Universality of local weak interactions and its application for interferometric alignment

Significance In this combined theoretical and experimental work, we describe and demonstrate a universal feature of local couplings of spatially preselected and postselected systems. All weak couplings, independently of their physical nature, are modified in the same universal manner, characterized by the weak value of local projection. We present a general analysis which neither relies on a specific form of the coupling nor is restricted to weak interactions. This allows us to define a consistent paradigm of the presence of a quantum particle, much more intricate than the concept of the presence of a classical particle. Furthermore, we discovered that Gaussian pointers remain reliable beyond the weak regime, which enabled a simple and efficient alignment method for interferometric setups.

P reselected and postselected systems are ubiquitous in quantum mechanics. In many quantum information schemes, the intended process is only realized by the interplay of preselection and postselection. The addition of postselection, often together with conditioned transformations, is the basis of protocols such as universal quantum computation within the Knill-Laflamme-Milburn scheme (1), entanglement swapping (2), and heralding in general (3).
The two-state vector formalism (TSVF) (4) provides a general framework for the description of preselected and postselected systems. It introduces a state evolving backward in time and thereby treats the postselection on equal footing as the preselection. The key element of the TSVF is the weak value of an observable. As long as the interaction is sufficiently weak or short, the observable effect on the external system is completely characterized by the weak value (5). For such interactions, the state of the external systems after the postselection can deviate significantly from the states expected by just considering the coupling to preselected systems (6). The concept of weak values became the basis of several successful applications in precision-measurement techniques (7,8). While there are theoretical controversies about the optimality of the weak value-based tomography and precision-measurement methods (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), a plethora of fruitful applications continues to emerge (21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34).
We take a step back and investigate the fundamental properties of preselected and postselected systems. We find that there exists a general universality principle characterizing how the effects of the interactions in one location of a spatially preselected and postselected quantum system are modified as a function of preselection and postselection. All these modifications are specified by a single complex number, the weak value of the spatial projection operator. One of the innovations of our approach is that it does not rely on the specific form of the interaction Hamiltonian. Instead, it expresses the change of the state via the complex amplitude of an orthogonal component, which emerges due to the interaction. If the weak value is a positive number, the size of the changes in every variable is multiplied by this number, and when it is negative, all modifications happen in the opposite direction. If the effect originally changed a particular variable, in the case of an imaginary weak value, the effect will occur in a variable conjugate to the initial one, and when the weak value is a complex number, both effects are combined together. This approach allows a formal definition of a quantum particle's presence.
Until now, most accounts considered the weak value to be limited to the case of weak interactions-e.g., refs. 35-39. It is another crucial innovation of our approach, however, that we explicitly apply the formalism to the case of much stronger interactions. We use an expression for the weak value which takes into account changes due to interactions of finite strength in the time interval between preselection and postselection. Besides incorporating the stronger interactions, we also account for decoherence or imperfections in the measurement system. We show experimentally that this weak value can in fact be measured by using weakly coupled pointers.
An interferometer, especially a Mach-Zehnder-type interferometer (MZI), can be seen as the iconic example for preselected and postselected systems. The reflectivity/transmittivity of the first beam splitter together with the phase shifter defines the preselected state of the system. The final beam splitter together

