Entanglement and Thouless times from coincidence measurements across disordered media
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Entanglement and Thouless times from coincidencemeasurements across disordered media
Nicolas Cherroret and Andreas Buchleitner
Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg,Hermann-Herder-Str. 3, D-79104 Freiburg, Germany (Dated: October 17, 2018)
Abstract
We show that the entanglement of an initially frequency-entangled photon pair propagatingthrough a disordered scattering medium can be read off in transmission from the coincidencecounting rate. The very same quantity also encodes the Thouless time of the medium.
PACS numbers: 42.25.Dd, 42.50.Lc, 42.65.Lm . INTRODUCTION The study of speckle patterns, irregular interference structures produced by light propa-gating through an inhomogeneous medium, is of fundamental importance in the understand-ing of coherent wave propagation in disordered media [1]. While, at first sight, a specklepattern looks perfectly random, closer analysis reveals a more complex picture. In thebackscattering direction, for instance, the light intensity is enhanced, due to the construc-tive interference of pairs of counter-propagating multiple scattering paths. This is knownas the coherent backscattering effect [2]. In transmission, speckle patterns exhibit long- andinfinite-range correlations between distant spots. The latter are, in particular, at the originof universal conductance fluctuations in electron transmission across mesoscopic conductors[3]. Once again, these highly nontrivial correlations originate from the interference betweenpairs of scattering paths which persists under disorder average, and have been studied ex-tensively during the last decades [1, 4].While our knowledge of the statistics of classical waves in disordered media is today fairlywell developed, little is known about the statistical properties of nonclassical light propa-gating in such media. Recently, an important step forward was made with the investigationof the interplay between quantum fluctuations of light, on the one hand, and the classicalfluctuations induced by a scattering medium with static disorder [5, 6] or moving scatterers[7], on the other hand. In this scenario, a given quantum state of light enters a disorderedmedium, and signatures of the quantum statistics are found in the photon number varianceanalyzed upon transmission. Later, the probability distribution of the coincidence countingrate, which defines a “two-photon speckle”, was analyzed on output of a disordered medium[8]. Two-photon speckles are relevant for studying entangled states of light, and were re-cently observed experimentally [9], but interferences between different scattering paths wereassumed to vanish under disorder average.Interference phenomena with nonclassical light in disordered media have so far only beenaddressed in the framework of one-dimensional tight-binding models [10] or in the transportof quantum states with independent photons [11]. In our present contribution, we considerthe propagation of a pair of spectrally entangled photons in an open, disordered mediumwith three-dimensional disorder. We show that the emerging interference effects bear all the information on the spectral entanglement between the two photons, and that, more2urprisingly, information on the dynamical properties of the disordered medium is also in-scribed in the interference signal. Finally, we suggest a method to access this informationexperimentally. To do so, we analyze the photon coincidence counting rate of a photon pairtransmitted through a disordered waveguide, within a continuous-mode approach (Secs. II,III). We show that interference contributions to the coincidence rate depend on two timescales, a time t η characteristic of the spectral, and, in particular, the entanglement propertiesof the photon pair, and the Thouless time t D , i.e. the typical time it takes for a photon totraverse the disordered sample by diffusion [12]. In Sec. IV, we examine how the interferencecontributions depend on these characteristic times. Finally, in Sec. V we propose a wayto measure these two times experimentally, by introducing a small delay between the twophotons before they enter the medium. In this situation, we show that interference leads toa peak in the coincidence counting rate, whose size gives direct access to t η and t D . II. MODEL
Our model is depicted in Fig. 1. A source produces pairs of photons which are scatteredfrom a disordered waveguide of length L and cross-section A . The mean free path, i.e. , theaverage distance between two consecutive scattering processes, is assumed to be small, ℓ ≪ L , such that each photon experiences multiple scattering before being collected on output bytwo detectors. In the most general situation considered in this paper (Sec. V), one of the twophotons is delayed by a time δτ with respect to the other. If we assume perfectly reflecting FIG. 1: (color online). Two photons, incident in spatial modes a and a ′ , are scattered from adisordered waveguide of length L , with a mean free path ℓ ≪ L . The coincidence counting ratebetween two outgoing modes b and b ′ is analyzed by photodetectors. The input-output relation (2)takes into account the coupling to all unoccupied (vacuum) modes α and β , which are depicted bydashed arrows. N ( ω ) =4 × k ( ω ) A/ (4 π ) of transverse spatial modes are supported, where k ( ω ) is the wave vectorat frequency ω . Note that N ( ω ) accounts for two possible propagation directions and twopolarization states of the electromagnetic field. Another important quantity characteristicof the disordered medium is the dimensionless conductance g = (4 / N ℓ/L . g is the singleparameter that controls the transport properties across the disordered waveguide, from theregime of Anderson localization ( g .
