Excess-noise suppression for a squeezed state propagating through random amplifying media via wave-front shaping
aa r X i v : . [ qu a n t - ph ] F e b Excess-noise suppression for a squeezed state propagatingthrough random amplifying media via wave-front shaping
Dong Li , , Song Sun , , Yao Yao , † Microsystems and Terahertz Research Center, China Academy of EngineeringPhysics, Chengdu Sichuan 610200, P. R. China Institute of Electronic Engineering, China Academy of Engineering Physics,Mianyang Sichuan 621999, P. R. China † yaoyao [email protected] 5, 2021 Abstract
After propagating through a random amplifying medium, a squeezed state com-monly shows excess noise above the shot-noise level. Since large noise can signifi-cantly reduce the signal-to-noise ratio, it is detrimental for precision measurement.To circumvent this problem, we propose a noise-reduction scheme using wavefrontshaping. It is demonstrated that the average output quantum noise can be effec-tively suppressed even beyond the shot-noise limit. Both the decrease on amplifica-tion strength and the increase on input squeezing strength can give rise to a decreasein the suppressed average quantum noise. Our results not only show the feasibilityof manipulating the output quantum noise of random amplifying media, but alsoindicate potential applications in quantum information processing in complex envi-ronments, such as, quantum imaging, quantum communication, and quantum keydistribution.
The random medium exhibits unusual transmission properties which couples light intodifferent channels randomly by multiple scattering. Previously, light scattering wasconsidered harmful, since it may distort the incident wavefront and result in a specklepattern. Later, it is shown that light scattering could also play a positive role in manyapplications. For instance, (1) in imaging, it can improve the resolution by overcomingthe traditional diffraction limit, owing to the increased effective aperture number bymultiple scattering [1, 2]; (2) in optical communication, it provides the possibility toincrease the capacity by the raising number of scattered modes that carry the information[3, 4]. In addition, light scattering can also be applied in other fields, such as, secureauthentication [5, 6], high-speed random number generator [7, 8], programmable opticalcircuit [9, 10]. Therefore, light transport through random media has become an activesubject from both theoretical and experimental perspectives.1n particular, the random amplifying media (RAMs) have attracted considerableattention because nonlinearity or amplification provides an additional degree of freedomfor coherent control of mesoscopic transport [11, 12]. By adjusting the amplificationstrength, one could conveniently manipulate the transmission properties of light whichcould benefit for many potential applications, such as, random laser [13, 14, 15].Recently, coherent-state light propagation through a RAM has been explored fromdifferent aspects. For example, Liew et al. [12] investigated the effect of amplificationon the reflection properties. It was revealed that the amplification could minimize thereflectance of the random medium by destructive interference. Burkov et al. [16] studiedthe correlation of scattered light. It was demonstrated that the angular correlation hasa power-law decay and exhibits oscillations. Patra et al. [17] analyzed the quantumnoise of the scattered light. It is found that the output shows excess noise related to thetransmission and reflection matrices of the medium for a coherent-state input.As a typical nonclassical state, the squeezed state is of importance since it possesseslower quantum noise in one quadrature component than that of the coherent state ( or equivalently the shot noise) [18, 19, 20, 21]. Therefore, the squeezed state can enhancesignal-to-noise ratio [22, 23, 24] and has been utilized in different applications rangingfrom quantum imaging [25, 26] to gravitational wave detection [27, 28, 29].However, the squeezed state suffers from an increase in output quantum noise afterpropagating through a RAM [30] (see Fig. 1(a)), which is induced by spontaneousemission and multiple scattering. It is worth pointing out that for a squeezed-stateinput, the input quantum noise is below the shot-noise level (SNL), whereas the outputnoise is always increased, even above the SNL. More interestingly, compared with thecoherent state, the squeezed state initially possessing lower noise will have larger noisein the output [30]. As is well known, the large noise leads to a decrease in the signal-to-noise ratio which is detrimental for precision detection (e.g., high-resolution imaging).Therefore, we wonder whether there exists a method to reduce the average output noisefor the squeezed-state input.Wavefront shaping (WFS) is a promising technology for optical focusing and imagingthrough random media [31, 32, 33, 34, 35, 36, 37, 38], which paves the way in manip-ulating the speckle pattern in an expected manner. Experimentally, the WFS can beperformed by a spatial light modulator (SLM), as shown in Fig. 1(b). The SLM acts asa reconfigurable matrix of pixels to imprint desired phases on the incident wavefront. Inrecent decades, it has been extensively used in various optical applications, for instance,quantum simulator [39], quantum data locking [40], and high-resolution imaging [41, 42].In particular, the