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Two-particle asynchronous quantum correlation: wavefunction collapse acting as a beamsplitter

A two-body quantum correlation is calculated for a particle reflecting from a moving mirror. Correlated interference results when the incident and reflected particle substates and their associated mirror substates overlap. Using the Copenhagen interpretation of measurement, an asynchronous joint probability density (PDF), which is a function both of the different positions and different times at which the particle and mirror are measured, is derived assuming that no interaction occurs between each measurement. Measurement of the particle first, in the correlated interference region, results in a splitting of the mirror substate into ones which have and have not reflected the particle. An analog of the interference from the Doppler effect for only measurements of the particle (a marginal PDF), in this two-body system, is shown to be a consequence of the asynchronous measurement. The simplification obtained for a microscopic particle reflecting from a mesoscopic or macroscopic mirror is used to illustrate asynchronous correlation interferometry. In this case, the small displacement between these mirror states can yield negligible environmental decoherence times. In addition, interference of these mirror states does not vanish in the limit of large mirror mass due to the small momentum exchange in reflecting a microscopic particle.


I. INTRODUCTION
Correlation and interference distinguish quantum from classical physics. The former is manifest in the measurement of many-body coincidences predicted by a quantum joint probability density function (PDF). The latter is most familiar as a one-body PDF for an outcome that can be achieved in at least two indistinguishable ways. However, interference can also be generated by superposing many-body states in indistinguishable ways [1]. Experimental confirmation of quantum correlation has involved photons [2,3], atoms [4], and Josephson phase qubits [5].
Analysis of correlated systems usually begins with an expression for the many-body state after it has been prepared by an interaction. The extremes of either a few or a large number of particles are most often treated. In the former case the analysis typically involves a simultaneous correlation between the positions [1] or angular momenta of two particles [6]. The formalism often deals only with identical particles The issue of non-simultaneous correlations is explored here in the context of perhaps the simplest quantum correlation possible: a particle reflecting from a mirror, both of which are distinguishable. The mirror and particle have non-zero rest mass and motion is in free space along one dimension with all states unbound. Measurements of particle reflection, but associated with neither correlated interference nor asynchronous measurement, have involved atoms reflecting from a solid surface [7], neutrons [8] and atoms [9] reflecting from vibrating mirrors, and atoms reflecting from a switchable mirror [10].
Consider first the particle and mirror in uncorrelated eigenstates of energy before reflection, referred to as the 'incident harmonic state'. The energy eigenstate after reflection, referred to as the 'reflected harmonic state', results in correlation between the particle and mirror via conservation of energy and momentum: an energy or momentum measurement of the reflected particle yields a correlated energy or momentum measurement of the mirror when given the incident harmonic state.
Superposing such states yields the incident and reflected particle-mirror wavegroups. These differ from the harmonic states in that a measurement of the energy (or position) of the particle does not uniquely constrain the energy (or position) of the mirror since the reflected particle-mirror wavegroup is not an eigenstate of the energy (or position) operator.
The incident and reflected wavegroup particle-mirror states interfere when they overlap. This is similar to the transient one-body interference of an electromagnetic wavegroup reflecting from a stationary mirror [11]. However, classically the mirror experiences only a continuous force due to radiation pressure.
Quantum mechanically, interference occurs since the incident and reflected states are indistinguishable for a measurement of position (but not for a momentum measurement). Interference is expected between the incident and reflected particle substates along with interference of the mirror substates which have and have not reflected the particle. Their correlation is perhaps not expected, being a consequence of the solution to the Schrödinger equation from which a joint PDF is constructed. The correlations in the two-body interference are manifest as coincidence rates, e.g. a correlation in the simultaneous measurement of particle and mirror positions. This two-body wavefunction is then modified to incorporate predictions for measurement of the particle first and then later that of the mirror using the Copenhagen interpretation of measurement in quantum mechanics. The resulting PDF is a function both of the different particle positions and different times at which each is measured. An assumption used is that between the times of the two measurements there is neither interaction between the particle and mirror nor with the environment.
The focus of the discussion is on synchronous and asynchronous correlated interference. The former is both spatially and temporally transient, occurring only when the incident and reflected particle-mirror wavegroups overlap during reflection. It is shown that in the latter case a measurement of only the particle in this correlated interference region splits the mirror substate, resulting in mirror interference which can endure for times much longer than that of the correlated interference in a synchronous measurement. Later measurement of the mirror reveals this interference.
Quantum correlation between the position of a particle and that of the center of mass (cm) of the potential well in which it is trapped or the cm of a barrier over which it traverses is also described. A single-body treatment of a particle in both of these potentials is familiar from a first exposure to quantum mechanics. However, the two-body quantum effects on the particle and cm of the well or barrier, including correlated interference and the "kinematic" effects (e.g. recoil) due to the interaction, have not been discussed in the literature as far as we know. They are of course not applicable in the more familiar single-body treatment. The asynchronous formalism is applied to the particle and well or barrier in a manner similar to that of the mirror.
For a particle traversing a barrier in a single-body treatment, a refractive index is used to parameterize the interaction. An example is in the momentum of a neutron traversing a barrier which consists of a slab of matter [12]. The analogous classical example is given by an electromagnetic wave traversing a slab of glass, where an index of refraction is similarly used. The controversy involved with the form of the electromagnetic stress-energy tensor in this case has a long history and involved many well known physicists of the twentieth century [13,14]. The analogous quantum result, presented here for the two-body solution to the Schrödinger equation, is shown to involve a simple division of kinetic energy and momentum between the particle and barrier.
A synopsis of this work is as follows. First the method to obtain a two-body solution of the Schrödinger equation for simultaneous measurement is outlined for the particle-mirror system by applying standard techniques to derive the energy eigenstates. This is used to construct the asynchronous result. Wavegroups are formed from a superposition of these solutions. Particularly emphasis is given to correlated interference or the overlap region of the incident and reflected particle-mirror wavegroups and in describing a two-body analog of the Doppler shift. Asynchronous transfer of coherence in reflection and correlated interference for a mirror with a macroscopic mass are also considered. To illustrate asynchronous predictions in a potential barrier or well, a similar development is then given. This is followed by an overview of the calculations and a discussion of the model assumptions, limitations, and its relation to other interpretations of measurement in quantum mechanics.
The foundations of this procedure are described in subsections II A, II B, and IV B. It may be useful to skim over these before proceeding with the details of the calculations.

II. PARTICLE REFLECTING FROM A MIRROR
A. Simultaneous measurement The particle-mirror interaction is modeled as a moving delta function potential where reflection is assumed to occur at the center of mass of the mirror with the Schrödinger equation given by where square brackets are used to indicate the argument of a function and x 1 and x 2 are the particle and mirror positions along the x-axis. The mirror reflectivity, related to β, goes to infinity for a lossless mirror. The standard separable solution to this equation results from a center of mass (cm) and relative (rel) transformation of the particle-mirror system (not to be confused with the cm of the particle or mirror). This does not change the total energy E = (hK) 2 /(2M ) + (hk) 2 /(2m) = E rel + E cm , where k and K are the wavevectors for the particle and mirror with k = mv/h, K = M V /h, masses m, M , and initial velocities v and V , respectively. The transformed Schrödinger equation becomes The next step in the derivation requires parsing the energy into E = E cm +E rel and assuming the separable solution which reduces the Schrödinger equation into two ordinary differential equations, −h 2µ The solution to these equations is then obtained from the initial values and boundary conditions. Finally, this solution is transformed back to the particle-mirror system yielding Ψ[x 1 , x 2 , t].
In subsection II C the infinite potential energy boundary condition at the mirror surface is used to obtain energy eigenstate solutions to eqns. 3 and 4. The full solution is then transformed to the particle-mirror system. In subsection II D wavegroups are formed via a superposition of these states. In section III eqn. 1 is modified for a barrier and potential well. The asynchronous procedure described next is similarly applied in these cases.

