3.2 Quantum states of light

3.2.1 Vacuum state

The quantum state of the travelling wave is a subtle structure, for the system it describes comprises a continuum of modes. However, each of these modes can be viewed at as a quantum oscillator with its own generalized coordinate ˆX = (ˆa + ˆa†)∕√2-- ω ω ω and momentum Yˆ = (ˆa − ˆa†)∕i√2-- ω ω ω. The ground state of this system, known as a vacuum state |vac⟩, is straightforward and is simply the direct product of the ground states |0 ⟩ω of all modes over all frequencies ω:

⊗ |vac⟩ ≡ |0⟩ω. (62 ) ω

By definition, the ground state of a mode with frequency ω is the state with minimum energy Evac = ℏ ω∕2 and no excitation:

ˆa |0⟩ = 0 and ⟨0| ˆa† = 0. (63 ) ω ω ω ω

Consider now statistical properties of the vacuum state. The mean values of annihilation and creation operators as well as any linear combination thereof that includes quadrature amplitudes, are zero:

⟨vac|ˆaω|vac⟩ ≡ ⟨ˆaω⟩ = ⟨ˆa†ω⟩ = 0 ⇒ ⟨ˆac(Ω )⟩ = ⟨ˆas(Ω )⟩ = 0.
Apparently, this also holds for time domain operators:
⟨vac|ˆa(t)|vac ⟩ ≡ ⟨ˆa (t)⟩ = ⟨ˆa†(t)⟩ = 0 ⇒ ⟨ˆac(t)⟩ = ⟨ˆas(t)⟩ = 0.
That the ground state of the oscillator is Gaussian is evident from its q-representation [136Jump To The Next Citation Point], namely
{ } -1-- X2ω- ψvac(X ω) ≡ ⟨X ω|0⟩ω = √4π exp − 2 .
It means that knowing the second moments of quadrature amplitudes suffices for full characterization of the state |vac ⟩. For this purpose, let us introduce a quadrature amplitudes matrix of spectral densities6 𝕊 (Ω ) according to the rule:
[ ′ ′ ] [ ] ⟨ˆac(Ω) ∘ ˆac(Ω )′⟩ ⟨ˆac(Ω ) ∘ ˆas(Ω )′⟩ = 2π δ(Ω + Ω ′) Scc(Ω ) Scs(Ω ) = 2π𝕊 δ(Ω + Ω ′). (64 &#x0029 ⟨ˆas(Ω) ∘ ˆac(Ω )⟩ ⟨ˆas(Ω ) ∘ ˆas(Ω )⟩ Ssc(Ω ) Sss(Ω )
where S (Ω ) ij (i,j = c,s) denote (cross) power spectral densities of the corresponding quadrature amplitudes ′ ⟨ˆai(Ω ) ∘ ˆaj(Ω )⟩ standing for the symmetrized product of the corresponding quadrature operators, i.e.:
⟨ˆai(Ω ) ∘ ˆaj(Ω′)⟩ ≡ 1⟨ˆai(Ω)ˆaj(Ω ′) + ˆaj(Ω ′)ˆai(Ω)⟩ ≡ 2πSij(Ω )δ(Ω − Ω ′). 2
For a vacuum state, this matrix of spectral densities can easily be obtained from the commutation relations (58View Equation) and equals to:
[1∕2 0 ] 𝕊vac(Ω) = , (65 ) 0 1∕2
which implies that the (double-sided) power spectral densities of the quadrature amplitudes as well as their cross-spectral density are equal to:
1- Scc(Ω) = Sss(Ω) = 2 and Scs(Ω ) = 0.
In time domain, the corresponding matrix of second moments, known as a covariance matrix with elements defined as 𝕍ijδ(t − t′) = ⟨ˆai(t) ∘ ˆaj(t′)⟩, is absolutely the same as 𝕊vac(Ω) :
[ ] 𝕍 = 1∕2 0 . (66 ) vac 0 1∕2

It is instructive to discuss the meaning of these matrices, 𝕊 and 𝕍, and of the values they comprise. To do so, let us think of the light wave as a sequence of very short square-wave light pulses with infinitesimally small duration 𝜀 → 0. The delta function of time in Eq. (66View Equation) tells us that the noise levels at different times, i.e., the amplitudes of the different square waves, are statistically independent. To put it another way, this noise is Markovian. It is also evident from Eq. (65View Equation) that quadrature amplitudes’ fluctuations are stationary, and it is this stationarity, as noted in [39Jump To The Next Citation Point] that makes quadrature amplitudes such a convenient language for describing the quantum noise of light in parametric systems exemplified by GW interferometers.

