{'fraction_non_alphanumeric': 0.05289761372986952, 'fraction_numerical': 0.024450452568473023, 'mean_word_length': 3.8533941814033086, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Demonstrating Additional Law of Relativistic Velocities based on Squeezed Light

Special relativity is foundation of many branches of modern physics, of which theoretical results are far beyond our daily experience and hard to realized in kinematic experiments. However, its outcomes could be demonstrated by making use of convenient substitute, i.e. squeezed light in present paper. Squeezed light is very important in the field of quantum optics and the corresponding transformation can be regarded as the coherent state of SU(1; 1). In this paper, the connection between the squeezed operator and Lorentz boost is built under certain conditions. Furthermore, the additional law of relativistic velocities and the angle of Wigner rotation are deduced as well.


Squeezed state operator
can be regarded as coset space of SU (1, 1), which is SU (1, 1)/U (1), where β is a complex number, * denotes the conjugate operation and a † and a are creation and annihilation operators respectively that satisfy the canonical commutation relation [a, a † ] = 1. In order to demonstrate this idea clearly, let us review the necessary knowledge about SU (1, 1) [7]. It's generators satisfy the following commutation relations, which are Via choosing another appropriate basis, the generators become Therefore, the commutation relations are transformed to be Furthermore, the definition of Perelomov's SU (1, 1) coherent state is given: Last but not least, the boson realization of SU (1, 1) is supplied, so the generators become operator, may have connection with the Lorentz boost, which will be proved in Sec. III. Moreover, the following paragraph will review the indispensable knowledge about special relativity. Let us consider about (2 + 1)-dimensional spacetime. There are two inertial frames of reference Σ and Σ ′ that are coincident at time t = 0 and the velocity of Σ ′ relative to Σ is v = (v 1 , v 2 ). An event is observed as (x 0 , x 1 , x 2 ) T in Σ and (y 0 , y 1 , y 2 ) T in Σ ′ . The two teams of coordinates is connected by a Lorentz boost L, i.e. (y 0 , y 1 , y 2 ) T = L(x 0 , x 1 , x 2 ) T , where x 0 = ct, T denotes the operation of transposition and L [8] takes the form where To simplify above Eq. (6), let us introduce the auxiliary variables which are cosh ρ = γ, sinh ρ = ηγ, cos ϕ = v 1 /v and sin ϕ = v 2 /v, so that Eq. (6) becomes where the formulae cosh 2 ρ − sinh 2 ρ = 1 and sinh 2 ρ/(1 + cosh ρ) = cosh ρ − 1 are useful. If ϕ = 0 or π/2, readers can check that the common Lorentz boost which always appears in the standard text book about special relativity is reduced.
Further, let us recall additional law of relativistic velocities. Σ, Σ ′ and Σ ′′ are arbitrary three inertial frames of reference that are coincident at the initial time. If the velocity of Σ ′ relative to Σ is u and the velocity of where ⊕ is the additional operation of relativistic velocities which is distinct from the parallelogram law of velocities. Its concrete formula [8] reads Observers at rest relative to Σ (relative to Σ ′ ) agree with observers at rest relative to Σ ′ (relative to Σ ′′ ) that their space time coordinate are linked by a pure Lorentz transformation without rotations. However, if the two successive Lorentz boost are in noncollinear directions, observers at rest relative to Σ agree with observers at rest relative to Σ ′′ that their coordinate are linked not only by a Lorentz transformation but also by a rotation, which is called Wigner rotation.

III. CONNECTING SQUEEZED TRANSFORMATION WITH LORENTZ BOOST
In this section, our main goal is to prove that the squeezed transformation corresponds to Lorentz boost. To begin with, we express a vector (x 0 , x 1 , x 2 ) in (2 + 1)-dimensional space time as whose basis are (K 0 , K 1 , K 2 ), which are the generators of SU (1, 1). Moreover, let us do this operation, In order to get specific connection between (y 0 , y 1 , y 2 ) and (x 0 , x 1 , x 2 ), we must calculate the r.h.s. of Eq. (10). During the calculation, the BCH formula [9] may be useful, which takes the form By using the above Eq. (11) and the commutations relations (2) of SU (1, 1), one can get where Substituting above Eq. (12) into Eq. (10), one gains that Because the linear independence of the generators, the above equation can be written as a matrix form, which is By comparison with Eq. (7), it is found that the above (3 × 3) coefficient matrix is identical with Lorentz boost (7). So it has been proved that S(β)xS † (β) is can be regarded as Lorentz boost under the condition that β = ρ 2 (sin ϕ+i cos ϕ). If we do the above calculations on the boson representation, the Lorentz boost can realized by the above operation of quantum optics.

