aa r X i v : . [ qu a n t - ph ] J un The Physics of = 1 + 1 Yanhua ShihDepartment of PhysicsUniversity of Maryland, Baltimore County,Baltimore, MD 21250
Abstract : One of the most surprising consequences of quantum mechanics is the entan-glement of two or more distant particles. In an entangled EPR two-particle system, thevalue of the momentum (position) for neither single subsystem is determined. However, ifone of the subsystems is measured to have a certain momentum (position), the other sub-system is determined to have a unique corresponding value, despite the distance betweenthem. This peculiar behavior of an entangled quantum system has surprisingly been ob-served experimentally in two-photon temporal and spatial correlation measurements, suchas “ghost” interference and “ghost” imaging. This article addresses the fundamental con-cerns behind these experimental observations and to explore the nonclassical nature oftwo-photon superposition by emphasizing the physics of 2 = 1 + 1. In quantum theory, a particle is allowed to exist in a set of orthogonal states simultaneously.A vivid picture of this concept might be Schr¨odinger’s cat, where his cat is in a state ofboth alive and dead simultaneously. In mathematics, the concepts of “alive” and “dead” areexpressed through the idea of orthogonality. In quantum mechanics, the superpositions of theseorthogonal states are used to describe the physical reality of a quantum object. In this respectthe superposition principle is indeed a mystery when compared with our everyday experience.In this article, we discuss another surprising consequence of quantum mechanics, namelythat of quantum entanglement. Quantum entanglement involves a multi-particle system ina coherent superposition of orthogonal states. Here again Schr¨odinger’s cat is a nice way ofcartooning the strangeness of quantum entanglement. Now imagine two Schr¨odinger’s catspropagating to separate distant locations. The two cats are nonclassical by means of thefollowing two criteria: (1) each of the cats is in a state of alive and dead simultaneously; (2)the two must be observed to be both alive or both dead whenever we observe them, despitetheir separation. There would probably be no concern if our observations were based on a largenumber of alive-alive or dead-dead twin cats, pair by pair, with say a 50% chance to observe adead-dead or alive-alive pair. However, we are talking about a single pair of cats with this singlepair being in the state of alive-alive and dead-dead simultaneously, and, in addition each of thecats in the pair must be alive and dead simultaneously. The superposition of multi-particlestates with these entangled properties represents a troubling concept to classical theory. Theseconcerns derive not only from the fact that the superposition of multi-particle states has noclassical counterpart, but also because it represents a nonlocal behavior which may never beunderstood classically.The concept of quantum entanglement started in 1935 [1]. Einstein, Podolsky and Rosen,suggested a gedankenexperiment and introduced an entangled two-particle system based onthe superposition of two-particle wavefunctions. The EPR system is composed of two distantinteraction-free particles which are characterized by the following wavefunction:Ψ( x , x ) = 12 π ¯ h Z dp dp δ ( p + p ) e ip ( x − x ) / ¯ h e ip x / ¯ h = δ ( x − x − x ) (1)where e ip ( x − x ) / ¯ h and e ip x / ¯ h are the eigenfunctions with eigenvalues p = p and p = − p ofthe momentum operators ˆ p and ˆ p associated with particles 1 and 2, respectively. x and x x is a constant. The EPR state is very peculiar. Although there is no interaction between thetwo distant particles, the two-particle superposition cannot be factorized into a product of twoindividual superpositions of two particles. Remarkably, quantum theory allow for such states.What can we learn from the EPR state of Eq. (1)?(1) In coordinate representation, the wavefunction is a delta function δ ( x − x − x ).The two particles are separated in space with a constant value of x − x = x , although thecoordinates x and x of the two particles are both unspecified.(2) The delta wavefunction δ ( x − x − x ) is the result of the superposition of planewavefunctions for free particle one, e ip ( x − x ) / ¯ h , and free particle two, e ip x / ¯ h , with a particulardistribution δ ( p + p ). It is δ ( p + p ) that made the superposition special. Although themomentum of particle one and particle two may take on any values, the delta function restrictsthe superposition to only those terms in which the total momentum of the system takes aconstant value of zero.Now, we transfer the wavefunction from coordinate representation to momentum represen-tation:Ψ( p , p ) = 12 π ¯ h Z dx dx δ ( x − x − x ) e − ip ( x − x ) / ¯ h e − ip x / ¯ h = δ ( p + p ) . (2)What can we learn from the EPR state of Eq. (2)?(1) In momentum representation, the wavefunction is a delta function δ ( p + p ). Thetotal momentum of the two-particle system takes a constant value of p + p = 0, although themomenta p and p are both unspecified.(2) The delta wavefunction δ ( p + p ) is the result of the superposition of plane wavefunc-tions for free particle one, e − ip ( x − x ) / ¯ h , and free particle two, e − ip x / ¯ h , with a particulardistribution δ ( x − x − x ). It is δ ( x − x − x ) that made the superposition special. Althoughthe coordinates of particle one and particle two may take on any values, the delta functionrestricts the superposition to only those terms in which x − x is a constant value of x .In an EPR system, the value of the momentum (position) for neither single subsystemis determined. However, if one of the subsystems is measured to be at a certain momentum(position), the other one is determined with a unique corresponding value, despite the distancebetween them. An idealized EPR state of a two-particle system is therefore characterized by∆( p + p ) = 0 and ∆( x − x ) = 0 simultaneously, even if the momentum and position ofeach individual free particle are completely undefined, i.e., ∆ p j ∼ ∞ and ∆ x j ∼ ∞ , j = 1 , superposition of two-particle states . The physics behind EPR states is far beyond the acceptablelimit of Einstein.Does a free particle have a defined momentum and position in the state of Eq. (1) andEq. (2), regardless of whether we measure it or not? On one hand, the momentum and positionof neither independent particle is specified and the superposition is taken over all possible valuesof the momentum and position. We may have to believe that the particles do not have anydefined momentum and position, or have all possible values of momentum and position withinthe superposition, during the course of their motion. On the other hand, if the measuredmomentum (position) of one particle uniquely determines the momentum (position) of the otherdistant particle, it would be hard for anyone who believes no action-at-a-distance to imaginethat the momenta (position) of the two particles are not predetermined with defined valuesbefore the measurement. EPR thus put us into a paradoxical situation. It seems reasonable forus to ask the same question that EPR had asked in 1935: “Can quantum-mechanical descriptionof physical reality be considered complete?” [1]2n their 1935 article, Einstein, Podolsky and Rosen argued that the existence of the entan-gled two-particle state of Eq. (1) and Eq. (2), a straightforward quantum mechanical superpo-sition of two-particle states, led to the violation of the uncertainty principle of quantum theory.To draw their conclusion, EPR started from the following criteria. Locality : there is no action-at-a-distance;
Reality : “if, without in any way disturbing a system, we can predict with certainty thevalue of a physical quantity, then there exist an element of physical reality corresponding tothis quantity.” According to the delta wavefunctions, we can predict with certainty the resultof measuring the momentum (position) of particle 1 by measuring the momentum (position)of particle 2, and the measurement of particle 2 cannot cause any disturbance to particle 1, ifthe measurements are space-like separated events. Thus, both the momentum and position ofparticle 1 must be elements of physical reality regardless of whether we measure it or not. This,however, is not allowed by quantum theory. Now consider:
Completeness : “every element of the physical reality must have a counterpart in the com-plete theory.” This led to the question as the title of their 1935 article: “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”EPR’s arguments were never appreciated by Copenhagen. Bohr criticized EPR’s criterion ofphysical reality [2]: “it is too narrow”. However, it is perhaps not easy to find a wider criterion.A memorable quote from Wheeler, “No elementary quantum phenomenon is a phenomenonuntil it is a recorded phenomenon”, summarizes what Copenhagen has been trying to teach us[3]. By 1927, most physicists accepted the Copenhagen interpretation as the standard view ofquantum formalism. Einstein, however, refused to compromise. As Pais recalled in his book,during a walk around 1950, Einstein suddenly stopped and “asked me if I really believed thatthe moon (pion) exists only if I look at it.” [4]There has been arguments considering ∆( p + p )∆( x − x ) = 0 a violation of the uncer-tainty principle. This argument is false. It is easy to find that p + p and x − x are notconjugate variables. As we know, non-conjugate variables correspond to commuting operatorsin quantum mechanics, if the corresponding operators exist. To have ∆( p + p ) = 0 and∆( x − x ) = 0 simultaneously, or to have ∆( p + p )∆( x − x ) = 0, is not a violation of theuncertainty principle. This point can easily be seen from the following two dimensional Fouriertransforms:Ψ( x , x ) = 12 π ¯ h Z dp dp δ ( p + p ) e ip ( x − x ) / ¯ h e ip x / ¯ h = 12 π ¯ h Z d ( p + p ) δ ( p + p ) e i ( p + p )( x ′ + x ) / h Z d ( p − p ) / e i ( p − p )( x ′ − x ) / h = 1 × δ ( x − x − x )where x ′ = x − x ;Ψ( p , p ) = 12 π ¯ h Z dx dx δ ( x − x − x ) e − ip ( x − x ) / ¯ h e − ip x / ¯ h = 12 π ¯ h Z d ( x ′ + x ) e − i ( p + p )( x ′ + x ) / h Z d ( x ′ − x ) / δ ( x ′ − x ) e − i ( p − p )( x ′ − x ) / h = δ ( p + p ) × . The Fourier conjugate variables are ( x + x ) ⇔ ( p + p ) and ( x − x ) ⇔ ( p − p ). Although itis possible to have ∆( x − x ) ∼ p + p ) ∼ x + x )∆( p + p ) ≥ ¯ h , and ∆( x − x )∆( p − p ) ≥ ¯ h ;with ∆( p − p ) ∼ ∞ and ∆( x + x ) ∼ ∞ . It is possible that no quantum mechanical operator is associated with a measurable variable, such as time t . From this perspective, an uncertainty relation based on variables rather than operators is more general.
3s a matter of fact, in their 1935 paper, Einstein-Podolsky-Rosen never questioned ∆( x − x ) ∆( p + p ) = 0 as a violation of the uncertainty principle. The violation of the uncertaintyprinciple was probably not Einstein’s concern at all, although their 1935 paradox was basedon the argument of the uncertainty principle. What really bothered Einstein so much? Forall of his life, Einstein, a true believer of realism, never accepted that a particle does not havea defined momentum and position during its motion, but rather is specified by a probabilityamplitude of certain a momentum and position. “God does not play dice” was the most vividcriticism from Einstein to refuse the Schr¨odinger’s cat. The entangled two-particle systemwas used as an example to clarify and to reinforce Einstein’s realistic opinion. To Einstein,the acceptance of Schr¨odinger’s cat perhaps means action-at-a-distance or an inconsistencybetween quantum mechanics and the theory of relativity, when dealing with the entangled EPRtwo-particle system. Let us follow Copenhagen to consider that each particle in an EPR pairhas no defined momentum and position, or has all possible momentum and position within thesuperposition state, i.e., imagine ∆ p j = 0, ∆ x j = 0, j = 1 ,
2, for each single-particle until themeasurement. Assume the measurement devices are particle counting devices able to identifythe position of each particle among an ensemble of particles. For each registration of a particlethe measurement device records a value of its position. No one can predict what value isregistered for each measurement; the best knowledge we may have is the probability to registerthat value. If we further assume no physical interaction between the two distant particles andbelieve no action-at-a-distance exist in nature, we would also believe that no matter how thetwo particles are created, the two registered values must be independent of each other. Thus,the value of x − x is unpredictable within the uncertainties of ∆ x and ∆ x . The abovestatement is also valid for the momentum measurement. Therefore, after a set of measurementson a large number of particle pairs, the statistical uncertainty of the measurement on p + p and x − x must obey the following inequalities:∆( p + p ) = p (∆ p ) + (∆ p ) > M ax (∆ p , ∆ p ) (3)∆( x − x ) = p (∆ x ) + (∆ x ) > M ax (∆ x , ∆ x ) . Eq. (3) is obviously true in statistics, especially when we are sure that no disturbance is possiblebetween the two independent-local measurements. This condition can be easily realized bymaking the two measurement events space-like separated events. The classical inequality ofEq. (3) would not allow ∆( p + p ) = 0 and ∆( x − x ) = 0 as required in the EPR state, unless∆ p = 0, ∆ p = 0, ∆ x = 0 and ∆ x = 0, simultaneously. Unfortunately, the assumptionof ∆ p = 0, ∆ p = 0, ∆ x = 0, ∆ x = 0 cannot be true because it violates the uncertaintyrelations ∆ p ∆ x ≥ ¯ h and ∆ p ∆ x ≥ ¯ h .In a non-perfect entangled system, the uncertainties of p + p and x − x may differ fromzero. Nevertheless, the measurements may still satisfy the EPR inequalities [5]:∆( p + p ) < min (∆ p , ∆ p ) (4)∆( x − x ) < min (∆ x , ∆ x ) . The apparent contradiction between the classical inequality Eq. (3) and the EPR inequalityEq. (4) deeply troubled Einstein. While one sees the measurements of p + p and x − x of thetwo distant individual free particles satisfying Eq. (4), but believing Eq. (3), one might easilybe trapped into concluding either there is a violation of the uncertainty principle or there existsaction-at-a-distance.Is it possible to have a realistic theory which provides correct predictions of the behaviorof a particle similar to quantum theory and, at the same time, respects the description ofphysical reality by EPR as “complete”? Bohm and his followers have attempted a “hiddenvariable theory”, which seemed to satisfy these requirements [6]. The hidden variable theorywas successfully applied to many different quantum phenomena until 1964, when Bell proved a4heorem to show that an inequality, which is violated by certain quantum mechanical statisticalpredictions, can be used to distinguish local hidden variable theory from quantum mechanics[7]. Since then, the testing of Bell’s inequalities became a standard instrument for the studyof fundamental problems of quantum theory [8]. The experimental testing of Bell’s inequalitystarted from the early 1970’s. Most of the historical experiments concluded the violation of theBell’s inequalities and thus disproved the local hidden variable theory [8][9][10].In the following, we examine a simple yet popular realistic model to simulate the behavior ofthe entangled EPR system. This model concerns an ensemble of classically correlated particlesinstead of the quantum mechanical superposition of a particle. In terms of “cats”, this model isbased on the measurement of a large number of twin cats in which 50% are alive-alive twins and50% are dead-dead twins. This model refuses the concept of Schr¨odinger’s cat which requires a cat to be alive and dead simultaneously, and each pair of cats involved in a joint detectionevent is in the state of alive-alive and dead-dead simultaneously.In this model, we may have three different states:(1) State one, each single pair of particles holds defined momenta p = constant and p =constant with p + p = 0. From pair to pair, the values of p and p may vary significantly.The sum of p and p , however, keeps a constant of zero. Thus, each joint detection of the twodistant particles measures precisely the constant values of p and p and measures p + p = 0.The uncertainties of ∆ p and ∆ p only have statistical meaning in terms of the measurementsof an ensemble. This model successfully simulated ∆( p + p ) = 0 based on the measurementof a large number of classically correlated particle pairs. This is, however, only half of the EPRstory. Can we have ∆( x − x ) = 0 simultaneously in this model? We do have ∆ x ∼ ∞ and ∆ x ∼ ∞ , otherwise the uncertainty principle will be violated. The position correlation,however, can never achieve ∆( x − x ) = 0 by any means.(2) State two, each single pair of particles holds a well defined position x = constantand x = constant with x − x = x . From pair to pair, the values of x and x may varysignificantly. The difference of x and x , however, maintains a constant of x . Thus, each jointdetection of the two distant particles measures precisely the constant values of x and x andmeasures x − x = x . The uncertainties of ∆ x and ∆ x only have statistical meaning interms of the measurements of an ensemble. This model successfully simulated ∆( x − x ) = 0based on the measurement of a large number of classically correlated particle pairs. This is,however, only half of the EPR story. Can we have ∆( p + p ) = 0 simultaneously in this model?We do have ∆ p ∼ ∞ and ∆ p ∼ ∞ , otherwise the uncertainty principle will be violated. Themomentum correlation, however, can never achieve ∆( p + p ) = 0 by any means.The above two models of classically correlated particle pairs can never achieve both ∆( p + p ) = 0 and ∆( x − x ) = 0. What would happen if we combine the two parts together? Thisleads to the third model of classical simulation.(3) State three, among a large number of classically correlated particle pairs, we assume50% to be in state one and the other 50% state two. The p + p measurements would have50% chance with p + p = 0 and 50% chance with p + p = random value. On the otherhand, the x − x measurements would have 50% chance with x − x = x and 50% chancewith x − x = random value. What are the statistical uncertainties on the measurements of( p + p ) and ( x − x ) in this case? If we focus on only these events of state one, the statisticaluncertainty on the measurement of ( p + p ) is ∆( p + p ) = 0, and if we focus on these events ofstate two, the statistical uncertainty on the measurement of ( x − x ) is ∆( x − x ) = 0; however,if we consider all the measurements together, the statistical uncertainties on the measurementsof ( p + p ) and ( x − x ), are both infinity: ∆( p + p ) = ∞ and ∆( x − x ) = ∞ .In conclusion, classically correlated particle pairs may partially simulate EPR correlation5ith three types of optimized observations:(1) ∆( p + p ) = 0 (100%) & ∆( x − x ) = ∞ (100%);(2) ∆( x − x ) = 0 (100%) & ∆( p + p ) = ∞ (100%);(3) ∆( p + p ) = 0 (50%) & ∆( x − x ) = 0 (50%);Within one setup of experimental measurements, only the entangled EPR states result in thesimultaneous observation of∆( p + p ) = 0 (100%) & ∆( x − x ) = 0 (100%)∆ p ∼ ∞ , ∆ p ∼ ∞ , ∆ x ∼ ∞ , ∆ x ∼ ∞ . We thus have a tool, besides the testing of Bell’s inequality, to distinguish quantum entangledstates from classically correlated particle pairs.