It is instructive to pay some attention to a pictorial representation of the quantum noise described by the covariance and spectral density matrices 𝕍 and 𝕊. With this end in view let us introduce quadrature operators for each short light pulse as follows:

1 ∫ t+𝜀∕2 1 ∫ t+𝜀∕2 ˆX 𝜀(t) ≡ √--- dτˆac(τ), and ˆY𝜀(t) ≡ √--- dτ ˆas(τ). (67 ) 𝜀 t−𝜀∕2 𝜀 t−𝜀∕2

These operators ˆ X 𝜀(t) and ˆ Y 𝜀(t) are nothing else than dimensionless displacement and momentum of the corresponding mode (called quadratures in quantum optics), normalized by zero point fluctuation amplitudes X0 and P0: ˆX𝜀(t) ≡ ˆx𝜀∕X0 and Xˆ𝜀(t) ≡ pˆ𝜀∕P0. This fact is also justified by the value of their commutator:

[ ] Xˆ (t),Yˆ(t) = i. 𝜀 𝜀
There is no difficulty in showing that diagonal elements of the covariance matrix 𝕍ii are equal to the variances of the corresponding mode displacement ˆ X 𝜀 and momentum ˆ Y𝜀:
ˆ2 ˆ 2 𝕍cc = ⟨X 𝜀(t)⟩ = 1∕2, and 𝕍ss = ⟨Y𝜀 (t)⟩ = 1∕2,
while non-diagonal terms represent correlations between these operators (zero in our case): 𝕍 = 𝕍 = ⟨ ˆX (t) ∘ ˆY (t)⟩ = 0 cs sc 𝜀 𝜀. At the same time, we see that there is no correlation between the pulses, justifying the Markovianity of the quantum noise of light in vacuum state:
ˆ ˆ ′ ˆ ˆ ′ ˆ ˆ ′ ′ ⟨X 𝜀(t)X 𝜀(t)⟩ = ⟨Y𝜀(t)Y𝜀(t)⟩ = ⟨X 𝜀(t) ∘Y𝜀(t )⟩ = 0, t ⁄= t.
An attempt to measure the light field amplitude as a function of time will give the result depicted in Figure 13View Image.
View Image

Figure 13: Light field in a vacuum quantum state |vac⟩. Left panel (a) features a typical result one could get measuring the (normalized) electric field strength of the light wave in a vacuum state as a function of time. Right panel (b) represents a phase space picture of the results of measurement. A red dashed circle displays the error ellipse for the state |vac⟩ that encircles the area of single standard deviation for a two-dimensional random vector ˆa of measured light quadrature amplitudes. The principal radii of the error ellipse (equal in vacuum state case) are equal to square roots of the covariance matrix 𝕍vac eigenvalues, i.e., to √ -- 1∕ 2.
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Figure 14: Wigner function W |vac⟩(X 𝜀,Y 𝜀) of a ground state of harmonic oscillator (left panel) and its representation in terms of the noise ellipse (right panel).

The measurement outcome at each instance of time will be a random variable with zero mean and variance defined by a covariance matrix 𝕍vac of Eq. (66View Equation):

Var[Eˆ(t)] = {cos ω t, sinω t}𝕍 {cosω t, sin ω t}T = 1-. 0 0 vac 0 0 2

In quantum mechanics, it is convenient to describe a quantum state in terms of a Wigner function, a quantum version of joint (quasi) probability distribution for particle displacement and momentum (X 𝜀 and Y 𝜀 in our case):

