Post-selected von Neumann measurement with Hermite-Gaussian and Laguerre-Gaussian pointer states

Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite--Gaussian (HG) and Laguerre--Gaussian (LG) modes as pointer states and calculate the average shift of the pointer states of the post-selected von Neumann measurement by assuming the system observable $\hat{A}$ with $\hat{A}^{2}=\hat{I}$ and $\hat{A}^{2}=\hat{A}$ for an arbitrary interaction strength, where $\hat{I}$ represents the identity operator. Our results show that the HG and LG pointer states for a given coupling direction have advantages and disadvantages over the fundamental Gaussian mode in improving the signal-to-noise ratio (SNR). We expect that our general treatment of the weak values will be helpful for understanding the connection between weak- and strong-measurement regimes and may be used to propose new experimental setups with higher-order Gaussian beams to investigate further the applications of weak measurement in optical systems such as the optical vortex.


I. INTRODUCTION
In a quantum measurement, the observable information in the measured system can be extracted from the statistical average shift of the pointer. Initially, von Neumann interaction is used with the standard model of quantum measurement by mathematically describing the coupling between the measured system and measuring devices [1]. However, such strong measurements are not time symmetric. When we consider time-symmetric quantum measurements, we need post-selection of the measured system after the measurement interaction [2]. On summing the post-selections, the statistical average shift of the pointer can be deduced to the standard model of quantum measurement. Therefore, throughout this work, measurements with post-selection are called generalized von Neumann quantum measurements. In particular, sufficiently weak coupling between the measuring device and measured system is called the weak measurement as proposed by Aharonov, Albert, and Vaidman (AAV) [3]. This statistical average shift of the pointer is characterized by the weak value of the observable on the measured system. A significant feature of the weak measurements is that the weak value of the measured quantity can be beyond the usual range of eigenvalues of an observable rather than within the range of the eigenvalues as in a standard quantum measurement [3]. The feature is usually referred to as the amplification effect for weak signals and is unlike a conventional quantum measurement that collapses a coherent superposition of quantum states [1]. A large weak value can amplify small unknown parameters to detect various properties such as beam deflection [4][5][6][7][8][9], frequency shifts [10], phase shifts [11], angular shifts [12,13], velocity shifts [14], and even temperature shift [15]. However, weak value amplification has purely technical advantages [16][17][18][19][20]. In general, the weak value is a complex number. Thus, the weak measurements can provide an ideal method for examining the fundamentals of quantum physics such as quantum paradoxes (Hardy's paradox [21][22][23][24] and the three-box paradox [25]), quantum correlation and quantum dynamics [26][27][28][29][30][31][32], and quantum state tomography [33][34][35][36][37][38], as well as the violation of the generalized Leggett-Garg inequalities [39][40][41][42][43][44] and the violation of the initial Heisenberg arXiv:1410.3189v1 [quant-ph] 13 Oct 2014 measurement-disturbance relationship [45,46].
Until now, most weak measurement studies use the zero-mean Gaussian state as an initial pointer state and expand the evolution unitary operator up to the first order because, in the weak measurement scheme, the coupling between the measured system and measuring device is very weak. However, when we consider the connection between weak and strong measurements, the amplification limit, and measurement back-action of the weak measurement scheme, the full-order effects of unitary evolution due to the von Neumann interaction between the measured system and measuring device is required. The measurements of arbitrary coupling strength beyond linear-order interaction have been previously discussed by Aharonov and Botero [47]. Di Lorenzo and Egues [48] investigated the von Neumann type measurement to clarify the detector dynamics in the weak measurement process. Wu and Li [49] proposed the general formulation of the weak measurement, which includes second-order effects of the unitary evolution due to the von Neumann interaction between the system and the detector, and they theoretically demonstrated that the back-action effect is important in the weak-value amplification on the basis of the second-order calculation. Most recently, several studies [50][51][52] analytically showed that an upper bound of the weak-value amplification exists in the generalized von Neumann measurement by assuming that the probe state wave function is Gaussian and the observableÂ satisfiesÂ 2 =Î, whereÎ is the identity operator.
