Balanced-heterodyne detection of sub-shot-noise optical signals
aa r X i v : . [ qu a n t - ph ] N ov Balanced-heterodyne detection of sub-shot-noise optical signals
Sheng Feng, ∗ Zehuan Lu, Jie Zhang, and Chenggang Shao
School of Physics, Huazhong University of Science and Technology,Wuhan 430074, People’s Republic of China (Dated: November 28, 2012)
Abstract
As part of an effort to make use of single-mode squeezing for detection of sub-shot-noise opticalsignals, we study the balanced-heterodyne scheme, for which the corresponding spectral density ofthe photocurrent fluctuations produced at the output of the detector is calculated as the Fouriertransform of their autocorrelation function. We show that, for maximal signal-to-noise ratio en-hancement by use of squeezed states of light, an optical signal to be measured in this scheme mustbe carried in the squeezed quadrature of the carrier field. Most importantly, our analysis indi-cates that “the additional heterodyne noise” can be eliminated in balanced-heterodyne detectionfor single-mode squeezing under some experimental conditions and, from this, we discuss how thisscheme may be exploited in gravitational-wave searching. To demonstrate its practical feasibility,we propose and study a phase-locking technique.
PACS numbers: 42.50.Dv, 42.50.Lc, 42.79.Sz, 42.50.Ar ∗ [email protected] . INTRODUCTION Of the essence is to sense extremely-weak optical signals in several ongoing precisionexperiments on fundamental physics [1, 2] and for practical applications [3], in which theshot noise of light resulting from zero-point vacuum fluctuations is becoming a major limitingfactor at the quantum level in further improvement of sensitivity. To suppress the shot noise,one can resort to the exotic properties of the squeezed states or the entangled states of light[4, 5]. As a matter of fact, large degree of squeezing of more than 10dB has been achievedexperimentally [6]. Depending on the nature of the optical beam to be measured, detection ofoptical squeezing may be carried out with different schemes. Direct detection is the simplestone and suited for bright amplitude-squeezed light, whereas homodyne detection schemefor quadrature-squeezed light. Another scheme using a cavity to rotate the quadratureorientation of optical sidebands in phase space relative to that of the carrier [7, 8] can be usedto measure the quadrature squeezing of bright light. The practical problem with these threeschemes is the intensity noise of light, which is classical and can be substantially suppressedin balanced-homodyne detection [9]. The great similarity of these schemes is their relying onthe detection of the beats between a strong optical mode, the local oscillator or the carrier,and noise sidebands that are correlated with each other and symmetrically located aroundthe strong mode in frequency spectrum.Although all the aforesaid detection schemes are playing crucial roles in experimentalstudy for quantum information processing, their applications to precision measurements fordetecting sub-shot-noise optical signals are restrained. This is because the beat note be-tween a slowly-varying sub-shot-noise optical signal and the local oscillator is a signal closeto DC and will be inevitably contaminated by the low-frequency dark-current noise of adetector. Actually, to circumvent the low-frequency electronic noise, optical squeezing isusually measured at frequencies off the carrier’s optical frequency in homodyne detection[6]. Apparently, this strategy does not apply to the case where optical squeezing must bemeasured together with an optical signal that is being under investigation. To shift thefrequency of the beat note beyond the low-frequency noise regime of the detector, one mayconsider the traditional heterodyne scheme with single local oscillator, which unfortunatelydoes not work for detection of squeezed light as proven, for example, by Collett et al. in[10]. Heterodyne detection schemes for gravitational-wave observation [11–14] have been2nvestigated, showing that an additional quantum-noise contribution may exist due to vac-uum fluctuations in frequency bands that are twice the heterodyne frequency away from thecarrier. In the present paper, we investigate a balanced-heterodyne scheme for detectingsub-shot-noise optical signals, in which the “additional heterodyne noise” disappears in cer-tain experimental circumstances, as one will see in the following. One should note that thescenario here is different from that of the theoretical work for two-mode squeezing detectionin [15], which does not deal with the detection of single-mode squeezing.The balanced-heterodyne scheme manifests itself by utilizing two local oscillators withequal strength that are symmetrically located in frequency spectrum around the opticalsignal to be measured, as depicted in Fig. 1 with detailed configurations to be described inthe next section. Section III is devoted to calculating the spectral density of the photocur-rent fluctuations produced by the detector. Unsurprisingly, our calculations show that thedetected noise level of the optical signal is phase sensitive, i.e., the noise level depends onsome relative phase of light, a feature also shared by homodyne schemes. For some givenrelative phase, observation of a sub-shot-noise signal is possible if the optical signal is carriedin the squeezed quadrature of the field. On the other hand, if the signal is carried in theanti-squeezed quadrature, it must be manipulated to be carried in the squeezed quadrature,using for example a single-ended cavity [16, 17], before heterodyne detection is performed.Moreover, we discuss how the additional heterodyne noise can be ruled out in balanced het-erodyne detection. What succeeds in Section IV is of how to lock the relative phase of lightfor balanced heterodyning. The locking technique is quantitatively analyzed to further showthe practical feasibility of the studied scheme. Finally, we discuss the potential applicationof the balanced-heterodyne scheme for practical gravitational-wave observation, followed bya short summary as conclusions to this work by the end with acknowledgements.
