JP2013108806A - Signal amplifier utilizing weak measurement - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an amplifier capable of amplifying a feeble signal.SOLUTION: The amplifier, when measuring an observation amount ^A of a measured system, makes a measuring instrument and the measured system whose initial state is previously selected interact with each other, performs post-selection of an end state of the measured system and measures a physical amount by using a wave function ξ(p) of the measuring instrument. When it is defined that respective vectors of a pre-selection state and a post-selection state of a quantum state of the measured system are |i>,|f>, a weak value Aof an equation (1) A=<f|^A|i>/<f|i>is known and interaction Hamiltonian H is given by an equation (2) H=g*δ(t-t)*^A*^p by using a momentum operator ^p and a coupling coefficient g (provided that g>0) of the measuring instrument, and when ^A=1 is formed, the wave function ξ(p) of the measuring instrument substantially becomes an equation (3) in moment indication.

Description

  The present invention relates to an amplifying apparatus that amplifies a weak signal using weak measurement.

  As a method for measuring a weak physical quantity, “Weak Measurement” has been proposed (Non-Patent Document 1). In “weak measurement”, the measured system is weakly coupled to the measuring device. When measuring the observable amount ^ A of a system under test, in weak measurement, first, the measurement system and the system under test whose initial state (Pre-selection) is pre-selected are interacted. . Next, the final state of the system to be measured is post-selected, and the physical quantity is measured using the wave function of the measuring instrument.

Preselection state of the quantum state of the system under measurement | i>, the post-selection state | when the f>, the amount A _W given by formula (1) is called the Yowachi.
A _W : = <f | ^ A | i> / <f | i>
This can be obtained as the difference between before and after the weak measurement of the center value of the wave function of the measuring instrument. The Yowachi A _W is | i> and | f> and increases when substantially orthogonal, weak signal obtained by the measurement is amplified.

Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988) O. Hosten, P. Kwiat, Science 319, 787 (2008) N. W. M. Ritchie, J. G. Story, R. G. Hulet, Phys. Rev. Lett. 66, 1107 (1991) P. B. Dixon, D. J. Starling, A. N. Jordan, J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009) S. Wu, Y. Li, Phys. Rev. A 83, 052106 (2011) K. Nakamura, A. Nishizawa, M. Fujimoto, arXiv: 1108.2114v2 [quant-ph] X. Zhu, Y. Zhang, S. Pang, C. Qiao, Q. Liu, S. Wu, arXiv: 1108.1608v2 [quant-ph] T. Koike, S. Tanaka, arXiv: 1108.2050v1 [quant-ph] A. M. Steinberg, Nature (London) 463, 890-891 (2010) A. Feizpour, X. Xing, A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011) D. J. Starling, P. B. Dixon, A. N. Jordan, J. C. Howell, Phys. Rev. A 80, 041803 (2009) Aya Furuta “Existence probability minus 1”, Nikkei Science, October 2009, pages 22-23

  The effect of weak measurement has been confirmed, for example, in experiments of Non-Patent Documents 2 to 4. Specifically, an amplifying apparatus using weak measurement can amplify a slight inclination of a mirror or a slide glass into a very large change in the position of a laser beam. The signal amplification degree due to the weak measurement can be increased as much as the degree of orthogonality of | i> and | f>, and it has been considered that there is no limit to amplification.

However, it is pointed out that this evaluation is not accurate because it uses a first-order approximation assuming that the interaction is small (Non-Patent Document 5). Further, it is shown that there is an upper limit in signal amplification in a case where the observation amount satisfies ^ A ² = 1 and the wave function of the measuring device is assumed to be Gaussian (Non-Patent Documents 6 to 8).

  SUMMARY An advantage of some aspects of the invention is that it provides an amplifying apparatus that can optimally amplify a weak signal.

One embodiment of the present invention relates to a signal amplifying apparatus using weak measurement. This amplifying apparatus includes a measuring instrument and a system to be measured. When measuring the observed quantity ^ A of the measured system, the measuring instrument and the measured system whose initial state is preselected are interacted. Next, the final state of the system to be measured is selected afterwards, and the physical quantity is measured using the wave function ξ (p) of the measuring instrument. When the ket vector in the preselected state of the quantum state of the system under measurement is | i> and the ket vector in the postselected state is | f>, the weak value A _W given by equation (1) is known,
A _W = <f | ^ A | i> / <f | i> (1)
The interaction Hamiltonian H between the measuring device and the system to be measured is given by equation (2) using the momentum operator ^ p and the coupling coefficient g (where g> 0) of the measuring device,
H = g · δ (t−t ₀ ) · ^ A · ^ p (2)
When ^ A ² = 1 holds, the measuring instrument is configured such that its wave function ξ (p) is substantially expressed by equation (3) in terms of momentum. In reality, it may be difficult to completely match the wave function with Equation (3) due to mechanical, physical, electrical, and optical constraints. “To be” means that a certain amount of error is allowed in this case. Even if there is such an error, the amplification degree can be greatly improved as compared with the case where no consideration is given to the wave function.