The entangled state of a two-particle system was mathematically formulated by Schr¨odinger[11]. Consider a pure state for a system composed of two distinguishable subsystems | Ψ i = X a,b c ( a, b ) | a i | b i (5)where {| a i} and {| b i} are two sets of orthogonal vectors for subsystems 1 and 2, respectively.If c ( a, b ) does not factor into a product of the form f ( a ) × g ( b ), then it follows that the statedoes not factor into a product state for subsystems 1 and 2:ˆ ρ = | Ψ ih Ψ | = X a,b c ( a, b ) | a i| b i X a ′ ,b ′ c ∗ ( a ′ , b ′ ) h b ′ |h a ′ | 6 = ˆ ρ × ˆ ρ , (6)where ˆ ρ is the density operator, the state was defined by Schr¨odinger as an entangled state.Following this notation, the first classic entangled state of a two-particle system, the EPRstate of Eq. (1) and Eq. (2), is thus written as: | Ψ i EP R = X x ,x δ ( x − x + x ) | x i| x i = X p ,p δ ( p + p ) | p i| p i , (7)where we have described the entangled two-particle system as the coherent superposition of themomentum eigenstates as well as the coherent superposition of the position eigenstates. The two δ -functions in Eq. (7) represent, respectively and simultaneously, the perfect position-positionand momentum-momentum correlation. Although the two distant particles are interaction-free,the superposition selects only the eigenstates which are specified by the δ -function. We mayuse the following statement to summarize the surprising feature of the EPR state: the values ofthe momentum and the position for neither interaction-free single subsystem is determinated.However, if one of the subsystems is measured to be at a certain value of momentum and/orposition, the momentum and/or position of the other one is 100% determined, despite thedistance between them .It should be emphasized again that Eq. (7) is true, simultaneously, in the conjugate spaceof momentum and position. This is different from classically correlated statesˆ ρ = X p ,p δ ( p + p ) | p i| p ih p |h p | , (8)6r ˆ ρ = X x ,x δ ( x − x + x ) | x i| x ih x |h x | . (9)Eq. (8) and Eq. (9) represent mixed states. Eq. (8) and Eq. (9) cannot be true simultaneously aswe have discussed earlier. Thus, we can distinguish entangled states from classically correlatedstates through the measurements of the EPR inequalities of Eq. (4). Two-photon state of spontaneous parametric down-conversion
The state of a signal-idler photon pair created in spontaneous parametric down-conversion(SPDC) is a typical EPR state [12][13]. Roughly speaking, the process of SPDC involvessending a pump laser beam into a nonlinear material, such as a non-centrosymmetric crystal.Occasionally, the nonlinear interaction leads to the annihilation of a high frequency pumpphoton and the simultaneous creation of a pair of lower frequency signal-idler photons formingan entangled two-photon state: | Ψ i = Ψ X s,i δ ( ω s + ω i − ω p ) δ ( k s + k i − k p ) a † s ( k s ) a † i ( k i ) | i (10)where ω j , k j ( j = s, i, p) are the frequency and wavevector of the signal (s), idler (i), and pump(p), a † s and a † i are creation operators for the signal and the idler photon, respectively, and Ψ isthe normalization constant. We have assumed a CW monochromatic laser pump, i.e., ω p and k p are considered as constants. The two delta functions in Eq. (10) are technically named asthe phase matching condition [12][14]: ω p = ω s + ω i , k p = k s + k i . (11)The names signal and idler are historical leftovers. The names perhaps came about due to thefact that in the early days of SPDC, most of the experiments were done with non-degenerateprocesses. One radiation was in the visible range (and thus easily observable, the signal), whilethe other was in the IR range (usually not measured, the idler). We will see in the followingdiscussions that the role of the idler is no any less important than that of the signal. The SPDCprocess is referred to as type-I if the signal and idler photons have identical polarizations, andtype-II if they have orthogonal polarizations. The process is said to be degenerate if the SPDCphoton pair has the same free space wavelength (e.g. λ i = λ s = 2 λ p ), and nondegenerate otherwise. In general, the pair exit the crystal non-collinearly , that is, propagate to differentdirections defined by the second equation in Eq. (11) and Snell’s law. In addition, the pair mayalso exit collinearly , in the same direction, together with the pump.The state of the signal-idler pair can be derived, quantum mechanically, by the first orderperturbation theory with the help of the nonlinear interaction Hamiltonian. The SPDC inter-action arises in a nonlinear crystal driven by a pump laser beam. The polarization, i.e., thedipole moment per unit volume, is given by P i = χ (1) i,j E j + χ (2) i,j,k E j E k + χ (3) i,j,k,l E j E k E l + ... (12)where χ ( m ) is the mth order electrical susceptibility tensor. In SPDC, it is the second or-der nonlinear susceptibility χ (2) that plays the role. The second order nonlinear interactionHamiltonian can be written as H = ǫ Z V d r χ (2) ijk E i E j E k (13)where the integral is taken over the interaction volume V .7t is convenient to use the Fourier representation for the electrical fields in Eq. (13): E ( r , t ) = Z d k [ E ( − ) ( k ) e − i ( ω ( k ) t − k · r ) + E (+) ( k ) e i ( ω ( k ) t − k · r ) ] . (14)Substituting Eq. (14) into Eq. (13) and keeping only the terms of interest, we obtain the SPDCHamiltonian in the interaction representation: H int ( t ) (15)= ǫ Z V d r Z d k s d k i χ (2) lmn E (+) p l e i ( ω p t − k p · r ) E ( − ) s m e − i ( ω s ( k s ) t − k s · r ) E ( − ) i n e − i ( ω i ( k i ) t − k i · r ) + h.c., where h.c. stands for Hermitian conjugate. To simplify the calculation, we have also assumedthe pump field to be a monochromatic plane wave with wave vector k p and frequency ω p .It is easily noticeable that in Eq. (15), the volume integration can be done for some simplifiedcases. At this point, we assume that V is infinitely large. Later, we will see that the finite sizeof V in longitudinal and/or transversal directions may have to be taken into account. For aninfinite volume V , the interaction Hamiltonian Eq. (15) is written as H int ( t ) = ǫ Z d k s d k i χ (2) lmn E (+) p l E ( − ) s m E ( − ) i n δ ( k p − k s − k i ) e i ( ω p − ω s ( k s ) − ω i ( k i )) t + h.c. (16)It is reasonable to consider the pump field to be classical, which is usually a laser beam, andquantize the signal and idler fields, which are both at the single-photon level: E ( − ) ( k ) = i r π ¯ hωV a † ( k ) , E (+) ( k ) = i r π ¯ hωV a ( k ) , (17)where a † ( k ) and a ( k ) are photon creation and annihilation operators, respectively. The stateof the emitted photon pair can be calculated by applying the first order perturbation | Ψ i = − i ¯ h Z dt H int ( t ) | i . (18)By using vacuum | i for the initial state in Eq. (18), we assume that there is no input radiationin any signal and idler modes, that is, we have a spontaneous parametric down conversion(SPDC) process.Further assuming an infinite interaction time, evaluating the time integral in Eq. (18) andomitting altogether the constants and slow (square root) functions of ω , we obtain the entangled two-photon state of Eq. (10) in the form of an integral [13]: | Ψ i = Ψ Z d k s d k i δ [ ω p − ω s ( k s ) − ω i ( k i )] δ ( k p − k s − k i ) a † s ( k s ) a † i ( k i ) | i (19)where Ψ is a normalization constant which has absorbed all omitted constants.The way of achieving phase matching, i.e., the delta functions, in Eq. (19) basically deter-mines how the signal-idler pair “looks”. For example, in a negative uniaxial crystal, one canuse a linearly polarized pump laser beam as an extraordinary ray of the crystal to generate asignal-idler pair both polarized as the ordinary rays of the crystal, which is defined as type-Iphase matching. One can alternatively generate a signal-idler pair with one ordinary polarizedand another extraordinary polarized, which is defined as type II phase matching. Fig. 1 showsthree examples of an SPDC two-photon source. All three schemes have been widely used fordifferent experimental purposes. Technical details can be found in text books and researchreferences in nonlinear optics.The two-photon state in the forms of Eq. (10) or Eq. (19) is a pure state, which math-ematically describes the behavior of a signal-idler photon pair. The surprise comes from thecoherent superposition of the two-photon modes:8 a)(cid:13) (b)(cid:13) (c)(cid:13) Figure 1: Three widely used SPDC setups. (a) Type-I SPDC. (b) Collinear degeneratetype-II SPDC. Two rings overlap at one region. (c) Non-collinear degenerate type-II SPDC.For clarity, only two degenerate rings, one for e-polarization and the other for o-polarization,are shown. Notice, the color rainbows represent the distribution function of a signal-idlerpair. One signal-idler pair yields the entire rainbow.Does the signal or the idler photon in the EPR state of Eq. (10) or Eq. (19) have a definedenergy and momentum regardless of whether we measure it or not? Quantum mechanicsanswers: No! However, if one of the subsystems is measured with a certain energy andmomentum, the other one is determined with certainty, despite the distance between them.It is indeed a mystery from a classical point of view. There has been, nevertheless, classicalmodels to avoid the surprises. One of the classical realistic models insists that the state ofEq. (10) or Eq. (19) only describes the behavior of an ensemble of photon pairs. In this model,the energy and momentum of the signal photon and the idler photon in each individual pair aredefined with certain values and the resulting state is a statistical mixture. Mathematically, itis incorrect to use a pure state to characterize a statistical mixture. The concerned statisticalensemble should be characterized by the following density operatorˆ ρ = Z d k s d k i δ ( ω p − ω s − ω i ) δ ( k p − k s − k i ) a † s ( k s ) a † i ( k i ) | ih | a s ( k s ) a i ( k i ) (20)which is very different from the pure state of SPDC. We will show later that a statistical mixtureof Eq. (20) can never have delta-function-like two-photon temporal and/or spatial correlationthat is shown by the measurement of SPDC.For finite dimensions of the nonlinear interaction region, the entangled two-photon state ofSPDC may have to be estimated in a more general format. Following the earlier discussions,we write the state of the signal-idler photon pair as | Ψ i = Z d k s d k i F ( k s , k i ) a † i ( k s ) a † s ( k i ) | i (21)where F ( k s , k i ) = ǫ δ ( ω p − ω s − ω i ) f (∆ z L ) h tr ( ~κ + ~κ ) f (∆ z L ) = Z L dz e − i ( k p − k sz − k iz ) z h tr ( ~κ + ~κ ) = Z A d~ρ ˜ h tr ( ~ρ ) e − i ( ~κ s + ~κ i ) · ~ρ (22)∆ z = k p − k sz − k iz where ǫ is named as the parametric gain index. ǫ is proportional to the second order electric sus-ceptibility χ (2) and is usually treated as a constant, L is the length of the nonlinear interaction,9he integral in ~κ is evaluated over the cross section A of the nonlinear material illuminated bythe pump, ~ρ is the transverse coordinate vector, ~κ j (with j = s, i ) is the transverse wavevectorof the signal and idler, and f ( | ~ρ | ) is the transverse profile of the pump, which can be treatedas a Gaussion in most of the experimental conditions. The functions f (∆ z L ) and h tr ( ~κ + ~κ )turn to δ -functions for an infinitely long ( L ∼ ∞ ) and wide ( A ∼ ∞ ) nonlinear interactionregion. The reason we have chosen the form of Eq. (22) is to separate the “longitudinal” andthe “transverse” correlations. We will show that δ ( ω p − ω s − ω i ) and f (∆ z L ) together can berewritten as a function of ω s − ω i . To simplify the mathematics, we assume near co-linearlySPDC. In this situation, | ~κ s,i | ≪ | k s,i | .Basically, the function f (∆ z L ) determines the “longitudinal” space-time correlation. Find-ing the solution of the integral is straightforward: f (∆ z L ) = Z L dz e − i ( k p − k sz − k iz ) z = e − i ∆ z L/ sinc (∆ z L/ . (23)Now, we consider f (∆ z L ) with δ ( ω p − ω s − ω i ) together, and taking advantage of the δ -function in frequencies by introducing a detuning frequency Ω to evaluate function f (∆ z L ): ω s = ω s + Ω ω i = ω i − Ω (24) ω p = ω s + ω i = ω s + ω i . Ω = ( ω s − ω i ) / . The dispersion relation k ( ω ) allows us to express the wave numbers through the frequencydetuning Ω: k s ≈ k ( ω s ) + Ω dkdω (cid:12)(cid:12)(cid:12) ω s = k ( ω s ) + Ω u s ,k i ≈ k ( ω i ) − Ω dkdω (cid:12)(cid:12)(cid:12) ω i = k ( ω i ) − Ω u i (25)where u s and u i are group velocities for the signal and the idler, respectively. Now, we connect∆ z with the detuning frequency Ω:∆ z = k p − k sz − k iz = k p − p ( k s ) − ( ~κ s ) − p ( k i ) − ( ~κ i ) ∼ = k p − k s − k i + ( ~κ s ) k s + ( ~κ i ) k i (26) ∼ = k p − k ( ω s ) − k ( ω i ) + Ω u s − Ω u i + ( ~κ s ) k s + ( ~κ i ) k i ∼ = D Ωwhere D ≡ /u s − /u i . We have also applied k p − k ( ω s ) − k ( ω i ) = 0 and | ~κ s,i | ≪ | k s,i | .The “longitudinal” wavevector correlation function is rewritten as a function of the detuningfrequency Ω = ( ω s − ω i ) / f (∆ z L ) ∼ = f (Ω DL ). In addition to the above approximations,we have inexplicitly assumed the angular independence of the wavevector k = n ( θ ) ω/c . Fortype II SPDC, the refraction index of the extraordinary-ray depends on the angle between thewavevector and the optical axis and an additional term appears in the expansion. Making theapproximation valid, we have restricted our calculation to a near-collinear process. Thus, for agood approximation, in the near-collinear experimental setup∆ z L ∼ = Ω DL = ( ω s − ω i ) DL/ . (27)10ype-I degenerate SPDC is a special case. Due to the fact that u s = u i , and hence, D = 0,the expansion of k ( ω ) should be carried out up to the second order. Instead of (27), we have∆ z L ∼ = − Ω D ′ L = − ( ω s − ω i ) D ′ L/ D ′ ≡ ddω ( 1 u ) (cid:12)(cid:12)(cid:12) ω . The two-photon state of the signal-idler pair is then approximated as | Ψ i = Z d Ω d~κ s d~κ i f (Ω) h tr ( ~κ s + ~κ i ) a † s ( ω s + Ω , ~κ s ) a † i ( ω i − Ω , ~κ i ) | i (29)where the normalization constant has been absorbed into f (Ω). EPR state is a pure state which characterizes the behavior of a pair of entangled particles.In principle, one EPR pair contains all information of the correlation. A question naturallyarises: Can we then observe the EPR correlation from the measurement of one EPR pair?The answer is no. Generally speaking, we may never learn any meaningful physics from themeasurement of one particle or one pair of particles. To learn the correlation, an ensemblemeasurement of a large number of identical pairs are necessary, where “identical” means thatall pairs which are involved in the ensemble measurement must be prepared in the same state,except for an overall phase factor. This is a basic requirement of quantum measurement theory.Correlation measurements are typically statistical and involve a large number of measure-ments of individual quanta. Quantum mechanics does not predict a precise outcome for ameasurement. Rather, quantum mechanics predicts the probabilities for certain outcomes. Inphoton counting measurements, the outcome of a measurement is either a “ yes ” (a count or a“click”) or a “ no ” (no count). In a joint measurement of two photon counting detectors, theoutcome of “ yes ” means a “ yes - yes ” or a “click-click” joint registration. If the outcome of ajoint measurement shows 100% “ yes ” for a certain set of values of a physical observable or acertain relationship between physical variables, the measured quantum system is correlated inthat observable. As a good example, EPR’s gedankenexperiment suggested to us a system ofquanta with perfect correlation δ ( x − x + x ) in position. To examine the EPR correlation, weneed to have a 100% “ yes ” when the positions of the two distant detectors satisfy x − x = x ,and 100% “ no ” otherwise, when x − x = x . To show this experimentally, a realistic approachis to measure the correlation function of | f ( x − x ) | by observing the joint detection countingrates of R , ∝ | f ( x − x ) | while scanning all possible values of x − x . In quantum optics,this means the measurement of the second-order correlation function, or G (2) ( r , t ; r , t ), inthe form of longitudinal correlation G (2) ( τ − τ ) and/or transverse correlation G (2) ( ~ρ − ~ρ ),where τ j = t j − z j /c , j = 1 ,
