# Low-loss polarization control in fiber systems for quantum computation

Tomohiro Nakamura  1 Takefumi Nomura  1 Mamoru Endo  1,2 He Ruofan  1 Takahiro
Kashiwazaki  3 Takeshi Umeki  3 Jun-ichi Yoshikawa  2 and Akira Furusawa1,2
1Department of Applied Physics, School of Engineering, The University of
Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
2Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing,
2-1 Hirosawa, Wako, Saitama 351-0198, Japan
3NTT Device Technology Labs, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi,
Kanagawa 243-0198, Japan<EMAIL_ADDRESS>
###### Abstract

Optical quantum information processing exploits interference of quantum light.
However, when the interferometer is composed of optical fibers, degradation of
interference visibility due to the finite polarization extinction ratio
becomes a problem. Here we propose a method to optimize interference
visibility by controlling the polarizations to a crosspoint of two circular
trajectories on the Poincaré sphere. Our method maximizes visibility with low
optical loss, which is essential for quantum light, by using fiber stretchers
as polarization controllers. We also experimentally demonstrate our method,
where the visibility was maintained basically above 99.9% for three hours
using fiber stretchers with an optical loss of 0.02 dB (0.5%). Our method
makes fiber systems promising for practical fault-tolerant optical quantum
computers.

††journal: oe

##  1 Introduction

Quantum computers are gathering more attentions as next-generation
technologies, and extensively researched with various physical systems.
Optical implementation is one of the promising candidates. Optical quantum
computer will be composed of beamsplitter networks by which quantum light
interacts, followed by detectors. In particular, cluster states, which are
entangled states that serve as resources for measurement-based quantum
computation, are generated by beamsplitter interactions of squeezed states. In
recent years, ultra-large-scale cluster states are experimentally demonstrated
by employing the time-domain multiplexing method, which utilizes Mach-Zehnder
interferometers with asymmetric lengths of arms[1, 2, 3, 4, 5]. The time-
domain multiplexing method takes advantage of the flying feature of light. The
demonstrated size of entanglement, therefore, is several orders of magnitude
larger than that with other quantum systems such as superconducting qubits,
showing the promise of optical quantum computing.

In order to exploit the strong quantumness of light, high interference
visibilities are required in the interferometers. For this purpose, both the
spatial mode and the polarization mode of the light beams should be matched.
In the case of free-space systems, even though it is possible to match the
spatial mode of all light beams at all interference points, it is time-
consuming. Furthermore, the beam positions shift over time to reduce
interference visibility, even though this problem can be solved by introducing
auto-alignment systems. Due to time constraints, visibilities are often
compromised, e.g., the experiment in Ref. [2] was performed with 98%
visibilities. On the other hand, optical fibers are suitable for constructing
maintenance-free systems. Using optical fibers, it is possible to achieve
almost 100% visibilities as long as the polarization in the fiber matches.

However, even if polarization-maintaining fibers are used in the fiber
systems, it is difficult to perfectly maintain polarizations due to the finite
polarization extinction ratio (PER) of the fiber components. Every fiber
component such as a fiber beamsplitter, a connector, or a connection point
with fiber fusion shows a finite PER. When an interferometer is constructed
using fibers, visibility, therefore, degrades due to polarization mismatch,
which is called visibility fading[6, 7].

For classical light, visibilities can be improved by using commercial inline-
type fiber polarization controllers. There are basically two types of fiber
polarization controllers. In one type, a non-polarization-maintaining fiber is
wound around spools and the angle of the spools are adjusted to cause
appropriate stress-induced birefringence, resulting in desired polarization
change[8]. In the other type, a non-polarization-maintaining fiber is stressed
by piezo actuators to cause birefringence for desired change of
polarization[9]. In these polarization controllers, non-polarization-
maintaining fibers are subjected to stresses, which generate microbends to
cause optical losses [typically 0.08 dB (1.8%) or higher][10]. However, for
classical light, these levels of optical losses are not serious problems.

For quantum light, polarization controllers are also needed to solve the
visibility fading problem. However, since the quantum light is vulnerable to
optical losses, the polarization controllers must have low optical losses. For
example, there is a study showing that $-10$ dB of squeezing is required for
fault-tolerant quantum computation[11]. In order to achieve this, the losses
of the entire systems must be sufficiently lower than 0.5 dB (10%).

In this paper, we propose a method to optimize visibility in a fiber
interferometer by controlling the polarization to a crosspoint of two circular
trajectories drawn on a Poincaré sphere (Fig. 1). We also experimentally
demonstrate visibility maximization with this method, where the visibility is
maintained basically above 99.9% for three hours. The circular trajectories
appear when the two light beams are off the polarization-maintaining axis of
polarization-maintaining fibers and when the fibers are stretched or heated.
We call this the circle-circle crosspoint (CCC) method in the following. The
CCC method is especially suitable for the quantum light because it can be
implemented with low losses. The CCC method can be implemented with fiber
stretchers or heaters, which do not cause severe microbends resulting in
optical losses. Another important point is that, by employing the CCC method,
fiber systems can be constructed solely with the polarization-maintaining
fibers. In contrast, the conventional methods employ non-polarization-
maintaining fibers for the polarization controlling parts, while other parts
are constructed with polarization-maintaining fibers. Because there is no
connection between the polarization-maintaining fibers and the non-
polarization-maintaining fibers, it is expected that the optical losses tend
to be lower. For the experimental demonstration, we made fiber stretchers and
measured the optical losses, which were as low as 0.02 dB (0.5%). As a result,
the visibility in three hours was basically maintained above 99.9% with the
polarization control, while it sometimes dropped to 98% without the
polarization control. Although this experiment is demonstrated for a single
Mach-Zehnder interferometer, the CCC method can be applied to more complex
interferometers having multiple interference points by maximizing visibilities
from the upstream to the downstream of the light paths.

