Experimental Quantum Enhanced Optical Interferometry
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Experimental Quantum Enhanced Optical Interferometry
M.Genovese1
Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135, Torino, Italy and INFN, sezione di Torino, via P. Giuria 1,
10125 Torino, Italy
(Dated: 11 January 2021)
Optical quantum interferometry represents the oldest example of quantum metrology and it is at the source of quan-
tum technologies. The original squeezed state scheme is now a significant element of the last version of gravitational
wave detectors and various additional uses have been proposed. Further quantum enhanced schemes, from SU(1,1)
interferometer to twin beam correlation interferometry, have also reached the stage of proof of principle experiments
enlarging the field of experimental quantum interferometry and paving the way to several further applications rang-
ing from Planck scale signals search to small effects detection. In this review paper I introduce these experimental
achievements, describing their schemes, advantages, applications and possible further developments.
arXiv:2101.02891v1 [quant-ph] 8 Jan 2021
I. INTRODUCTION Further applications of quantum interferometry addressing
quantum imaging, absorption measurement etc. are beyond
In the last two decades a second quantum revolution based the puropose of this paper and can be found, for instance, in
on exploiting peculiar properties of single quantum states Ref.8,96
prompted the emerging of quantum technologies, ranging
from quantum information to quantum metrology1 . Sub-
stantial progresses have been made in the development of II. GENERAL INTRODUCTION TO QUANTUM
ENHANCED INTERFEROMETRY
quantum-enhanced measurement in recent years: Quantum
metrology is the new discipline addressed to overcome the
limits of classical measurements, in particular shot noise2 , by Nowadays interferometers represent probably the most
exploiting specific properties, and in particular peculiar cor- sensitive instruments we dispose of. Nevertheless, sev-
relations, of quantum systems, as entanglement4 , discord195 eral kinds of noise must be coped for reaching the highest
or squeezing197 . As other quantum technologies (e.g. quan- sensitivity18 . Just for mentioning the ones considered by
tum information, quantum computation, quantum communi- LIGO collaboration19 : Thermal noise, Seismic noise, Newto-
cation, . . . ) it had an exponential growth in the last 20 years, nian noise, Laser frequency noise, Laser intensity noise, Aux-
leading to the realization of several interesting proof of prin- iliary length control noise, Actuator noise, Alignment control
ciple experiments, mostly in quantum optics, and approach- noise, Beam jitter noise, Scattered light noise, Residual gas
ing now practical applications ranging from biology to re- noise, Photodetector dark noise,...
mote detection7–12 . As demonstrated potentialities of this dis- Anyway even when all these sources of noise are tamed, in-
cipline one can mention, without any pretension to be ex- trinsic limits remain. In√particular the classical interferometer
haustive, the possibility of enhancing the performances of sensitivity scales as 1/ N where N is number of photons (i.e.
interferometers42–50 , of phase estimation53 , of object detec- scales with the square root of the inputed light intensity), the
tion or testing55,67 , of superresolution57,58 , of spectroscopy56 so called shot noise.
and of beating shot noise in imaging8,51,52 or absorption Surprisingly enough this limit can eventually be beaten by
measurements59,60 . exploiting peculiar quantum properties of light22 , i.e. by in-
In particular, the application of quantum enhanced schemes puting light states with specific properties of quantum systems
to optical interferometry, a technique of huge widespread ap- (as squeezing or entanglement)23,24 . In this last case the phase
plication ranging from basic science13,14 to computation15 , sensitivity ∆θ can eventually reach the so called Heisenberg
represents the oldest example of quantum metrology and it limit, i.e. a 1/N scaling. A limit generically imposed by the
is at the very source of quantum technologies. Nowadays, laws of quantum mechanics, namely, the generalized Heisen-
the advantage of using squeezed light in interferometers has berg uncertainty ∆N∆θ ≥ 127–37 . It is worth noticing that
found application in enhancing the sensitivity of the upgraded even this limit can eventually be beaten in presence of prior
version of gravitational wave experiments as GEO 60016,17 , information38,39 or should be refined in specific situations, in-
LIGO13 and VIRGO14 , while new ideas are reaching a proof deed Hofmann40 suggested to consider the limit
of principle stage. In this review paper I will introduce the
main elements of Experimental Quantum Enhanced Optical 4θ 2 ≥ 1/hn2 i (1)
Interferometry, touching the following arguments:
where one considers the average of the squared photon num-
- General introduction to quantum enhanced interferometry
ber hn2 i. This allows for a better sensitivity in the case of large
- Squeezed light enhanced interferometers photon number fluctuations, 4n2 = hn2 i−hni2 > 0. However,
- SU(1,1) interferometers the effective possibility of using this advantage is still under
- Quantum enhanced correlation interferometers discussion186 .
- New ideas and perspectives The possibility of beating classical limits in interferometry
- Conclusions represents one of the most intriguing chances offered by quan-
2
√
Xθ = 1/ 2(a0 e−iθ + a†0 eiθ ) being the quadrature variable
(in the following X0 = X).
If, on the same line, we evaluate the variance of n5 we arrive
to
h∆n25 i = (sin(θ /2) cos(θ /2)|α|)2 h∆X 2 i (4)
If the port 0 is unused, i.e. the input is the vacuum state,
|vacuumi, then h∆X 2 i = 1/4. This represents the shot noise.
