The non-measurement based coherent feedback control (CFC) is a control method without introducing any backaction noise into the controlled system, thus is specially suitable to manipulate various quantum optical systems for preparing nonclassical states of light. By simply tuning the transmissivity of an optical controller in a CFC loop attached to a non-degenerate optical parametric amplifier (NOPA), the quantum entanglement degree of the output optical entangled state of the system is improved. At the same time, the threshold pump power of the NOPA is reduced also. The experimental results are in reasonable agreement with the theoretical expectation.
The non-measurement based coherent feedback control (CFC) is a control method without introducing any backaction noise into the controlled system, thus is specially suitable to manipulate various quantum optical systems for preparing nonclassical states of light. By simply tuning the transmissivity of an optical controller in a CFC loop attached to a non-degenerate optical parametric amplifier (NOPA), the quantum entanglement degree of the output optical entangled state of the system is improved. At the same time, the threshold pump power of the NOPA is reduced also. The experimental results are in reasonable agreement with the theoretical expectation.
In recent years, quantum control theory has been widely used in various protocols of
quantum optics research12345. The control methods are usually
divided into two kinds: measurement-based feedback control123 and
non-measurement based coherent feedback control (CFC)45. CFC is a
control approach which never introduces any backaction noise into the controlled
system678910, and thus it extensively attracts interest since
the initial concept was proposed by H. M. Wiseman et al.4. R.
Hamerly et al. pointed out that CFC could outperform the best possible linear
quadratic Gaussian measurement-based schemes in the quantum regime of low steady-state
excitation number6. A CFC system for testing basic principles of linear
quantum stochastic control theory was experimentally realized by H. Mabuchi7. The simple and general algebraic methods for describing a series of
connections in quantum networks were presented11. J. Kerckhoffhave et
al. designed a novel approach for the quantum error correction based on CFC12.Nonclassical states of light, such as squeezed and entangled states are important
resources for the fundamental study of quantum optics and continuous variable (CV)
quantum information processing13141516. It is necessary to find
effective schemes of manipulating quantum states of light. Firstly, the manipulation of
optical nonclassical states was realized by the phase-sensitive operations. G. S.
Agarwal theoretically studied the interferences of the quantum fluctuations from a
cavity driven by a quantized squeezed state of light17 and then the
quantum manipulation phenomenon was experimentally demonstrated by injecting a squeezed
state of light into an optical cavity18. Successively, the
phase-sensitive manipulation of entangled state of light was theoretically proposed and
experimentally realized192021. Because there is no measurement
element in CFC, i.e. no the access noise, the CFC is a excellent approach to implement
the manipulation of the noise-sensitive optical squeezed or entangled states. The
enhancement of squeezing of optical field based on CFC was proposed and experimentally
realized, which proved the applicability of the CFC in nonclassical regime1122232425. Then a scheme of manipulating the multipartite
entangled states of light through CFC was theoretically proposed and detailedly analyzed
by our group26. Very recently, the CFC feedback-optimized extraordinary
optical transmission of CV entangled states through a hexagonal metal-hole array was
experimentally realized27. In this paper, we present the experimental
demonstration of the theoretical proposal26. The experiment shows that
the entanglement degree of the entangled state produced by a non-degenerate optical
parametric amplifier (NOPA) can be manipulated by a tunable beam splitter in CFC loop
and the entanglement up to −6.0 dB is obtained.
Especially, the threshold pump power of the NOPA is decreased and thus the same
entanglement can be achieved under a relative lower pump level for a given NOPA. The
experimental measurements reasonably agree with the theoretical calculations. The
advantages and the restrictions of the CFC scheme are discussed.