∫ ∞ dξ W |vac⟩(X 𝜀,Y𝜀) = ---exp {− iξY 𝜀} ⟨X 𝜀 + ξ∕2 |vac⟩⟨vac|X 𝜀 − ξ∕2⟩ −∞ 2π { } -----1------ 1- T − 1 1- { 2 2 } = 2π √det-𝕍--- exp − 2{X 𝜀,Y 𝜀} 𝕍vac{X 𝜀,Y𝜀} = π exp − (X𝜀 + Y 𝜀 ) , (68 ) vac
where ξ is simply the variable of integration. The above Wigner function describes a Gaussian state, which is simply the ground state of a harmonic oscillator represented by a mode with displacement ˆX 𝜀 and momentum Yˆ𝜀. The corresponding plot is given in the left panel of Figure 14View Image. Gaussian states are traditionally pictured by error ellipses on a phase plane, as drawn in the right panel of Figure 14View Image (cf. right panel of Figure 13View Image). Here as well as in Figure 13View Image, a red line in both plots circumscribes all the values of X 𝜀 and Y 𝜀 that fall inside the standard deviation region of the Wigner function, i.e., the region where all pertinent points are within 1 standard deviation from the center of the distribution. For a vacuum state, such a region is a circle with radius √ ---- √ ---- √ -- 𝕍cc = 𝕍ss = 1 ∕ 2. The area of this circle, equal to 1∕2 in dimensionless units and to ℏ∕2 in case of dimensional displacement and momentum, is the smallest area a physical quantum state can occupy in a phase space. This fact yields from a very general physical principle, the Heisenberg uncertainty relation, that limits the minimal uncertainty product for canonically conjugate observables (displacement X 𝜀 and momentum Y 𝜀, in our case) to be less than 1∕2 in ℏ-units:
( ) ( ) Var[Xˆ ] 1∕2 Var[ˆY ] 1∕2 ≥ 1. 𝜀 𝜀 2
The fact that for a ground state this area is exactly equal to 1∕2 is due to the fact that it is a pure quantum state, i.e., the state of the particle that can be described by a wave function |ψ⟩, rather than by a density operator ˆρ. For more sophisticated Gaussian states with a non-diagonal covariance matrix 𝕍, the Heisenberg uncertainty relation reads:
det𝕍 ≥ 1-, (69 ) 4
and noise ellipse major semi-axes are given by the square root of the matrix 𝕍 eigenvalues.

Note the difference between Figures 13View Image and 14View Image; the former features the result of measurement of an ensemble of oscillators (subsequent light pulses with infinitesimally short duration 𝜀), while the latter gives the probability density function for a single oscillator displacement and momentum.

3.2.2 Coherent state

Another important state of light is a coherent state (see, e.g., [163Jump To The Next Citation Point, 136, 99, 132Jump To The Next Citation Point]). It is straightforward to introduce a coherent state |α ⟩ of a single mode or a harmonic oscillator as a result of its ground state |0⟩ shift on a complex plane by the distance and in the direction governed by a complex number α = |α |eiarg(α). This can be caused, e.g., by the action of a classical effective force on the oscillator. Such a shift can be described by a unitary operator called a displacement operator, since its action on a ground state |0⟩ inflicts its shift in a phase plane yielding a state that is called a coherent state:

† ∗ |α⟩ = ˆD [α]|0⟩ ≡ eαˆa −α ˆa|0⟩,
or, more vividly, in q-representation of a corresponding mode of the field [132]
{ √ -- } -1-- (X-ω-−---2α-)2 ψcoh(X ω) ≡ ⟨Xω |α ⟩ = 4√ π exp − 2 .
The shift described by Dˆ[α] is even more apparent if one writes down its action on an annihilation (creation) operator:
† ( † † † ∗) Dˆ [α]ˆa ˆD [α ] = ˆa + α, ˆD [α]ˆa ˆD [α ] = ˆa + α .
Moreover, a coherent state is an eigenstate of the annihilation operator:
ˆa|α ⟩ = α|α⟩.