B. Overview of the asynchronous method
This solution, Ψ[x 1 , x 2 , t], is next used to construct the wavefunction which predicts the outcome of asynchronous measurements. For illustrative purposes, let the particle be measured first. The assumption is made that once the particle is measured there is then and thereafter no interaction between the particle and mirror. While consequences of this assumption are simple to interpret classically for point particles, they are not as straightforward for reflection of a wavegroup. An example is with measurement in the interference region where it is not known if the particle has or has not reflected from the mirror. Measurement is of the position and not direction (i.e. it is not a momentum measurement) of the particle. Experimental realizations of such a measurement procedure are discussed in subsection IV B.
To account for different temporal measurements of the particle and mirror, respective time parameters t 1 and t 2 are used instead of t in Ψ[x 1 , x 2 , t]. Coefficients of these time parameters in the phase are then the energies of the particle and mirror, respectively. This is a parsing of the energy in the solution to the Schrödinger equation for the particle-mirror system, similar to that applied after eqn. 1 to transform it into the cm and rel system. Since that transformation yielded a simultaneous PDF there was no need for the two-time notation. To modify it for asynchronous predictions, relative and cm times are introduced in subsection II C, which then both account for different temporal measurements of the cm and relative positions and isolate the cm and rel energy terms.
However, transformation to the cm-rel system is simply a mathematical technique to facilitate a solution to the Schrödinger equation. For example, there exists no observable particle with a reduced mass as described in the cm-rel system. Once a solution, which satisfies the initial conditions and boundary values, is obtained in the cm-rel system the inverse transformation is applied to describe the system that can be measured.
These time parameters provide labels for the particle and mirror energies in the wavefunction's phase and also label the different times at which the particle and mirror are measured. They assist in isolating the energy of the particle from that of the mirror, just as x 1 and x 2 isolate the effect of the particle and mirror wavevectors in the phase of the wavefunction. This then allows all particle parameters to be fixed in the two-body wavefunction when the particle is measured. Such labeling, while simple in a separable system, is not trivial in the correlated particle-mirror system.
The different time labels do not indicate evolution of the subsystems via different Hamiltonians as do the different time variables used by McGuire [15] nor are they used to generate a manifestly Lorentz invariant theory as done by Petrat [16]. They neither tick at different rates nor are they out of phase, but rather act as only a label, just as x 1 and x 2 label the particle and mirror spatial coordinates along the x axis. Such notation then results in the wavefunction derived in subsection II A being expressed as Ψ[x 1 , x 2 , t] → Ψ[x 1 , t 1 , x 2 , t 2 ]. Now let the particle be measured first, at position x 1 = x 10 at time t 10 . This then collapses the particle substate into an eigenstate of that operator, Ψ[x 1 , t 10 ] = δ[x 1 − x 10 ]. The first measurement forces the two-body correlated wavefunction into an uncorrelated state which is, at the time that the particle is measured, a product of the particle substate with the mirror substate, given by Ψ[x 1 , t 10 ]Ψ[x 10 , t 10 , x 2 , t 10 ].
The particle substate after measurement is irrelevant to the subsequent time evolution of the mirror substate since they are no longer correlated and no longer interact. The mirror substate is determined by fixing all the particle coordinates in the two-body wavefunction at the time of the measurement of the particle. The two time formalism facilitates this procedure by fixing the particle parameters while allowing only those of the mirror to time evolve in the mirror substate. The coordinates of this measurment of the particle, (x 10 , t 10 ), then determine the initial substate of the mirror from the two-body wavefunction as Ψ[x 10 , t 10 , x 2 , t 10 ]. This one-body wavefunction for the mirror substate then continues to time evolve with Ψ[x 10 , t 10 , x 2 , t 2 ] for t 2 > t 10 .
For example, the phase of the mirror substate evolves in time only with the mirror's energy times the time t 2 in the phase. The particle's energy times the time t 1 term in the phase should not influence the time evolution of the mirror substate after the particle has been measured. It does not in this formalism since all of the energy terms associated with the particle are fixed at t 1 = t 10 . Similarly, the particle's wavevector times the position x 1 is fixed by setting x 1 = x 10 while the mirror's wavevector times its position x 2 is allowed to vary.
The probability of simultaneously measuring the particle and mirror within a small region ∆x 1 ∆x 2 is PDF∆x The probability of measuring the particle first and the mirror later is given by the product of two one-body probabilities. The first, the probability of measuring the particle within a region ∆x 10 at time t 10 , is determined from the average of the two-body PDF, associated with Ψ[x 10 , t 10 , x 2 , t 10 ], over the mirror coordinate x 2 (since the position of the mirror has yet to be measured and is therefore not known). This averaging over the two-body PDF results in a one-body PDF which when multiplied by ∆x 10 becomes the probability of measuring only the particle within a region ∆x 10 . The probability of measuring the mirror at a later time t 20 , within a small region ∆x 20 , is given by PDF∆x 20 with PDF= Ψ[x 10 , t 10 , x 20 , t 20 ]Ψ * [x 10 , t 10 , x 20 , t 20 ]. This one-body PDF times ∆x 20 is the probability of measuring the mirror within a region ∆x 20 after having measured the particle. A product of these two probabilities is then the probability of first measuring the particle and then the mirror. The particle substate into which it collapses after measurement does not influence this probability due to the assumption of no interaction. The effects of relaxing this assumption are discussed in subsection IV B.
The predictions for synchronous measurement presented below are plots of the two-body PDF as a function of x 1 and x 2 for snapshots at t 1 = t 2 = t. The probability of measuring both the particle and mirror at the same time in a small region of these plots is Ψ[ In the asynchronous predictions presented, it is assumed that the particle has been measured and therefore the plots are of the one-body PDF for the mirror as a function of both x 2 and the position x 10 and time t 10 at which the particle was measured. These are snapshots at t 2 . The probability of measuring the mirror in a small region at time t 2 , having already measured the particle at x 10 and t 10 , is then Ψ[x 10 , t 10 , x 2 , t 2 ]Ψ * [x 10 , t 10 , x 2 , t 2 ]∆x 2 . From these figures, only the probability of measuring the mirror after the particle has been measured (not the probability of first measuring the particle and then the mirror) can be determined. Conservation of probability for this one-body mirror wavefunction is addressed in the appendix.
While the validity of this asynchronous model is limited by the assumption of no interaction after the first measurement it also makes subtle assumptions about collapse of a two-body wavefunction: a measurement of the particle substate influences neither the mirror substate at t 10 nor the subsequent evolution of the mirror substate apart from fixing the parameters of the particle substate in the two-body wavefunction. A more detailed discussion is found in subsection IV B. The interaction of a particle with a finite well and barrier is used as an example of this parsing in unbound states with finite potentials where the total energy then includes a finite potential energy term. The infinite well is included to illustrate the method with a bound state. This finite barrier/well discussion begins in section III.