Significance
In this combined theoretical and experimental work, we describe and demonstrate a universal feature of local couplings of spatially preselected and postselected systems. All weak couplings, independently of their physical nature, are modified in the same universal manner, characterized by the weak value of local projection. We present a general analysis which neither relies on a specific form of the coupling nor is restricted to weak interactions. This allows us to define a consistent paradigm of the presence of a quantum particle, much more intricate than the concept of the presence of a classical particle. Furthermore, we discovered that Gaussian pointers remain reliable beyond the weak regime, which enabled a simple and efficient alignment method for interferometric setups. with detection of the particle in one output of the interferometer sets the postselected state. The effect of weak interactions of the particle with external systems, which can be seen as a trace the particle leaves inside the interferometer, is characterized by the weak value of the projection operator on the corresponding arm. Surprisingly, we also found that for Gaussian states of the external system, the weak value characterizes the modification of the trace for arbitrary strength of the interaction.
The interferometer enables straightforward experimental implementation, where we consider a preselected and postselected photon passing through. We experimentally characterize the various effects of multiple interactions in one of the interferometer's arms using the mode and the polarization of the propagating photon as the external systems coupled to the photon's path. We find that the modifications of the weak effects on the photon can be described by the weak value of the projection operator on the corresponding arm for various types and strengths of couplings.
We can now turn the picture upside-down and view any coupling to the external degrees of freedom as being due to misalignment of the interferometer. For example, a tilted mirror in one of the beams now becomes an interaction deflecting the Gaussian mode of the beam from its ideal direction. This analogy directly leads us to an efficient alignment technique for interferometers, where our analysis provides a simple model for the image observable at the output of an interferometer. More precisely, by measuring the phase-dependent trajectory of the centroid of the output mode on only a single spatially resolving detector, we can extract the misalignment parameters in one go. This technique harnesses the benefits of the weak amplification method (6) to improve precision.
Weak Value of Local Projection and its Connection to the Trace Let us first consider the effect of a quantum particle on external systems due to all kinds of local interactions in the channel through which it passes. The interactions might be caused by various properties of the particle-e.g., charge, mass, magnetic moment, etc.-but we assume that the particle passing through the channel does not change its quantum state. * If the quantum particle is not present in the channel, the state of the external systems at a particular time is |χ . When the quantum particle is localized in the channel, as shown in Fig. 1A, the interactions change the total state of the external systems as where |χ ⊥ denotes the component of |χ , which is orthogonal to |χ . By definition, we choose the phase of |χ ⊥ such that > 0. For simplicity, but without loss of generality, we also disregard the global phase and consider the coefficient η to be positive, The trace left by the particle is manifested by the presence of the orthogonal component |χ ⊥ and is quantified by the parameter .
Next, let this channel be an arm of an MZI (Fig. 1B). We assume that the arm B of the MZI is ideal-i.e., the particle leaves no trace there. † For the creation of the preselected state * To deal with cases where the states of some degrees of freedom of the particle change, we treat them equivalently to the external degrees of freedom. In fact, this is the case in our experiment (Observing the Universality Property). † This assumption is made for simplicity of presentation. The main results of the paper about the universality of modification of interactions are easily transformed to the case when some weak traces are left in all parts of the interferometer. Fig. 1. Comparison between effect of the local coupling of the particle when it passes through a single channel and when it passes through an interferometer. (A) The particle interacts with external systems in a single channel originally in the state |χ . (B) The particle passes through an MZI with arm A identical to the channel described in A with all its local interactions, while it is assumed that there are no local couplings in arm B.

A B
inside the interferometer |ψ , the unbalanced input beam splitter is followed by a phase shifter, resulting in where |A and |B represent the eigenstates of the path degree of freedom, and α and ϕ are the two real parameters of the state. The second beam splitter is balanced, so its operation can be modeled as We collect photons in output port C , which corresponds to a postselection of the state |φ = 1 √ 2 (|A + |B ).

[4]
Accounting for the interactions in arm A (Fig. 1B), the composite state |Ψ of the particle and the external systems before the second beam splitter is where here and in the rest of the paper, we use a shorthand notation for tensor products with |A |χ ≡ |A ⊗ |χ . After detection of the particle in arm C -i.e., postselection of the particle in state [4]-the state of the external systems becomes where N is the normalization factor. Here and in the rest of the paper, we use the accent symbol "∼" to denote situations with preselection and postselection. We start by considering interactions which are sufficiently small, with 1. In the case of a single channel, the particle passing through it leads to the change of the state of the external systems, |χ → |χ = |χ + |χ ⊥ + O( 2 ), [7] which is just an expansion of [1] in orders of .

PHYSICS
For the particle that has passed through the corresponding MZI and has been detected in C , we observe a different change of the state of the external systems. Expanding Eq. 6 in orders of , we can see that the weak effect of the interaction is modified relative to [7] by a single parameter, the weak value of projection on arm A, where, for defining the weak value, we neglect the coupling to external systems [9] The design of the interferometer allows the full range of weak values of projection onto arm A, by varying the parameters tan α and ϕ. Note that we did not restrict the number of interactions as long as their combined effect is sufficiently weak. When the trace left in the interferometer is small, 1, the weak value can be considered neglecting the effect of the interactions, as in [9]. In the next section, we will turn toward scenarios with stronger couplings for which the interactions cannot be neglected.