Figure 1: Principle of the CCC method. (a) Interferometer with fiber
stretchers inserted. (b) Two circular trajectories drawn by polarization A and
B on the Poincaré sphere when the voltages applied to the piezo actuators of
the fiber stretchers are varied. In the CCC method, polarization states are
controlled to a crosspoint of the two circles to optimize visibility. Poincaré
sphere is explained in Section 2. V, H, R and L correspond to vertical,
horizontal, right-handed circular and left-handed circular polarizations,
respectively.

Section 2 describes the Poincaré sphere representation of the polarization
states. Section 3 discusses the change in polarization states at the output of
polarization-maintaining fibers. Section 4 illustrates the effect of a fiber
stretcher on polarization. Section 5 shows low-loss polarization control using
fiber stretchers. In Section 6, the fabrication method of the fiber stretchers
for polarization controllers are described. In Section 7, the experimental
method for polarization control using the CCC method is described. Section 8
demonstrates the experimental result. Section 9 shows the comparison of the
CCC method and the conventional methods. Section 10 presents the conclusion of
this study.

##  2 Poincaré sphere representation

We introduce the Poincaré sphere, on which we can describe polarization
states. Let $x$ and $y$ components of the electric field of monochromatic
light propagating in the $z$ direction be

$\displaystyle E_{x}$ $\displaystyle=a_{x}\cos(kz-\omega_{c}t-\Gamma_{x}),$
(1a) $\displaystyle E_{y}$
$\displaystyle=a_{y}\cos(kz-\omega_{c}t-\Gamma_{y}),$ (1b)

where $a_{x}$ and $a_{y}$ are $x$ and $y$ components of the electric field
amplitudes, $\omega_{c}$ is the angular frequency of light, $k$ is the wave
number, and $\Gamma_{x}$ and $\Gamma_{y}$ are the phases of $x$ and $y$
components. Stokes parameters are defined as follows:

$\displaystyle S_{0}$ $\displaystyle=a_{x}^{2}+a_{y}^{2},$ (2a) $\displaystyle
S_{1}$ $\displaystyle=a_{x}^{2}-a_{y}^{2},$ (2b) $\displaystyle S_{2}$
$\displaystyle=2a_{x}a_{y}\cos\Gamma,$ (2c) $\displaystyle S_{3}$
$\displaystyle=2a_{x}a_{y}\sin\Gamma,$ (2d)

where $\Gamma=\Gamma_{y}-\Gamma_{x}$. Since $S_{0}$ corresponds to the optical
power and $S_{1}^{2}+S_{2}^{2}+S_{3}^{2}=S_{0}^{2}$, a set of vectors
$(S_{1},S_{2},S_{3})$ forms a sphere called the Poincaré sphere when the
optical power is constant (Fig. 2(a))[12]. Also, Stokes parameters
$S_{1},S_{2},S_{3}$ are expressed as follows:

$\displaystyle S_{1}$ $\displaystyle=S_{0}\cos 2\chi\cos 2\psi,$ (3a)
$\displaystyle S_{2}$ $\displaystyle=S_{0}\cos 2\chi\sin 2\psi,$ (3b)
$\displaystyle S_{3}$ $\displaystyle=S_{0}\sin 2\chi,$ (3c)

where $\psi$ is the azimuth and $\chi$ is the ellipticity [13]. For elliptical
polarization states, the azimuth $\psi$ and the ellipticity $\chi$ are
expressed in Fig. 2(b).

Figure 2: Polarization states. (a) Poincaré sphere. (b) The azimuth $\psi$ and
the ellipticity $\chi$ in elliptical polarization states.

##  3 Polarization states through polarization-maintaining fibers

The polarization-maintaining fiber has a core and cladding with a stress-
applying part to generate a birefringence as shown in Fig. 3(a), and realizes
an anisotropic refractive index distribution. The direction in which the
stress-applying part exists is called the slow axis, and the direction
orthogonal to the slow axis is called the fast axis. In this paper, we call
the linear polarization state parallel to the slow axis as the polarization
state of the point V on the Poincaré sphere. When the linearly polarized input
light is parallel to either the slow axis or the fast axis, the polarization
is maintained through the fiber. Now we consider the output polarization state
from the polarization maintaining fiber when the input linear polarization
state is $\theta$ rotated from the slow axis. Because the slow-axis component
and the fast-axis component undergo different phase changes, the electric
field of the output light from a fiber, corresponding to Eq. (1), becomes,

