# Control Scheme for Polarization Circulation Speed Meter Using a Dual-
Retardation Waveplate

Yohei Nishino1,2<EMAIL_ADDRESS>Tomotada Akutsu2 Yoichi Aso2,3
Takayuki Tomaru1,2,3,4,5 1Department of Astronomy, University of Tokyo,
Bunkyo, Tokyo 113-0033, Japan  2Gravitational Wave Science Project, National
Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan
3The Graduate University for Advanced Studies (SOKENDAI), Mitaka City, Tokyo
181-8588, Japan  4Accelerator Laboratory, High Energy Accelerator Research
Organization (KEK), Tsukuba City, Ibaraki 305- 0801, Japan  5Institute for
Cosmic Ray Research (ICRR), KAGRA Observatory, The University of Tokyo,
Kashiwa City, Chiba 277-8582, Japan

###### Abstract

In interferometric gravitational wave detectors, quantum radiation pressure
noise, which is a back action of the measurement, limits their sensitivity at
low frequencies. Speed meters are one of the techniques to reduce the back
action noise and improve the sensitivity. Furthermore, a speed meter detector
can surpass the standard quantum limit over a wide range of frequencies. The
Polarization Circulation Speed Meter (PCSM) is the latest incarnation of the
speed meter concept that requires only a modest modification to the
conventional interferometer design. However, its control scheme has not been
developed. The main difficulty is the length and alignment control of the
cavity formed by the polarization circulation mirror and the input test
masses, whose round-trip phase shift should be kept at $\pi$. In this article,
we propose a new control scheme for the PCSM using a dual-retardation
waveplate, called Dual-Retardance Control (DRC). We also compare the shot
noise level of the DRC to another simpler scheme by using mirror dithering.
Finally, we design the experimental setup to demonstrate the feasibility of
the DRC and show the expected results through transfer function measurements.

††preprint: APS/123-QED

##  I Introduction

The sensitivity of the gravitational wave (GW) detectors is fundamentally
limited by quantum noise. Especially at low frequencies, after the seismic
noise and thermal noise are well suppressed and with the use of a high-power
laser, it will be limited by quantum radiation pressure noise. This low-
frequency-limiting noise gives rise to the standard quantum limit (SQL), which
is one of the consequences of Heisenberg’s uncertainty relation [1]. The SQL
is a fundamental limit that we cannot overcome by conventional methods, and
many techniques to beat it, so-called quantum non-demolition (QND)
measurement, have been studied [1, 2].
Speed meters are one of the QND measurements. The concept was first proposed
by Braginsky and Khalili [3], and many practical implementations have been
investigated [4, 5, 6, 7, 5, 8, 7, 9, 10, 11, 12, 13]. The amplitude
fluctuations of vacuum fields entering from the anti-symmetric (AS) port of an
interferometer are coupled with the pump laser and kick the mirrors randomly,
which appears as the quantum radiation pressure noise [2]. In speed meters,
the vacuum field interacts with the mirror twice with opposite signs. Taking
into account the sloshing time $\tau$, which is an interval of two
measurements, the back-action force applied on the mirror is [14]:

$\displaystyle\hat{F}_{\mathrm{b.a.}}(\Omega)\simeq-i\Omega\tau\frac{2\hat{I}_{c}(\Omega)}{c}$
(1)

at low frequencies $\Omega\ll 1/\tau$. $\hat{I}_{c}$ is the fluctuation part
of the circulating laser power via the coupling between vacuum fluctuation and
the pump laser and $c$ is the speed of light. The signal is proportional to
the mean velocity ($\bar{v}$) at low frequencies

$\displaystyle\phi(t)$ $\displaystyle\propto\hat{x}(t+\tau)-\hat{x}(t)$ (2)
$\displaystyle\sim\tau\bar{v},$ (3)

where $\phi(t)$ is the phase modulation of light and $x(t)$ is the
displacement of the mirrors. Note that the velocity measurement reduces the
amount of gravitational wave signal, but in terms of the signal-to-noise
ratio, it shows better performance than position measurements at low
frequencies.
The advantage of speed meters is a broadband-sensitivity improvement at low
frequencies. Also by combination with a balanced homodyne detection (BHD), it
can go beyond the free mass SQL. It is worth noting that it does not need
frequency-dependent homodyne angles, which means we do not need additional
filter cavities [2]. All the noise-reduction processes happen inside the
interferometer, so it is more robust to losses [15].
The Polarization Circulation Speed Meter (PCSM) is a new speed meter design
proposed by Danilishin et al [11] (see Fig. 1). The PCSM only needs a small
modification in the AS port; there is no need to modify the central
interferometer. Under the current situation that all the large-scale GW
detectors are based on the Dual-Recycled Fabry-Pérot Michelson Interferometer
(DRFPMI) and an assumption that it will not be largely changed soon, this
design is the most promising candidate for the practical implementation of a
speed meter. However, the control scheme has not been investigated yet.
There are two main issues to be solved to achieve PCSM. The first issue is
that DC component of differential arm signal is reduced to zero in an ideal
speed meter, making it hard to control the interferometer. This is a generic
problem in speed meters since they reduce the back action noise by
deliberately decreasing the mirror motion signal. To obtain the DC signal for
the differential arm (DARM) control, one needs to add loss to the
interferometer that deteriorates its performance from the ideal case (detailed
analysis has been done in ref. [16] in the case of the Sagnac-type speed
meter). The DARM control for speed meters has already been demonstrated in the
proof-of-principle experiment of the Sagnac speed meter in Glasgow group [17].
The second issue is inherent to the PCSM scheme, i.e. we need to keep the
round-trip phase shift from the input test masses (ITMs) to the polarization
circulation mirror (PCM) at $\pi$ to flip the sign of the second interaction.
In this paper, we focus on this issue and propose a new scheme to control the
phase shift using a dual-retardation waveplate and an auxiliary laser. These
components enable us to obtain a Pound-Drever-Hall (PDH) signal [18] of the
cavity length formed by the PCM and ITMs (Polarization Circulation Cavity:
PCC). The PDH method is commonly used in the GW detectors and gives us high
stability of the PCC length/alignment control. Also, this scheme is well
compatible with the balanced homodyne detection (BHD), which is a signal
detection scheme planned to be used in the future.
The outline of this paper is as follows: in Section II we show the details of
the PCSM and the difficulties of its control. In Section III, we propose a new
control scheme, DRC, and in Section IV we analyze the theoretical performance
of the DRC and compare the shot noise levels of the control signals between
the DRC and another candidate, the dithering control. In Section V, we show a
possible experimental setup for the layout for the demonstration of the DRC.
Finally, we give discussions in Section VI and conclusions in Section VII.

##  II Background and issues

In this section, we review the mechanism of the PCSM, whose detailed study is
shown in Danilishin et al., 2018 [11], and explain the inherent difficulties
in the PCC control.

###  II.1 PCSM

The schematic design of the PCSM is shown in Fig. 1. The main interferometer
is the same as the conventional Fabry-Perot Michelson style position meter,
but the AS port has two polarization optics, a quarter-wave plate (QWP) and a
polarization beam splitter (PBS). These components together with the PCM are
collectively called the polarization circulator (PC). The cavity formed by the
PCM and the input test masses (ITMs) is called the polarization circulation
cavity (PCC). The linear $p$-polarized ($p$-pol) vacuum fluctuation entering
from the AS port is converted into the left-polarization vacuum ($l$-pol,
denoted by $\hat{a}_{l}$) by the QWP, couples with the pump laser and kicks
the mirror randomly. Then the vacuum field (denoted by $\hat{b}_{l}$) returns
to the AS port and is converted into $s$-polarization ($s$-pol). This $s$-pol
beam is reflected by the PBS, thus making a round trip to the PCM. Then it is
converted into right-polarization ($r$-pol) by the QWP before coming back to
the BS as a field denoted by $a_{r}$. The $r$-pol beam kicks the mirror again,
comes back to the AS port (denoted by $\hat{b}_{r}$), and finally goes through
the PBS. The round-trip phase shift between the ITMs and PCM is kept to $\pi$,
so the radiation pressure forces given by $\hat{a}_{r}$ and $\hat{a}_{l}$ have
opposite signs and cancel out each other.

Figure 1:  Configuration of the PCSM [11]. The QWP converts the polarization
state of the vacuum so that it experiences the interferometer twice. The PC is
a set of the QWP, PBS, and PCM, and the PCC is a cavity formed by the PCM and
the ITMs with the QWP and PBS inside. (E)ITMs stand for (end) input test
masses.

###  II.2 Difficulties in PCC control

The PDH method is a commonly used scheme to stabilize the distance between two
mirrors can be stabilized on a nanometer scale by the PDH method [18]. All the
second-generation GW detectors make full use of this technique to control many
degrees of freedom, including the signal recycling cavity (SRC). To control
the SRC in the resonant sideband extraction configuration, radio-frequency
(RF) sidebands generated by an electro-optic modulator (EOM) are used to sense
the length fluctuation of the SRC.