Results
Schematic of CFC-NOPA system
The schematic of the system is shown in Fig. 1. The NOPA is
an optical cavity involving a type-II phase matching nonlinear crystal. The pump
field at the harmonic-wave frequency
(2ω) and a weak signal (idler) beam at the subharmonic-wave frequency
(ω) are coupled into the NOPA. Via an intracavity
frequency-down conversion of the pump field inside the NOPA, the produced two
nondegenerate subharmonic-wave modes form an Einstein-Podolsky-Rosen (EPR)
optical entangled state with quantum correlations of both amplitude and phase
quadratures282930. The output entangled state from the NOPA is injected in a CFC control loop
composed of two mirrors (M and M0) and a control beam splitter (CBS)
with the transmissivity T at ω. The total extra loss in the
CFC loop is regarded as a vacuum noise coupled
from the mirror M with a transmissivity L at ω. The input
signal (idler) mode of the entire CFC-NOPA system is a weak optical field of
coherent state at ω
which is coupled into the system from CBS. The
output optical field of the system is made up of
two parts: one part is the reflected input field of the CBS and the other part is the transmitted field of the
optical mode in the CFC loop 26. The transmissivity of the input
coupler of the NOPA for the pump field and the subharmonic optical field are
expressed by γ10 and γ1,
respectively. All the unwanted other losses in NOPA can be thought as the intracavity losses of the signal (idler) mode
(γ2) and the pump field (γ20),
respectively.
Figure 1
The principle schematic of CFC-NOPA system.
The pump field and a weak signal (idler) beam
are coupled into the NOPA. The output
entangled state from the NOPA is injected in
a CFC control loop. The output optical field
of the system is made up of two parts, one part is the reflected input field
of the CBS and the other part is the transmitted field of the optical mode
in the CFC loop.
Theoretical analysis of the pump threshold and correlation variances of
CFC-NOPA system
The quantum Langevin motion equations of the pump optical field and the parametric down-conversion optical field
when the NOPA and the CFC loop resonate
are313233:where is the round trip time of light in the NOPA,
χ is the parametric coupling constant of the crystal. In the
linearized description of fields, the operators
can be expressed by the sum of an average steady state value
(a) and a fluctuating component , that is, ,
(j = p, s and i). From
the stationary equation and the relationship between the input and output field
of the NOPA system31, the oscillation threshold
(P) of the CFC-NOPA can be easily
calculated:The dependence of the threshold of the CFC-NOPA system on the transmissivity of
CBS (T) is shown in Fig. 2. The trace (1) (2) and
(3) are the calculated threshold pump power when γ1 is
selected as 0.07, 0.05 and 0.03, respectively. It can be seen that the threshold
of the CFC-NOPA system reduces a lot comparing with the NOPA without the CFC
control (T = 1). When T is 1, the CFC-NOPA
system works as an usual NOPA. If T < 1,
the CFC loop feeds back a part of its output optical field into the NOPA. The
lower the T is, the stronger the intensity of the feedback light is.
Since the feedback subharmonic-wave field increases the intracavity intensity of
the signal and idler modes in the NOPA, its threshold pump power for the
frequency-down conversion is reduced naturally.
Figure 2
The dependence of the threshold of the CFC-NOPA cavity on the transmissivity
of CBS for subharmonic optical field.
The traces (1) (2) and (3) are the calculated threshold pump power when
γ1 is selected as 0.07, 0.05 and 0.03,
respectively. The star is the experimentally measured result when
γ1 = 0.05.
The amplitude quadrature and the phase quadrature
of the optical mode equal to: and
, respectively. According to the
relationship between the input and output field of NOPA31 and
Eq. (1), the correlation variances of the quadrature
amplitude and quadrature phase of the output field for the CFC-NOPA system are
calculated:where,
m=(−κ + γ1 − γ2 − iΩ)/(κ + γ1 + γ2 + iΩ);
and
are the quadrature amplitude and quadrature phase
of the output signal (idler) mode
, Ω is the analysis frequency and
κ stands for the non-linear conversion coefficient which
is related to the parametric coupling constant χ19.
In the calculation, we have assumed that the signal and the idler modes are
totally balanced, which is easily realized in usual experiments.
Experimental setup and results
Figure 3 shows the experimental setup. The laser source is
a solid-state single-frequency and stable-frequency continuous wave Nd: YAP/LBO
laser with two output wavelengths at 540 nm and
1080 nm (CDPSSFG-VIB made by YuGuang company). The
maximal output power are 4 watt for 540 nm and 1 watt for
1080 nm, respectively. The two output lasers are
separated into two parts by a dual-color mirror with a high-reflection coated
for 1080 nm and an anti-reflection coated for
540 nm. The mode cleaner (MC1) is a three-mirror ring
cavity that provides the spatio-temporal filtering of the
1080 nm laser for the downstream experiment and is used
as the reference cavity for stabilizing the frequency of the laser source also.