2, and ~ρ j is the transverse coordinate of the jth point-like photoncounting detector.Now, we study the two-photon correlation of the entangled photon pair of SPDC. Theprobability of jointly detecting the signal and idler at space-time points ( r , t ) and ( r , t ) isgiven by the Glauber theory [15]: G (2) ( r , t ; r , t ) = h E ( − ) ( r , t ) E ( − ) ( r , t ) E (+) ( r , t ) E (+) ( r , t ) i (30)where E ( − ) and E (+) are the negative-frequency and the positive-frequency field operators ofthe detection events at space-time points ( r , t ) and ( r , t ). The expectation value of the joint11etection operator is calculated by averaging over the quantum states of the signal-idler photonpair. For the two-photon state of SPDC, G (2) ( r , t ; r , t ) = | h | E (+) ( r , t ) E (+) ( r , t ) | Ψ i | = | ψ ( r , t ; r , t ) | (31)where | Ψ i is the two-photon state, and Ψ( r , t ; r , t ) is named the effective two-photon wave-function. To evaluate G (2) ( r , t ; r , t ) and ψ ( r , t ; r , t ), we need to propagate the fieldoperators from the two-photon source to space-time points ( r , t ) and ( r , t ).In general, the field operator E (+) ( r , t ) at space-time point ( r , t ) can be written in termsof the Green’s function, which propagates a quantized mode from space-time point ( r , t ) to( r , t ) [16][17]: E (+) ( r , t ) = X k g ( k , r − r , t − t ) E (+) ( k , r , t ) . (32)where g ( k , r − r , t − t ) is the Green’s function, which is also named the optical transfer function.For a different experimental setup, g ( k , r − r , t − t ) can be quite different. To simplify thenotation, we have assumed one polarization. c(cid:13) (2)(cid:13) Pump(cid:13) S i g n a l (cid:13) I d l e r (cid:13) D D G (cid:13) (2)(cid:13) k(cid:13) s(cid:13)0(cid:13) k(cid:13) i(cid:13)0(cid:13)
Figure 2: Collinear propagated signal-idler photon pair, either degenerate or non-degenerate,are received by two distant point photo-detectors D and D , respectively, for longitudinal G (2) ( τ − τ ) and transverse G (2) ( ~ρ − ~ρ ) correlation measurements. To simplify the math-ematics, we assume paraxial approximation is applicable to the signal-idler fields. The z and z are chosen along the central wavevector k s and k i .Considering an idealized simple experimental setup, shown in Fig. 2, in which collinearpropagated signal and idler pairs are received by two point photon counting detectors D and D , respectively, for longitudinal G (2) ( τ − τ ) and transverse G (2) ( ~ρ − ~ρ ) correlation measure-ments. To simplify the mathematics, we further assume paraxial experimental condition. It isconvenient, in the discussion of longitudinal and transverse correlation measurements, to writethe field E (+) ( r j , t j ) in terms of its longitudinal and transversal space-time variables under theFresnel paraxial approximation: E (+) ( ~ρ j , z j , t j ) (33) ∼ = Z dω d~κ g ( ~κ, ω ; ~ρ j , z j ) e − iωt j a ( ω, ~κ ) ∼ = Z dω d~κ γ ( ~κ, ω ; ~ρ j , z j ) e − iωτ j a ( ω, ~κ )where g ( ~κ, ω ; ~ρ j , z j ) = γ ( ~κ, ω ; ~ρ j , z j ) e iωz j /c is the spatial part of the Green’s function, ~ρ j and z j are the transverse and longitudinal coordinates of the jth photo-detector and ~κ is the trans-verse wavevector. We have chosen z = 0 and t = 0 at the output plane of the SPDC. Forconvenience, all constants associated with the field are absorbed into g ( ~κ, ω ; ~ρ j , z j ).12he two-photon effective wavefunction Ψ( ~ρ , z , t ; ~ρ , z , t ) is thus calculated as followsΨ( ~ρ , z , t ; ~ρ , z , t )= h | Z dω , d~κ , g ( ~κ , , ω , ; ~ρ , z ) e − iω , t a ( ω , , ~κ , ) × Z dω ,, d~κ ,, g ( ~κ ,, , ω ,, ; ~ρ , z ) e − iω ,, t a ( ω ,, , ~κ ,, ) × Z d Ω d~κ s d~κ i f (Ω) h tr ( ~κ s + ~κ i ) a † s ( ω s + Ω , ~κ s ) a † i ( ω i − Ω , ~κ i ) | i = Ψ e − i ( ω s τ + ω i τ ) × Z d Ω d~κ s d~κ i f (Ω) h tr ( ~κ s + ~κ i ) e − i Ω( τ − τ ) γ ( ~κ s , Ω; ~ρ , z ) γ ( ~κ i , − Ω; ~ρ , z ) . (34)Although Eq. (34) cannot be factorized into a trivial product of longitudinal and transverseintegrals, it is not difficult to measure the temporal correlation and the transverse correlationseparately by choosing suitable experimental conditions.Experiments may be designed for measuring either temporal (longitudinal) or spatial (trans-verse) correlation only. Thus, based on different experimental setups, we may simplify thecalculation to either the temporal (longitudinal) part:Ψ( τ ; τ ) = Ψ e − i ( ω s τ + ω i τ ) Z d Ω f (Ω) e − i Ω( τ − τ ) = Ψ e − i ( ω s τ + ω i τ ) F τ − τ (cid:8) f (Ω) (cid:9) (35)or the spatial part:Ψ( ~ρ , z ; ~ρ , z ) = Ψ Z d~κ s d~κ i h tr ( ~κ s + ~κ i ) g ( ~κ s , ω s ; ~ρ , z ) g ( ~κ i , ω i ; ~ρ , z ) . (36)In Eq. (35), F τ − τ (cid:8) f (Ω) (cid:9) is the Fourier transform of the spectrum amplitude function f (Ω).In Eq. (36), we may treat h tr ( ~κ s + ~κ i ) ∼ δ ( ~κ s + ~κ i ) by assuming certain experimental conditions. Two-photon temporal correlation
To measure the two-photon temporal correlation of SPDC, we select a pair of transversewavevectors ~κ s = − ~κ i in Eq. (34) by using appropriate optical apertures. The effective two-photon wavefunction is thus simplified to that of Eq. (35)Ψ( τ ; τ ) ∼ = Ψ e − i ( ω s τ + ω i τ ) Z d Ω f (Ω) e − i Ω( τ − τ ) (37)= (cid:2) Ψ e − i ( ω s + ω i )( τ + τ ) (cid:3) (cid:2) F τ − τ (cid:8) f (Ω) (cid:9) e − i ( ω s − ω i )( τ − τ ) (cid:3) where, again, F τ − τ (cid:8) f (Ω) (cid:9) is the Fourier transform of the spectrum amplitude function f (Ω).Eq. (37) indicates a 2-D wavepacket: a narrow envelope along the τ − τ axis with constantamplitude along the τ + τ axis. In certain experimental conditions, the function f (Ω) ofSPDC can be treated as constant from −∞ to ∞ and thus F τ − τ ∼ δ ( τ − τ ). In this case,for fixed positions of D and D , the 2-D wavepacket means the following: the signal-idlerpair may be jointly detected at any time; however, if the signal is registered at a certain time t , the idler must be registered at a unique time of t ∼ t − ( z − z ) /c . In other words,although the joint detection of the pair may happen at any times of t and t with equalprobability (∆( t + t ) ∼ ∞ ), the registration time difference of the pair must be a constant∆( t − t ) ∼
0. A schematic of the two-photon wavepacket is shown in Fig. 3. It is a non-factorizeable 2-D wavefunction indicating the entangled nature of the two-photon state. Thelongitudinal correlation function G (2) ( τ − τ ) is thus G (2) ( τ − τ ) ∝ | F τ − τ (cid:8) f (Ω) (cid:9) | , (cid:13) 0 Figure 3: A schematic envelope of a two-photon wavepacket with a Gaussian shape along τ − τ corresponding to a Gaussian function of f (Ω). In the case of SPDC, the envelopeis close to a δ -function in τ − τ corresponding to a broad-band f (Ω) = constant. Thewavepacket is uniformly distributed along τ + τ due to the assumption of ω p = constant.which is a δ -function-like function in the case of SPDC. Thus, we have shown the entangledsignal-idler photon pair of SPDC hold a typical EPR correlation in energy and time:∆( ω s + ω i ) ∼ t − t ) ∼ ω s ∼ ∞ , ∆ ω i ∼ ∞ , ∆ t ∼ ∞ , ∆ t ∼ ∞ . Now we examine a statistical model of SPDC for temporal correlation. As we have discussedearlier, realistic statistical models have been proposed to simulate the EPR two-particle state.Recall that for a mixed state in the form ofˆ ρ = X j P j | Ψ j ih Ψ j | where P j is the probability for specifying a given set of state vectors | Ψ j i , the second-ordercorrelation function of fields E ( r , t ) and E ( r , t ) is given by G (2) ( r , t ; r , t )= T r [ ˆ ρ E ( − ) ( r , t ) E ( − ) ( r , t ) E (+) ( r , t ) E (+) ( r , t ) ]= X j P j h Ψ j | E ( − ) ( r , t ) E ( − ) ( r , t ) E (+) ( r , t ) E (+) ( r , t ) | Ψ j i = X j P j G (2) j ( r , t ; r , t ) , which is a weighted sum over all individual contributions of G (2) j . Considering the followingsimplified version of Eq. (20) to simulate the state of SPDC as a mixed state:ˆ ρ = Z d Ω | f (Ω) | a † ( ω s + Ω) a † ( ω i − Ω) | ih | a ( ω i − Ω) a ( ω s + Ω) , (38)with | Ψ Ω i = a † ( ω s + Ω) a † ( ω i − Ω) | i , P j = d Ω | f (Ω) | . (39)14t is easy to find G (2)Ω ( τ − τ ) = constant, and thus G (2) ( τ − τ ) = constant. This means that theuncertainty of the measurement on t − t for the mixed state of Eq. (38) is infinite: ∆( t − t ) ∼∞ . Although the energy (frequency) or momentum (wavevector) for each photon may be definedwith constant values pair by pair, the corresponding temporal correlation measurement of theensemble can never achieve a δ -function-like relationship. In fact, the correlation is undefined,i.e., taking an infinite uncertainty. Thus, the statistical model of SPDC cannot satisfy the EPRinequalities of Eq. (4). Two-photon spatial correlation
Similar to that of the two-photon temporal correlation, as an example, we analyze theeffective two-photon wavefunction of the signal-idler pair of SPDC. To emphasize the spatialpart of the two-photon correlation, we choose a pair of frequencies ω s and ω i with ω s + ω i = ω p .In this case, the effective two-photon wavefunction of Eq. (34) is simplified to that of Eq. (36)Ψ( ~ρ , z ; ~ρ , z ) = Ψ Z d~κ s d~κ i δ ( ~κ s + ~κ i ) g ( ~κ s , ω s , ~ρ , z ) g ( ~κ i , ω i , ~ρ , z )where we have assumed h tr ( ~κ s + ~κ i ) ∼ δ ( ~κ s + ~κ i ), which is reasonable by assuming a largeenough transverse cross-session laser beam of pump.We now design a simple joint detection measurement between two point photon countingdetectors D and D located at ( ~ρ , z ) and ( ~ρ , z ), respectively, for the detection of the signaland idler photons. We have assumed that the two-photon source has a finite but large transversedimension. Under this simple experimental setup, the Green’s function, or the optical transferfunction describing arm- j , j = 1 ,
2, in which the signal and the idler freely propagate to photo-detector D and D , respectively, is given by Eq. ( A −
5) of the Appendix. Substitute the g j ( ω, ~κ ; z j , ~ρ j ), j = 1 ,
2, into Eq. (36), the effective wavefunction is then given byΨ( ~ρ , z ; ~ρ , z ) (40)= Ψ Z d~κ s d~κ i δ ( ~κ s + ~κ i ) (cid:0) − iω s πcz e i ωsc z (cid:1) (cid:0) − iω i πcz e i ωic z (cid:1) × Z A d~ρ s d~ρ i G ( | ~ρ − ~ρ s | , ω s cz ) e i~κ s · ~ρ s G ( | ~ρ − ~ρ i | , ω i cz ) e i~κ i · ~ρ i where ~ρ s ( ~κ s ) and ~ρ i ( ~κ i ) are the transverse coordinates (wavevectors) for the signal and theidler fields, respectively, defined on the output plane of the two-photon source. The integral of d~ρ s and d~ρ i is over area A , which is determined by the transverse dimension of the nonlinearinteraction. The Gaussian function G ( | ~α | , β ) = e i ( β/ | ~α | represents the Fresnel phase factorthat is defined in the Appendix. The integral of d~κ s and d~κ i can be evaluated easily with thehelp of the EPR type two-phonon transverse wavevector distribution function δ ( ~κ s + ~κ i ): Z d~κ s d~κ i δ ( ~κ s + ~κ i ) e i~κ s · ~ρ s e i~κ i · ~ρ i ∼ δ ( ~ρ s − ~ρ i ) . (41)Thus, we have shown that the entangled signal-idler photon pair of SPDC holds a typicalEPR correlation in transverse momentum and position while the correlation measurement is onthe output plane of the two-photon source, which is very close to the original proposal of EPR:∆( ~κ s + ~κ i ) ∼ ~ρ s − ~ρ i ) ∼ ~κ s ∼ ∞ , ∆ ~κ i ∼ ∞ , ∆ ~ρ s ∼ ∞ , ∆ ~ρ i ∼ ∞ . In EPR’s language, we may never know where the signal photon and the idler photon areemitted from the output plane of the source. However, if the signal (idler) is found at a certainposition, the idler (signal) must be observed at a corresponding unique position. The signal andthe idler may have also any transverse momentum. However, if the transverse momentum of the15ignal (idler) is measured at a certain value in a certain direction, the idler (signal) must be ofequal value but pointed to a certain opposite direction. In collinear
SPDC, the signal-idler pairis always emitted from the same point in the output plane of the two-photon source, ~ρ s = ~ρ i ,and if one of them propagates slightly off from the collinear axes, the other one must propagateto the opposite direction with ~κ s = − ~κ i .The interaction of spontaneous parametric down-conversion is nevertheless a local phe-nomenon. The nonlinear interaction coherently creates mode-pairs that satisfy the phase match-ing conditions of Eq. (11) which are also named as energy and momentum conservation. Thesignal-idler photon pair can be excited to any of these coupled modes or in all of these coupledmodes simultaneously, resulting in a particular two-photon superposition. It is this superposi-tion among those particular “selected” two-photon states which allows the signal-idler pair tocome out from the same point of the source and propagate to opposite directions with ~κ s = − ~κ i .The two-photon superposition becomes more interesting when the signal-idler is separatedand propagated to a large distance, either by free propagation or guided by optical componentssuch as a lens. A classical picture would consider the signal photon and the idler photonindependent whenever the pair is released from the two-photon source because there is nointeraction between the distant photons in free space. Therefore, the signal photon and theidler photon should have independent and random distributions in terms of their transverseposition ~ρ and ~ρ . This classical picture, however, is incorrect. It is found that the signal-idler two-photon system would not lose its entangled nature in the transverse position. Thisinteresting behavior has been experimentally observed in quantum imaging by means of an EPRtype correlation in transverse position. The sub-diffraction limit spatial resolution observed inthe “quantum lithography” experiment and the nonlocal correlation observed in the “ghostimaging” experiment are both the results of this peculiar superposition among those “selected”two-photon amplitudes, namely that of two-photon superposition, corresponding to differentyet indistinguishable alternative ways of triggering a joint photo-electron event at a distance.Two-photon superposition does occur in a distant joint detection event of a signal-idler photonpair. There is no surprise that one has difficulties facing this phenomenon. The two-photonsuperposition is a nonlocal concept in this case. There is no counterpart for such a concept inclassical theory and it may never be understood classically.Now we consider propagating the signal-idler pair away from the source to ( ~ρ , z ) and( ~ρ , z ), respectively, and taking the result of Eq. (41), i.e., ~ρ s = ~ρ i = ~ρ on the output planeof the SPDC source, the effective two-photon wavefunction becomesΨ( ~ρ , z ; ~ρ , z ) (42)= − ω s ω i (2 πc ) z z e i ( ωsc z + ωic z ) Z A d~ρ G ( | ~ρ − ~ρ | , ω s cz ) G ( | ~ρ − ~ρ | , ω i cz )where ~ρ is defined on the output plane of the two-photon source. Eq. (42) indicates that thepropagation-diffraction of the signal and the idler cannot be considered as independent. Thesignal-idler photon pair are created and diffracted together in a peculiar entangled manner.This point turns out to be both interesting and useful when the two photodetectors coincided,or are replaced by a two-photon sensitive material. Taking z = z and ~ρ = ~ρ , Eq. (42)becomes Ψ( ~ρ, z ; ~ρ, z ) = − ω s ω i (2 πcz ) e i ( ωpc z ) Z A d~ρ G ( | ~ρ − ~ρ | , ω p cz ) (43)where ω p is the pump frequency, which means that the signal-idler pair is diffracted as ifthey have twice the frequency or half the wavelength. This effect is named as “two-photondiffraction”. This effect is useful for enhancing the spatial resolution of imaging.16 Quantum imaging