Weak Value Considering Finite Coupling Strength and Imperfections
Calculating the weak value as in Eq. 9, we have implicitly assumed that it only depends on the preselection and postselection states at the boundaries of the considered time interval. This is correct in the limit of weak coupling, which is considered in most works about weak measurements. However, sometimes even in scenarios with coupling of finite strength, the weak value has been treated as if there was no coupling-i.e., using formula [9] (20, [35][36][37][38][39][40]. To correctly account for couplings of finite strength, we turn to the proper definition of the weak value in the framework of the TSVF, which refers to a single point in time t, at which the particular forward-and backward-evolving quantum states have to be evaluated (41). All interactions of finite strength and imperfections of optical devices between preselection and t, as well as between t and postselection, must be considered. Thus, Eq. 2 correctly describes the forward-evolving state only immediately after the first beam splitter, and Eq. 4 describes the backwardevolving state only immediately before the second beam splitter. Since all evolutions due to imperfections or interactions with the different external systems are local-i.e., they have the common eigenstates |A and |B -the time ordering of the evolutions is of no consequence. Therefore, the weak value (PA)w stays constant in time, and we are free to choose any moment in time to calculate it. For convenience, we calculate the weak value immediately before postselection on state [4] and modify only the forwardevolving state to account for the evolution due to interactions inside the interferometer.
Due to the interactions, the system becomes entangled with the external systems as described by Eq. 5. Thus, the particle is in the mixed state described by the density matrix in the basis {|A , |B } ρ = cos 2 α cos α sin αe −iϕ η cos α sin αe iϕ η sin 2 α . [10] The weak value in the case of mixed states has been derived in ref. 5 (equation 32 therein), In our case, this formula is not applicable for arbitrary time between the preselection and postselection due to entanglement in both forward-and backward-evolving states with the same external systems (section VI of ref. 5), but it can be used to calculate the weak value immediately before the last beam splitter since the backward-evolving state is not entangled (see also refs. 35, 42, and 43). As we explained above, the weak value of the projector PA is constant between preselection and postselection, so it can be calculated as From Eq. 10, we see that the overlap η quantifies the loss of coherence between the two arms of the interferometer due to interactions and imperfections, which consequently leads to a reduction of the maximally achievable weak value. The dependence of the weak value [12] on η as well as on α and ϕ is presented in Fig. 2. Fig. 2 A and B show the case with ideal overlap η = 1, while Fig. 2 C-F illustrate the dependence for the nonideal case with reduced overlap and thus smaller (PA)w . The weak value [12] which accounts for multiple and even strong interactions is not useful to describe the whole of the external systems when inserted into expansion [8] because is large. However, Eq. 12 can be used to describe the modification for those interactions which are weak, even if some of the other interactions or all of them together are arbitrarily strong. We will show this now.
In our scenario, we neglect the interactions of external systems in arm A among themselves. If between some particular systems, the interaction cannot be neglected, they are considered as a single composite system. Thus, the interactions [1] in a single channel (Fig. 1A) can be decomposed as [13] Here, as for [1], we absorbed the phases in the definitions of states, such that j and ηj are positive numbers.
In the case where the coupling to the, say, k -th system, is weak, the change of the state of the system can be also expressed by using density-matrix language in the |χ k , |χ ⊥ k basis as For a particle passing through the MZI, when both the preselection as well as the postselection state are superpositions of |A and |B , several interactions in A (Fig. 1B) will lead to entanglement between the various external systems. Thus, each of the systems will be described by a mixed state. The modified evolution of the weakly coupled k -th system is Again, the modification of the effect of the weak interaction is characterized by the weak value (PA)w .

Manifestation of the Trace as Shifts in Pointer States
In the previous sections, we described the trace a particle leaves as the appearance of an orthogonal component in the quantum state of external systems. Another language, frequently closer to experimental evidence, is the change in the expectation values of the external systems. Given the small change due to interactions in Fig. 1A, expressed in [7], every observable O of the external system changes its expectation value as [16] Then, using [8] (or [15], respectively), we see that for the preselected and postselected particle (Fig. 1B), the change in the expectation value of O is modified according tõ [

