Quantum Measurement of Broadband Nonclassical Light Fields
QQuantum Measurement of Broadband Nonclassical Light Fields
P. Grünwald, ∗ D. Vasylyev, J. Häggblad, and W. Vogel AG Theoretische Quantenoptik Institut für Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock, Germany Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Metrolohichna 14-b, UA-03680 Kiev, Ukraine. Department of Numerical Analysis, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (Dated: October 24, 2018)Based on the measurement of quantum correlation functions, the quantum statistical properties of spectralmeasurements are studied for broadband radiation fields. The spectral filtering of light before its detectionis compared with the direct detection followed by the spectral analysis of the recorded photocurrents. As anexample, the squeezing spectra of the atomic resonance fluorescence are studied for both types of filteringprocedures. The conditions for which the detection of the nonclassical signatures of the radiation is possibleare analyzed. For the considered example, photocurrent filtering appears to be the superior option to detectnonclassicality, due to the vacuum-noise effects in the optical filtering.
PACS numbers: 42.50.Ar, 42.79.Ci, 84.30.Vn
I. INTRODUCTION
Filtering of optical signals plays an important role in ex-perimental quantum optics. The optical field under investi-gation always includes unwanted components, or noise, con-tributing to the signal due to the imperfections and losses inconstituents of the optical setup or due to the environment sur-rounding this setup. From the quantum optical point of view,it is impossible to have no loss at all [1]. The task of theexperimenter is to minimize the inaccuracies, caused by thepresence of such noise, by proper filtering.Optical filtering is a process in which certain spectral partsof the signal are suppressed due to convolution with a filterfunction, which represents the selecting device. The mostcommon filters are glasses, specifically designed to transmitsome definite wavelengths, which are placed in the input portsof the detectors [2]. Alternatively, electric current filters canbe used in the detectors output channels [3]. These filters arerealized mainly as simple electronic pass-band filters, whichcan be easier controlled than optical filters. From the view-point of classical optics the application of both filtering tech-niques is equivalent. Thus electronic filtering techniques havebeen applied in many modern experimental setups [4]. Thisdoes not only include optical experiments, but virtually everysignal analysis in which a frequency dependent input is trans-formed into an electric current signal, e.g. in geophysics [5],acoustics [6], and electronic devices themselves [7].In the quantum domain the equivalence of optical and elec-tronic filtering is by no means obvious. On one hand the dis-parity arises since the relation between the optical and thephotoelectric current spectra strongly depends on the statis-tical properties of the optical signal field to be measured [8].On the other hand, the filtering convolutions occur at very dif-ferent stages of the light-analyzing process. Optical filters acton the quantum light itself, before the detector records theradiation field. As a consequence, such a filtering process un-avoidably introduces additional quantum noise effects into the ∗ Electronic address: [email protected] signal before it is measured. The current filtering, is a purelyclassical procedure, which is implemented after the comple-tion of the detection process of the radiation. Hence the cur-rent filtering does not add quantum noise to the data. How-ever, when broadband fields are measured, in general the de-tectors may only integrate over parts of the radiation spectrum,so that information may be lost before the currents are spec-trally analyzed. Thus, it is a cumbersome problem to iden-tify the optimal strategy for spectral measurements of quan-tum light fields of a broad spectral bandwidths. It is note-worthy that the optimal type of filtering may also depend onthe physical situation under study. For example, the opticalspectral filtering may be the preferential choice for the ex-traction of entangled photon pairs, which could be generatedin the biexciton-radiative cascade process [9] or by V-typethree-level systems in microcavities [10]. However, electronicphotocurrent filters have been useful for the measurements ofthe signal to noise ratio of light with a Gaussian statistics [8]and for the quadrature-fluctuation spectroscopy with squeezedlight [11]. In the following we shall focus on spectral corre-lation measurements, for which both types of filtering may beapplied.The theory of passive optical filters and their influence oncorrelation properties of filtered quantum light was developedin Ref. [12–16]. This topic has become of interest more re-cently since methods were developed and set up to measurearbitrary field correlation functions [17, 18]. The theoreticalconcepts, however, have proven difficult to analyze for higherorder moments. Therefore, alternative descriptions have alsobeen studied [19, 20]. Based on the above argumentation thecurrent-filtering procedure includes the implicit filtering bythe detector, which acts in a similar manner as a spectral fil-ter, as well as the classical filtering of the current signals afterdetection. The latter is a purely classical process.The aim of the present paper is to compare the spectralmeasurements of broadband radiation, based on optical andelectronic filtering. We provide a consistent theoretical ap-proach to treat the quantum noise effects in both techniques.Furthermore, detecting normally and time ordered field cor-relation functions via balanced correlation homodyning withfilters preserves the ordering from the original fields. Thus, a r X i v : . [ qu a n t - ph ] N ov the filtered fields can be used to detect nonclassicality in thesame way as for the original fields. To illustrate the results,we analyze the elementary example of the squeezing spectrumof the atomic resonance fluorescence. Our finding is that thespectral filtering light limits the ability to detect the squeezingto a greater extent than the current filtering, making the latterpreferential for this setup.The paper is organized as follows. In Sec. II we will de-scribe the techniques for measuring the spectral correlationfunctions of an optically filtered radiation field. In Sec. III theprocedure of current filtering will be analyzed. Both kinds offiltering techniques are compared in Sec. IV for the exampleof the squeezing spectrum in the resonance fluorescence of atwo-level atom. A summary and some conclusions are givenin Sec. V. II. CORRELATION PROPERTIES OF SPECTRALLYFILTERED LIGHT