C. Details of the asynchronous method
Before reflection an uncorrelated solution to the Schrödinger equation is given by An incident wavegroup constructed from this then leads to uncorrelated predictions about the probability of finding the particle at (x 1 , t 1 ) and mirror at (x 2 , t 2 ). For the reflected wavefunction, the solution to eqn. 4 must vanish at x 1 = x 2 to satisfy the boundary condition at the mirror and not exist for x rel < 0 (or x 1 > x 2 ) since the particle cannot move through the mirror (for the uncorrelated incident state, however, there is no interaction and the particle does move past the mirror). In this transformed system, a solution to eqn. 4 can be constructed from the superposition of incident and "reflected" wavefunctions in the cm-rel system (in much the same way as is the solution for a wave traveling along a string toward a rigidly clamped boundary is constructed from free string solutions traveling in opposite directions), where θ[x rel ] is the unit step function. The only difference between the arguments of the two exponentials is the sign of the relative wavevector K rel corresponding to reflection in the relative coordinate. That is, where the initial velocities must allow reflection to occur. Relative and center of mass times t rel and t cm are introduced and associated with the relative and center of mass energies. These time variables satisfy the same properties as do t 1 and t 2 but in this case provide the notation needed in separating the energies associated with the relative and center of mass subsystems. The solution for reflection to eqn. 3 is given by The complete solution for an eigenstate of energy in reflection is then Ψ[x cm , t cm , x rel , t rel ] = ψ cm ψ rel . The particle-mirror system has now been partitioned into separable center of mass and relative coordinate subsystems. This separable solution, for the two uncorrelated substates, can be used to construct a wavegroup whose substates yield uncorrelated wavegroups associated with the cm and relative coordinates which satisfies initial and/or boundary conditions. A measurement of the cm position effects neither the time evolution of the wavegroup associated with the relative motion nor introduces any correlation between the cm and relative positions.
While the energy and momentum of the center of mass subsystem are unaffected by reflection, that is not the case for the relative subsystem where the relative wavevector changes sign upon reflection. Therefore, in this separable system interference of incident and reflected wavefunctions, determined by the PDF is associated only with the relative coordinate subsystem. The cm-rel transformation does not yield a solution for a physically realizable system. There exists no particle with a reduced mass, for example. Nevertheless, its utility lies in inverting the solution from cm and relative substates into particle and mirror substates. This then yields a correlated state in the parameters which can be measured. An example of this procedure is found in the solution to the hydrogen atom where the Schrödinger equation is first transformed from the laboratory to the cm-relative coordinates yielding uncorrelated substates. Transforming back to the electron-proton system yields correlated electron and proton substates [17].
This change from cm-rel to particle-mirror systems is accomplished by using the following relations in the separable solutions given by equations 2, 6, 7, and 8: rel /2µ, and E cm =h 2 K 2 cm /2(m + M ). These relations, however, do not address the two-time labeling issue in the particle-mirror system. To do so, note that the energy of the reflected particle, given by p 2 ref /2m with p ref =h∂φ ref /∂x 1 , is associated with the temporal coordinate t 1 . Similarly, the energy for the mirror is P 2 ref /2M with P ref =h∂φ ref /∂x 2 and is associated with the temporal coordinate t 2 . Both of these energies and momenta are consistent with those of a classical particle reflecting from a moving mirror. These are manifest in the two-body wavefunctions, however, as a Doppler shift.
Application of these transformation relations then changes e iφin in equation 6 into equation 5. The two-time expression for e iφ ref in the particle-mirror system, although simple to calculate using the procedure just outlined, is too large to present here. This complexity is a consequence of the correlations generated in reflection.
Correlated interference of these incident and reflected wavefunctions in the particle and mirror subsystems is then given by (transforming eqn. 9) This is similar to Gottfried's joint or simultaneous PDF for the interference obtained in the correlation between two particles produced in a momentum-conserving decay after each has traversed separate double slits [1] when t 1 = t 2 .
Note also that interference of the mirror and particle are coupled in eqn. 10, illustrating how many-body systems interfere with 'themselves' rather than only the particle and mirror each interfering with itself (in which case there would be no correlation) [18].
To gain familiarity with this result, consider a simultaneous measurement. For fixed x 1 , the approximation m/M << 1 leads to interference for the mirror which varies from maximum to minimum through a distance Similarly for fixed x 2 this approximation leads to interference for the particle which varies from maximum to minimum through a distance ∆x 1 = ∆x 2 . For a static mirror both the mirror and particle fringes are spaced at half the deBroglie wavelength of the particle, which can be up to 10 −6 m for ultra cold atoms [19]. The time dependence in equation 10 is determined by the time components of the phase of the incident wavefunction, . For simultaneous measurements (t 1 = t 2 = t) this phase difference is zero since the time variable factors from all energy terms and the total energy before and after reflection does not change: That, however, is not the case for non-simultaneous measurements since the times of measurement of the particle and mirror differ and are therefore no longer a common factor of all energy terms in the phase. This leads to the Doppler effect which is described next.
Consider an ensemble of identically prepared particle-mirror systems. Let a measurement on every member of the ensemble be made at both fixed particle-mirror positions, x 1 , x 2 , and time t 2 while the particle is measured at different times t 1 for different members of the ensemble. The time dependent interference pattern from eqn. 10 emerges from these ensemble measurements as the expected "beat frequency" Ω [21] associated with interference of the incident and reflected particle substates (the superposition of states with different energies commensurate with the energy exchanged in reflection). If instead, x 1 , x 2 , and t 1 are fixed while the mirror position is measured at different times t 2 , the "beat frequency" is that associated with superposing mirror substates differing in energy. These particle and mirror beat frequencies are identical due to the same energy being exchanged between the particle and mirror in reflection and are given by

D. Wavegroups: Simultaneous measurement
To better understand the experimental consequences of these results, wavegroups are next formed from a superposition of the incident and reflected 'energy eigenstates' (given by eqn. 6) expressed in terms of the correlated particle and mirror substates rather than the cm and relative substates. An analytic expression for such wavegroups can be obtained for a Gaussian distribution in wavevector components k and K (or velocities v and V ). It is assumed that the initial particle and mirror Gaussian substates are sufficiently separated that any probability of the particle initially being on the "wrong" side of the mirror is negligible. For the mirror this is proportional to exp[−(K − K 0 ) 2 ]/(2∆K 2 ) where the peak of the distribution is at K 0 and ∆K is its width while for the particle this is proportional to exp[−(k − k 0 ) 2 ]/(2∆k 2 ) where the peak of the distribution is at k 0 and ∆k is its width. The incident wavegroup propagates in the (x 1 , x 2 ) plane along a line whose slope is determined by a ratio of the group velocities of each substate and spreads due to dispersion independently in each direction.
Consider first simultaneous measurement of the particle and mirror. In fig. 1 snapshots of such a two-body joint PDF are shown at three times, t 1 = t 2 = (−τ, 0, τ ), for M/m = 100, ∆K/∆k = 2, and K/k = 60. The incident wavegroup propagates in the (x 1 , x 2 ) plane along a line whose slope is determined by a ratio of the group velocities of each substate and spreads due to dispersion independently in each direction. The particle and mirror initially move in the positive direction for the parameters chosen. During reflection the incident and reflected wavegroups overlap resulting in interference. After reflection the speed of the mirror increases while the particle continues moving in the positive direction with decreased speed (see the insets for the classical analog). Careful inspection of the lower and middle snapshots just to the left of the diagonal white line confirms the prediction that the spatial location of the interference maxima and minima do not depend on time as given in eqn. 10 when t 1 = t 2 . Wavegroup distortion shown in the upper reflected snapshot is discussed below. In the cm-rel system (not shown) the "fringes" are aligned parallel to the cm coordinate illustrating the result given in eqn. 9.
A slice of fig : Two-body joint probability density snapshots for three sequential times vs coordinates (x2, x1) for a particle reflecting from a mirror. The lower PDF waveform moves toward the diagonal white line, corresponding to x1 = x2, then reflects in the middle snapshot where the incident and reflected two-body wavefunctions 'overlap', and finally it moves away from the diagonal in the upper snapshot. The correlated interference fringes are spaced by about half the deBroglie wavelength of the particle. The upper left inset is a schematic of the 'classical' analog before reflection while the upper right inset is that after reflection with initial and final particle and mirror velocities v, V , v f , and V f respectively. There is no classical analog for the middle snapshot. spacing for the particle and mirror substates with narrow bandwidth wavegroups, as discussed for the approximation M/m >> 1 following eqn. 11. The minima of the interference shown in fig. 1 correspond to positions where the particle and mirror can never simultaneously be found. Verification of this result requires simultaneous cm measurement of the particle and mirror with instruments which have a spatial resolution that is smaller than this fringe spacing along both coordinates.

E. Wavegroups: fixed t10 while t2 varies
Next consider non-simultaneous measurements of the particle at position x 10 and time t 10 while allowing t 2 to vary until a measurement is made of the mirror's cm position. The differences between simultaneous and non-simultaneous : Regime A one-body PDF plots for the mirror at t2 = τ, 2τ, 3τ when the particle has been measured at t10 = τ and x10. Since t2 = t10 = τ for the rightmost PDF, it is similar to the upper snapshot in fig. 1. The diagonal white line is the same as that in fig. 1.
measurements are a consequence of fixing particle parameters in the phase while allowing the time evolution of mirror parameters. These differences can be categorized into two regimes. Measurement of the particle either occurs in the region where the incident and reflected wavegroups do not overlap, regime A, or in the region where the interference similar to that shown in fig. 1 occurs, regime B.
To illustrate such asynchronous measurements, the particle is measured at a particular x 10 and t 10 while the mirror substate then evolves with Ψ[x 10 , t 10 , x 2 , t 2 ]. Rather than graph the time evolution of the mirror using a 2-D plot of x 2 vs. t 2 for various values of x 10 , one-body PDF plots of snapshots at different times t 2 are shown on a 3-D graph of x 2 vs fixed values of x 10 .