Although questions regarding fundamental issues of quantum theory still exist, quantumentanglement has started to play important roles in practical engineering applications. Quan-tum imaging is one of these exciting areas [18]. Taking advantage of entangled states, Quantumimaging has so far demonstrated two peculiar features: (1) enhancing the spatial resolution ofimaging beyond the diffraction limit, and (2) reproducing ghost images in a “nonlocal” man-ner. Both the apparent “violation” of the uncertainty principle and the “nonlocal” behaviorof the momentnm-momentum position-position correlation are due to the two-photon coherenteffect of entangled states, which involves the superposition of two-photon amplitudes, a non-classical entity corresponding to different yet indistinguishable alternative ways of triggering ajoint-detection event in the quantum theory of photodetection. In this section, we will focusour discussion on the physics of imaging resolution enhancement. The nonlocal phenomenon ofghost imaging will be discussed in the following section.The concept of imaging is well defined in classical optics. Fig. 4 schematically illustrates astandard imaging setup. A lens of finite size is used to image the object onto an image plane
Source Image PlaneImaging LensObjectPlane S o(cid:13) f(cid:13) S i(cid:13) Figure 4: A lens produces an image of an object in the plane defined by the Gaussian thinlens equation 1 /s i + 1 /s o = 1 /f . The concept of an image is based on the existence of apoint-to-point relationship between the object plane and the image plane.which is defined by the “Gaussian thin lens equation”1 s i + 1 s o = 1 f (44)where s o is the distance between object and lens, f is the focal length of the lens, and s i is thedistance between the lens and image plane. If light always follows the laws of geometrical optics,the image plane and the object plane would have a perfect point-to-point correspondence, whichmeans a perfect image of the object, either magnified or demagnified. Mathematically, a perfectimage is the result of a convolution of the object distribution function f ( ~ρ o ) and a δ -function.The δ -function characterizes the perfect point-to-point relationship between the object planeand the image plane: F ( ~ρ i ) = Z obj d~ρ o f ( ~ρ o ) δ ( ~ρ o + ~ρ i m ) = f ( ~ρ o ) ⊗ δ ( ~ρ o + ~ρ i m ) (45)where ~ρ o and ~ρ i are 2-D vectors of the transverse coordinate in the object plane and the imageplane, respectively, and m is the magnification factor. The symbol ⊗ means convolution.Unfortunately, light behaves like a wave. The diffraction effect turns the point-to-pointcorrespondence into a point-to-“spot” relationship. The δ -function in the convolution of Eq. (45)will be replaced by a point-spread function. F ( ~ρ i ) = Z obj d~ρ o f ( ~ρ o ) somb (cid:2) Rs o ωc (cid:12)(cid:12) ~ρ o + ~ρ i m (cid:12)(cid:12)(cid:3) = f ( ~ρ o ) ⊗ somb (cid:2) Rs o ωc (cid:12)(cid:12) ~ρ o + ~ρ i m (cid:12)(cid:12)(cid:3) (46)17here somb ( x ) = 2 J ( x ) x , and J ( x ) is the first-order Bessel function, R is the radius of the imaging lens. R/s o is named asthe numerical aperture of the imaging system. The finite size of the spot, which is defined by thepoint-spread function, determines the spatial resolution of the imaging setup, and thus, limitsthe ability of making demagnified images. It is clear from Eq. (46), the use of a larger imaginglens and shorter wavelength light of source will result in a narrower point-spead function. Toimprove the spatial resolution, one of the efforts in the lithography industry is the use of shorterwavelengths. This effort is, however, limited to a certain level because of the inability of lensesto effectively work beyond a certain “cutoff” wavelength. D D r r r o s o Light(cid:13) Source s i r l Figure 5: Typical imaging setup. A lens of finite size is used to produce a demagnified imageof a object with limited spatial resolution. Replacing classical light with an entangled N-photon system, the spatial resolution can be improved by a factor of N, despite the Rayleighdiffraction limit.Eq. (46) imposes a diffraction limited spatial resolution on an imaging system while theaperture size of the imaging system and the wavelength of the light source are both fixed. Thislimit is fundamental in both classical optics and in quantum mechanics. Any violation wouldbe considered as a violation of the uncertainty principle.Surprisingly, the use of quantum entangled states gives a different result: by replacingclassical light sources in Fig. 5 with entangled N-photon states, the spatial resolution of theimage can be improved by a factor of N, despite the Rayleigh diffraction limit. Is this a violationof the uncertainty principle? The answer is no! The uncertainty relation for an entangled N-particle system is radically different from that of N independent particles. In terms of theterminology of imaging, what we have found is that the somb ( x ) in the convolution of Eq. (46)has a different form in the case of an entangled state. For example, an entangled two-photonsystem has x = Rs o ωc (cid:12)(cid:12) ~ρ o + ~ρ i m (cid:12)(cid:12) . Comparing with Eq. (46), the factor of 2 ω yields a point-spread function half the width of thatfrom Eq. (46) and results in a doubling spatial resolution for imaging.It should be further emphasized that one must not confuse a “projection” with an image.A projection is the shadow of an object, which is obviously different from the image of anobject. Fig. 6 distinguishes a projection shadow from an image. In a projection, the object-shadow correspondence is essentially a “momentum” correspondence, which is defined only bythe propagation direction of the light rays.We now analyze classical imaging. The analysis starts with the propagation of the fieldfrom the object plane to the image plane. In classical optics, such propagation is described byan optical transfer function h ( r − r , t − t ), which accounts for the propagation of all modes ofthe field. To be consistent with quantum optics calculations, we prefer to work with the single-mode propagator g ( k , r − r , t − t ), and to write the field E ( r , t ) in terms of its longitudinal18 rojections ObjectPlaneSource
Figure 6: Projection: a light source illuminates an object and no image forming system ispresent, no image plane is defined, and only projections, or shadows, of the object can beobserved.( z ) and transverse ( ~ρ ) coordinates under the Fresnel paraxial approximation: E ( ~ρ, z, t ) = Z dω d~κ ˜ E ( ~κ, ω ) g ( ~κ, ω ; ~ρ, z ) e − iωt (47)where ˜ E ( ω, ~κ ) is the complex amplitude of frequency ω and transverse wave-vector ~κ . In Eq. (47)we have taken z = 0 and t = 0 at the object plane as usual. To simplify the notation, wehave assumed one polarization.Based on the experimental setup of Fig. 5, g ( ~κ, ω ; ~ρ, z ) is found to be g ( ~κ, ω ; ~ρ i , s o + s i )= Z obj d~ρ o Z lens d~ρ l n A ( ~ρ o ) e i~κ · ~ρ o o n − iω πc e i ωc s o s o G ( | ~ρ l − ~ρ o | , ωcs o ) o × n G ( | ~ρ l | , − ωcf ) o n − iω πc e i ωc s i s i G ( | ~ρ i − ~ρ l | , ω cs i ) o (48)where ~ρ o , ~ρ l , and ~ρ i are two-dimensional vectors defined, respectively, on the object, the lens,and the image planes. The first curly bracket includes the object-aperture function A ( ~ρ o ) andthe phase factor e i~κ · ~ρ o contributed to the object plane by each transverse mode ~κ . Here wehave assumed a far-field finite size source. Thus, a phase factor e i~κ · ~ρ o appears on the objectplane of z = 0. If a collimated laser beam is used, this phase factor turns out to be a constant.The terms in the second and the fourth curly brackets describe free-space Fresnel propagation-diffraction from the source/object plane to the imaging lens, and from the imaging lens tothe detection plane, respectively. The Fresnel propagator includes a spherical wave function e i ωc ( z j − z k ) / ( z j − z k ) and a Fresnel phase factor G ( | ~α | , β ) = e i ( β/ | ~α | = e iω | ~ρ j − ~ρ k | / c ( z j − z k ) .The third curly bracket adds the phase factor, G ( | ~ρ l | , − ωcf ) = e − i ω cf , which is introduced bythe imaging lens.Applying the properties of the Gaussian function, Eq. (48) can be simplified into the fol-lowing form g ( ~κ, ω ; ~ρ i , z = s o + s i )= − ω (2 πc ) s o s i e i ωc ( s o + s i ) G ( | ~ρ i | , ωcs i ) Z obj d~ρ o A ( ~ρ o ) G ( | ~ρ o | , ωcs o ) e i~κ · ~ρ o × Z lens d~ρ l G ( | ~ρ l | , ωc [ 1 s o + 1 s i − f ]) e − i ωc ( ~ρoso + ~ρisi ) · ~ρ l . (49)The image plane is defined by the Gaussian thin-lens equation of Eq. (44). Hence, the secondintegral in Eq. (49) simplifies and gives, for a finite sized lens of radius R , the so called point-spread function of the imaging system: somb ( x ) = 2 J ( x ) /x , where x = [ Rs o ωc | ~ρ o + ρ i /m | ], J ( x )is the first-order Bessel function and m = s i /s o is the magnification of the imaging system.19ubstituting the result of Eqs. (49) into Eq. (47) enables one to obtain the classical self-correlation of the field, or, equivalently, the intensity on the image plane I ( ~ρ i , z i , t i ) = h E ∗ ( ~ρ i , z i , t i ) E ( ~ρ i , z i , t i ) i (50)where h ... i denotes an ensemble average. We assume monochromatic light for classical imagingas usual. Case (I): incoherent imaging.
The ensemble average of h ˜ E ∗ ( ~κ, ω ) ˜ E ( ~κ ′ , ω ) i yields zerosexcept when ~κ = ~κ ′ . The image is thus I ( ~ρ i ) ∝ Z d~ρ o (cid:12)(cid:12) A ( ~ρ o ) (cid:12)(cid:12) (cid:12)(cid:12) somb [ Rs o ωc | ~ρ o + ~ρ i m | ] (cid:12)(cid:12) . (51)An incoherent image, magnified by a factor of m , is thus given by the convolution betweenthe squared moduli of the object aperture function and the point-spread function. The spatialresolution of the image is thus determined by the finite width of the | somb | -function.Case (II): coherent imaging. The coherent superposition of the ~κ modes in both E ∗ ( ~ρ i , τ )and E ( ~ρ i , τ ) results in a wavepacket. The image, or the intensity distribution on the imageplane, is thus I ( ~ρ i ) ∝ (cid:12)(cid:12)(cid:12) Z obj d~ρ o A ( ~ρ o ) e i ω cso | ~ρ o | somb [ Rs o ωc | ~ρ o + ~ρ i m | ] (cid:12)(cid:12)(cid:12) . (52)A coherent image, magnified by a factor of m , is thus given by the squared modulus of theconvolution between the object aperture function (multiplied by a Fresnel phase factor) andthe point-spread function.For s i < s o and s o > f , both Eqs. (51) and (52) describe a real demagnified inverted image.In both cases, a narrower somb -function yields a higher spatial resolution. Thus, the use ofshorter wavelengths allows for improvement of the spatial resolution of an imaging system.To demonstrate the working principle of quantum imaging, we replace classical light withan entangled two-photon source such as spontaneous parametric down-conversion (SPDC) andreplace the ordinary film with a two-photon absorber, which is sensitive to two-photon transitiononly, on the image plane. We will show that, in the same experimental setup of Fig. 5, anentangled two-photon system gives rise, on a two-photon absorber, to a point-spread functionhalf the width of the one obtained in classical imaging at the same wavelength. Then, withoutemploying shorter wavelengths, entangled two-photon states improve the spatial resolution ofa two-photon image by a factor of 2 [19][20]. We will also show that the entangled two-photonsystem yields a peculiar Fourier transform function as if it is produced by a light source with λ/ z = s o + s i to an arbitrary plane of z = s o + d , where d may take any values for differentexperimental setups: g ( ~κ j , ω j ; ~ρ k , z = s o + d )= Z obj d~ρ o Z lens d~ρ l A ( ~ρ o ) { − iω j πcs o e i~κ j · ~ρ o e i ωjc s o G ( | ~ρ o − ~ρ l | , ω j cs o ) }× G ( | ~ρ l | , − ω j cf ) { − iω j πcd e i ωjc d G ( | ~ρ l − ~ρ k | , ω j cd ) } , (53) Even if assuming a perfect lens without chromatic aberration, Fresnel diffraction is wavelength dependent.Hence, large broadband (∆ ω ∼ ∞ ) would result in blurred images in classical imaging. Surprisingly, the situationis different in quantum imaging: no aberration blurring. ~ρ o , ~ρ l , and ~ρ j are two-dimensional vectors defined, respectively, on the (transverse) outputplane of the source (which coincide with the object plane), on the transverse plane of the imaginglens and on the detection plane; and j = s, i , labels the signal and the idler; k = 1 ,
2, labelsthe photodetector D and D . The function A ( ~ρ o ) is the object-aperture function, while theterms in the first and second curly brackets of Eq. (53) describe, respectively, free propagationfrom the output plane of the source/object to the imaging lens, and from the imaging lens tothe detection plane.Similar to the earlier calculation, by employing the second and third expressions given inEq. ( A − g ( ~κ j , ω j ; ~ρ k , z = s o + d )= − ω j (2 πc ) s o d e i ωjc ( s o + d ) G ( | ~ρ k | , ω j cd ) Z obj d~ρ o A ( ~ρ o ) G ( | ~ρ o | , ω j cs o ) e i~κ j · ~ρ o × Z lens d~ρ l G ( | ~ρ l | , ω j c [ 1 s o + 1 d − f ]) e − i ωjc ( ~ρoso + ~ρkd ) · ~ρ l . (54)Substituting the Green’s functions into Eq. (34), the effective two-photon wavefunctionΨ( ~ρ , z ; ~ρ , z ) is thusΨ( ~ρ , z ; ~ρ , z ) = Ψ Z d Ω f (Ω) G ( | ~ρ | , ω s cd ) G ( | ~ρ | , ω i cd ) × Z obj d~ρ o A ( ~ρ o ) G ( | ~ρ o | , ω s cs o ) Z obj d~ρ ′ o A ( ~ρ ′ o ) G ( | ~ρ ′ o | , ω i cs o ) × Z lens d~ρ l G ( | ~ρ l | , ω s c [ 1 s o + 1 d − f ]) e − i ωsc ( ~ρoso + ~ρ d ) · ~ρ l × Z lens d~ρ ′ l G ( | ~ρ ′ l | , [ ω i c [ 1 s o + 1 d − f ]) e − i ωic ( ~ρ ′ oso + ~ρ d ) · ~ρ ′ l × Z d~κ s d~κ i δ ( ~κ s + ~κ i ) e i ( ~κ s · ~ρ o + ~κ i · ~ρ ′ o ) (55)where we have absorbed all constants into Ψ , including the phase e i ωsc ( s o + d ) e i ωic ( s o + d ) = e i ωpc ( s o + d ) . The double integral of d~κ s and d~κ i yields a δ -function of δ ( ~ρ o − ~ρ ′ o ), and Eq. (55) is simplifiedas: Ψ( ~ρ , z ; ~ρ , z )= Ψ Z d Ω f (Ω) G ( | ~ρ | , ω s cd ) G ( | ~ρ | , ω i cd ) Z obj d~ρ o A ( ~ρ o ) G ( | ~ρ o | , ω p cs o ) × Z lens d~ρ l G ( | ~ρ l | , ω s c [ 1 s o + 1 d − f ]) e − i ωsc ( ~ρoso + ~ρ d ) · ~ρ l × Z lens d~ρ ′ l G ( | ~ρ ′ l | , [ ω i c [ 1 s o + 1 d − f ]) e − i ωic ( ~ρoso + ~ρ d ) · ~ρ ′ l . (56)We consider the following two cases:Case (I) on the imaging plane and ~ρ = ~ρ = ~ρ .In this case, Eq. (56) is simplified asΨ( ~ρ, z ; ~ρ, z ) ∝ Z obj d~ρ o A ( ~ρ o ) G ( | ~ρ o | , ω p cs o ) Z d~ρ l e − i ωp c ( ~ρoso + ~ρsi ) · ~ρ l Z d~ρ ′ l e − i ωp c ( ~ρoso + ~ρsi ) · ~ρ ′ l × n Z d Ω f (Ω) e − i Ω[( ~ρocso + ~ρcsi ) · ( ~ρ l − ~ρ ′ l )] o (57)21here we have used ω s = ω p / ω s = ω p / − Ω following ω s + ω i = ω p . The integralof d Ω gives a δ -function of δ [( ~ρ o cs o + ~ρcs i )( ~ρ l − ~ρ ′ l )] while taking the integral to infinity with aconstant f (Ω). This result indicates again that the propagation-diffraction of the signal andthe idler are not independent. The “two-photon diffraction” couples the two integrals in ~ρ o and ~ρ ′ o as well as the two integrals in ~ρ l and ~ρ ′ l and thus gives the G (2) function G (2) ( ~ρ, ~ρ ) ∝ (cid:12)(cid:12)(cid:12) Z obj d~ρ o A ( ~ρ o ) e i ωp cso | ~ρ o | J (cid:16) Rs o ω p c (cid:12)(cid:12) ~ρ o + ~ρm (cid:12)(cid:12)(cid:17)(cid:16) Rs o ω p c (cid:12)(cid:12) ~ρ o + ~ρm (cid:12)(cid:12)(cid:17) (cid:12)(cid:12)(cid:12) (58)which indicates that a coherent image (see Eq. (52)) magnified by a factor of m = s i /s o isreproduced on the image plane by joint-detection or by two-photon absorption.In Eq. (58), the point-spread function is characterized by the pump wavelength λ p = λ s,i / J ( x ) /x , which is much narrower than the somb ( x ).It is interesting to see that, different from the classical case, the frequency integral over∆ ω s ∼ ∞ does not give any blurring, but rather enhances the spatial resolution of the two-photon image.Case (II): on the Fourier transform plane and ~ρ = ~ρ = ~ρ .The detectors are now placed in the focal plane, i.e., d = f . In this case, the spatial effectivetwo-photon wavefunction Ψ( ~ρ, z ; ~ρ, z ) becomes:Ψ( ~ρ, z ; ~ρ, z ) ∝ Z d Ω f (Ω) Z obj d~ρ o A ( ~ρ o ) G ( | ~ρ o | , ω p cs o ) Z lens d~ρ l G ( | ~ρ l | , ω s cs o ) e − i ωsc ( ~ρoso + ~ρf ) · ~ρ l × Z lens d~ρ ′ l G ( | ~ρ ′ l | , ω i cs o ) e − i ωic ( ~ρoso + ~ρf ) · ~ρ ′ l . (59)We will first evaluate the two integrals over the lens. To simplify the mathematics we approxi-mate the integral to infinity. Differing from the calculation for imaging resolution, the purposeof this evaluation is to determine the Fourier transform. Thus, the approximation of an infinitelens is appropriate. By applying Eq. ( A − ~ρ o to the integral of d~ρ o in Eq. (59): C G ( | ~ρ o | , − ω p cs o ) e − i ωpcf ~ρ o · ~ρ where C absorbs all constants including a phase factor G ( | ~ρ | , − ω p cf /s o ). Replacing the twointegrals of d~ρ l and d~ρ ′ l in Eq. (59) with this result, we obtain:Ψ( ~ρ, z ; ~ρ, z ) ∝ Z d Ω f (Ω) Z obj d~ρ o A ( ~ρ o ) e − i ωpcf ~ρ · ~ρ o ∝ F [ ωpcf ~ρ ] (cid:8) A ( ~ρ o ) (cid:9) , (60)which is the Fourier transform of the object-aperture function. When the two photodetectorsscan together (i.e., ~ρ = ~ρ = ~ρ ), the second-order transverse correlation G (2) ( ~ρ, z ; ~ρ, z ), where z = s o + f , is reduced to: G (2) ( ~ρ, z ; ~ρ, z ) ∝ (cid:12)(cid:12) F [ ωpcf ~ρ ] (cid:8) A ( ~ρ o ) (cid:9)(cid:12)(cid:12) . (61)Thus, by replacing classical light with entangled two-photon sources, in the double-slit setup ofFig. 