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 square-wave 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):
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:
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Using the definitions of the mode quadrature operators and
(dimensionless oscillator
displacement and momentum normalized by zero-point oscillations amplitude) given above, one
immediately obtains for their mean values in a coherent state:
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Further calculation shows that quadratures variances:
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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)):
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 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., [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 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 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):
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where two-mode 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 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 on the vacuum state
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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:
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and for a covariance matrix one can get the following expression:
where we introduced squeezing parameter 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:
The covariance matrix (78) refers to a unique error ellipse on a phase plane with semi-major
axis
and semi-minor 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 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):
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Let us now generalize the results of a two-mode 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:
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 two-mode squeezed states. Figure 19
exemplifies the state of a ponderomotively squeezed light that would leave the speed-meter type of the
interferometer (see Section 6.2).
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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