WFS is also a common technique in the optical authentication schemebased on scattering medium [5, 6].In this work, we propose a noise-reduction scheme using WFS for the case of squeezed-state light propagating through RAMs. Comparing with Ref. [30], we exploit the tech-nique of WFS to reduce the output noise. In addition, the comparison between the linearand amplifying cases is performed. It is found that the amplifying media always havelarger average quantum noise than that of the linear ones regardless of WFS. Besides,unlike the linear case where the suppressed quantum fluctuation always reaches below2igure 1: Quadrature fluctuation detection of beams transmitted through a randomamplifying medium (a) in the absence of WFS, (b) in the presence of WFS. ˆ a in a ′ (ˆ a in b ′ )represent the annihilation operators of the input modes and ˆ a a (ˆ a b ) the output modes.ˆ c in † c ′ denote the creation operators of the spontaneous emission modes symbolized by thewavy-dotted lines inside the media. The random amplifying medium, with the transportmean free path l , the thickness L , the amplification length L a , and the number of trans-mission channels N , is comprised by randomly distributed small active particles for lightscattering and amplification. In (a), when the beams are injected, the medium amplifiesand separates the light into different optical channels randomly. As a consequence, theoutput is in a speckle pattern. In (b), the medium amplifies and couples the beams intothe desired optical paths. Hence the output presents an ordered pattern. The WFS,performed by a spatial light modulator (SLM) in (b), controls the incident phase oflight. In our scheme, the focus is on the quadrature of the scattered mode, monitoredby homodyne detection.the SNL, the reduced quantum fluctuation can be either below or above the SNL for theamplifying case. Moreover, we provide the condition for the reduced average quantumfluctuation to reach below the SNL.This paper is organized as follows: in Sec. 2, we briefly describe the model of prop-agation of quantized light through a RAM. Sec. 3 elucidates how the WFS suppressesthe average quantum fluctuation of output modes. In Sec. 4, we compare the cases ofamplifying and linear media. Sec. 5 is devoted to the conclusion of the main results. Fig. 1(a) illustrates the propagation of quantized light through a RAM. Generally, aRAM consists of randomly distributed scattering particles with amplification either inthe background medium or in the particles themselves. When light propagates througha RAM, it would be multiple scattered and amplified.To quantitatively characterize the property of a RAM, three kernel factors are in-troduced: the transport mean free path l , the thickness L , and the amplification length L a [17]. Note that different from the linear media with only two primary parameters( L and l ) [43, 44, 45, 46], the RAMs require an extra amplification length L a = √ Dτ a to account the nonlinearity [17], where 1 /τ a is the amplification rate and D = cl/ c the velocity of light in the medium).3 .1 Propagation of quantized light through a random amplifying medium After propagating through a RAM, the scattered mode b can be expressed as [47]ˆ a b = N X a ′ =1 t a ′ b ˆ a in a ′ + N X b ′ = N +1 r b ′ b ˆ a in b ′ + X c ′ v ∗ c ′ b ˆ c in † c ′ , (1)where ˆ a b indicates the annihilation operator of scattered mode b , ˆ a in a ′ (ˆ a in b ′ ) the annihilationoperators of input modes on the left-hand side (right-) of the RAM, ˆ c in † c ′ the creationoperators of spontaneous emission modes inside the RAM, t a ′ b ( r b ′ b ) the transmission(reflection) coefficients from the input modes a ′ ( b ′ ) to the output mode b , v ∗ c ′ b theconnection between the spontaneous emission modes and the output mode b , N thenumber of transmission channels. Noticeably, the last term on the right-hand side inEq. (1) quantifies the spontaneous emission inside the RAM, with c ′ running over“objects” (e.g., atoms or molecules) and the operator ˆ c in † c ′ fulfilling the commutationrelation [ˆ c in i , ˆ c in † j ] = δ ij .Unlike the random linear medium with only transmission and reflection coefficients( t a ′ b , r b ′ b ), the RAM involves an additional spontaneous emission coefficient ( v ∗ c ′ b ), whichare subject to a constraint P a ′ | t a ′ b | + P b ′ | r b ′ b | − P c ′ | v ∗ c ′ b | = 1 (see Appendix A).The ensemble-averaged transmission, reflection, and spontaneous-emission coefficientsare given by [47] T a ′ b = 1 N sin( l/L a )sin( L/L a ) , (2) R b ′ b = 1 N sin[( L − l ) /L a ]sin( L/L a ) , (3) V b = sin( l/L a ) + sin[( L − l ) /L a ]sin( L/L a ) − , (4)where T a ′ b = | t a ′ b | , R b ′ b = | r b ′ b | , V b = P c ′ V c ′ b = P c ′ | v c ′ b | , and the overline standsfor the average over the ensemble of disorder realizations. Note that T a ′ b , R b ′ b , and V c ′ b diverge at L/L a = π which is identify as a threshold for random-laser emission. Clearly,this analysis method based on Eq. (1) can only be applied below the laser threshold (i.e., L/L a < π ). If L/L a is infinite small (ie., L/L a → T a ′ b = [1 / ( L/l )] /N , R b ′ b = [1 − / ( L/l )] /N , and V b = 0, respectively, which are exactlythe same as the coefficients for the linear case [48]. Evidently, this generalized model issuitable for both the amplifying and the linear cases.The quadrature operators are introduced as ˆ x = ˆ a † + ˆ a and ˆ p = i (ˆ a † − ˆ a ). According4o Eq. (1), the quadrature operators of scattered mode b are then written asˆ x b = P a ′ √ T a ′ b [cos φ a ′ b ˆ x in a ′ − sin φ a ′ b ˆ p in a ′ ]+ P b ′ √ R b ′ b [cos φ b ′ b ˆ x in b ′ − sin φ b ′ b ˆ p in b ′ ]+ P c ′ √ V c ′ b [cos φ c ′ b ˆ x in c ′ − sin φ c ′ b ˆ p in c ′ ] , (5)ˆ p b = P a ′ √ T a ′ b [sin φ a ′ b ˆ x in a ′ + cos φ a ′ b ˆ p in a ′ ]+ P b ′ √ R b ′ b [sin φ b ′ b ˆ x in b ′ + cos φ b ′ b ˆ p in b ′ ] − P c ′ √ V c ′ b [sin φ c ′ b ˆ x in c ′ + cos φ c ′ b ˆ p in c ′ ] , (6)where we have defined t a ′ b = √ T a ′ b e iφ a ′ b , r b ′ b = √ R b ′ b e iφ b ′ b , v ∗ c ′ b = √ V c ′ b e − iφ c ′ b , ˆ x in c ′ =ˆ c in † c ′ + ˆ c in c ′ , and ˆ p in c ′ = i (ˆ c in † c ′ − ˆ c in c ′ ). In this work we consider the situation of optical focusing through a random medium withWFS. In such a case, the expected phases, φ SLM a ′ = − φ a ′ b ( a ′ = 1 , , ..., N ) are imprintedon the incident wavefront via WFS where the output mode b corresponds to the focusedbeam. This phase modulator exactly compensates the phase retardation in the RAMfor each transmission channel which leads to a constructive interference in the outputmode b . Correspondingly, the initial input-output relation in Eq. (1) is modified asˆ a w b = N X a ′ =1 | t a ′ b | ˆ a in a ′ + N X b ′ = N +1 r b ′ b ˆ a in b ′ + X c ′ v ∗ c ′ b ˆ c in † c ′ , (7)where the superscript w stands for WFS and | t a ′ b | takes the place of the original complextransmission coefficient t a ′ b .By taking into account the WFS, based on Eq. (7), the quadrature operators of thescattered mode b now becomesˆ x w b = X a ′ p T a ′ b ˆ x in a ′ + X b ′ p R b ′ b [cos φ b ′ b ˆ x in b ′ − sin φ b ′ b ˆ p in b ′ ]+ X c ′ p V c ′ b [cos φ c ′ b ˆ x in c ′ − sin φ c ′ b ˆ p in c ′ ] , (8)ˆ p w b = X a ′ p T a ′ b ˆ p in a ′ + X b ′ p R b ′ b [cos φ b ′ b ˆ p in b ′ + sin φ b ′ b ˆ x in b ′ ]+ X c ′ p V c ′ b [cos φ c ′ b ˆ p in c ′ + sin φ c ′ b ˆ x in c ′ ] . (9)Note in passing that our scheme can be realized with a similar experimental setupas shown in Refs. [49, 50, 51]. Nevertheless, those works mainly focusing on the en-hanced intensity in the speckle pattern, whereas our work will concentrate on the reducedquantum noise of scattered modes. 5 .0 0.3 0.6 0.9 1.2 1.50.00.51.01.52.0 r A m p li f i c a t i on L / L a (cid:1) ( (cid:2) x b ) sqz - (cid:0)(cid:3)(cid:4) x bw (cid:6) (cid:5) sqz ( ) L / l A m p li f i c a t i on L / L a ( (cid:7) x b ) (cid:8) sqz - (cid:9)(cid:10)(cid:11) x bw (cid:12) (cid:13) sqz Figure 2: The difference between h (∆ˆ x b ) i and h (∆ˆ x w b ) i as a function of (a) r and L/L a ,(b) L/l and
L/L a , in the presence of a squeezed-state input ( | Ψ in i = [ ˆ D ( α ) ˆ S ( r ) | i ] ⊗ N ),with the number of transmission channels N , displacement operator ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a ,and squeezing operator ˆ S ( r ) = e ( r/ a † − ˆ a ) (the complex number α being the amplitudeand the real number r the squeezing parameter). Parameters used are (a) L/l = 6 and(b) r = 1 . The variance of operator ˆ O is defined as h (∆ ˆ O ) i ≡ h ˆ O i − h ˆ O i , (10)where ˆ O = ˆ x w b , ˆ p w b , and h ˆ O i denotes the expectation value of ˆ O . That is to say, to obtainthe variances, it requires to calculate h ˆ x w b i , h ˆ p w b i , h (ˆ x w b ) i , and h (ˆ p w b ) i .Assuming that the light is only injected on the left-hand side of the RAM [see Fig.1(b)] and the input beams of the other side are vacuum states (i.e., h ˆ x in b ′ i = h ˆ p in b ′ i = 0).According to Eqs. (8) and (9), the expectation values of ˆ x w b and ˆ p w b can be obtained h ˆ x w b i = X a ′ p T a ′ b h ˆ x in a ′ i , (11) h ˆ p w b i = X a ′ p T a ′ b h ˆ p in a ′ i . (12)Note that h ˆ x w b i ( h ˆ p w b i ) is only related to the transmitted modes h ˆ x in a ′ i ( h ˆ p in a ′ i ).Similarly, from Eqs. (8) and (9), the expectation values of (ˆ x w b ) and (ˆ p w b ) are found6o be h (ˆ x w b ) i = X a ′ a ′′ p T a ′ b T a ′′ b [ h ˆ x in a ′ ˆ x in a ′′ i ]+ X b ′ R b ′ b [cos φ b ′ b h ˆ x in2 b ′ i + sin φ b ′ b h ˆ p in2 b ′ i− cos φ b ′ b sin φ b ′ b h ˆ x in b ′ ˆ p in b ′ + ˆ p in b ′ ˆ x in b ′ i ]+ X c ′ V c ′ b [cos φ c ′ b h ˆ x in2 c ′ i + sin φ c ′ b h ˆ p in2 c ′ i− cos φ c ′ b sin φ c ′ b h ˆ x in c ′ ˆ p in c ′ + ˆ p in c ′ ˆ x in c ′ i ] , (13) h (ˆ p w b ) i = X a ′ a ′′ p T a ′ b T a ′′ b [ h ˆ p in a ′ ˆ p in a ′′ i ]+ X b ′ R b ′ b [cos φ b ′ b h ˆ p in2 b ′ i + sin φ b ′ b h ˆ x in2 b ′ i + cos φ b ′ b sin φ b ′ b h ˆ x in b ′ ˆ p in b ′ + ˆ p in b ′ ˆ x in b ′ i ]+ X c ′ V c ′ b [cos φ c ′ b h ˆ p in2 c ′ i + sin φ c ′ b h ˆ x in2 c ′ i + cos φ c ′ b sin φ c ′ b h ˆ x in c ′ ˆ p in c ′ + ˆ p in c ′ ˆ x in c ′ i ] , (14)which are universal for arbitrary input state.Inserting Eqs. (11) and (13) into Eq. (10) yields the variance h (∆ˆ x w b ) i as h (∆ˆ x w b ) i = X a ′ T a ′ b h (∆ˆ x in a ′ ) i + X a ′ = a ′′ p T a ′ b T a ′′ b [cov(ˆ x in a ′ , ˆ x in a ′′ )]+ X b ′ R b ′ b [cos φ b ′ b h (∆ˆ x in b ′ ) i + sin φ b ′ b h (∆ˆ p in b ′ ) i− φ b ′ b sin φ b ′ b cov(ˆ x in b ′ , ˆ p in b ′ )]+ X c ′ V c ′ b [cos φ c ′ b h (∆ˆ x in c ′ ) i + sin φ c ′ b h (∆ˆ p in c ′ ) i− φ c ′ b sin φ c ′ b cov(ˆ x in c ′ , ˆ p in c ′ )] , (15)where the covariance function is defined as cov( ˆ Y , ˆ Z ) ≡ ( h ˆ Y ˆ Z i + h ˆ Z ˆ Y i ) − h ˆ Y ih ˆ Z i .7y averaging over the ensemble of RAMs, we