In one embodiment, the system under measurement may include an interferometer and a light source that makes a light beam incident on the interferometer. The signal to be measured may be a slight inclination of an optical element provided in the interferometer. The amplifying device may amplify the tilt to a lateral displacement of the light beam output from the interferometer in response to the post selection. In the light source, the wave function of the light beam is substantially expressed by Equation (3), and the spatial coordinate display of the complex amplitude of the cross section of the light beam is obtained by inverse Fourier transform of Equation (3). It may be generated.
The inclined “optical element” may be a mirror constituting an interferometer, or may be an optical element such as a slide glass inserted in the interferometer.

  Note that any combination of the above-described components, or a conversion of the expression of the present invention between methods, apparatuses, and the like is also effective as an aspect of the present invention.

  According to the amplification device of the present invention, a weak signal can be amplified with a large gain.

It is a figure which shows the amplifier which concerns on a 1st structural example. It is a figure which shows the amplifier which concerns on a 2nd structural example. 3A to 3D are diagrams showing the real part, the imaginary part, the square of the absolute value, and the phase G (p) of the wave function ξ (p) of momentum display. 4A to 4D are diagrams showing a real part, an imaginary part, a square of an absolute value, and a phase of a wave function ξ (q) of coordinate display.

  The present invention will be described below based on preferred embodiments with reference to the drawings. The same or equivalent components, members, and processes shown in the drawings are denoted by the same reference numerals, and repeated descriptions are omitted as appropriate. The embodiments do not limit the invention but are exemplifications, and all features and combinations thereof described in the embodiments are not necessarily essential to the invention.

(principle)
In the present embodiment, a signal amplifying apparatus using weak measurement will be described. In general, an amplifying apparatus includes a measuring instrument and a system to be measured. When measuring the observed amount ^ A of the system under measurement in this amplifying apparatus, first, the measuring instrument and the system under measurement whose initial state is preselected are interacted. Next, the final state of the system to be measured is selected afterwards, and the physical quantity is measured using the wave function ξ (p) of the measuring instrument.

The packet vector preselected state of the quantum state of the system under measurement | i>, the packet vector of post-selection state | when the f>, Yowachi A _W given by formula (1) is known.
A _W = <f | ^ A | i> / <f | i> (1)
The interaction Hamiltonian H between the measuring instrument and the system to be measured is given by Equation (2) using the momentum operator ^ p and the coupling coefficient g (where g> 0) of the measuring instrument.
H = g · δ (t−t ₀ ) · ^ A · ^ p (2)
Assume that ^ A ² = 1 holds.

Ｒは実部を表しており、ＲＡ_Ｗは、弱値Ａ_Ｗの実部を表す。 In the conventional amplifying apparatus using weak measurement, no particular consideration is given to the wave function ξ (p), and the Gaussian type wave function ξ (p) that can be used most simply is used. On the other hand, in the amplifying apparatus according to the present embodiment, the wave function ξ (p) given by Expression (3) is used.

R represents the real part, RA _W represents a real part of Yowachi _{A W.}

As will be described in detail later, the wave function ξ (p) of the equation (3) is obtained by using the position operator <^ q> _before of the weak measurement and the position operator <^ of the measurement apparatus after the weak measurement. _{q> after} the difference, in other words the shift amount _<^ q> after _- give the maximum value to the _{<^ q> before.} That is, the wave function ξ to (p), not just Gaussian, by optimizing a function in response to Yowachi A _W, it is possible to maximize the gain.

When the wave function ξ (p) of the equation (3) is used, the expected value <^ q> _after of the position operator of the measuring device is given by the following equation (4).

That is, when the pre-selection and post-selection states are orthogonal, R (A _W ) becomes infinite and the right side of the above equation diverges, indicating that there is no upper limit for amplification.

  For example, in an amplifier that uses an interferometer and laser light to amplify the tilt of an optical element provided in the interferometer into a lateral displacement of the laser beam, the expected value <^ q> of the position operator of the measuring instrument Corresponds to an expected value of the displacement amount of the laser beam.