However, in optical experiments, we encounter higherorder Gaussian beams such as Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams. They are higherorder solutions of the paraxial wave equation with rectangular and cylindrical symmetry about their axes of propagation, respectively. Both HG and LG beams are widely used in the theory of lasers and resonators [54,55]. In fact, the zero-mean Gaussian beam is a special case of HG and LG beams. Weak measurement with the higherorder Gaussian beam pointer state has been discussed in Refs. [56][57][58][59]. In particular, de Lima Bernardo et al. [59] gave the simplified algebraic description of the weak measurements with HG and LG pointer states. In Ref. [59], they only consider the unitary evolution operator up to the first order, raising an intriguing question whether the higher-order Gaussian beams are advantageous in quantum measurement compared to the fundamental Gaussian beam.
In this study, we determine the generalized von Neumann quantum measurement for an arbitrary coupling strength with HG and LG mode pointer states under the assumption that the system observableÂ satisfiesÂ 2 =Î andÂ 2 =Â (projection operator). To clarify the advantages of the higher-order Gaussian beams in practical use, we investigate the signal-to-noise ratio (SNR), which is defined by Here, . f denotes the expectation value of the measuring system operator under the final state of the pointer, andX = |x x| (x is the coupling direction of the von Neumann measurement) andŶ = |y y| (y is the orthogonal coupling direction). To verify our general formulas, we take two special limits. If we use the zero-mean Gaussian pointer as the initial state, we found that our general expectation values reduce to the results given in Refs. [49,52]. On the other hand, if we only consider the evaluation up to the first order, our general expectation values reproduce all results given in Ref. [59]. The remainder of this paper is organized as follows. In Section II, we give the model setup for the generalized von Neumann measurement. In Sections III and IV, we first give the expressions of HG and LG mode pointer states in Fock state representation according to de Lima Bernardo et al. [59]. We then give general forms of the expectation values and discuss the SNRs with HG and LG mode pointer states for systems operatorÂ witĥ A 2 =Î andÂ 2 =Â. In section V, for check the validity of our general results we consider some special initial pointer states and approximated treatments was used in previous works and found that our general formulas can reproduce all the related results were given in those previous works [49,52,59]. We give conclusion and remarks to our paper in final section VI. Throughout this paper, we use the unit = 1.

II. MODEL SETUP
For the generalized von Neumann measurement, the coupling interaction between the system and detector is taken to be the standard von Neumann Hamiltonian: where g is a coupling constant andP x is the conjugate momentum operator to the position operatorX of the measuring device, i.e., [X,P x ] = iÎ. We have taken the interaction to be impulsive at time t = t 0 for simplicity. For this kind of impulsive interaction, the time evolution operator becomes e −igÂ⊗Px . The generalized von Neumann measurement is characterized by the pre-and post-selection of the system state. If we prepare the initial state |ψ i of the system and the pointer state, after some interaction time t 0 , we postselect a system state |ψ f and obtain information about a physical quantityÂ from the pointer wave function by the following weak value: In general, the weak value is a complex number. From Eq.
(3), we know that when the pre-selected state |ψ i and the post-selected state |ψ f are almost orthogonal, the absolute value of the weak value can be arbitrarily large. This feature leads to weak value amplification.
From these definitions, we notice that unitary evolution operator e −igÂ⊗Px for operatorÂ satisfies the prop-ertyÂ 2 =Î as follows: (4) Similarly, for the propertyÂ 2 = A, the evolution operator satisfies Here, we use the position operatorsX andŶ as well as their corresponding momentum operatorsP x andP y , which can be written in terms of the annihilation (creation) operators,â i (â † i ) with i = x, y as [60] Here, σ is the width of the fundamental Gaussian beam. These annihilation (creation) operators obey the commutation relations â i ,â † j = δ ijÎ with i, j = x, y. The parameter s defined by s :≡ g/σ, and D (ξ) is a displacement operator with complex ξ defined by Note that s characterizes the coupling strength. Thus, we can say that the coupling between system and pointer is weak (strong) if s 1(s 1). In the next sections, we consider the generalized von Neumann measurement with HG and LG mode pointer states for arbitrary coupling strength s for the system operatorÂ withÂ 2 =Î andÂ 2 =Â, respectively.