Caves demonstrated42 that inputing on the port 0 a squeezed
state D(α)S(z)|vacuumi, where S(z) = exp[1/2(a2 z∗ −
(a† )2 z)], z = re(iφ ) , e.g. the squeezed vacuum S(z)|vacuumi,
then
h∆X 2 i = e−(2r) /4 , (5)
i.e. the noise is reduced respect to the shot noise (a detailed
FIG. 1. Schematics of a Mach Zehnder interferometer. mathematical analysis of shot noise, and in general noise in
interferometers, can be found in Ref.18 ).
This result represented a fundamental progress, showing as
tum metrology, that now is overpassing the stage of theoreti- the use of quantum resources, in the case the squeezing, can
cal proposals41 , reaching experimental applications (in some lead to an improvement respect to classical limits, in this case
case beyond or proof of principle demonstrations), whose a the shot noise.
few examples will be discussed in the following. This upshot is related to the specific quantum properties
of squeezed
√ states. When considering
√ quadrature variables,
X = 1/ 2(a + a† ) and Y = 1/( 2i)(a − a† ), for the vacuum
III. SQUEEZED LIGHT ENHANCED INTERFEROMETERS ∆X = ∆Y = 1/2, the same is for coherent states, which in
the complex-amplitude plane are just a translated vacuum (see
As mentioned the first proposal of quantum enhanced mea- fig.2). On the other hand, for squeezed states the variance in
surement was the idea of Caves42 concerning the use of one quadrature is "squeezed", e.g. ∆X = 1/2e−r , while for
squeezed states for improving the performances of interferom- the other increases, ∆Y = 1/2er , keeping the product fixed
eters under the shot noise. In real situations eventually other to 1/4. By combining a squeezed and a coherent state on a
technical noises can be more important than shot noise, but, beam splitter antibunched light can be produced, where even-
as we will see, this proposal finally found an important appli- tually there are increased amplitude fluctuations and better de-
cation in gravitational wave detectors. fined phases42 . Incidentally, by combining two of them a pho-
The idea is relatively simple. In a "traditional" interferome- ton number entangled state is obtained (a two mode squeezed
ter a laser beam (well approximated by a coherent state) enters state)63 .
a port of the system, propagates inside the interferometer split For several years this proposal remained of theoretical in-
in two (or eventually more) components that at end are recom- terest, or had proof of principle experiments72,73 . Nonethe-
bined and interfere according to the relative acquired phase θ . less, in the last years the situation strongly changed. After
Let us consider for example a Mach-Zehnder interferome- that in Ref.45 was experimentally demonstrated a 2.3 dB im-
ter (but the same considerations apply mutate mutandis to any provement in a power recycled Michelson with squeezed light
other interferometer), see fig.1. Let call 0 and 1 the two modes injected into the dark port, this method found a large inter-
entering the initial beam splitter. After propagation in the two est with the development of extremely sensitive interferome-
arms, acquiring a relative phase θ , they recombine on a sec- ters for gravitational wave search. Following the use in Geo
ond beam splitter, whose outputs we denote with modes 4 and 60016,17 , it found also application in LIGO13 , where it now
5. The photon destruction operator corresponding to mode 5 represents a significant component19 .
can thus be written in terms of the operators relative to the Respect to previous versions, now in LIGO the squeezed
input modes 0,1 as: vacuum source (an optical parametric oscillator, OPA) is
placed inside the vacuum envelope on a separate suspended
platform, reducing in this way the squeezing ellipse phase
a5 = ieiθ /2 · (a1 sin (θ /2) + a0 cos (θ /2)) (2)
noise and backscattered light noise. Furthermore, the
Thus, if the mode 1 is a coherent state |αi = D(α)|vacuumi squeezer has been fully integrated into the automated lock ac-
(much more intense than the state inputing mode 0), where quisition sequence. Thanks to this improvements, now above
D(α) = exp(αa − α ∗ a† ), the output photon number in mode 50 Hz the interferometer sensitivity is increased by 2.0 dB and
5, hn5 i = ha†5 a5 i is: 2.7 dB at LIGO Hanford Observatory and LIGO Livingston
Observatory, respectively. Thus, while increasing interferom-
eter sensitivity by increasing the input power to the interfer-
hn5 i = sin(θ /2)2 |α|2 +sin(θ /2) cos(θ /2)|α|ha0 e−iθ +a†0 eiθ i ometer, which grows by enhancing the gravitational-wave sig-
(3) nal, is limited by radiation pressure inducing instabilities and
3
absorption of the test masses, injecting squeezed vacuum im-
proves the signal-to-noise ratio by decreasing the interferom-
eter noise. In this sense 3 dB of squeezing is equivalent to
doubling the intracavity power to 450kW. For instance, this
provides a 12% and 14% increase in binary neutron star in spi-
ral range at each respective site19 . Similar improvements have
also been obtained into the other big gravitational wave detec-
tor, Virgo14 . Nonetheless, below 50 Hz, injecting frequency-
independent squeezed vacuum increases the quantum radia-
tion pressure noise This effect in LIGO limits at the moment
a further increasing of the current squeezing level. Frequency
dependent squeezing could alleviate this problem.