From a mathematical point of view, the intrinsic spectralproperties of a light field under study are recovered by aFourier analysis of the signals obtained in the time domain. Inclassical optics this procedure is straightforward. In quantumoptics, however, the application of the spectral analysis is amore sophisticated problem, because of the time- and normal-ordering prescriptions of the field operators in the measuredcorrelation functions together with the related quantum noiseeffects [15].In order to recover the information about the spectral prop-erties of measured light, one may send the light beam througha frequency sensitive device, before detection. In classicalphysics, the spectrally filtered field is expressed by a convo-lution integral of the signal field with a filter response func-tion. The quantum theory of photodetection of optically fil-tered light contains additional difficulties, due to the quantumnoise effects introduced by the filtering procedure [12–16].Therefore, a careful analysis of correlation properties must beperformed for the filtered optical radiation fields.In Ref. [17] a universal measurement scheme has been pro-posed to measure the quantum correlation functions of light.We will briefly recall the results and refer to the paper for de-tails. A simple example of such a setup is shown in Fig. 1 ifone neglects the spectral filter (SF). The scheme can be ex-tended by adding more beamsplitters and detectors. It recordsnormally-ordered intensity correlation functions Γ ( k ) (cid:96) of thelight field ˆ E , superimposed with the local oscillator (LO). Thespecific form of these correlation functions in our scenariowill be discussed later on. These correlations are then com-bined in a binomial sum F ( k ) = k (cid:88) (cid:96) =0 ( − k − (cid:96) (cid:18) k(cid:96) (cid:19) Γ ( k ) (cid:96) ∝ (cid:104) : ˆ X kϕ : (cid:105) , (1)which is proportional to the k -th moment of the field quadra-ture ˆ X ϕ . Herein, k is the total number of detectors and (cid:96) is the number of detectors chosen on the left side of the firstbeamsplitter (BS). In this section we extend this scheme by a spectral filter (SF), thereby changing the signal field from ˆ E to ˆ E , in order to describe the measurement of filtered broadbandlight fields. SFBSBS BS
FIG. 1: The setup for four-detector correlation measurements. Thesignal field ˆ E is filtered by passing through the spectral filter ( SF ) andthen it is mixed with the local oscillator ( LO ) by a beamsplitter ( BS ).The resulting field components ˆ E ± pass through two beamsplitters BS (cid:48) and BS (cid:48)(cid:48) , and are detected by four photodetectors ( PD (cid:48) , . . . ). A. Spectral filtering of light with a single filter
Let us consider the measurement scheme proposed in [17]and restrict it to the case of four photodetectors, see Fig. 1.In this case one measures the second-order intensity correla-tion functions of the signal field superimposed with the LO,cf. Eq. (1) for k =2 . This is sufficient for the detection ofthe squeezing spectrum. The filter in this scheme must becarefully chosen; when we add more spectral filters we needto make sure that we preserve the possibility to combine themeasured data in a binomial form as in Eq. (1).The original signal field will be labeled ˆ E . After transmis-sion through the SF the resulting field ˆ E is a convolution of theunfiltered field with the filter function T f plus some (vacuum)noise field ˆ E n . Afterwards, the filtered field is superimposedwith the LO via the BS and reads as [15]: ˆ E (+) ± ( t ) = e iφ ± √ (cid:20) (cid:90) dt (cid:48) T f ( t − t (cid:48) ) ˆ E (+) ( t (cid:48) )+ ˆ E (+)n ( t ) ± i ˆ E (+) LO ( t ) (cid:21) , (2) ˆ E ( − ) ± ( t ) = (cid:104) ˆ E (+) ± ( t ) (cid:105) † , (3)where the upper indices +( − ) refer to positive(negative) fre-quency components of the fields, whereas the lower indices +( − ) refer to transmitted(reflected) parts of the incident lightby the first beamsplitter (cf. Fig. 1). The two phases φ ± thatcorrespond to the fields ˆ E ± satisfy the constraint φ + − φ − = π/ .Finally, after propagation through the other two beamsplit-ters BS (cid:48) and BS (cid:48)(cid:48) , the fields at the photodetectors are ˆ E ( ± ) (cid:48) j = e ± iφ j √ (cid:16) ˆ E ( ± )+ + ˆ E vac1 (cid:17) , i = 1 , (4) ˆ E ( ± ) (cid:48)(cid:48) j = e ± iφ j √ (cid:16) ˆ E ( ± ) − + ˆ E vac2 (cid:17) , i = 1 , , (5)where φ , are the phase differences caused by the beamsplit-ters. The terms ˆ E vac1,2 describe the vacuum contributions inthe unused input ports, which are eliminated by the normal-and time-ordering of the field correlation functions [14]. Hereit has been assumed that all the beamsplitters are symmetric,50:50 ones. As usual in homodyne measurements, the LOis a strong coherent field with amplitude E LO , such that theoperator nature of the LO -field plays no role in the observedcorrelation functions. Hence, the result is the same if we usea classical approximation for the LO, ˆ E ( − ) LO ( t )= E LO e i ( ω LO t − φ LO ) , ˆ E (+) LO = (cid:104) ˆ E ( − ) LO (cid:105) ∗ . (6)Consequently, only the signal field shows quantum effects inthe measured quantities.Let us define the following analogs of the photon numberoperator (cf. [17]): ˆ N ± = ˆ E ( − ) ± ˆ E (+) ± = 12 (cid:34) (cid:90) dt (cid:48) dt (cid:48) T ∗ f ( t − t (cid:48) ) T f ( t − t (cid:48) ) ˆ E ( − ) ( t (cid:48) ) ˆ E (+) ( t (cid:48) )+ ˆ E ( − )n ˆ E (+)n + ˆ E ( − ) ˆ E (+)n + ˆ E ( − )n ˆ E (+) + E LO ± E LO (cid:0) ˆ X ϕ + ˆ X n ,ϕ (cid:1)(cid:35) , (7)where ϕ = ϕ LO + π/ and ˆ X ϕ = ˆ˜ E (+) e − iϕ + ˆ˜ E ( − ) e iϕ , (8) ˆ X n ,ϕ = ˆ˜ E (+)n e − iϕ + ˆ˜ E ( − )n e iϕ , ˆ˜ E ( ± ) = ˆ E ( ± ) e ± iω LO t , ˆ˜ E ( ± )n = ˆ E ( ± )n e ± iω LO t , (9) ˆ E (+) = (cid:90) dt (cid:48) T f ( t − t (cid:48) ) ˆ E (+) ( t (cid:48) ) + ˆ E (+) n . (10)Here and in the following we indicate the slowly varying fieldamplitudes via a tilde. Using the definition (7), we calcu-late the field correlation functions similar to those in [17], cf.Eq. (1) with k = 2 . For (cid:96) ( ≤ (cid:96) ≤ ) photodetectors on the leftside of the setup in Fig. 1 and − (cid:96) on the right side, we getthe correlation functions Γ (2) (cid:96) = 2 − (cid:68) ◦◦ ˆ N (cid:96) + ˆ N − (cid:96) − ◦◦ (cid:69) ≤ (cid:96) ≤ . (11)Combining Eqs. (1), (7) and (11) we obtain for the spectralfiltered version of the quantity F (2) defined in Eq. (1) the ex- pression F (2) spectral = 2 − (cid:88) (cid:96) =0 ( − − (cid:96) (cid:18) (cid:96) (cid:19) (cid:68) ◦◦ ˆ N (cid:96) + ˆ N − (cid:96) − ◦◦ (cid:69) = 12 (cid:68) ◦◦ (cid:16) ˆ N + − ˆ N − (cid:17) ◦◦ (cid:69) = E LO (cid:68) ◦◦ ˆ X ϕ ◦◦ (cid:69) . (12)Here ◦◦ . . . ◦◦ denotes the normal and time ordering prescrip-tion [21]. The ordering allows the application of the binomialsummation, which leads to higher order moments of ˆ X ϕ . Us-ing Eq. (8), we may write Eq. (12) explicitly as F (2) spectral = E LO (cid:90) dt (cid:48) (cid:90) dt (cid:48) × (cid:28) ◦◦ (cid:89) i =1 (cid:20) T f ( t − t (cid:48) i ) ˆ E (+) ( t (cid:48) i ) e i ( ω LO t − ϕ ) + T ∗ f ( t − t (cid:48) i ) ˆ E ( − ) ( t (cid:48) i ) e − i ( ω LO t − ϕ ) (cid:21) ◦◦ (cid:29) . (13)This formula generalizes the result of Ref. [17] for the case ofspectrally filtered radiation fields.Performing the Fourier transformation of Eq. (12) with re-spect to the phase ϕ , we are able to reconstruct the momentsof field operators according to (cid:90) π dϕF ( n + m ) spectral e − i ( n − m ) ϕ ∝ (cid:68) ◦◦ ˆ˜ E ( − ) n ˆ˜ E (+) m ◦◦ (cid:69) , (14)with m and n being integers. For the case k = 2 Eq. (14)yields (cid:90) π dϕF (2) spectral e − i ϕ = π E LO (cid:68) ◦◦ ˆ˜ E ( − )2 ◦◦ (cid:69) , (15) (cid:90) π dϕF (2) spectral = πE LO (cid:68) ◦◦ ˆ˜ E ( − ) ˆ˜ E (+) ◦◦ (cid:69) , (16) (cid:90) π dϕF (2) spectral e i ϕ = π E LO (cid:68) ◦◦ ˆ˜ E (+)2 ◦◦ (cid:69) . (17)These moments, when expressed in terms of the signal fields,are for the case of Eq. (17) of the form (cid:68) ◦◦ ˆ˜ E (+)2 ◦◦ (cid:69) = (cid:90) dt (cid:90) dt T f ( t − t ) T f ( t − t ) × e iω LO t (cid:68) ◦◦ ˆ E (+) ( t ) ˆ E (+) ( t ) ◦◦ (cid:69) . (18)Hence, we obtained the connection between the incident lightfields, the filter functions and the fields at the detector. B. Spectral filtering of light with two filters
Let us turn to the case of two optical filters applied withinthe measurement setup. Calculating the correlations of opti-cal fields with different frequencies allows us to resolve thesqueezing spectrum. Again, the filters must be configured ina manner to allow the binomial summation.The setup is given in Fig. 2. The signal field ˆ E is split in twoequal parts and each one passes one of two different homodyn-ing setups. At the spectral filters SF and SF the signal field ˆ E transforms into the fields ˆ E and ˆ E . These fields are thenmixed with two LO s with different phases ϕ and ϕ and thenimpinge on the four detectors. The detected fields are ˆ E (+) j, ± = e iφ ± √ (cid:16) ˆ E (+) j ± i ˆ E (+) j, LO (cid:17) , (19)where each detector is numbered by the index { j, ±} , j =1 , , which refers to the corresponding subdevice in Fig. 2.The filtered fields ˆ E j are related to the unfiltered ones as ˆ E (+) j = (cid:90) dt (cid:48) j T f j ( t − t (cid:48) j ) ˆ E (+) ( t (cid:48) j ) + E (+) j, n , (20)where the response functions T f j ( t − t (cid:48) j ) describe the actionof the filter devices. For the local oscillator field in a coherentstate, the photon number operators read as ˆ N j, ± = ˆ E ( − ) j, ± ˆ E (+) j, ± = 12 (cid:16) ˆ E ( − ) j ˆ E (+) j + ˆ E ( − ) j, n ˆ E (+) j, n + ˆ E ( − ) j ˆ E (+) j, n + ˆ E ( − ) j, n ˆ E (+) j + E j, LO ± E j, LO ˆ X j,ϕ (cid:17) , (21)with ˆ X j,ϕ = (cid:0) ˆ˜ E (+) j + ˆ˜ E (+) j, n (cid:1) e − iϕ j + H.c. (22)and tilde denotes the slowly-varying field component, e.g. ˆ˜ E ( ± ) j = ˆ E ( ± ) j e ± iω j, LO t . Note also that ϕ j = ϕ j, LO + π/ . SF SF
FIG. 2: The four-detector measurement scheme for correlations ofelectromagnetic waves of different frequencies and phases. The sig-nal field ˆ E in the j -th arm of the setup ( j = 1 , ) is passing throughthe spectral filter SF j . Then it is mixed with the phase-controlled LO . The resulting beams are detected by the photodetectors PD ± .The outcomes of the photodetectors are correlated. Now we need to correlate the detected signals from bothfilter arms. Consequently, we may chose two indices (cid:96) and m with ≤ (cid:96), m ≤ k for the left and the right setup, re-spectively. The normally ordered correlation functions of thephotodetectors are Γ (1 , (cid:96),m = (cid:68) ◦◦ ˆ N (cid:96) , + ˆ N − (cid:96) , − ˆ N m , + ˆ N − m , − ◦◦ (cid:69) . (23) The upper double indices of Γ ( d ,d ) (cid:96),m indicate the depth lev-els of the homodyning measurement in each arm of the setupand are equal to half of the numbers k j ( j =1 , ) of detectorsplaced in each arm of the setup. Since in our case both indicesare equal to one we can use this setup to measure second ordercorrelation functions of two frequencies.Using Eq. (23), analogously to Eq. (1), we define F (1 , spectral = (cid:88) (cid:96) =0 1 (cid:88) m =0 ( − − (cid:96) ( − − m Γ (1 , (cid:96),m . (24)Applying the binomial formula we obtain F (1 , spectral = (cid:68) ◦◦ (cid:16) ˆ N , + − ˆ N , − (cid:17) (cid:16) ˆ N , + − ˆ N , − (cid:17) ◦◦ (cid:69) = E LO (cid:68) ◦◦ ˆ X ,ϕ ˆ X ,ϕ ◦◦ (cid:69) = E (cid:90) dt (cid:48) (cid:90) dt (cid:48) × (cid:42) ◦◦ (cid:89) j =1 (cid:34) T f j ( t − t (cid:48) j ) ˆ E (+) ( t (cid:48) j ) e i ( ω j, LO t − ϕ j ) + T ∗ f j ( t − t (cid:48) j ) ˆ E ( − ) ( t (cid:48) j ) e − i ( ω j, LO t − ϕ j ) (cid:35) ◦◦ (cid:43) , (25)which can be used for the reconstruction of the field opera-tor moments, similarly as in Eq. (14). After performing twodimensional Fourier-transformation, we arrive at (cid:68) ◦◦ ˆ˜ E ( ± )1 ˆ˜ E ( ± )2 ◦◦ (cid:69) = (cid:90) dt (cid:90) dt T ( ± ) f ( t − t ) T ( ± ) f ( t − t ) × e i ( ± ω , LO ± ω , LO ) t (cid:68) ◦◦ ˆ E ( ± ) ( t ) ˆ E ( ± ) ( t ) ◦◦ (cid:69) , (26)where T + f ( t )= T f ( t ) and T − f =[ T + f ] ∗ . This formula can becompared with Eq. (18) for the corresponding expression forone filter frequency. III. PHOTOCURRENT FILTERING
The other major technique of spectral detection used in ex-periments is based on current filtering. In this method the pho-toelectric current generated from the light field incident on thedetector is filtered. The obvious advantages are that the lightfield itself is not modified by the filter and the technical pro-cess of current filtering is much easier controlled than opticalselective devices. Furthermore, as we have mentioned above,current filtering is a classical process, since the current is al-ready the output of the detection.