1) to the regime of diffusive multiple scattering g > g > | ψ i = R dωdω ′ S ( ω, ω ′ )ˆ a † a ( ω )ˆ a † a ′ ( ω ′ ) | vac i , wherethe continuous-mode operator ˆ a † α ( ω ) creates a photon with frequency ω in the spatial mode α . The two-photon wave-packet amplitude S ( ω, ω ′ ) contains all spectral information aboutthe photon pair. In analogy with [15], we choose S ( ω, ω ′ ) = " π p − η /t c / exp " − ( ω − ω ) − ( ω ′ − ω ) + 2 η ( ω − ω )( ω ′ − ω )4 p − η /t c e − iω ′ δτ , (1)which provides a simple and general model of a two-photon wave-packet, characterized by acentral photon frequency ω , and a single-photon coherence time t c . Experimentally, such atwo-photon wave-packet can be produced by means of parametric down conversion, in whichtwo entangled photons originate from a nonlinear crystal irradiated by a “pump” photon.The dimensionless parameter η ∈ ] − ,
1[ determines the frequency entanglement of thetwo down-converted photons, and depends on the coherence time of the pump photon, thelength of the crystal, and the phase velocities of the pump and the down-converted photonsin the crystal [16]. Different types of entanglement are distinguished by the cases η < η > η = 0. These correspond to anticorrelated, correlated and uncorrelated photons,respectively. The amplitude | η | gives the “strength” of entanglement, the limits η → ± η → −
1) correspondto S ( ω, ω ′ ) ∝ δ ( ω + ω ′ − ω ) and are produced by standard parametric-down conversionwith a monochromatic pump [17]. The possibility to generate fully correlated ( η → S ( ω, ω ′ ) ∝ δ ( ω − ω ′ ), or completely uncorrelated ( η = 0) photons, for which S can be written as the product of two independent wave-packets, has also been discussed416, 18]. Recently, an original experimental method was devised to produce photon pairswith arbitrary values of η , with the help of a superlattice of nonlinear crystals [19]. Let usfinally note that the Gaussian form of Eq. (1) is chosen for convenience, since the resultsderived herafter do not qualitatively change for slightly different shapes of S .In order to study the transport properties of the photon pair in the disordered medium,we make use of standard input-output relations between incoming and outgoing annihilationoperators [20]: ˆ a b ( ω ) = N ( ω ) X α =1 t αb ( ω )ˆ a α ( ω ) + N ( ω ) X β = N ( ω )+1 r βb ( ω )ˆ a β ( ω ) , (2)where t αb and r βb are transmission and reflection coefficients from the incoming modes α and β , respectively, to the outgoing mode b . Note that Eq. (2) fully accounts for the coupling toall vacuum states α = a, a ′ and β = b, b ′ (see Fig. 1). Creation and annihilation operatorsobey the usual bosonic commutation relations [ˆ a i ( ω ) , ˆ a † j ( ω ′ )] = δ ij δ ( ω − ω ′ ). III. COINCIDENCE COUNTING RATE FOR δτ = 0 In this section, we assume that there is no time delay between the incident photons. Wefirst examine the mean photon number h I b i = R T dt h ˆ a † b ( t )ˆ a b ( t ) i registered by a photodetectorin a given outgoing mode b , during a sampling time T . This quantity gives the probabilityto obtain a “click” from the photodetector, meaning that it collected one photon of the pair. h· · · i refers to quantum mechanical averaging over the two-photon state, and · · · to classicalensemble averaging. These two types of averaging highlight that the photon number fluctu-ates due to both, the quantum nature of the light, and the stochastic properties of the disor-dered medium. The time-dependent operators ˆ a b ( t ) = (1 / √ π ) R ω +∆ ω/ ω − ∆ ω/ dω ˆ a b ( ω )exp( − iωt )are defined via a Fourier integral over the bandwidth ∆ ω ≪ ω of the photodetector aroundthe central photon frequency ω . ∆ ω is typically much larger than the spectral bandwidth1 /t c of individual photon wave-packets. In addition, since usual photodetectors are not ableto resolve photon wave-packets, the detection time T exceeds any other time scale of theproblem, such that we have 1 /T ≪ /t c ≪ ∆ ω ≪ ω . (3)This separation of time scales allows to extend the range of frequency and time integrationsfrom −∞ to ∞ , and to approximate the number of transverse spatial