Beyond these technical limits (e.g. contribution to radiation
pressure noise), also losses are a significant problem64 , since
they reduce the squeezing65,66 . The conundrum of keeping a
strong squeezing coping losses without increasing the squeez-
ing level, can eventually be solved by amplifying the squeez-
ing quadrature where the information is encoded70 . A proof FIG. 2. Uncertainty circle in complex-amplitude state for vacuum,
of principle of this scheme has been recently realised with a coherent state |αi and squeezed state.
Mach Zehnder like interferometer71 , demonstrating a 6dB sub
shot noise sensitivity even for 50% detection efficiency.
Coming back to the tradeoff between radiation pressure about the true value of the phase shift is contained in the quan-
noise and shot noise reduction, some idea for improving tum fluctuations of the number of particles measured at the
both by realising frequency dependent squeezing has recently output ports. In summary, the combination of coherent and
emerged. The point is that quantum fluctuations in the am- squeezed light at input beam splitter creates an entangled state
plitude quadrature of the light generate a fluctuating radiation that a suited measure could further exploit. Theoretical con-
pressure on mirrors, that in turn couples amplitude and phase siderations, based on Cramér-Rao bound, lead to a bound (p
quadratures21 being the number of measurements and F(θ ) the Fisher infor-
mation)
phase phase
aout = ain − Kaampl
in + signal (6)
where the Kimble factor K depends on the frequency (beyond 1 1
∆θ = p =q (7)
the line width of detector, the light power, the mass of mirror). pF(θ )
The radiation pressure noise (dominating at low frequency) p(|α|2 e2r + sinh2 (r))
can be reduced by squeezing the amplitude quadrature, the
shot noise (dominating the high frequency sensitivity) de- that reaches the Heisenberg scaling
mands for squeezing the phase quadrature. In Ref.20 it was
1
suggested a method for realising this situation, based on ∆θ = √ (8)
generating EPR entangled beams23 through a detuned OPA. pN
The method takes advantage of the entanglement between
fields around the half of the pump frequency: measuring on when |α|2 ' sinh2 (r) ' N/2 and p, N
4
signal) of the first PA are recombined in the second PA. This
scheme was then demonstrated to be able to reach the Heisen-
berg scaling 1/N (in terms of photon number N), Ref.98 .
Indeed, when defining the generators of SU(2) in terms of
input mode annihilation operators
J1 = (a†0 a1 + a†1 a0 )/2 (9)
J2 = −i(a†0 a1 − a†1 a0 )/2 (10)
J3 = (a†0 a0 − a†1 a1 )/2 (11)
the beam splitter with transmissivity cos(α/2) corresponds to
the related group transformation U = e−iαJ1 .
On the other hand, by defining the SU(1,1) algebra genera-
tors
K1 = (a†0 a†1 + a1 a0 )/2 (12) FIG. 3. Schematics of a SU(1,1) interferomter.
K2 = −i(a†0 a†1 − a1 a0 )/2 (13)
K3 = (a†0 a0 + a1 a†1 )/2 (14) where θ1 and θ2 are the phases acquired on the arm 1 and 2 of
the interferometer, respectively.
the PA is described by the relative group element U = e−2irK2 . One interesting property that differentiates a SU(1,1) inter-
By considering the sequence of transformations describing ferometer from a usual Mach-Zehnder is, as one immediately
a traditional interferometer (Mach-Zehnder,
√ Fabry-Perot or evinces form Eq.17, that the interference fringes depends on
whatelse) one obtains the usual v 1/ N sensitivity98 ; while, the sum of θ1 and θ2 (instead of depending on the difference).
as the squeezed state interferometers, also SU(1,1) interfer- More interesting the fringe size depends quadratically (thanks
ometers can achieve a phase sensitivity of 1/N, but in this case to amplification) on the phase sensing field strength to be com-
with only vacuum fluctuations entering the input ports and co- pared to a linear dependence of the linear interferometer. Fi-
herent states pumping the active devices. nally, another significant difference is that for the SU(1,1) in-
The original proposal was based on spontaneous emission terferometer making the difference between the two outputs
(i.e. the parametric amplifier had the input modes as vac- the phase dependence is cancelled.
uum states): this is not very effective due to the limitation However, the most interesting property of a SU(1,1) inter-
in the intensity (i.e. in N) of the output modes. Thus, experi- ferometer is that the noise is not amplified as much as the
mental realisations follow a scheme where coherent states are signal, increasing the signal to noise ratio (an amplifier at the
injected99,100 . output of a usual interferometer would amplify in the same
The experimental set-up, and the possible eventual applica- way signal and noise).
tions, depend on the medium chosen as parametric amplifier. Indeed, if one measures the quadrature variable99 , the result
At the moment three different configurations have been real- is
ized: exploiting four-wave mixing101–105 , a bulk crystal106 or
non-linear optical fibers107–109 . h∆2 Xi = |G1 G2 + (G21 − 1)(G22 − 1)ei(θ1 +θ2 ) |2 (18)
The general idea of a SU(1,1,) interferometer is, as men-
tioned, to substitute the beam splitters with parametric ampli- reaching a minimum at the dark fringe (θ1 + θ2 ) = π for all
fiers, see fig17, whose Hamiltonian is (in parametric approx- quadrature variables (at variance with squeezed state interfer-
imation where the pump field is approximated to a classical ometry, where only one quadrature experiences noise reduc-
field) tion).