$\displaystyle E_{x}$
$\displaystyle=a\cos\theta\cos(kz-\omega_{c}t-\Gamma_{x}),$ (4a)
$\displaystyle E_{y}$
$\displaystyle=a\sin\theta\cos(kz-\omega_{c}t-\Gamma_{y}),$ (4b)

where $a^{2}=a_{x}^{2}+a_{y}^{2}$. In this case, the Stokes parameters Eq. (2)
are

$\displaystyle S_{0}$ $\displaystyle=a^{2},$ (5a) $\displaystyle S_{1}$
$\displaystyle=a^{2}\cos(2\theta),$ (5b) $\displaystyle S_{2}$
$\displaystyle=a^{2}\sin(2\theta)\cos\Gamma,$ (5c) $\displaystyle S_{3}$
$\displaystyle=a^{2}\sin(2\theta)\sin\Gamma.$ (5d)

In the Poincaré sphere, $\theta$ and $\Gamma$ are depicted in Fig. 3(b).

Figure 3: Polarization-maintaining fibers and polarization changes. (a) Cross
section of polarization-maintaining fiber. (b) $\Gamma$ and $\theta$ in the
Poincaré sphere.

The phase difference $\Gamma$ changes depending on the length of the fiber and
the refractive index difference between the slow axis and the fast axis, while
$\theta$ does not change. The fiber length where $\Gamma$ changes by $2\pi$ is
called the beat length. The beat length is about 4 mm with ordinary fibers for
the communication wavelength band[14]. Changes in the temperature or stress
applied to the fiber slightly alter the fiber length and the refractive index
difference, resulting in changes in the phase difference $\Gamma$ at the fiber
exit. With this changes of $\Gamma$, ($S_{2},S_{3}$) draws a circular
trajectory from Eq. (5c) and Eq. (5d). The radius of the circle depends on
$\theta$. At $\theta=0$, the trajectory of the polarization state converges to
the point V in the Poincaré sphere and the polarization is no longer affected
by the disturbance.

Now we consider the polarization state of light propagating through multiple
polarization-maintaining fibers connected by connectors with misalignments as
shown in Fig. 4(a). Suppose that the light propagating through fiber 0 is in a
linearly polarized state and the polarization plane is coincident with the
polarization-maintaining axis. Fiber 0 is sequentially connected to Fiber 1,
Fiber 2, and Fiber 3 by Connector 1, Connector 2, and Connector 3,
respectively. The angles between slow axes of two connected fibers at
Connector 1, Connector 2, and Connector 3 are expressed by
$\theta_{1},\theta_{2},\theta_{3}$, respectively. Although connectors are
supposed here, in actual situations, there are various components that cause
polarization-axis mismatches, such as incomplete fiber fusion splices. Let
$\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}$ be the phase differences between
the slow axis and the fast axis which light receives through Fiber 1, Fiber 2,
and Fiber 3, respectively. When disturbances are added to Fiber 1, Fiber 2,
and Fiber 3, $\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}$ shift, respectively.
The output light from Fiber 3 is measured with a polarimeter. The Poincaré
sphere for the output light is defined so that the slow axis of Fiber 3
corresponds to the point V. When any of $\Gamma_{1}$, $\Gamma_{2}$, and
$\Gamma_{3}$ changes, the polarization state draws a circular trajectory on
the Poincaré sphere. In particular, in Fig. 4(b), the circular trajectory by
changing $\Gamma_{2}$ more than $2\pi$ is shown by a blue circle for various
$\Gamma_{1}$ and $\Gamma_{3}$. In this figure, $\theta_{1}$, $\theta_{2}$, and
$\theta_{3}$ are set to 2, 2, and 5 degrees for the sake of visual
understandability. These are typical amounts of misalignments when connected
by connectors. When $\Gamma_{1}$ is slightly altered, the radius of the circle
changes. When $\Gamma_{3}$ is slightly altered, the center of the circle
changes. Thus, the radius and the center point of the circle can be changed by
adding disturbances before and after the fiber.

Figure 4: Polarization changes through polarization-maintaining fibers with
multiple connectors. (a) Setup with misalignments $\theta_{1}$, $\theta_{2}$,
$\theta_{3}$ at connectors. (b) Circular trajectories when a disturbance of
$2\pi$ is added to $\Gamma_{2}$. Here $S_{0}=1$. A slight modification in
$\Gamma_{1}$ changes the radius of the circle, and a slight change in
$\Gamma_{3}$ shifts the center of the circle.

##  4 Phase and Polarization changes by fiber stretchers

Fiber stretchers pull fibers to change the optical path lengths which are
typically used for controlling the phases of light. A fiber stretcher has a
structure, for example, where a fiber is wound around a cylindrical object
with a variable radius. The radius of the cylindrical object is typically
changed by a piezo actuator, which is controlled electrically. Other phase
control devices for fiber systems include Electro-Optical Modulator (EOM)[15],
which uses the electro-optic effect. EOMs have the advantage of a wide
bandwidth of tens of gigahertz. However, fiber EOMs have the disadvantage of
severe optical losses of typically 3 dB (50%)[16]. In contrast, fiber
stretchers have a low bandwidth of about 100 kHz, which is determined by the
bandwidth of the piezo actuator. However, it has the advantage that the
optical loss is typically less than 0.2 dB (4.5%)[17]. Fiber stretchers are
chosen when phase controls are needed for quantum light, which is vulnerable
to optical losses. In optical quantum computers, quantum light is combined
with classical light for measurement. If appropriate ancillary states can be
prepared[18, 19], universal quantum computation will be realized by fast
measurement switching on the cluster state[20, 21, 22]. Fast phase controls,
therefore, are only needed for classical light, which is done by EOMs, and
slow phase controls are sufficient for quantum light, which is done by fiber
stretchers.