III. GENERALIZED VON NEUMANN MEASUREMENTS WITH HG MODE POINTER STATES
The general HG modes can be generated from the fundamental Gaussian mode, |0, 0 HG , and can be defined as These modes are complete sets of solutions to the paraxial wave equation with rectangular coordinates. Any arbitrary paraxial wave can be described as a superposition of HG modes with the appropriate weighting and the phase factors. Practically, the higher-order HG modes can be simply generated by inserting cross wires into the laser cavity with the wires aligned with the nodal lines of the desired HG mode [61,62]. However, a more convenient way for generating higher-order modes is the use of computer-generated holograms or a spatial light modulator (SLM) [63], which allows reprogrammable waveform generation under the computer control.
In this paper we take the initial state of the HG mode pointer as |φ i = |n, m HG . Note that the HG modes are factorable in functions that depend on x and y directions. In our standard von Neumann measurement Hamiltonian (2), we only have x-direction interaction; thus, we will omit the y-direction quantum number m in the HG mode calculations.
In what follows, we will discuss the generalized von Neumann measurement for the system operatorÂ that satisfy the propertiesÂ 2 =Î andÂ 2 =Â.

A.Â 2 =Î case
After unitary evolution given in Eq. (4), the system state is post-selected to |ψ f . Then, we obtain the following normalized final pointer states: where the normalization coefficient is given by Here, the Laguerre polynomials are defined as The explicit expression of Eq. (9) can be obtained using the displaced Fock states defined by [64,65] Here, the generalized Laguerre polynomials are defined as for integer η. Using Eqs. (9) and (12), we can calculate the general forms of the expectation values of conjugate momentumP x and the position operatorX under the final pointer states |φ f1 , which are given by and respectively. Eqs. (14,15) are the general forms of expectation values for the system operatorÂ satisfyingÂ 2 =Î, and they are valid for an arbitrary value of the coupling parameter s.
To investigate the advantages of the higher-order Gaussian modes in practical use, we check the signalto-noise ratio (SNR) in the following two cases. We consider the two-dimensional quantum (qubit) state. We assume that the operatorÂ to be observed is the spin x-component of a spin-1/2 particle through the von Neumann interaction (2) A =σ Here, |↑ z and |↓ z are eigenstates ofσ z with the corresponding eigenvalues 1 and −1, respectively. We select the pre-and post-selected states as and respectively. Then, we can obtain the weak value by substituting these states into Eq. (3): where θ ∈ [0, π] and φ ∈ [0, 2π). In Fig. 1, the behavior of the SNR is shown as a function of the coupling parameter s and the pre-selection angle θ. When φ = 0, the weak value becomes tan θ 2 . We can see that the SNR decreases as n increases (higherorder mode case). We find a ridge around θ = π/2, which is a result of strong measurement; when θ = π/2, the preselection state is the eigenstate of the operatorσ x with the corresponding eigenvalue +1. From Fig. 1, we can also identify a bridge between the weak coupling regime (s 1) and the strong coupling regime (s 1). We also check the SNR with some specific weak values, and the numerical results are given in Fig. 2. As shown in Figs. 1 and 2, the higher-order HG modes have no practical advantages for improving the SNR. We also notice that the imaginary part of the weak value has no role in improving the SNR in the x-direction.