Figure 2:  Conceptual illustration of the dithering control. (a) describes the
AS port of the PCSM. $r$-pol beam from the interferometer transmits through
the PBS as $p$-pol as denoted by \scriptsize1⃝. $l$-pol beam is recycled by
the PC and transmits through the PBS after the second circulation (see Section
II.1). A local oscillator is connected to a PZT behind the PCM to modulate the
position of the PCSM. It generates modulation sidebands around the carrier as
denoted by \scriptsize2⃝. (b) is a phaser diagram for the PBS transmission. A
sum of the carrier \scriptsize1⃝ and \scriptsize2⃝ has a phase component when
the PCM is shifted from the best position. Taking an interference between the
carrier and the generated sidebands, one can get an error signal of the PCC
length.

However, PCC cannot be locked in the same manner, because the IR beam can
circulate inside the PCC twice at most due to the QWP and PBS. It means the
finesse of the PCC is almost $0$. This is a serious problem since one cannot
effectively amplify signal sidebands. In short, one cannot use the PDH method.
As shown above, the PCSM needs a control scheme for the PCC. One simple
solution is modulating the PCM position to generate sidebands on the carrier
and demodulating the output from the AS port. It is what we call ’dithering’
(see Fig. 2). Detuning the arm cavities and leaking some amount of the DC
light (DC offset, see ref. [19]), the DC value of the output is zero if the
round-trip phase shift in the PCC is kept at $\pi$. Then if the position of
the PCM is shifted from the optimal position, non-zero DC signals appear.
Taking a beat between the sidebands and the DC offset, one can obtain an error
signal.
This method is simple but has several problems. In the first place, mechanical
modulation onto the PCM adds noises to the signal sideband, since the
modulation frequency of the mirror dithering is $\sim 10$ kHz at most.
Secondly, one cannot expect a high signal-to-noise ratio (SNR) in the error
signal as shown in Section IV.2. The amount of light reaching the AS port is
limited, so DC offsets are needed to increase the SNR. However, in future GW
detectors, one might not need DC offsets thanks to balanced homodyne detection
(BHD), which is also critical for speed meters. To make full use of this
advantage, a scheme without DC offsets is preferable. Lastly, it is not
sensitive to alignment fluctuations. For these reasons, we propose an
alternative scheme for the PCC that can also yield alignment control signals.

##  III Dual-Retardance Control

###  III.1 Idea

The main obstacle is that the QWP changes the polarization so that the PBS can
transmit half of the IR light. If the waveplate does not change the
polarization state in one-way transmission or keep the state so that the PBS
does not discard any light, one can form a cavity with the PCM and ITMs. It
can be achieved by a dual-retardance waveplate that works as a QWP for IR but
as a HWP for a green (GR) laser.
The retardance of a waveplate is described as:

$\displaystyle\phi_{\mathrm{ret}}=2\pi\frac{(n_{s}-n_{f})d}{\lambda_{0}},$ (4)

where $n_{s}$ and $n_{f}$ are refractive indices along the slow and fast axes
respectively, $d$ is the thickness of the waveplate and $\lambda_{0}$ is the
wavelength of the light. In this simple assumption, when it works as a QWP at
the wavelength of $\lambda_{0}$, it should work as a HWP at the wavelength of
$\lambda_{0}/2$. A HWP does not change the polarization state by round-trip
transmission (see Fig. 3). If we inject a laser with half the wavelength of
the main interferometer beam from behind the PCM with $s$-pol, it can resonate
inside the PCC. Practically, the refractive index has a wavelength dependence,
so it is critical to manufacture the dual-retardation waveplate. We call this
scheme the dual-retardance control (DRC).
The DRC solves the issues discussed in the previous section. With DRC, we can
extract a high-SNR error signal for the PCC without using mechanical
modulation. We can also obtain alignment signals through wave-front sensing.

Figure 3:  Conceptual illustration of the DRC. We prepare a waveplate that
works as a QWP for the carrier and HWP for GR. It changes the polarization
from $s$-pol to $p$-pol or $p$-pol to $s$-pol, but it can be kept to only
$s$-pol between the PCM and the HWP.

###  III.2 Setup

As shown in Fig. 3, the GR laser frequency ($\omega_{\mathrm{GR}}$) should be
phase-locked to the main IR frequency ($\omega_{0}$), with a tunable frequency
offset of $\omega_{\mathrm{off}}$:

$\displaystyle\omega_{\mathrm{GR}}=2(\omega_{0}+\omega_{\mathrm{off}}).$ (5)

Also, the IR and GR beams have to be co-aligned. Then one needs an additional
cavity outside the PCC to make them share the same beam path. For example,
there is a ring cavity to co-align the IR and GR beam paths in Fig. 7. The arm
cavity can also be used for the path-sharing process. The transmissivity of
the BS for GR is set to $\sim 1$ for simplicity.
Even though the paths seem to completely overlap, the optical path length of
the PCC for the IR ($l_{\mathrm{PCC}}^{\mathrm{IR}}$) may not exactly be the
same as that for the GR ($l_{\mathrm{PCC}}^{\mathrm{GR}}$) due to the
dispersion of materials:

$\displaystyle
l_{\mathrm{PCC}}^{\mathrm{GR}}=l_{\mathrm{PCC}}^{\mathrm{IR}}+\delta
l_{\mathrm{PCC}},$ (6)

where $\delta l_{\mathrm{PCC}}$ the difference of the optical path lengths.
Adding a frequency offset $\omega_{\mathrm{off}}$, the round-trip phase shift
in the PCC for the GR is:

$\displaystyle\phi_{\mathrm{GR}}$
$\displaystyle=2\omega_{\mathrm{GR}}l_{\mathrm{PCC}}^{\mathrm{GR}}/c$ (7)
$\displaystyle=2\left[2(\omega_{0}+\omega_{\mathrm{off}})+\delta\omega\right](l_{\mathrm{PCC}}^{\mathrm{IR}}+\delta
l_{\mathrm{PCC}})/c$ (8)
$\displaystyle=\frac{4\omega_{0}l_{\mathrm{PCC}}^{\mathrm{IR}}}{c}+\frac{4\omega_{0}\delta
l_{\mathrm{PCC}}}{c}+\frac{4\omega_{\mathrm{off}}l_{\mathrm{PCC}}^{\mathrm{GR}}}{c}+\frac{2\delta\omega
l_{\mathrm{PCC}}^{\mathrm{GR}}}{c}$ (9)

The first term is a phase shift if there is no dispersion. The second term is
a shift due to the dispersion, and the third term is a phase compensation by
the frequency offset. The fourth term is a phase noise in the Phase Locked
Loop (PLL), which results in the average PCC fluctuation (see $\epsilon$ in
Eq. (7) in ref. [11]).

###  III.3 Lock acquisition

In order to draw the PCC into the operational condition ($\phi_{0}=\pi$), we
need to follow a set of certain steps. We call the procedure ”lock
acquisition”.
The conceptual figure of the lock acquisition is shown in Fig. 4. First, using
the dithering method, the round-trip phase shift of the IR
$\phi_{\mathrm{IR}}$ is locked to $\pi$ (see the denotation (i) in Fig. 4).
This is realized when the first term in Eq. (9) satisfies the following
condition:

$\displaystyle
2\phi_{\mathrm{IR}}=\frac{4\omega_{0}l_{\mathrm{PCC}}^{\mathrm{IR}}}{c}\equiv
0\ (\mathrm{mod}\ 2\pi).$ (10)

One needs to detune the arm cavities to obtain enough DC signals if necessary.
The dithering signal is fed-back to the mechanical actuator on the PCM (a PZT,
for example).
Second, the GR beam is kept at the resonance of the PCC by adding the offset
frequency (see the denotation (ii) in Fig. 4). It corresponds to the round
trip phase shift for the GR in the PCC satisfying below:

$\displaystyle\phi_{\mathrm{GR}}\equiv 0\ (\mathrm{mod}\ 2\pi).$ (11)

The GR PDH signal is fed-back to the frequency actuator on the GR (an acousto-
optic modulator, for example).
Lastly, one can hand over the error signals to the GR PDH which is steeper
than the dithering signal (see the denotation (iii) in Fig. 4). Given the
absolute frequency of the main IR ($\omega_{0}$) and the optical path
difference ($\delta l_{\mathrm{PCC}}$) is stable enough, the round-trip phase
fluctuations for the GR are proportional to the length fluctuation of the PCC.
In this final stage, the GR PDH is fed-back to the PCM. Note that the last
term in Eq. (9):

$\displaystyle\delta\phi_{\mathrm{PCC}}=\frac{2\delta\omega
l_{\mathrm{PCC}}^{\mathrm{GR}}}{c}$ (12)

contributes to the noise of the PCC length. After the handover, the dithering
and DC offset can be removed.

