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 and momentum . The ground state of this system, known as a vacuum state , is straightforward and is simply the direct product of the ground states of all modes over all frequencies :
By definition, the ground state of a mode with frequency is the state with minimum energy and no excitation:
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:
It is instructive to discuss the meaning of these matrices, and , and of the values they comprise. To do so, let us think of the light wave as a sequence of very short squarewave light pulses with infinitesimally small duration . The delta function of time in Eq. (66) 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. (65) that quadrature amplitudes’ fluctuations are stationary, and it is this stationarity, as noted in [39] 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 and . With this end in view let us introduce quadrature operators for each short light pulse as follows:
These operators and are nothing else than dimensionless displacement and momentum of the corresponding mode (called quadratures in quantum optics), normalized by zero point fluctuation amplitudes and : and . This fact is also justified by the value of their commutator:
The measurement outcome at each instance of time will be a random variable with zero mean and variance defined by a covariance matrix of Eq. (66):
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 ( and in our case):
where 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 and momentum . The corresponding plot is given in the left panel of Figure 14. Gaussian states are traditionally pictured by error ellipses on a phase plane, as drawn in the right panel of Figure 14 (cf. right panel of Figure 13). Here as well as in Figure 13, a red line in both plots circumscribes all the values of and 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 . The area of this circle, equal to in dimensionless units and to 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 and momentum , in our case) to be less than in units:Note the difference between Figures 13 and 14; the former features the result of measurement of an ensemble of oscillators (subsequent light pulses with infinitesimally short duration ), while the latter gives the probability density function for a single oscillator displacement and momentum.
Another important state of light is a coherent state (see, e.g., [163, 136, 99, 132]). It is straightforward to introduce a coherent state of a single mode or a harmonic oscillator as a result of its ground state shift on a complex plane by the distance and in the direction governed by a complex number . 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 inflicts its shift in a phase plane yielding a state that is called a coherent state:

Using the definitions of the mode quadrature operators and (dimensionless oscillator displacement and momentum normalized by zeropoint oscillations amplitude) given above, one immediately obtains for their mean values in a coherent state:

Further calculation shows that quadratures variances:

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 . It is instructive to calculate a Wigner function for the coherent state using a definition of Eq. (68):
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. (62)):
where is the coherent state that the mode of the field with frequency is in, and is the distribution of complex amplitudes over frequencies . Basically, is the spectrum of normalized complex amplitudes of the field, i.e., . For example, the state of a free light wave emitted by a perfectly monochromatic laser with emission frequency and mean optical power will be defined by , which implies that only the mode at frequency will be in a coherent state, while all other modes of the field will be in their ground states.Operator is unitary, i.e., with 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. (60). 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 15.
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 Heisenberglimited. 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., [90] 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 twophoton 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 give birth to pairs of correlated photons with frequencies and (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 with strong classical amplitude (see, e.g., Section 5.2 of [163] for details):
where describe annihilation operators for the photons of the signal and idler modes and is the complex coupling constant that is proportional to the secondorder susceptibility of the crystal and to the pump complex amplitude. Worth noting is the meaning of 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 , 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 . It is straightforward to obtain the evolution of the two modes in the interaction picture (leaving apart the obvious free evolution time dependence ) solving the Heisenberg equations: Let us then assume the signal and idler mode frequencies symmetric with respect to the half of pump frequency : and ( and ). Then the electric field of a twomode 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):

where twomode quadrature amplitudes and are defined along the lines of Eqs. (57), keeping in mind that only idler and signal components at the frequencies should be kept in the integral, which yields:
This geometric representation is rather useful, particularly for the characterization of a squeezed state. If the initial state of the twomode 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 on the vacuum state
This operator is no more and no less than the evolution operator for the PDC process in the interaction picture, i.e., Action of this operator on the twophoton quadrature amplitudes is fully described by Eqs. (76):

while annihilation operators of the corresponding modes are transformed in accordance with Eqs. (72).
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 and and their covariance matrix . Let us calculate these values:

and for a covariance matrix one can get the following expression:
where we introduced squeezing parameter and used a short notation for the symmetrized outer product of vector with itself:The squeezing parameter 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 through the following simple formula:
For example, 10 dB squeezing corresponds to .The covariance matrix (78) refers to a unique error ellipse on a phase plane with semimajor axis and semiminor axis rotated by angle clockwise as featured in Figure 16.
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:
where the error ellipse refers to the level where the Wigner function value falls to of the maximum. The corresponding plot and phase plane picture of the squeezed vacuum Wigner function are featured in Figure 17.Another important state that arises in GW detectors is the displaced squeezed state 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):
The light leaving a GW interferometer from the signal port finds itself in such a state, if a classical GWlike force changes the difference of the arm lengths, thus displacing a ponderomotively squeezed vacuum state in phase quadrature 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 and by some constant values dependent on shift complex amplitude :

Let us now generalize the results of a twomode consideration to a continuous spectrum case. Apparently, quadrature operators and are similar to and for the traveling wave case. Utilizing this similarity, let us define a squeezing operator for the continuum of modes as:
where and are frequencydependent 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. (76). 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 18. Quadrature amplitudes for each frequency transform in accordance with Eqs. (73). 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 in Eq. (64) and using Eq. (74) one immediately gets: Note that entries of might be frequency dependent if squeezing parameter and squeezing angle 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 according to the Wiener–Khinchin theorem: Of course, the exact shape of could be obtained only if we specify and . Note that the noise described by is stationary since all the entries of the covariance matrix (correlation functions) depend on the difference of times .The spectral density matrix allows for pictorial representation of a multimode squeezed state where an error ellipse is assigned to each sideband frequency . This effectively adds one more dimension to a phase plane picture already used by us for the characterization of a twomode squeezed states. Figure 19 exemplifies the state of a ponderomotively squeezed light that would leave the speedmeter type of the interferometer (see Section 6.2).
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Living Rev. Relativity 15, (2012), 5
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