17]
This formula is universal-it is valid for every system which was coupled weakly in arm A to the particle passing through the interferometer.
Eq. 17 represents a result in a very general scenario. Let us now focus on the less general but very common measurement situation, which is usually considered when treating weak values (41). There, a single observable O is the pointer variable Q, the pointer wavefunction χ(Q) is real, and the interaction with the particle in the channel shifts the wave function in the pointer variable representation as Obviously, this also shifts the expectation value In this scenario, χ ⊥ (Q) is also real, as well as χ|Q|χ ⊥ . Then, a positive weak value (PA)w just tells us how the effect of the interaction is amplified or reduced according tõ If (PA)w is negative, it tells us that the pointer will be shifted in the opposite direction.
If the weak value is imaginary, the expectation value of the pointer position will not be changed. However, an orthogonal component in the quantum state of the pointer will still appear. It will manifest itself in the shift of the expectation value of the momentum PQ conjugate to Q where (∆PQ ) 2 = χ|P 2 Q |χ − χ|PQ |χ 2 and = 1. Eqs. 20 and 21 were obtained from [16] and [17] on the assumption of weak coupling when higher orders of can be neglected. In the measurement situation [18] with a Gaussian pointer, χ = e −Q 2 /4(∆Q) 2 (we omit normalization), the usual range of validity of the weak value formalism is extended. Even when the coupling is strong and the pointer distribution is significantly distorted during the measurement, the expressions for the shifts of the expectation values of Q, [16] and [17], remain exact, with Indeed, for the Gaussian pointer χ|Q|χ = 0, χ |Q|χ = δQ, and also the following expressions are easily calculated as Then, [22a] is proven by substituting [19] and [1], while including [6] and [12] proves [22b].
If the pointer is a Gaussian in the position variable Q, it is of course also a Gaussian in the conjugate momentum PQ representation. Therefore, [20] and [21], in analogy to the above, become exact formulas with ∆PQ = 1 2∆Q . There are corresponding exact formulas for the effect of a shift in momentum δPQ with see also ref. 44. Direct substitution shows that the expressions remain correct for Gaussians in the regime of strong interactions also in the case of combinations of shifts in Q and PQ , such that [25b] These equations are the basis of the alignment method presented in Alignment Method.

Observing the Universality Property
We use an optical MZI to experimentally visualize our central claim-namely, that all kinds of small effects of spatially preselected and postselected systems taking place at a specific location are modified in a universal manner characterized by the weak value of spatial projection. In the experiment, we demonstrate the universal change for three different couplings. In every case, the effect is modified in the same manner.
There are proposals and actual experiments where the photon couples to other particles in one arm of the interferometer (45)(46)(47)(48)(49). In ref. 49, one arm of the interferometer is a Kerr medium, and the photon passing through this arm changes the quantum state of the pointer by introducing a shift in the relative phase between the wave packets of the pointer photons. As it is done in most weak-measurement experiments, instead of coupling to external particles, we, rather, study interactions of the photons in an arm of the interferometer by observing the effect on other degrees of freedom of the photons itself. We also used a (weak) laser beam, so all of the results can be explained by using Maxwell equations (although in a much more difficult way), but the observations would not change by using single photons, since intensity measurements are in one-to-one correspondence to single-photon probability distributions.
The interactions in arm A are realized by introducing controlled changes of spatial and polarization degrees of freedom. The initial state of the position degree of freedom can be well approximated by a Gaussian along the x as well as the y coordinates. The interaction is implemented by shifting the center of the Gaussian intensity distribution of the light beam going through arm A by δx , compared with the beam going through arm B , where w0 denotes the waist of the beam and normalization factors are omitted. Another degree of freedom is the spatial state in the y direction of the light beam, which we modified by changing the angle of the beam around the x axis, which for small angles corresponds to the momentum shift δpy = 2π λ δθx . The resulting modification in arm A can be expressed by χy (py ) = e −w 2 0 p 2 y /4 → χ y (py ) = e −w 2 0 (py −δpy ) 2 /4 .

[27]
As a third external system, we use the photon polarization. The interaction parameter here is the rotation of polarization by the angle δΘ, where the states |H and |V are defined via σz |H = |H and σz |V = −|V for the Pauli matrix σz .
All other properties of the photon are expressed in the state |χO . Any imperfections of the interferometer can be understood to lead to a change of the initial state of these properties in arm It is a good approximation to assume that there are no interactions between the external degrees of freedom we consider, and thus we can express the quantum state of the photon in arm B just before reaching the final beam splitter of the interferometer as |B |χ = |B |χx |χy |χσ |χO , while in arm A it is To test the universality of modifications of effects for various degrees of freedom, one could either perform complete tomographies of the final pointer states [14] and [15] or, more clearly, show the modification of the effects of the three couplings according to [20] and [21]. We follow the second approach. More explicitly, we test the differences between effects of the interactions on the expectation values in three degrees of freedom when the particle passes through the single arm (expressed by δ) and when the particle passes through both arms (expressed byδ). ‡ Because of the linear relation between θy and px , as well as θx and py , one obtainsδ δ y = −zRδ θx Im[(PA)w ].