Figure 4: Toy picture of the lock acquisition. a) The DC output of the PBS
transmission. b) GR intra-cavity power. c) The solid red line is the dithering
signal by dithering and the solid green line is the GR PDH signal. After
adding offsets, one can hand over the error signals to the GR PDH which is
steeper than the dithering signal.

##  IV Theoretical performance

###  IV.1 Error signal

Figure 5: PCC as seen by the green field. The HWP is represented in the Jones
matrix, $\hat{J}^{\mathrm{GR}}$. The reflectivity of the PBS is also
represented in the reflectivity matrix, $\hat{\rho}^{\mathrm{GR}}$. We assume
the BS is transparent for the GR for simplicity

In this section, we analyze the electric fields of a cavity with an HWP and
PBS inside to derive the GR PDH signal. We define bases of $p$\- and $s$-pol
electric fields as:

$\displaystyle\mathbf{e}_{p}=\begin{pmatrix}1\\\ 0\end{pmatrix},\
\mathbf{e}_{s}=\begin{pmatrix}0\\\ 1\end{pmatrix}.$ (13)

Symbols used in the analysis are shown in Fig. 5. The reflectivity matrix of
the PBS is:

$\displaystyle\hat{\rho}^{\mathrm{GR}}=\begin{pmatrix}\sqrt{R_{p}^{\mathrm{GR}}}&0\\\
0&\sqrt{R_{s}^{\mathrm{GR}}}\end{pmatrix}$ (14)

where $R_{s}^{\mathrm{GR}}$ and $R_{p}^{\mathrm{GR}}$ is the power
reflectivity for $s$-pol and $p$-pol of the PBS.
$r_{0}^{\mathrm{GR}},t_{0}^{\mathrm{GR}}$ are the amplitude reflectivity and
transmissivity of the PCM. The Jones matrix for the $45^{\circ}$ rotated HWP
can be written as:

$\displaystyle\hat{J}^{\mathrm{GR}}=\frac{1}{2}\begin{pmatrix}1+e^{-2i\delta\phi}&1-e^{-2i\delta\phi}\\\
1-e^{-2i\delta\phi}&1+e^{-2i\delta\phi}\end{pmatrix},$ (15)

where $\delta\phi$ is the retardation error.
Relations between the fields are written as:

$\displaystyle{\bf\it E}_{1}$ $\displaystyle=t_{0}^{\mathrm{GR}}{\bf\it
E}_{0}+r_{0}^{\mathrm{GR}}{\bf\it E}_{3},$ (16) $\displaystyle{\bf\it E}_{2}$
$\displaystyle=e^{i\Phi/2}\hat{J}^{\mathrm{GR}}\hat{\rho}^{\mathrm{GR}}{\bf\it
E}_{1},$ (17) $\displaystyle{\bf\it E}_{3}$
$\displaystyle=e^{i\Phi/2}\hat{\rho}^{\mathrm{GR}}\hat{J}^{\mathrm{GR}}{\bf\it
E}_{2},$ (18) $\displaystyle{\bf\it E}_{r}$
$\displaystyle=-r_{0}^{\mathrm{GR}}{\bf\it E}_{0}+t_{0}^{\mathrm{GR}}{\bf\it
E}_{3},$ (19)

where $\Phi$ is the round-trip phase shift in the PCC. Solving those
equations, the reflectivity for the $s$-pol at the anti-reflection side of the
PCM is:

$\displaystyle r_{s\rightarrow
s}(\Phi^{{}^{\prime}})=\frac{E_{0,s}}{E_{\mathrm{r},s}}$ (20)
$\displaystyle=-r_{0}^{\mathrm{GR}}+\frac{(t_{0}^{\mathrm{GR}})^{2}(R_{s}^{\mathrm{GR}}\cos\delta\phi
e^{i\Phi^{{}^{\prime}}}-r_{0}^{\mathrm{GR}}R_{p}^{\mathrm{GR}}R_{s}^{\mathrm{GR}}e^{2i\Phi^{{}^{\prime}}})}{\det
M}$ (21)

where

$\displaystyle\Phi^{{}^{\prime}}=\Phi-\delta\phi,$ (22)

and

$\displaystyle\det M$
$\displaystyle=1-r_{0}^{\mathrm{GR}}(R_{s}^{\mathrm{GR}}+R_{p}^{\mathrm{GR}})\cos\delta\phi
e^{i\Phi^{{}^{\prime}}}$
$\displaystyle\qquad\qquad+(r_{0}^{\mathrm{GR}})^{2}R_{s}^{\mathrm{GR}}R_{p}^{\mathrm{GR}}e^{2i\Phi^{{}^{\prime}}}.$
(23)

Here we assumed the reflectivity of the ITMY and the transmissivity of the BS
are $\sim 1$ for simplicity. While optical components inside the PCC may have
their own losses, here we assumed that all the loss is concentrated in the PCM
(denoted $\mathcal{L}_{\mathrm{GR}}$ in Fig. 6). We set $R_{p}^{\mathrm{GR}}$
to 0, which means $p$-pol generated by the retardation error is discarded from
the PBS. The imperfection of the $s-$pol reflectivity is counted as a loss on
the PCM:

$\displaystyle\mathcal{L}_{s}^{\mathrm{GR}}=1-R_{s}^{\mathrm{GR}}.$ (24)

We show the imaginary part of Eq. (21) in Fig. 6 with various round trip
losses, which decrease the slope of the error signals.

Figure 6:  Imaginary part of the PCC reflectivity. Red curves show the
imaginary part of Eq. 21 with various round-trip losses with retardation error
of $\lambda_{0}/300$. The black curve is an error signal without any
retardation error and loss.

###  IV.2 Estimation of the shot noise level of the DRC

We compare the shot noise levels of two methods, the dithering control and
DRC. The detailed analysis is shown in Appendix A. Using Eq. (51) and choosing
realistic parameters (see Table 1), the ratio of the shot noises of the two
methods becomes:

$\displaystyle\frac{S_{L}^{\mathrm{Dither}}}{S_{L}^{\mathrm{DRC}}}\sim
7.4\times 10^{4}.$ (25)

This large ratio comes from the advantages of using a cavity: the
amplification of the phase change by a factor of the finesse of the cavity and
the amount of the local oscillator power that can be used for control.

##  V Experimental demonstration of DRC

Figure 7:  Design of an experimental setup for demonstrating the DRC. The
basic configuration is a FPMI with 15 cm arm cavities. The GR laser is phase-
locked to the main IR laser and injected from the AR side of the PCM.

An experimental setup to demonstrate the DRC is shown in Fig. 7. Possible
parameters for IR and GR optics are shown in Table 1 and 2, respectively. We
aim to confirm the DRC works. The GR laser is generated by the second harmonic
generation and phase-locked to the main IR laser. The basic design is a Fabry-
Pérot Michelson Interferometer (FPMI) with $15$-cm-long rigid arm cavities
with flat ITMs and curved ETMs. The FPMI part is controlled by the pre-
modulation method as used in all the current GW detectors. The error signal of
the PCC obtained by the GR PDH is fed back to the PCM. The GR laser frequency
is tunable by changing the frequency offset in the PLL.
A small fraction of the main IR is picked off after the EOM and injected from
the AR side of the ETMY. This light gets phase-modulated through an EOM to
generate sidebands that play the role of pseudo-GW signals.
The expected transfer function from the phase modulation to the DARM output is
shown in Fig. 8. Given the carrier is resonant in the arm cavities, the
amplitude reflectivity of a single arm cavity can be written as:

$\displaystyle r(\Omega)$
$\displaystyle\simeq\frac{\gamma_{1}-\gamma_{2}+i\Omega}{\gamma_{1}+\gamma_{2}-i\Omega},$
(26)

where $\Omega$ is the frequency of an audio sideband. $\gamma_{1}$ and
$\gamma_{2}$ are defined as:

$\displaystyle\gamma_{1}$
$\displaystyle\equiv\frac{cT_{\mathrm{ITM}}}{4l_{\mathrm{arm}}}\
(\mathrm{cavity\ pole}),$ (27) $\displaystyle\gamma_{2}$
$\displaystyle\equiv\frac{c\mathcal{L}_{\mathrm{arm}}}{4l_{\mathrm{arm}}}.$
(28)

$T_{\mathrm{ITM}}$ is the power transmissivity of the input mirror.
$\mathcal{L}_{\mathrm{arm}}$ is the round-trip power loss of the arm cavity
and $l_{\mathrm{arm}}$ is the length of the arm cavity. Denoting the round-
trip power loss in the PCC as $\mathcal{L}_{\mathrm{PCC}}^{{}^{\prime}}$, the
transfer function is proportional to:

$\displaystyle\mathrm{(Output)}$
$\displaystyle\propto\frac{1-\cos\delta\phi_{\mathrm{PCC}}(1-\mathcal{L}_{\mathrm{PCC}}^{{}^{\prime}})r(\Omega)}{2}$
$\displaystyle\simeq\frac{\gamma_{2}+\mathcal{L}_{\mathrm{PCC}}\gamma_{1}/2-i\Omega}{\gamma_{1}-i\Omega},$
(29)

where $\mathcal{L}_{\mathrm{PCC}}$ is the effective-total PCC loss including
the round-trip phase fluctuation in the PCC $\delta\phi_{\mathrm{PCC}}$:

$\displaystyle\mathcal{L}_{\mathrm{PCC}}$
$\displaystyle\simeq\mathcal{L}_{\mathrm{PCC}}^{{}^{\prime}}+\frac{(\delta\phi_{\mathrm{PCC}})^{2}}{2}$
(30)
$\displaystyle=2(\mathcal{L}_{\mathrm{BS}}+\mathcal{L}_{\mathrm{QWP}}+\mathcal{L}_{\mathrm{PBS}}+T_{\mathrm{SPBS}}+R_{\mathrm{PPBS}})$
$\displaystyle\qquad\qquad+\mathcal{L}_{\mathrm{PCM}}+\mathcal{L}_{\mathrm{align}}+\mathcal{L}_{\mathrm{mis}}+\frac{(\delta\phi_{\mathrm{PCC}})^{2}}{2}.$
(31)

In Eq. (30), we assume that $\delta\phi_{\mathrm{PCC}}$ is so small that its
cosine is approximated as:

$\displaystyle\cos(\delta\phi_{\mathrm{PCC}})\simeq
1-\frac{(\delta\phi_{\mathrm{PCC}})^{2}}{2}.$ (32)

Definitions of each term in Eq. (31) are shown in Table 3. Note that losses in
the PCC optical path are doubled due to the polarization circulation, except
for the PCM. The mode-mismatching due to the PCM misalignment and the Schnupp
asymmetry is also counted as losses. The final term
$\delta\phi_{\mathrm{PCC}}$ is the length fluctuation of the PCC.
Eq. (29) means the PCC loss generates a zero at:

$\displaystyle\gamma_{\mathrm{cut}}$
$\displaystyle=\gamma_{2}+\frac{\mathcal{L}_{\mathrm{PCC}}\gamma_{1}}{2}$ (33)
$\displaystyle=\frac{c}{4l_{\mathrm{arm}}}\left[\mathcal{L}_{\mathrm{arm}}+\frac{\pi\mathcal{L}_{\mathrm{PCC}}}{\mathcal{F}}\right],$
(34)

where $\mathcal{F}$ is the finesse of the arm cavity:

$\displaystyle\mathcal{F}\equiv\frac{2\pi}{T_{\mathrm{ITM}}}.$ (35)

In Fig. 8, we show transfer functions of both lossless and loss-included
cases. The cutoffs at low frequencies are generated by the losses. The
$\propto 1/f$ structure above the cavity pole is due to the first-order low-
pass nature of the arm cavities. Note that even in the lossless case (gray
line), we still see a cutoff. It is caused by the transmission of the ETM that
is necessary to inject the artificially phase-modulated light.
The $\propto f$ structure in the transfer function will be observed if we
realize the proposed experiment with the designed parameters. Through the
measurement, we can evaluate the performance of the DRC.

Figure 8:  Expected transfer functions for the DARM noise injection. The red
curve shows the transfer function when the PCC is assumed to have a loss. The
gray curve shows the lossless case.

##  VI Discussion

One of the potential issues is the long-term stability of the dispersion of
the QWP. It might change due to the heat effect of the laser or environmental
temperature fluctuation. Also, beam jitters might also be a source of the
noise. It is necessary to test the stability of the PCC control and check how
frequently the dithering control needs to be used to ensure the round trip
phase of the PCC to be $\pi$.
From the perspective of practical implementation, the DRC might conflict with
the lock acquisition scheme of the existing detectors. KAGRA, for example,
injects auxiliary GR lasers from the center part of the interferometer. To
avoid the GR leaking and resonating inside the arm cavity, the DRC sets the
ITM transmissivity for the GR as small as possible. Hence it is necessary to
find a compromise between them.

##  VII Conclusion

In this article, we propose a feasible control scheme for the PCSM using a
dual-retardation waveplate and auxiliary laser. We name it DRC. The DRC makes
it possible to control the PCC length and alignment. In the DRC, we can get
error signals with a higher SNR than the dithering control. Also, the DRC is
compatible with the BHD because we do not need the DC offset anymore after the
full PCC lock is achieved. After the experimental demonstration of the DRC
with rigid cavities, we will proceed to the fully-suspended systems to realize
the PCSM in the future GW detectors such as the Einstein Telescope [20].

###### Acknowledgements.

We thank Stefan Danilishin, Marc Eisenmann, Kentaro Komori, and Kentaro Somiya
for fruitful discussions.

Table 1:  Parameters for IR used for the design of the experiment. Parameters | value | Note
---|---|---
$\lambda_{0}$[nm] | 1064 | Nd:YAG
$P_{0}$ [mW] | 50 | IR laser intensity
$P_{\mathrm{pick}}$ [$\mu$W] | 125 | Pick-off laser intensity111for phase-noise injection
$T_{\mathrm{ITM}}$ | 0.004 | ITM transmissivity222Fused Silica substrate
$T_{\mathrm{ETM}}$ | 30 ppm | ETM transmissivity
$T_{\mathrm{PCM}}$ | $1\ \%$ | PCM transmissivity
$R_{\mathrm{ITM}}$ [m] | $\infty$ | ITM radius of curvature
$R_{\mathrm{ETM}}$ [m] | 1.5 | ETM radius of curvature
$R_{\mathrm{PCM}}$ [m] | 1 | PCM radius of curvature
$l_{\mathrm{arm}}$ [m] | 0.15 | Arm cavity length
$l_{\mathrm{michx}}$ [m] | 0.075 | Length from the BS to ITMX
$l_{\mathrm{michy}}$ [m] | 0.125 | Length from the BS to ITMY
$l_{\mathrm{PCC}}$ [m] | 0.307 | Mean PCC length
$f_{\mathrm{m}}$ [MHz] | 47.5 | RF sideband frequency333used for the FPMI control
$A$ [nm] | 0.1 | Modulation amplitude 444see Appendix A
$P_{c}$ [$\mathrm{\mu}$W] | 100 | DC offset power at the AS port444see Appendix A
$\mathcal{F}$ | $\sim 1500$ | Finesse
$f_{c}$ |  $3.2\times 10^{5}$ [Hz] | Cavity pole
$f_{\mathrm{cut}}$ |  $1.7\times 10^{4}$ [Hz] | Cutoff frequency
Table 2:  Parameters for GR. Parameters | value | Note
---|---|---
$\lambda_{\mathrm{GR}}$ [nm] | 532 | Wavelength of the GR laser
$P_{\mathrm{GR}}$ [mW] | 20 | GR laser intensity
$T_{\mathrm{PCM}}$ | $0.01$ | PCM transmissivity
$T_{\mathrm{ITM}}$ [m] |  $<10$ ppm | ITM transmissivity
$l_{\mathrm{PCC}}$ [m] | 0.332 | Length from the PCM to ITMY
$\beta_{\mathrm{m}}$ | 0.2 | Modulation index111see Appendix A
$\delta\phi^{\mathrm{GR}}_{\mathrm{ret}}$ | $2\pi\lambda_{\mathrm{GR}}/300$ | QWP retardation error for GR
$\mathcal{L}_{\mathrm{GR}}$ | 3 % | Total losses in the PCC
$\mathcal{F}_{\mathrm{GR}}$ | 150 | Finesse of the PCC

##  Appendix A Shot noise estimation

In the case of the PDH method, the reflection power before demodulation can be
written as [21]:

$\displaystyle P=P_{\mathrm{DC}}+D\delta
l\sin\omega_{\mathrm{m}}t+\mathcal{O}(2\omega_{\mathrm{m}}).$ (36)

$\delta l$ is the length fluctuation of the cavity we want to control and
$\omega_{\mathrm{m}}$ is the frequency of sidebands generated by an EOM. $D$
[W/m] corresponds to the slope of the error signal, which is proportional to
the carrier and sideband power $P_{\mathrm{c}},P_{\mathrm{s}}$ and the
imaginary part of the reflectivity $\mathrm{Im}[r(\omega)]$:

$\displaystyle
D\propto\sqrt{P_{\mathrm{c}}P_{\mathrm{s}}}\mathrm{Im}[r(\omega)].$ (37)

$P_{\mathrm{DC}}$ is the DC power, which is the source of the shot noise. The
shot noise can be written in the single-sided form as:

$\displaystyle S_{\mathrm{shot}}=\sqrt{2e\frac{e\eta
P_{\mathrm{DC}}}{\hbar\omega_{0}}}\ \mathrm{[A/\sqrt{Hz}]},$ (38)

where $e$ is the elementary charge and $\eta$ is the quantum efficiency of the
photo detector [A/W]. Hence the shot-noise-equivalent length noise is:

$\displaystyle S_{L}=\frac{S_{\mathrm{shot}}}{D}.$ (39)

In the case of the DRC, one can calculate its shot noise level in the same
manner. The imaginary part of the reflectivity $r_{s\rightarrow s}$ can be
expressed around $\Phi^{{}^{\prime}}$ as:

$\displaystyle\left.\mathrm{Im[r_{s\rightarrow
s}(\Phi^{{}^{\prime}})]}\right|_{\Phi^{\prime}=0}$
$\displaystyle\simeq\left.\frac{\mathrm{d}\mathrm{Im}[r_{s\rightarrow
s}(\Phi^{\prime})]}{\mathrm{d}\Phi^{\prime}}\right|_{\Phi^{\prime}=0}\times\delta\Phi^{{}^{\prime}}$
(40) $\displaystyle=\left.\frac{\mathrm{d}\mathrm{Im}[r_{s\rightarrow
s}(\Phi^{\prime})]}{\mathrm{d}\Phi^{\prime}}\right|_{\Phi^{\prime}=0}\times\frac{4\omega_{0}\delta
l_{\mathrm{PCC}}}{c}.$ (41)

The slope amplitude $D^{\mathrm{DRC}}$ can be written as:

$\displaystyle D^{\mathrm{DRC}}\delta l_{\mathrm{PCC}}$
$\displaystyle=4\sqrt{P_{c}P_{s}}\left.\mathrm{Im[r_{s\rightarrow
s}(\Phi^{{}^{\prime}})]}\right|_{\Phi^{\prime}=0}$
$\displaystyle=\frac{8\beta_{\mathrm{m}}\omega_{0}P_{\mathrm{GR}}}{c}\left.\frac{\mathrm{d}\mathrm{Im}[r_{s\rightarrow
s}(\Phi^{\prime})]}{\mathrm{d}\Phi^{\prime}}\right|_{\Phi^{\prime}=0}\delta
l_{\mathrm{PCC}}$
$\displaystyle=\frac{8\beta_{\mathrm{m}}\omega_{0}P_{\mathrm{GR}}\xi}{c}\delta
l_{\mathrm{PCC}},$ (42)

where

$\displaystyle\xi$
$\displaystyle\equiv\left.\frac{\mathrm{d}\mathrm{Im}[r_{s\rightarrow
s}(\Phi^{\prime})]}{\mathrm{d}\Phi^{\prime}}\right|_{\Phi^{\prime}=0}.$ (43)

$\beta_{\mathrm{m}}$ is the modulation index of the EOM, $P_{\mathrm{GR}}$ is
the GR laser power. The DC power can be written as:

$\displaystyle P_{\mathrm{DC}}^{\mathrm{DRC}}=|r_{s\rightarrow
s}(0)|^{2}P_{\mathrm{GR}}.$ (44)

Substituting Eq. (42) and (44), for (39), one can obtain the shot noise level
of the DRC:

$\displaystyle S^{\mathrm{DRC}}$ $\displaystyle=\sqrt{2e\frac{e\eta
P_{\mathrm{DC}}^{\mathrm{DRC}}}{2\hbar\omega_{0}}}/D^{\mathrm{DRC}}.$ (45)

Also in the case of the dithering control, one can use the same approach as
[21]. The slope amplitude $D^{\mathrm{Dither}}$ is:

$\displaystyle D^{\mathrm{Dither}}\delta l_{\mathrm{PCC}}$ $\displaystyle\sim
2J_{1}(\beta)P_{c}\mathrm{Im}[1-e^{-i\delta\phi_{\mathrm{PCC}}}]$ (46)
$\displaystyle=\frac{8\pi\beta}{\lambda_{0}}P_{c}\delta l_{\mathrm{PCC}},$
(47)

where $P_{c}$ is the carrier power leaking from the BS to the QWP by the DC
offset, $A$ is the amplitude of the PCM modulation, $\lambda_{0}$ is the
wavelength of the main laser, $J_{n}$ is the $n$-th order Bessel functions and
$\beta$ is the modulation index. For the transformation from Eq. (46) to Eq.
(47) we have used

$\displaystyle J_{1}(\beta)$ $\displaystyle=\frac{\beta}{2}\equiv\frac{2\pi
A}{\lambda_{0}},$ $\displaystyle\delta\phi_{\mathrm{PCC}}$
$\displaystyle=\frac{4\pi\delta l_{\mathrm{PCC}}}{\lambda_{0}}.$

$P_{\mathrm{DC}}^{\mathrm{Dither}}$ can be written as:

$\displaystyle P_{\mathrm{DC}}^{\mathrm{Dither}}=|1-F(\psi_{0})|^{2}P_{c},$
(48)

where $F$ is the arm cavity reflectivity:

$\displaystyle
F(\psi_{0})=-r_{1}+\frac{t_{1}^{2}r_{1}e^{-i\psi_{0}}}{1-r_{1}r_{2}e^{-i\psi_{0}}}$
(49)

and $\psi_{0}$ is the round-trip phase shift of the arm cavity for the IR.
Substituting Eq. (47) and (48) for (39), one can obtain the shot noise level
of the dithering control:

$\displaystyle S^{\mathrm{Dither}}$ $\displaystyle=\sqrt{2e\frac{e\eta
P_{\mathrm{DC}}^{\mathrm{Dither}}}{\hbar\omega_{0}}}/D^{\mathrm{Dither}}.$
(50)

The ratio of the two control methods can be written as:

$\displaystyle\frac{S_{L}^{\mathrm{Dither}}}{S_{L}^{\mathrm{DRC}}}$
$\displaystyle=\frac{4\beta_{m}}{\beta}\xi\left|\frac{1-F(\psi_{0})}{r_{s\rightarrow
s}(0)}\right|\sqrt{\frac{P_{\mathrm{GR}}}{P_{c}}}.$ (51)

Table 3:  Losses and errors used for the simulation. Parameters | value | Note
---|---|---
$\mathcal{L}_{\mathrm{arm}}$ | 130 ppm | Arm cavity round-trip loss including ETM transmissivity, $30$ ppm
$\mathcal{L}_{\mathrm{BS,QWP,PBS,PCM}}$ | 50 ppm | PCC optics losses
$T_{\mathrm{SPBS}}$ | 1 % | PBS $s$-pol transmissivity
$R_{\mathrm{PPBS}}$ | 1 % | PBS $p$-pol reflectivity
$\mathcal{L}_{\mathrm{align}}$ | 1 % | Loss by PCM misalignment
$\mathcal{L}_{\mathrm{mis}}$ | 0.15 % | Mode mismatch between X and Y arm
$\delta\phi_{\mathrm{PCC}}$ [rad] | $10^{-7}$ | PCC round-trip phase fluctuation by the PLL noise

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##  Appendix A Nomenclature

Symbols used in the manuscript are summarized in the following tables.