5, a Young’s double-slit interference/diffraction pattern with twice the interference modu-lation and half the pattern width, compared to that of classical light at wavelength λ s,i = 2 λ p ,22 CC i n s e c (cid:13) (a)(cid:13) -6(cid:13) -4(cid:13) -2(cid:13) 0 2 4 6 (cid:13) Angle (mrad)(cid:13) C oun t s pe r s e c (cid:13) (b)(cid:13) Figure 7: (a) Two-photon Fourier transform of a double-slit. The light source was a collineardegenerate SPDC of λ s,i = 916 nm . (b) Classical Fourier transform of the same double-slit.A classical light source of λ = 916 nm was used.is observed in the joint detection. This effect has also been examined in a recent “quantumlithography” experiment [20].Due to the lack of two-photon sensitive material, the first experimental demonstration ofquantum lithography was measured on the Fourier transform plane, instead of the image plane.Two point-like photon counting detectors were scanned jointly, similar to the setup illustratedin Fig. 5, for the observation of the interference/diffraction pattern of Eq. (61). The publishedexperimental result is shown in Fig. 7 [20]. It is clear that the two-photon Young’s double-slit interference-diffraction pattern has half the width with twice the interference modulationcompared to that of the classical case although the wavelengths are both 916 nm .Following linear Fourier optics, it is not difficult to see that, with the help of another lens(equivalently building a microscope), one can transform the Fourier transform function of thedouble-slit back onto its image plane to observe its image with twice the spatial resolution.The key to understanding the physics of this experiment is again through entangled natureof the signal-idler two-photon system. As we have discussed earlier, the pair is always emittedfrom the same point on the output plane of the source, thus always passing the same slittogether if the double-slit is placed close to the surface of the nonlinear crystal. There is nochance for the signal-idler pair to pass different slits in this setup. In other words, each pointof the object is “illuminated” by the pair “together” and the pair “stops” on the image plane“together”. The point-“spot” correspondence between the object and image planes are based onthe physics of two-photon diffraction, resulting in a twice narrower Fourier transform functionin the Fourier transform plane and twice the image resolution in the image plane. The unfoldedschematic setup, which is shown in Fig. 8, may be helpful for understanding the physics. It isnot difficult to calculate the interference-diffraction function under the experimental conditionindicated in Fig. 8. The non-classical observation is due to the superposition of the two-photonamplitudes, which are indicated by the straight lines connecting D and D . The two-photondiffraction, which restricts the spatial resolution of a two-photon image, is very different fromthat of classical light. Thus, there should be no surprise in having an improved spatial resolutioneven beyond the classical limit. 23 (cid:13) D(cid:13)
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Figure 8: Unfolded experimental setup. The joint measurement is on the Fourier transformplane. Each point of the object is “illuminated” by the signal-idler pair “together”, result-ing in twice narrower interference-diffraction pattern width in the Fourier transform planethrough the joint detection of the signal-idler pair, equivalent to the use of classical light of λ/ imaging setups only; sub-wavelength interference in a Mach-Zehnder type interferometer, for instance, does notnecessarily lead to an image.(2) In the imaging setup, it is the peculiar nature of the entangled N-photon system that allowsone to generate an image with N-times the spatial resolution: the entangled photons come outfrom one point of the object plane, undergo N-photon diffraction, and stop in the image planewithin a N-times narrower spot than that of classical imaging. The historical experiment byD’Angelo et al , in which the working principle of quantum lithography was first demonstrated,has taken advantage of the entangled two-photon state of SPDC: the signal-idler photon paircomes out from either the upper slit or the lower slit that is in the object plane, undergoestwo-photon diffraction, and stops in the image plane within a twice narrower image than thatof the classical one. It is easy to show that a second Fourier transform, by means of the useof a second lens to set up a simple microscope, will produce an image on the image plane withdouble spatial resolution.(3) Certain “clever” tricks allow the production of doubly modulated interference patterns byusing classical light in joint photo-detection. These tricks, however, may never be helpful forimaging. Thus, they may never be useful for lithography.24 Ghost imaging
The nonlocal position-position and momentum-momentum EPR correlation of the entan-gled two-photon state of SPDC was successfully demonstrated in 1995 [21] inspired by thetheory of Klyshko [22] The experiment was immediately named as “ghost imaging” in thephysics community due to its surprising nonlocal nature. The important physics demonstratedin the experiment, however, may not be the so called “ghost”. Indeed, the original purposeof the experiment was to study the EPR correlation in position and in momentum and totest the EPR inequality of Eq. (4) for the entangled signal-idler photon pair of SPDC [18][23].The experiments of “ghost imaging” [21] and “ghost interference” [24] together stimulated thefoundation of quantum imaging in terms of geometrical and physical optics.
BBO(cid:13) lens(cid:13)filter(cid:13) filter(cid:13)collection(cid:13)lens(cid:13)D(cid:13) D(cid:13) signal(cid:13) idler(cid:13) prism(cid:13) aperture(cid:13)laser(cid:13)pum(cid:13)p(cid:13) polarizing(cid:13)beam(cid:13)splitter(cid:13)
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Gated(cid:13)N(cid:13)
Figure 9: Schematic set-up of the “ghost” image experiment.The schematic setup of the “ghost” imaging experiment is shown in Fig. 9. A CW laser isused to pump a nonlinear crystal, which is cut for degenerate type-II phase matching to producea pair of orthogonally polarized signal (e-ray of the crystal) and idler (o-ray of the crystal)photons. The pair emerges from the crystal as collinear, with ω s ∼ = ω i ∼ = ω p /
2. The pump isthen separated from the signal-idler pair by a dispersion prism, and the remaining signal andidler beams are sent in different directions by a polarization beam splitting Thompson prism.The signal beam passes through a convex lens with a 400 mm focal length and illuminates achosen aperture (mask). As an example, one of the demonstrations used the letters “UMBC”for the object mask. Behind the aperture is the “bucket” detector package D , which consistsof a short focal length collection lens in whose focal spot is an avalanche photodiode. D ismounted in a fixed position during the experiment. The idler beam is met by detector package D , which consists of an optical fiber whose output is mated with another avalanche photodiode.The input tip of the fiber is scanned in the transverse plane by two step motors. The outputpulses of each detector, which are operating in photon counting mode, are sent to a coincidencecounting circuit for the signal-idler joint detection.By recording the coincidence counts as a function of the fiber tip’s transverse plane coordi-nates, the image of the chosen aperture (for example, “UMBC”) is observed, as reported in Fig.10. It is interesting to note that while the size of the “UMBC” aperture inserted in the signalbeam is only about 3 . mm × mm , the observed image measures 7 mm × mm . The imageis therefore magnified by a factor of 2. The observation also confirms that the focal length ofthe imaging lens, f , the aperture’s optical distance from the lens, S o , and the image’s optical25igure 10: (a) A reproduction of the actual aperature “UMBC” placed in the signal beam.(b) The image of “UMBC”: coincidence counts as a function of the fiber tip’s transverseplane coordinates. The step size is 0.25mm. The image shown is a “slice” at the halfmaximum value.distance from the lens, S i (which is from the imaging lens going backward along the signalphoton path to the two-photon source of the SPDC crystal then going forward along the pathof idler photon to the image), satisfy the Gaussian thin lens equation. In this experiment, S o was chosen to be S o = 600 mm , and the twice magnified clear image was found when the fibertip was on the plane of S i = 1200 mm . While D was scanned on other transverse planes notdefined by the Gaussian thin lens equation, the images blurred out.The measurement of the signal and the idler subsystem themselves are very different. Thesingle photon counting rate of D was recorded during the scanning of the image and was foundfairly constant in the entire region of the image. This means that the transverse coordinateuncertainty of either signal or idler is considerably large compared to that of the transversecorrelation of the entangled signal-idler photon pair: ∆ x (∆ y ) and ∆ x (∆ y ) are muchgreater than ∆( x − x ) (∆( y − y )).The EPR δ -functions, δ ( ~ρ s − ~ρ i ) and δ ( ~κ s + ~κ i ) in transverse dimension, are the key tounderstanding this interesting phenomenon. In degenerate SPDC, although the signal-idlerphoton pair has equal probability to be emitted from any point on the output surface of thenonlinear crystal, the transverse position δ -function indicates that if one of them is observedat one position, the other one must be found at the same position. In other words, the pairis always emitted from the same point on the output plane of the two-photon source. Thetransverse momentum δ -function, defines the angular correlation of the signal-idler pair: thetransverse momenta of a signal-idler amplitude are equal but pointed in opposite directions: ~κ s = − ~κ i . In other words, the two-photon amplitudes are always existing at roughly equalyet opposite angles relative to the pump. This then allows for a simple explanation of theexperiment in terms of “usual” geometrical optics in the following manner: we envision thenonlinear crystal as a “hinge point” and “unfold” the schematic of Fig. 9 into that shown inFig. 11. The signal-idler two-photon amplitudes can then be represented by straight lines (but26 EPR(cid:13)Source(cid:13)
S (cid:13)
S (cid:13) f (cid:13)
D(cid:13)
Imaging lens(cid:13)lens(cid:13)Collection(cid:13) o(cid:13) i(cid:13)
D(cid:13)
Figure 11: An unfolded setup of the “ghost” imaging experiment, which is helpful for un-derstanding the physics. Since the two-photon “light” propagates along “straight-lines”, itis not difficult to find that any geometrical light point on the subject plane corresponds toan unique geometrical light point on the image plane. Thus, a “ghost” image of the subjectis made nonlocally in the image plane. Although the placement of the lens, the object, anddetector D obeys the Gaussian thin lens equation, it is important to remember that thegeometric rays in the figure actually represent the two-photon amplitudes of an entangledphoton pair. The point to point correspondence is the result of the superposition of thesetwo-photon amplitudes.keep in mind the different propagation directions) and therefore, the image is well producedin coincidences when the aperture, lens, and fiber tip are located according to the Gaussianthin lens equation of Eq.(5). The image is exactly the same as one would observe on a screenplaced at the fiber tip if detector D were replaced by a point-like light source and the nonlinearcrystal by a reflecting mirror.Following a similar analysis in geometric optics, it is not difficult to find that any geometri-cal “light spot” on the subject plane, which is the intersection point of all possible two-photonamplitudes coming from the two-photon light source, corresponds to a unique geometrical “lightspot” on the image plane, which is another intersection point of all the possible two-photonamplitudes. This point to point correspondence made the “ghost” image of the subject-aperturepossible. Despite the completely different physics from classical geometrical optics, the remark-able feature is that the relationship between the focal length of the lens, f , the aperture’soptical distance from the lens, S o , and the image’s optical distance from the lens, S i , satisfythe Gaussian thin lens equation: 1 s o + 1 s i = 1 f . Although the placement of the lens, the object, and the detector D obeys the Gaussian thin lensequation, it is important to remember that the geometric rays in the figure actually representthe two-photon amplitudes of a signal-idler photon pair and the point to point correspondenceis the result of the superposition of these two-photon amplitudes. The “ghost” image is arealization of the 1935 EPR gedankenexperiment .Now we calculate G (2) ( ~ρ o , ~ρ i ) for the “ghost” imaging experiment, where ~ρ o and ~ρ i are thetransverse coordinates on the object plane and the image plane. We will show that there existsa δ -function like point-to-point relationship between the object plane and the image plane, i.e.,if one measures the signal photon at a position of ~ρ o on the object plane the idler photon can befound only at a certain unique position of ~ρ i on the image plane satisfying δ ( m~ρ o − ~ρ i ), where m = − ( s i /s o ) is the image-object magnification factor. After demonstrating the δ -function, weshow how the object-aperture function of A ( ~ρ o ) is transfered to the image plane as a magnified27mage A ( ~ρ i /m ). Before showing the calculation, it is worthwhile to emphasize again that the“straight lines” in Fig. 11 schematically represent the two-photon amplitudes belonging to apair of signal-idler photon. A “click-click” joint measurement at ( r , t ), which is on the objectplane, and ( r , t ), which is on the image plane, in the form of an EPR δ -function, is the resultof the coherent superposition of all these two-photon amplitudes. EPR(cid:13)Source(cid:13) f (cid:13)