II. HETERODYNE DETECTION SCHEME
According to the uncertainty principle, when one quadrature of an optical field issqueezed, its canonical conjugate should be anti-squeezed. In the conventional homodynedetection scheme, one may choose to measure either the squeezed quadrature or its conju-gate by controlling the phase of the local oscillator relative to some reference defined by themeasured optical mode. Changing this relative phase actually orients the direction of the os-3illator along that of the detected quadrature in phase space (Fig. 2a). Only when the localoscillator is parallel to the squeezed quadrature, could one observe the most noise-reductionbelow the shot-noise level in the homodyne detection. In contrast, the heterodyne schemewith single local oscillator has proven unsuitable for squeezing detection [10], because therelative phase and, hence, the relative orientation in phase space, of the local oscillator withrespect to the squeezed quadrature continuously varies with time and therefore is out ofexperimental control. As a remedy to this, the balanced-heterodyne scheme takes advantageof two local oscillators with equal strength, the superposition of which oscillates but remainspointing in a fixed direction in phase space, selecting the corresponding quadrature to bemeasured (Fig. 2b). To see this, we may write the quantity of light intensity at the outputport of the beamsplitter in the following form (see Fig. 1 for reference)ˆ I ( t ) = [ ˆ E ( − ) ( t ) + E ( − )1 ( t ) + E ( − )2 ( t )] × [ ˆ E (+) ( t ) + E (+)1 ( t ) + E (+)2 ( t )]= [ e iω t (ˆ a † + E e i Ω t − iφ + E e − i Ω t − iφ )] × [ e − iω t (ˆ a + E e − i Ω t + iφ + E e i Ω t + iφ )]= ˆ a † ˆ a + 2 E cos (Ω t + δφ ) ˆ X ( ¯ φ ) + 4 E cos (Ω t + δφ ) . (1)Here we adopted units in which the light intensity is expressed in photons per second.ˆ E (+) ( t ) = ˆ ae − iω t , E (+)1 ( t ) = E e − i ( ω +Ω) t + iφ , and E (+)2 ( t ) = E e − i ( ω − Ω) t + iφ represent theoptical field to be detected and the two oscillators, respectively. And we assumed that thetwo oscillators are strongly-excited coherent modes with the same amplitude E ( E a positivereal number) and can be approximated by classical fields. ˆ X ( ¯ φ ) = ˆ X cos ¯ φ + ˆ P sin ¯ φ , whereˆ X ≡ ˆ ae − iβ + ˆ a † e iβ and ˆ P ≡ ˆ ae − i ( β + π/ + ˆ a † e i ( β + π/ are two quadrature amplitudes of theoptical field ˆ E ( − ) ( t ). ¯ φ ≡ ( φ + φ ) / − β , and β is an arbitrary phase that is used to defineˆ X and ˆ P , thereby the major axis of the distribution ellipse of ˆ X and ˆ P in phase space pointsin the direction of the ˆ X or ˆ P axis [18]. δφ ≡ ( φ − φ ) / A. Photocurrent fluctuations
The first term on the right-hand side of Eq.(1) is the DC signal of the detected field, whichis weak compared to other terms and, hence, negligible in strong-oscillator approximation.The last term contributes a classical signal that does not fluctuate when the oscillators4re in highly coherent states [19]. Accordingly, the middle term dominates in the photonfluctuations: ∆ ˆ I ( t ) ≈ E cos (Ω t + δφ )∆ ˆ X ( ¯ φ ) , (2)which shows phase sensitivity on ¯ φ and differs from the homodyne case by only a modula-tion function cos (Ω t + δφ ). In other words, through controlling the relative phase ¯ φ , onecan choose to measure the noise of any quadrature of the detected field in the balanced-heterodyne scheme, which is analogous to the conventional homodyne scheme. Since themodulation function cos (Ω t + δφ ) in Eq. (2) leaves unchanged the relative orientation ofthe measured quadrature of the detected field and the superposition of the two oscillators(Fig. 2b), its only effects are to split the spectral density of the photocurrent fluctuationsinto two parts and to shift each part along the frequency axis by +Ω or − Ω, as will be shownin the following.