The finesse of the MC1 is 500 for 1080 nm. The mode cleaner
(MC2) which has the identical structure with MC1 is used for reducing amplitude
and phase fluctuations of the pump field and its finesse is 650 for
540 nm. The lengths of MC1 and MC2 are servo-controlled
to resonate with their carrier field by lock-in amplifiers. The cleaned pump
laser (540 nm) is injected into the NOPA cavity from M3 as
the pump field. One part of the cleaned infrared laser
(1080 nm) serves as the local oscillators of the balanced
homodyne detectors (BHD1 and BHD2) and the other part is used as the seed beam
of the CFC-NOPA system. The NOPA cavity is a four-mirror ring cavity consisting
of two plat mirrors (M1, M2) and two spherical mirrors (M3, M4). A
α-cut type-II phase matching KTP (KTiOPO4,
potassium titany1 phosphate) crystal is placed between M3 and M4. The plat
mirror M1 is used as the input and output coupler of the NOPA, which is coated
with anti-reflection of the pump field and
partial transmission of the subharmonic optical field with a designated transmissivity of
γ1 = 0.05. The mirrors (M2, M3,
M4) are all coated with highly reflecting of the subharmonic modes and
anti-reflecting of the pump field. The CFC loop is also a four-mirror ring
cavity which includes two plat mirrors (M1, M7) and two spherical mirrors (M5,
M6). The transmissivity T of the plat mirror M7 for the subharmonic optical
field is changeable and M7 is used as the controller of the CFC-NOPA system. M5
and M6 are highly reflecting for the subharmonic optical field. The radius of
curvature of all four spherical mirrors (M3, M4, M5, M6) equals to
100 mm. The type-II non-critical phase matching KTP
crystal of dimensions 3*3*10 mm3 is placed in
a copper-made oven and is temperature-controlled around
63 °C with a peltier element for achieving
the optimal phase matching. M4 is mounted on a piezoelectric transducer (PZT1)
to scan actively the cavity length of the NOPA or lock it on the resonance with
the injected signal as needed by the Pound-Drever-Hall technique. The geometric
length and the waist size of the NOPA cavity are 540 mm and
62.8 μm, respectively. In order to
effectively manipulate the EPR entangled states generated by the NOPA, PZT2 is
mounted on M6 to actively scan the cavity length of CFC loop.
Figure 3
Experimental setup of manipulating and enhancing the EPR entangled states
with the CFC loop.
The input coherent optical beam at ω
is equally distributed into two orthogonal
intrinsic polarizations of the type-II phase matching KTP by a λ/2
wave plate (HWP4) placed ahead of CBS (M7 in Fig. 3). The
two orthogonally polarized optical beams serve as the seed beams of the
subharmonic-wave signal and idler modes in the NOPA, respectively. The measured
threshold power of the CFC-NOPA is 30 mW,
60 mW, 130 mW and
1150 mW for T = 0.3, 0.5,
0.7, 1, respectively, which is marked by a star in Fig. 2.
It can be seen that the measured threshold power of the CFC-NOPA cavity is a
little higher than the theoretically calculated value, that is because some
extra loss in the experimental system is not accounted in Eq.(2). In the experiment, the power of the injected subharmonic-wave
is kept at 200 μW. When the transmissivity of CBS
is chosen as T = 0.7 and the pump power is
60 mW, the classical gain of 4 times is observed if only
scanning the length of NOPA. However, if the lengths of both NOPA and CFC loop
are scanned simultaneosly, the maximal classical gain of the NOPA can reach to
35 times. Locking the relative phase between the pump field and the injected
seed optical beam to (2n + 1)π
(n is an integer), the NOPA operates at the state of deamplification.
In this case, the EPR entangled states of light with the anti-correlation of the
quadrature amplitudes and the correlation of the quadrature phases are
generated20. The output entangled beams with orthogonal
polarizations are separated by a polarizing beam splitter (PBS3) and then
detected by two sets of BHDs (BHD1 and BHD2), each of which consists of a 50/50
beam splitter and two high quantum efficiency InGaAs photo diodes. Meanwhile, we
lock the relative phase between the local beam and the injected signal (idler)
to 0 or π/2 to measure its amplitude or phase quadrature by
BHD1 (2), respectively. The noise powers of the amplitude or phase quadratures
measured by BHD1 and BHD2 are combined by the positive (negative) power combiner
(+/−) and then the correlation variances of the amplitude-sum
(phase-difference) are measured by a spectrum analyzer. The measured noise power
spectra of the amplitude-sum (a) and the phase-difference (b) between the output
signal and idler optical fields from 1.5 MHz to
6.5 MHz are shown in Fig. 4. Trace
(1) is the QNL, which is obtained by blocking the output beam of the CFC-NOPA.