Regime A: the particle is measured when there is no incident and reflected wavefunction overlap
All three joint PDF snapshot results for regime A shown in fig. 3 are for t 10 = τ . The one farthest to the right is similar to the upper PDF waveform snapshot in fig. 1 since it occurs at t 2 = t 10 = τ . The other two snapshots are for times, t 2 = 2τ and 3τ with M/m = 3, ∆K/∆k = 2, and K/k = 1.8. Note that this mirror wavegroup moves only along the x 2 axes and disperses.
It is useful to compare the physical interpretation of the joint PDF shown in fig. 1 with that of the one-body PDF of fig. 3. In the former case the probability of finding the particle and cm of the mirror in a region centered around x 1 = a and x 2 = b simultaneously at time t is given by fig. 3 the particle is measured at t 10 = τ while the snapshots are given for different values of when and where the cm of the mirror is measured. For the leftmost PDF in this figure the probability of measuring just the mirror, once the particle has been measured, is given by 2. Regime B: the particle measured in the overlap region All three joint PDF snapshot results for regime B, shown in fig. 4, are for t 10 = 0. The one farthest to the right is similar to the middle PDF waveform snapshot in fig. 1 since it occurs at t 2 = t 10 = 0. The middle snapshot of fig. 4 occurs predominately between x 2 = 4 and 9 for t 2 = τ , while the leftmost snapshot consists of two "bumps" encompassing the region between x 2 = 9 and 18 for t 2 = 2τ . As in the previous figure M/m = 3, ∆K/∆k = 2, and K/k = 1.8.
The complexity of the joint PDF in fig. 4, in comparison with that of regime A, is a consequence of measuring the particle in the interference region. That is, there are two indistinguishable ways that the particle could have reached the interference region. It could have come from the incident or reflected particle wavegroup substates. This lack of knowledge about the particle is manifest in the subsequent mirror wavegroup which then consists of a superposition in which the mirror has yet to reflect and has already reflected the particle. These two mirror states have different speeds due to the mirror recoil in one but not the other. To see this one need only change the speed of the incident particle wavegroup. The speed of the mirror state which has not reflected the particle does not change while the speed of the mirror state which reflected the particle increases. Overlap of these two mirror wavegroup states is shown in fig. 4 from complete overlap (right hand snapshot) to partial overlap (middle snapshot) and finally to virtually complete separation (left hand snapshot) due to the differing speeds of the two mirror states.
Although these results illustrate important issues in non-simultaneous measurement they have explored only a limited set of parameters and interferometric geometries. Rather than present a comprehensive treatment of nonsimultaneous measurement, three examples are used to further probe the consequences of asynchronous measurements. First the beamsplitting effect that a measurement on the particle has on the mirror is explored in more depth using a particle whose mass is equal to that of the mirror. Second, a manifestation of the Doppler effect is illustrated using a time scale shorter than that used in the figures above. Finally, a specific example of reflection of a microscopic particle from a mirror which has a macroscopic mass is described.
3. Regime B: measurement functioning as a beamsplitter using a coherence transfer example After reflection the spatial width of the mirror wavegroup substate is exchanged with that of the particle wavegroup substate when M = m. This is most easily seen by constructing a particle-mirror wavefunction with different bandwidths for the particle and mirror wavegroup substates and is shown in fig. 5, which is a contour plot of joint PDF's similar to figure 1 but without the snapshot in the interference region. The solid and dashed contours correspond to M/m = 1 and M/m = 20 respectively with the spread in velocities given by ∆V /∆v = 10.
This result can be understood by comparing classical and quantum reflection. In a one-dimensional classical collision, conservation of energy and momentum result in the exchange of particle-mirror velocities independent of either velocity for m = M . This is manifest quantum mechanically in the exchange of commensurate substate parameters k and K between the incident and reflected two-body wavefunctions. If an incident particle substate, consisting of only one harmonic component (corresponding to speed v) reflects from a mirror substate with many velocity components, then each harmonic component of the mirror substate (corresponding to different values of V ) reflects the particle substate and therefore acquires velocity v while the reflected particle substate acquires different velocity values for each reflected component of the mirror wavegroup. This results in the reduction of the mirror bandwidth and an increase in the particle bandwidth, which is manifest in fig. 5 as the exchange of incident and reflected wavegroup shapes. It also is responsible for the distortion of the reflected wavegroup shape in fig. 1.  Consider next non-simultaneous joint PDF's for equal particle and mirror masses. However, rather than choosing measurement of the particle at t 10 = ±τ as shown in fig. 5 the particle is measured at t 10 = 0 to illustrate asynchronous measurement in the interference region. The results are shown in fig. 6. This again splits the mirror state into two parts. Now these states differ dramatically in their shapes and in their splitting ratios. That is, depending on the position that the particle is measured at, a slice of the joint PDF along the x 2 axis can reveal one or two peaks of unequal height. The long PDF waveform parallel to the x 10 axis is that of the mirror substate associated with no reflection while the narrow peak corresponds to the mirror substate which reflected the particle. The differing speeds of these two waveforms, due to mirror recoil in one but not the other, is evident in the figure. One-body PDF snapshots of the mirror after the particle was measured at x10 = 0 and t10 = 0 starting at time t2 = 0 (which is a slice of fig. 1) as the leftmost plot. The remaining plots to the right progressively increase the time t2 when the mirror is measured by 0.04τ .
The time order of measuring the particle and then the mirror can be reversed. In regime B the splitting then generates two states consisting of a superposition in which the particle has yet to reflect and has already reflected from the mirror. If the mirror is measured when it has not reflected the particle then the particle can be found later at x 1 > x 2 , since after the mirror measurement the particle is assumed to no longer interact with the mirror and the particle. The particle, which moves faster than does the mirror, then can move past the mirror. 4. Regime B: measurement of the particle resulting in a beat frequency for the mirror Variation of t 2 on a temporal scale smaller than that shown in fig. 4 reveals the energy differences between the superposition of these two mirror substates in regime B as a "beat frequency" in the joint PDF. This is related to the result for harmonic states given in eqn. 12 and is illustrated for wavegroups in fig. 7. A slice, taken from fig. 1, along the x 2 axis for x 10 = 0 and t 10 = t 2 = 0 is shown in fig. 7 as the leftmost plot. Plots to the right of this are shown at x 10 = 0 and t 10 = 0 for t 2 = 0.04τ, 0.08τ , and 0.12τ respectively. All other parameters are the same as used in fig. 1. This illustrates the expected beat frequency from the "Doppler shift" in reflection, although it is shrouded in two-body interference. Equation 10 predicts no such beat frequency for simultaneous measurements (t 1 = t 2 ) which is discussed in more detail in subsection IV C.