One can then evaluate the relative signal to noise ratio
HPA = i(ga†1 a†2 − g∗ a1 a2 ) (15) (SNR) for a quadrature measurement for a small change δ in
Introducing the gain G j = cosh(kg), k being a constant, with the phase in one arm is (in the case (G22 − 1) >> (G21 − 1))110
j = 1, 2 denoting the first or second PA, when a coherent state
hY i2
q
|αi is inputed in mode 1 (while the other input mode is in the SNR = = 2[G 1 + (G21 − 1)] · |α|2 δ 2 (19)
vacuum) the output intensities are: h∆2Y i
I4 =|α|2 (G21 G22 + (G21 − 1)(G22 − 1)+
q
q q which provides a [G1 + (G21 − 1)]/2 enhancement respect to
2G1 G2 G21 − 1 G22 − 1 cos (θ1 + θ2 ) (16) a traditional interferometer. A result that is substantially un-
affect by detection inefficiencies (or other losses outside the
interferometer)102,106,111 . On the other hand, it is not insensi-
I5 =|α|2 (G21 · (G21 − 1) + G22 · (G22 − 1)+ tive to losses inside the interferometer99 .
The first experimental realization of a SU(1,1) interferom-
q q
2G1 G2 G21 − 1 G22 − 1 cos (θ1 + θ2 ) (17)
eter was achieved in Ref.101 , where the two parametric ampli-5
The idea of a dual beam interferometer (collinear, but non
degenerate PDC with two independent measurements) was
then considered theoretically in Ref.115 and experimentally in
Ref.116 . Since the SU(1,1) interferometer fringes depend on
the sum of the phases acquired by the two beams, this configu-
ration allows optimising the measurement achieving the same
SNR of squeezed state interferometry.
In order to point toward real applications of SU(1,1) in-
terferometers, to increase intensity is mandatory. In Ref.117
a pumped up version of SU(1,1) interferometer using all
particles (suited for atomic interferometers) was proposed,
while in Ref.127 a bright seeded SU(1,1) interferometer,
where the pump beams generated by a Titanium sapphire
laser were injected in 85 Ru vapor cells. This last ex-
periment led to a 1.15 dB advantage respect to standard
FIG. 4. Reproduced on permission from Ref.106 : Best phase sensi- noise
p limit, where losses lowered the expected
√ advantage,
tivity normalized to shot noise against the detection transmission η. G2 (1 − G)2 /(G2 + (1 − G)2 ) (with 1/ N scaling), of 2.6
The curves correspond to different parametric gains. dB.
An alternative for overcoming the limited intensity power,
fiers were based on a four-wave mixing process in 85 Rb va- that remains the most severe drawback of this kind of inter-
por cells 12 mm long. They were respectively illuminated by ferometers, was recently suggested and realized considering
an intense vertically polarized pump beam (400 mW) from a a configuration where a SU(2) interferometer is nested in a
frequency-stabilized Ti:Sapphire laser locked to a Fabry-Perot SU(1,1) one121 . In more detail, the signal beam of the first PA
reference cavity. Furthermore, a horizontally polarized seed is fed into the dark port of a SU(2) (a Mach-Zehnder) interfer-
beam was combined with the pump beam at an angle of 0.7° ometer, while an intense coherent state is fed on the other port.
in the center of the first vapor cell. The pump laser frequency The light emerging from the dark port of this SU(2) interfer-
was about 0.8 GHz blue detuned from the 85 Rb F = 2 → F 0 ometer is then recombined with the idler beam in the second
transition at 795 nm and the seed signal beam was about 3.04 PA. The experiment, with two 85 Rb cells generating four wave
GHz red shifted from the pump with an acoustic optic mod- mixing, led to a 2.2(5) dB advantage, that was substantially
ulator. The amplified signal and the conjugate idler beams of maintained even with a 1mW power.
the first cell were addressed into the second vapor cell where
they were symmetrically crossed with the pump , similar to the Concluding this section it is worth mentioning a few further
first vapor cell. The two output beams from the second vapor proposals suggesting "variations" on the theme of the SU(1,1)
cell were then measured demonstrating a quadratic fringe in- interferometer122–126,128 .
tensity dependence on the intensity of the phase sensing field
In Ref.122 the second PA is substituted by a beam splitter
at high gain.