B. Photoelectrical signal
At the output of the detector, the photoelectrical signal, with photon fluctuations ne-glected, is proportional to the light intensity averaged over the initial states of the opticalfields. Using Eq. (1), we may calculate the averaged light intensity as < ˆ I ( t ) > = < ˆ a † ˆ a > +2 E cos (Ω t + δφ ) < ˆ X ( ¯ φ ) > +4 E cos (Ω t + δφ ) . (3)Apparently, the heterodyne signal of interest at frequency Ω has an amplitude 2 E < ˆ X ( ¯ φ ) > that exhibits a dependence on ¯ φ , in other words, a photoelectrical signal appearing in < ˆ X ( ¯ φ ) > for a given ¯ φ may not show up in the conjugate < ˆ X ( ¯ φ + π/ > .Consequently, a naive picture that one may imagine is as follows: To take advantage ofoptical squeezing to achieve maximal enhancement of signal-to-noise ratio in precision mea-surements, one must find a ¯ φ , for which the photoelectrical signal at heterodyne frequencyΩ is maximized and the photon fluctuations are suppressed the most at the same time. Toput it another way, for sub-shot-noise detection of an optical signal in balanced-heterodynescheme, the quadrature ˆ X ( ¯ φ ) in which the optical signal is carried must be a squeezed one.True or not, we expect to find the answer in the subsequent section after calculating thespectral density of the photocurrent fluctuations produced at the output of the detector.5 II. SPECTRAL DENSITY OF PHOTOCURRENT FLUCTUATIONS
We will follow the approach of Ou, Hong and Mandel [18] to calculate the spectral densityof the photocurrent fluctuations. Nonetheless, stationary photocurrents can be assumed onlyat the time scale of Ω − , beyond which the measured spectral density should be treated asan average over measurement time T (see Ref. [10, 20] for similar treatments). Usually T ∼ Ω r − >> Ω − (Ω r the detection bandwidth. To resolve the heterodyne beat noteat Ω, one must set Ω r << Ω). In consequence, the spectral density of the photocurrentfluctuations ∆ J ( t ) ≡ J ( t ) − < J ( t ) > reads χ ( ω ) = 1 T Z T dt Z + ∞−∞ dτ e iωτ < ∆ J ( t )∆ J ( t + τ ) >, (4)where < ∆ J ( t )∆ J ( t + τ ) > is the auto-correlation function of the photocurrent fluctuations.Supposing that every photoelectron emitted at time t ′ gives rise to a definite photoelectricalcurrent pulse j ( t − t ′ ) for t > t ′ , then the total photocurrent is J ( t ) = X i j ( t − t i ) , (5)wherein the sum is taken over the various random-emission times t i . When t < t i , j ( t − t i ) = 0because no photoelectrical pulses exist yet prior to the emission of electrons. With Eq. (5),the auto-correlation function of the photocurrents can be computed as [18, 21] < J ( t ) J ( t + τ ) > = X l,m < j ( t − t l ) j ( t + τ − t m ) > = X l < j ( t − t l ) j ( t + τ − t l ) > + X l = m < j ( t − t l ) j ( t + τ − t m ) > = X l j ( t − t l ) j ( t + τ − t l ) P ( t l )∆ t l + X l = m j ( t − t l ) j ( t + τ − t m ) P ( t l , t m )∆ t l ∆ t m , (6)where P ( t )∆ t and P ( t, t ′ )∆ t ∆ t ′ are respectively the probability of photodetection registeredat time t within time interval ∆ t and joint probability that two photodetections are enrolledat t within ∆ t and at t ′ within ∆ t ′ , respectively. The first term is attributable to theshot noise of the photoelectrical currents, while the second term depends on the fluctuationnature of the optical fields at the photoreceiver. These probabilities are related to the light6ntensity as [22, 23] P ( t )∆ t = η < ˆ I ( t ) > ∆ tP ( t, t ′ )∆ t ∆ t ′ = η < T : ˆ I ( t ) ˆ I ( t ′ ) : > ∆ t ∆ t ′ , (7)in which η is a parameter characterizing the response of the detector to the incident light.The symbol T :: stands for time- and normal-ordering of the field operators. Convertingthe summation into integrand in Eq. (6) and using Eq. (7), one arrives at < ∆ J ( t )∆ J ( t + τ ) > = η Z ∞ dt ′ < ˆ I ( t − t ′ ) > j ( t ′ ) j ( t ′ + τ )+ η Z Z ∞ dt ′ dt ′′ λ ( t − t ′ , τ + t ′ − t ′′ ) j ( t ′ ) j ( t ′′ ) , (8)where introduced is the correlation function of light-intensity fluctuations λ ( t, ι ) = < T : ∆ ˆ I ( t )∆ ˆ I ( t + ι ) : > . (9)One should recall that < ˆ I ( t ) > in heterodyne detection is not time-independent, in contrastto the homodyne case