Table 1: Nomenclature $G$ | Gravitational constant
---|---
$k_{B}$ | Boltzmann constant
$M_{\odot}$ | Solar mass
$r$ | Orbital radius from the Sun
$z$ | Distance above disk midplane
$v_{K}$ | Keplerian velocity, $\sqrt{GM_{\odot}/r}$
$\Omega_{K}$ | Keplerian angular velocity, $v_{K}/r$
$\rho_{g}$ | Protosolar disk gas density at midplane
$\Sigma_{g}$ | Gas surface density of protosolar disk
$P_{g}$ | Gas pressure in midplane of protosolar disk
$T_{g}$ | Temperature of gas in protosolar disk
$c_{g}$ | Sound speed in protosolar disk
$h_{g}$ | Vertical scale height of gas in protosolar disk
$\lambda_{g}$ | Mean free path in gas
$\eta_{g}$ | Pressure parameter (equation 2)
$\alpha_{g}$ | Alpha parameter for a turbulent disk
$\gamma_{g}$ | Adiabatic index for gas
$\bar{m}$ | Mean molecular mass
$\nu_{g}$ | Kinematic gas viscosity
$\sigma_{H2}$ | Collision cross section for molecular hydrogen
$g_{\Sigma},g_{T}$ | Coefficients describing disk model
$u$ | Gas velocity
$u_{hw}$ | Headwind velocity
$u_{turb}$ | Turbulent eddy velocity, $\sqrt{\alpha_{g}}c_{g}$
$s$ | Grain radius, particle in disk or splash particle
$\rho_{s}$ | Grain density
$t_{stop}$ | Stopping time
$St$ | Stokes number, dimensionless stopping time
|  $St=t_{stop}\Omega_{K}$
${\bf F}_{D}$ | Drag force
$a_{D}$ | Acceleration on a particle from drag
$R_{a}$ | Radius of planetesimal
$D_{a}$ | Diameter of planetesimal
$M_{a}$ | Mass of planetesimal
$\rho_{a}$ | Density of planetesimal
$g_{a}$ | Surface gravitational acceleration, $GM_{a}/R_{a}^{2}$
$v_{esc,a}$ | Escape velocity, $\sqrt{2GM_{a}/R_{a}}$
$v_{a}$ | Gravitational velocity $\sqrt{GM_{a}/R_{a}}$
$\Omega_{a}$ | Inverse gravitational timescale $v_{a}/R_{a}$
$Y_{a}$ | Material strength
${\bm{\Omega}}_{spin,a}$ | Spin vector of planetesimal
$t_{cross}$ | Crossing time, $D_{a}/u_{hw}$
$F_{coh}$ | Cohesive pull-off force
${\cal B}$ | Parameter describing surface roughness
$\gamma_{c}$ | Cohesive contact energy per unit area
Table 1: Nomenclature-continued $\rho_{pj}$ | Projectile density
---|---
$a_{pj}$ | Projectile radius
$m_{pj}$ | Projectile mass
$u_{pj}$ | Projectile velocity
$v_{ej}$ | Ejecta velocity
$R_{cr}$ | Crater radius
$M()$ | Mass ejecta function
$M_{ej}$ | Total ejecta mass
$x$ | Distance to impact site
$C_{1},H_{2},k$ | Coefficients for crater scaling
$\mu,\nu$ | Exponents for crater scaling
$Re$ | Reynolds number
$M\\!a$ | Mach number
$Fr$ | Froude number
$K\\!n$ | Knudsen number
$\pi_{2},\pi_{3},\pi_{4}$ | Dimensionless crater scaling parameters
$U$ | Wind velocity distant from planetesimal
$t$ | Time
$w$ | Impact parameter
$\tilde{\bf r}_{s}$ | Particle position in units of $R_{a}$
$\tilde{\bf v}_{s}$ | Particle velocity in units of $v_{a}$
$\tilde{\bf a}_{s}$ | Particle acceleration in units of $g_{a}$
$\tilde{\bf u}$ | Wind velocity in units of $v_{a}$
$\tilde{\bf u}_{\rm pf}$ | Flow field of potential flow
$\tilde{\bf u}_{\delta}$ | Flow field within boundary layer
$\tau$ | Dimensionless time, $\tau=t\Omega_{a}$
$\tilde{t}_{s}$ | Dimensionless stopping time; $t_{stop}\Omega_{a}$
$\tilde{\bm{\omega}}_{a}$ | Planetesimal spin vector in units of $\Omega_{a}$
$\theta_{sw}$ | Angle between spin axis and wind direction
$\delta$ | Boundary layer thickness
$\tilde{x}_{b}$ | Distance from stagnation point along surface
$\tilde{y}_{b}$ | Distance from surface
$\tilde{\bf u}_{pj}$ | Projectile velocity in frame
|  rotating with the surface
$\tilde{\bf v}_{ej}$ | Ejecta velocity in frame rotating
|  with the surface
$\hat{\bf d}$ | Unit vector pointing downrange
|  with respect to the impact velocity
$\hat{\bf n}$ | Surface normal unit vector
$\zeta_{ej}$ | Elevation angle of ejecta velocity
$\beta$ | Angle describing how far a splash particle
|  moves with respect to the surface
Table 1: Nomenclature-continued $D_{erode}$ | Diameter of a planetesimal with
---|---
|  a critical erosion rate
$D_{a,vw}$ | Diameter of a planetesimal with
|  $v_{esc,a}=u_{hw}$
$f_{p}$ | Mean mass density in disk particles
|  divided by gas density
$\xi_{p}$ | Mass fraction of particles that can
|  impact a planetesimal
$F_{m}$ | Particle flux
$h$ | Depth
$\frac{dh_{erode}}{dt},\frac{dh_{acc}}{dt}$ | Accretion and erosion rates (depth/time)
$\delta_{h}$ | Depth of redistributed material divided
|  by planetesimal radius


of all finite subsets $t$ of $\text{Reg}\times\omega\times\text{Ord}$ such
that for every $\alpha\in\text{dom}(t)$,
$t(\alpha):=\\{(m,\beta)\mid(\alpha,m,\beta)\in t\\}$ is an injective function
from a finite subset of $\omega$ to $\alpha$.

######  Definition 4.22.

Let $P_{2}$ denote the set of all $t\in P_{1}$ such that:

  1. (a)

For every $\alpha\in\text{dom}(t)$,
$\mathrm{cof^{\prime}}\alpha\in\text{dom}(t)$ and
$\text{dom}\bigl{(}t(\mathrm{cof^{\prime}}\alpha)\bigr{)}\supseteq\text{dom}\bigl{(}t(\alpha)\bigr{)}$.

  2. (b)

If $(\alpha_{0},\ldots,\alpha_{n-1})$ is the increasing sequence enumeration
of $\text{dom}(t)\setminus\kappa_{0}$, then there exist $m\geq 1$ and $j\leq
n-1$ such that:

    1. (i)

For each $k<j$, $\text{dom}\bigl{(}t(\alpha_{k})\bigr{)}=m+1$.

    2. (ii)

For each $k$ with $j\leq k<n$, $\mathrm{dom}\bigl{(}t(\alpha_{k})\bigr{)}=m$.

The values $m$ and $\alpha_{j}$ in (b) are unique for $t$ and can thus be
denoted $m(t):=m$ and $\alpha(t):=\alpha_{j}$. Note that
$\bigl{(}\alpha(t),m(t)\bigr{)}$ is the point that needs to be filled in next
in order to extend $t$. For any $t\in T\subseteq P_{2}$, let

$\mathrm{Suc}_{T}(t):=\\{\beta\mid
t\cup\\{\bigl{(}\alpha(t),m(t),\beta\bigr{)}\\}\in T\\}.$

######  Definition 4.23.

Let $P_{3}$ denote the set of all pairs $(s,T)$ such that $s\in T\subseteq
P_{2}$ and:

  1. (a)

For every $t\in T$:

    1. (i)

Either $t\supseteq s$ or $t\subseteq s$.

    2. (ii)

$\text{dom}(t)=\text{dom}(s)$.

    3. (iii)

_(tree-like)_ If $t=r\cup\\{\bigl{(}\alpha(r),m(r),\beta)\\}$, then $r\in T$.

  2. (b)

For every $t\in T$ with $s\subseteq t$:

    1. (i)

If $\alpha(t)$ is of type 1, then $\mathrm{Suc}_{T}(t)\in\Phi_{\alpha(t)}.$

    2. (ii)

If $\alpha(t)$ is of type 2 and
$m(t)\in\text{dom}\bigl{(}t(\mathrm{cof^{\prime}}(\alpha(t)))\bigr{)}$, then

$\text{Suc}_{T}(t)\in\Phi_{\alpha(t),t\bigl{(}\mathrm{cof^{\prime}}(\alpha(t))\bigr{)}\,(m(t))}.$

$s$ is called the _stem_ of $(s,T)$.

Let $\mathbb{P}$ denote $P_{3}$ with the following partial order. For
$(r,R),(t,T)\in P_{3}$, let $(r,R)\leq(t,T)$ if
$r{\upharpoonright}\kappa_{0}\supseteq t{\upharpoonright}\kappa_{0}$,
$R{\upharpoonright}\bigl{(}\text{dom}(t)\setminus\kappa_{0}\bigr{)}\subseteq
T$ and $\mathrm{dom}(r)\supseteq\mathrm{dom}(t)$. Let $I$ denote the class of
finite subsets of $\mathsf{Reg}$ closed under $\mathrm{cof^{\prime}}$. For any
$s\in I$, let

$\mathbb{P}_{s}:=\\{(t,T)\in P_{3}\mid\text{ dom}(t)\subseteq s\\}.$

We have
$(t,T){\upharpoonright}\mathbb{P}_{s}:=\bigl{(}t{\upharpoonright}s,\\{u{\upharpoonright}s\mid
u\in T\\}\bigr{)}\in P_{3}$ for all $(t,T)\in P_{3}$ and $s\in I$ [7, Lemma
2.4]. The next lemma shows how $\mathbb{P}_{s}$ can be factored.

######  Lemma 4.24.

[1, Theorem 2.5] For any $s\in I$ and any strongly compact $\kappa_{\xi}\in
s$, $\mathbb{P}_{s}$ is forcing equivalent313131I.e., the Boolean completions
are isomorphic. to a forcing of the form
$\mathbb{P}_{s\cap\kappa_{\xi}}\ast\dot{\mathbb{Q}}$, where
$\mathbb{P}_{s\cap\kappa_{\xi}}$ forces that $\dot{\mathbb{Q}}$ does not add
any bounded subset of $\kappa_{\xi}$.