(Specific example of amplification device)
Prior to the derivation of the wave function, a specific amplifying apparatus 1 will be described in order to help understanding. FIG. 1 is a diagram illustrating an amplifying apparatus 1 according to a first configuration example. The amplifying apparatus 1 includes a laser light source 10, an interferometer 12, a light receiving device 14, and a slide glass SG.

  The laser light source 10 outputs laser light having a certain wave function. As will be described later, the laser light source 10 is configured to be capable of adjusting the momentum or beam profile of the laser light.

  Waveform-shaped laser light is input to a double type Mach-Zehnder interferometer (hereinafter simply referred to as an interferometer) 12. The interferometer 12 includes a first mirror M1 to a fourth mirror M4, and a first beam splitter BS1 to a third beam splitter BS3.

  Laser light from the laser light source 10 is input to the first beam splitter BS1. The beam that has passed through the first beam splitter BS1 is reflected by the first mirror M1. The beam reflected by the first beam splitter BS1 is reflected by the second mirror M2. A slide glass SG is provided on the path of the beam reflected by the second mirror M2. In the amplifying apparatus 1 of FIG. 1, the inclination φ of the slide glass SG is a minute signal to be amplified.

  The beams reflected by the first mirror M1 and the second mirror M2 are incident on the second beam splitter BS2. The beam that has passed through the second beam splitter BS2 in the first direction is reflected by the third mirror M3. The beam that has passed through the second beam splitter BS2 in the second direction is reflected by the fourth mirror M4. The beams reflected by the third mirror M3 and the fourth mirror M4 are input to the third beam splitter BS3.

  The beam that has passed through the third beam splitter BS3 in the first direction is input to the light receiving device. The beam that has passed through the third beam splitter BS3 in the second direction is discarded.

  All optical elements except the second beam splitter BS2 and the slide glass SG are arranged to receive the laser beam at an incident angle of 45 °. The inclination of the second beam splitter BS2 is θ.

  The light receiving device 14 is an imaging device such as a CCD (Charge Coupled Device) camera, and measures the intensity distribution in the lateral direction of the incident laser beam. The intensity distribution of the laser beam incident on the light receiving device 14 varies depending on the inclination φ of the slide glass SG. The amplifying apparatus 1 converts and amplifies the inclination φ of the slide glass SG into a lateral displacement amount of the laser beam.

In the amplifying apparatus 1 of FIG. 1, the incidence of a laser beam on the interferometer 12 corresponds to the interaction between the measuring device and the system under measurement whose initial state is preselected.
Further, measuring one laser beam output from the third beam splitter BS3 by the interferometer 12 and discarding the other beam corresponds to the subsequent selection of the final state of the system under measurement. The measurement by the light receiving device 14 is nothing but measuring the expected value <^ q> of the position operator.

In the amplifying apparatus 1, each state of the laser beam is denoted as A to D.
The state | i> of the system under test whose initial state is preselected is given by the following equation.
| I> = | A> = 1 / √2 × (| B> + | C>)
| B> = cos θ · | E> + sin θ · | F>
| C> = − sin θ · | E> + cos θ · | F>
Further, the state | f> of the system under measurement whose final state is selected afterwards is given by the following equation.
| F> = | D> = 1 / √2 × (| E> − | F>)
= 1 / √2 × [(cos θ−sin θ) | B> − (cos θ−sin θ) | C>]

In the amplifying apparatus 1 in FIG. 1, the observation amount ^ K is C × C (A = 2K−1). Therefore, the weak value K _W of the observable C × C is
<F | ^ K | i> = <f | C × C | i> = − (sin θ + cos θ) / 2
<F | i> = − sin θ
Is given by the following equation.
K _W = (sin θ + cos θ) / (2 sin θ)
= 1 / √2 × sin (θ + π / 4) / sinθ

In other words, the amplification device 1 of FIG. 1 has a Yowachi K _W corresponding to the inclination θ of the second beam splitter BS2. The weak value K _W is 1 when θ = 3π / 4, and diverges infinitely as θ approaches zero.

  FIG. 2 is a diagram illustrating an amplifying apparatus 1a according to the second configuration example. As in FIG. 1, the amplifying apparatus 1a includes a laser light source 10, an interferometer 12, a light receiving device 14, and an optical element (slide glass SG).