Regime B: interference for a mirror of macroscopic mass
One constraint for regime B is that the fringe visibility function must be non-zero. That is, the incident and reflected particle-mirror wavegroups must 'overlap.' The interference fringes are then determined predominately by a superposition of 'energy eigenstates' [22]. For example, the interference shown in fig. 1 is determined predominately by eqn. 10 when the wavegroups 'overlap' in the center snapshot, where the longitudinal coherence lengths for both the particle and mirror are greater than the fringe spacing. In the upper snapshot of fig. 1 there is neither 'overlap' nor such interference.
The fringe "visibility function" is non-zero if each wavegroup substate 'overlaps' within approximately a coherence length [23], which is given by l c ≈ λ 2 /∆λ = λV /∆V [24]. For particle substates this can be l particle c = 10000Å for ultracold atoms [19] or l particle c = 790Å for slow neutrons [25]. This longitudinal mirror coherence length can be estimated from the uncertainty in the mirror velocity. If it is in thermal equilibrium with the environment then ∆V thermal ≈ 2k B T /M yielding l thermal c ≈ h/ √ 2M k B T , which is consistent with results for atoms in a Bose-Einstein condensate [19]. Fig. 8 illustrates how variation in the longitudinal coherence length of the mirror substate effects the particlemirror interference for fixed particle coherence length and simultaneous measurements. Part (a) shows a longer mirror coherence length than is used in fig. 1 while parts (b) through (d) progressively reduce the coherence length of only the mirror substate. In fig. 8 (d) the coherence length of the mirror is so small that overlap is prevented over a range of x 1 values where it was present before. Nevertheless, a slice along the x 2 axis for measurement of the particle at a particular x 1 indicates a splitting of the mirror substate into two states which do not overlap and are therefore distinguishable. This is a consequence of two ways that the particle could have reached x 1 . It could have come from the incident or reflected particle wavegroup substates due to the large particle coherence length. As the mirror's coherence length increases, these two ways overlap and generate correlated interference as shown in fig. 8 (a). As the mirror's coherence length decreases, the position of the mirror before reflection is distinguishable from that after reflection resulting in no interference, as shown in part of fig. 8 (d).
One might expect that the small coherence length associated with a macroscopic mirror mass would not allow the interference shown in fig. 8 (a). This is the case in fig. 9 where snapshots of two-body PDF's are shown at three times for a rubidium atom with m = 1.4 × 10 −25 kg reflecting from a mirror with M = 10 −8 kg. The spread in wavevectors is determined by thermal equilibrium using T = 10 −7 K for the atom (which is released from a Bose-Einstein condensate) and T = 1 K for the mirror. The snapshots are for t 10 = 0 with t 2 = 10 −16 , 2 × 10 −15 , and 4 × 10 −15 s. This mirror longitudinal coherence length, determined by thermal equilibrium at T = 1 K, corresponds to the interference in reflection shown in fig. 8 (c) for simultaneous measurement. For non-simultaneous measurements, Fig. 9 shows only snapshots over a small range of times t 2 when the mirror is measured while the particle has been measured at t 10 = 0.
However, varying t 2 over a much larger range, for a mirror initially at rest, does not change the character of the joint PDF shown in this figure. Initial mirror and particle velocities of V = 1/100 and v = 3/100 m/s are used in fig. 9 which results in a beat frequency of 3 × 10 5 Hz and changes the joint PDF in a manner similar to that shown in fig. 10 along the x 2 axis. Although these two interfering mirror states (which either reflect or do not reflect the particle) have different speeds they do not completely separate as do those shown in the leftmost plot of fig. 4. This is a consequence of the small difference in mirror speeds due to the large M/m ratio, resulting in a mirror offset which is small compared with the size of the wavepacket.
These results illustrate that a measurement of the particle at one of the minima of the interference pattern in the joint PDF of fig. 9 yields no possibility of measuring the cm of the mirror for any value of x 2 and subsequent time t 2 (within a duration limited by the beat frequency). This is to be contrasted with the result shown in fig. 1 where simultaneous measurement of the particle and mirror at an interference minimum is constrained temporally and spatially as follows. First, the duration of the simultaneous interference is finite. For a static mirror it is essentially determined by the coherence length of the particle divided by its speed, ≈ l particle c /v when l particle confirmation of the interference in fig. 1 therefore requires both a small spatial and a short temporal resolving power of the measuring instrument. Asynchronous correlated destructive interference, on the other hand, is not similarly constrained. The interference minimum in fig. 9 occurs over a range of x 2 much larger than the mirror coherence length. It is also robust over a wide range of mirror masses. 6. Regime B: measurement of the particle but not the mirror Correlated measurements are more difficult to perform than only a measurement on the particle while making none on the mirror. Predictions of such effects are given by an average of the PDF over the mirror coordinate converting it into a 'one-body' PDF [1].
In the context of this work, the probability for detecting the particle without measuring the mirror for incident and reflected energy eigenstates states is then manifest as an integral of eqn. 10 over the mirror coordinate (or given by the reduced density matrix obtained as a trace over the mirror coordinates [17]). Such a procedure, applied to eqn. 10, "washes out" the one-body interference. Gottfried carries out a similar averaging on the two-body PDF for simultaneous measurement in a momentum-conserving decay after each particle traverses separate double slits with a similar result [1].
However, a one-body standing wave interference pattern is certainly expected for the particle reflecting from a static mirror. For a moving mirror the Doppler shift introduces a time dependence in this pattern. Therefore, the prediction that there is no such interference between incident and reflected particle states, when the mirror is not measured, seems surprising.
This prediction, however, must be re-evaluated for wavegroups. Consider the averaging procedure when applied to the four PDF's shown in fig. 8. It is apparent that the one-body interference 'washes out' in graph 8(a) since many cycles in the mirror coordinate of the joint PDF are averaged over, while that is not the case for fig. 8(c). As shown in fig. 9, the macroscopic mirror has a coherence length smaller than the fringe spacing determined by an ultracold atom reflecting from it and therefore most closely resembles fig. 8(c). It is apparent from this figure that averaging over x 2 does not 'wash out' the one-body interference. However, this averaging of fig. 8(c) is only for simultaneous measurement and does not change with time even if the mirror moves (see subsection IV C for more details).
For a measurement of the particle at x 10 and t 10 , the joint PDF must be averaged over x 2 if the mirror is not measured, yet at what time t 2 must this occur for asynchronous measurements? This is addressed in fig. 10 where four sequential plots are shown for different t 10 values while within each plot snapshots at different times t 2 are shown. The bandwidth parameters used in fig. 10 have been adjusted to model those of a macroscopic mirror mass. It is apparent from this figure that averaging over x 2 does not 'wash out' the one-body interference for any value of t 2 . That is, the result of averaging over x 2 is independent of t 2 . Yet variation of the one-body PDF with t 10 at fixed x 10 (the Doppler shift) is evident in fig. 10. As mentioned above, the initial mirror and particle velocities of V = 1/100 and v = 3/100 m/s that are used in fig. 9 result in a beat frequency for the particle of 3 × 10 5 Hz.
This effect of the Doppler shift on the particle PDF (shown along the x 1 axis) differs from that in fig. 7 which illustrates the effect of the Doppler shift in the mirror's PDF (shown along the x 2 axis). There is no Doppler effect for the mirror in fig. 10 due to its coherence length being much shorter than the fringe spacing.
The choice of short coherence length macroscopic mirror and a long coherence length particle substates yields the expected interference for a particle reflecting from a mirror when the mirror is not measured but requires the use of the two-time formalism (again see subsection IV C for more details). However, such a prediction is limited to a special case of a more general result. For example, this averaging procedure applied to fig. 8(d) for simultaneous measurement does not yield the expected particle interference in reflection when the mirror is not measured since there is no interference over a large range of x 1 where it exists in fig. 8(c).
Consider the opposite case of measurement of the mirror without a measurement of the particle using the assumptions that generated fig. 10. In this case the averaging is over x 1 , which washes out the fringes. Therefore measurement of just the mirror results in no interference. This interference for the particle but not the mirror is that expected for a classical wave reflecting from a mirror. Yet it is a special case rather than a general result since it depends on the coherence length assumptions.

III. MULTIPLE REFLECTIONS
When the particle is trapped in an infinite well or traverses a finite barrier or well, the added complication of multiple reflections occurs. The finite potential energy also introduces the issue of how to treat the parsing of the particle-barrier energies and times when the total energy contains a constant finite potential energy term. We show below that such a potential energy contribution is manifest only parametrically within the momenta and kinetic energies of the particle and barrier. The asynchronous results, which demonstrate typical quantum effects, are then illustrated.

A. Energy eigenstates
As in the treatment of the particle reflecting from a mirror, the calculation of the two-body PDF begins with energy eigenstate solutions to the Schrödinger equation in the cm-rel system. The two-time formalism is then applied to obtain the two-body two-time particle-well or particle-barrier wavefunctions. Each potential is of width D, mass M , initial velocity V and wavevector K while the particle has mass m, initial velocity v and wavevector k. Wavegroups are then formed from a superposition of these states.

Infinite-potential well energy eigenstates
The boundary condition for the infinite potential well is ψ[x rel ± D] → 0. The relative wavefunction does not exist outside the well for x rel < −D and x rel > D. A solution [20] to eqn. 4 for a particle which has reduced mass is where ) has value one within the well and zero everywhere else with n being the number of nodes. Expanding the sine function in eqn. 13 into exponential form results in wavefunctions traveling in opposite directions. Transforming back to the particle-well coordinates, the momenta and energies of the particle and well differ in these two directions.
The solution to eqn. 3 for this potential is again given by The modes of the square well, characterized by the number of nodes n, are determined in the cm-rel coordinates by K rel = nπ/2D. In the particle-well coordinates this constrains the incident values of particle and well velocities. That is, given an initial well velocity V only the following initial particle velocities are allowed, The complete solution is then ψ[x cm , t cm , x rel , t rel ] ∝ ψ cm ψ rel . A transformation from the rel-cm system to the particle-well system is then needed and involves the relations given in subsection II C. The temporal part of the phase for the incident wavefunction can be easily parsed into the two-time notation. For the reflected wavefunction's phase, φ ref , this is again done by associating the temporal coordinate t 1 with the particle's kinetic energy, given by Similarly, the energy for the well is P 2 ref /2M with P ref =h∂φ ref /∂x 2 and is associated with the temporal coordinate t 2 . The resulting expression for the eigenstate of energy, Ψ[x 1 , t 1 , x 2 , t 2 ], is too large to present here.

Potential barrier energy eigenstates
The barrier has potential energy P E and extends over a distance D. This divides space into three regions: before the barrier or "before", in the barrier region or "barrier," and after the barrier or "after." Solutions are first obtained for these three regions in the cm and rel coordinates by solving eqns. 3 and 4.
The solution to eqn. 4 in the region before the barrier consists of incident and reflected wavefunctions given by where K bef ore = √ 2µE rel /h. The solution in the barrier region also consists of incident and reflected wavefunctions given by where K barrier = 2µ(E rel − P E)/h. The solution after the barrier consists only of a transmitted wavefunction given by where K af ter = √ 2µE rel /h. The boundary conditions are continuity of the wavefunctions and their derivatives with respect to x rel at x rel = ±D. These then constrain the coefficients B, F , G, and H with A = 1.
Again, the solution to eqn. 3 is given by eqn. 14. The complete solution is then ψ[x cm , t cm , x rel , t rel ] ∝ ψ cm ψ rel . Since we are interested in predictions about measurements of the particle and well rather than measurements of total mass and non-existent reduced mass "objects," a transformation from the rel-cm to the particle-well systems is utilized.
The incident and reflected wavefunction's phase, φ, has its temporal part parsed using t 1 and t 2 in the same manner described above for the mirror and infinite well. These have to be calculated separately in each region and wave propagation direction.
Since this procedure does not assign a time variable to the potential energy in the barrier region, a time t P E is used for this purpose. Such a term has no measurable effect as will be shown next. The wavefunction in the barrier, expressed in terms of the particle-well coordinates, is then where Φ right spatial and Φ lef t spatial contain the spatial terms in the phase and are functions of m, M, v, V, P E, x 1 , x 2 , andh. The temporal terms contain the kinetic energy for the particle and barrier moving to the right and left, KE right The potential energy term e iP E t P E /h is a common factor of both the incident and reflected wavefunctions in the barrier for all harmonic components used in generating a wavegroup. Since the PDF is generated from the wavefunction multiplied by its complex conjugate, such common factors have no effect on the PDF. There is then no need to associate the potential energy part of the total energy with either the particle, as a coefficient of t 1 , or the barrier, as a coefficient of t 2 , or as a separate term in the phase, P E t P E /h. The potential energy part of the total energy has observable consequences only parametrically within the momenta and kinetic energies of the particle and barrier. This simple division of the momentum and energy of the particle in the barrier can be contrasted with that of the stress-energy tensor for an electromagnetic wave traversing a dielectric slab [14].