where < ˆ I ( t ) > can be assumed to be stationary. The calculation pro-cedure of the intensity correlation function λ ( t, ι ) is trivial, and the treatment is resemblantto that based on the approach of Ou, Hong and Mandel [18]. The key idea is to establish aconnection between the spectral density of the photocurrent fluctuations and the normallyordered, time-ordered correlation functions of the quadrature fluctuations of the optical fielddetected.For the heterodyne configuration as in Fig. 1, the correlation function of intensity fluc-tuations is found to be different from that for the traditional homodyne configuration (seethe appendix for detailed calculations): λ ( t, ι ) = < T : ∆ ˆ I ( t )∆ ˆ I ( t + ι ) : > = E { Γ (1 , ( ι ) (cid:2) e i Ω ι + e − i Ω ι + e − i Ω(2 t + ι ) − i δφ + e i Ω(2 t + ι )+2 iδφ (cid:3) + Γ (2 , ( ι ) (cid:2) e i Ω ι + i ( φ + φ ) + e − i Ω ι + i ( φ + φ ) + e − i Ω(2 t + ι )+2 iφ + e i Ω(2 t + ι )+2 iφ (cid:3) + c.c. } + O ( E ) , (10)wherein Γ (1 , ( ι ) ≡ < ∆ ˆ E ( − ) ( t )∆ ˆ E (+) ( t + ι ) > e iω ι and Γ (2 , ( ι ) ≡ < ∆ ˆ E ( − ) ( t )∆ ˆ E ( − ) ( t + ι ) >e − iω (2 t + ι ) . Eq. (10) obviously contains terms that vary with a temporal period of (2Ω) − .Because the observed spectral density of the photocurrent fluctuations are averaged over a7ime period T ∝ Ω − r >> Ω − , these time-dependent terms will be ruled out of Eq. (4) bythe temporal integrand and not show up in the spectral density. For this reason, we willkeep only the time-independent terms in Eq. (10) and replace the correlation function λ ( t, ι )with a t -independent function λ ′ ( τ ) ≈ E cos (Ω τ ) { Γ (1 , ( τ ) + Γ (2 , ( τ ) e i ( φ + φ ) + c.c. } , (11)in the succeeding calculations. Here the strong-oscillator approximation is utilized such thatthe terms in E in Eq. (10) dominate over all others, which have henceforth been discarded.Using Eq. (3), similar treatment can be applied the < ˆ I ( t ) > term on the right-hand side ofEq. (8), resulting in T R T dt < ˆ I ( t ) > = 2 E . Plugging Eqs. (8) and (10) into Eq. (4) withthese manipulations leads to χ ( ω ) = Z + ∞−∞ dτ e iωτ (cid:20) E η Z ∞ dt ′ j ( t ′ ) j ( t ′ + τ ) + η Z Z ∞ dt ′ dt ′′ j ( t ′ ) j ( t ′′ ) λ ′ ( τ + t ′ − t ′′ ) (cid:21) . (12)In comparison to the spectral density of the photocurrent fluctuations in conventional ho-modyne scheme [18, 21], Eq. (12) differs by a global factor of two and a cos (Ω τ ) function(see Eq. (11)) in the second term. A factor of two means that the spectral density in ourheterodyne scheme is 3dB higher, which is obviously due to the usage of two local oscillators.As for the function cos (Ω τ ), as one will see, it plays a crucial role in the heterodyne schemein splitting squeezing spectrum into two parts and shifting their centers off the carrier fre-quency, leading to the “traditional heterodyne noise” for certain experimental parameters.The spectral density given by Eq. (12) can be easily related to the time-ordered, normallyordered second-order correlation functions of the quadrature fluctuations of the detected fieldthrough [21] ReΓ (1 , ( τ ) = [Γ ( τ ) + Γ ( τ )] / (1 , ( τ ) = [Γ ( τ ) − Γ ( τ )] / (2 , ( τ ) e iβ = [Γ ( τ ) − Γ ( τ )] / (2 , ( τ ) e iβ = − [Γ ( τ ) + Γ ( τ )] / , (13)where Γ mn ( τ ) ≡ < T : ∆ ˆ E m ( t )∆ ˆ E n ( t + τ ) : >, ( m, n = 1 , E ( t ), ˆ E ( t ) are thequadrature operators defined asˆ E ( t ) = ˆ E (+) ( t ) e i ( ω t − β ) + ˆ E ( − ) ( t ) e − i ( ω t − β ) ˆ E ( t ) = ˆ E (+) ( t ) e i ( ω t − β − π/ + ˆ E ( − ) ( t ) e − i ( ω t − β − π/ . (14)8 is the arbitrary phase associated with the field quadrature ˆ E ( − ) ( t ), and ∆ ˆ E ≡ ˆ E − < ˆ E > .After substituting Eqs. (13) into Eq. (11), one achieves λ ′ ( τ ) ≈ E cos (Ω τ ) { Γ ( τ )(1 + cos 2 ¯ φ ) + Γ ( τ )(1 − cos 2 ¯ φ ) + [Γ ( τ ) + Γ ( τ )] sin 2 ¯ φ } , (15)¯ φ = ( φ + φ ) / − β again. Let K ( ω ) be the Fourier transform of the photoelectrical currentpulse j ( t ), K ( ω ) = Z ∞ dτ j ( τ ) e iωτ , (16)which may be interpreted as the frequency response of the detector, and let Φ mn ( ω ) theFourier transform of the correlation functions Γ mn ( τ ) of the field quadrature fluctuations:Φ mn ( ω ) = Z + ∞−∞ dτ Γ mn ( τ ) e iωτ ( m, n = 1 , . (17)With the help of Eqs. (15), (16), and (17), one can readily rewrite the spectral density ofthe photocurrent fluctuations Eq. (12) as χ ( ω ) ≈ η E | K ( ω ) | { η/