When a polarization-maintaining fiber is stretched by a fiber stretcher, the
phase difference $\Gamma$ between the slow and fast axes changes. If the
polarization state of the input light, therefore, does not coincide with the
polarization-maintaining axes, the polarization state changes, and draws a
circular trajectory on the Poincaré sphere as discussed in Section 3. Note
that similar polarization changes can be also induced by temperature changes
using such as heaters. We will describe in Section 5 how to use these
polarization changes for maximizing visibility.

##  5 Maximizing visibility by the circle-circle crosspoint method

Figure 5: The CCC method in an interferometer with multiple finite PER
components. (a) An interferometer setup for considering the changes in
polarization states. (b) Polarization changes on the Poincaré sphere when
light passes through the finite PER components in the interferometer. (c) Two
circular trajectories drawn by A4 and B4 when voltages applied to Fiber
stretcher 1 and Fiber stretcher 2 are changed. In the CCC method, A4 and B4
are controlled to a crosspoint of the two circles.

So far, we have considered a single optical path system with multiple fiber
elements connected. In this section, we will consider the case of an optical
interferometer as shown in Fig. 5(a). The Mach-Zehnder interferometer consists
of two beamsplitters. In fiber systems, these beamsplitters have finite PERs
(typically 25 dB or more). In addition, these paths of the interferometer
often have fiber stretchers for phase locking and tapping fiber beamsplitters
for monitoring the state of the light, which also have finite PERs.
Furthermore, when connecting these fiber elements, if the fibers are fused
together, the PER of fusion splices is typically about 35 dB, and if they are
connected with connectors, there are typically axis misalignments of 2 degrees
(PER = 29 dB) to 5 degrees (PER = 21 dB).

The interferometer has two optical paths, and the polarization state of each
optical path changes independently as shown in Fig. 5(b). We suppose that the
polarization state A0 of the input to the interferometer is at point V on the
Poincaré sphere. As the light passes through Beamsplitter 1, the light splits
into two paths, but due to the finite PER of the fiber beamsplitter, the
polarization state shifts slightly to A1 and B1 on the Poincaré sphere,
respectively. Then, due to the finite PER of the elements in each path, the
polarization state of one changes from A1 to A2, A3, and A4, while the other
changes from B1 to B2, B3, and B4. When light enters Beamsplitter 2, the
visibility decreases as the distance between polarization states A4 and B4 on
the Poincaré sphere increases.

In the CCC method, both paths of the interferometer have fiber stretchers (or
heaters) for polarization control. By driving the fiber stretchers, the
polarizations of A4 and B4 draw two different circular trajectories as shown
in Fig. 5(c). In most cases, these two circular trajectories have crosspoints.
By adjusting the fiber stretchers to the crosspoint of these circles,
polarization states A4 and B4 are matched. Visibility, therefore, is maximized
by this polarization control.

If the circular trajectories do not have any crosspoint, visibility does not
reach 100%. However, as explained in Section 4, the radius and the center of
the circle trajectory can be moved by placing a device that causes a phase
difference between the slow and fast axes before and after the fiber stretcher
is inserted. The trajectories of the two circles, therefore, can be changed to
have crosspoints.

If the polarization control is performed by heaters or Peltier devices instead
of fiber stretchers, larger dynamic ranges are realized. This larger dynamic
range may be useful in some situations.

The bandwidth of the fiber stretcher is typically up to about 100 kHz, and
heaters or Peltier devices are much slower. But since the polarization control
for visibility maximization only needs to follow the slow drift of
polarization, high speed is not necessary and these slow polarization
controllers are sufficient to achieve visibility maximization.

In addition to visibility maximizations, phase lock is necessary for optical
quantum information processing. However, polarization control in the CCC
method causes a significant phase drift. Polarization control and phase
control, therefore, should be performed in two steps. Firstly, polarization
control is performed with phase control off, and the polarization controllers
are held at the optimal polarization states. Secondly, the phase control is
turned on. Phase control can be performed by connecting a small fiber
stretcher in series with the polarization controller. Optical quantum
information processing is performed after the preparation by this two-step
control (Fig. 6). In principle, it is also possible to perform polarization
control and phase control in a single fiber stretcher. This is because the
voltage scales applied to the fiber stretcher for polarization control and
phase control are significantly different. It is, therefore, possible to first
perform polarization control on a larger voltage scale and then perform phase
control on a smaller voltage scale.

Figure 6: Processes for operating a quantum computer.