Using the definitions of the mode quadrature operators ˆ ˆ X ω ≡ X and ˆ ˆ Yω ≡ Y (dimensionless oscillator displacement and momentum normalized by zero-point oscillations amplitude) given above, one immediately obtains for their mean values in a coherent state:

√ -- √ -- ⟨α|Xˆ|α⟩ = ⟨0| ˆD †[α ]X ˆDˆ[α]|0 ⟩ = 2Re [α], ⟨α |Yˆ|α⟩ = ⟨0| ˆD †[α]ˆY ˆD [α ]|0⟩ = 2Im [α &#x005

Further calculation shows that quadratures variances:

ˆ ˆ2 ( ˆ )2 1- ˆ ˆ2 ( ˆ )2 1- Var[X ] = ⟨α |X |α⟩ − ⟨α |X |α ⟩ = 2, Var[Y ] = ⟨α |Y |α⟩ − ⟨α|Y |α ⟩ = 2

have the same values as those for a ground state. These two facts unequivocally testify in favour of the statement that a coherent state is just the ground state shifted from the origin of the phase plane to the point with coordinates √ -- (⟨ ˆX ⟩α,⟨ˆY⟩α ) = 2(Re [α],Re[α]). It is instructive to calculate a Wigner function for the coherent state using a definition of Eq. (68View Equation):

∫ ∞ d-ξ W |α⟩(X, Y ) = −∞ 2π exp {− iξY }⟨X + ξ∕2|α⟩⟨α |X − ξ∕2⟩ = { √ -- √ -- √ -- √ -- } --√--1------ exp − 1{X − 2Re [α],Y − 2Im [α]}T𝕍 −1{X − 2Re [α ],Y − 2Im [α ]} = 2π det 𝕍vac 2 vac 1 { [ √ -- 2 √ -- 2]} --exp − (X − 2Re[α]) + (Y − 2Im [α ]) , π
which once again demonstrates the correctness of the former statement.

Generalization to the case of continuum of modes comprising a light wave is straightforward [14] and goes along the same lines as the definition of the field vacuum state, namely (see Eq. (62View Equation)):

⊗ ⊗ { ∫ ∞ d ω } |α (ω )⟩ ≡ |α ⟩ω = ˆD[α(ω )]|0⟩ω = exp ---(α (ω)ˆa†ω − α∗(ω )ˆaω) |vac⟩, (70 ) ω ω −∞ 2π
where |α ⟩ω is the coherent state that the mode of the field with frequency ω is in, and α(ω) is the distribution of complex amplitudes α over frequencies ω. Basically, α(ω ) is the spectrum of normalized complex amplitudes of the field, i.e., α(ω ) ∝ ℰ (ω). For example, the state of a free light wave emitted by a perfectly monochromatic laser with emission frequency ωp and mean optical power ℐ0 will be defined by ∘ ---- α (ω) = π 2ℐ0δ(ω − ωp) ℏωp, which implies that only the mode at frequency ωp will be in a coherent state, while all other modes of the field will be in their ground states.
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Figure 15: Light field in a coherent quantum state |α (ω )⟩. Left panel a) features a typical result one could get measuring the (normalized) electric field strength of the light wave in a coherent state as a function of time. Right panel b) represents a phase space picture of the results of measurement. The red dashed line in the left panel marks the mean value ⟨Eˆ(t)⟩. The red arrow in the right panel features the vector of the mean values of quadrature amplitudes, i.e., A, while the red dashed circle displays the error ellipse for the state |α(ω )⟩ that encircles the area of single standard deviation for a two-dimensional random vector ˆa of quadrature amplitudes. The principle radii of the error ellipse (equal in the coherent state case) are equal to square roots of the covariance matrix 𝕍 coh, i.e., to √ -- 1∕ 2.

Operator Dˆ[α] is unitary, i.e., Dˆ†[α] ˆD [α ] = Dˆ[α]Dˆ†[α ] = ˆI with ˆI the identity operator, while the physical meaning is in the translation and rotation of the Hilbert space that keeps all the physical processes unchanged. Therefore, one can simply use vacuum states instead of coherent states and subtract the mean values from the corresponding operators in the same way we have done previously for the light wave classical amplitudes, just below Eq. (60View Equation). The covariance matrix and the matrix of power spectral densities for the quantum noise of light in a coherent state is thus the same as that of a vacuum state case.

The typical result one can get measuring the electric field strength of light emitted by the aforementioned ideal laser is drawn in the left panel of Figure 15View Image.