  The laser light source 10 outputs laser light having a certain wave function. This laser beam includes an H polarization component and a V polarization component. It is input to a single type Mach-Zehnder interferometer (hereinafter simply referred to as an interferometer) 12a. The interferometer 12a includes a first mirror M1, a second mirror M2, a first polarizing beam splitter PBS1, a second polarizing beam splitter PBS2, a polarizing plate PL, and a slide glass SG.

  Laser light from the laser light source 10 is input to the first polarization beam splitter PBS1. The H polarization component passes through the first polarization beam splitter PBS1 and is subsequently reflected by the first mirror M1. The V-polarized component is reflected by the first polarizing beam splitter PBS1 and subsequently reflected by the second mirror M2.

  A slide glass SG is provided on the path of the beam reflected by the second mirror M2. Also in the amplifying apparatus 1a of FIG. 2, the inclination φ of the slide glass SG becomes a minute signal to be amplified.

  The beams reflected by the first mirror M1 and the second mirror M2 are incident on the second polarizing beam splitter PBS2. The beam that has passed through the second polarizing beam splitter PBS2 in the first direction passes through the polarizing plate PL and enters the light receiving device. The linearly polarizing plate PL passes a polarized component inclined by (θ + π / 4) with respect to the horizontal direction. Conversely, the beam that has passed through the second polarizing beam splitter PBS2 in the second direction is discarded.

  The light receiving device 14 measures the intensity distribution in the lateral direction of the incident laser beam. The intensity distribution of the laser beam incident on the light receiving device 14 varies depending on the inclination φ of the slide glass SG. The amplifying apparatus 1 converts and amplifies the inclination φ of the slide glass SG into a lateral displacement amount of the laser beam.

Also in the amplifying apparatus 1a of FIG. 2, the incidence of the laser beam on the interferometer 12a corresponds to the interaction between the measuring instrument and the system under test whose initial state is preselected.
Further, measuring one laser beam output from the second polarization beam splitter PBS2 by the interferometer 12a and discarding the other beam corresponds to post-selection of the final state of the system under measurement. The measurement by the light receiving device 14 is nothing but measuring the expected value <^ q> of the position operator.

Weak value K _W of the amplifier 1a can be derived as follows.
The initial state | ψ _int > and the final state | ψ _post > are respectively defined as follows.
| Ψ _int >: = 1 / √2 × (| H> + | V>)
| Ψ _post >: = cos (θ + π / 4) | H> + sin (θ + π / 4) | V>
However,
| H> = (1, 0) ^T ,
| V> = (0,1) ^T ,
T represents a transposed matrix.

By passing through the polarization beam splitters PBS1 and PBS2, the preselected state | i> and the postselected state | f> are given as follows.
| I> = 1 / √2 × (| B> | H> + | C> | V>)
| F> = cos (θ + π / 4) | B> | H> + sin (θ + π / 4) | C> | V>

Also in the amplifying apparatus 1a of FIG. 2, the observation amount ^ K is C × C (A = 2K−1). Therefore, the weak value K _W of the observable C × C is
<F | ^ K | i> = <f | C × C | i> = − 1 / √2 × sin (θ + π / 4)
<F | i> = 1 / √2 × {cos (θ + π / 4) + sin ((θ + π / 4))}
= Sin (θ + π / 4) + cos (θ + π / 4)
Is given by the following equation.
K _W = <f | ^ K | i> / <f | i> = (1 + tan θ) / 2

In other words, the amplification device 1a of Figure 2 has a Yowachi K _W corresponding to the polarization direction of the polarizer PL. The weak value K _W becomes 0 when θ = −π / 4 and diverges as it approaches θ = ± π / 2.

(Derivation of optimal wave function)
From the description so far with reference to FIGS. 1 and 2, it is understood that the amplifying apparatus 1 (1 a) has a weak value K _W corresponding to the configuration. Subsequently, the wave function ξ (p) capable of realizing the maximum amplification factor is derived by the amplification device 1.

Maximizing the amplification factor is nothing but maximizing the shift amount <^ q> _after- <q> _before of the horizontal coordinate of the intensity of the laser beam measured by the light receiving device 14. In the following description, the coordinate axis q and the momentum p are assumed to be parallel to the beam shift direction.

ここで、ξ（ｐ）＝＜ｐ｜φ＞とする。測定器と被測定系の相互作用ハミルトニアンは、測定器の運動量演算子＾ｐおよび結合係数ｇを用いて、上述した式（２）で与えられる。
Ｈ＝ｇ・δ（ｔ−ｔ_０）・＾Ａ・＾ｐ …（再２） 1. Lagrange Multiplier Method When the quantum state of the measuring instrument before the weak measurement is written as | φ>, the expected value <^ q> _before of the measuring operator before the weak measurement is given by the equation (5).