2) [Φ ( ω + Ω) + Φ ( ω − Ω)] (1 + cos 2 ¯ φ )+ ( η/
2) [Φ ( ω + Ω) + Φ ( ω − Ω)] (1 − cos 2 ¯ φ )+ ( η/
2) [Φ ( ω + Ω) + Φ ( ω + Ω) + Φ ( ω − Ω) + Φ ( ω − Ω)] sin 2 ¯ φ } . (18)Particularly, in the special case that ¯ φ = kπ ( k any integer), χ ( ω ) ≈ η E | K ( ω ) | { η/
2) [Φ ( ω + Ω) + Φ ( ω − Ω)] } . (19)It follows that, if Φ <
0, the measured spectral density χ ( ω ) is to fall below the vacuumlevel for ¯ φ = kπ , which is similar to the case of the conventional homodyne detection [18, 21].When ¯ φ = kπ ± π/ k any integer), χ ( ω ) ≈ η E | K ( ω ) | { η/
2) [Φ ( ω + Ω) + Φ ( ω − Ω)] } . (20)Again, akin to the homodyne scheme, the spectral density will be lower than the vacuumlevel for ¯ φ = kπ ± π/
2, if Φ <
0. Whether Φ or Φ is to be observed is conditioned on thespecific values of ¯ φ , confirming the intuitive conclusion drawn upon Eq. (2) together withEq. (3): To achieve sub-shot-noise detection of an optical signal with the best signal-to-noise9atio enhancement in the balanced-heterodyne scheme, the optical signal must appear in thecarrier’s quadrature < ˆ X ( ¯ φ ) > that is squeezed.Let one consider an optical signal carried in the amplitude of a field. The heterodynesignal, proportional to < ˆ X ( ¯ φ ) > according to Eq. (3), at frequency Ω is maximizedwhen ¯ φ = kπ ( k integer) because it is carried in the amplitude of the field. In this case,noise reduction below the shot noise level is available in the photocurrent only if Φ < <
0, one cannotobserve amplitude-maximized signal with noise reduction, since maximized heterodyne signaldemands ¯ φ = kπ and, on the other hand, to observe Φ < φ = kπ ± π/ <
0, before the balanced-heterodyne scheme is appliedto yield a sub-shot-noise signal, one needs to rotate the axis of the squeezed quadratureby an angle of 90 ◦ in phase space, for instance, with a single-ended cavity [16], such thata quadrature-phase squeezing Φ < < ii (1 = 1 , ± Ω, as a direct consequence of the existenceof cos(Ω τ ) in Eq. (11), which is nothing but our enthusiastic argument previously made onEq. (2). This will cause an issue in practice when applying balanced-heterodyne scheme inprecision measurements: The spectrum of squeezing Φ ( ω ) or Φ ( ω ) can put a limit to thedegree of noise reduction in balanced-heterodyne detection. For a conceptual illustration, letone consider a squeezed-vacuum field generated from an optical parametric down-converterin a cavity. Suppose that this squeezed field is to be heterodyned with two local oscillatorsin the configuration as depicted in Fig. 1. The spectrum of squeezing at the output of thecavity is [18] Φ = − η ǫγ ( γ/ ǫ ) + ω , (21)where γ ≡ (1 − R ) c/l ( R is the cavity mirror reflectivity, c the speed of light in vacuum,and l the cavity length) is the cavity damping rate and ǫ is a measure of the effectivepump intensity of the down-converter. Obviously, γ/ ǫ is a key parameter that determinesthe spectrum of squeezing. To achieve sub-shot-noise detection with signal-to-noise ratio10mproved the most, one must control the relative phase ¯ φ = kπ , so that the spectral density χ ( ω ) ≈ η E | K ( ω ) | (cid:20) − ǫγ ( γ/ ǫ ) + ( ω + Ω) − ǫγ ( γ/ ǫ ) + ( ω − Ω) (cid:21) . (22)Numerical simulations based on this equation (Fig. 3) show that, to fully take advantageof the degree of squeezing of the incident signal, the heterodyne frequency must satisfyΩ << γ . This is the condition under which “the additional heterodyne noise” can becompletely removed in the studied scheme, at least, in principle. This may find interestingapplications in precision measurements, such as in the observation of gravitational waves.In the limit of Ω >> γ , one can obtain a noise reduction by 3dB at most, because ofthe apparent fact that Φ ( ω + Ω) makes no contribution to noise squeezing at frequencieswhere Φ ( ω − Ω) does its best, the physics behind which is the vacuum fluctuations twicethe heterodyne frequency away from the carrier, an important subject of previous works[11–14].