A. Photocurrent filtering with one filter frequency
Let us consider the four-detector setup, shown in Fig. 3. In-stead of the optical spectral filters, four electronic filters acton the photocurrents. In the following, we analyze the mea-surement scheme in more detail.Following the procedure given in Ref. [21], that describesthe detector operation based on quantum and classical statis-tics, we introduce the ˆΓ -functions ˆΓ( t, ∆ t ) = N (cid:90) t +∆ tt dτ (cid:90) t +∆ tt dτ (cid:48) S ( τ − τ (cid:48) ) ˆ E ( − ) ( τ ) ˆ E (+) ( τ (cid:48) ) . (27)It corresponds to the observable measured by a single detec-tor. These functions hold for N identical atoms in a point-like detector setup irradiated by light within the time inter-val t, t + ∆ t ; S ( τ ) is the detector response function. In thesituation where the bandwidth of the field is much narrowerthan the detector bandwidth, the detector response is usuallyapproximated by a delta function, the so called ‘broad-band-detector approximation’. Here this is not justified and we keep S ( τ ) in the form of a general function. FIG. 3: Four-detector setup with current filtering. The outcomes ofthe photodetection measurement are filtered by the current filters T c . Now we implement the results of [21], and calculate thecorrelation of two detectors, indicated by 1 and 2, measuringover the same interval ∆ t , but from different initial times t and t , n ( t , ∆ t ) n ( t , ∆ t ) = ∞ (cid:88) m , =0 m m P m ,m ( t , ∆ t, t , ∆ t )= (cid:68) ◦◦ ˆΓ (1) ( t , ∆ t )ˆΓ (2) ( t , ∆ t ) ◦◦ (cid:69) , (28)where n ( t j , ∆ t ) is the number of ‘clicks’ in the detector j .Here, P m ,m ( t , ∆ t, t , ∆ t ) is the joint probability of emis-sion of m photoelectrons within the time interval t , t + ∆ t in detector 1 and m photoelectrons within t , t + ∆ t in de-tector . Eq. (28) is equivalent to the corresponding expres-sion for one detector in a case of non-overlapping time inter-vals t , t + ∆ t and t , t + ∆ t . For two detectors such anoverlap is not relevant. The only correlation stems from thefact, that the same light field is incident on both detectors,given by the ˆΓ -operators. The photocurrent generated in an electron multiplying de-tector can be described as i ( t ) = ge n ( t, ∆ t ) / ∆ t with g beingthe gain factor, which we assume to be constant. We thus ob-tain two-time current-correlation function of the form i ( t ) i ( t ) = g e (∆ t ) (cid:68) ◦◦ ˆΓ (1) ( t , ∆ t )ˆΓ (2) ( t , ∆ t ) ◦◦ (cid:69) . (29)The correlation function for the filtered currents, i f ( t )= (cid:90) dt (cid:48) T c ( t − t (cid:48) ) i ( t (cid:48) ) , (30)is calculated to be i f ( t ) i f ( t ) = g e (∆ t ) (cid:90) dt (cid:48) T c ( t − t (cid:48) ) (cid:90) dt (cid:48) T c ( t − t (cid:48) ) × (cid:68) ◦◦ ˆΓ (1) ( t (cid:48) , ∆ t )ˆΓ (2) ( t (cid:48) , ∆ t ) ◦◦ (cid:69) . (31)Now we can turn to the special scheme in Fig. 3 and definethe appropriate ˆΓ -operators as ˆΓ (cid:48) j ( t, ∆ t )= N (cid:90) t +∆ tt dτ (cid:90) t +∆ tt dτ (cid:48) S ( τ − τ (cid:48) ) ˆ E ( − ) (cid:48) j ( τ ) ˆ E (+) (cid:48) j ( τ (cid:48) ) , (32) ˆΓ (cid:48)(cid:48) j ( t, ∆ t )= N (cid:90) t +∆ tt dτ (cid:90) t +∆ tt dτ (cid:48) S ( τ − τ (cid:48) ) ˆ E ( − ) (cid:48)(cid:48) j ( τ ) ˆ E (+) (cid:48)(cid:48) j ( τ (cid:48) ) . (33)One prime denotes here the left arm of the detector setup,whereas double prime denotes the right arm (cf. Fig. 3).The detected fields ˆ E (cid:48) j and ˆ E (cid:48)(cid:48) j are expressed through linearcombinations of fields ˆ E − and ˆ E + and vacuum contributions.Defining ˆΓ ± ( t (cid:48) j , ∆ t )= N t (cid:48) j +∆ t (cid:90) t (cid:48) j dτ t (cid:48) j +∆ t (cid:90) t (cid:48) j dτ (cid:48) S ( τ − τ (cid:48) ) ˆ E ( − ) ± ( τ ) ˆ E (+) ± ( τ (cid:48) ) , (34)as a correlation function for the field, that would be detectedright after the signal and LO fields pass the first beamsplit-ter, one can show, after some straightforward algebra, that (cid:104) ◦◦ ˆΓ (cid:96) + ˆΓ − (cid:96) − ◦◦ (cid:105) = (cid:104) ◦◦ ˆΓ (cid:48) (cid:96) i ˆΓ (cid:48)(cid:48) − (cid:96) j ◦◦ (cid:105) , with i, j = 1 , and (cid:96) =0 , , . Then it is easy to see, that the (equal time) currentcorrelation functions for our system can be written as i + ( t ) (cid:96) i − ( t ) − (cid:96) = g e (∆ t ) (cid:90) dt (cid:48) T c ( t − t (cid:48) ) (cid:90) dt (cid:48) T c ( t − t (cid:48) ) (cid:68) ◦◦ ˆΓ (cid:96) + ˆΓ − (cid:96) − ◦◦ (cid:69) , and the subscript ± refers to the correspondingfields/detectors on the left ( + ) and right ( − ) side of thefirst beamsplitter.Having obtained the expression for the correlation func-tions we are interested in, we construct the F ( k ) current function[cf. Eq. (1)] F ( k ) current = k (cid:88) (cid:96) =0 ( − k − (cid:96) (cid:18) k(cid:96) (cid:19) n (cid:96) + n k − (cid:96) − , n ± = ∆ tge i ± . (35)(36)For our setup with k = 2 this expression reduces to F (2) current = (cid:88) (cid:96) =0 ( − − (cid:96) (cid:18) (cid:96) (cid:19) n (cid:96) + n − (cid:96) − = (cid:90) dt T c ( t − t ) (cid:90) dt T c ( t − t ) × (cid:42) ◦◦ (cid:89) i =1 (cid:16) ˆΓ + ( t i ) − ˆΓ − ( t i ) (cid:17) ◦◦ (cid:43) . (37)In turn, the fields ˆ E ± in Eq. (34) for a symmetric beamsplitterare linear combinations of signal and LO fields, leading to ˆ E ( − ) ± ˆ E (+) ± = 12 (cid:20) ˆ E ( − ) ˆ E (+) + ˆ E ( − ) LO ˆ E (+) LO ± i ˆ E ( − ) ˆ E (+) LO ∓ i ˆ E ( − ) LO ˆ E (+) (cid:21) . With the help of this relation, the difference of two correlationfunctions in Eq. (37) can be reduced to ˆΓ + ( t i ) − ˆΓ − ( t i )= N E LO (cid:90) t i +∆ tt i dτ (cid:90) t i +∆ tt i dτ (cid:48) S ( τ − τ (cid:48) ) × (cid:104) ˆ E ( − ) ( τ ) e − iω LO τ (cid:48) + iϕ + ˆ E (+) ( τ (cid:48) ) e iω LO τ − iϕ (cid:105) , (38)where Eq. (6) has been used and ϕ = ϕ LO + π/ . Thus, thefull expression for the F (2) current function becomes F (2) current = N E LO (cid:90) dt T c ( t − t ) (cid:90) dt T c ( t − t ) × (cid:28) ◦◦ (cid:89) j =1 (cid:90) t j +∆ tt j dτ j (cid:90) t j +∆ tt j dτ (cid:48) j S ( τ j − τ (cid:48) j ) × (cid:104) ˆ E ( − ) ( τ j ) e − iω LO τ (cid:48) j + iϕ + ˆ E (+) ( τ (cid:48) j ) e iω LO τ j − iϕ (cid:105) ◦◦ (cid:29) . (39)The obtained result can be compared with Eq. (13) for theradiation filtering case. One should note, that the essential dif-ference between radiation and current filtering now becomesobvious. Namely, the spectral filtering process is performedbefore the quantum mechanical averaging procedure, whereasthe current filtering acts on the averaged light field. At thesame time, one should note that the detector response func-tion acts similar to an optical spectral filter now. Hence, forboth methods, a certain degree of optical filtering is unavoid-able when dealing with broadband fields. B. Filtered current using two filter frequencies
