(ηωl)e2rB=cε02ωlB|αs|2.

    Using Eqs. (3), (4), and (30), one finally obtains the NF of the heterodyne detector, NF=10log10SNRinSNRout=10log10ωlωs=0dB,where the approximation in the last step is based on the fact that |ωlωs|/ωl0 for heterodyne detection.

    4. Discussions

    The result of Eq. (31) proves that the noise performance of a heterodyne detector can be enhanced by the quantum correlation between the image band vacuum and the signal mode, without beating the quantum noise limit though. The price to pay is the change of the phase sensitivity of the detector: the output signal becomes sensitive to the LO phase. The good news is that, no matter what the input phase θs is, the amplifier will automatically amplify the cos(ΩtΔθ) quadrature component of the detected signal with a gain of G=er, as shown by Eq. (13). Therefore, the practical difficulty in the phase control for the input signal imposed by space-based GW experiments does not put any fundamental limit to the implementation of heterodyne detection enhanced by quantum correlation.

    Another interesting feature in the studied heterodyne detection scheme revealed by Eq. (31) is that the NF of the detector is independent of imperfect quantum efficiency η. It has been known for decades that the NF of a regular detector with imperfect quantum efficiency is[8,2527]NF=10log10(ξ1),where ξ=ηωs<1 in practice. In other words, a usual detector with lower η entails higher NF and poorer noise performance[28]. For the direct detection scheme, optical amplification prior to signal detection may serve to conquer the NF degradation due to non-perfect quantum efficiency[8]. Here, in this work, we have shown the same effect for heterodyne detection, i.e., the “beamsplitter noise” due to optical loss not affecting the SNR of a signal amplified by an amplifier without extra noises such as amplified spontaneous emission[29]. In the high-gain limit, Eq. (31) shows that the NF of the heterodyne detector approaches its best value of 0 dB despite imperfect quantum efficiency, from which space-based GW experiments will surely benefit.

    5. Conclusion

    We have studied a detector’s noise performance enhancement by quantum correlation in heterodyne detection. The SNR degradation of the output signal from the detector can be overcome by correlating the image band vacuum with the signal mode using a linear high-gain amplifier. We have shown that the studied heterodyne detection scheme requires no phase control for the input signal, which is of essence for space-borne GW experiments. The presented work paves the way to overcome vacuum-induced SNR degradation for optical precision measurements with heterodyne detectors, and the achieved results should be of great interest to space-borne experiments for low-frequency GW signal searching.

    References

    [1] M. Armano, H. Audley, G. Auger, J. T. Baird, M. Bassan, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt, M. Caleno, L. Carbone, A. Cavalleri, A. Cesarini, G. Ciani, G. Congedo, A. M. Cruise, K. Danzmann, M. de Deus Silva, R. De Rosa, M. Diaz-Aguiló, L. Di Fiore, I. Diepholz, G. Dixon, R. Dolesi, N. Dunbar, L. Ferraioli, V. Ferroni, W. Fichter, E. D. Fitzsimons, R. Flatscher, M. Freschi, A. F. García Marín, C. García Marirrodriga, R. Gerndt, L. Gesa, F. Gibert, D. Giardini, R. Giusteri, F. Guzmán, A. Grado, C. Grimani, A. Grynagier, J. Grzymisch, I. Harrison, G. Heinzel, M. Hewitson, D. Hollington, D. Hoyland, M. Hueller, H. Inchauspé, O. Jennrich, P. Jetzer, U. Johann, B. Johlander, N. Karnesis, B. Kaune, N. Korsakova, C. J. Killow, J. A. Lobo, I. Lloro, L. Liu, J. P. López-Zaragoza, R. Maarschalkerweerd, D. Mance, V. Martín, L. Martin-Polo, J. Martino, F. Martin-Porqueras, S. Madden, I. Mateos, P. W. McNamara, J. Mendes, L. Mendes, A. Monsky, D. Nicolodi, M. Nofrarias, S. Paczkowski, M. Perreur-Lloyd, A. Petiteau, P. Pivato, E. Plagnol, P. Prat, U. Ragnit, B. Raïs, J. Ramos-Castro, J. Reiche, D. I. Robertson, H. Rozemeijer, F. Rivas, G. Russano, J. Sanjuán, P. Sarra, A. Schleicher, D. Shaul, J. Slutsky, C. F. Sopuerta, R. Stanga, F. Steier, T. Sumner, D. Texier, J. I. Thorpe, C. Trenkel, M. Tröbs, H. B. Tu, D. Vetrugno, S. Vitale, V. Wand, G. Wanner, H. Ward, C. Warren, P. J. Wass, D. Wealthy, W. J. Weber, L. Wissel, A. Wittchen, A. Zambotti, C. Zanoni, T. Ziegler, P. Zweifel. Sub-femto-g free fall for space-based gravitational wave observatories: LISA pathfinder results. Phys. Rev. Lett., 116, 231101(2006).

    [2] J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, Z.-B. Zhou. TianQin: a space-borne gravitational wave detector. Class. Quantum Grav., 33, 035010(2016).

    [3] S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sopuerta, C. P. L. Berry, E. Berti, P. Amaro-Seoane, A. Petiteau, A. Klein. Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals. Phys. Rev. D, 95, 103012(2017).

    [4] J. H. Shapiro, H. P. Yuen, J. A. Machado Mata. Optical communication with two-photon coherent states-part II: photoemissive detection and structured receiver performance. IEEE Trans. Information Theory, 25, 179(1979).

    [5] H. P. Yuen, V. W. S. Chan. Noise in homodyne and heterodyne detection. Opt. Lett., 8, 177(1983).

    [6] Y. Yamamoto, H. A. Haus. Preparation, measurement and information capacity of optical quantum states. Rev. Mod. Phys., 58, 1001(1986).

    [7] C. M. Caves, P. D. Drummond. Quantum limits on bosonic communication rates. Rev. Mod. Phys., 66, 481(1994).

    [8] K. Bencheikh, O. Lopez, I. Abram, J. A. Levenson. Improvement of photodetection quantum efficiency by noiseless optical preamplification. Appl. Phys. Lett., 66, 399(1995).

    [9] A. Sesana. Prospects for multiband gravitational-wave astronomy after GW150914. Phys. Rev. Lett., 116, 23110(2016).

    [10] H. P. Yuen, J. H. Shapiro. Optical communication with two-photon coherent states–part Ill: quantum measurements realizable with photoemissive detectors. IEEE Transactions on Information Theory, 26, 78(1980).

    [11] Y.-Q. Li, D. Guzun, M. Xiao. Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator. Phys. Rev. Lett., 82, 5225(1999).

    [12] A. A. M. Marino, C. R. Stroud, V. Wong, R. S. Bennink, R. W. Boyd. Bichromatic local oscillator for detection of two-mode squeezed states of light. J. Opt. Soc. Am. B, 24, 335(2007).

    [13] W. Li, X. D. Yu, J. Zhang. Measurement of the squeezed vacuum state by a bichromatic local oscillator. Opt. Lett., 40, 5299(2015).

    [14] S. Feng, D. C. He, B. Y. Xie. Quantum theory of phase-sensitive heterodyne detection. J. Opt. Soc. Am. B, 33, 1365(2016).

    [15] F. Liu, Y. Zhou, J. Yu, G. Guo, Y. Wu, S. Xiao, D. Wei, Y. Zhang, X. Jia, M. Xiao. Squeezing-enhanced fiber Mach–Zehnder interferometer for low-frequency phase measurement. Appl. Phys. Lett., 110, 021106(2017).

    [16] B. Y. Xie, S. Feng. Squeezing-enhanced heterodyne detection of 10 Hz atto-watt optical signals. Opt. Lett., 43, 6073(2018).

    [17] J. Kong, F. Hudelist, Z. Y. Ou, W. P. Zhang. Cancellation of internal quantum noise of an amplifier by quantum correlation. Phys. Rev. Lett., 111, 033608(2013).

    [18] C. M. Caves. Quantum limits on noise in linear amplifiers. Phys. Rev. D, 26, 1817(1982).

    [19] Z. Y. Ou, S. F. Pereira, H. J. Kimble. Quantum noise reduction in optical amplification. Phys. Rev. Lett., 70, 3239(1993).

    [20] Y. J. Wang, Y. H. Tian, X. C. Sun, L. Tian, Y. H. Zheng. Noise transfer of pump field noise with analysis frequency in a broadband parametric downconversion process. Chin. Opt. Lett., 19, 052703(2021).

    [21] R. J. Glauber. Coherent and incoherent states of the radiation field. Phys. Rev., 131, 2766(1963).

    [22] Z. Y. Ou, C. K. Hong, L. Mandel. Coherence properties of squeezed light and the degree of squeezing. J. Opt. Soc. Am. B, 4, 1574(1987).

    [23] L. Mandel, E. Wolf. Optical Coherence and Quantum Optics(1995).

    [24] R. J. Glauber. The quantum theory of optical coherence. Phys. Rev., 130, 2529(1963).

    [25] A. Mosset, F. Devaux, E. Lantz. Spatially noiseless optical amplification of images. Phys. Rev. Lett., 94, 223603(2005).

    [26] R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, P. D. Lett. Low-noise amplification of a continuous-variable quantum state. Phys. Rev. Lett., 103, 010501(2009).

    [27] N. V. Corzo, A. M. Marino, K. M. Jones, P. D. Lett. Noiseless optical amplifier operating on hundreds of spatial modes. Phys. Rev. Lett., 109, 043602(2012).

    [28] X. C. Sun, Y. J. Wang, L. Tian, Y. H. Zheng, K. C. Peng. Detection of 13.8 dB squeezed vacuum states by optimizing the interference efficiency and gain of balanced homodyne detection. Chin. Opt. Lett., 17, 072701(2019).

    [29] T. Torounidis, P. A. Andrekson, B.-E. Olsson. Fiber-optical parametric amplifier with 70-dB gain. IEEE Photon. Tech. Lett., 18, 1194(2006).

    Boya Xie, Sheng Feng. Heterodyne detection enhanced by quantum correlation[J]. Chinese Optics Letters, 2021, 19(7): 072701
    Download Citation