Following the process used forÂ 2 =Î case, we can obtain the normalized final pointer states after the unitary evolution given in Eq. (5). The post-selection to |ψ f is given as follows: where γ is the normalization coefficient given by (21) Thus, using Eqs. (12) and (20), we can calculate the general forms of expectation values of the conjugate mo-mentumP x and the position operatorX under the final pointer sates |φ f2 . The obtained results are given by respectively. In these calculations, we use the following properties of the displaced Fock states [66]: We know that the operatorÂ satisfying the propertŷ A 2 =Â can be a projection operatorÂ = |C C| that can also be taken asÂ = (Î ±B)/2 withB 2 =Î. This type of operator has numerous applications in the weak measurement theory such as the three-box paradox problem [25], and quantum tomography [33,34]. Here, we simply verify the SNR for this case with some specific weak values, and the numerical results are shown in Fig.  3. As indicated in Fig. 3, the higher-order HG modes have no practical advantages for improving the SNR for the operatorÂ satisfying the propertyÂ 2 =Â, and the imaginary part of the weak values has no role in increasing the SNR as mentioned above.

IV. GENERALIZED VON NEUMANN MEASUREMENTS WITH LG MODE POINTER STATES
The general LG modes can be defined as where ν and µ are integers. Here, the indices α = (µ + ν)/2 and β = (µ − ν)/2 are related to the usual radial p and azimuthal l indices by the relations p = min (α, β) and l = |α − β|, respectively. We let |0, 0 HG denote the HG mode fundamental Gaussian state. If we use the binomial formula for Eq. (25), we can find the more explicit form of LG mode pointer states as a sum of HG modes: (26) Here, we note C α,j;β,k is given by In this paper, we take the initial state of the LG mode pointer as |ϕ i = |µ, ν LG . The LG modes are a complete set of solutions to the paraxial wave equation with cylindrical coordinates characterized by a radial and azimuthal indexes p and l [55]. Physically, the LG modes have been created using various experimental setups such as spatial light modulators [67] and reflection from a conical mirror [68]. Furthermore, the LG modes have a zero-intensity point at the center, which is called the optical vortex. The relationship between the optical vortex and the weak value has been investigated from different viewpoints [12,58,[69][70][71][72]. Thus, a general treatment of the generalized von Neumann measurements with LG mode pointer states will provide an efficient method for further exploration of weak value applications in higher-order optical beams and optical vortices. Next, we will give the explicit treatment of the generalized von Neumann measurements with LG mode pointer state for the system operatorÂ that satisfies the propertiesÂ 2 =Î andÂ 2 =Â.
A.Â 2 =Î case Using the same process as that used in the HG mode cases, after the unitary evolution given in Eq. (4) and the post-selection of the system to |ψ f , we can obtain the normalized final pointer states as where the normalization coefficient is given by Using Eq. (28) and the displaced Fock states, i.e., Eq. (12), we can obtain the expectation value of the position operatorX under the final pointer states |ϕ f1 as Likewise, the expectation value of the momentum oper-atorP x under the final pointer states |ϕ f1 is given by From the definitions of the HG and LG modes in the Fock state representation, i.e., Eqs. (8) and (26), we can see that, contrary to the HG modes, the LG modes are not factorable into functions depending only on x and y. This feature of the LG mode causes the coupling of the system observableÂ with the x-and y-dimension of the pointer. Thus, the pointer also shift values in the y-direction. The pointer value is given by {iC α,j;β,k C * α,j ;β,k }δ k +j ,k+j+1 These expectation values are the general forms of the desired values in generalized von Neumann measurements with LG pointer states for the system operatorÂ satis-fying the propertyÂ 2 =Î. For the weak value (19) with fixed value φ = 0, the SNR is determined to be a function of the coupling parameter s and the pre-selection angle θ for lower radial and azimuthal indices p and l, as shown in Fig. 4 (Here, we only plot figures for p = 0, 1, 2 and the corresponding l = 0, 1, 2 cases). Furthermore, by selecting specific weak values, we plot the SNR as a function of the coupling parameter s, as shown in Fig. 5. From Figs. 4 and 5, we can see that the higher-order LG modes have no advantages for improving the SNR over the fundamental Gaussian mode case (corresponding to the p = 0, l = 0 case). From Fig. 5, we can also see that the imaginary part of the weak value has no role in improving the SNR in the x-direction.