B. Wavegroup results
Wavegroups are next formed from a Gaussian superposition of the two-body energy eigenstates for the particle and well or barrier described above. Correlated interference is a consequence of any such superposition. However, we focus the following discussion on the subsets of correlated interference effects which deal with the superposition of two such wavegroups. One example is the interference when the incident and/or multiple two-body reflected wavegroups overlap. These types of PDF's are first illustrated for simultaneous measurement of the particle and well or barrier, followed by ones for asynchronous measurements.

Particle-barrier or finite well wavegroups: theory
The wavegroup for the particle and barrier or finite well is calculated using a Gaussian superposition of the energy eigenstates given in subsection III A 2. Unfortunately, the coefficients B, F , G, and H of eqns. 16, 17, and 18 depend in a non-trivial manner on the variables of integration. The resulting integrals cannot be determined in closed form. To facilitate the calculation, the following sums will replace these integrals: where the peak of the barrier velocity distribution is at V 0 , ∆V is its width, and the sum is from an initial barrier velocity V i to a final velocity V f . Summing over the particle velocity distribution yields the wavefunction for the wavegroup given by where the peak of the particle velocity distribution is at v 0 , ∆v is its width, and the sum is from an initial particle velocity v i to a final velocity v f . 2. Simultaneous measurement of finite well wavegroups: KE relative initial > P E Consider next a particle interacting with a finite well. Their sum of initial kinetic energies in the relative coordinate system is greater than the well P E. The size of the particle substate wavegroup is chosen to be a few times larger than the finite well width D. Fig. 11 shows results of the two-body PDF's for three sequential snapshots taken at equal time intervals progressing along the dashed line from the lower left to upper right and upper left. The analogous classical positions of the particle and finite well for particular snapshots are illustrated in the insets. The barrier boundaries occur at the diagonal white lines, corresponding to x 1 = x 2 ± D. The parameters used in fig. 11 are v 0 /V 0 = 6, ∆v/∆V = 1.5, and M/m = 5. Using the average value of KE relative initial for the wavegroup particle and well distributions, KE relative initial − P E/ | P E |= 1.4 One category of correlated interference, which we call type I, occurs when the incident and reflected two-body wavefunctions 'overlap' when traveling in opposite directions in the (x 2 , x 1 ) plane. This is illustrated in fig. 1 by the fringes of the middle snapshot. A similar effect is found in fig. 11 just to the left of the barrier or line x 2 = x 1 − D when the reflectivity is so small that multiple reflections from the other barrier interface can be neglected. These fringes are spaced by about half the deBroglie wavelength of the particle for M >> m and v >> V .
However, the well generates another form of correlated interference when the reflected wavegroups, one from each barrier surface, interfere as they travel in the same direction in the (x 2 , x 1 ) plane. This new category of correlated interference, which is referred to as type II, is illustrated in fig. 11 by the peak labeled b and is similar to the classical interference of a pulse of light reflecting from a thin film.
Changing only the barrier spacing generates an oscillation in the two-body PDF for peak b analogous to that found in the interference of a pulse of light reflecting from a thin film whose thickness varies. That is, this peak goes through constructive and destructive interference from the two barrier reflections when the wavegroup size is much larger than that of the barrier and the spacing D is varied. As time progresses peak b maintains this interference as it travels in the (x 2 , x 1 ) plane, whereas the type I correlated interference associated with the fringes shown in the middle snapshot of fig. 11 is localized to a small temporal and spatial region.
3. Simultaneous measurement of finite barrier wavegroups: KE relative initial ≈ P E Consider next wavegroups for which some Fourier components of the particle and barrier substates have a total initial kinetic energy in the relative coordinate system which exceeds the barrier potential energy while other components have a total relative initial kinetic energy which is less than the barrier potential. To illustrate the resulting PDF's the size of the particle and barrier substate wavegroups are chosen to be somewhat larger than the barrier width D. Fig. 12 shows the PDF's from such an interaction using three sequential snapshots, progressing along the dashed line from the lower left to upper right. The speed of the particle and well are illustrated for a classical system of two FIG. 12: Three two-body PDF snapshots for sequential times vs coordinates (x2, x1) for the particle traversing the barrier. The only difference between the parameters used here and in fig. 11 is the P E which forms a barrier rather than a well. The first snapshot generates peak a while peaks b, c, and d comprise the third snapshot.
such particles in the insets next to each snapshot. Again the diagonal white lines correspond to x 1 = x 2 ± D. The parameters used in fig. 12 are v 0 /V 0 = 6, ∆v/∆V = 1.5, M/m = 5. Using the average value of the KE relative initial for the wavegroup distributions KE relative initial − P E/ | P E |= 0.3. Although multiple reflections contribute to both type I and II interference, they are particularly evident within the barrier or well. A somewhat artificial separation, referred to as type III interference, is then used to label this in the classification scheme. Such interference is illustrated as peak d in fig. 12 (and the fringes barely visible between peaks b and c in fig. 11) along with the unlabeled interference in the PDFs located between both x 1 = x 2 ± D in both figures. This is analogous to the buildup and decay of electromagnetic energy in a optical cavity. Later snapshots (not shown) illustrate their decay.
The position of peak c can be compared between figs. 11 and 12 since all parameters are the same except the P E. The location of this peak indicates the effect of the interaction on the relative transit times for the particle and finite well or barrier wavegroup substates. 4. Asynchronous results: particle-finite well wavegroups KE relative initial > P E Next consider asynchronous measurements of the particle at a particular x 10 and t 10 while the finite well substate then evolves with Ψ[x 10 , t 10 , x 2 , t 2 ]. Once the particle has been measured there is no interaction between the particle and well. Asynchronous predictions are described first for type II and then type I correlated interference. As an example of the former we fix x 10 and t 10 on peak (b) in fig.11 while for the latter x 10 = −2 and t 10 is fixed during the middle snapshot time shown in fig. 11. The most straightforward way to illustrate asynchronous measurement in type II correlated interference is to graph the well's PDF vs. x 2 for different snapshots in t 2 while fixing x 10 and t 10 on peak (b) in fig. 11. This is shown in fig.  13. The lowest snapshot, labeled (b), is for t 2 = t 10 and is therefore the same as peak (b) in fig. 11 while the other snapshots sequentially increase only t 2 .
To illustrate asynchronous measurement in type I correlated interference, two 3-D plots of the barrier's PDF vs. x 2 and D are shown in fig. 14 while fixing x 10 and t 10 to be in the region associated with the type I correlated inteference of the middle snapshot in fig. 11 (x 10 = −2 and t 10 is time associated with this snapshot). While both plots are for the same time t 2 , fig. 14 (a) is for a smaller initial particle velocity than is fig. 14 (b).
The parameters used in fig. 14 (a) are v 0 /V 0 = 4.6, ∆v/∆V = 1.5, M/m = 5 while in fig. 14 (b) they only differ by the increased particle speed v 0 /V 0 = 6.1. Differences in the PDF sizes and shapes of these figs. are associated with the states being expressed as sums, the transmissions and reflection coefficients depending on velocity, and the Asynchronous one-body mirror PDF snapshots for equal sequential times t2 when the particle is measured during type II correlated interference for the finite well at position x10 and the time associated with peak (b) in fig. 11. The snapshot labeled (b) is the same as peak (b) in that fig. dependence of the type I fringe pattern on velocity (the PDF values for a given x 10 and t 10 depend on the particle speed).
Since the particle is measured in the region associated with type I correlated inteference, the particle could have come from the incident or reflected wavegroups. The largest peak in fig. 14 corresponds to the barrier state when the measured particle did not interacted with the barrier (the measured particle came from the incident wavegroup moving to the right). The position of this peak is then the same for different particle speeds as is shown in fig. 14 (a) and (b), which is consistent with the expected position of the barrer for no interaction.
The smaller peak corresponds to the barrier state due to the particle being measured after it interacted (the measured particle came from the reflected wavegroup moving to the left in the middle snapshot of fig. 12). The position of this smaller barrier PDF peak therefore increases with increased particle speed due to recoil of the barrier, as shown by the different x 2 positions of the smaller peaks between fig. 14 (a) and (b). These positions are consistent with those expected from "classical recoil" of the barrier from the particle moving at these different speeds. This "classical recoil" corresponds to the particle reflecting only once from the barrier even though the wave reflects from both the front and back boundaries of the barriers.
Since the particle could have reflected from either barrier boundary, the barrier is in a superposition of both these possibilities. The PDF as a function of barrier width along the D axis illustrates interference due to the barrier being put into a superposition in which the particle either reflected from the front or rear boundaries of the barrier. For larger particle speeds the wavelength decreases, resulting in the increased number of fringes along the D axis as shown in this figure.