Let
$D:=\\{(\alpha,n,\beta)\in\text{Reg}\times\omega\times\mathsf{Ord}\mid\beta<\alpha\\}$.
In the next definition, we work with proper class functions to simplify the
notation. One can obtain a formally correct definition by restricting each
function to its support.

######  Definition 4.25.

  1. (1)

Let $\mathcal{G}$ be the group of permutations $g\colon D\rightarrow
D$323232We use the domain $D$ instead of
$\text{Reg}\times\omega\times\mathsf{Ord}$ to ensure that $g$ is supported on
a set. such that there exists a sequence of permutations $g_{\alpha}$ of
$\alpha\in\mathsf{Reg}$ with:

    1. (a)

$g(\alpha,n,\beta)=\bigl{(}\alpha,n,g_{\alpha}(\beta)\bigr{)}$ for all
$\alpha\in\mathsf{Reg}$, $\beta<\alpha$ and $n\in\omega$.

    2. (b)

$\mathrm{supp}(g_{\alpha}):=\\{\beta<\alpha\mid g_{\alpha}(\beta)\neq\beta\\}$
is finite for all $\alpha\in\mathsf{Reg}$.

    3. (c)

$\mathrm{supp}(g):=\\{\alpha\in\mathsf{Reg}\mid
g_{\alpha}\neq\operatorname{id}\\}$ is finite.

  2. (2)

For each $g\in\mathcal{G}$, let $\mathbb{P}^{g}\subseteq P_{3}$ be the set of
all $(t,T)\in P_{3}$ such that $\mathrm{dom}(t)\supseteq\mathrm{supp}(g)$ and
for all $\alpha\in\mathrm{dom}(t)$:

    1. (a)

$\mathrm{dom}\bigl{(}t(\alpha)\bigr{)}=\text{dom}\bigl{(}t(\mathrm{cof^{\prime}}(\alpha))\bigr{)}$.

    2. (b)

If $\alpha\geq\kappa_{0}$, then
$\mathrm{ran}\bigl{(}t(\alpha)\bigr{)}\supseteq\\{\beta\in\text{supp}(g_{\alpha})\mid\exists
r\in T\,\bigl{(}\beta\in\mathrm{ran}(r(\alpha))\bigr{)}\\}.$

  3. (3)

For each $g\in\mathcal{G}$ and $(t,T)\in P_{3}$, let
$g^{\prime}(t,T):=(g[t],\\{g[t^{\prime}]\mid t^{\prime}\in T\\})$.

######  Lemma 4.26.

[7, Lemma 3.2] For each $g\in\mathcal{G}$, $\mathbb{P}^{g}$ is a dense
subclass of $\mathbb{P}$ and $g^{\prime}\colon\mathbb{P}^{g}\to\mathbb{P}^{g}$
is an automorphism of $\mathbb{P}^{g}$.

Since the restriction of $\mathbb{P}$ to any ordinal is a complete subforcing,
the Boolean algebra $\mathbb{B}:=\mathbb{B}(\mathbb{P})$ of regular open
subsets of $\mathbb{P}$ is complete.333333A class Boolean algebra is called
_complete_ if every subset has a supremum. Let
$\iota\colon\mathbb{P}\rightarrow\mathbb{B}$ denote the canonical $\leq$\- and
$\perp$-homomorphism. It is easy to see that every $g\in\mathcal{G}$ induces
an automorphism $\bar{g}$ of $\mathbb{B}$ defined by
$\bar{g}(U)=g^{\prime}[U\cap\mathbb{P}^{g}]_{\mathrm{down}}$, where
$U\in\mathbb{B}$ is a regular open subset of $\mathbb{P}$ and
$U^{\prime}_{\mathrm{down}}$ denotes the downward closure of any
$U^{\prime}\subseteq\mathbb{P}$. Using density of $\mathbb{P}^{g}$, it can be
show that one can recover $g$ from $\bar{g}$ and $\overline{g\circ
h}=\bar{g}\circ\bar{h}$ for all $g,h\in\mathcal{G}$. Thus
$\bar{\mathcal{G}}:=\\{\bar{g}\mid g\in\mathcal{G}\\}\cong\mathcal{G}$.

######  Definition 4.27.

  1. (1)

Let $\mathrm{fix}(s):=\\{\bar{g}\in\bar{\mathcal{G}}\mid g\in\mathcal{G},\
\forall\alpha\in s\ g_{\alpha}=\operatorname{id}_{\alpha}\\}$ for $s\in I$.

  2. (2)

Let $\bar{\mathcal{F}}$ be the normal filter of subgroups of
$\bar{\mathcal{G}}$ generated by $\mathrm{fix}(s)$ for all $s\in I$.

As usual, for any $\mathbb{B}$-name $\dot{x}$, let

$\mathrm{sym}(\dot{x}):=\\{\bar{g}\in\bar{\mathcal{G}}\mid\bar{g}(\dot{x})=\dot{x}\\}.$

A $\mathbb{B}$-name $\dot{x}$ is called _symmetric_ if $\mathrm{sym}(\dot{x})$
is in $\bar{\mathcal{F}}$. Let $\mathrm{HS}_{\bar{\mathcal{F}}}$ be the class
of all hereditarily symmetric $\mathbb{B}$-names. We say that $s\in I$
_supports_ a name $\dot{x}\in\mathrm{HS}_{\bar{\mathcal{F}}}$ if
$\mathrm{fix}(s)\subseteq\mathrm{sym}(\dot{x})$. The next lemma is important
in the proof.

######  Lemma 4.28.

[7, Lemma 3.3] Suppose that $\varphi$ is a formula with $n$ free variables and
$\mathrm{fix}(s)$ supports
$\dot{x}_{0},\ldots,\dot{x}_{n-1}\in\mathrm{HS}_{\bar{\mathcal{F}}}$, where
$s\in I$. Then for every $(t,T)\in P_{3}$:

$(t,T)_{\iota}\Vdash\varphi(\dot{x}_{0},\ldots,\dot{x}_{n-1})\iff((t,T){\upharpoonright}\mathbb{P}_{s})_{\iota}\Vdash\varphi(\dot{x}_{0},\ldots,\dot{x}_{n-1}).$

Fix a $\mathbb{B}$-generic filter $G$ over $V$. The symmetric model is defined
as $V(G)=\\{\dot{x}^{G}\mid\dot{x}\in\text{HS}_{\bar{\mathcal{F}}}\\}$. Note
that for any $s\in I$,
$G{\upharpoonright}\mathbb{P}_{s}:=\\{(t,T){\upharpoonright}\mathbb{P}_{s}\mid(t,T)_{\iota}\in
G\\}$ is a $\mathbb{P}_{s}$-generic filter over $V$.

######  Theorem 4.29.

  1. (1)

[1, Lemma 2.4] For any set of ordinals $X\in V(G)$, there exists some $s\in I$
with $X\in V[G\upharpoonright\mathbb{P}_{s}]$.

  2. (2)

[1, Corollary 2.10] $V(G)$ is a model of ${\sf ZF}$ where every infinite
cardinal has cofinality $\omega$. Moreover, the closure of $\vec{\kappa}$
equals the class of all uncountable cardinals of $V(G)$.

####  4.4.2. Absoluteness fails over Gitik’s model

The following variants of $\mathbf{c}$ will play a key role. For any cardinal
$\kappa$, let

$\displaystyle\mathbf{c}_{\kappa}=\sup\\{\lambda\in{\sf
Card}\mid\lambda\leq_{i}\kappa^{\omega}\\}.$

We will see that each of $\mathsf{A}_{\mathbb{C}^{*}}$ and
$\mathsf{A}_{\mathbb{R}_{*}}$ implies that $\mathbf{c}_{\kappa}>\kappa$ for
some infinite cardinal $\kappa$, while in Gitik’s model,
$\mathbf{c}_{\kappa}=\kappa$ for all infinite cardinals $\kappa$. Recall the
notation $\mathbf{c}^{\mathbb{C}^{\kappa}}$ from Lemma 4.12.

######  Lemma 4.30.

$\mathbf{c}_{\kappa}=\mathbf{c}_{\omega}^{\mathbb{C}^{\kappa}}$ for all
infinite cardinals $\kappa$.