In the CCC method, optimized polarization states are shifted from the slow
axis of the polarization-maintaining fiber. Hence, the polarization dependence
of the beamsplitter becomes the nonideality of the interferometer, which
degrades the quality of the quantum computation. In order to assess these
problems, we conducted some measurements. When various polarizations of light
are input to a fiber beamsplitter 954P (Evanescent optics) with a coupling
ratio of 49:51, the variation of the coupling ratio was $\pm$0.3%, which is
usually negligible.

Although the CCC method is explained for a simple Mach-Zehnder interferometer,
the CCC method can be applied to systems that have multiple interference
points, by maximizing visibilities from the upstream to the downstream of the
system.

##  6 Making of fiber stretchers

The fiber stretchers for the CCC method should be able to change the phase
difference between the slow and fast axes larger than $2\pi$. Due to the small
difference of refractive indices between slow and fast axes of the
polarization-maintaining fiber, shifts of the phase difference by $2\pi$
empirically correspond to a few hundred wavelengths of phase shifts. Hence,
larger fiber stretchers are required in comparison with those for phase
control which changes the phase by a few wavelengths.

Although fiber stretchers are commercially available, in this study we made
our own fiber stretchers because we wanted to customize the number of fiber
windings and the radius at which the fibers are wound. The fiber stretcher we
fabricated utilizes a 40 mm diameter cylindrical piezo actuator PT140.70
(PI)[23], around which a polarization-maintaining fiber for communication
wavelengths SM15-PS-U25D (Fujikura) is wrapped about 10 times, as shown in
Fig. 7. The wound fiber is fixed with Polyimide tape. A voltage of up to 1000
V can be applied to the piezo actuator and a maximum diameter contraction is
12 um. Hence, if all the piezoelectric contraction contributed to fiber
stretching, the fiber would be stretched by 0.37 mm. The fiber stretcher we
fabricated draws a single circle at about 600 V, which corresponds to about 7
um contraction. This contraction causes the change of the fiber length by
about 200 um, which is much smaller than the beat length about 4 mm of the
polarization-maintaining fiber[24, 14]. Hence, the change of the phase
difference is mainly due to stress-induced birefringence, rather than the
change in the fiber length.

The measured insertion loss of the fiber stretcher was 0.02 dB (0.5%), which
is acceptable for quantum light. Furthermore, this optical loss is almost
unchanged when the voltage applied to the fiber stretcher is changed from 0 V
to 1000 V.

Figure 7: Fiber stretcher for polarization control. (a) Structure of the fiber
stretcher. (b) Photo of the fiber stretcher.

The actual circular trajectories are measured by the setup in Fig. 8 and
obtained polarization trajectories are shown in Fig. 9. Firstly, only Light 1
was fed and the voltage applied to Fiber stretcher 1 was changed, resulting in
Circular trajectories (i) in Fig. 9. Secondly, only Light 2 was fed and the
voltage applied to Fiber stretcher 2 was varied, resulting in Circular
trajectories (ii) in Fig. 9. In the CCC method, polarizations are controlled
to the crosspoints of the two trajectories. Note that these circular
trajectories are not obtained from the experimental setup in Sec. 7, where all
fiber components are connected by fusion splices and thus we cannot see the
polarization trajectory of each optical path like Fig. 9.

Figure 8: Setup for the polarimeter measurement of the circular trajectories
of polarization drawn on the Poincaré sphere.  Figure 9: Two circular
trajectories obtained by the setup in Fig. 8, which have cross points. Here
$S_{0}=1$.

##  7 Experimental Method

We experimentally demonstrate visibility maximization by the CCC method with
the setup shown in Fig. 10. The polarization state of Light 1 is adjusted as
much as possible so that the light is linearly polarized to match the slow
axis of the polarization-maintaining fiber. Light 1 is split into Light 3 and
Light 4 by passing through Beamsplitter 1, which has a coupling ratio of
50:50. Light can also be input from the Light 2 path, although Light 2 is
blocked in this experiment. Light 3 and Light 4 interfere at Beamsplitter 2
with a coupling ratio of 50:50 to generate Light 5 and Light 6. Beamsplitter 3
with the coupling ratio of 0.5:99.5 is inserted in the path of Light 3 so that
the interference phase between Light 1 and Light 2 can be monitored by
Photodetector 1 and locked by feedback control. A mini fiber stretcher is
inserted in the Light 4 path to scan the phase, by which an interference
signal between Light 3 and Light 4 is generated to calculate the visibility.
In both the Light 3 and Light 4 paths, Fiber stretcher 1 and Fiber stretcher
2, which are explained in Section 6, are inserted to control the polarization.
To monitor the interference signal, Light 5 is input to Beamsplitter 4 with
the coupling ratio of 0.5:99.5, and the output of 0.5% is detected by
Photodetector 2. Light 1 has an optical power of about 1.2 mW. Beamsplitter 1,
Beamsplitter 2, Beamsplitter 3, and Beamsplitter 4 are 954P (Evanescent
optics). The individual difference in the coupling ratio for a 50:50
beamsplitter is $\pm$2%. The voltages are applied to Fiber stretcher 1 and
Fiber stretcher 2 with a piezo actuator driver, SVR 1000/3 (piezo
actuatormechanik GmbH), which can apply up to 1000 V. The mini fiber stretcher
is FS20 (IDIL). With this mini fiber stretcher, the phase can be scanned by
about 5 wavelengths in the voltage range of 0 V to 70 V. Triangle wave signal
with a frequency of 100 Hz is applied to this mini fiber stretcher. The
triangle wave signal is generated by an arbitrary wave generator (33500B
Keysight) whose ±0.8 V signal is amplified to the range between 20 V and 50 V
by a homemade amplifier. Photodetector 2 is homemade and the photodiode is
G8195-11 (Hamamatsu Photonics). The transimpedance gain is 100 k$\Omega$, the
reverse bias is 15 V, and the flat bandwidth is about 30 MHz. A motor-driven
optical shutter is placed in the Light 1 path to periodically block the light
and retake the voltage level of Photodetector 2 without light input, which may
drift during the experiment.