Extending the concept of current filtering to the case of twocurrent filters tuned on different frequencies, we adopt thescheme in Fig. 4. In order to construct the F current -functionwe note the following useful relation for the field momentsbeing detected (cid:88) (cid:96) =0 1 (cid:88) m =0 ( − − (cid:96) ( − − m n (cid:96) , + n − (cid:96) , − n m , + n − m , − = n , − n , − − n , + n , − − n , − n , + + n , + n , + . (40) Here the indices , refer to photons detected in differentarms of the setup. The F current -function which involves fil-tered currents can be expressed with the help of Eq. (40) interms of the ˆΓ -operators as F (1 , current = (cid:90) dt T c ( t − t ) (cid:90) dt T c ( t − t ) × (cid:68) ◦◦ ˆΓ , − ˆΓ , − − ˆΓ , + ˆΓ , − − ˆΓ , − ˆΓ , + +ˆΓ , + ˆΓ , + ◦◦ (cid:69) . (41) FIG. 4: Modified scheme of Fig. 2 without radiation filtering butwith the current filtering devices T c j for the j -th arm of the setup( j = 1 , ). The sum inside the normal- and time-ordering in Eq. (41)can be evaluated analogously to Eq. (38), yielding (cid:89) j =1 (cid:104) ˆΓ j, + ( t j ) − ˆΓ j, − ( t j ) (cid:105) = N E LO (cid:89) j =1 (cid:90) t j +∆ tt j dτ (cid:90) t j +∆ tt j dτ (cid:48) S ( τ − τ (cid:48) ) × (cid:104) ˆ E ( − ) ( τ ) e − i ( ω j, LO τ (cid:48) − ϕ j ) + ˆ E (+) ( τ (cid:48) ) e i ( ω j, LO τ − ϕ j ) (cid:105) , (42)where ϕ j = ϕ j, LO + π/ . The full expression for F (1 , current follows as F (1 , current = N E LO (cid:90) dt T c ( t − t ) (cid:90) dt T c ( t − t ) × (cid:42) ◦◦ (cid:89) j =1 (cid:90) t j +∆ tt j dτ j (cid:90) t j +∆ tt j dτ (cid:48) j S ( τ j − τ (cid:48) j ) × (cid:104) ˆ E ( − ) ( τ j ) e − i ( ω j, LO τ (cid:48) j − ϕ j ) + ˆ E (+) ( τ (cid:48) j ) e i ( ω j, LO τ j − ϕ j ) (cid:105) ◦◦ (cid:43) . (43)Again Eq. (43) can be compared with the corresponding ex-pression (25) for the radiation filtering case.One should note, that both methods of spectral filtering canbe applied in one experimental setup as well. For the descrip-tion of this case, one will have to combine the formalisms forthe two cases above. This calculation is straightforward butlengthy. Otherwise it is interesting to compare the two meth-ods and raise the question under which conditions the differentfilterings are useful from the viewpoint of an experiment. IV. APPLICATION TO RESONANCE FLUORESCENCE
As an example for our calculations, let us now considerthe resonance fluorescence from a driven two-level atom as asource field. We are interested in nonclassical properties of theresonance fluorescence, namely the squeezing phenomenonpredicted in Refs [22], [23] and then verified experimentally[24]. Based on the influence of the two filtering processes, wediscuss, which method is preferable for this specific quantumoptical problem.The light field of interest is emitted by a free two-level atom(with the transition frequency ω ), which in turn is irradiatedwith a resonant laser field of the same frequency ω L = ω .The total emission field can be written as ˆ (cid:126) E ( (cid:126)r, t ) = ˆ (cid:126) E (+) ( (cid:126)r, t ) + ˆ (cid:126) E ( − ) ( (cid:126)r, t ) , (44) ˆ (cid:126) E (+) ( (cid:126)r, t ) = ˆ (cid:126) E (+) free ( (cid:126)r, t ) + ˆ (cid:126) E (+) s ( (cid:126)r, t ) , (45) ˆ (cid:126) E ( − ) ( (cid:126)r, t ) = (cid:16) ˆ (cid:126) E (+) ( (cid:126)r, t ) (cid:17) † , (46)where the source field is given by ˆ (cid:126) E (+) s ( (cid:126)r, t ) = (cid:126)g ( (cid:126)r − (cid:126)r A ) ˆ A ( t − | (cid:126)r − (cid:126)r A | /c ) . (47)Herein, ˆ A ab = | a (cid:105)(cid:104) b | are the atomic flip operators ( { a, b } =1 , refer to ground and excited state of an atom, respectively)evaluated at the retarded times t R = t −| (cid:126)r − (cid:126)r A | /c , with (cid:126)r A be-ing the position of the atom and (cid:126)g relates atomic operators tothe field quantities. We assume, that the free field is in the vac-uum state at the detectors. Hence, only the source field part ofEq. (47) is observed in measurements of time- and normally-ordered correlation functions. For simplicity, in the followingwe shall denote it by ˆ (cid:126) E . A. The Bloch equations
In order to evaluate the correlation functions for our filteredcorrelations as in Eqs. (25),(43) we need explicit informationabout the incident field ˆ (cid:126) E . For the basic methods to deal withatomic resonance fluorescence we refer to [21]. We start withthe optical Bloch equations that describe the time evolution ofthe radiating atom, ˙ σ = − Γ σ − i Ω R ˜ σ + 12 i Ω R ˜ σ , (48) ˙ σ = Γ σ + 12 i Ω R ˜ σ − i Ω R ˜ σ , (49) ˙˜ σ = − Γ ˜ σ + 12 i Ω R ( σ − σ ) , (50) ˙˜ σ = − Γ ˜ σ − i Ω R ( σ − σ ) , (51) where σ ab = (cid:104) ˆ A ba (cid:105) are the density matrix elements withslowly varying diagonal elements. The off-diagonal elementsare split into a fast oscillating term ∝ exp( ± iω L t ) and aslowly varying term ˜ σ ab , a (cid:54) = b . Moreover, Ω R is the Rabifrequency and Γ a , a = 1 , are the energy and phase dampingrates, respectively.Using the quantum regression theorem [21], [25], we define G ab ( τ ) = (cid:68) ˆ A ba ( τ ) ˆ A (0) (cid:69) , τ ≥ . (52)The correlation functions G ab obey the same Bloch equationsas σ ab , but the initial conditions are given by G ab (0) = δ a σ b . (53)As we deal with a continuous-wave scenario, the explicit val-ues of the initial conditions for G ab follow from the steadystate values of σ ab . The system of differential equations (48)-(51) can be solved more easily by reformulating the correla-tion functions as Laplace integrals. We define ˜ S ab ( s ) = (cid:90) + ∞ dτ e − sτ ˜ G ab ( τ ) (54)as the Laplace-transform of the slowly varying ˜ G ab func-tions (cf. [21]), which leads