obtain h (∆ˆ x w b ) i = X a ′ T a ′ b h (∆ˆ x in a ′ ) i + X a ′ = a ′′ p T a ′ b T a ′′ b cov(ˆ x in a ′ , ˆ x in a ′′ )+ X b ′ R b ′ b [ h (∆ˆ x in b ′ ) i + h (∆ˆ p in b ′ ) i ]+ X c ′ V c ′ b [ h (∆ˆ x in c ′ ) i + h (∆ˆ p in c ′ ) i ] , (16)where we have used sin φ b ′ b = sin φ c ′ b = cos φ b ′ b = cos φ c ′ b = 1 / φ b ′ b sin φ b ′ b =cos φ c ′ b sin φ c ′ b = 0 [52].Consider squeezed states as input ( | Ψ in i = [ ˆ D ( α ) ˆ S ( r ) | i ] ⊗ N ), with the number oftransmission channels N , displacement operator ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a , and squeezing oper-ator ˆ S ( r ) = e ( r/ a † − ˆ a ) (complex number α being the amplitude and real number r the squeezing parameter). One can obtain h (∆ˆ x in a ′ ) i = e − r , h (∆ˆ x in b ′ ) i = h (∆ˆ p in b ′ ) i = h (∆ˆ x in c ′ ) i = h (∆ˆ p in c ′ ) i = 1, cov(ˆ x in a ′ , ˆ x in a ′′ ) | a ′ = a ′′ = 0. Straightforwardly, Eq. (16) can besimplified to h (∆ˆ x w b ) i sqz = 2 V b + 1 − T b (1 − e − r ) , (17)since T b + R b − V b = 1 (see Appendix A). It is obvious that with the increase of r , themodified average quantum fluctuation in Eq. (17) decreases for a given RAM.Consider r = 0 (i.e., the coherent-state input), Eq. (17) is then reduced to h (∆ˆ x w b ) i sqz → coh =2 V b + 1. For convenience, one can rewrite Eq. (17) as h (∆ˆ x w b ) i sqz = h (∆ˆ x w b ) i coh − T b (1 − e − r ) , (18)where h (∆ˆ x w b ) i coh ≡ V b + 1. One can find that h (∆ˆ x w b ) i sqz < h (∆ˆ x w b ) i coh alwayssucceeds when r >
0, which indicates that with WFS, the squeezed state has loweraverage output noise than that of the coherent state.For comparison, we calculate the average quantum fluctuation in the absence of WFS h (∆ˆ x b ) i sqz = h (∆ˆ x b ) i coh + T b [cosh(2 r ) − , (19)where h (∆ˆ x b ) i coh ≡ V b + 1 represents the average quantum fluctuation of the scatteredlight in the absence of WFS with the coherent-state input. The detailed derivation ispresent in Appendix B.Comparing Eqs. (18) and (19), one can extract the difference between the averagequantum fluctuations with and without WFS h (∆ˆ x b ) i sqz − h (∆ˆ x w b ) i sqz = T b sinh(2 r ) , (20)8 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ Without WFS, L / l = ▲ Without WFS, 6 ▲ Without WFS, 2With WFS, 10With WFS, 6With WFS, 2Coherent r R θ ( a ) L / L a = ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ Without WFS, L / L a = ▲ Without WFS, 2.5 ▲ Without WFS, 2With WFS, 3With WFS, 2.5With WFS, 2Coherent <=> r R θ ( b ) L / l = ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ Without WFS, r = ▲ Without WFS, 0.7 ▲ Without WFS, 1.0With WFS, 0.3 With WFS, 0.7With WFS, 1Coherent ?@A
BCD
EFG L / L a R θ ( c ) L / l = ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ Without WFS, L / l = ▲ Without WFS, 8 ▲ Without WFS, 10With WFS, 6 With WFS, 8With WFS, 10Coherent
HIJ L / L a R θ ( d ) r = Figure 3: The rescaled average quantum fluctuations R θ ( θ = ˆ x b , ˆ x w b ) versus r [(a), (b)]and L/L a [(c), (d)]. The curves with the triangle marks denote the situation withoutWFS while the ones without the triangle marks (except for the dashed-gray line) repre-sent the situation with WFS. The gray-dashed line stands for the average output noisewith the coherent-state input. Parameters used are: (a) L/L a = 2 .
5, (b)
L/l = 10, (c)
L/l = 10, and (d) r = 1.since h (∆ˆ x b ) i coh = h (∆ˆ x w b ) i coh is used. Fig. 2(a) [2(b)] depicts the difference between h (∆ˆ x b ) i and h (∆ˆ x w b ) i as a function of L/L a and r [ L/L a and L/l ]. It is found that thedifference is always larger than zero for a squeezed-state input ( r >
0) which implies thatthe WFS can reduce the average quantum fluctuation in the presence of a squeezed-stateinput.For convenience, we introduce the rescaled average quantum fluctuation, R θ = h (∆ θ ) i sqz / h (∆ θ ) i coh , (21)where θ = ˆ x b , ˆ x w b . Fig. 3 compares the rescaled average quantum fluctuations R θ withand without WFS as a function of r [(a), (b)] and L/L a [(c), (d)]. In Figs. 3(a)-3(d),the curves with the triangle marks denote the situations without WFS whereas thosewithout the triangle marks (except for the gray-dashed line) represent the cases withWFS. The gray-dashed line stands for the average output noise for the coherent-stateinput.As shown in Figs. 3(a)-3(d), the purple-solid, red-dashed, and blue-dashed lineswithout triangle marks are always below their corresponding lines with triangle markswhen r >
0. This implies that the WFS can always reduce the average quantum noisefor the squeezed-state input. 9
Comparison and discussion ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■▲
Without WFS, θ= x KL ■ Without WFS, p MN With WFS, x OP w With WFS, p Q bw Coherent
RST UVW XYZ [\] ^_‘d r ( Δ θ ) s q z ( e ) L / L a = L / l = ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■ ■ ■ ■ fgh ijk lmn opq L / L a ( Δ θ ) s q z ( b ) r = L / l = Figure 4: The average quantum fluctuations h (∆ θ ) i ( θ = ˆ x b , ˆ p b , ˆ x w b , or ˆ p w b ) versus (a) r and (b) L/L a . The curves with the triangle (square) marks denote the situation of θ = ˆ x b ( θ = ˆ p b ) while the blue-dashed-dotted (red-dashed) line represents the situation of θ = ˆ x w b ( θ = ˆ p w b ). The gray-dashed line labeled “coherent” stands for the average outputnoise for the coherent-state input. Parameters used are: (a) L/L a = 2 . L/l = 10, and(b) r = 0 . L/l = 10.