Figure 10: Experimental setup for visibility maximization with the CCC method
using Fiber stretcher 1 and Fiber stretcher 2. FS: fusion splice.

Interference voltage signals which are obtained by Photodetector 2 are sent to
analog-to-digital converters on an FPGA board, STEMlab 125-14 (RedPitaya). The
FPGA board has an analog input voltage range of ±1V, a voltage resolution of
14 bits, and a sampling frequency of 125 MHz. The data are then downsampled to
1/256 in the FPGA board and finally acquired at a sampling frequency of 488
kHz. In the downsampling, 256 points are averaged in the FPGA board with
14-bit accuracy.

The digital data acquired on the FPGA board is transferred to a personal
computer. Data of 16384 points are acquired at one time on the FPGA board,
which we call one frame of data. The time width of the one frame is 33 ms,
taking account of the sampling frequency of 488 kHz. The data are transmitted
to the personal computer via an Ethernet cable using socket communications.
The FPGA board has two input channels. Ch. 1 acquires the interference signal
from Photodetector 2, and Ch. 2 acquires the triangle wave signal applied to
the mini fiber stretcher, which is appropriately attenuated. Fig. 11(a) shows
a plot of the 2-channel data acquired by the FPGA board.

As shown in Fig. 12, the visibility is calculated on the personal computer
using the transferred data. First, the visibility value is calculated from
each frame. Then, the visibility of one time is obtained by averaging the five
visibility values from five frames, and its error bar is obtained by the
standard deviation of them. Due to the time required for data transfer and
real-time processing, a visibility value of one time is obtained about every
2.5 seconds.

Visibility $v$ is calculated from the maximum (max) and minimum (min) of the
interference signal as

$v=\frac{\rm{max}-\rm{min}}{\rm{max}+\rm{min}}.$ (6)

Here, max and min are the voltages that are measured from the voltage level
without light.

The min and max of the interference signal are obtained from the area where
the triangle wave signal is monotonically increasing (Fig. 11(a)). Hence, the
area is narrowed down to $\pm$3 ms around the time when the triangle wave
signal crosses 0 V. Fig. 11(b) and (c) show the min and max of the
interference signal in Fig. 11(a). In order to compensate for the slight DC
level without light, the output voltage level without light is measured by
blocking Light 1 with a motor-driven optical shutter (gray points in Fig.
11(b)). This voltage level without light is acquired in one frame and the
average value is used as the reference voltage. The process to obtain voltage
level without light is repeated every 20 time points of visibility
measurements.

Figure 11: Example of an interference signal to obtain visibility. (a)
Interference signal acquired in Ch. 1 of the FPGA board (blue points) and
triangle wave signal applied to the mini fiber stretcher acquired in Ch. 2
(red points). Min and max of the interference signal obtained in the green
area, where the triangle wave signal monotonically increases. (b) Magnified
interference signal (blue points) around the min (red point), together with
the detector signal without light (gray points) as a reference. (c) Magnified
interference signal (blue points) around the max (red point).  Figure 12:
Processes to obtain visibilities.

The visibility is calculated in real time, and the voltages applied to Fiber
stretcher 1 and Fiber stretcher 2 are changed step by step to maximize the
visibility. The visibility is maximized by trial and error as follows: First,
Fiber stretcher 1 is moved in a certain direction by an appropriate step, and
the visibility before and after the movement is compared. If the visibility
improves, Fiber stretcher 1 is moved further, and this process is repeated
until the visibility decreases. Next, the voltage applied to Fiber stretcher 2
is changed in the same way. Then, the driving target is back to Fiber
stretcher 1, while the direction is opposite to the previous direction. By
repeating these procedures, we maximize visibility. These processes are
controlled by Python programs. In the experiment, we demonstrate not only the
increase of visibilities by the trial-and-error processes but also the keeping
of high visibilities by compensating for the drifts of polarizations by
continuing the trial-and-error processes.

The step widths of the trial-and-error process explained above are set as
follows: 30 mV when visibility is less than 99%, 20 mV when visibility is
between 99% and 99.5%, and 10 mV when visibility is greater than 99.5%. Note
that the FPGA board outputs DC voltages between $\pm$0.8 V. The $\pm$0.8 V
outputs are amplified to about $\pm$3 V with op-amp ADA4898-1, and further
amplified with a piezo actuator driver, adding about 400 V offset. Finally,
the voltage applied to Fiber stretcher 1 ranges from 34 V to 772 V, and the
voltage applied to Fiber stretcher 2 ranges from 110 V to 702 V.