modes by N ( ω ) ≃ ( ω ). We further assume that the properties of the disordered medium do not change atthe scale of ∆ ω , such that quantities like the mean free path are constant upon integration.Under these premises, the input-output relations (2) and standard diffusion theory [4] leadto h I b i = (4 /N )( ℓ/L ). Note that h I b i is independent of t c and η , i.e. the mean photonnumber does not contain any spectral information on the two-photon state [8]. We thereforeturn to the coincidence number h I b I b ′ i between two outgoing modes b and b ′ (with possibly b = b ′ ), which will be the figure of merit from now on. This quantity is defined as h I b I b ′ i = Z T dt Z T dt ′ h ˆ a † b ( t )ˆ a † b ′ ( t ′ )ˆ a b ′ ( t ′ )ˆ a b ( t ) i . (4)After ensemble averaging, this turns into h I b I b ′ i = Z ∞−∞ dω Z ∞−∞ dω ′ { K aa ′ bb ′ ( ω, ω ′ )Re [ S ∗ ( ω, ω ′ ) S ( ω ′ , ω )] } , (5)with Eqs. (2) and (3). The kernel K aa ′ bb ′ accounts for fourth-order interference betweenphoton path amplitudes multiply scattered by the disorder. These interferences are describedby combinations of products of four transmission coefficients, averaged over disorder, andrepresented by Feynmann diagrams [4]. In the diffusive limit g > K aa ′ bb ′ can be expanded in powers of 1 /g . Performing this expansion up to the first order in 1 /g ,the coincidence counting rate R reads R = h I b I b ′ ih I b i × h I b ′ i = 12 (cid:26) g C (2) (cid:18) t η t D (cid:19) + δ bb ′ (cid:20) C (1) (cid:18) t η t D (cid:19) + 23 g C (2) (cid:18) t η t D (cid:19)(cid:21)(cid:27) , (6)which is the main result of the present contribution. In Eq. (6) we introduce two char-acteristic times, t η = t c / √ − η and the Thouless time t D = L / ( π D ), with D being thephoton diffusion coefficient. Whereas t η contains the spectral information on the photonpair (through the coherence time t c and the entanglement parameter η ), t D is a quantitycharacteristic of the disordered medium, giving the typical time it takes for a photon todiffuse through the sample. The Thouless time was initially introduced by Thouless in thecontext of electronic conduction in metals [12, 21], and is, in essence, a quantity character-izing the dynamics of the disordered medium, due to its proportionality to the inverse ofthe diffusion coefficient [22–24]. The dependence of R on t η and t D is discussed in detail inthe next section. Here we first comment on the physical content of Eq. (6): its right-handside contains four terms. The first, constant term describes the independent propagation ofthe two photons in the disordered medium, and corresponds to the diagram in Fig. 2a. The6erm denoted C (1) is depicted in Fig. 2b. It results from a fourth-order interference, saidto be local , because the exchange of photon path amplitudes (the “crossing”) occurs closeto the boundary of the medium. The two terms ∝ C (2) and ∝ δ bb ′ C (2) in Eq. (6) encodeanother type of fourth-order interference, in which photon path amplitudes change partnersat some point in the multiple scattering sequence. They are represented by the diagramsof Fig. 2c and 2d, respectively [25]. Note that here the photon path amplitudes recombineand then propagate again after the crossing, such that the underlying interference processis nonlocal . FIG. 2: Leading order diagrams that contribute to the kernel K aa ′ bb ′ , and give rise to the fourterms in the right-hand side of Eq. (6). In each diagram, the two parallel lines connected bydotted “ladders” symbolize averages of products of transmission coefficients, t ij ( ω ) t i ′ j ′ ( ω ′ ), withincoming i, i ′ = a or a ′ and outgoing j, j ′ = b or b ′ directions [25]. We now discuss the structure of Eq. (6). When the coincidence counting rate is measuredin one outgoing mode, b = b ′ , such that R ≃ / C (1) +(2 / g ) C (2) ] ≃ / C (1) ], and thelocal interference is the main interference contribution to R . On the contrary, when b = b ′ , R ≃ / / g ) C (2) ] and the nonlocal interference is the main interference contributionto R . Therefore, C (1) is a “short-range” interference effect unlike C (2) , which is spatiallylong-range. These properties are reminiscent of classical optics in disordered media, wherelocal interferences explain the granularity of speckle patterns, whereas nonlocal interferences7re responsible for long-range correlations in the speckle pattern [1]. IV. INTERFERENCE, SPECTRAL INFORMATION AND THOULESS TIME