The SNR for the y-direction shift is shown in Figs. 6 and 7. In Fig. 6, we find the SNRs for the lowerorder cases of LG modes, while in Fig.7, we plot the SNR in the y-direction as a function of the coupling parameter s for some specific weak values with fixed radial index p, i.e., p = 0, with increasing azimuthal index values l. We can see that the SNR for the y-direction is related to the azimuthal indices l, while the SNR de-creases as the radial indices p are increased (See Fig. 6). Thus, when l = 0, there is no information about the ydirection. We should emphasize that while the maximum of the y-direction shift is very small compared to that of the x-direction, in the weak measurement regime, the y-direction shift is sufficiently large compared to the xdirection shift. From Fig. 7, we can also see that the real part of the weak value has no role in improving the SNR in the y-direction. Because there is no direct interaction between the pointer and the measured system along the y-direction, the strong measurement regime (s 1) includes only the x-direction shift. In the weak measurement regime, however, the pointer state can be shifted along not only the x-direction but also the y-direction because the unfactorability of the LG modes induces ydirection interference in the x-direction interaction.
Using a similar process to that in the previous section, we can find the normalized final state of the LG mode pointer states as follows: for the normalization coefficient The expectation values of the position operatorsX,Ŷ , and the momentum operatorP x under the final state |ϕ f2 are given by and respectively.
For LG mode pointer states with system operatorÂ satisfying the propertyÂ 2 =Â, we verify the SNR in the x-and y-direction as function of coupling parameter s LG mode pointer states with the operatorÂ that satisfies the propertŷ A 2 =Î is plotted with respect to the coupling parameter s and the pre-selection angle θ with fixed value φ = 0. In these figures, we give the SNRs for the lowest-order LG modes.
with some specific weak values, and the numerical results are given in Fig. 8 and Fig. 9, respectively. For the SNR in x-direction, we reach the same conclusions as before: the higher-order LG modes and imaginary parts of the weak value have no advantages in improving the SNR in the x-direction (See Fig. 8).
As shown in Fig. 9, the SNR in the y-direction was investigated as function of coupling parameter s. Here, we plot the SNR curves with a fixed radial index p, i.e., p = 0, and changing azimuthal indices l. From Fig. 9, we can see that in the weak coupling regime (s 1), there is an improvement of the SNR in y-direction compared with theÂ 2 =Î case shown in Fig. 7. We found that the maximum value of SNR in the weak coupling regime occurred when A w = 0.5 + 5i, as shown in Fig.9(e). The maximum condition of this SNR corresponds to the minimum condition for Eq. (34). Furthermore, from Fig. 9, we also can see that when the azimuthal indices l increase, the SNR in y-direction also increases for fixed radial indices p. When the coupling between the system (x-direction) and the pointer devices is sufficiently strong, the SNR in y-direction gradually vanishes. From Fig. 9, we can further deduce that the real part of the weak value has no role in improving the SNR in y-direction.
If we take the fundamental Gaussian beam as the initial pointer state (this corresponds to taking m = n = 0 and α = β = 0 in Eq. (8) and Eq. (25), respectively), the general expectation values for position operatorX, i.e., Eqs. (14) and (30), and momentum operatorP , i.e., Eqs. (15) and (31), are reduced to LG mode pointer states with the operatorÂ satisfying the propertyÂ 2 =Î is plotted with respect to the coupling parameter for some specific weak values: (a) A w = 0.5, (b) and respectively, where These results were also given in Ref. [52]. Furthermore, under the weak coupling regime (s 1), if we only consider the evolution up to the first order, our general expectation values reproduce the results given in Ref. [59]. In this case, the HG and LG pointer states are shifted along x−direction with the same value, i.e., The expectation value in the y-direction, i.e., Eq.(32), is reduced to The expectation value of momentum operator for the HG mode pointer states, i.e., Eq. (15), is reduced to while for the LG mode pointer states, Eq. (31) is reduced to LG f1,f irst = g A w 2σ 2 (2p + |l| + 1) .
