The concept of weak measurement was first proposed by Aharonov, Albert, and Vaidman (AAV),^[1] which involves weak coupling of the system and the meter. The weak values, which could be complex and exceed the eigenvalue spectrum of observable, have been demonstrated to have important applications in exploring fundamental problems of quantum physics in Refs. [2–10]. As an important extension, the weak value amplification technique acts an essential role in precision metrology^[11–13] and has been exploited to amplify minute physical quantities.^[14–23,23] Recently it was also used to realize highly sensitive optical shifts in position or time domains by weak measurement with a small weak value.^[24,25]

Among the practical applications of the weak measurement, it is necessary to establish an exact relationship, which was often expressed by the standard weak value formalism proposed by AAV, among the pointer shift, weak value and the coupling constant. For the standard weak value formalism (AAV formalism), which is the first order approximation in coupling strength, Jozsa gave a physical interpretation of the weak values in terms of the pointer shift.^[26] The mean position shift of the pointer is divided two parts: one is proportional to the real part of the weak value and the other is proportional to the pointer spreading rate in space. Obviously, the influence of the latter is less intuitive than that of the former. It was also used to describe numerous physical phenomenons, which has resembled the optical beam propagation enhancement.^[14,27] The AAV formalism related to the imaginary part of weak value was also expressed as a shift induced by varying wave function phase^[28] or by covariance between the meter’s input and output variables.^[12,29]

Furthermore, some other investigations beyond the AAV formalism were discussed and analyzed, such as higher-order expansions for nearly orthogonal pre- and post-selected states,^[30,31] nonperturbative pre- and post-selected measurement,^[29] weak measurement with the orbital angular momentum state,^[32–34] the fast oscillations of large pointer shifts^[35,36] and the amplification limitations of pointer shifts.^[37–40]

In fact, the pointer shift is related not only to the weak value of system and the coupling constant, but to the meter’s characteristics. The previous discussions were mainly concentrated on uncertainty of meter’s input variable,^[1,30,41] varying phase of wave function^{[12,14,26–29]} and more in Refs. [32–36]. However, in some occasions the pointer shift cannot be simply expressed by the AAV formalism as shown in Refs. [2,3,18,20,23,42–48] for the dependence of the pointer shift on the central frequency. Furthermore, a series of validity conditions of the weak measurement were explored, which showed that the mean value of meter’s input variable has the same influence as the meter’s uncertainty on validity conditions of the weak measurement.^[29] Hence, the effect of the mean value of meter’s input variable cannot be simply neglected because it not only affects the pointer shift but also constrains the validity of weak measurement.

Several treatments in two level systems were presented, such as the redefined system states with a rotation of the polarization of the input field according to frequency,^[2,3,44,46] directly integrating the probability distributions of the final state to calculate the central frequency shifts without using the weak value formalisms.^[45,48,49] Moreover, the inverse of central frequency was regarded as the interaction strength to realize phase estimation.^[43] Unfortunately, a general theoretical solution has not been provided so far.

In this paper, we analyze the role of the mean value of the meter’s input variable in such a process. In order to settle this doubt, we present a modified formulas for the pointer shift, coming from the modified “standard” procedure, and a modified weak value is meanwhile introduced, related to the mean value of meter’s input variable and interaction constant, as there are also other theory about the modular value^[50–52] and general approaches.^[53,54] By the modified formulas, the relationship among the pointer shift, the mean value of the meter’s input variable and the validity conditions of weak measurement is clarified. Such a kind of treatment is helpful for extending the validity of weak measurement, in particular, making it capable to go on exploiting the weak value formalism with the modified weak value to precisely represent the pointer shift when the mean value of the meter’s input variable is large and necessary to be taken into account for the pointer shift. Therefore, it is of great benefit to give guidance of practical experiments and to simplify calculations whenever the mean value of input meter variable is nonzero.

The system and meter are initially prepared in the states |ψ〉 and |ψ_m〉, respectively. In the Von-Neumann measurement scheme, the interaction between the system and meter is described by the coupling Hamiltonian , where γ is the coupling strength, is the operator corresponding to the measured system observable A, and is the operator corresponding to the meter’s input variable F. The unitary operator is expressed as with the free Hamiltonian of the system and meter being neglected. The total initial state undergoes unitary transformation and becomes joint , through which the information about the parameter to measure is encoded in the meter. After post-selection on the state of the system |ϕ〉, the final state of the meter is without normalization. Through measurement on the meter’s output observable (then called the pointer variable), in which for continuous variables the sum becomes integral, the pointer shift is given by

In general, the pointer shift can be rewritten as^[29,30]

Under the weak validity condition^[29]

In the linear regime,^[29] i.e. the first order approximation in γ,

According to the weak condition (3) and linear regime condition (5), it is found that the average value and uncertainty ΔF have the same influence on the validity condition for the nonlinear formula and linear formula of the pointer shift. The occasions, or corresponding to γ|A_ϕψ|ΔF ≪ 1 or γ|A_w|ΔF ≪ 1 respectively, may sometimes occur, showing the corresponding total validity Eq. (3) or Eq. (5) is violated but the conditions related to ΔF are satisfied. Hence, the normal weak value cannot be used to represent the pointer shift in Eq. (4) or Eq. (6).

In order to settle this doubt, the modified weak value is introduced,

Based on the pointer shift form shown in Eq. (2), we perform the following transformations:^[29]

By the transformations, the initial state and the postselected state of the system become ρ′ and , respectively. The meter’s input variable becomes with its mean value . For the purpose of eliminating the influence of mean value on the validity conditions (3) an (5), the arbitrary number is replaced by with a new modified operator

For deducing the general nonlinear formula (4), three replacements need be made, of which the with , the initial state of system |ψ〉 with and the final state of system |ϕ〉 with . The corresponding modified weak value expressed by Eq. (8) is obtained, depending on the coupling constant γ and the meter’s parameter .