(in this case of course idler and signal beam must have the
As a second example of experimental implementation, I
same wave length) followed by a homodyne detection leading
will present in some detail the one of Ref.106 . Here the in-
substantially to the same SNR of Eq.19,
terferometer consisted of two cascaded 3 mm long β -barium
borate (BBO) crystals (the first placed on a translational stage In Ref.123 it was theoretically studied the phase sensitivity
for controlling the interferometer phase) cut for collinear fre- of a SU(1,1) interferometer with a coherent state in one input
quency degenerate type-I phase matching. The pump was port and a squeezed-vacuum state in the other input port using
the second harmonic of a 400 nm laser with a 5 kHz repe- the method of homodyne detection, showing the potentiality
tition rate, 1.5 ps pulses. The multimode parametric down- of achieving the Heisenberg limit.
converted (PDC) light generated by the SU(1,1) interferome-
ter was then spatially and spectrally filtered before being de- Finally, in Ref.124 it was demonstrated, theoretically and
tected. In this realisation of the SU(1,1) interferometer both experimentally, that the second PA can eventually be sub-
the outputs of the first PA cross the same medium producing stituted by two homodyne detections (with the same local
the same phase change. An advantage of this configuration is oscillator) combining the relative photocurrents, while in
that, since the three fields (idler, signal and pump) copropa- Ref.125 application to stochastic phase estimation was stud-
gate, random phase fluctuations (e.g. caused by air flow, tem- ied, in Ref.126 a theoretical analysis of a multimode integrated
perature fluctuations,...) wipe out. One of the main results of SU(1,1) interferometer was performed, in Ref.127 the effect of
this work is the demonstration of the insensitivity to losses: additional external resources was investigated and in the very
for specific gain choices the phase estimation was under shot recent Ref.119 it was shown the possibility of combining the
noise up to 80% losses, see Fig. 4. This kind of interfer- two photodetector outputs with an optimal weight factor for
ometer was then further investigated112,113 and, in particular, beating the shot noise by the same amount regardless of the
extended to a wide field SU(1,1) interferometer114 . phase shift in the interferometer.6
V. QUANTUM ENHANCED CORRELATION sured at the output of the holometer is described by an ap-
INTERFEROMETERS propriate operator C(φ b 1 , φ2 ), φk being the phase shift (PS) de-
tected by the interferometer Ik , k = 1, 2, with expectation value
One more significant application of quantum light concerns hC(φ
b 1 , φ2 )i = Tr[ρ12C(φ b 1 , φ2 )], where ρ12 is the overall den-
the possibility of enhancing the performances of correlation sity matrix associated with the state of the two light beams
interferometry. injected in I1 and I2 .
In particular, in the last years the interest for correlation in- In the Fermilab scheme the idea for observing the eventual
terferometry grew in connection to the possible application to existence of the HN is comparing hC(φ b 1 , φ2 )i in two differ-
the search of quantum gravity effects. Indeed, the dream of ent experimental configurations of I1 and I2 , namely when the
building a theory unifying general relativity and quantum me- arms are parallel, “k”, or perpendicular, “⊥”132 . Indeed, ac-
chanics, the so called quantum gravity (QG), has been a key cording to Ref.132 , one expects that HN is correlated in the
element in theoretical physics research for the last 60 years: first case (being the arms in the same light cone), while it
several attempts in this sense have been considered. How- is not in the second case. Thus, on the one hand, in the
ever, for many years, no testable prediction emerged from configuration“k” the correlation of the interference fringes
these studies, leading to the common wisdom that this kind would highlight the presence of the HN. On the other hand, the
of research was more properly a part of mathematics than configuration “⊥” serves as a reference measurement, namely,
of physics, being by construction unable to produce experi- it corresponds to the situation where the correlation due to
mentally testable predictions as required by Galilean scien- HN is absent, in other words, it is equivalent to estimating the
tific method. In the last few years this common wisdom was “background”.
challenged129–134 . In particular, it has been proposed that In general, being the statistical properties of the PSs
Planck scale effects, connected to non-commutativity of po- fluctuations due to HN described by a suitable proba-
sition variables in different directions135,136 , could be mea- bility density function, fx (φ1 , φ2 ), x =k, ⊥, the expec-
sured, for instance in cavities with microresonators133 or in tation of any operator O(φ b 1 , φ2 ) is obtained by aver-
two coupled interferometers132 , the so called “holometer”.
h i
aging over fx , namely, hO(φ b 1 , φ2 )i → Ex O(φ b 1 , φ2 ) ≡
This possibility led to the building of the 40m Fermilab
holometer√147–149 , that recently started to pose limits (2.1 ·
R
hO(φ
b 1 , φ2 )i fx (φ1 , φ2 ) dφ1 dφ2 .
10−20 m/ Hz) on the possible noise stemming from these Since in the holometer the HN can be observed through a
effects147,149 in 1-6 MHz region. correlation between two phases, the appropriate function to be
In synthesis, a holometer is a device consisting of two estimated is their covariance:
Michelson interferometers (MI), that are spatially close. The h i h i
purpose of the holometer is to search for a particular type of Ek C(φ b 1 , φ2 ) − E⊥ C(φ b 1 , φ2 )
correlated noise, which is conjectured to arise from gravita- Ek [δ φ1 δ φ2 ] ≈ , (δ φ1 , δ φ2 1)
h∂φ21 ,φ2 C(φ
b 1,0 , φ2,0 )i
tional effects at the Planck scale, affecting the two spatially
(20)
close interferometers as relative phase noise. A little more
δ φk = φk − φk,0 , φk,0 being the mean PS value measured by
in detail, the idea at the basis of these studies is that non-
Ik , k = 1, 2 reducing as much as possible the associated uncer-
commutativity at the Planck scale (l p = 1.616 × 10−35 m) of
tainty:
position variables in different directions, predicted by several
Planck scale physics models137,138 , generates an additional r h i
very weak phase noise, referred to as holographic noise (HN). 2 Var C(φ b 1,0 , φ2,0 )
In a single interferometer this noise substantially confounds U (0) = , (21)
with other sources of noise, even when the most sensible grav- h∂φ21 ,φ2 C(φ
b 1,0 , φ2,0 )i
itational wave interferometers are considered132 , since by con-
struction their HN resolution is worse than their resolution to
h i
where Var C(φ b 1,0 , φ2,0 )2 i − hC(φ
b 1,0 , φ2,0 ) = hC(φ b 1,0 , φ2,0 )i2
gravitational-wave at low frequencies. Nevertheless, when the
two equal interferometers of the holometer have overlapping does not depend on the PSs fluctuations due to the HN, but it
space–time volumes, then the HN between them is correlated represents the intrinsic quantum fluctuations of the measure-
and can be identified easier132 . Since the ultimate limit for ment described by the operator C(φ b 1 , φ2 ) and depends on the
holometer sensibility, as for any classical-light based appara- optical (quantum) states entering the holometer.