The analysis of the previous section reveals that interference, i.e. , terms proportional to C (1) and C (2) in Eq. (6), depend on the ratio t η /t D . Therefore, they contain informationabout the spectral properties of the pair, through the characteristic time t η = t c / √ − η ,and information about the dynamics of the disordered medium, through the Thouless time t D . In this section, we analyze more closely the dependence of C (1) and C (2) on this ratio.As shown in Fig. 3, both C (1) and C (2) increase with t η /t D , before they saturate at the FIG. 3: (color online). Interference contributions C (1) and C (2) to the coincidence counting rate R [see Eq. (6)], plotted as a function of t η /t D . Both saturate at unity for t η /t D ≫ asymptotic value C (1) ≃ C (2) ≃
1, for t η ≫ t D . In this limit, the disordered medium isinsensitive to the spectral properties of the photon pair, and the coincidence rate reachesa constant value. This is the monochromatic limit, in which propagation of the pair canbe treated within a single-mode approach [5, 11]. According to the definition of t η , theincrease of C (1) and C (2) can be induced in two different manners: by an increase of thecoherence time t c of individual photons, or of the entanglement parameter η . While the firstcase simply means that interference between multiply scattered photons can only occur ifthe individual photon wave-packets overlap ( i.e. , if t c is not too small), the second is lessobvious. To clarify the role of entanglement in the interference terms in Eq. (6), we show8 IG. 4: (color online). Functions C (1) (solid curves) and C (2) (dashed curves) plotted as a functionof the entanglement parameter η , for t c /t D = 0 . t c /t D = 1 (green curves) and t c /t D = 8 (blue curves). All curves saturate at unity for strongly correlated photons ( η → in Fig. 4 C (1) and C (2) as a function of the parameter η , for three different values of t c /t D .All curves increase with η and saturate at unity when η →
1. This can be understood asfollows: when η <
0, a given frequency component of one photon tends to be entangled witha different frequency component of the other (quantum anticorrelation ), and therefore doesnot interfere with it, leading to a reduction of the interference signal after averaging overdifferent realizations of the disorder. Conversely, when η >
0, a given frequency componentof the first photon tends to be entangled with the corresponding same frequency componentof the other (quantum correlation ), and therefore interferes constructively with it, leadingto an enhancement of the interference signal after averaging. From Fig. 4, we note, however,that the behavior of anticorrelated and correlated photons is not symmetric. In particular,in the limit η → t c is small [15]. V. COINCIDENCE COUNTING RATE WITH DELAYED PHOTONS
We have shown in the previous section that interference contributions to the coincidencecounting rate (6) carry information (encoded in the ratio t η /t D ) on the entanglement prop-erties of the photon pair and on the dynamics of the disordered medium. We now propose9n experimental scheme giving individual access to these two characteristic times. For thispurpose, we assume that one of the photons arrives with a small delay δτ with respect tothe other (see Fig. 1). For δτ = 0, the coincidence counting rate (6) becomes R = h I b I b ′ ih I b i × h I b ′ i = 12 (cid:26) g C (2) (cid:18) t η t D , δτt η (cid:19) + δ bb ′ (cid:20) C (1) (cid:18) t η t D , δτt η (cid:19) + 23 g C (2) (cid:18) t η t D , δτt η (cid:19)(cid:21)(cid:27) , (7) i.e. , all interference contributions acquire a dependence on δτ . This dependence originatesfrom the fact that the occurence of interference is only possible if the time intervals duringwhich the two photons propagate in the medium overlap. C (1) and