Imaging lens(cid:13)lens(cid:13)Collection(cid:13) d(cid:13) d(cid:13) (cid:13) s(cid:13) o(cid:13) D(cid:13) i(cid:13) s(cid:13) f(cid:13) coll(cid:13)
Figure 12: In arm-1, the signal propagates freely over a distance d from the output planeof the source to the imaging lens, then passes an object aperture at distance s o , and then isfocused onto photon counting detector D by a collection lens. In arm-2, the idler propagatesfreely over a distance d from the output plane of the source to a point-like photon countingdetector D .We follow the unfolded experimental setup shown in Fig. 12 to establish the Green’s func-tions g ( ~κ s , ω s , ~ρ o , z o ) and g ( ~κ i , ω i , ~ρ , z ). In arm-1, the signal propagates freely over a distance d from the output plane of the source to the imaging lens, then passes an object aperture atdistance s o , and then is focused onto photon counting detector D by a collection lens. Wewill evaluate g ( ~κ s , ω s , ~ρ o , z o ) by propagating the field from the output plane of the two-photonsource to the object plane. In arm-2, the idler propagates freely over a distance d from theoutput plane of the two-photon source to a point-like detector D . g ( ~κ i , ω i , ~ρ , z ) is thus a freepropagator.(I) Arm-1 (source to object):The optical transfer function or Green’s function in arm-1, which propagates the field fromthe source plane to the object plane, is given by: g ( ~κ s , ω s ; ~ρ o , z o = d + s o )= e i ωsc z o Z lens d~ρ l Z A d~ρ S n − iω s πcd e i ~κ s · ~ρ S G ( | ~ρ S − ~ρ l | , ω s cd ) o × n G ( | ~ρ l | , ω s cf ) o n − iω s πcs o G ( | ~ρ l − ~ρ o | , ω s cs o ) o , (62)where ~ρ S and ~ρ l are the transverse vectors defined, respectively, on the output plane of thesource and on the plane of the imaging lens. The terms in the first and third curly bracketsin Eq. (62) describe free space propagation from the output plane of the source to the imaginglens and from the imaging lens to the object plane, respectively. The function G ( | ~ρ l | , ωcf ) in thesecond curly brackets is the transformation function of the imaging lens. Here, we treat it as athin-lens: G ( | ~ρ l | , ωcf ) ∼ = e − i ω cf | ~ρ l | . 28II) Arm-2 (from source to image):In arm-2, the idler propagates freely from the source to the plane of D , which is also theplane of the image. The Green’s function is thus: g ( ~κ i , ω i ; ~ρ , z = d ) = − iω i πcd e i ωic d Z A d~ρ ′ S G ( | ~ρ ′ S − ~ρ | , ω i cd ) e i~κ i · ~ρ ,S (63)where ~ρ ′ S and ~ρ are the transverse vectors defined, respectively, on the output plane of thesource, and on the plane of the photo-dector D .(III) Ψ( ~ρ o , ~ρ i ) (object plane - image plane):To simplify the calculation and to focus on the transverse correlation, in the followingcalculation we assume degenerate ( ω s = ω i = ω ) and collinear SPDC. The transverse two-photon effective wavefunction Ψ( ~ρ o , ~ρ ) is then evaluated by substituting the Green’s functions g ( ~κ s , ω ; ~ρ o , z o ) and g ( ~κ i , ω ; ~ρ , z ) into the expression given in Eq. (36):Ψ( ~ρ o , ~ρ ) ∝ Z d~κ s d~κ i δ ( ~κ s + ~κ i ) g ( ~κ s , ω ; ~ρ o , z o ) g ( ~κ i , ω ; ~ρ , z ) ∝ e i ωc ( s + s i ) Z d~κ s d~κ i δ ( ~κ s + ~κ i ) Z lens d~ρ l Z A d~ρ S e i ~κ s · ~ρ S G ( | ~ρ S − ~ρ l | , ωcd ) × G ( | ~ρ l | , ωcf ) G ( | ~ρ l − ~ρ o | , ωcs o ) Z A d ~ρ ,S e i~κ i · ~ρ ,S G ( | ~ρ ,S − ~ρ | , ωcd ) (64)where we have ignored all the proportional constants. Completing the double integral of d~κ s and d~κ s Z d~κ s d~κ i δ ( ~κ s + ~κ i ) e i ~κ s · ~ρ S e i~κ i · ~ρ ,S ∼ δ ( ~ρ S − ~ρ ,S ) , (65)Eq. (64) becomes:Ψ( ~ρ o , ~ρ ) ∝ Z lens d~ρ l Z A d~ρ S G ( | ~ρ − ~ρ S | , ωcd ) G ( | ~ρ S − ~ρ l | , ωcd ) G ( | ~ρ l | , ωcf ) G ( | ~ρ l − ~ρ o | , ωcs o ) . We then apply the properties of the Gaussian functions of Eq. ( A −
3) and complete the integralon d~ρ S by assuming the transverse size of the source is large enough to be treated as infinity.Ψ( ~ρ o , ~ρ ) ∝ Z lens d~ρ l G ( | ~ρ − ~ρ l | , ωcs i ) G ( | ~ρ l | , ωcf ) G ( | ~ρ l − ~ρ o | , ωcs o ) . (66)Although the signal and idler propagate to different directions along two optical arms, Inter-estingly, the Green function in Eq. (66) is equivalent to that of a classical imaging setup, ifwe imagine the fields start propagating from a point ~ρ o on the object plane to the lens andthen stop at point ~ρ on the imaging plane. The mathematics is consistent with our previousqualitative analysis of the experiment.The integral on d~ρ l yields a point-to-point relationship between the object plane and theimage plane that is defined by the Gaussian thin-lens equation: Z lens d~ρ l G ( | ~ρ l | , ωc [ 1 s o + 1 s i − f ]) e − i ωc ( ~ρoso + ~ρisi ) · ~ρ l ∝ δ ( ~ρ o + ~ρ i m ) (67)where the integral is approximated to infinity and the Gaussian thin-lens equation of 1 /s o +1 /s i = 1 /f is applied. We have also defined m = s i /s o as the magnification factor of the imaging29ystem. The function δ ( ~ρ o + ~ρ i /m ) indicates that a point ~ρ o on the object plane correspondsto a unique point ~ρ i on the image plane. The two vectors point in opposite directions and themagnitudes of the two vectors hold a ratio of m = | ~ρ i | / | ~ρ o | .If the finite size of the imaging lens has to be taken into account (finite diameter D ), theintegral yields a point-spread function of somb ( x ): Z lens d~ρ l e − i ωc ( ~ρoso + ~ρisi ) · ~ρ l ∝ somb (cid:16) Rs o ωc [ ~ρ o + ~ρ i m ] (cid:17) (68)where somb ( x ) = 2 J ( x ) /x , J ( x ) is the first-order Bessel function and R/s o is named as thenumerical aperture. The point-spread function turns the point-to-point correspondence betweenthe object plane and the image plane into a point-to-“spot” relationship and thus limits thespatial resolution. This point has been discussed in detail in the last section.Therefore, by imposing the condition of the Gaussian thin-lens equation, the transversetwo-photon effective wavefunction is approximated as a δ functionΨ( ~ρ o , ~ρ i ) ∝ δ ( ~ρ o + ~ρ i m ) (69)where ~ρ o and ~ρ i , again, are the transverse coordinates on the object plane and the imageplane, respectively, defined by the Gaussian thin-lens equation. Thus, the second-order spatialcorrelation function G (2) ( ~ρ o , ~ρ i ) turns out to be: G (2) ( ~ρ o , ~ρ i ) = | Ψ( ~ρ o , ~ρ i ) | ∝ | δ ( ~ρ o + ~ρ i m ) | . (70)Eq. (70) indicates a point to point EPR correlation between the object plane and the imageplane, i.e., if one observes the signal photon at a position ~ρ o on the object plane, the idler photoncan only be found at a certain unique position ~ρ i on the image plane satisfying δ ( ~ρ o + ~ρ i /m )with m = s i /s o .We now include an object-aperture function, a collection lens and a photon counting de-tector D into the optical transfer function of arm-1 as shown in Fig. 9.We will first treat the collection-lens- D package as a “bucket” detector. The “bucket” de-tector integrates all Ψ( ~ρ o , ~ρ ) which passes the object aperture A ( ~ρ o ) as a joint photo-detectionevent. This process is equivalent to the following convolution : R , ∝ Z obj d~ρ o (cid:12)(cid:12) A ( ~ρ o ) (cid:12)(cid:12) (cid:12)(cid:12) Ψ( ~ρ o , ~ρ i ) (cid:12)(cid:12) ≃ (cid:12)(cid:12) A ( − ~ρ i m ) (cid:12)(cid:12) (71)where, again, D is scanning in the image plane, ~ρ = ~ρ i . Eq. (71) indicates a magnified (or de-magnified) image of the object-aperture function by means of the joint-detection events betweendistant photodetectors D and D . The “-” sign in A ( − ~ρ i /m ) indicates opposite orientation ofthe image. The model of the “bucket” detector is a good and realistic approximation.Now we consider a detailed evaluation by including the object-aperture function, the col-lection lens and the photon counting detector D into arm-1. The Green’s function of Eq. (62)becomes: g ( ~κ s , ω s ; ~ρ , z = d + s o + f coll )= e i ωsc z Z obj d~ρ o Z lens d~ρ l Z A d~ρ S n − iω s πcd e i ~κ s · ~ρ S G ( | ~ρ S − ~ρ l | , ω s cd ) o × G ( | ~ρ l | , ω s cf ) n − iω s πcs o G ( | ~ρ l − ~ρ o | , ω s cs o ) o A ( ~ρ o ) × G ( | ~ρ o | , ω s cf coll ) n − iω s πcf coll G ( | ~ρ o − ~ρ | , ω s cf coll ) o (72)30here f coll is the focal-length of the collection lens and D is placed on the focal point ofthe collection lens. Repeating the previous calculation, we obtain the transverse two-photoneffective wavefunction:Ψ( ~ρ , ~ρ ) ∝ Z obj d~ρ o A ( ~ρ o ) δ ( ~ρ o + ~ρ m ) = A ( ~ρ o ) ⊗ δ ( ~ρ o + ~ρ m ) (73)where ⊗ means convolution. Notice, in Eq. (73) we have ignored the phase factors which haveno contribution to the formation of the image. The joint detection counting rate, R , , betweenphoton counting detectors D and D is thus: R , ∝ G (2) ( ~ρ , ~ρ ) ∝ (cid:12)(cid:12) A ( ~ρ o ) ⊗ δ ( ~ρ o + ~ρ m ) (cid:12)(cid:12) = (cid:12)(cid:12) A ( − ~ρ m ) (cid:12)(cid:12) (74)where, again, ~ρ = ~ρ i .As we have discussed earlier, the point-to-point EPR correlation is the result of the coherentsuperposition of a special selected set of two-photon states. In principle, one signal-idler paircontains all the necessary two-photon amplitudes that generate the ghost image - a nonclassicalcharacteristic which we name as a two-photon coherent image. In quantum mechanics, one can never expect to measure both the precise position and mo-mentum of a particle simultaneously. It is prohibited. We say that the quantum observable“position” and “momentum” are “complementary” because the precise knowledge of the po-sition (momentum) implies that all possible outcomes of measuring the momentum (position)are equally probable.Karl Popper, being a “metaphysical realist”, however, took a different point of view. In hisopinion, the quantum formalism could and should be interpreted realistically: a particle musthave a precise position and momentum [25]. This view was shared by Einstein. In this regard, heinvented a thought experiment in the early 1930’s aimed to support his realistic interpretationof quantum mechanics [26]. What Popper intended to show in his thought experiment is that aparticle can have both precise position and momentum simultaneously through the correlationmeasurement of an entangled two-particle system.Similar to EPR’s gedankenexperiment , Popper’s thought experiment is also based on thefeature of two-particle entanglement : if the position or momentum of particle 1 is known, thecorresponding observable of its twin, particle 2, is then 100% determined. Popper’s originalthought experiment is schematically shown in Fig. 13. A point source S, positronium as Poppersuggested, is placed at the center of the experimental arrangement from which entangled pairsof particles 1 and 2 are emitted in opposite directions along the respective positive and negative x -axes towards two screens A and B. There are slits on both screens parallel to the y -axis andthe slits may be adjusted by varying their widths ∆ y . Beyond the slits on each side stand anarray of Geiger counters for the joint measurement of the particle pair as shown in the figure.The entangled pair could be emitted to any direction in 4 π solid angles from the point source.However, if particle 1 is detected in a certain direction, particle 2 is then known to be in theopposite direction due to the momentum conservation of the pair.First, let us imagine the case in which slits A and B are both adjusted very narrowly. In thiscircumstance, particle 1 and particle 2 experience diffraction at slit A and slit B, respectively,and exhibit greater ∆ p y for smaller ∆ y of the slits. There seems to be no disagreement in thissituation between Copenhagen and Popper.Next, suppose we keep slit A very narrow and leave slit B wide open. The main purposeof the narrow slit A is to provide the precise knowledge of the position y of particle 1 andthis subsequently determines the precise position of its twin (particle 2) on side B throughquantum entanglement. Now, Popper asks, in the absence of the physical interaction with an31 a)(cid:13) S(cid:13)Slit A(cid:13) Slit B(cid:13)1(cid:13) 2(cid:13)(b)(cid:13) S(cid:13)Slit A(cid:13) Slit B(cid:13)removed(cid:13)1(cid:13) 2(cid:13) X(cid:13)Y(cid:13) Figure 13: Popper’s thought experiment. An entangled pair of particles are emitted froma point source with momentum conservation. A narrow slit on screen A is placed in thepath of particle 1 to provide the precise knowledge of its position on the y -axis and this alsodetermines the precise y -position of its twin, particle 2, on screen B. (a) Slits A and B areboth adjusted very narrowly. (b) Slit A is kept very narrow and slit B is left wide open.actual slit, does particle 2 experience a greater uncertainty in ∆ p y due to the precise knowledgeof its position? Based on his beliefs, Popper provides a straightforward prediction: particle 2must not experience a greater ∆ p y unless a real physical narrow slit B is applied . However,if Popper’s conjecture is correct, this would imply the product of ∆ y and ∆ p y of particle 2could be smaller than h (∆ y ∆ p y < h ). This may pose a serious difficulty for Copenhagen andperhaps for many of us. On the other hand, if particle 2 going to the right does scatter like itstwin, which has passed though slit A, while slit B is wide open, we are then confronted withan apparent action-at-a-distance !The use of a “point source” in Popper’s proposal has been criticized historically as thefundamental mistake Popper made [27]. It is true that a point source can never produce a pairof entangled particles which preserves the EPR correlation in momentum as Popper expected.However, notice that a “point source” is not a necessary requirement for Popper’s experiment.What is required is a precise position-position EPR correlation: if the position of particle 1 isprecisely known, the position of particle 2 is 100% determined. As we have shown in the lastsection, “ghost” imaging is a perfect tool to achieve this.In 1998, Popper’s experiment was realized with the help of two-photon “ghost” imaging[28]. Fig. 14 is a schematic diagram that is useful for comparison with the original Popper’sthought experiment. It is easy to see that this is a typical “ghost” imaging experimental setup.An entangled photon pair is used to image slit A onto the distant image plane of “screen” B. Inthe setup, s o is chosen to be twice the focal length of the imaging lens LS , s o = 2 f . Accordingto the Gaussian thin lens equation, an equal size “ghost” image of slit A appears on the two-photon image plane at s i = 2 f . The use of slit A provides a precise knowledge of the positionof photon 1 on the y -axis and also determines the precise y -position of its twin, photon 2, onscreen B by means of the two-photon “ghost” imaging. The experimental condition specified inPopper’s experiment is then achieved. When slit A is adjusted to a certain narrow width andslit B is wide open, slit A provides precise knowledge about the position of photon 1 on the y -axis up to an accuracy ∆ y which equals the width of slit A, and the corresponding “ghost32 a)(cid:13)(b)(cid:13) D (cid:13) BBO(cid:13)LS(cid:13)Slit A(cid:13) Slit B(cid:13)removed(cid:13) D (cid:13) Coincidence(cid:13)Circuit(cid:13)
Scan(cid:13) D (cid:13) BBO(cid:13)LS(cid:13)Slit A(cid:13) Slit B(cid:13) D (cid:13) D(cid:13) p(cid:13) y(cid:13)
D(cid:13) p(cid:13) y(cid:13) ?(cid:13)
Scan(cid:13)
X(cid:13) Y(cid:13)Y(cid:13)