tus, is dictated by the shot noise, the possibility of going be- In151,152 it was demonstrated that, when the two input
yond this limit by exploiting quantum optical states is of the modes of each interferometer Ik , k = 1, 2, are excited in a co-
utmost interest139–141 . herent state and a squeezed vacuum state with mean number
In addition to this application, measuring relative phase of photons µ and λ , respectively:
noise could find application in the detection of the gravita- √
tional wave background142–145 , or finding traces of primordial √ λ + µ 1 + 2λ − 2 λ + λ 2
(0)
black holes146,150 . USQ (µ, λ ) ≈ 2 . (22)
(λ − µ)2
The use of quantum light in correlation interferometry was
firstly investigated in151,152 . As expected, in analogy with the PS measurement for a sin-
For this purpose it was considered that the observable mea- gle interferometer42 , if µ λ 1, then we have the opti-7
(0) √
mal accuracy USQ ≈ (2 2λ µ)−1 . This represents an evi-
dent advantage in terms of uncertainty reduction (of the order
(0) √
(4λ )−1 ) with respect to classical case UCL ≈ 2/µ when
only coherent states are employed. An important difference
arises between the single interferometer PS measurement, in-
volving a first order moment of the photon number distribu-
tion, and the covariance estimation, involving the second or-
der moments: whilst in the first case the uncertainty scales as
the usual standard quantum limit one, ∝ µ −1/2 , in the second
case it scales ∝ µ −1 (neglecting the little relative contribution
of the squeezed state to the intensity).
Nonetheless, even a larger advantage was demonstrated
when considering a configuration where second port input
modes of I1 and I2 , a1 and a2 , are the component of a photon-
number entangled beam, in particular a twin beam (or a two-
mode squeezed vacuum state)154
FIG. 5. Analysis of the time series obtained from the independent
|TWBii = (1 + λ )−1/2 ∑(1 + λ −1 )−n/2 |nia1 |nia2 (23) squeezer configuration from the experiment of Ref.155 , with added
n white noise. The red curves correspond to the squeezing injection
configuration and the blue curves to the coherent case. The plot
while two coherent states still enter into the other two ports. shows the cross-correlation of the data in the time domain versus
This correlation property leads to the amazing result that the the number of samples. The correlation peak is hidden in the noise
contribution to the uncertainty coming from the photon num- for small sample numbers, equivalent to short acquisition time, and
(0)
ber fluctuation noise is UTWB = 0 (when λ , µ 6= 0) in the ideal it emerges when the number of samples is increased. The use of
situation of no losses, representing an ideal accuracy of the in- squeezing allows clearly for an earlier detection of the peak.
terferometric scheme to the PSs covariance due to HN.
This advantage is conserved even in presence of losses
in the case of high quantum resources exploited, i.e. µ
(0) (0)
λ 1, where one finds USQ /UCL ≈ (1 − η) + η/(4λ )
(0) (0) √
and UTWB /UCL ≈ 2 5 (1 − η) when the total detection effi-
ciency η is sufficiently large.
These theoretical ideas were then experimentally demon-
strated in155 .
Both the two independent squeezed state and a twin-beam
like injection were considered. Each Michelson interferom-
eter (MI), with arm length of 0.92m, was fed with 1.5 mW
of 1064 nm light from a low noise Nd:YAG laser source.
This laser source was also used to seed the optical paramet-
ric oscillators generating the squeezing. The light for the
MIs was spatially cleaned with an optical fiber. Then, each
interferometer had two piezo-actuated high-reflectivity end
mirrors (RM = 99.9 %), while the partially reflecting mir-
rors for the two power recycling cavities had a reflectivity of
RPRM = 90 %.