Infinite well-particle wavegroup: theory
This calculation differs from that of the barrier due both to the particle and barrier velocities being constrained by the resonance condition given in eqn. 15 and by the lack of coefficients, such as the B, F , G, and H used in the previous section, which depend on the variables of integration.
A closed form expression for the well substate wavegroup can be obtained from a Gaussian distribution of the energy eigenstates parametrized in terms of velocity components, V , of the well, given by where the peak of the distribution is at V 0 and ∆V is its width. This is then summed over integral values of n (the number of nodes) using the gaussian distribution [20] Ψ FIG. 14: One-body barrier PDF snapshots when the particle is measured during type I correlated interference. The two peaks indicate a splitting of the barrier due to measurement of the particle at an earlier time.
where the peak of the distribution is at n 0 and ∆x is its width.

Simultaneous measurement: infinite well wavegroups
Using these relations, the PDF is plotted first with particle and well substate wavegroups whose spatial widths are less than the well spacing D. Fig. 15 shows such PDF results for six snapshots taken at equal time intervals progressing from the lower left to upper right along the dashed line. 'Reflection' occurs at the diagonal white lines, corresponding to x 1 = x 2 ± D. The particle and well "reflect" from each other twice, in the second and fourth snapshots. The classical analogs for the particle and well positions for some snapshots are shown in the insets, labeled by a, b, and c. These correspond to the snapshots of the wavegroups labeled with the respective letters. While the insets are schematics of the 'classical' analog between the wave and particle pictures, there is nothing similar for the correlated interference snapshots. The parameters used in fig. 15 are n 0 = 50, ∆x/D = 1/15, ∆V /V 0 = 1/30, and M/m = 10.
Type I correlated interference occurs when the incident and reflected two-body wavefunctions 'overlap' and is shown in higher spatial resolution for the first reflection in the right side inset of fig. 15. The fringes are spaced by about half the deBroglie wavelength of the particle for M >> m and v >> V , as are similar correlated interference fringes in two-body reflection of a particle from a mirror described above.
Next, particle and well substate wavegroups are chosen so that the spatial width of the particle substate fills the well spacing D by using only the ground state n = 1 while the well substate width is much less than D. Fig. 16 (a) shows such PDF results of three snapshots taken at equal time intervals progressing from the lower left to upper right. 'Reflection' again occurs at the diagonal white line, corresponding to x 1 = x 2 ± D. This particle substate is a superposition of waves traveling in opposite directions with different energies, resulting in the PDF appearing as 'traveling wave' that matches the boundary conditions at x 1 = x 2 ± D. The parameters used in fig. 16

Asynchronous measurement: infinite well wavegroups
To illustrate asynchronous measurements, the particle is measured at a particular x 10 , with t 10 being the time used for the lower snapshot in fig. 16 (a), while the well substate then evolves with Ψ[x 10 , t 10 , x 2 , t 2 ]. Rather than graph this time evolution using a 2-D plot of the well's one-body PDF vs. x 2 for different t 2 and various values of x 10 with t 10 fixed, PDF plots of snapshots at different times t 2 are shown on a 3-D one-body PDF graph of x 2 vs fixed values of x 10 in fig. 16 (b). To obtain the aforementioned plot from fig. 16 (b) one must take a slice along the x 2 axis for a fixed value of x 10 . Both figures 16 (a) and (b) use the same parameters. Although the time intervals per snapshot are the same for both figures, more snapshots are shown as a vertical progression in part (b). The resulting time dependent one-body PDF along the x 2 axis for fixed x 10 and t 10 is a consequence of these two well states interfering, each of which has reflected the particle moving in opposite directions with different energies.

IV. DISCUSSION
A. Prolonging interference: microscopic particle reflecting from a macroscopic mirror Types I and III synchronous correlated interference in reflection are temporally and spatially limited by the coherence lengths of both the particle and reflector (mirror, barrier, or well) wavegroups. We have presented a simple model illustrating the magnitude of these effects.
This limitation is mitigated in type II synchronous correlated interference which generates two-body wavegroups in a manner similar to that of an electromagnetic pulse reflecting from the two surfaces of a moving thin film. That is, the interfering two-body wavegroups, one from interaction with each of the moving barrier surfaces, travel in the same direction and maintain overlap, thereby eliminating the transient behavior found in type I and III interference. Since these two interfering wavegroups have the same frequency spectrum (reflection occurs from both surfaces which move at the same speed) there is no 'beat' or time dependent fringe pattern.
A different method to prolong interference, but only of the reflector, is the focus of this paper. We refer to this asynchronous effect as type IV interference. It involves first measuring the position of the particle and then later that of the reflector. This requires that the particle be measured while the two-body wavefunction is in a state of correlated interference. If the particle is measured first, the reflector state is thereafter split into ones in which the particle has and has not interacted with the reflector. These two subsequent one-particle reflector states then travel in the same direction but with momenta and energy differing by that exchanged with the particle. If the coherence length for the reflector is small then such interference persists only for a small momentum exchange. Otherwise, these two reflector wavegroups quickly separate beyond their coherence lengths due to their different average momenta. Since these states have different energies their interference results in a 'beat' or time dependent fringe pattern. However, a microscopic particle reflecting from a macroscopic reflector at rest is well suited to prolong this oneparticle interference of the two reflector states for two main reasons. First the two reflector wavegroups maintain overlap longer since their difference in speeds is ∆V ≈ 4mv/M while their coherence length is l thermal c . This results in an overlap time ≈ h √ M /(4mv √ 2k B T ). Second their energy difference is negligible for a stationary reflector. That is, the "beat" frequency goes to zero in the limit of large reflector mass while the difference in momenta of these reflector states remains finite.
It is this difference in reflector momenta which leads to the phase difference ∆K x 2 when superposing these two mirror wavefunctions. That is, the reflector fringe spacing is only determined by the change in particle momentum since conservation of momentum in reflection requires ∆K = −∆k. The microscopic particle then imposes a fringe structure from its small change in momentum on the macroscopic reflector.
Double slit interference with a massive particle, on the other hand, superposes two one-body states with the same momentum whose difference in phase, K∆x, is due to the difference in path lengths ∆x from each slit to the measurement point times an extremely large wavevector (rather than the small ∆K for the mirror reflecting a microscopic particle). This results in an imperceptible fringe spacing on the macroscopic object traversing the double slits.
This interference of the macroscopic reflector states does not require utilizing standard division of amplitude nor division of wavefront interferometric methods. It does however, use measurement acting as a beamsplitter to generate two reflector states which interfere. In addition, path lengths need not be carefully matched for interference to be manifest. These are distinct experimental advantages over division of amplitude or wavefront interferometers, particularly for a macroscopic reflector mass.