Figure 14: Modified version of Popper’s experiment. An entangled photon pair is generatedby SPDC. A lens and a narrow slit A are placed in the path of photon 1 to provide theprecise knowledge of its position on the y -axis and also to determine the precise y -positionof its twin, photon 2, on screen B by means of two-photon “ghost” imaging. Photon countingdetectors D and D are used to scan in y -directions for joint detections. (a) Slits A and Bare both adjusted very narrowly. (b) Slit A is kept very narrow and slit B is left wide open.image” of pinhole A at screen B determines the precise position y of photon 2 to within thesame accuracy ∆ y . ∆ p y of photon 2 can be independently studied by measuring the width ofits “diffraction pattern” at a certain distance from “screen” B. This is obtained by recordingcoincidences between detectors D and D while scanning detector D along its y -axis, whichis behind screen B at a certain distance.Figure 15 is a conceptual diagram to connect the modified Popper’s experiment with two-photon “ghost” imaging. In this unfolded “ghost” imaging setup, we assume the entangledsignal-idler photon pair holds a perfect transverse momentum correlation with ~k s + ~k i ∼ s o = s i = 2 f . Thus,an equal size “ghost” image of slit A is expected to appear on the image plane of screen B.The detailed experimental setup is shown in Fig.16 with indications of the various distances.A CW Argon ion laser line of λ p = 351 . nm is used to pump a 3 mm long beta barium borate(BBO) crystal for type-II SPDC to generate an orthogonally polarized signal-idler photon pair.The laser beam is about 3 mm in diameter with a diffraction limited divergence. It is importantto keep the pump beam a large size so that the transverse phase-matching condition, ~k s + ~k i ∼ ~k p = 0), is well reinforced in the SPDC process, where ~k j ( j = s, i ) is the transverse wavevectorof the signal (s) and idler (i), respectively. The collinear signal-idler beams, with λ s = λ i =702 . nm = 2 λ p are separated from the pump beam by a fused quartz dispersion prism, andthen split by a polarization beam splitter PBS. The signal beam (photon 1) passes through theconverging lens LS with a 500 mm focal length and a 25 mm diameter. A 0 . mm slit is placedat location A which is 1000 mm (= 2 f ) behind the lens LS. A short focal length lens is usedwith D for focusing the signal beam that passes through slit A. The point-like photon countingdetector D is located 500 mm behind “screen B”. “Screen B” is the image plane defined by theGaussian thin lens equation. Slit B, either adjusted as the same size as that of slit A or openedcompletely, is placed to coincide with the “ghost” image. The output pulses from the detectorsare sent to a coincidence circuit. During the measurements, detector D is fixed behind slit Awhile detector D is scanned on the y -axis by a step motor.33 (cid:13) a(cid:13) b = b1+b2(cid:13)BBO(cid:13)LS(cid:13) D (cid:13) Y(cid:13)
Collection(cid:13)Lens(cid:13) Screen B(cid:13)
Figure 15: An unfolded schematic of ghost imaging. We assume the entangled signal-idlerphoton pair holds a perfect momentum correlation δ ( k s + k i ) ∼
0. The locations of the slitA, the imaging lens LS, and the “ghost” image must be governed by the Gaussian thin lensequation. In this experiment, we have chosen s o = s i = 2 f . Thus, the “ghost” image of slitA is expected to be the same size as that of slit A. Measurement 1 : Measurement 1 studied the case in which both slits A and B wereadjusted to be 0 . mm . The y -coordinate of D was chosen to be 0 (center) while D wasallowed to scan along its y -axis. The circled dot data points in Fig. 17 show the coincidence counting rates against the y -coordinates of D . It is a typical single-slit diffraction pattern with∆ y ∆ p y = h . Nothing is special in this measurement except that we have learned the widthof the diffraction pattern for the 0 . mm slit and this represents the minimum uncertaintyof ∆ p y . We should emphasize at this point that the single detector counting rate of D as afunction of its position y is basically the same as that of the coincidence counts except for ahigher counting rate. Measurement 2 : The same experimental conditions were maintained except that slit Bwas left wide open. This measurement is a test of Popper’s prediction. The y -coordinate of D was chosen to be 0 (center) while D was allowed to scan along its y -axis. Due to the entanglednature of the signal-idler photon pair and the use of a coincidence measurement circuit, onlythose twins which have passed through slit A and the “ghost image” of slit A at screen B withan uncertainty of ∆ y = 0 . mm (which is the same width as the real slit B we have used inmeasurement 1) would contribute to the coincidence counts through the joint detection of D and D . The diamond dot data points in Fig. 17 report the measured coincidence countingrates against the y coordinates of D . The measured width of the pattern is narrower thanthat of the diffraction pattern shown in measurement 1. It is also interesting to notice that thesingle detector counting rate of D keeps constant in the entire scanning range, which is verydifferent from that in measurement 1. The experimental data has provided a clear indicationof ∆ y ∆ p y < h in the joint measurements of the entangled photon pairs.Given that ∆ y ∆ p y < h , is this a violation of the uncertainty principle? Does quantummechanics agree with this peculiar experimental result? If quantum mechanics does provide asolution with ∆ y ∆ p y < h for photon 2. We would indeed be forced to face a paradox as EPRhad pointed out in 1935.Quantum mechanics does provide a solution that agrees with the experimental result. How-ever, the solution is for a joint measurement of an entangled photon pair that involves bothphoton 1 and photon 2, but not just for photon 2 itself .We now examine the experimental results with the quantum mechanical calculation byadopting the formalisms from the ghost image experiment with two modifications:Case (I): slits A = 0 . mm , slit B = 0 . mm .34 (cid:13) D (cid:13) Coincidence(cid:13)(3nsec)(cid:13)
BBO(cid:13) PBS(cid:13) LS(cid:13) Slit A(cid:13) a (cid:13) b1 (cid:13) b2 (cid:13) mm (cid:13) Slit B(cid:13)
Ar Laser (351.1nm)(cid:13) Y(cid:13)
Collection(cid:13)Lens(cid:13)
Figure 16: Schematic of the experimental setup. The laser beam is about 3 mm in diameter.The “phase-matching condition” is well reinforced. Slit A (0 . mm ) is placed 1000 mm = 2 f behind the converging lens, LS ( f = 500 mm ). The one-to-one “ghost image” (0 . mm ) ofslit A is located at B. The optical distance from LS in the signal beam taken as backthrough PBS to the SPDC crystal ( b = 255 mm ) and then along the idler beam to “screenB” ( b = 745 mm ) is 1000 mm = 2 f ( b = b + b ).This is the experimental condition for measurement one: slit B is adjusted to be the same asslit A. There is nothing surprising about this measurement. The measurement simply providesus with the knowledge for ∆ p y of photon 2 caused by the diffraction of slit B (∆ y = 0 . mm ).The experimental data shown in Fig. 17 agrees with the calculation. Notice that slit B is about745 mm away from the 3 mm two-photon source, the angular size of the light source is roughlythe same as λ/ ∆ y , ∆ θ ∼ λ/ ∆ y , where λ = 702 nm is the wavelength and ∆ y = 0 . mm isthe width of the slit. The calculated diffraction pattern is very close to that of the “far-field”Fraunhofer diffraction of a 0 . mm single-slit.Case (II): slit A = 0 . mm , slits B ∼ ∞ (wide open).Now we remove slit B from the ghost image plane. The calculation of the transverse effectivetwo-photon wavefunction and the second-order correlation is the same as that of the ghost imageexcept the observation plane of D is moved behind the image plane to a distance of 500 mm .The two-photon image of slit A is located at a distance s i = 2 f = 1000 mm ( b + b ) from theimaging lens, in this measurement D is placed at d = 1500 mm from the imaging lens. Themeasured pattern is simply a “blurred” two-photon image of slit A. The “blurred” two-photonimage can be calculated from Eq. (75) which is a slightly modified version of Eq. (66)Ψ( ~ρ o , ~ρ ) ∝ Z lens d~ρ l G ( | ~ρ − ~ρ l | , ωcd ) G ( | ~ρ l | , ωcf ) G ( | ~ρ l − ~ρ o | , ωcs o ) ∝ Z lens d~ρ l G ( | ~ρ l | , ωc [ 1 s o + 1 d − f ]) e − i ωc ( ~ρoso + ~ρid ) · ~ρ l (75)where d is the distance between the imaging lens and D . In this measurement, D was placed500 mm behind the image plane, i.e., d = s i + 500 mm . The numerically calculated “blurred”image, which is narrower then that of the diffraction pattern of the 0 . mm slit B, agrees withthe measured result of Fig. 17 within experimental error.The measurement does show a result of ∆ y ∆ p y < h . The measurement, however, has35 D2 position (mm) N o r m a li z ed C o i n c i den c e s With Slit BWithout Slit B
Figure 17: The observed coincidence patterns. The y -coordinate of D was chosen to be 0(center) while D was allowed to scan along its y -axis. Circled dot points: Slit A = Slit B = . mm . Diamond dot points: Slit A = . mm , Slit B wide open . The width of the sinc function curve fitted by the circled dot points is a measure of the minimum ∆ p y diffractedby a 0 . mm slit.nothing to do with the uncertainty relation, which governs the behavior of photon 2 (the idler).Popper and EPR were correct in the prediction of the outcomes of their experiments. Popperand EPR, on the other hand, made the same error by applying the results of two-particle physicsto the explanation of the behavior of an individual subsystem.In both the Popper and EPR experiments, the measurements are “joint detection” betweentwo detectors applied to entangled states. Quantum mechanically, an entangled two-particlestate only provides the precise knowledge of the correlations of the pair . The behavior of“photon 2” observed in the joint measurement is conditioned upon the measurement of itstwin. A quantum must obey the uncertainty principle but the “conditional behavior” of aquantum in an entangled two-particle system is different in principle. We believe paradoxes areunavoidable if one insists the conditional behavior of a particle is the behavior of the particle.This is the central problem in the rationale behind both Popper and EPR. ∆ y ∆ p y ≥ h is notapplicable to the conditional behavior of either “photon 1” or “photon 2” in the cases of Popperand EPR.The behavior of photon 2 being conditioned upon the measurement of photon 1 is wellrepresented by the two-photon amplitudes. Each of the straight lines in the above discus-sion corresponds to a two-photon amplitude. Quantum mechanically, the superposition ofthese two-photon amplitudes are responsible for a “click-click” measurement of the entangledpair. A “click-click” joint measurement of the two-particle entangled state projects out certaintwo-particle amplitudes, and only these two-particle amplitudes are featured in the quantumformalism. In the above analysis we never consider “photon 1” or “photon 2” individually .Popper’s question about the momentum uncertainty of photon 2 is then inappropriate.Once again, the demonstration of Popper’s experiment calls our attention to the importantmessage: the physics of an entangled two-particle system must be inherently very different fromthat of individual particles. The entangled EPR two-particle state is a pure state with zero entropy. The precise corre-lation of the subsystems is completely described by the state. The measurement, however, is36ot necessarily always on the two-photon system. It is an experimental choice to study a singlesubsystem and to ignore the other. What can be learn about a subsystem from these kindsof measurements? Mathematically, it is easy to show that by taking a partial trace of a two-particle pure state, the state of each subsystem is in a mixed state with entropy greater thanzero. One can only learn statistical properties of the subsystems in this kind of measurement.In the following, again, we use the signal-idler pair of SPDC as an example to study thephysics of a subsystem. The two-photon state of SPDC is a pure state that satisfiesˆ ρ = ˆ ρ, ˆ ρ ≡ | Ψ i h Ψ | where ˆ ρ is the density operator corresponding to the two-photon state of SPDC. The singlephoton states of the signal and idlerˆ ρ s = tr i | Ψ i h Ψ | , ˆ ρ i = tr s | Ψ i h Ψ | are not pure states. To calculate the signal (idler) state from the two-photon state, we take apartial trace, as usual, summing over the idler (signal) modes.We assume a type II SPDC. The orthogonally polarized signal and idler are degeneratein frequency around ω s = ω i = ω p /
2. To simplify the discussion, by assuming appropriateexperimental conditions, we trivialize the transverse part of the state and write the two-photonstate in the following simplified form: | Ψ i = Ψ Z d Ω Φ(DLΩ) a † s ( ω s + Ω) a † i ( ω i − Ω) | i where Φ(DLΩ) is a sinc -like function:Φ(DLΩ) = 1 − e − i DLΩ i DLΩwhich is a function of the crystal length L, and the difference of inverse group velocities of thesignal (ordinary) and the idler (extraordinary), D ≡ /u o − /u e . The constant Ψ is calculatedfrom the normalization tr ˆ ρ = h Ψ | Ψ i = 1. It is easy to calculate and to find ˆ ρ = ˆ ρ for thetwo-photon state of the signal-idler pair.Summing over the idler modes, the density matrix of signal is given byˆ ρ s = Ψ Z d Ω | Φ(Ω) | a † s ( ω s + Ω) | i h | a s ( ω s + Ω) (76)with | Φ(Ω) | = sinc DLΩ2where all constants coming from the integral have been absorbed into Ψ . First, we findimmediately that ˆ ρ s = ˆ ρ s . It means the state of the signal is a mixed state (as is the idler).Second, it is very interesting to find that the spectrum of the signal depends on the groupvelocity of the idler. This, however, should not come as a surprise, because the state of thesignal photon is calculated from the two-photon state by summing over the idler modes.The spectrum of the signal and idler has been experimentally verified by Strekalov et al using a Michelson interferometer in a standard Fourier spectroscopy type measurement [29]. Themeasured interference pattern is shown in Fig. 18. The envelope of the sinusoidal modulations(in segments) is fitted very well by two “notch” functions (upper and lower part of the envelope).The experimental data agrees with the theoretical analysis of the experiment.The following is a simple calculation to explain the observed “notch” function. We firstdefine the field operators: E (+) ( t, z d ) = E (+) ( t − z c , z ) + E (+) ( t − z c , z )37igure 18: Experimental data indicated a “double notch” envelope. Each of the doted singlevertical line contines many cycles of sinusoidal modulation.where z d is the position of the photo-detector, z is the input point of the interferometer, t = t − z c and t = t − z c , respectively, are the early times before propagating to the photo-detector at time t with time delays of z /c and z /c , where z and z are the optical paths inarm 1 and arm 2 of the interferometer. We have defined a very general field operator whichis a superposition of two early fields propagated individually through arm 1 and arm 2 of anytype of interferometer. The counting rate of the photon counting detector is thus R d = tr (cid:2) ˆ ρ s E ( − ) ( t, z d ) E (+) ( t, z d ) (cid:3) = Ψ Z d Ω | Φ(Ω) | (cid:12)(cid:12) h | E (+) ( t, z d ) a † s ( ω s + Ω) | i (cid:12)(cid:12) = Ψ Z d Ω | Φ(Ω) | (cid:12)(cid:12) h | (cid:2) E (+) ( t − z c , z ) + E (+) ( t − z c , z ) (cid:3) a † s ( ω s + Ω) | i (cid:12)(cid:12) ∝ Re h e − iω τ Z d Ω sinc DLΩ2 e − i Ω τ i (77)where τ = ( z − z ) /c . The Fourier transform of sinc (DLΩ /
2) has a “notch” shape. It isnoticed that the base of the “notch” function is determined by parameter DL of the SPDC,which is easily confirmed from the experiment.Now we turn to another interesting aspect of physics, namely the physics of entropy. Inclassical information theory, the concept of entropy, named as Von Neuman entropy, is definedby [30] S = − tr (ˆ ρ log ˆ ρ ) (78)where ˆ ρ is the density operator. It is easy to find that the entropy of the entangled two-photonpure state is zero. The entropy of its subsystems, however, are both greater than zero. The valueof the Von Neuman entropy can be numerically evaluated from the measured spectrum. Notethat the density operator of the subsystem is diagonal. Taking its trace is simply performing anintegral over the frequency spectrum with the measured spectrum function. It is straightforwardto find the entropy of the subsystems S s >