[34]
Here, we have used the Rayleigh range zR ≡ πw 2 0 λ as the characteristic parameter of the Gaussian beam.
The conjugate variable to the angle Θ defining polarization changes in the σx − σz plane is an angle Υ describing polarization rotations in the σy − σz plane relative to the initial state |H . For small deviations, these angles relate linearly to σx and σy , respectively, and are given bỹ δ Υ = −δ Θ Im[(PA)w ].

[36]
The test was performed for the full range of ϕ and thus for a large range of values (PA)w ; see violet lines on the graphs of Fig. 2. The parameters for the calculation of (PA)w necessary for testing relations [31][32][33][34][35][36] were also obtained from measurements. The signals from separate arms (when the other arm was blocked) provided tan α. The phase ϕ and the overlap η were obtained from the intensity of the interference signal and visibility measurements, respectively. The relation between the visibility V and the overlap η for the phase-dependent output intensity I ∝ φ|ρ|φ ∝ 1 + tan 2 α + 2 tan α η cos ϕ is given by The experiment is shown schematically in Fig. 3. After propagation through a single-mode fiber for spatial filtering, the horizontally polarized light from a laser diode (λ = 780 nm) is split by ‡ We chose this method since our measurements of the shifts δ x , δ θx , and δ Θ in a single channel are more precise than our control of the shifts δx, δθx , and δΘ via manual stages. a nonpolarizing beam splitter. The moduli of the amplitudes of the preselection state [2] are controlled by means of rotating the polarization using a half-wave plate in arm A, followed by a horizontal polarization filter. The relative phase between the arms ϕ is set by an optical trombone system with retroreflecting prisms moved by a piezoelectric crystal (not shown).
This setup enables us to directly implement the three desired interactions along beam A and simultaneously measure their effect. Fig. 3 depicts the setup. The spatial displacement δx , which is schematically depicted as a shift of the mirror, was achieved by lateral movement of the prism from the trombone system. Instead of a vertical tilt of this mirror, we incorporate the vertical rotation δθx by tilting the second beam splitter. The polarization rotation δΘ is controlled by rotating a half-wave plate in arm A. Detecting light only from the output port C provides the post-selection onto state |φ , Eq. 4.
The photons at port C are distributed onto several detectors using beam splitters for position and polarization analysis. A position-sensing detector PSD1 placed near the interferometer and a detector PSD2 placed farther away allows the estimation of position and angle in x and y directions. We perform tomography of the polarization state using half-and quarter-wave plates in combination with polarizing beam splitters, as shown in Fig. 3.
A measurement run consists of three steps-namely, first a measurement of light propagating in arm A alone, second of arm B alone, and last a measurement of the interference signal. The six expectation values obtained from measurements of arm B are used as a reference for the subsequent analysis.
The measurement with only beam A shows the effect of the interactions when the photons pass through a single channel, as in Fig. 1A. The results are indicated in the graphs of Fig. 4 as red dashed horizontal lines since they exhibit no dependence on the phase. § The universality is clearly shown by the similarity of the results for the three couplings (Fig. 4). Of course, in all graphs, the observed values are different and have different units. For demonstration purposes, we arranged the scales of the graphs § Please contact J.D. if you desire access to the raw experimental data for this plot as well as for all other plots.
in Fig. 4, Upper such that the signals of all interactions, x A, θy A, and Θ A, have the same size. We were trying to avoid shifts in conjugate variables as much as possible. Our measurement results, red dashed lines in the plots from the Fig. 4, Lower, show that the tuning was good, although not perfect.
Continuous violet lines on these graphs provide theoretical predictions based on the weak value (PA)w given by [12], and the interactions in the single arms are presented as red and blue dashed lines in the graphs. The intensities obtained by measuring arm A and arm B alone yield tan α = 1.3323 ± 0.0002. From the visibility measurement, V = 95.09 ± 0.02%, we obtained η = 0.9904 ± 0.0003. For these parameters, we observed amplifications with factors up to 4 and −3. The very good agreement between experimental data and theoretical predictions, shown in Fig. 4, demonstrates the universality of the modification of several fundamentally distinct forms of interactions for couplings with a preselected and postselected system.
To evaluate the dependence of the weak value on the coherence between the two arms parametrized by η, we measured the effect of the displacement in x on the output beam. For this run, we kept the phase fixed at ϕ = π and varied the amplitude ratio tan α covering another region of the parameter space from Fig. 2. We changed the coherence by varying the polarization misalignment, leading to a smaller overlap between the photon states passing through the two arms. The modification of the shift in the x direction presented in Fig. 5 follows nicely the weak value [12].