One important aspect neglected so far is the quadrature ˆ p w b . Similar to ˆ x w b , the averagequantum fluctuation of ˆ p w b can be cast into h (∆ˆ p w b ) i sqz = h (∆ˆ p w b ) i coh + T b ( e r − , (22)where h (∆ˆ p w b ) i coh = 2 V b + 1 means the average output quantum fluctuation for thecoherent-state input. Meanwhile, the average quantum fluctuation of ˆ p b in the absenceof WFS is found to be h (∆ˆ p b ) i sqz = h (∆ˆ p b ) i coh + T b [cosh(2 r ) − , (23)where h (∆ˆ p b ) i coh = 2 V b + 1 represents the case of the coherent-state input.Figs. 4(a) and 4(b) compare the average quantum fluctuations between ˆ x and ˆ p with and without WFS. Fig. 4(a) plots the average quantum fluctuations as a functionof r . The blue-dotted-dashed line with (without) triangle marks represents the caseof ˆ x b (ˆ x w b ) while the red-dashed line with (without) square marks denotes the case ofˆ p b (ˆ p w b ). The gray-dashed line stands for the average quantum noise for the coherent-state input. It is easy to find that with the increase of r , h (∆ˆ x w b ) i decreases whereas h (∆ˆ x b ) i , h (∆ˆ p b ) i , and h (∆ˆ p w b ) i increase. In the absence of WFS, the average quantumfluctuations of ˆ x b and ˆ p b coincide with each other. Intriguingly, in the presence of WFS,the average quantum fluctuation of ˆ x w b is smaller than that of ˆ x b whereas the averagequantum fluctuation of ˆ p w b becomes larger than that of ˆ p b , which yields that the WFS10eads to a decrease in the average quantum fluctuation of ˆ x but an increase in that of ˆ p .In the absence of WFS, the squeezed light experiences random phases when propagatingthrough the RAM. This means that both ˆ x b and ˆ p b will be the mixture of the squeezing,anti-squeezing, and quadrature component in various orientation of the original squeezedlight, which leads to the same fluctuation for both quadrature. On the other hand, theWFS removes these random phases, which results in ˆ x w b and ˆ p w b retaining the originalquadrature of the squeezed light, with additional noise added from the spontaneousemission.Fig. 4(b) depicts the average quantum fluctuations as a function of L/L a . It canbe seen that as L/L a increases, h (∆ˆ x b ) i , h (∆ˆ x w b ) i , h (∆ˆ p b ) i , and h (∆ˆ p w b ) i increase.Notably, h (∆ˆ x w b ) i is still below the gray-dashed line, which indicates that the squeezedstate has lower average output quantum noise than that of the coherent state with WFS. ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ Without WFS, L / L a = ▲ Without WFS, L / L a = ▲ Without WFS, L / L a = L / L a = L / L a = L / L a = r ( Δ x ) s q z ( a ) L / l = ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ L / l ( Δ x ) s q z ( b ) r = Figure 5: The average output quantum fluctuations h (∆ˆ x ) i sqz (ˆ x = ˆ x b , ˆ x w b ) versus (a) r and (b) L/l . The blue triangle-marked dashed, blue dashed, red triangle-marked solid,red solid, and gray dashed-dotted curves denote the cases of amplifying media withoutWFS, amplifying ones with WFS, linear ones without WFS, linear ones with WFS, andthe shot noise, respectively. Parameters used are: (a)
L/L a = 1 , L/l = 2 for amplifyingmedia and L/L a = 0 , L/l = 2 for linear ones, (b) L/L a = 1 , r = 1 for amplifying onesand L/L a = 0 , r = 1 for linear ones. To give insight into the effects of nonlinearity on the suppressed quantum noise, wecompare the amplifying and linear situations. By setting L a → ∞ (i.e., L/L a → V b = 0, amplifying effects vanishing), based on Eqs. (17) and (19), the average quantumfluctuations in linear cases with and without WFS can be expressed as h (∆ˆ x w b ) i sqz , lin = 1 − T b (1 − e − r ) , (24) h (∆ˆ x b ) i sqz , lin = 1 + T b [cosh(2 r ) − , (25)respectively, which is consistent with our previous work [46].Fig. 5 shows the average output quantum fluctuations h (∆ˆ x ) i sqz (ˆ x = ˆ x b , ˆ x w b ) versus(a) r and (b) L/l . The blue-triangle-marked-dashed, blue-dashed, red-triangle-marked-11olid, red-solid, and gray-dotted-dashed curves denote the cases of amplifying mediawithout WFS, amplifying ones with WFS, linear ones without WFS, linear ones withWFS, and the SNL, respectively.In Fig. 5(a), with the increasing of r , the average output quantum noise withoutWFS increases whereas the one with WFS decreases. This is due to the fact that theaverage output quantum noise without WFS is related to the input quantum noise,which encompasses not only the noise of squeezed quadrature ( h (∆ˆ x in a ′ ) i = e − r ) butalso the noise of anti-squeezed quadrature ( h (∆ˆ p in a ′ ) i = e r ). When r becomes large,the maximum noise ascends steeply, which provokes the increase of the average outputquantum noise. By contrast, the average output quantum noise with WFS is relatedto the input quantum noise, which includes only the squeezed noise ( h (∆ˆ x in a ′ ) i = e − r ).The noise of the anti-squeezed quadrature h (∆ˆ p in a ′ ) i = e r disappears owning to thedestructive interference of quantum noise [46, 53]. With the increase of r , the squeezednoise ( h (∆ˆ x in a ′ ) i = e − r ) decreases, which gives rise to a decrease in the average outputquantum noise. Fig. 5(b) shows the average output noise as a function of L/l . It isobvious that, with the increase of
L/l , the average output noise without WFS decreases,whereas the average output noise with WFS increases.From Figs. 5(a) and 5(b), it is found that WFS can reduce the average quantumfluctuation in both linear and amplifying cases. Nevertheless, unlike the linear case(
L/l = 0, solid lines) where the suppressed quantum noise can always reach below theSNL, the reduced average quantum noise can be either below or above the SNL for theamplifying case.The phenomenon can be explained as follows: initially, the linear situation withoutWFS shows the excess noise resulting from multiple scattering. On the contrary, theamplifying case without WFS presents the excess noise induced by not only multiplescattering but also nonlinear amplification. The WFS can effectively reduce the excessnoise from multiple scattering rather than amplification. Resultantly, for the linearcase, the excess noise is well suppressed below the SNL via WFS. Nevertheless, for theamplifying case, the excess noise can be reduced below the SNL only if the multiplescattering dominates (i.e., weak amplification strength). It is worth noting that theexcess noise from amplification can be categorized into two types: one from spontaneousemission and the other one from stimulated emission. Although the WFS is not ableto reduce the noise due to spontaneous emission, it can still effectively suppress the onefrom stimulated emission. Therefore, the WFS could still reduce the output averagenoise below the SNL in some certain condition for the amplifying case.