0. This is an expected result due to the statisticalmixture nature of the subsystem. Considering that the entropy of the two-photon system is zeroand the entropy of the subsystems are both greater than zero, does this mean that negativeentropy is present somewhere in the entangled two-photon system? According to classical“information theory”, for the entangled two-photon system, S s + S s | i = 0, where S s | i is theconditional entropy. It is this conditional entropy that must be negative, which means that38 iven the result of a measurement over one particle, the result of a measurement over the othermust yield negative information [31]. This paradoxical statement is similar and, in fact, closelyrelated to the EPR “paradox”. It comes from the same philosophy as that of the EPR. Summary
The physics of an entangled system is very different from that of either classically inde-pendent or correlated systems. We use 2 = 1 + 1 to emphasize the nonclassical behavior of anentangled two-particle system. The entangled system is characterized by the properties of anentangled state which does not specify the state of an individual system, but rather describesthe correlation between the subsystems. An entangled two particle state is a pure state whichinvolves the superposition of a set of “selected” two-particle states, or two-particle quantummechanical amplitudes. Here, the term “selection” stems from the physical laws which governthe creation of the subsystems in the source, such as energy or momentum conservation. In-terestingly, quantum mechanics allows for the superposition of these local two-particle stateswhich have been observed in nature. However, the most surprising physics arises from the jointmeasurement of the two particles when they are released form the source and propagated a largedistance apart. The two well separated interaction-free particles do not lose their entangledproperties, i.e., they maintain their “selected” set of two-particle superposition. In this sensequantum mechanics allows for the two-particle superposition of well separated particles whichhas, remarkably, also been observed to exist in nature.The two-photon state of SPDC is a good example. The nonlinear interaction of sponta-neous parametric down-conversion coherently creates a set of mode in pairs that satisfy thephase matching conditions of Eq. (11) which is also characteristic of energy and momentumconservation. The signal-idler photon pair can be excited to any or all of these coupled modessimultaneously, resulting in a superposition of these coupled modes inside of the nonlinear crys-tal. The physics behind the two-photon superposition becomes even more interesting whenthe signal-idler pair is separated and propagated a large distance apart outside the nonlinearcrystal, either through free propagation or guided by optical components. Remarkably the en-tangled pair does not lose its entangled properties once the subsystems are interaction free. Asa result the properties of the entangled two-photon system, such as the EPR correlation or theEPR inequalities, are still observable in the joint detection counting rate of the pair, regardlessof the distance between the two photons as well as the two individual photo-detection events.In this situation the superposition of the two-photon amplitudes, corresponding to different yetindistinguishable alternative ways of triggering a joint photo-electron event at any distance canbe regarded as nonlocal. There is no counterpart to such a concept in classical theory and thisbehavior may never be understood in any classical sense. It is with this intent that we use2 = 1 + 1 to emphasize that the physics of a two-photon is not the same as that of two photons. A statement from the author
This article was originally prepared as lecture notes for my students a few years ago. It wasalso used in 2006 for a conference. My colleagues, friends and students have urged me to includeit in this archive. They believe that this article is helpful for the general physics and engineeringcommunity. Truthfully, I have been hesitant because I cannot forget my terrible experience in1996 as a result of Pittman’s experiment: “Can tow-photon interference be considered theinterference of two photons?” [21] My email account was bombarded for months. Of course, Iwas happy to have scientific discussions on the subject, but certain types of messages causedheadaches. For example, an individual attempted to force my laboratory to pay a visit for aface-to-face condemnation on my guilt for saying 1 + 1 = 2. (I truly believe what I said was2 = 1 + 1 and anyone would be able to see the difference by reading this article). Another39ndividual expressed their interests in a law suit because we did not acknowledge that theywere the first to show “Dirac was mistaken”. (I am definitely sure that we have nothing todo with their “discovery”. What we said was “Dirac was correct”.) I decided to keep quiet. Iunderstood that it takes time for people to recognize the truth.I have to break my silence now, because we are experiencing the same problem again. Mystudent Scarcelli published a lens-less ghost imaging experiment of chaotic light and raised areasonable question: “Can two-photon correlation of chaotic light be considered as correla-tion of intensity fluctuations?” [32]. The lens-less ghost imaging setup of Scarcelli et al . isa straightforward modification of the historical Hanbury-Brown and Twiss experiment (HBT)[33]. Advancing from HBT to the fundamentally interesting and practically useful lens-lessghost imaging, what one needs to do is simply move the two HBT photodetectors from far-fieldto near-field. We cannot but stop to ask: What has been preventing this simple move for 50years (1956-2006)? Some aspect must be terribly misleading to give us such misled confidencenot to even try the near-field measurement in half a century. As we know, unlike the first-ordercorrelation of radiation that is considered as the interference effect of the electromagnetic waves,the second-order correlation of light is treated as statistical correlation of intensity fluctuations.Scarcelli et al . pointed out that although the theory of statistical correlation of intensity fluc-tuations gives a reasonable explanation to the far-field HBT phenomena, it does not work innear-field and consequently does not work for their lens-less near-field ghost imaging experiment[34]. It was the idea of statistical correlation of intensity fluctuation that has prevented thisfrom happening for 50 years. On the other hand, under the framework of Glauber’s theory ofphotodetection, Scarcelli et al . proved a successful interpretation based on the quantum pictureof two-photon interference. This successes indicates that although the concept of multi-photoninterference, or the superposition of multi-photon amplitudes, was benefited from the researchof entangled states, the concept is generally true and applicable to any radiation, including“classical” thermal light. Unfortunately, this concept has no counterpart in classical electro-magnetic theory of light. Now, we are back to 1996. My student and I have been chargedwith “guilt” again because we have told the physics community a simple truth of the failure ofa classical idea and adapted the quantum mechanical concept of two-photon superposition to“classical” light.It was a mistake to keep silence. I have finally resolved to speak about the subject. Theconcept of multi-photon coherence, or the superposition principle of multi-photon amplitudes,is important and worthwhile to do, even if I might be burned at the stake. Appendix: Fresnel propagation-diffraction
In Fig. A −
1, the field is freely propagated from the source plane σ to an arbitrary plane σ . It is convenient to describe such a propagation in the form of Eq. (33). We now evaluate g ( ~κ, ω ; ~ρ, z ), namely the Green’s function for free-space Fresnel propagation-diffraction.According to the Huygens-Fresnel principle, the field at a space-time point ( ~ρ, z, t ) is theresult of a superposition of the spherical secondary wavelets originated from each point on the σ plane, see Fig. A − E (+) ( ~ρ, z, t ) = Z dω d~κ a ( ω, ~κ ) Z σ d~ρ ˜ A ( ~ρ ) r ′ e − i ( ωt − kr ′ ) ( A − A ( ~ρ ) is the complex amplitude, or distribution function, in terms of the transversecoordinate ~ρ , which may be a constant, a simple aperture function, or a combination of thetwo. In Eq. ( A − z = 0 and t = 0 on the source plane of σ as usual.In a paraxial approximation, we take the first-order expansion of r ′ in terms of z and ~ρr ′ = p z + | ~ρ − ~ρ | ≃ z (1 + | ~ρ − ~ρ | z ) . (cid:13) r(cid:13) zr’ k( k,w)(cid:13) r r’’ r(cid:13) A( )(cid:13) s(cid:13) s(cid:13) E( r, z )(cid:13) Figure A −
1: Schematic of free-space Fresnel propagation. The complex amplitude ˜ A ( ~ρ ) iscomposed by a real function A ( ~ρ ) and a phase e − i~κ · ~ρ associated with each of the transversewavevector ~κ on the plane of σ . Notice: only one mode of wavevector k ( ~κ, ω ) is shown inthe figure. E (+) ( ~ρ, z, t ) is thus approximated as E (+) ( ~ρ, z, t ) ≃ Z dω d~κ a ( ω, ~κ ) Z d~ρ ˜ A ( ~ρ ) z e i ωc z e i ω cz | ~ρ − ~ρ | e − iωt where e i ω cz | ~ρ − ~ρ | is named as the Fresnel phase factor.Assuming the complex amplitude ˜ A ( ~ρ ) is composed of a real function A ( ~ρ ) and a phase e − i~κ · ~ρ , associated with the transverse wavevector and the transverse coordinate on the planeof σ , which is reasonable for the setup of Fig. A − E ( ~ρ, z, t ) can be written in the followingform E (+) ( ~ρ, z, t ) = Z dω d~κ a ( ω, ~κ ) e − iωt e i ωc z z Z d~ρ A ( ~ρ ) e i~κ · ~ρ e i ω cz | ~ρ − ~ρ | . The Green’s function g ( ~κ, ω ; ~ρ, z ) for free-space Fresnel propagation is thus g ( ~κ, ω ; ~ρ, z ) = e i ωc z z Z σ d~ρ A ( ~ρ ) e i~κ · ~ρ G ( | ~ρ − ~ρ | , ωcz ) . ( A − A −
2) we have defined a Gaussian function G ( | ~α | , β ) = e i ( β/ | α | , namely the Fresnelphase factor. It is straightforward to find that the Gaussian function G ( | ~α | , β ) has the followingproperties: G ∗ ( | ~α | , β ) = G ( | ~α | , − β ) ,G ( | ~α | , β + β ) = G ( | ~α | , β ) G ( | ~α | , β ) ,G ( | ~α + ~α | , β ) = G ( | ~α | , β ) G ( | ~α | , β ) e iβ~α · ~α , Z d~α G ( | ~α | , β ) e i~γ · ~α = i πβ G ( | ~γ | , − β ) . ( A − A −
3) is the Fourier transform of the G ( | ~α | , β ) function.As we shall see in the following, these properties are very useful in simplifying the calculationsof the Green’s functions g ( ~κ, ω ; ~ρ, z ).Now, we consider inserting an imaginary plane σ ′ between σ and σ . This is equivalenthaving two consecutive Fresnel propagations with a diffraction-free σ ′ plane of infinity. Thus,the calculation of these consecutive Fresnel propagations should yield the same Green’s function41s that of the above direct Fresnel propagation shown in Eq. ( A − g ( ~κ, ω ; ~ρ, z )= C e i ωc ( d + d ) d d Z σ ′ d~ρ ′ Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ ′ − ~ρ | , ωcd ) G ( | ~ρ − ~ρ ′ | , ωcd )= C e i ωc z z Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ − ~ρ | , ωcz ) ( A − C is a necessary normalization constant for a valid Eq. ( A − z = d + d . Thedouble integral of d~ρ and d~ρ ′ in Eq. ( A −
4) can be evaluated as Z σ ′ d~ρ ′ Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ ′ − ~ρ | , ωcd ) G ( | ~ρ − ~ρ ′ | , ωcd )= Z σ d~ρ ˜ A ( ~ρ ) G ( ~ρ , ωcd ) G ( ~ρ, ωcd ) Z σ ′ d~ρ ′ G ( ~ρ ′ , ωc ( 1 d + 1 d )) e − i ωc ( ~ρ d + ~ρd ) · ~ρ ′ = i πcω d d d + d Z σ d~ρ ˜ A ( ~ρ ) G ( ~ρ , ωcd ) G ( ~ρ, ωcd ) G ( | ~ρ d + ~ρd | , ωc ( d d d + d ))= i πcω d d d + d Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ − ~ρ | , ωc ( d + d ) )where we have applied Eq. ( A − d~ρ ′ has been taken to infinity. Substitutingthis result into Eq. ( A − g ( ~κ, ω ; ~ρ, z ) = C i πcω e i ωc ( d + d ) d + d Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ − ~ρ | , ωc ( d + d ) )= C e i ωc z z Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ − ~ρ | , ωcz ) . Therefore, the normalization constant C must take the value of C = − iω/ πc. The normalizedGreen’s function for free-space Fresnel propagation is thus g ( ~κ, ω ; ~ρ, z ) = − iω πc e i ωc z z Z σ d~ρ ˜ A ( ~ρ ) G ( | ~ρ − ~ρ | , ωcz ) . ( A − References [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. , 777 (1935).[2] N. Bohr, Phys. Rev. , 696, (1935).[3] Quantum Theory and Measurement , J.A. Wheeler and W.H. Zurek, eds., Princeton Uni-versity Press, Princeton, 1983.[4] A. Pais, ‘Subtle is the lord...’ The Science and the Life of Albert Einstein , Oxford UniversityPress, Oxford and New York, 1982.[5] M. D’Angelo, Y.H. Kim, S. P. Kulik, and Y.H. Shih, Phys. Rev. Lett. , 233601 (2004).[6] D. Bohm, Phys. Rev. , 166, 180 (1952); D. Bohm, Causality and Chance in ModernPhysics , D. Van Nostrand Co., Inc., Princeton, 1957; D. Bohm and Y. Aharonov, Phys.Rev. , 1070 (1957). 427] J.S. Bell, Physics, , 195 (1964); Speakable and Unspeakable in Quantum Mechanics , Cam-bridge University Press, New York, 1987.[8] J.F. Clauser and A. Shimony, Rep. Prog. Phys. , 1883 (1978).[9] A. Aspect, et al ., Phys. Rev. Lett. , 460 (1981); A. Aspect, et al ., Phys. Rev. Lett. ,91 (1982); A. Aspect, et al ., Phys. Rev. Lett. , 1804 (1982).[10] Y.H. Shih and C.O. Alley, Phys. Rev. Lett. , 2921, (1988); Z.Y. Ou and L. Mandel,Phys. Rev. Lett. , 50 (1988); T.E. Kiess, Y.H. Shih, A.V. Sergienko, and C.O. Alley,Phys. Rev. Lett. , 3893 (1993); P.G. Kwiat et al ., Phys. Rev. Lett. , 4337 (1995).[11] E.Schr¨odinger, Naturwissenschaften , 807, 823, 844 (1935); English translations appearin ref. [3].[12] D.N. Klyshko, Photon and Nonlinear Optics , Gordon and Breach Science, New York, 1988.[13] Y.H. Shih, IEEE J. of Selected Topics in Quantum Electronics, , 1455 (2003).[14] A. Yariv, Quantum Electronics , John Wiley and Sons, New York, (1989).[15] R.J. Glauber, Phys. Rev. , 2529 (1963); Phys. Rev. , 2766 (1963).[16] M. H. Rubin, Phys. Rev. A , 5349, (1996).[17] J. W. Goodman, Introduction to Fourier Optics , McGraw-Hill Publishing Company, NewYork, NY, 1968.[18] Y.H. Shih, IEEE J. of Selected Topics in Quantum Electronics, (2007).[19] This effect was first proposed for lithography application, namely quantum lithography, byA.N. Boto et al ., Phys. Rev. Lett. , 2733 (2000).[20] M. D’Angelo, M.V. Chekhova, and Y.H. Shih, Phys. Rev. Lett. , 013603 (2001). M.D’Angelo, M.V. Chekhova, and Y.H. Shih, Phys. Rev. Lett., , 013603 (2001). Note: Dueto the lack of a two-photon absorber, the joint-detection measurement in this experimentwas on the Fourier transform plane rather than on the image plane. It was implicit inRef.[20] that a second Fourier transform, by inserting a second lens in that experimentalsetup, would transfer the Fourier transform of the object onto its image plane, thus givingan image with doubled spatial resolution despite the Rayleigh diffraction limit. It might behelpful to point out that the observation of sub-wavelength interference in a Mach Zehndertype interferometer cannot lead to sub-diffraction-limited images, except a set of doublemodulated interference pattern. The Fourier transform argument works only for imagingsetups as is in Ref.[20][21] T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, Phys. Rev. A , R3429(1995).[22] D.N. Klyshko, Usp. Fiz. Nauk, , 133 (1988); Sov. Phys. Usp, , 74 (1988); Phys. Lett.A, , 299 (1988).[23] M. D’Angelo, A. Valencia, M.H. Rubin, and Y.H. Shih, Phys. Rev. A , 013810 (2005).[24] D.V. Strekalov, A.V. Sergienko, D.N. Klyshko and Y.H. Shih, Phys. Rev. Lett. , 3600(1995).[25] K.R. Popper, in Open Questions in Quantum Physics , G. Tarozzi and A. van der Merwe,eds., D. Reidel Publishing Co., Dordrecht, 1985; K.R. Popper, in
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