The validity conditions for Eqs. (41)- (44) are for the HG mode pointer states, and for the LG pointer states.
In the SNR case, the SNRs are directly related to coupling parameter s. Thus, in the strong measurement regime, if we take the limit s → ∞, we notice that the SN R X becomes as a function of the weak value If we take the fundamental Gaussian beam as the initial pointer states, the general expectation values for the position operatorX, i.e., Eqs. (22) and (35), and momentum operatorP x , i.e., Eqs. (23) and (37), are reduced to and respectively, where Furthermore, under the weak coupling condition (s 1), if we only consider the evolution up to the first order, our general expectation values are reduced to the below form: In the position operatorX case, the HG mode and LG mode have the same value. The expectation value in the y-direction, Eq. (36), is reduced to For the momentum operatorP x , the expectation values for the HG mode pointer states, Eq. (23), and for the LG mode pointer states, Eq. (37), reduce to Figure 6. (Color online) The SNR in the y-direction for LG mode pointer states with the operatorÂ satisfying the propertŷ A 2 =Î is plotted with respect to the coupling parameter. We take φ = π 2 in Eq. (19). and P x LG respectively. The validity conditions for Eqs. (51)-(54) are for the HG mode pointer states and for the LG mode pointer states. For the SNR in the strong coupling regime (s 1), if we consider the limit case of s → ∞, we notice the SN R X becomes a function of the weak value We can observe this limiting trend from Figs. 3 and 8.
We have to emphasize that the limit values given in Eqs. (47) and (57) are valid in the x-direction SNR for the HG and LG mode pointer states in corresponding lower-order modes.

VI. CONCLUSION AND REMARKS
In summary, we studied the generalized von Neumann measurement with the HG and LG mode pointer states for the system operatorÂ withÂ 2 =Î andÂ 2 =Â. Our general expectation formulas are not only valid in the weak measurement regime but also in the strong measurement regime. If we only consider the evaluation up to the first order, our general results reproduce all results given in Ref. [59]. Moreover, if we let the initial pointer state be a fundamental Gaussian state, our general results show the full evaluation values given in Ref. [52].
To clarify the advantages of high-order Gaussian beams in practical use, we verified the SNR and found that the higher-order HG and LG modes have no advantages for improving the SNR process over that of the fundamental Gaussian mode case. Moreover, we found that the imaginary part of the weak values has no role in improving the SNR in the x-direction in the HG and LG mode pointer state cases. For the SNR in y-direction with the LG mode case, we also found that the SNR is related to the azimuthal indices l, and the real part of the weak value has no role in improving the SNR in the y-direction. However, in theÂ 2 =Î case, the SNR in the y-direction has an upper bound even for increasing azimuthal indices l. In theÂ 2 =Â case, we find SNR improvement in the y-direction in the weak coupling regime because the SNR increases with increasing azimuthal indices l . However, we found that the SNR in the y-direction gradually vanishes when the coupling strength between the system (x-direction) and pointer devices is increased.
These methods can provide a new technique for calculating the expectation values of the momentum and the position operator generation functions. Thus, our results are useful for investigating applications of the weak measurement theory in quantum dynamics and quantum correlations with higher-order optical beams.
We expect that our general treatment of the weak values will be helpful for understanding the connection between weak and strong measurement regimes and may be used to propose new experimental setups using higherorder Gaussian beams to further investigate the applications of weak measurement in optical systems such as the optical vortex. In this work, we only consider the pure higher-order HG and LG modes as initial pointer states and investigate the corresponding SNRs. However, en-tanglement of the initial pointer states is useful for weak value amplification [73]. Thus, our setup may provide another scheme for improving the SNR if we consider the initial state of the pointer as a coherent superposition state of higher-order Gaussian beams.