Here, ξ (p) = <p | φ>. The interaction Hamiltonian between the measuring instrument and the system to be measured is given by the above equation (2) using the momentum operator ^ p and the coupling coefficient g of the measuring instrument.
H = g · δ (t−t ₀ ) · ^ A · ^ p (2)

After the interaction, the quantum state of the measuring instrument after the weak measurement is obtained by the following equation (6) through the post-selection of the system to be measured.

The condition of ^ A ² = 1 is used in the expression modification in the second row. As a result, Equations (7) to (11) are obtained as expected values of the position operator of the measuring instrument after the weak measurement.

1.1 Variational Problem For Lagrangian L as follows, ξ (p) when <^ q> _after takes an extreme value is obtained. Here, for simplicity, <^ q> _before = 0. After the optimum wave function ξ (p) of the measuring device is obtained, adjustment is performed so that <^ q> _before = 0.

  Here, λ is an undetermined multiplier, and the constraint condition is a normalization condition of ξ (p). First, when the variation is performed by the undetermined multiplier λ, Expression (13) is obtained. Next, when ξ * (p) is changed, Equation (14) is obtained. When this is indefinitely integrated with respect to p, equations (15) and (16) are obtained. From these, equation (17) is obtained. C is a real number normalization constant.

1.2 Determination of the undetermined multiplier Z, C, and C ₀ are calculated from Equation (17). Z is obtained from equation (8) as equation (18), and C is obtained from equation (13) as equation (19). Also, C ₀ can be obtained from the equation (10 to the equation (20). When D is calculated using these, the value of D becomes indefinite but λ = 0 is derived.

2. Signal amplification 2.1 Expected value shift Expected value <^ q> _{before before} weak measurement is calculated. Equation (21) is obtained from Equation (4).

Here, as an integration interval in which the first item is 0 and C = 0, since F (p) is a periodic function of π / g, [−π / 2g, π / 2g] for one period. Is adopted. At this time, F (π / 2g) = F (−π / 2g) = | A _W | ² , and the first item is 0. Further, C is calculated as an integral of F-1 (p) from Equation (19) to become Equation (22), and it is confirmed that C ≠ 0.

In order to satisfy this, a condition R (A _W ) ≠ 0 is imposed that the real part of A _W is not zero. Therefore, the expected value shift before and after the weak measurement is obtained as equation (23) by calculating the integral of F ⁻² (p) from equation (21).

2.2 Expected value of position operator before and after weak measurement Adjust so that <^ q> _before = 0 in the measurement using ξ (p). Now, from the periodicity F (p + π / g) = F (p), the equation (24) is obtained from the equation (17). k is an arbitrary real number. Now, G (p) is given by equation (25) from equation (16).

Taking the logarithm of equation (24), the periodicity of the wave function is
G (p + π / g) = πk + G (p) (26)
It is expressed. Thus, Expression (27) is obtained.
<^ Q> _after = g (± 1-k) (27)
The sign of <^ q> _after is positive when R (A _W )> 0, and negative when R (A _W ) <0. At this time, <^ q> _before is expressed by equation (28) from equation (21). If the arbitrary constant k is selected from the equation (29), <^ q> _before = 0.

Therefore, Expression (27) can be rewritten to Expression (30).

Equation (30) shows that the shift amount diverges infinitely when the limit where | i> and | f> are orthogonal, that is, when | A _W | ² | is infinite. This indicates that there is no theoretical upper limit for signal amplification by weak measurement.

In addition, since Equation (31) holds, there is a lower limit to the shift. The minimum value is g, which is when R (A _W ) = 1 and I (A _W ) = 0. In this case, from the equation (1), ^ A is projected and measured on the system to be measured. At this time, the unitary transformation e ^{± ig · ^ p} shifts the coordinates of the measuring instrument by ± g. That is, if measurement is performed using the optimal wave function ξ (p) of the measuring instrument, the signal can always be amplified as compared with the case where the signal is detected by projection measurement. Also, when calculating the expected value of the square of the position operator after weak measurement,
<^ Q ² > _after = (<^ q> _after ) ² (32)
It can be seen that it is. Therefore, since the variance is 0, it can be seen that the wave function of the measuring instrument after the weak measurement behaves like a delta function in coordinate display.