IV. PHASE LOCKING TECHNIQUE
To enforce the balanced-heterodyne scheme for sub-shot-noise detection, the optical signalto be measured should appear in the squeezed quadrature of the carrier field, as discussedearlier. Meanwhile, the relative phase ¯ φ = ( φ + φ ) / − β must be well controlled atsome values for maximal < ˆ X ( ¯ φ ) > according to Eq. (3), requiring an electrical lockingtechnique for this scheme. A well known technique of quantum noise locking [24] proposed forhomodyne scheme apparently is not applicable to the balanced-heterodyne scheme entailingtwo local oscillators. We propose to phase-modulate the two oscillators simultaneously withthe same modulator and utilize the interference of the detected optical signal with two of thefour sidebands of the oscillators, see Fig. 4. The beat notes of the detected signal and thosemodulation-created sidebands are to be demodulated, then low-pass-filtered by a loop-filterto obtain an error signal for the locking loop. This phase-locking scheme is actually a versionof the coherent-modulation-locking technique, whose stability is much better than the noiselocking scheme for homodyne detection [24].At the output of the photodetector, the photocurrent is [21] < J ( t ) > = Z + ∞−∞ dt ′ P ( t ′ ) j ( t − t ′ )11 Z + ∞−∞ dt ′ h η < ˆ I ( t − t ′ ) > i j ( t ′ ) ≈ ηq < ˆ I ( t ) >, (23)in which q ≡ R + ∞−∞ dt ′ j ( t ′ ) is the total electrical charge delivered by the current pulse resultingfrom one photoelectron. Here one assumed that the electrical current pulse is much shorterthan the oscillation period of < ˆ I ( t ) > , in another word, < ˆ I ( t ) > varies slowly compared toelectrical pulse j ( t ), which is usually the case in practice. Under this assumption, < ˆ I ( t ) > may be considered approximately constant as long as the electrical pulse lasts and does notcontribute to the integral in Eq. (23).After passing the same electro-optical modulator, the oscillator fields become E (+)1 , ( t ) = E e − i ( ω ± Ω) t + i ( φ , + θ sin Ω ′ t ) , (24)wherein Ω ′ < Ω is the phase-modulation frequency and θ the modulation depth. Then < J ( t ) > ≈ ηq < [ ˆ E ( − ) ( t ) + E ( − )1 ( t ) + E ( − )2 ( t )] × [ ˆ E (+) ( t ) + E (+)1 ( t ) + E (+)2 ( t )] > = ηq < (ˆ a † + E e i Ω t − iφ ( t ) + E e − i Ω t − iφ ( t ) ) × (ˆ a + E e − i Ω t + iφ ( t ) + E e i Ω t + iφ ( t ) ) > ≈ ηq E J ( θ ) < ˆ a > e − iφ e i (Ω − Ω ′ ) t − ηq E J ( θ ) < ˆ a > e − iφ e − i (Ω − Ω ′ ) t + ηq E J ( θ ) < ˆ a † > e iφ e − i (Ω − Ω ′ ) t − ηq E J ( θ ) < ˆ a † > e iφ e i (Ω − Ω ′ ) t + ..., (25)where an appropriate modulation depth is assumed so that e iθ sin Ω ′ t ≈ J ( θ ) + J ( θ ) e i Ω ′ t − J ( θ ) e − i Ω ′ t with J , ( θ ) the zero- and the first-order Bessel functions. In the last step,explicitly written out are the terms oscillating at frequency Ω − Ω ′ and the reason is thata demodulation signal at this frequency is to be used for error-signal pick up. Hence, theerror signal to be demodulated reads S ( ¯ φ ) = 4 ηq E J ( θ ) h ∂∂ ¯ φ < ˆ X ( ¯ φ ) > i sin [(Ω − Ω ′ ) t + δφ ] . (26)The term ∂∂ ¯ φ < ˆ X ( ¯ φ ) > in this equation ensures that ¯ φ can be readily locked at any valuesfor which < ˆ X ( ¯ φ ) > is maximal, i.e., the peak of the photoelectrical signal. Nevertheless,one must note that the relative phase δφ of the two oscillators should also be kept steadyfor the locking scheme to work well. 12 . BALANCED HETERODYNING FOR GRAVITATIONAL-WAVE OBSERVA-TION The existence of the additional heterodyne noise makes the previously-investigatedschemes [12–14] less competitive than homodyne schemes in practical applications for pre-cision measurements. However, our analysis on the balanced-heterodyne scheme based onthe quantum theory of optical coherence shows the possibility of eliminating this additionalnoise under certain experimental circumstance. Therefore, the balanced-heterodyne schememay be a promising choice for photoelectrical readout in gravitational-wave observationexperiments. The price that one pays is that the heterodyne frequency Ω is not free tochoose. Instead, the choice of Ω must guarantee that the degree of squeezing at opticalfrequencies ω ±