##  8 Experimental Result

Figure 13: (a) Time variation of visibility when visibility maximization by
polarization control is turned on (red points) and turned off (blue points).
Black dashed lines represent 100.0% and 99.9% visibilities. (b) Voltages
applied to Fiber stretcher 1 (orange points) and Fiber stretcher 2 (green
points) when visibility maximization is activated.

The visibility maximization experiment is performed continuously for three
hours, and the results are as shown in Fig. 13(a). The visibilities are red
points for the case where visibility maximization by polarization control is
activated and blue points for the case where it is deactivated for comparison.
The horizontal axis is elapsed time and the vertical axis is visibility. Red
points and blue points cannot be acquired at the same time. Hence, red points
are firstly obtained for three hours, and then blue points for three hours.
During the visibility maximization activated, visibilities are maintained
basically above 99.9%. On the other hand, during the visibility maximization
deactivated, even though the initial visibility is set to almost 100%, in
three hours it drifts and temporarily drops to around 98%. The error bars for
each time are $\pm$0.02% on average. These results show that the stability of
the visibility is drastically improved by the CCC method.

A decrease in visibility can be evaluated as the angular mismatch of two
linear polarizations. For the case the visibility maximization is activated,
the visibility of 99.9% corresponds to $0.9$ degrees of polarization mismatch.
For the case the visibility maximization is deactivated, the minimum
visibility of 98% corresponds to $1.3$ Degrees of polarization mismatch. Note
that the estimated worst case of visibility is 87%, corresponding to the
polarization mismatch of $\pm$10.3 Degrees, which are obtained by considering
the finite PERs of used components. The experimental lowest visibility of 98%
is much above the estimated worst case of 87%.

Fig. 13(b) shows the voltages applied to Fiber stretcher 1 and Fiber stretcher
2, for the case where the visibility maximization is activated. The orange
points represent the voltage applied to Fiber stretcher 1, and the green
points represent the voltage applied to Fiber stretcher 2. The voltages
constantly change to maximize visibility. Since the voltage to draw the
circular trajectory is 500 V, about 100 V change during the three-hour
measurement shows that the polarization drifts by about one-fifth of the
circular trajectories are compensated by the continuous trial-and-error
processes.

The dynamic ranges of the fiber stretchers are finite. Thus, during long-term
experiments to compensate for drifts of the polarizations, the voltages
applied to the fiber stretchers may exceed the dynamic ranges, though such
cases did not occur during the three-hour experiment in Fig. 13. Our control
program is written such that, when the voltage exceeds the dynamic range, the
voltage is reset to the center of the dynamic range and then the visibilities
are recovered by the trial-and-error processes. Fig. 14 shows the case where
the voltage exceeds the dynamic range and the visibility is recovered. Fig.
14(a) shows the visibilities, and Fig. 14(b) shows the voltages applied to
Fiber stretcher 1 (orange points) and Fiber stretcher 2 (green points). The
voltage applied to Fiber stretcher 2 reaches the lower limit of 110 V at 50
seconds, and the voltage is reset to 400 V. Then the voltage gradually
approaches an optimal point by the trial-and-error processes. The visibility
decreases to about 98% when the voltage is reset, and then gradually recovers
to 99.9%. It is also expected that this discontinuous visibility change can be
rarer by substituting fiber stretchers with heaters with larger dynamic
ranges.

Figure 14: Reset process of visibility maximization when the voltage applied
to Fiber stretcher 2 reaches the lower limit. (a) Visibilities before and
after the reset process. (b) Voltages applied to Fiber stretcher 1 (orange
points) and Fiber stretcher 2 (green points). When the voltage applied to
Fiber stretcher 2 reaches the lower limit, it is reset to 400 V.

We discuss the accuracy of the acquired visibilities. As is shown in Fig. 11,
the maximum value of the interference signal is about 600 mV. From Eq. (6),
visibility decreases at a rate of 0.33% when the minimum value increases by 1
mV. Since visibility is sensitive to the minimum value of the interference
signal, it is important to obtain the minimum precisely. The FPGA board can
acquire voltage signals with a range of ± 1 V and a resolution of 14 bits,
where 1 bit corresponds to 0.12 mV. The circuit noise of the FPGA board is
0.10 $\text{mV}_{\text{rms}}$ when downsampled to 1/256. When Photodetector 2
is connected to the FPGA board, the circuit noise is 0.17
$\text{mV}_{\text{rms}}$. When light is incident and a DC voltage is 600 mV,
the sum of optical shot noise and circuit noise is 0.18
$\text{mV}_{\text{rms}}$. Thus, the optical shot noise is almost negligible
compared with the circuit noise. From the accuracy of the minimum values
determined by circuit noises, it is assumed that the accuracy of the
visibilities obtained from one frame data is about 0.06%.

It is theoretically shown that fault-tolerant quantum computation is possible
with more than 10 dB of squeezing levels[11]. A high squeezing level of 15 dB
is reported where the visibility of homodyne measurements was 99.6%[25]. Thus,
the visibility of 99.9% with the CCC method is good enough to realize fault-
tolerant quantum computation in the future.