to algebraic equations in placeof Eqs. (48)-(51).The relevant solutions for the ˜ S ab -functions are ˜ S ( s ) = σ ( ∞ ) s + Γ − ˜ S ( s ) (55)and ˜ S ( s ) = i Ω R s ( s + Γ ) (cid:104) ˜ σ ( ∞ ) − is Ω R σ ( ∞ ) + Ω R ˜ σ ( ∞ )[( s + Γ )( s + Γ ) + Ω R ] (cid:105) , (56)which are expressed by the steady-state solutions of the den-sity matrix elements, σ ( ∞ ) = 12 Ω R Γ Γ + Ω R , (57) ˜ σ ( ∞ ) = i Ω R Γ Γ + Ω R . (58)The solutions of the system of Bloch-equations are furtherused for the calculation of the electromagnetic field correla-tion functions. Here we intend to calculate the normally or-dered squeezing spectrum as defined in [26], S sq ( ω ) = 12 π (cid:90) dτ e iωτ (cid:28) ◦◦ ∆ ˆ −→E ( τ )∆ ˆ −→E (0) ◦◦ (cid:29) , (59)where we use ∆ ˆ −→E = ˆ −→E −(cid:104) ˆ −→E (cid:105) . The squeezing spectrum (59)follows from Eqs. (55) and (56) by inserting Eqs. (47), (52),and (54). We shall now discuss the squeezing spectrum forboth spectral and current filtering of resonance fluorescencelight. B. The squeezing spectrum of filtered light
As special filters used in the detection scheme, we chooseLorentz-type filter functions with different filter frequencies ω f i ( i =1 , ), but equal pass bandwidths Γ f . For details onthe filters we refer to the Appendices A, B. Using the mea-surement scheme of Fig. 2 with Lorentzian filters SF and SF , we reconstruct the spectral function F (1 , by meansof Eq. (25), which can be related to the field moments (26) bytwo-dimensional Fourier transform. By Eq. (59), we can ex-press the squeezing spectrum as a function of ∆ ω = ω f − ω f .We characterize squeezing in the form S maxsq (∆ ω ) = 2 π Γ f | g | (cid:90) dτ Re (cid:110) (cid:68) ◦◦ ˆ E ( − )1 ( τ ) ˆ E (+)2 (0) ◦◦ (cid:69) − (cid:68) ◦◦ ˆ E (+)1 ( τ ) ˆ E (+)2 (0) ◦◦ (cid:69) (cid:111) e i ∆ ωτ (60) = 2 π Re (cid:20) σ ( ∞ )Γ + Γ f − i ∆ ω − ˜ S (Γ f − i ∆ ω ) (cid:21) , (61)which is considered for those phases of the field, for whichsqueezing is maximally pronounced.
1. Idealized filtering of light
Unless mentioned otherwise, we will in the following con-sider the atom in the purely radiative damping regime, that is Γ = 2Γ . For the special case when the pass bandwidth ofthe spectral filter goes to zero ( Γ f → ), the detected squeez-ing spectrum in Eq. (61) coincides with the one calculated inRef. [26]. This spectrum is shown in Fig. 5 for various valuesof the Rabi frequency. Squeezing is present when S maxsq < .In the ∆ ω region where this condition holds true, the fluctua-tions of the field are below the vacuum noise level. (cid:45) (cid:68)Ω (cid:144) (cid:71) S s q m a x (cid:72) (cid:68) Ω (cid:76) FIG. 5: The spectrum S maxsq for the maximally squeezed phase,for different values of the Rabi frequency Ω R / Γ : 1/2 (dotted), 1/4(dashed), 1/12 (solid). For small values of Ω / Γ (cf. with solid line inFig. 5) the term (cid:104) ◦◦ ˆ E (+)1 ˆ E (+)2 ◦◦ (cid:105) contributes stronger than (cid:104) ◦◦ ˆ E ( − )1 ˆ E (+)2 ◦◦ (cid:105) to the squeezing spectrum, resulting in a Lorentzian dip below the vacuum level. The maximal squeez-ing is obtained for Ω / Γ = in agreement with the re-sults of Refs. [22], [27]. For increasing excitations (dottedline in Fig. 5) the spectrum of inelastically scattered lightshows a pronounced peak of half-width Γ centered on thedriving frequency ω L . This peak is superimposed with theLorentzian dip of half-width Ω (cid:48) R = (cid:113) Ω + Γ . In the strong-driving limit ( Ω (cid:29) Γ ) the main contribution to S maxsq stemsfrom (cid:104) ◦◦ ˆ E ( − )1 ˆ E (+)2 ◦◦ (cid:105) . The spectrum S maxsq (∆ ω ) shows twopeaks situated at frequencies ∆ ω = ± Ω (cid:48) R that correspond to thesideband peaks of the Mollow triplet [28]. While squeezingfor small filter detuning ∆ ω is absent in this case, we alwaysfind some squeezing at higher detuning if Γ > Γ holds.For Γ = Γ the radiationless dephasing becomes as large asthe energy relaxation rate, destroying all squeezing [29]. For Γ > Γ , we obtain a negative squeezing spectrum for (∆ ω ) > R Γ Γ − Γ − Γ . (62)
2. Realistic filtering of light
We now turn to the more realistic case of non-zero filterwidth. Fig. 6 depicts the squeezing spectra for different val-ues of Γ f / Γ in the case of weak pumping Ω R / Γ = 1 / .One can clearly see, that, in contrast to the idealized filter-ing ( Γ f = 0 ), realistic values of the spectral filter band-widths significantly reduce the accessible squeezing effect, es-pecially for small filter detunings ∆ ω . For Γ f / Γ = 1 / ,the squeezing effect is preserved for ∆ ω > Γ , whereas forhigher values of the filterwidth only a small squeezing ef-fect can be observed. Increasing the filter bandwidths fur-ther quickly destroys the squeezing effect, which almost dis-appears already for Γ f / Γ = 1 / . However, similar to thecase of idealized filtering, we find some squeezing for suffi-ciently large filter detuning. The former condition for nega-tivity generalizes to Γ > Γ + Γ f . (63)Hence,with respect to the possibility of detecting squeezing,the filter bandwidth acts like a radiationless dephasing. Notealso, that squeezing, which is lost through dephasing, cannotbe recovered by optical filtering.It should be noted at this point, that in all calculations weneglect the effect of back action of light reflected by the opti-cal filter, compare [15]. This means, we assumed the spectralfilters to be slightly tilted with respect to the light to be mea-sured, to suppress effects of the fields reflected from the filterto interfere with the original signal field. In turn, the reasonfor the reduction of squeezing is not due to back action in thisscenario. Spectral filters, which are narrow compared to thesqueezing spectrum of the signal field, act like delta functionsreproducing the original field under convolution. Therefore,narrow optical filters, while diminishing the intensity of thelight field substantially, are better for detecting squeezing. (cid:45) (cid:68)Ω (cid:144) (cid:71) S s q m a x (cid:72) (cid:68) Ω (cid:76) FIG. 6: The spectrum S maxsq for Ω R / Γ = 1 / and different valuesof the pass bandwidth Γ f / Γ : 1/3 (dotted), 1/10 (dashed), 1/100 (dash-dotted), 0 (solid). (cid:45) (cid:87) R (cid:144) (cid:71) S s q m a x (cid:72) (cid:68) Ω (cid:76) FIG. 7: The spectrum S maxsq for ∆ ω = 0 and different values of thepass-band width Γ f / Γ : / (dotted), / (dashed), (solid). In Fig. 7 we show the squeezing spectrum at ∆ ω = 0 . Ap-plying Eq. (61) to the case of idealized filters, we obtain forthe maximal squeezing at ∆ ω = 0 S maxsq (0) = 2 σ Γ π Γ Γ + 2Ω R − Γ [Γ Γ + Ω R ] . (64)For Γ > Γ , there are values of the driving Ω R , for whichsqueezing can be observed. However, if we include a nonzerofilter width Γ f , no squeezing occurs at all at ∆ ω = 0 , as S maxsq (0) = 2 σ π Γ f (Γ + Γ f ) (Γ + Γ f )(Γ + Γ f ) + Ω R . (65)As it is also seen from Fig. 6, for a nonzero filter bandwidthone also needs nonzero ∆ ω values to observe some nonclas-sical effect.The peak arising at ∆ ω = 0 for nonzero Γ f can also beseen in the strong-driving-field limit Ω (cid:29) Γ in Fig. 8. Ad-ditionally, the two Mollow sideband peaks are visible in sucha scenario. As discussed in Ref. [30] a very similar effectwas observed in the emission spectrum from a two-level atomdriven by a strong coherent field to which an appropriate noisehas been added. (cid:45) (cid:45) (cid:45) (cid:68)Ω (cid:144) (cid:71) S s q m a x (cid:72) (cid:68) Ω (cid:76) FIG. 8: The squeezing spectrum S maxsq for the high-driving-field limit( Ω / Γ =4 ) for different values of the optical filter pass-band width Γ f / Γ : / (dotted), / (dashed), (solid). C. The squeezing spectrum for current filtering
We now turn to the discussion of the current filtering proce-dure for squeezed light from resonance fluorescence. We usethe scheme of Fig. 4 with Lorentz-type filters, for which thefilter frequencies are chosen symmetrical relative to the laserfrequency ω L or, equivalently, to the resonance frequency ω of the signal field. The squeezing spectrum is calculated asfollows. One first calculates the total squeezing spectrum S by using Eq. (43), for the filtering of the photocurrents as inthe scheme of Fig. 4. To compare with typical experimen-tal procedures, in a next step the signal field is switched offand the corresponding correlations give the photon shot noisespectrum S sn . The difference S sq = S−S sn is the squeezingspectrum of resonance fluorescence as it would be determinedin an experiment. It is derived by specifying Eq. (43) forLorentzian filter functions with equal bandwidths Γ c and set-ting frequencies ω c j ( j = 1 , , cf. Appendix B. By varyingthe phases φ and φ of the local oscillator, one can reachthe maximum squeezing effect, S sq = S maxsq . Here we considerthe case of resonance between the fluorescence and the meandetection frequency.Unfortunately, the filtered squeezing spectrum cannot begiven in a closed form for the current filtering as it was pos-sible for the optical filtering in Eq. (61). From the analysis ofFig. 9 it is evident that the current filtering with narrow-bandfilters is more suitable for the detection of squeezing than theradiation filtering with the same bandwidth parameters. Thisis especially evident for frequencies ∆ ω close to zero (com-pare the dashed-dotted with the dotted line). We conclude thatthe current filtering procedure is more suitable for analyzingthe squeezing properties of light than the schemes involvingthe spectral filters.For broadband current filters we obtain a flat squeezingspectrum, indicating a small observed squeezing effect only.Thus, we conclude that the narrow-band current filtering pro-cedure is the most appropriate among other possibilities con-sidered in the present paper. We also note that the relativepositions of the local oscillator and current filter frequencies0 FIG. 9: The squeezing spectrum S maxsq obtained from filtered pho-tocurrents (dashed-dotted: Γ c = 1 / , dashed: Γ c = 10Γ ),compared to optical filtering with Γ f = 1 / γ (dotted) andideal squeezing spectrum (solid). The spectra are calculated for Ω R / Γ =1 / . The inset shows the position of the central filter fre-quencies ω c j and local oscillator frequencies ω j, LO , j = 1 , withrespect to the laser frequency ω L . as indicated in the inset of Fig. 9 are optimized for the detec-tion of squeezing. For other possible frequencies one obtainssqueezing effects in a very small ∆ ω range close to zero. V. SUMMARY AND CONCLUSIONS
Based on the method of balanced homodyne correlationmeasurements we have studied the influences of the radia-tion field- and the photocurrent filtering on spectral correlationmeasurements of general quantum correlations of light. Wehave considered in detail the two different spectral measure-ment schemes for second-order field correlation functions andderived the connection between the original signal fields andthe filtered field correlations, which are eventually detected.The theory has been formulated for normal- and time-orderedcorrelation functions of second order in the field operators.The general results have been illustrated for the example ofthe squeezing spectra of the resonance fluorescence of a two-level atom. Both the filtering of the radiation field and thefiltering of the photocurrent have been analyzed. For the lat-ter technique the optimal setting of the local oscillator and thecurrent filter frequencies have been determined. Optical fil-tering substantially limits the available squeezing, that can bedetected. Only for different filter resonance frequencies andvery small filter bandwidth, squeezing can be observed. Onthe other hand, the current filtering is a powerful technique toanalyze spectral correlation effects for the considered exampleof squeezing in atomic resonance fluorescence. In particular,it has been demonstrated that the current filtering scheme isbetter suited for the measurement of squeezing than the setupwith optical spectral filters. This feature, together with thelower costs and better controllability of current filters in com-parison with the spectral ones, makes current filtering more favorable for the experimental study of nonclassical light.
Acknowledgments
PG, DV and WV gratefully acknowledge support by theDeutsche Forschungsgemeinschaft (DFG) through SFB 652.DV has benefited from discussions with D. Karnaushenko,M. de Oliveira and A.A. Semenov. DV also acknowledgesthe support by the project N 0113U001093 of the NationalAcademy of Sciences of Ukraine.