IV. SQUEEZED TRANSFORMATION AND ADDITIONAL LAW OF RELATIVISTIC VELOCITIES
Furthermore, the next task is to demonstrate the additional law of relativistic velocities by use of squeezed operator in this section. We start out from this formula where From the two previous sections, we get the idea that Eq. (15) means that two Lorentz boosts is equal to not only a boost but also a rotation. In order to represent β 3 and δ by use of β 1 and β 2 , let's do the following operations on Eq. (15), i.e. , With the help of the equations l.h.s. and r.h.s. of Eq. (16) can be written as linear combination of a and a † respectively. As a and a † are linear independent, the coefficient of a of l.h.s. of Eq. (16) coincides with counterpart of r.h.s., and the same is true for a † . Hence, we obtain a : cosh and Moreover, the above two complex Eq. (17) and Eq. (18) can be converted into four real equations, which are and By use of Eq. (19) and Eq. (20), one can reach the following results and We should note that Eq. (23) is just the formula for Wigner angle [10]. Substituting the following formulae cosh 2x = cosh 2 x + sinh 2 x, cosh 2 x − sinh 2 x = 1 and sinh 2x = 2 sinh x cosh x into Eq.(24), we can get a more elegant and pragmatic expression about cosh ρ 3 , which is In addition, let us introduce the following auxiliary variables whereû,v andŵ are unit vectors along the direction of u, v and w respectively and c is the velocity of light propagating in the vacuum. Substituting Eq. (26) into Eq. (25), we can get a more meaningful formula, which is Now, we concentrate on another two equations (21) and (22), by substituting cos δ/2 from Eq. (19) and sin δ/2 from Eq. (20) into them, then they compose a set of equations about unknown elements cos ϕ 3 and sin ϕ 3 , i.e. , where A = −D = tanh ρ3 2 sin(ϕ 2 − ϕ 1 ) sinh ρ1 2 sinh ρ2 2 , B = C = tanh ρ3 2 [cosh ρ1 2 cosh ρ2 2 + cos(ϕ 2 − ϕ 1 ) sinh ρ1 2 sinh ρ2 2 ], E = cos ϕ 1 sinh ρ1 2 cosh ρ2 2 + cos ϕ 2 sinh ρ2 2 cosh ρ1 2 and F = sin ϕ 1 sinh ρ1 2 cosh ρ2 2 + sin ϕ 2 sinh ρ2 2 cosh ρ1 2 . According to Cramer's rules, the above Eq. (28) can be solved, hence we can obtain and In order to disclose the meaning of additional law of relativistic velocities. Let us introduce three unit vectors, which areû = cos ϕ 1î + sin ϕ 1ĵv = cos ϕ 2î + sin ϕ 2ĵ andŵ = cos ϕ 3î + sin ϕ 3ĵ .
Depending on the analysis above, the additional law of relativistic velocities has been successfully achieved from squeezed optics. Furthermore, the theoretical results can be possibly verified by the following experiment. In practice, it is much easier to realize the two-mode squeezed state than one-mode one, where subscripts 1 and 2 denote the two different photons and superscript "tm" represents twomode, thus we choose two-mode process. Because the generators of two-mode squeezed operator obey the same Lie algebra as the one-mode squeezed operator, our analysis above also applies. At first, a beam of light was splitted into two beams. Second, beam 1 goes through a nondegenerate optical parametric amplifier (NOPA) which could generate two-mode squeezed vacuum state [11]. As a result, beam 1 becomes S tm (β 2 )|00 . Moreover, it continues to pass an equipment which makes the light become S tm (β 1 )S tm (β 2 )|00 . Up to beam 2, it is changed to be S tm (β 3 )|00 by NOPA, where β 3 is calculated from β 1 and β 2 according to Eq. (25), (29) and (30). After all the transformations on beams 1 and 2, let them interfere with each other. Then the dark and bright fringes can be observed. The separation of adjacent bright fringes gives δ ′ . If δ ′ coincides with δ of Eq. (23), then the additional law of relativistic velocities is demonstrated.
Moreover, a possible experimental test on the additional law of relativistic velocities is also discussed.