The squeezed-light (6.5 dB relative to the shot noise), pro-
duced by a parametric down-conversion in a potassium titanyl
phosphate (PPKTP) crystal placed in a semi-monolithic linear
cavity, was injected in the two MIs via their output ports. FIG. 6. Simplified schematic of the double-interferometer setup of
Ref155 in the twin beams configuration. The two spatially close (∼
The signal-to-noise ratio with squeezing injected was con-
10 cm) interferometer input ports were fed 1064 nm laser light from
sistently higher, by a factor of 2, of the classical interferome- a low noise Nd:YAG laser source. Each interferometer had a power
ter. In the spectral domain, correlated signals were extracted recycling mirror (PRM) in the input port, to form a cavity around the
by the cross-linear spectral density of the two interferometers, interferometer. The input beams were split at a beamsplitter (BS) in
demonstrating an improvement of a factor 1.35 from the in- each interferometer, and subsequently impinged on piezo controlled
jection of squeezed states. A result that also poses
√ an upper end mirrors (M). A Faraday isolator in each output port allowed for
limit on HN in the 13 MHz region (3 · 10−17 m/ Hz). measuring the output while twin-beam (TWB) modes were injected
When a twin-beam like signal was injected (see Fig.6), the into the output ports.
correlation between the two modes led to a noise reduction of
2.5 dB with respect to the SNL.8
entangled states of the form
|Ni|0i + |0i|Ni
√ (24)
2
dubbed NOON states.
In synthesis, while using N independent photons in a two
arms (a,b) interferometer one has
|ai + eiθ |bi ⊗N
[ √ ] (25)
2
√
leading to a as 1/ N phase uncertainty scaling, with a NOON
state the phase cumulates as
|Nia |0ib + eiNθ |0ia |Nib
√ (26)
FIG. 7. Comparison of the PSDs of the photocurrent difference be- 2
tween the MI outputs of the experiment of Ref.155 , with and without
TWB. Green and black traces have uncorrelated white noise intro- leading to a 1 + cos(Nθ ) interference dependence, which im-
duced in the MIs, with and without TWB respectively. The white plies a 1/N scaling163 . Albeit of large theoretical interest,
noise clearly emerges in the TWB case demonstrating the efficiency unluckily no effective idea exists for creating high N photon
of this method. NOON states with present technology, and only a few proof
of principle experiments were realised with N=2164–169 (eas-
ily created by Hong-Ou-Mandel interference) or N=4,5170,171
This enhancement was also observed in the power spectral (with postselection that, due to low probability success, limits
density(PSD), fig.7, of the subtracted interferometer outputs, an eventual practical use). Furthermore, the advantage offered
demonstrating that the presence of faint uncorrelated noise by these states is very sensitive to losses182 .
can be more easily detected by twin beam-like correlations Also the use of twin photon Fock states |N, Ni, would
and, therefore, the possible advantage offered by this tech- lead to asymptotically approaching Heisenberg limit175–177 .
nique in experiments of correlation interferometry. Nonetheless also high N photon Fock states are far to be
Following this proof of principle experiment, very recently produced173 (postselected heralded N=8 Fock states are the
in Ref.156 a detailed description was presented of a table top present limit174 ), as well as maximally correlated states de-
quantum enhanced holometer (in the version based on the in- fined in Ref.178 , which would allow exactly reaching Heisen-
jection of two independent squeezed beams), with the purpose berg limit.
of realising a new experiment on Planck scale effects search. In order to overpass these difficulties, on the one hand the
Furthermore, interest emerged in investigating possible use of twin beams coupled to photo-count measurements was
advantages of using non-gaussian (as photon subtracted) suggested179 and, in a simplified version, implemented180 .
states157 , while in Ref.158 decoherence effects in this kind of In this proof of principle experimental realisation the pho-
experiments were analyzed. ton counting detection was realised through a multiplexed
scheme; the measured data were then compared with what
expected by a different phase shift impressed on the two com-
VI. NEW IDEAS AND PERSPECTIVES ponents of the twin beam and a final detection described by
a specific POVM keeping into account the multiplexed mea-
Finally, in this section we consider new schemes and ideas surement. The shot noise limit was surpassed despite a real-
that still far from practical applications, could eventually lead istic level of losses. Very recently181 a new experiment en-
to interesting developments in a more or less next future. hanced further the advantage by using NIST transition edge
A first line of research, somehow connected to SU(1,1) in- sensors.
terferometers, is inserting PA in the arms of an interferome- On the other hand new detection methods, suited for reach-
ter for enhancing its performances. A first example was ex- ing Heisenberg limit with quantum states experimentally real-
perimentally realised in Ref.159 , where the performances of a isable, were considered. Heisenberg limit in phase estimation
Mach-Zehnder interferometer operating with single photons was also reached in Mach-Zehnder interferometers by exploit-
probes were enhanced by inserting a parametric amplifier in ing a generalized Kitaev’s algorithm, reaching a 10dB advan-
√
one arm, still scaling as 1/ N. Very recently a new experi- tage respect to standard quantum limit with highest number
ment with two PAs, one per arm, was realised160 , achieving a of resources (378 photons)183 . Nevertheless, the extension to
5.6 dB squeezed noise floor under shot noise limit and a 4.9 practical uses (i.e. a much larger number of photons) still re-
dB enhancement of signal to noise ratio, reaching Heisenberg mains far from reach.