Appendix A: Lorentzian radiation filter
Let us consider the measurement scheme in Fig. 2 with twofilters SF j , j = 1 , . In order to simulate the action of thefilter on the incident light field, we apply the special case of aLorentzian filter function, which is a very typical elementaryfilter type. The shape of the function for the j th filter in thetime domain reads as T f j ( t ) = Θ( t )Γ f e − Γ f t − iω f j t , (A1)where ω f j is the characteristic frequency of the j th filter and Γ f the pass bandwidth, which is the same for both filters. Theunit step function Θ( t ) ensures causality. One obtains afterthe substitution in Eq. (25) the following expression for thefunction F (1 , spectral F (1 , spectral = E Γ (cid:90) t dt (cid:48) (cid:90) t dt (cid:48) e − Γ f (2 t − t (cid:48) − t (cid:48) ) × (cid:42) ◦◦ (cid:89) j =1 (cid:34) ˆ E (+) ( t (cid:48) j ) e − iω f j t (cid:48) j e i ( ω j, LO − ω f j ) t − iϕ j + ˆ E ( − ) ( t (cid:48) j ) e iω f j t (cid:48) j e − i ( ω j, LO − ω f j ) t + iϕ j (cid:35) ◦◦ (cid:43) . (A2)For the case of both filters having the same central passfrequency as the respective phase shifted LO -fields, ω j, LO = ω f j , we obtain (cid:68) ◦◦ ˆ˜ E (+)1 ˆ˜ E (+)2 ◦◦ (cid:69) = Γ f (cid:90) t dt (cid:48) (cid:90) t dt (cid:48) e − Γ f (2 t − t (cid:48) − t (cid:48) ) × e i ( ω f t (cid:48) + ω f t (cid:48) ) (cid:28) ◦◦ ˆ E (+) ( t (cid:48) ) ˆ E (+) ( t (cid:48) ) ◦◦ (cid:29) . (A3)Further factorizing the incoming fields into slowly varyingamplitude and fast oscillating term with mean frequency ω , ˆ E ( ± ) = ˆ˜ E ( ± ) e ∓ iω t , (A4)and denoting the frequency difference by ∆ ω = ω f − ω f weget in terms of new variables τ j = t − t (cid:48) j for stationary fields thefollowing expression (cid:68) ◦◦ ˆ˜ E (+)1 ˆ˜ E (+)2 ◦◦ (cid:69) = Γ f (cid:90) t dτ (cid:90) t dτ e − Γ f ( τ + τ ) × e − i ∆ ω ( τ − τ ) (cid:68) ◦◦ ˆ˜ E (+) ( τ − τ ) ˆ˜ E (+) (0) ◦◦ (cid:69) . (A5)1Denoting τ = τ − τ and integrating this expression over τ (cid:48) = τ + τ yields (cid:68) ◦◦ ˆ˜ E (+)1 ˆ˜ E (+)2 ◦◦ (cid:69) = Γ f − e − Γ f t ) (cid:90) t dτ e − i ∆ ωτ/ × (cid:68) ◦◦ ˆ˜ E (+) ( τ ) ˆ˜ E (+) (0) ◦◦ (cid:69) . (A6)The negative-negative frequency correlation function is ob-tained from (A6) by conjugation. In the case of negative-positive correlation we obtain (cid:68) ◦◦ ˆ˜ E ( − )1 ˆ˜ E (+)2 ◦◦ (cid:69) = Γ f (cid:90) t dt (cid:48) (cid:90) t dt (cid:48) e − Γ f (2 t − t (cid:48) − t (cid:48) ) × e i ∆ ω ( t (cid:48) + t (cid:48) ) (cid:68) ◦◦ ˆ˜ E ( − ) ( t (cid:48) ) ˆ˜ E (+) ( t (cid:48) ) ◦◦ (cid:69) . (A7)Using Eqs (A6), (A7) one can calculate the squeezing spec-trum of the filtered light. Appendix B: Lorentzian current filter
We consider the two filter frequency setup for current fil-tering from Fig. 4. For the current filter and for the detectorresponse functions the same Lorentz-type functions as for thespectral filter [cf. (A1)] are used. Namely, we assume, that T c j ( t ) = Θ( t )Γ c e − Γ c t − iω c j t , (B1) S ( t ) = Θ( t )Γ s e − Γ s t − iω s t , (B2) where ω c j ( j = 1 , ), and ω s are correspondingly current filterand detector response frequencies and Γ c , Γ s are the pass-bandwidths of the current filter and detector, respectively. Substi-tuting Eq. (B1) into Eq. (43) we arrive at F (1 , current = Γ c Γ s N E (cid:90) t dt (cid:48) (cid:90) t dt (cid:48) × e − Γ c (2 t − t (cid:48) − t (cid:48) ) e − iω c ( t − t (cid:48) ) − iω c ( t − t (cid:48) ) × (cid:28) ◦◦ (cid:89) j =1 (cid:90) t (cid:48) j +∆ tt j dτ j (cid:90) t (cid:48) j +∆ tt (cid:48) j dτ (cid:48) j e − Γ s ( τ j − τ (cid:48) j ) × (cid:20) ˆ˜ E ( − ) ( τ ) e − i ( ω s − ω ) τ j e i ( ω s − ω j, LO ) τ (cid:48) j + iϕ + ˆ˜ E (+) ( τ (cid:48) ) e i ( ω s − ω ) τ (cid:48) j e − i ( ω s − ω j, LO ) τ j − iϕ (cid:21) ◦◦ (cid:29) , (B3)where we have used the slowly varying amplitudes, ˆ E ( ± ) = ˆ˜ E ( ± ) e ∓ iω t . For the resonance condition ω = ω s Eq. (B3) can be further simplified. [1] H. A. Haus,
Electromagnetic Noise and Quantum Optical Mea-surements (Wiley, 2000).[2]
One may look into any current catalogue of scientific opticalcomponents to study the various optical filters. For example:Edmund Optics, Newport, etc. [3] A. I. Zverev,
Handbook of filter synthesis (Wiley, 1976).[4] S. Darlington, IEEE Transactions on Circuits and Systems–I:Fundamental Theory and Applications , 4 (1999).[5] D. P. Ghosh, Geophysical Prospecting , 192 (1971).[6] J. Bodenschatz, IEEE International Conference on Acoustics,Speech, and Signal Processing , 1945 (1997).[7] A. Fettweis, Proceedings of IEEE , 270 (1986).[8] H. Z. Cummins and H. L. Swinney, Progress in Optics, Vol. 8 edited by E. Wolf (North-Holland Publ., 1970), pp. 134-201.[9] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D.Gershoni, B. D. Gerardot, and P. M. Petroff, Phys. Rev. Lett. , 130501 (2006).[10] H. Ajiki, H. Ishihara, and K. Edamatsu, New J. Phys. ,033033 (2009).[11] D. V. Kuprianov and I. M. Sokolov, Zh. Eksp. Teor. Fiz. ,837 (1996) [JETP , 460 (1996)].[12] L. Knöll, W. Vogel, and D.-G. Welsch, J. Opt. Soc. Am. B ,1315 (1986).[13] J. D. Cresser, J. Phys. B: At. Mol. Phys. , 4915 (1987).[14] L. Knöll, W. Vogel, and D.-G. Welsch, Phys. Rev. A , 3803(1987). [15] L. Knöll, W. Vogel, and D.-G. Welsch, Phys. Rev. A , 503(1990).[16] U. Leonardt, J. Mod. Opt. , 1123 (1993).[17] E. Shchukin and W. Vogel, Phys. Rev. Lett. , 200403 (2006).[18] S. Gerber, D. Rotter, L. Slodi˘cka, J. Eschner, H. J. Carmichael,and R. Blatt, Phys. Rev. Lett. , 183601 (2009).[19] E. del Valle, A. Gonzalez-Tudela, F. P. Laussy, C. Tejedor, andM. J. Hartmann, Phys. Rev. Lett. , 183601 (2012).[20] A. Christ, C. Lupo, M. Reichelt, T. Meier, and C. Silberhorn,Phys. Rev. A , 023823 (2014).[21] W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH,2006).[22] D. F. Walls and P. Zoller, Phys. Rev. Lett. , 709 (1981).[23] L. Mandel, Phys. Rev. Lett. , 136 (1982).[24] Z. H. Lu, S. Bali, and J. E. Thomas. Phys. Rev. Lett. , 3635(1998).[25] M. Lax, Phys. Rev. , 2342 (1963).[26] M. J. Collett, D. F. Walls, and P. Zoller, Opt. Comm. , 145(1984).[27] S. Swain in Quantum Squeezing edited by P. D. Drummond, Z.Ficek (Springer, 2004) pp. 263-309.[28] B. R. Mollow, Phys. Rev. , 1969 (1969).[29] P. Grünwald and W. Vogel, Phys. Rev. A , 023837 (2013).[30] P. Zhou and S. Swain, Phys. Rev. Lett.82