3.2.3 Squeezed state

One more quantum state of light that is worth consideration is a squeezed state. To put it in simple words, it is a state where one of the oscillator quadratures variance appears decreased by some factor compared to that in a vacuum or coherent state, while the conjugate quadrature variance finds itself swollen by the same factor, so that their product still remains Heisenberg-limited. Squeezed states of light are usually obtained as a result of a parametric down conversion (PDC) process [92, 172] in optically nonlinear crystals. This is the most robust and experimentally elaborated way of generating squeezed states of light for various applications, e.g., for GW detectors [149, 152, 141], or for quantum communications and computation purposes [31]. However, there is another way to generate squeezed light by means of a ponderomotive nonlinearity inherent in such optomechanical devices as GW detectors. This method, first proposed by Corbitt et al. [47], utilizes the parametric coupling between the resonance frequencies of the optical modes in the Fabry–Pérot cavity and the mechanical motion of its mirrors arising from the quantum radiation pressure fluctuations inflicting random mechanical motion on the cavity mirrors. Further, we will see that the light leaving the signal port of a GW interferometer finds itself in a ponderomotively squeezed state (see, e.g., [90Jump To The Next Citation Point] for details). A dedicated reader might find it illuminating to read the following review articles on this topic [133, 101].

Worth noting is the fact that generation of squeezed states of light is the process that inherently invokes two modes of the field and thus naturally calls for usage of the two-photon formalism contrived by Caves and Schumaker [39, 40]. To demonstrate this let us consider the physics of a squeezed state generation in a nonlinear crystal. Here photons of a pump light with frequency ωp = 2ω0 give birth to pairs of correlated photons with frequencies ω1 and ω2 (traditionally called signal and idler) by means of the nonlinear dependence of polarization in a birefringent crystal on electric field. Such a process can be described by the following Hamiltonian, provided that the pump field is in a coherent state |α⟩ω p with strong classical amplitude |αp| ≫ 1 (see, e.g., Section 5.2 of [163] for details):

Hˆ ( ) --PDC-= ω1ˆa†1ˆa1 + ω2ˆa†2ˆa2 + i χˆa†1ˆa†2e− 2iω0t − χ ∗ˆa1 ˆa2e2iω0t , (71 ) ℏ
where ˆa1,2 describe annihilation operators for the photons of the signal and idler modes and χ = ρe2iϕ is the complex coupling constant that is proportional to the second-order susceptibility of the crystal and to the pump complex amplitude. Worth noting is the meaning of t in this Hamiltonian: it is a parameter that describes the duration of a pump light interaction with the nonlinear crystal, which, in the simplest situation, is either the length of the crystal divided by the speed of light c, or, if the crystal is placed between the mirrors of the optical cavity, the same as the above but multiplied by an average number of bounces of the photon inside this cavity, which is, in turn, proportional to the cavity finesse ℱ. It is straightforward to obtain the evolution of the two modes in the interaction picture (leaving apart the obvious free evolution time dependence −iωs,it e) solving the Heisenberg equations:
† 2iϕ † 2iϕ ˆa1(t) = ˆa1cosh ρt + ˆa2e sinh ρt, ˆa2 (t) = ˆa2 coshρt + ˆa1e sinh ρt. (72 )
Let us then assume the signal and idler mode frequencies symmetric with respect to the half of pump frequency ω0 = ωp∕2: ω1 → ω+ = ω0 + Ω and ω2 → ω − = ω0 − Ω (ˆa1 → ˆa+ and ˆa2 → ˆa−). Then the electric field of a two-mode state going out of the nonlinear crystal will be written as (we did not include the pump field here assuming it can be ruled out by an appropriate filter):
ˆ [ ˆ ˆ ] E (t) = 𝒞0 X (t)cos ω0t + Y (t)sinω0t ,

where two-mode quadrature amplitudes ˆ X (t) and ˆ Y(t) are defined along the lines of Eqs. (57View Equation), keeping in mind that only idler and signal components at the frequencies ω ± − ω0 = ± Ω should be kept in the integral, which yields:

[ ] ˆX (t) = 1√---ˆa+ (t)eiΩt + ˆa†(t)e−iΩt + ˆa− (t)e−iΩt + ˆa†(t)eiΩt = ˆasqze−iΩt + ˆasqz†eiΩt, 2 + − c c 1 [ † † ] ˆY (t) = √--- ˆa+ (t)eiΩt − ˆa+ (t)e− iΩt + ˆa− (t)e−iΩt − ˆa − (t)eiΩt = ˆassqze −iΩt + ˆassqz†eiΩt, i 2
where √ -- ˆascqz = (ˆa+ (t) + ˆa†− (t))∕ 2 and √ -- ˆasqsz= (ˆa+(t) − ˆa†− (t))∕(i 2) are the spectral quadrature amplitudes of the two-mode field at sideband frequency Ω (cf. Eqs. (52View Equation)) after it leaves the nonlinear crystal. Substituting Eqs. (72View Equation) into the above expressions yields transformation rules for quadrature amplitudes:
ˆasqz= ˆa (coshρt + cos2 ϕsinh ρt) + ˆa sin 2ϕ sinh ρt, c c s ˆassqz= ˆac sin 2ϕ sinh ρt + ˆas(cosh ρt − cos2ϕ sinh ρt), (73 )
where ˆa = (ˆa + ˆa†)∕√2-- c + − and ˆa = (ˆa − ˆa† )∕(i√2-) s + − stand for initial values of spectral quadrature amplitudes of the two-mode light wave created in the PDC process. A close look at these transformations written in the matrix form reveals that it can be represented as the following sequence:
( sqz) ˆasqz = ˆacsqz = 𝕊sqz[ρt,ϕ]ˆa = ℙ [ϕ ]𝕊sqz[ρt,0 ]ℙ [− ϕ]ˆa (74 ) ˆas
[ ] [ ] coshρt + cos 2ϕsinh ρt sin2ϕ sinhρt eρt 0 𝕊sqz[ρt,ϕ ] ≡ sin 2ϕ sinh ρt cosh ρt − cos2ϕ sinh ρt =⇒ 𝕊sqz[ρt,0] = 0 e−ρt (75 )
are squeezing matrices in general and in special (ϕ = 0) case, while ℙ[ϕ] stands for a counterclockwise 2D-rotation matrix by angle ϕ defined by (17View Equation). Therefore, the evolution of a two-mode light quadrature amplitude vector aˆ in a PDC process described by the Hamiltonian (71View Equation) consists of a clockwise rotation by an angle ϕ followed by a deformation along the main axes (stretching along the ac-axis and proportional squeezing along the as-axis) and rotation back by the same angle. It is straightforward to show that vector { } ˆXsqz = ˆX (t),Yˆ(t) T = aˆsqze−iΩt + ˆasqz∗eiΩt transforms similarly (here ˆasqz∗ = {ˆasqz†,ˆasqz†}T c s ).

This geometric representation is rather useful, particularly for the characterization of a squeezed state. If the initial state of the two-mode field is a vacuum state then the outgoing field will be in a squeezed vacuum state. One can define it as a result of action of a special squeezing operator Sˆ[ρt,ϕ ] on the vacuum state

|sqz (ρt,ϕ)⟩ = ˆS [ρt,ϕ ]|vac⟩. (76 ) 0
This operator is no more and no less than the evolution operator for the PDC process in the interaction picture, i.e.,
{ } ˆS[ρt,ϕ] ≡ exp ρt(ˆa+ˆa − e−2iϕ − aˆ†ˆa† e2iϕ) . (77 ) + −
Action of this operator on the two-photon quadrature amplitudes is fully described by Eqs. (76View Equation):
sqz ˆ† ˆ ˆa = S [ρt,ϕ]ˆaS [ρt,ϕ ] = ℙ [ϕ ]𝕊sqz[ρt,0]ℙ[− ϕ]ˆa,

while annihilation operators of the corresponding modes ˆa ± are transformed in accordance with Eqs. (72View Equation).

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Figure 16: Schematic plot of a vacuum state transformation under the action of the squeezing operator Sˆ[r,ϕ ]. Eqs. (74View Equation) demonstrate the equivalence of the general squeezing operator ˆS[r,ϕ] to a sequence of phase plane counterclockwise rotation by an angle ϕ (transition from a) to b)), phase plane squeezing and stretching by a factor r e (transition from b) to c)) and rotation back by the same angle ϕ (transition from c) to d)). Point P ′ tracks how transformations change the initial state marked with point P.