Alignment Method
In the previous sections, we considered a scenario in which the path state of a photon in an arm of an interferometer is coupled to its other degrees of freedom, in particular, its spatial degrees of freedom in the x and y direction. This scenario exactly represents a situation encountered in real experimental interferometric setups-namely, when the arms of the interferometer are misaligned. The differences in position δ r ≡ (δx , δy) and angle δθ ≡ (δθx , δθy ) between the photons passing through distinct arms of the interferometer can be considered as results of interactions in one arm, which change the initially identical spatial states of the particle. The introduced displacements of arm A in x direction, angle around the x axis, and angle of polarization Θ (δx, δθx, and δΘ) can be seen from the single red datapoints plotted at an arbitrary phase position. The blue datapoints corresponding to arm B are taken as a reference and thus show zero shift. The axes are scaled such that the readings of A agree for the three external systems. For each of these three, the same behavior of the interference signal (black datapoints) is observed for the shifts of the variablesδ x ,δ θx , andδ Θ : The effect seen from the measurement of the single arm is multiplied with the phase-dependent real part of the weak value. (Lower) The analogous plots for the shift of the respective conjugate variables represented byδ θy ,δ y , andδ Υ show nicely the dependence on the imaginary part of the weak value. The violet theoretical curves represent the rescaled real and imaginary parts of the weak value (no fit).
It is well known that the picture generated by the interference of the beams from a misaligned interferometer displays a strong phase dependence. Fig. 6A shows the centroid trajectory during the phase scan of a misaligned interferometer. We demonstrate that it is possible to quantitatively determine the exact misalignment parameters of the interferometer by analyzing this Fig. 5. Modification of weak value due to decoherence. The colored dots represent the measured values for the modification of the shift δx in the interference signal when varying the weak value via the relative amplitudes of the paths A and B (tan α in Eq. 12) and fixed ϕ = π. The four datasets correspond to four different values of the overlap η, which quantifies the coherence between the states of the external systems from the two arms. The lines are theoretical curves as highlighted by the colored lines in Fig. 2 C and E. Respective average error bars are shown for each η on one of the first data points. For comparison, also the theoretical line with η = 1 ( Fig. 2A) is shown. phase-dependent movement. In fact, the misalignment parameters δ r and δθ could be calculated from measurements described in the previous section. Disregarding the polarization analysis, it was a measurement of the misalignment parameters based on position measurements of centroids of the beams on two detectors at different locations. But the method is more powerful and can be implemented with only a single position-sensitive detector as well.
The basis for our alignment method are Eqs. 25A and 25B which, somewhat surprisingly, remain precise, even for large misalignment. The shift observed on the single detectorδ R is the sum of the position shiftδ r and the position shift due to the shift in direction δθ × L, where L = (0, 0, z ) is the vector parallel to the beam with the length equal to the distance z along the beam between the waist and the detector. Thus, the position shift of the centroid on the detectorδ R is given bỹ The weak value is given by [12]. The parameters tan α, η, z , and zR are found experimentally as in the previous section. The function [38] corresponds to the trajectory of the beam centroid on the detector, as shown in Fig. 6A. Even small misalignments, which otherwise might be difficult to resolve, become detectable due to the effect of weak amplification. Fig. 6B shows the x and y components ofδ R as functions of ϕ. A least-squares fit of this function provides the four unknown misalignment parameters δ r and δθ. It is remarkable that a fit function with so few parameters accurately models the experimental results. For the data shown, the fit provided δ r = (49 ± 2, 7 ± 2) µm and δθ = (12.7 ± 0.4, 0.2 ± 0.4) µrad.
In our method to obtain the misalignment parameters, we rely on the knowledge of the beam parameters-i.e., Rayleigh range zR and longitudinal position of the detector relative to the waist z . In some situations, the reversed task might be of interest. If Trajectories of beam centroids after one alignment step. It can be clearly seen how the size of the ellipse and the distance between the centroids of the single beams A (red) and B (blue) are significantly reduced in comparison with Fig. 6A.