Firstly, we measure the entangled degree of the EPR entangled beams directly
generated by the NOPA through changing the transmissivity of CFC controller (M7)
to T = 1.00 i.e, the CFC loop does not work. The
measured correlation variances of the amplitude-sum and the phase-difference are
about −3.5 dB below the corresponding QNL as
shown in trace (5) in Fig. 4. By means of replacing the
mirror M7 with mirrors having different transmissivities, the controller CBS
(M7) can be tuned to T = 0.72, 0.50 and 0.31,
respectively. When the cavity length of the CFC loop is swept by PZT2, the
correlation variance of the output beams from the CFC-NOPA fluctuates rapidly,
and the entanglement even disappears sometimes. To realize the stable
entanglement enhancement and manipulation, the cavity length of CFC loop is
locked to resonate with the injected seed beams during the experiment. Traces
(2–4) in Fig. 4 are the measured correlation
variances of the amplitude-sum and the phase-difference corresponding to the
T = 0.31, 0.50 and 0.72 respectively. We can
see that the optimal entanglement enhancement appears at lower analysis
frequency as theoretically expected26. Because the cavity
bandwidth of CFC-NOPA is smaller than that of NOPA. The delay of the light in
the feedback loop will affect the control performance at some extent, the
influence becomes stronger at the region of higher frequencies, so the
performance of entanglement enhancement of the CFC is limited by the bandwidth
of the CFC-NOPA system, which results in that the correlation variances of
output beams from the CFC-NOPA are higher than that of the beams coming directly
from the NOPA at higher frequencies. For the optimal transmissivity (0.72) of
CBS (trace (4) in Fig. 4), the entanglement is no longer
increased by the CFC and some extra noises deriving from the CFC will raise the
correlation variances of the output field from the NOPA when the frequency is
higher than 7 MHz. The entanglement of the output optical field can
be manipulated by tuning the transmissivity of CBS. From Fig.
1, we can see that both the output optical field of the CFC-NOPA
system and the field fed back into the NOPA include two parts. One part is the
EPR entangled state of light, which plays the positive role for the entanglement
enhancement. The other part is input coherent optical field and the extra vacuum
noise resulting from the CFC loop, which reduces the entanglement and thus plays
the negative role for the entanglement enhancement. When T is lower, the
coherent light directly reflected by CBS increases and the positive role of the
entanglement enhancement possibly becomes smaller than the negative influence of
the input coherent light, which results in the quantum correlation variances of
the output field of the CFC-NOPA system being higher than that without using the
CFC loop. For the given system, the optimal entanglement enhancement is obtained
at T = 0.72, which agrees with the theoretical
expectation from Eq.3.
Figure 4
The measured amplitude-sum (a) and phase-difference (b) correlation variances noise powers
of the output beams, where (1) is the normalized QNL; (2–5) is
the correlation noise powers with T = 0.31, 0.50,
0.72 and 1.00, respectively. The measurement parameters of SA: RBW
10 kHz; VBW 100 Hz.
Discussion
For the conclusion, we design and experimentally demonstrate a CFC-NOPA linearly
optical system for the generation and manipulation of optical entangled state. Not
only the entanglement between the output optical modes can be enhanced but also the
threshold pump power of the NOPA can be reduced under appropriate conditions. By
simply adjusting the transmissivity T of the CBS in the CFC loop, the quantum
feature of the output entangled state can be manipulated. The presented CFC scheme
based on linear optics is able to be applied in other systems for the generation and
control of the entangled states, such as nonlinear optical fiber or nanophotonic
devices34353637.
Additional Information
How to cite this article: Zhou, Y. et al. Quantum Coherent Feedback
Control for Generation System of Optical Entangled State. Sci. Rep.
5, 11132; doi: 10.1038/srep11132 (2015).
Authors: Amir H Safavi-Naeini; Simon Gröblacher; Jeff T Hill; Jasper Chan; Markus Aspelmeyer; Oskar Painter Journal: Nature Date: 2013-08-08 Impact factor: 49.962
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