2Ω is more or less the same as at ω (Fig. 3).Since the balanced-heterodyne scheme makes use of dual local oscillators with equalstrength, the unbalanced Schnupp sidebands generated by phase modulation in AdvancedLIGO [14] cannot be used as the balanced local-oscillators. However, one might pick up partof the light from the bright port of the interferometer to serve as oscillators for balancedheterodyning after some appropriate manipulations. If this is the choice, the proposed phase-locking technique in the preceding section may be exploited to control the fluctuations ofthe involved phases of light. VI. CONCLUSIONS
Although homodyne detection scheme has been widely used to measure the quantumnoise of light below the shot-noise level in the field of quantum information processing, it canhardly be exploited to detect slowly-varying sub-shot-noise optical signals due to the intrinsicelectrical noise of the photodetector at low frequencies. To fulfill the need of noise reductionin precision experiments where the shot noise of light is becoming a substantial restrictivefactor for further sensitivity improvement, we have investigated the balanced-heterodyne de-tection scheme, where observation of noise reduction below the shot-noise level is sensitive tosome relative phase defined by the signal-carrying optical field and the oscillators, a trait akinto the classical homodyne schemes. An optical signal carried in the squeezed quadrature ofthe carrier field can be measured with the best signal-to-noise ratio enhancement, whereas13hat carried by the quadrature conjugate to the squeezed one may be manipulated by asingle-ended cavity before a balanced-heterodyne detection is performed. Most importantly,we have shown that, under certain experimental circumstances, the additional heterodynenoise existing in previously-studied schemes may disappear in balanced-heterodyne detec-tion, which may be particularly interesting in applications for gravitational-wave searching.A phase-locking technique has been analyzed to yonder demonstrate the feasibility of thescheme for practical implementation.
Appendix
According to the definition, the correlation function of light-intensity fluctuations reads λ ( t, ι ) = < T : ∆ ˆ I ( t )∆ ˆ I ( t + ι ) : > = < T : ˆ I ( t ) ˆ I ( t + ι ) : > − < ˆ I ( t ) >< ˆ I ( t + ι ) >, (A.1)where ˆ I ( t ) = [ ˆ E ( − ) ( t ) + E ( − )1 ( t ) + E ( − )2 ( t )] × [ ˆ E (+) ( t ) + E (+)1 ( t ) + E (+)2 ( t )] in conformity withEq. (1). One may expand the two terms on the right-hand side of Eq. (A.1) separately asfollows: < T : ˆ I ( t ) ˆ I ( t + ι ) : > = < [ ˆ E ( − ) ( t ) + E ( − )1 ( t ) + E ( − )2 ( t )] × [ ˆ E ( − ) ( t + ι ) + E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] × [ ˆ E (+) ( t + ι ) + E (+)1 ( t + ι ) + E (+)2 ( t + ι )] × [ ˆ E (+) ( t ) + E (+)1 ( t ) + E (+)2 ( t )] > = [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )]+ [ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E (+) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E (+) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )] < ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) > + [ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t + ι ) ˆ E (+) ( t ) > + [ E (+)1 ( t ) + E (+)2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E ( − ) ( t + ι ) >
14 [ E ( − )1 ( t ) + E ( − )2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E (+) ( t + ι ) ˆ E (+) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )] < ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) ˆ E (+) ( t ) > + [ E (+)1 ( t ) + E (+)2 ( t )] < ˆ E ( − ) ( t ) ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E (+) ( t + ι ) ˆ E (+) ( t ) > + [ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E ( − ) ( t + ι ) ˆ E (+) ( t ) > + < ˆ E ( − ) ( t ) ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) ˆ E (+) ( t ) >, (A.2) < ˆ I ( t ) >< ˆ I ( t + ι ) > = < [ ˆ E ( − ) ( t ) + E ( − )1 ( t ) + E ( − )2 ( t )] × [ ˆ E (+) ( t ) + E (+)1 ( t ) + E (+)2 ( t )] > × < [ ˆ E ( − ) ( t + ι ) + E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] × [ ˆ E (+) ( t + ι ) + E (+)1 ( t + ι ) + E (+)2 ( t + ι )] > = [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )]+ [ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E (+) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E (+) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t ) + E (+)2 ( t )] < ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) > + [ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E ( − ) ( t ) >< ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t + ι ) >< ˆ E (+) ( t ) > + [ E (+)1 ( t ) + E (+)2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) >< ˆ E ( − ) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E (+) ( t + ι ) >< ˆ E (+) ( t ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )] < ˆ E (+) ( t ) >< ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) > + [ E (+)1 ( t ) + E (+)2 ( t )] < ˆ E ( − ) ( t ) >< ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E (+) ( t ) >< ˆ E (+) ( t + ι ) > + [ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ˆ E ( − ) ( t ) ˆ E (+) ( t ) >< ˆ E ( − ) ( t + ι ) > + < ˆ E ( − ) ( t ) ˆ E (+) ( t ) >< ˆ E ( − ) ( t + ι ) ˆ E (+) ( t + ι ) > . (A.3)Since the seven leading terms in the expansion of < T : ˆ I ( t ) ˆ I ( t + ι ) : > are identical to thosein < ˆ I ( t ) >< ˆ I ( t + ι ) > , they are canceled out when Eqs. (A.2) and (A.3) are plugged into15q. (A.1). Therefore, λ ( t, ι ) = [ E (+)1 ( t ) + E (+)2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ∆ ˆ E ( − ) ( t )∆ ˆ E (+) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ∆ ˆ E ( − ) ( t + ι )∆ ˆ E (+) ( t ) > + [ E (+)1 ( t ) + E (+)2 ( t )][ E (+)1 ( t + ι ) + E (+)2 ( t + ι )] < ∆ ˆ E ( − ) ( t )∆ ˆ E ( − ) ( t + ι ) > + [ E ( − )1 ( t ) + E ( − )2 ( t )][ E ( − )1 ( t + ι ) + E ( − )2 ( t + ι )] < ∆ ˆ E (+) ( t + ι )∆ ˆ E (+) ( t ) > + O ( E ) , (A.4)where O ( E ) represents the terms in E and those without E in Eqs. (A.2) and (A.3), with E being the field amplitude of the two oscillators. Substituting E (+)1 ( t ) = E e − i ( ω +Ω) t + iφ ,and E (+)2 ( t ) = E e − i ( ω − Ω) t + iφ into Eq. (A.4), one can arrive at Eq. (10) after some simplemathematical manipulations. ACKNOWLEDGMENTS
This work is supported by Huazhong University of Science and Technology through theStartup Funding for New Faculty. The authors would like to thank Mr. D.C. He for helpingpreparing Fig. 1, Fig. 2, and Fig. 4 and Ms. Y. Xiao for Fig. 3. [1] A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoe-maker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, andM. E. Zucker, Science , 325 (1992).[2] the VIRGO Collaboration, Class. Quant. Grav. , 1421 (2002).[3] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A , R4649(1996).[4] C. M. Caves, Phys. Rev. D , 1693 (1981).[5] D. F. Walls, Nature , 141 (1983).[6] H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Go β ler,K. Danzmann, and R. Schnabel, Phys. Rev. Lett. , 033602 (2008).[7] M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. , 514 (1985).
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05. Almost perfect squeezing is present in the photocurrent fluctuations,indicating the absence of the additional heterodyne noise present in other heterodyne schemes [12–14] due to vacuum fluctuations in frequency bands that are twice the heterodyne frequency Ω awayfrom the carrier. (b) Heterodyning with Ω /γ = 0 .
5. The degree of squeezing shows a tendencyof degradation due to increased quantum noise at ω ± /γ = 5. Thegreatest degree of squeezing is only 3dB (50% noise reduction), showing vacuum fluctuations at ω ± P SAA D DC mixer
A EOM P Z T
W ¢ W wW- w W+ wW¢+W- wW¢-W- w W¢-W+ w W¢+W+ w w BS(T ≈ ﹪ ) W+ wW¢-W- w W¢+W- w W¢-W+ w W¢+W+ wW- w W+ wW- w W¢-W
PLL mixer
FIG. 4. (color online) Phase locking scheme for balanced-heterodyne detection of sub-shot-noiseoptical signals. The two oscillators are phase-modulated by the same modulator at a RF frequencyΩ ′ < Ω with a modulation depth θ . The error signal is fed into a piezo-electrical transducerto change the global phase of the oscillators, after the photoelectrical signal is demodulated atfrequency Ω − Ω ′ . Obviously, the two RF signals at Ω and Ω ′ must be phase-locked to eachother. The control part for the relative phase of the two oscillators is omitted in the figure, albeitalso important as explained in the text. EOM: electro-optical modulator. PZT: piezo-electricaltransducer. BS: beamsplitter. D: photodetector. DC: electrical directional coupler. SA: spectralanalyzer. LP: loop filter. A: amplifiers. Ω and Ω ′ : signal generators. PLL: electrical phase-lockingloop.: signal generators. PLL: electrical phase-lockingloop.