limit. A further class of proposals is based on using parity
Another very interesting seminal proposal161,162 , much dis- measurements184 , i.e. on measuring the parity operator Π =
†
cussed in the theoretical literature, is the use of photon number eiπa a , which substantially requires photocounting detection.9
A first proposal in this sense appeared in Ref.185 : here it is was demonstrated that they can be applied to reduce system-
suggested to use collinear degenerate twin beams, Eq. 23, atic uncertainties in phase estimation. Specifically, it was con-
i.e.two mode squeezed vacuum (TMSV), entering one port of sidered the case of a Sagnac interferometer where the beam
the interferometer, e.g. 1 in Fig.1, and performing a parity splitter is substituted by a polarizing beam splitter and the
measurement to one output port, e.g. 4 in Fig.1. This leads to: phase is encoded in one of the polarizations. The experiment
demonstrated a 30 times phase resolution improvement. A
1 further approach exploiting weak values appeared in Ref.210 .
hΠiθ +π/2 = q (27)
1 + n̄(n̄ + 2) sin2 θ Here it is considered a Michelson (or Mach - Zehnder) inter-
ferometer with a "stable" polarization dependent phase shift
where n̄ is the average photon number of the in one arm. In this case the preselected state is (in terms
p TMSV, which, of Horizontal, H, and vertical, V, polarization states) of the
through Cramér - Rao bound, leads to a 1/ n̄(n̄ + 2) depen-
dence for the phase uncertainty in proximity of θ = 0, that √ i) and the postselection is (cosα|Hi + sinα|V i),
form (|Hi+|V2
even beats the Heisenberg limit: here is an example of appli- while the polarization measurement operator is, for instance,
cation of Hofmann limit40 , since 4n2 = hn2 i − hni2 = n̄2 + 2n̄ b = |HihH|. The uncertainty is amplified as the signal by the
A
for TMSV. weak value hAib w , but a small weak value attenuates phase and
Heisenberg limit can eventually be beaten also through non- uncertainty at the same time. Thus, the reduction of the uncer-
linear transformations187 , i.e. this limit can be overcome when tainty is achieved by exploiting a feedback control: a ∼ 10−3
the generator of the transformation is proportional to the pho- reduction factor was experimentally demonstrated.
† 2
ton number, e.g. in188 a Kerr type eiθ (a a) transformation was Thus, in summary, several new ideas are emerging for fur-
considered leading to a 4θ ∼ 1/N 3/2 scaling, besides a fi- ther improving optical (and not only) interferometry. Many of
nite detection efficiency only represents a multiplicative fac- them already were demonstrated in proof of principle experi-
tor to this scaling. More in general a Hamiltonian including ments. However, if and which of them will really find practi-
all possible k-body couplings allows a 1/N k scaling for en- cal applications, overcoming present limitations, still remains
tangled probe states, 1/N k−1/2 for a probe initially in a prod- to be clarified.
uct state187–190 , representing in this last case also an inter-
esting example of quantum enhanced measurement without
entanglement191,192 . Further theoretical optical schemes with
non linear phase shifts can be found in Ref.193,194 .
Finally, different ideas emerged on using weak values or VII. CONCLUSIONS
weak values-inspired methods195,196 in interferometry. Weak
value measurements were introduced195 in 1988 Aharonov,
In the panorama of quantum technologies, quantum metrol-
Albert and Vaidman, and they represent a new paradigm of
ogy represents one of the most significant candidates to prac-
quantum measurement where so little information is extracted
tical applications in an advanced stage. In particular quan-
from a single measurement that the state does not collapse.
tum enhanced optical interferometry has already found signif-
In little more detail, the weak value of an observable A b is de-
icant use in gravitational wave detectors. Furthermore, several
hψ f | b ii
A|ψ
fined as hAi
bw=
hψ f |ψi i , where the key role is symmetrically other methods have already overpassed the stage of theoreti-
played by the pre-selected (|ψi i) and post-selected (|ψ f i) cal proposals having been realised at least in proof of princi-
quantum states. When considering a von Neumann coupling ple experiments, that are prompting new applications of these
between the observable A b and a pointer observable P,
b accord- methods, for example in correlation interferometry.
ing to the unitary transformation Ub = exp(−igA b ⊗ P),
b in the The theoretical and experimental studies in this area are
weak interaction regime the evolution of this system is further promoted by new proposal of applications that range
from technological needs, in sensing or computation or other,
hψ f |e−igA⊗P |ψi i ' hψ f |ψi i(1 − ighAi
b b b w P).
b (28) to fundamental physics, some already significant nowadays,
as gravitational wave detection, other addressing more vision-
Thus, weak values are measurable quantities and can provide ary ideas that could be of interest in a next or farther future,
an "amplification" in measuring a small parameter g (e.g. a ranging from Planck scale effects148,211–213 up to wormhole
phase)195,197–205 , which can be significantly useful in presence search214 , ...
of technical noise206 .
For example, in Ref.207 it was demonstrated (both theoret-
ically and experimentally in a Sagnac configuration) that it is
possible reaching the same sensitivity of balance homodyne
ACKNOWLEDGMENTS
detection measuring the dark port only. This could allow a
general improvement of sensitivity since allowing a large re-
duction of intensity at the detector. This method was recently This work has received funding from the European Com-
extended in Ref.208 . mision’s PATHOS EU H2020 FET-OPEN grant no. 828946
As mentioned, weak value amplification finds a significant and Horizon 2020, from the EMPIR Participating States in
use in presence of technical noise. On this line, in Ref.209 it the context of the project 17FUN01 “BeCOMe”.10
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