By the modified procedure, the weak validity condition becomes

The linear regime condition becomes

The weak validity condition and the linear regime condition become and , respectively, showing a possibility of going on the using weak value formulas to represent the pointer shift with the modified weak value, particularly when the value makes the validity condition (3) or (5) unsatisfied but the corresponding condition or satisfied respectively.

Moreover, the roles of value in the formulas and the corresponding validity conditions make a change that being as a multiplier factor becomes being as a phase factor. Meanwhile, it reveals a deep truth that the uncertainty of the meter acts more essential roles in the validity of weak measurement.

We take an example to show the superiority of the weak value formalisms with the modified weak value, to represent pointer shift. Here we choose a qubit system with the operator of the system observable , the initial state of the system and postselected state . The modified weak value by Eq. (8) becomes

In the following, three treatments are made to obtain the average frequency shift. First the frequency shift is calculated by direct integral ∫Ψ(ω)(ω – ω₀) d ω / ∫Ψ(ω) d ω, a treatment used as comparison. The exact expression is given by

The third treatment is to represent the pointer shift with the normal weak value. The nonlinear formula (4) is given by

In order to analyze the validity of the latter two treatments, the frequency shift calculated by four formulas are shown in Fig. 1 with the specific parameters σ = 1, τ = 0.01, φ = 0.1. As the dependence of the pointer shifts on the central frequency is shown in Fig. 1, it is not enough to take the linear formula 2τ Δω²ImA_w to represent the pointer shift in more general occasion, showing its necessity of considering the effect of on the pointer shift. Then the blue dashed curve represented by the linear formula (24) is partially, just on the two sides of the curve, fitted well with the exact red curve (Eq. (22)), while the middle part is not. The reason is that the linear validity condition shown in Fig. 2 on the two sides of the curve is no longer satisfied.

Fig. 1.

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Fig. 1. The average frequency shift versus ω0 for the four formulas. Parameters σ = 1, τ = 0.01, φ = 0.1. The red curve represented by direct integral shown in Eq. (22), the green dashed curve represented by nonlinear formula with the modified weak value shown in Eq. (23), the black dot-dashed curve represented by the nonlinear formula with the normal weak value shown in Eq. (25) and the blue curve represented by linear formula shown in Eq. (24).

Fig. 2.

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	Fig. 2. The value for the linear validity condition with the modified value versus ω0. Parameters σ = 1, τ = 0.01, φ = 0.1.

The green dashed curve represented by nonlinear formula, with the modified weak value, shown in Eq. (23) and the black dot-dashed curve represented by the nonlinear formula, with the normal weak value, shown in Eq. (25) are fitted well with the exact red curve represented by direct integral Eq. (22). This is because the nonlinear conditions τΔ ωcos(ω₀τ – φ/2)≤ τΔ ω≪ 1 and for Eqs. (23) and (25), where is the maximum coordinate value of ω₀ in Fig. 1, are satisfied under their given parameters.

For larger values ω₀, the curves for the two kinds of nonlinear formulas are shown in Fig. 3. It is shown that the green dashed curve representing the nonlinear formula (23) is still fitted well with the exact red curve, while the black dashed curve representing the nonlinear formula (25) is not. This is because the nonlinear condition τΔ ωcos(ω₀τ – φ/2)≪ 1 for the formula (23) is still satisfied but the condition τ(Δ ω + ω₀)cos(–φ/2)≪ 1 for the formula (25) is obviously violated.

Fig. 3.

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	Fig. 3. The average frequency shift versus the larger central frequency ω0 for the three formulas.

Moreover, the periodicity of variation of the pointer shift with the value can be interestingly observed, whereas this phenomenon would not occur through the formula (4). This is because the value in the modified formula (15) is indeed a phase factor rather than a multiplier factor.

By the modified “standard” procedure of the weak measurement, the extended validity of weak measurement has been verified in our simulation. Since the frequency shift would be influenced by central frequency, the former nonlinear formula (4) and nonlinear formula (7) with normal weak value cannot be simply used to represent the pointer shift, particularly as the central frequency makes the validity of the two formulas violated. Under this occasion, the two formulas (15) and (18) with the modified weak value works better. Through the above analysis, the simulation is coincident well with the theoretical treatment.

By the modified “standard” procedure for the pointer shift, general nonlinear formulas with the modified weak value related to the mean value of the meter’s input variable and the coupling constant have been presented. The mean value , treated as a multiplier factor under the validity conditions and the former formulas, becomes a phase influence on the modified formulas. The relationship between the mean value of meter’s input variable, validity conditions and the pointer shift has been clarified. Such a kind of treatment helps extend the validity of weak measurement, especially for the occasions with a large value . It is of great benefit in giving guidance of practical experiments and simplifying calculations whenever the mean value of the meter’s input variable is nonzero.

As the importance of the mean value of the meter’s input variable in weak measurements had been shown in the earlier work,^[29] we give a way how to deal with it. For the role of the mean value as a multiplier factor in the validity conditions becomes the one as phase factor, it reveals a deep truth that the uncertainty of the meter’s input variable may play more essential role than the meter’s input variable in the validity of weak measurement.

Our work unveils the deep connection between weak measurement and meter’s input variable, thereby making progress toward understanding and solving the problems when the mean value of meter’s input variable need be taken into account. Furthermore, we may leave a talk about if the system phase can influence the mean value of the meter’s input variable, or if the system phase can influence the meter input variable.