The linearity of the squeezing transformations implies that the squeezed vacuum state is Gaussian since it is obtained from the Gaussian vacuum state and therefore can be fully characterized by the expectation values of operators ˆX and ˆY and their covariance matrix 𝕍 sqz. Let us calculate these values:

⟨Xˆ⟩sqz = ⟨sqz0(r,ϕ)|Xˆ|sqz0(r,ϕ )⟩ = ⟨vac|ˆS †[r,ϕ]XˆˆS[r,ϕ]|vac⟩ = ⟨vac|X ˆ(t)|vac⟩ = 0, ⟨ˆY ⟩sqz = 0,

and for a covariance matrix one can get the following expression:

𝕍sqz = ⟨sqz0(r,ϕ)|Xˆ ∘XˆT |sqz0(r,ϕ)⟩ = ℙ [− ϕ]𝕊sqz[r,0]𝕍vac𝕊sqz[r,0]ℙ[ϕ] = 1 1[ cosϕ sinϕ ] [e2r 0 ] [cos ϕ − sin ϕ] --ℙ[− ϕ]𝕊sqz[2r,0]ℙ[ϕ] = -- −2r , (78 ) 2 2 − sin ϕ cosϕ 0 e sin ϕ cosϕ
where we introduced squeezing parameter r ≡ ρt and used a short notation for the symmetrized outer product of vector ˆ X with itself:
[Xˆ ∘ ˆX Xˆ ∘ ˆY] Xˆ ∘XˆT ≡ ˆ ˆ ˆ ˆ . Y ∘ X Y ∘Y

The squeezing parameter r is the quantity reflecting the strength of the squeezing. This way of characterizing the squeezing strength, though convenient enough for calculations, is not very ostensive. Conventionally, squeezing strength is measured in decibels (dB) that are related to the squeezing parameter r through the following simple formula:

rdB = 10log e2r = 20r log e ⇐ ⇒ r = rdB∕(20 log e). (79 ) 10 10 10
For example, 10 dB squeezing corresponds to r ≃ 1.15.

The covariance matrix (78View Equation) refers to a unique error ellipse on a phase plane with semi-major axis √ -- er∕ 2 and semi-minor axis √ -- e−r∕ 2 rotated by angle ϕ clockwise as featured in Figure 16View Image.

It would be a wise guess to make, that a squeezed vacuum Wigner function can be obtained from that of a vacuum state, using these simple geometric considerations. Indeed, for a squeezed vacuum state it reads:

{ } 1 1 T −1 W |sqz⟩(X, Y ) = --∘---------exp − 2{X, Y } 𝕍sqz{X, Y} , (80 ) 2π det 𝕍sqz
where the error ellipse refers to the level where the Wigner function value falls to -- 1∕√ e of the maximum. The corresponding plot and phase plane picture of the squeezed vacuum Wigner function are featured in Figure 17View Image.
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Figure 17: Left panel: Wigner function of a squeezed vacuum state with squeeze parameter r = 0.5 (5 dB) and rotation angle ϕ = − π∕4. Right panel: Error ellipse corresponding to that Wigner function.

Another important state that arises in GW detectors is the displaced squeezed state |sqzα(r,ϕ )⟩ that is obtained from the squeezed vacuum state in the same manner as the coherent state yields from the vacuum state, i.e., by the application of the displacement operator (equivalent to the action of a classical force):

|sqz (r,ϕ)⟩ = ˆD [α ]|sqz (r,ϕ)⟩ = Dˆ[α]ˆS[r,ϕ]|vac⟩. (81 ) α 0
The light leaving a GW interferometer from the signal port finds itself in such a state, if a classical GW-like force changes the difference of the arm lengths, thus displacing a ponderomotively squeezed vacuum state in phase quadrature Y by an amount proportional to the magnitude of the signal force. Such a displacement has no other consequence than simply to shift the mean values of ˆ X and ˆ Y by some constant values dependent on shift complex amplitude α:
√ -- √ -- ⟨Xˆ⟩sqz = 2Re [α], ⟨ˆY ⟩sqz = 2Im [α ].