Sub-shot noise of light belongs to the most prominent nonclassical trait. Nevertheless,the suppressed average quantum noise can not always reach below the SNL for theamplifying case. We shall now discuss the condition for the suppressed average quantumnoise to achieve below the SNL. Assuming that the suppressed average quantum noise12 r s L / l L / L a Δ x bw The region for Δ x bw below the SNL tuv Figure 6: The region for the suppressed average quantum fluctuation to reach below theSNL. Parameter: e − r → h (∆ˆ x w b ) i sqz < V b − T b (1 − e − r ) < . (26)Combined with Eqs. (2) and (4), Eq. (26) can be rewritten assin( lL a )(1 + e − r ) + 2 sin( L − lL a ) − LL a ) < . (27)The solution is found to be L/L a < arcsin M + , (28)where the detailed derivation is shown in Appendix C, M + = [ mn + √ m − n + 4] / [2( m +1)] with n = 1 + e − r , m = (1 − p − p ) /p and p = sin( l/L a ). Fig. 6 intuitively il-lustrates the solution in Eq. (28) where for simplicity we consider the situation of thelarge squeezing strength ( e − r →
0) [54]. The colored region allows h (∆ˆ x w b ) i sqz to reachbelow the SNL. It is obvious that this condition requires a weak amplification strength.This is because the WFS can effectively reduce the excess noise induced by multiplescattering but not amplification. In summary, we investigate the effect of wavefront shaping on the average quantumnoise of scattered modes in the random amplifying media. It is demonstrated that thewavefront shaping offers the ability to reduce the average output quantum noise for asqueezed-state input. Particularly, the wavefront shaping can effectively suppress theexcess noise resulted from multiple scattering but not amplification. This reduction isowing to the destructive interference of quantum noise. In addition, both the decreaseon amplification strength and the increase on the input squeezing strength can lead toa decrease in the suppressed average noise.13t is found that unlike the linear media where the suppressed average quantum noiseis always below the shot-noise level, the reduced average quantum noise can be eitherbelow or above the shot-noise level for the amplifying case. Moreover, we provide thecondition for the suppressed noise to achieve below the shot-noise level which requiresthe amplification strength to be weak. Our results may have potential implications inquantum information processing, such as high-resolution imaging and optical authenti-cation. For instance, in the authentication system based on scattering medium [5, 6],the most vital process involves light focusing through a random medium with the helpof WFS. Our work might contribute to design this kind of authentication schemes withsqueezed-state input.
A Derivation of the summation of transmission, reflection, and spon-taneous emission coefficients
The input-output relation of a random amplifying medium is given byˆ a † b = X a ′ t ∗ a ′ b ˆ a in † a ′ + X b ′ r ∗ b ′ b ˆ a in † b + X c ′ v ∗ c ′ b ˆ c in c ′ , (29)ˆ a b = X a ′ t a ′ b ˆ a in a ′ + X b ′ r b ′ b ˆ a in b ′ + X c ′ v ∗ c ′ b ˆ c in † c ′ , where t ∗ a ′ b ( r ∗ b ′ b , v ∗ c ′ b ) is the conjugate of t a ′ b ( r b ′ b , v c ′ b ). According to the commutationrelation [ˆ a b , ˆ a † b ] = 1, one can easily obtain X a ′ | t a ′ b | + X b ′ | r b ′ b | − X c ′ | v c ′ b | = 1 , (30)where [ˆ a in i , ˆ a in † j ] = δ ij ( i, j = a ′ , b ′ ) has been used. Let T b = P a ′ | t a ′ b | , R b = P b ′ | r b ′ b | ,and V b = P c ′ | v c ′ b | . Eq. (30) can be then rewritten as T b + R b − V b = 1. B Average quantum fluctuation of the scattered light in the absenceof WFS
The variance of ˆ x b without WFS is given by h (∆ˆ x b ) i = h ˆ x b i − h ˆ x b i . (31)To obtain the variance h (∆ˆ x b ) i , it is necessary to calculate h ˆ x b i and h ˆ x b i .In the absence of WFS, the mean value of ˆ x b in Eq. (5) is found to be h ˆ x b i = P a ′ √ T a ′ b [cos φ a ′ b h ˆ x in a ′ i − sin φ a ′ b h ˆ p in a ′ i ] , (32)14ccording to Eq. (5), ˆ x b can be obtainedˆ x b = X a ′ a ′′ p T a ′ b T a ′′ b [cos φ a ′ b cos φ a ′′ b ˆ x in a ′ ˆ x in a ′′ + sin φ a ′ b sin φ a ′′ b ˆ p in a ′ ˆ p in a ′′ − cos φ a ′ b sin φ a ′′ b ˆ x in a ′ ˆ p in a ′′ − sin φ a ′ b cos φ a ′′ b ˆ p in a ′ ˆ x in a ′′ ]+ X b ′ b ′′ p R b ′ b R b ′′ b [cos φ b ′ b cos φ b ′′ b ˆ x in b ′ ˆ x in b ′′ + sin φ b ′ b sin φ b ′′ b ˆ p in b ′ ˆ p in b ′′ − cos φ b ′ b sin φ b ′′ b ˆ x in b ′ ˆ p in b ′′ − sin φ b ′ b cos φ b ′′ b ˆ p in b ′ ˆ x in b ′′ ]+ X c ′ c ′′ p V c ′ b V c ′′ b [cos φ c ′ b cos φ c ′′ b ˆ x in c ′ ˆ x in c ′′ + sin φ c ′ b sin φ c ′′ b ˆ p in c ′ ˆ p in c ′′ − cos φ c ′ b sin φ c ′′ b ˆ x in c ′ ˆ p in c ′′ − sin φ c ′ b cos φ c ′′ b ˆ p in c ′ ˆ x in c ′′ ]+ X a ′ b ′ p T a ′ b R b ′ b { [cos φ a ′ b ˆ x in a ′ − sin φ a ′ b ˆ p in a ′ ] × [cos φ b ′ b ˆ x in b ′ − sin φ b ′ b ˆ p in b ′ ] } + X a ′ c ′ p T a ′ b V c ′ b { [cos φ a ′ b ˆ x in a ′ − sin φ a ′ b ˆ p in a ′ ] × [cos φ c ′ b ˆ x in c ′ − sin φ c ′ b ˆ p in c ′ ] } + X b ′ c ′ p R b ′ b V c ′ b { [cos φ b ′ b ˆ x in b ′ − sin φ b ′ b ˆ p in b ′ ] × [cos φ c ′ b ˆ x in c ′ − sin φ c ′ b ˆ p in c ′ ] } . (33)Then h ˆ x b i is found to be h ˆ x b i = X a ′ a ′′ p T a ′ b T a ′′ b [cos φ a ′ b cos φ a ′′ b h ˆ x in a ′ ˆ x in a ′′ i + sin φ a ′ b sin φ a ′′ b h ˆ p in a ′ ˆ p in a ′′ i− cos φ a ′ b sin φ a ′′ b h ˆ x in a ′ ˆ p in a ′′ i− sin φ a ′ b cos φ a ′′ b h ˆ p in a ′ ˆ x in a ′′ i ]+ X b ′ R b ′ b [cos φ b ′ b h ˆ x in2 b ′ i + sin φ b ′ b h ˆ p in2 b ′ i− cos φ b ′ b sin φ b ′ b h ˆ x in b ′ ˆ p in b ′ + ˆ p in b ′ ˆ x in b ′ i ]+ X c ′ V c ′ b [cos φ c ′ b h ˆ x in2 c ′ i + sin φ c ′ b h ˆ p in2 c ′ i− cos φ c ′ b sin φ c ′ b h ˆ x in c ′ ˆ p in c ′ + ˆ p in c ′ ˆ x in c ′ i ] . (34)15he variance is then expressed as h (∆ˆ x b ) i = X a ′ T a ′ b [cos φ a ′ b h (∆ˆ x in a ′ ) i + sin φ a ′ b h (∆ˆ p in a ′ ) i− φ a ′ b sin φ a ′ b cov(ˆ x in a ′ , ˆ p in a ′ ))]+ X b ′ R b ′ b [cos φ b ′ b h (∆ˆ x in b ′ ) i + sin φ b ′ b h (∆ˆ p in b ′ ) i− φ b ′ b sin φ b ′ b cov(ˆ x in b ′ , ˆ p in b ′ )]+ X c ′ V c ′ b [cos φ c ′ b h (∆ˆ x in c ′ ) i + sin φ c ′ b h (∆ˆ p in c ′ ) i− φ c ′ b sin φ c ′ b cov(ˆ x in c ′ , ˆ p in c ′ )]+ X a ′ = a ′′ { p T a ′ b T a ′′ b [2 cos φ a ′ b cos φ a ′′ b × cov(ˆ x in a ′ , ˆ x in a ′′ ) + 2 sin φ a ′ b sin φ a ′′ b cov(ˆ p in a ′ , ˆ x in a ′′ ) − φ a ′ b sin φ a ′′ b cov(ˆ x in a ′ , ˆ p in a ′′ )] } , (35)where the covariance fuction is defined as cov( ˆ Y , ˆ Z ) ≡ ( h ˆ Y ˆ Z i + h ˆ Z ˆ Y i ) − h ˆ Y ih ˆ Z i .Consider the squeezed state as input ( | Ψ in i = [ ˆ D ( α ) ˆ S ( r ) | i ] ⊗ N ), by averaging overrealizations of disorder, Eq. (35) can be simplified as h (∆ˆ x b ) i sqz = 12 T b [ h (∆ˆ x in a ′ ) i + h (∆ˆ p in a ′ ) i ] + R b + V b = 2 V b + 1 + T b [cosh(2 r ) − . (36)When the input is the coherent state (i.e., r = 0), Eq. (36) can be cast into h (∆ˆ x b ) i coh = 2 V b + 1 . (37) C Derivation of Eq. (28)
Inserting Eqs. (2) and (4) into (26) arrives atsin( lL a )(1 + e − r ) + 2 sin( L − lL a ) − LL a ) < . (38)By using trigonometric formulas, Eq. (38) could be expressed as[1 + e − r − LL a )] sin( lL a ) + [cos( lL a ) − LL a ) < . (39)16q. (39) could be further cast into1 + e − r − − p − p p M < p − M , (40)where we have defined M = sin( L/L a ) and p = sin( l/L a ) for simplicity.From Eq. (40), one can obtain( m + 1) M − mnM − (4 − n ) / < , (41)where n = 1 + e − r and m = (1 − p − p ) /p .It is easy to verify that there always exists a solution for Eq. (41) M − < M < M + , (42)where M ± = mn ± √ m − n + 42( m + 1) . (43)It is worthy pointing out that M − < M + >
0. However, in our scheme, M =sin( L/L a ) > L/L a < π corresponding to the case below thelaser threshold [47]). Therefore, the solution is found to be0 < M < M + . (44)Accordingly, one can obtain L/L a < arcsin( M + ) . (45) References [1] E. Van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. Mosk,Scattering lens resolves sub-100 nm structures with visible light, Phys. Rev. Lett. , 193905 (2011).[2] C. Park, J.-H. Park, C. Rodriguez, H. Yu, M. Kim, K. Jin, S. Han, J. Shin, S. H.Ko, K. T. Nam, Y.-H. Lee, Y.-H. Cho, and Y. Park, Full-field subwavelengthimaging using a scattering superlens, Phys. Rev. Lett. , 113901 (2014).[3] S. H. Simon, A. L. Moustakas, M. Stoytchev, and H. Safar, Communication in adisordered world, Phys. Today , 38 (2001).[4] J. Tworzyd lo and C. Beenakker, Quantum optical communication rates throughan amplifying random medium, Phys. Rev. Lett. , 043902 (2002).[5] S. A. Goorden, M. Horstmann, A. P. Mosk, B. ˇSkori´c, and P. W. Pinkse, Quantum-secure authentication of a physical unclonable key, Optica , 421 (2014).176] Y. Yao, M. Gao, M. Li, and J. Zhang, Quantum cloning attacks against PUF-basedquantum authentication systems, Quantum Inf. Process. , 3311 (2016).[7] A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, Implementationof 140 gb/s true random bit generator based on a chaotic photonic integratedcircuit, Opt. Express , 18763 (2010).[8] D. Xiang, P. Lu, Y. Xu, S. Gao, L. Chen, and X. Bao, Truly random bit generationbased on a novel random brillouin fiber laser, Opt. Lett. , 5415 (2015).[9] S. R. Huisman, T. J. Huisman, T. A. Wolterink, A. P. Mosk, and P. W. Pinkse,Programmable multiport optical circuits in opaque scattering materials, Opt. Ex-press , 3102 (2015).[10] G. Marcucci, D. Pierangeli, P. W. Pinkse, M. Malik, and C. Conti, Programmingmulti-level quantum gates in disordered computing reservoirs via machine learning,Opt. Express , 14018 (2020).[11] L. Renthlei, H. Wanare, and S. A. Ramakrishna, Enhanced propagation of photondensity waves in random amplifying media, Phys. Rev. A , 043825 (2015).[12] S. F. Liew and H. Cao, Minimum reflection channel in amplifying random media,Phys. Rev. B , 224202 (2015).[13] H. Cao, Lasing in random media, Waves in Random Media , R1 (2003).[14] F. Luan, B. Gu, A. S. Gomes, K.-T. Yong, S. Wen, and P. N. Prasad, Lasing innanocomposite random media, Nano Today , 168 (2015).[15] D. V. Churkin, S. Sugavanam, I. D. Vatnik, Z. Wang, E. V. Podivilov, S. A. Babin,Y. Rao, and S. K. Turitsyn, Recent advances in fundamentals and applications ofrandom fiber lasers, Adv. Opt. Photon. , 516 (2015).[16] A. Burkov and A. Y. Zyuzin, Correlations in transmission of light through a dis-ordered amplifying medium, Phys. Rev. B , 5736 (1997).[17] M. Patra and C. Beenakker, Excess noise for coherent radiation propagatingthrough amplifying random media, Phys. Rev. A , 4059 (1999).[18] D. F. Walls, Squeezed states of light, Nature , 141 (1983).[19] D. F. Walls and G. J. Milburn, Quantum Optics (Springer Science & BusinessMedia, 2007).[20] S. Barnett and P. M. Radmore,
Methods in Theoretical Quantum Optics (OxfordUniversity, 2002).[21] A. I. Lvovsky, Squeezed light, in