C (2) are shown in themain plot of Fig. 5, as a function of δτ /t η , for several values of t η /t D . We recall that C (1) is the dominant interference contribution to R when b = b ′ (since R ≃ / C (1) ]), and C (2) when b = b ′ (since R ≃ / / g ) C (2) ]). Two different regimes can be identified FIG. 5: (color online). Main plot: functions C (1) (solid curves) and C (2) (dashed curves) plottedas a function of the time delay δτ /t η , for t η /t D = 0 . t η /t D = 2 (green curves)and t η /t D = 7 (blue curves). Inset: mean square size ( σ/t η ) of the functions C (1) ( δτ /t η ) and C (2) ( δτ /t η ), as a function of t η /t D ( σ is defined in the text). For t η /t D ≫ σ ∼ t η , whereas for t η /t D ≪ σ ∼ t D . from Fig. 5: first, when t η ≫ t D , C (1) ≃ C (2) ≃ exp[ − ( δτ /t η ) ] are peaked functions of δτ , and the mean square size σ = R ∞−∞ δτ C (1 / ( δτ ) d ( δτ ) / R ∞−∞ C (1 / ( δτ ) d ( δτ ) ≃ t η /
2, asobvious from the asymptotic behavior in the inset of Fig. 5. In this limit, measurement of10 for different time delays makes the joint coherence time t η of the photon pair accessible.In the opposite limit t η ≪ t D , C (1) and C (2) are broad functions of δτ , with a reducedvisibility. The inset of Fig. 5 clearly shows that σ ∝ t D for both C (1) ( δτ ) and C (2) ( δτ ) inthis limit. The coincidence counting rate then gives access to the Thouless time, and thusto the diffusion coefficient.In the diffusive regime considered here, nonlocal interference [which accounts for the C (2) terms in the coincidence counting rate (6)] is of weak magnitude, and is, therefore, moredifficult to measure than local interference. When b = b ′ however, R ≃ / / g ) C (2) ][see Eq. (7)], and the observation of nonlocal interference may be greatly facilitated bytaking advantage of its dependence on δτ , which allows us to distinguish it from the constantbackground of 1 /
2. An interesting problem concerns the regime where g ∼ i.e. , whenAnderson localization sets in. It was shown recently that nonlocal interferences persist inthis regime [11], where they are of the order of unity and acquire a dependence on the ratioof the sample size L and on the localization length. However, in the context of the presentwork, their dependence on the spectral properties of the photon pair and on the dynamicalproperties of the medium is less clear, in particular since the Thouless time is no longer arelevant quantity when g ∼ VI. SUMMARY AND CONCLUDING REMARKS
We have calculated the coincidence counting rate that results from the propagation ofa pair of entangled photons through a disordered medium, and we have analyzed its de-pendence on the fourth-order interference that occurs between multiple scattering photontrajectories. We have shown that these interferences convey information on the spectral en-tanglement of the pair and on the Thouless time of the disordered medium. Finally, for thepurpose of measuring this information, we have proposed an experimental scheme in whichthe photons of the pair are incident on the disordered medium with a time delay. In classicaloptics, the measurement of the Thouless time usually requires a dynamical experiment, suchas time-resolved transmission of short pulses [22, 23] or spectral correlation mesurements[24]. In the method we propose, no dynamics are involved, only the counting of photoncoincidences in the framework of a quantum optics experiment.Let us conclude with a discussion of the relevance of these results for state of the art11xperiments. Recently, multiple scattering experiments [6, 26] were performed on disorderedmedia of titania powders with thicknesses from 5 to 20 µ m, and a mean free path ℓ ≃ . µ m. For an average refractive index n ≃ .
34 [26], this implies Thouless times t D = ( c/n ) ℓ/ . × − s to 6 . × − s, with c the speed of light in vacuum. Thus, photonswith subpicosecond or subnanosecond coherence time will allow us to explore both scenariosoutlined above, t η ≪ t D and t η ≫ t D , and to quantify photon entanglement as well as thetransport characteristics of the medium. Acknowledgments
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