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Figure 18: Light field in a squeezed state |sqzα(r,ϕ )⟩. Upper row features time dependence of the electric field strength E (t) in three different squeezed states (10 dB squeezing assumed for all): a) squeezed vacuum state with squeezing angle ϕ = π ∕4; b) displaced squeezed state with classical amplitude Ac = 5 (mean field strength oscillations ⟨ ˆE (t)⟩sqz are given by red dashed line) and amplitude squeezing (ϕ = π∕2); c) displaced squeezed state with classical amplitude Ac = 5 and phase squeezing (ϕ = 0). Lower row features error ellipses (red dashed lines) for the corresponding plots in the upper row.

Let us now generalize the results of a two-mode consideration to a continuous spectrum case. Apparently, quadrature operators ˆX (t) and ˆY(t) are similar to ˆac(t) and ˆas(t) for the traveling wave case. Utilizing this similarity, let us define a squeezing operator for the continuum of modes as:

{ ∫ ∞ dΩ [ ] } Sˆ[r(Ω ),ϕ(Ω )] ≡ exp ---r(Ω) ˆa+ˆa− e−2iϕ(Ω) − ˆa†+ˆa†− e2iϕ(Ω) , (82 ) −∞ 2π
where r(Ω) and ϕ (Ω) are frequency-dependent squeezing factor and angle, respectively. Acting with this operator on a vacuum state of the travelling wave yields a squeezed vacuum state of a continuum of modes in the very same manner as in Eq. (76View Equation). The result one could get in the measurement of the electric field amplitude of light in a squeezed state as a function of time is presented in Figure 18View Image. Quadrature amplitudes for each frequency Ω transform in accordance with Eqs. (73View Equation). Thus, we are free to use these formulas for calculation of the power spectral density matrix for a traveling wave squeezed vacuum state. Indeed, substituting sqz ˆac,s(Ω) → ˆac,s(Ω ) in Eq. (64View Equation) and using Eq. (74View Equation) one immediately gets:
𝕊sqz(Ω) = ℙ [− ϕ(Ω )]𝕊sqz[r(Ω),0]𝕊vac(Ω )𝕊sqz[r(Ω ),0 ]ℙ [ϕ (Ω )] = 𝕊sqz(r(Ω ),ϕ (Ω)). (83 &
Note that entries of 𝕊sqz(Ω) might be frequency dependent if squeezing parameter r(Ω) and squeezing angle ϕ(Ω ) are frequency dependent as is the case in all physical situations. This indicates that quantum noise in a squeezed state of light is not Markovian and this can easily be shown by calculating the the covariance matrix, which is simply a Fourier transform of 𝕊(Ω ) according to the Wiener–Khinchin theorem:
∫ ∞ ′ dΩ- −iΩ(t−t′) 𝕍sqz(t − t) = − ∞ 2π 𝕊sqz(Ω )e . (84 )
Of course, the exact shape of ′ 𝕍sqz(t − t) could be obtained only if we specify r(Ω) and ϕ (Ω). Note that the noise described by 𝕍sqz(t − t′) is stationary since all the entries of the covariance matrix (correlation functions) depend on the difference of times t − t′.

The spectral density matrix allows for pictorial representation of a multimode squeezed state where an error ellipse is assigned to each sideband frequency Ω. This effectively adds one more dimension to a phase plane picture already used by us for the characterization of a two-mode squeezed states. Figure 19View Image exemplifies the state of a ponderomotively squeezed light that would leave the speed-meter type of the interferometer (see Section 6.2).

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Figure 19: Example of a squeezed state of the continuum of modes: output state of a speed-meter interferometer. Left panel shows the plots of squeezing parameter r (Ω ) dB and squeezing angle ϕ (Ω ) versus normalized sideband frequency Ω∕ γ (here γ is the interferometer half-bandwidth). Right panel features a family of error ellipses for different sideband frequencies Ω that illustrates the squeezed state defined by rdB(Ω) and ϕ(Ω) drawn in the left panel.

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