B. Measurement theory
The formalism for asynchronous measurement contains some subtile issues. Foremost among them is what constitutes a measurement of the particle after which no interaction with the mirror occurs. An experimental realization of such a measurement might consist of the particle-mirror reflection occurring only along the x-axis while another particle called the "probe" moves along the y-axis. A measurement then involves the probe absorbing, being absorbed by, being scattered by, or scattering from the particle. The operational definition of such a measurement, when the particle is in the correlated interference region, is then that which causes splitting of the mirror state. This could result in the observation of prolonged (type IV) interference of the mirror states or observation of these states when they no longer overlap. Note that the probe need not interact with the particle at any particular location apart from it occurring within this correlated interference region.
Related issues involve modeling this sequential measurement. In the Copenhagen interpretation, a measurement of the particle's position collapses its substate into an eigenstate of that position operator. The assumption of no interaction after the first measurement then requires that the particle does not reflect a second time from the mirror's substate. The state of the particle after measurement is therefore irrelevant to the subsequent evolution of the mirror substate and is treated as such in the formalism present above by fixing the particle coordinates in the two-body function at its measurement position and time. A more realistic model would account for a measurement which occurs over a distance ∆x 1 rather than at a point in space. Such a treatment is beyond the scope of this work.
This assumption of the lack of interaction can be relaxed to further study the implications of the collapse postulate. To illustrate this let Ψ 1 [x 1 , t 1 , x 2 , t 2 ] be the solution to the Schrödinger equation for the incident uncorrelated plus the reflected correlated wavegroups. Also let these overlap to form correlated interference. Next, let the first measurement be of the particle at x 1 = x 10 and time t 10 . The particle substate then collapses into an eigenstate of the position operator and therefore generates the two-body uncorrelated wavefunction To determine the effects of allowing interaction (a second reflection) before a later measurement of the mirror, the solution to the Schrödinger equation Ψ 2 [x 1 , t 1 , x 2 , t 2 ] for t 1 > t 10 and t 2 > t 10 is needed. This is determined from the initial condition Ψ 20 along with the boundary condition at the reflector surface. However, the particle must not be absorbed in the measurement process since it is allowed to interact again with the mirror after its initial measurement.
As a particular example, consider measuring the particle at a position in the correlated interference region given by the middle snapshot in fig. 1. The particle-mirror state just after this measurement is a cut along the x 2 axis at x 1 = x 10 of the particle-mirror wavefunction associated with this PDF, due to the delta function nature of this collapse. This wavefunction in the (x 1 , x 2 ) plane is then the initial condition used in solving the Schrödinger equation for the subsequent particle-mirror state. The reflection boundary condition at x 1 = x 2 must also be satisfied. The narrow particle substate along the x 1 axis will lead to its "diffusion" and subsequent "re-reflection." Note that if the particle is measured a second time, before a measurement of the mirror, then the procedure just mentioned must be repeated. Measurement-induced multiple reflections then occur and are, in themselves, of fundamental interest in testing the collapse postulate.
This use of wavefunction collapse can be contrasted with that of the "appearance of collapse" via decoherence of the wavefunction with the environment. In such a model, measurement of the particle does not fix the particle coordinates at the time and position of its measurement in the subsequent many-body wavefunction. The coordinates for each degree of freedom in this wavefunction simply continue to time evolve in the decoherence model. It is beyond the scope of this work to apply this and other interpretations of measurement in quantum mechanics to such asynchronous measurements.
C. Asynchronous measurement: prediction of the Doppler effect A time dependent interference pattern is expected classically in the overlap region when a wavegroup reflects from a moving mirror. For a stationary mirror the incident and reflected waves form a standing wave which is measured by placing a detector at a fixed position where the incident and reflected waves overlap [11]. Motion of the mirror introduces a Doppler shift in the reflected wave which generates time dependence in this interference. The detector fixed in space then measures a time dependent intensity variation.
Consider the analogous process for the particle-mirror system. The incident and reflected two-body wavefunctions have the same total energy in reflection from a moving mirror, although it is parsed differently. Interference in the synchronous formalism involves superposing these two wavefunctions, each with the same frequency. The result, given by eqn. 10 with t 1 = t 2 , is then independent of time.
To illustrate this, consider the middle snapshot of fig. 1. For different snapshots (not shown), all of which still partially overlap spatially with this middle snapshot, the envelope of the PDF waveform moves through the (x 1 , x 2 ) plane yet the maxima and minima of the fringes for these different snapshots remains in the same location.
The Doppler shift for the particle-mirror system, analogous to a classical wave reflecting from a moving mirror, involves only measuring the particle flux at a fixed position with no measurement of the mirror. Not measuring the mirror involves averaging the two-body PDF over the mirror coordinates. The mirror coherence length determines if this averaging "washes-out" the interference, which is the case shown in fig. 8 (a). However, a smaller mirror coherence length results in a PDF which when averaged over x 2 maintains the interference along x 1 and is illustrated in fig. 8 (c). The PDF for simultaneous correlation, with a coherence length associated with that shown in fig. 8 (c), then generates only static interference and not the expected time dependent effect associated with the Doppler shift of the particle wavefunction from the moving mirror.
However, as shown in subsections II E 4 and II E 6 the two-time formalism predicts a Doppler shift, similar to that classically, when measuring only the particle. This is a consequence of using the two-time formalism to parse the total energy and therefore the frequency of the wavefunction into terms associated with the particle and mirror rather than only using an expression for the total energy. It is beyond the scope of this work to apply other interpretations of measurement theory in quantum mechanics to this issue of the Doppler shift.

V. SUMMARY
Asynchronous measurement in quantum mechanics has been treated for a particle reflecting from a mirror or interacting with a well or barrier. The results presented assume measurement of the particle first, collapse of the particle substate, and a later measurement of the mirror (or barrier-well) while no interaction occurs between the times of these measurements. The two-body Schrödinger equation, solved with standard techniques, is parsed to completely separate the variables associated with the particle from those of the mirror so that when the particle is measured only its parameters are fixed while those of the mirror can continue to evolve in time until the mirror is measured. The resulting interference effects, such as two mirror states being generated when it is not known if the measured particle did or did not reflect, are in many ways familiar from examples in quantum mechanics in which an outcome can be achieved in indistinguishable ways. Yet this process of splitting differs by using measurement as its mechanism.
Some issues of asynchronous measurements presented include: comparison of asynchronous correlated interference with division of amplitude and wavefront interferometers, correlated interference in the bound state of a particle and infinite well or unbound state of a finite barrier, transfer of coherence between the particle and mirror, the effects of substate coherence on correlated interference, distortion of the two-body wavegroups due to reflection, the Doppler shift for two-body states, interference effects on macroscopic reflectors, the division of energy and momentum as the particle traverses a barrier, interference of multiple two-body wavefunctions from the two surfaces of a barrier, and the prolonging of interference of the mirror substate when the particle is measured first, in the region of correlated interference.
Some consequences of asynchronous measurements not presented include: multiple sequential particle measurements before a measurement of the mirror (two-body quantum Zeno effect), treatment of asynchronous measurement in other interpretations of measurement in quantum mechanics, two-body effects for a spatially varying potentials, and both synchronous and asynchronous two-body treatment of quantum tunneling.
Analogies have been made with classical interference even though there is no direct classical correspondence with correlated interference or asynchronous measurements. One analogy is between the particle traversing a finite well and an electromagnetic wavegroup interacting a slab of glass. Another is between reflection of a particle from a mirror and a similar reflection of an electromagnetic wavegroup.
Since two-bodies are required for correlated interference, one might expect that by not measuring one there would be no interference in measurements of only the other. Such one body interference is predicted from an average over the two-body PDF along the coordinate of the unmeasured body, reducing it to a one-body PDF. One-particle interference can survive this averaging, but this depends on the coherence lengths of the particle and mirror substates as discussed in subsection II E 6.
The language used to describe the two-body system implies that these results are valid only for a microscopic particle interacting with a macroscopic mirror or barrier. However, tests of these predictions will most likely first occur in microscopic systems.
Measurement in quantum mechanics plays a role different from that in any other physical theory and its interpretation is controversial. Asynchronous measurement in a two-body system is modeled here by collapse of the particle substate while the mirror substate continues to time evolve until it is measured. Verification of the predictions presented here will support the Copenhagen interpretation of measurement.
These results are far from comprehensive as indicated by the discussion in the previous section. This work may stimulate further research in understanding correlation and measurement in quantum mechanics.

VI. APENDIX
Without collapse the two-body wavefunction, expressed in terms of the two times, evolves and conserves probability. The probability of measuring the particle at (x 1 , t 1 ) and the mirror at (x 2 , t 2 ) is given by P DF [x 1 , t 1 , x 2 , t 2 ]dx 1 dx 2 with the joint PDF determined by the solution of equation 1 as ΨΨ * but expressed in terms of the two time variables in the particle-mirror or particle-well system. Using this equation, conservation of probability can then be expressed locally as, where j 1 [x 1 , t 1 , x 2 , t 2 ] =h(Ψ * ∂ x1 Ψ − Ψ∂ x1 Ψ * )/(2im) and j 2 [x 1 , t 1 , x 2 , t 2 ] =h(Ψ * ∂ x2 Ψ − Ψ∂ x2 Ψ * )/(2iM ). While the expressions for these current densities appear similar to that for one particle systems there are subtle but important differences for a two body system [26]. Multiplying equation 19 by dx 1 dx 2 , integrating over the segment from a to b along the x-axis (a ≤ x 1 ≤ b and a ≤ x 2 ≤ b), and then rearranging terms yields a solution to equation 19 if These equations indicate that the time rate of change of probability within the ab segment on the x-axis is determined separately by a net change in particle and mirror probability fluxes in that region, which is similar to conservation of probability in a one-body system. Now, however, probability of the two-body system is conserved for the mirror even when x 1 and t 1 are fixed. The asynchronous procedure described here therefore maintains conservation of probability for the mirror after the particle has been measured.