3. Optimal Wave Function Up to now, ξ (p) is written as Expression (33) from Expressions (11), (15), (22), (25), and (30).

Here, if α is defined by equation (34), equations (35)-(37) are established.

Using these, the wave function of equation (33) is rewritten into equation (38), which is consistent with equation (3).

Finally, a specific example of the wave function ξ (p) is shown. 3A to 3D are diagrams showing the real part, the imaginary part, the square of the absolute value, and the phase G (p) of the wave function ξ (p) of momentum display. The calculation is performed based on the weak value A _W = 1.7 + 3.2i and g = 0.1. 4A to 4D are diagrams showing a real part, an imaginary part, a square of an absolute value, and a phase of a wave function ξ (q) of coordinate display. The wave function ξ (q) of coordinate display can be obtained by performing inverse Fourier transform on the wave function ξ (p) of momentum display. The square of the absolute value of the wave function ξ (q) in the coordinate display corresponds to a cross-sectional profile in the lateral direction (shift direction) of the beam intensity.

  The laser light source 10 can be designed according to the wave function ξ (p) of momentum display. It is difficult to directly measure whether an actually generated beam satisfies the wave function ξ (p) of momentum display. Therefore, by measuring the axial profile of the beam intensity and comparing it with the square of the absolute value of the wave function ξ (q) of the coordinate display, it can be confirmed whether an optimum beam is generated.

Laser beam shaping technology has recently become standard at the laboratory level. Thus, the wave function of the measuring instrument xi] (p) or xi] (q), you are sufficiently possible to the optimum profile depending on Yowachi A _W. Therefore, the present invention is expected to be widely applied to the amplification of weak signals in general.

  As an example, the amplification device 1 is expected to be used for measuring extremely weak signals such as gravity waves.

  The present invention has been described based on the embodiments. This embodiment is an exemplification, and it will be understood by those skilled in the art that various modifications can be made to combinations of the respective constituent elements and processing processes, and such modifications are within the scope of the present invention. is there. Hereinafter, such modifications will be described.

The configuration of the amplifying apparatus 1 is not limited to that exemplified in the embodiment, and the configurations described in the enumerated non-patent literature may be used, or other configurations may be used, that is, weak values. a _W may be adopted any as long as theoretically predictable system under measurement.

  Although the present invention has been described using specific terms based on the embodiments, the embodiments only illustrate the principles and applications of the present invention, and the embodiments are defined in the claims. Many variations and modifications of the arrangement are permitted without departing from the spirit of the present invention.

DESCRIPTION OF SYMBOLS 1 ... Amplifying apparatus, 10 ... Laser light source, 12 ... Interferometer, 14 ... Light receiving device, PBS1 ... 1st polarizing beam splitter, PBS2 ... 2nd polarizing beam splitter, M1 ... 1st mirror, M2 ... 2nd mirror, M3 ... Third mirror, M4, fourth mirror, BS1, first beam splitter, BS2, second beam splitter, BS3, third beam splitter, SG, slide glass, PL, polarizing plate.

Claims (2)

A signal amplifying device using weak measurement,
A measuring instrument;
A system to be measured;
With
When measuring the observed quantity ^ A of the measured system, the measuring instrument and the measured system whose initial state is preselected are allowed to interact, and then the final state of the measured system is post-selected, It is configured to measure a physical quantity using the wave function ξ (p) of the measuring device,
When the ket vector in the preselected state of the quantum state of the measured system is | i> and the ket vector in the postselected state is | f>, the weak value A _W given by the equation (1) is known,
A _W = <f | ^ A | i> / <f | i> (1)
The interaction Hamiltonian H between the measuring device and the system under test is given by equation (2) using the momentum operator ^ p and the coupling coefficient g (where g> 0) of the measuring device,
H = g · δ (t−t ₀ ) · ^ A · ^ p (2)
And when ^ A ² = 1 holds, the measuring device is configured such that its wave function ξ (p) is substantially expressed by the equation (3) in momentum display. .

The system under measurement has an interferometer and a light source that makes a light beam incident on the interferometer,
The signal to be measured is a slight inclination of an optical element provided in the interferometer,
The amplifying apparatus amplifies the inclination to a lateral displacement of a light beam output from the interferometer in response to a subsequent selection,
In the light source, the wave function of the light beam is substantially expressed by Equation (3), and the spatial coordinate display of the complex amplitude of the cross section of the light beam is obtained by inverse Fourier transform of Equation (3). The amplification device according to claim 1, wherein the light beam is generated.

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