PHYSICS
we control the misalignment parameters, we can also use our algorithm with the fit to obtain the beam parameters. In fact, the general idea of alignment using weak values was already used in alignment of the interferometer demonstrating the past of a particle in nested interferometers (50), and, since then, it was significantly developed and improved (51,52) until it reached the efficiency presented in the present work when a single scan led to a very good alignment.

Trace and Presence
A generic property of weak measurements is the possibility to perform several weak measurements on the same system. Thus, we can interpret our experiment as multiple weak measurements of the projection operator which all yield the same result, the weak value of the projection on the arm of the interferometer. However, it also implies a broader meaning with respect to the discussion of the local presence of quantum particles.
A classical particle can either be in a particular location or not. The presence of a quantum particle in a certain location, however, is a subtle issue, and its analysis strongly depends on the adopted interpretation of quantum mechanics. To avoid controversial interpretational issues, we do not discuss ontological aspects of the concept of the presence of a particle and instead argue within the operational approach.
When the wavefunction of a quantum particle is well localized in a particular location, the trace is specified in a unique way by the local interaction in that location in analogy to the trace of a classical particle when it is present (Eq. 1). Given that there are only local interactions in nature, there is no trace when the wavefunction vanishes. Similarly, there is no trace in a classical channel when the particle is absent.
Scenarios when the wavefunction does not vanish, but is also not fully localized at this location, are no longer understandable from a classical perspective. The universal relation between the trace in these scenarios and the trace of a fully localized particle which we found in our work can be considered as a basis of an operational concept of presence of a particle. It goes beyond defining the particle as present when it leaves a trace and not present when it does not (53).
According to our operational approach, the "presence" of a preselected and postselected particle in the arm A of an interferometer is defined according to the way it affected the external systems to which it was coupled and is quantified by the complex number (PA)w . This definition yields presence 1 when the forward-evolving wavefunction of the particle is solely inside the arm A independently of postselection, but presence 1 can happen also when neither the preselected nor the postselected states are eigenstates of the local projection on arm A. The presence 0 or "absence" of the particle is ensured when the forward-evolving wavefunction of the particle vanishes in arm A, but it is not a necessary condition. The postselected particle might have been absent in arm A (no effect on local external systems can be observed), even when the forward-evolving wavefunction did not vanish there.
Our concept provides a quantification and characterization of presence by describing the modification of effects of the par-ticle's interactions with external systems. It can be increased, decreased, or changed in a particular, well-defined way, and this change is the same for all local interactions-it is universal.

Conclusions
We have analyzed theoretically and experimentally the modifications of the effect of weak interactions on preselected and postselected particles. We have shown that there is a universal description of the modification of these couplings for all weak interactions given by a single complex number, the weak value of the projection on the corresponding location.
Our approach is based on expressing the effect on external systems in terms of the orthogonal components which appear due to the interactions. This allows us to formalize the meaning of the weak value without reference to a specific form of coupling. The weak value not only modifies a shift of expectation values as usual, but also the relative amplitudes of the orthogonal components of all external quantum systems interacting with the particle.
The experiment shows for three different couplings that each of the effects is modified in exactly the same way. This is shown for not just a few cases of preselected and postselected particles, but for a continuum of parameters with a large range of weak values of projection.
The approach derives the general expression [12] which allows us to apply the concept of weak values for several couplings which are not necessarily weak. These findings enable one to understand seemingly complicated dependencies seen in experiments-for example, ref. 37-and can facilitate multiparameter precision measurements in the future.
We define an operational paradigm for the presence of a preselected and postselected particle according to the trace it leaves. It is more intricate than the dichotomic concept of the presence of a classical particle, which can only be present or not. This complexity is surprising in light of the fact that in all scenarios, the external systems are in a superposition or a mixture of the undisturbed state with a single particular orthogonal component.
Our demonstration of the universality of the modification of the interactions led us to an alignment method. Its effectiveness relies on the unexpected robustness of the modification of Gaussian pointers, where the weak value expressions remain precise, even for strong couplings. In our method, a single phase scan suffices to recover all misalignment parameters from the analysis of the position of the centroid of a single output beam, clearly reducing the effort in an often-tedious task, while at the same time potentially harnessing the benefits of weak value amplification.