###### Proof.

To show $\mathbf{c}_{\omega}^{\mathbb{C}^{\kappa}}\leq\mathbf{c}_{\kappa}$,
suppose that $\dot{f}$ is a $\mathbb{C}^{\kappa}$-name and
$p\in\mathbb{C}^{\kappa}$ forces that $\dot{f}\colon\gamma\to 2^{\omega}$ is
injective. Since $\mathbb{C}^{\kappa}$ is nice, we have
$\gamma\leq_{i}\kappa^{\omega}$ and thus $\gamma\leq\mathbf{c}_{\kappa}$, as
desired. To show
$\mathbf{c}_{\omega}^{\mathbb{C}^{\kappa}}\leq\mathbf{c}_{\kappa}$, it
suffices to construct an injective function from $\kappa^{\omega}\cap V$ into
the set of Cohen reals over $V$ in a $\mathbb{C}^{\kappa}$-generic extension
$V[G]$. Let $\langle i_{\alpha}\mid\alpha<\kappa\rangle$ denote the sequence
of the first digits of the Cohen reals added by $G$. Let $S$ denote a set of
size $\kappa^{\omega}$ of injective functions in $\kappa^{\omega}$ such that
for any $f,g\in S$ with $f\neq g$, there exist infinitely many $n\in\omega$
with $f(n)\neq g(n)$. For each $f\in S$, consider the Cohen real $x_{f}$ over
$V$ defined by $x_{f}(n)=i_{f(n)}$. These reals are pairwise distinct by the
choice of $S$. ∎

######  Lemma 4.31.

In Gitik’s model, $\mathbf{c}_{\kappa}=\kappa$ for all infinite cardinals
$\kappa$.

###### Proof.

Suppose that $\kappa$ is an infinite cardinal in $V(G)$ and
$f\colon\gamma\to{}^{\omega}\kappa$ is an injective function in $V(G)$. It
suffices to show $\gamma<(\kappa^{+})^{V(G)}$. By Theorem 4.29 (2),
$(\kappa^{+})^{V(G)}=\kappa_{\xi}$ for some $\xi$, where $\kappa_{\xi}$ is a
strongly compact cardinal in $V$. We will show $\gamma<\kappa_{\xi}$. By
Theorem 4.29 (1), there is some $s\in I$ with $f\in
V[G{\upharpoonright}\mathbb{P}_{s}]$. We may assume $\kappa_{\xi}\in s$. By
Lemma 4.24, $\mathbb{P}_{s}$ is forcing equivalent to a forcing of the form
$\mathbb{P}_{s\cap\kappa_{\xi}}\ast\dot{\mathbb{Q}}$, where
$\mathbb{P}_{s\cap\kappa_{\xi}}$ forces that $\dot{\mathbb{Q}}$ does not add
new bounded subsets of $\kappa_{\xi}$. Let $\lambda$ be an inaccessible
cardinal in $V$ with $\max(s\cap\kappa_{\xi})<\lambda<\kappa_{\xi}$. Since
$|\mathbb{P}_{s\cap\kappa_{\xi}}|<\lambda$ and
$\mathbb{P}_{s\cap\kappa_{\xi}}$ forces that $\dot{\mathbb{Q}}$ does not add
new bounded subsets of $\kappa_{\xi}$, $\lambda$ remains inaccessible in
$V[G{\upharpoonright}\mathbb{P}_{s}]$. Since $f\in
V[G{\upharpoonright}\mathbb{P}_{s}]$, we have
$\gamma<\lambda<\kappa_{\xi}=(\kappa^{+})^{V(G)}$, as desired. ∎

Note that $\mathbf{c}_{\omega}=\omega$ implies that no cardinal
characteristics of the reals exist. Using the previous lemma, we obtain the
failure of the above absoluteness principles in Gitik’s model.

######  Theorem 4.32.

$\mathsf{A}_{\mathbb{C}^{*}}$, $\mathsf{A}_{\mathbb{R}_{*}}$ and
$\mathsf{A}_{\mathbb{H}_{*}}$ fail in Gitik’s model.

###### Proof.

Each of $\mathsf{A}_{\mathbb{C}^{*}}$ and $\mathsf{A}_{\mathbb{R}_{*}}$
implies $\mathbf{c}_{\kappa}>\kappa$ for all $\omega$-strong limit cardinals
$\kappa$ by Theorem 4.14 (1) and Lemma 4.30. But in Gitik’s model,
$\mathbf{c}_{\kappa}=\kappa$ for all such $\kappa$ by Lemma 4.31. Moreover,
$\mathsf{A}_{\mathbb{H}_{*}}$ implies that the bounding number equals
$\omega_{1}$ by Theorem 4.19. But in Gitik’s model, the bounding number does
not exist, since $\mathbf{c}_{\omega}=\omega$ by Lemma 4.31. ∎

##  5\. Open problems

The absoluteness principle $\mathsf{A}_{\mathrm{Col}(\omega,*)}$ for the class
of collapse forcings $\mathrm{Col}(\omega,\kappa)$ for arbitrary cardinals
$\kappa$ is consistent even with ${\sf ZFC}$ [9]. Our main open problem is:

######  Problem 5.1.

Are the principles $\mathsf{A}_{\mathbb{C}^{*}}$,
$\mathsf{A}_{\mathbb{R}_{*}}$ and $\mathsf{A}_{\mathbb{H}^{(*)}}$ consistent?

Our results in Sections 4.2-4.4 indicate some properties that a model of these
principles must have. In particular, we argued that they fail in Gitik’s model
[7]. One could aim to show that they also fail in all set generic extensions
of this model. For $\mathsf{A}_{\mathbb{C}^{*}}$ and
$\mathsf{A}_{\mathbb{R}_{*}}$, it suffices that set forcing preserves
$\mathbf{c}_{\lambda}=\lambda$ for sufficiently large cardinals $\lambda$, but
the argument for Lemma 4.30 does not work for forcings which are not
wellordered. Can the construction of Gitik’s model [7] or Gitik’s alternative
construction of a model where all uncountable cardinals are singular from an
almost huge cardinal [8] be adapted to obtain models of the above absoluteness
principles? If these principles are consistent, we would like to understand
the structure of their models. For instance, does the ${\sf HOD}$ of a model
of $\mathsf{A}_{\mathbb{C}^{*}}$ contain large cardinals?

Section 4.1 shows that Cohen and random extensions have different theories.
This suggests to study the same problem for other classical c.c.c. forcings.
For instance:

######  Problem 5.2.

Do extensions by sufficiently large forcings in $\mathbb{C}^{*}$ and
$\mathbb{H}^{(*)}$ necessarily have different theories?

The problem of differentiating theories for different forcing extensions is
interesting from the viewpoint of the modal logic of forcing [10]. Switches
$\varphi_{0},\dots,\varphi_{n}$ are called _independent_ if any choice of
truth values for these sentences can be realised in the relevant generic
extensions.

######  Problem 5.3.

Is it provable in ${\sf ZF}$ for each natural number $n$ that there exist $n$
independent switches?

Regarding preservation of cardinals in Section 3, we ask whether all
$(\theta,1)$-narrow forcings already preserve $\theta^{+}$.343434A recent
argument of Karagila, Schilhan and the second-listed author shows that this is
the case for $\theta=\omega$. This would improve Lemma 3.2. Is every
$(\theta,1)$-narrow forcing $\theta$-narrow? Is every $\theta$-narrow forcing
uniformly $\theta$-narrow? Is every wellordered $\theta^{+}$-c.c. forcing
uniformly $\theta$-narrow? Cardinal preservation is also a problem for many
classical forcings. For example, we do not know whether Sacks forcing
preserves $\omega_{1}$ even if one assumes that $\omega_{1}$ is inaccessible
to reals. The problem is to show that every new real has a name in
$H_{\omega_{1}}$. If this is the case, then one can show that $\omega_{1}$ is
preserved using capturing as in [4, 21]. Regarding random algebras in Section
3.4, note that we would have a shortcut for some of our results if
$\mathbb{R}_{\kappa}$ is $\mathbb{C}^{\kappa}$-linked. However, it is open
whether even ${\sf ZFC}$ provides a negative answer to the next problem.

######  Problem 5.4.

Is $\mathbb{R}_{\omega_{1}}$ $\sigma$-linked?

A further natural question regarding $\mathbb{R}_{\kappa}$ is whether it is
provable in ${\sf ZF}$ that $\mathbb{R}_{\kappa}$ does not have uncountable
antichains for any infinite cardinal $\kappa$. Regarding Section 3.5.2, one
can ask for a finer understanding of the effect of ${\rm Add}(\kappa,1)$. For
instance, is it consistent that ${\rm Add}(\omega_{3},1)$ collapses
$\omega_{1}$ and $\omega_{3}$ while preserving $\omega_{2}$? It would further
be interesting to extend the results about Hechler forcing in Section 4.3 to
obtain a precise characterisation of the bounding number in finite support
iterations of Hechler forcing or arbitrary ordinal length even in ${\sf ZFC}$.

Gitik’s model in Section 4.4 is a useful test case to study forcing over
choiceless models. What are the answers to the following questions for Gitik’s
model: Does every nonatomic $\sigma$-closed forcing collapse $\omega_{1}$?
Does the dominating number exist in some generic extension? Is the bounding
number $\omega_{1}$ in any generic extension where it exists? Does Sacks
forcing preserve $\omega_{1}$? Can one increase ${\sf HOD}$ by forcing?

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