Photonics: Scientific Foundations, Technologyand Applications , Vol. , pp. 121–163, (2015).1822] C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D ,1693 (1981).[23] B. Yurke, S. L. McCall, and J. R. Klauder, SU(2) and SU(1, 1) interferometers,Phys. Rev. A , 4033 (1986).[24] M. Xiao, L.-A. Wu, and H. J. Kimble, Precision measurement beyond the shot-noise limit, Phys. Rev. Lett. , 278 (1987).[25] V. N. Beskrovnyy and M. I. Kolobov, Quantum limits of super-resolution in re-construction of optical objects, Phys. Rev. A , 043802 (2005).[26] I. V. Sokolov and M. I. Kolobov, Squeezed-light source for superresolving mi-croscopy, Opt. Lett. , 703 (2004).[27] LIGO Scientific Collaboration, Enhanced sensitivity of the ligo gravitational wavedetector by using squeezed states of light, Nat. Photon. , 613 (2013).[28] L. Barsotti, J. Harms, and R. Schnabel, Squeezed vacuum states of light for grav-itational wave detectors, Rep. Prog. Phys. , 016905 (2018).[29] M. Mehmet and H. Vahlbruch, High-efficiency squeezed light generation for grav-itational wave detectors, Classical Quantum Gravity , 015014 (2019).[30] M. Patra and C. Beenakker, Propagation of squeezed radiation through amplifyingor absorbing random media, Phys. Rev. A , 063805 (2000).[31] I. M. Vellekoop and A. Mosk, Focusing coherent light through opaque stronglyscattering media, Opt. Lett. , 2309 (2007).[32] I. M. Vellekoop and A. Mosk, Phase control algorithms for focusing light throughturbid media, Opt. Commun. , 3071 (2008).[33] S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Image transmissionthrough an opaque material, Nat. Commun. , 1 (2010).[34] A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Controlling waves in spaceand time for imaging and focusing in complex media, Nat. Photon. , 283 (2012).[35] P. Hong and G. Zhang, Heisenberg-resolution imaging through a phase-controlledscreen, Opt. Express , 22789 (2017).[36] P. Hong, Two-photon imaging assisted by a thin dynamic scattering layer, Appl.Phys. Lett. , 101109 (2018).[37] O. Tzang, E. Niv, S. Singh, S. Labouesse, G. Myatt, and R. Piestun, Wavefrontshaping in complex media with a 350 khz modulator via a 1d-to-2d transform,Nat. Photon. , 788 (2019). 1938] B. Blochet, K. Joaquina, L. Blum, L. Bourdieu, and S. Gigan, Enhanced stabilityof the focus obtained by wavefront optimization in dynamical scattering media,Optica , 1554 (2019).[39] D. Pierangeli, G. Marcucci, and C. Conti, Large-scale photonic Ising machine byspatial light modulation, Phys. Rev. Lett. , 213902 (2019).[40] D. J. Lum, J. C. Howell, M. Allman, T. Gerrits, V. B. Verma, S. W. Nam, C. Lupo,and S. Lloyd, Quantum enigma machine: experimentally demonstrating quantumdata locking, Phys. Rev. A , 022315 (2016).[41] M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi,H. Ruan, A. Faraon, and C. Yang, Wavefront shaping with disorder-engineeredmetasurfaces, Nat. Photon. , 84 (2018).[42] X. Chen, W. Liu, B. Dong, J. Lee, H. O. T. Ware, H. F. Zhang, and C. Sun,High-speed 3D printing of millimeter-size customized aspheric imaging lenses withsub 7 nm surface roughness, Adv. Mater. , 1705683 (2018).[43] Y. Xu, H. Zhang, Y. Lin, and H. Zhu, Light transport in quasi-one-dimensionaldisordered waveguides composed of locally two-dimensional random square lattices,J. Mod. Opt. , 1215 (2017).[44] Y. Xu, H. Zhang, Y. Lin, and H. Zhu, Light transmission properties ininhomogeneously-disordered random media, Ann. Phys. , 1600225 (2017).[45] D. Li, Y. Yao, and M. Li, Statistical distribution of quantum correlation inducedby multiple scattering in the disordered medium, Opt. Commun. , 106 (2019).[46] D. Li and Y. Yao, Modulating quantum fluctuations of scattered light in disorderedmedia via wavefront shaping, J. Opt. Soc. Am. B , 3290 (2019).[47] V. Y. Fedorov and S. Skipetrov, Photon noise in a random laser amplifier withfluctuating properties, Phys. Rev. A , 063822 (2009).[48] P. Lodahl, Quantum correlations induced by multiple scattering of quadraturesqueezed light, Opt. Express , 6919 (2006).[49] Y. Qiao, Y. Peng, Y. Zheng, F. Ye, and X. Chen, Second-harmonic focusing bya nonlinear turbid medium via feedback-based wavefront shaping, Opt. Lett. ,1895 (2017).[50] Y. Peng, Y. Qiao, T. Xiang, and X. Chen, Manipulation of the spontaneous para-metric down-conversion process in space and frequency domains via wavefrontshaping, Opt. Lett. , 3985 (2018).[51] G. Osnabrugge, L. V. Amitonova, and I. M. Vellekoop, Blind focusing throughstrongly scattering media using wavefront shaping with nonlinear feedback, Opt.Express , 11673 (2019). 2052] In the limit of large N , coefficients t a ′ b , r b ′ b , and v c ′ b can be approximated by a com-plex Gaussian distribution, where their corresponding phase angle φ a ′ b , φ b ′ b , and φ c ′ c are uniformly distributed in the region of ( − π, π ). According to the propertyof random variables of uniform distribution [55], one can easily obtain sin φ b ′ b =sin φ c ′ b = cos φ b ′ b = cos φ c ′ b = 1 / φ b ′ b sin φ b ′ b = cos φ c ′ b sin φ c ′ b = 0.[53] F. Elste, S. Girvin, and A. Clerk, Quantum noise interference and backactioncooling in cavity nanomechanics, Phys. Rev. Lett. , 207209 (2009).[54] In fact, to the best of our knowledge, the maximum achievable value for thesqueezing parameter is around r ≈ . e − r ≈ .
03 close to zero. Thereupon, it is reasonable to assume that e − r → Statistical Optics (John Wiley & Sons, 2015).[56] H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Detection of 15 dBsqueezed states of light and their application for the absolute calibration of pho-toelectric quantum efficiency, Phys. Rev. Lett.117