##  9 Comparison with conventional polarization controllers

There are various types of polarization controllers, but basically, they
control polarization by applying stress to a non-polarization-maintaining
fiber to induce birefringence. In this section, we compare the optical losses
of the two types of manual fiber polarization controllers FPC032
(Thorlabs)[26], HFPC-11-1300/1500-S-9/125-3A3A (OZ Optics)[27] and two types
of motorized fiber polarization controllers MPC320 (Thorlabs), PCD-M02
(Luna)[28]. The comparisons are summarized in Table 1.

Product Name | Fiber type | Manual/Electric | Loss
---|---|---|---
FPC032 | SM | Manual | 0.4 dB (8%)
HFPC-11-1300/1500-S-9/125-3A3A | SM | Manual | 0.15 dB (3.4%)
MPC320 | SM | Electric | 3 dB (50%)
PCD-M02 | SM | Electric | 0.08 dB (1.8%)
Homemade (used in this demonstration) | PM | Electric | 0.02 dB (0.5%)
Table 1: Comparison of the performance of various polarization controllers.

FPC032 has three spools around which a non-polarization-maintaining fiber is
wound several times to produce stress-induced birefringence. Then, by
adjusting the angle of the spools, the direction of birefringence is changed
to control polarization. This polarization controller requires about 5 m of
non-polarization-maintaining fiber. If the fiber is wound tightly, the optical
loss increases, but if the fiber is wound weakly, the birefringence becomes
smaller, making polarization change difficult. The measured optical loss was
0.4 dB (8%). In this product, microbends are considered to be generated at the
base of the spool where the fiber is twisted, resulting in large optical
losses. Instead of FPC032 where the angle of the spool is changed manually,
MPC320 can be used where the angle of the spool can be changed electrically.
However, MPC320 has a smaller spool radius, resulting in large optical losses.
The actual optical loss of MPC320 was measured to be 3 dB (50%).

HFPC-11-1300/1500-S-9/125-3A3A clamps a non-polarization-maintaining fiber to
produce birefringence. The polarization is changed by the strength of the
clamping force and the rotation angle of the clamp. The actual optical loss
was measured to be 0.15 dB (3.4%). Microbends are considered to be generated
at the part where the single mode fiber is pressed and twisted by the clamp.
In this product, the clamp is controlled manually.

PCD-M02 uses piezo actuators to press a non-polarization-maintaining fiber
from various directions to produce birefringence. The non-polarization-
maintaining fiber is pressed in a short distance of only a few millimeters,
which causes microbends. The data sheet value of PCD-M02(Luna)[28]
polarization controller is less than 0.05 dB (1.1%), and the measured value
was 0.08 dB (1.8%). This product controls polarization electrically.

In our homemade fiber stretcher used in the demonstration of the CCC method,
birefringence is generated by stretching a 1.5 m fiber uniformly. Thus,
microbends are considered to be less likely to occur and thus optical loss is
reduced. In the demonstration, polarizations are controlled electrically.

##  10 Conclusion

In this paper, we propose a method to maximize the interference visibility in
a polarization-maintaining fiber system by controlling the polarizations to a
crosspoint of the two circular trajectories on the Poincaré sphere, which we
call the CCC method. Using fiber stretchers as polarization controllers,
optical losses can be reduced. We also experimentally demonstrated the CCC
method for three hours, and visibility was basically kept above 99.9%. In
contrast to the conventional methods where polarization is controlled by
applying stresses to a non-polarization-maintaining fiber, the CCC method
controls polarizations by pulling polarization-maintaining fibers with fiber
stretchers, resulting in smaller microbends. Hence, the optical losses of the
fiber stretchers were measured to be as low as 0.02 dB (0.5%). Furthermore, in
contrast to the conventional methods, we do not have to combine non-
polarization-maintaining fibers with a polarization-maintaining fiber system.

The CCC method has broad applications in situations where optical losses in
fiber interferometer systems are undesirable. In particular, we can adapt this
method to interference of loss-sensitive squeezed states, such as cluster
states used for one-way quantum computation[20, 2, 29, 5, 22]. Fiber systems
do not require spatial alignment and are expected to improve stability over
long periods of time. The CCC method can achieve polarization control
electrically, which allows for automatic control. Furthermore, even if fiber
systems have multiple interference points, visibilities can be optimized at
all interference points by applying the CCC method from upstream to downstream
of the systems.

## Funding

Japan Science and Technology Agency (JPMJMS2064, JPMJPR2254); Japan Society
for the Promotion of Science KAKENHI (18H05207, 20K15187).

## Acknowledgments

This work was partly supported by the UTokyo Foundation and donations from
Nichia Corporation of Japan. T.N acknowledges financial support from Forefront
Physics and Mathematics Program to Drive Transformation (FoPM). M.E.
acknowledges supports from Research Foundation for Opto-Science and
Technology.

## Disclosures

T.N and A.F. are inventors on a patent related to this work filed by the
University of Tokyo on a priority date of 2 December 2021 with the Japan
Patent Office (P).

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