Table Of ContentAn Entanglement-Enhanced Microscope
Takafumi Ono1,2, Ryo Okamoto1,2, and Shigeki Takeuchi1,2∗
1Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0812, Japan and
2The Institute of Scientific and Industrial Research,
Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
Among the applications of optical phase measurement, the differential interference contrast mi-
croscope is widely used for the evaluation of opaque materials or biological tissues. However, the
signal to noise ratio for a given light intensity is limited by the standard quantum limit (SQL),
which is critical for the measurements where the probe light intensity is limited to avoid damag-
ing the sample. The SQL can only be beaten by using N quantum correlated particles, with an
improvement factor of √N. Here we report the first demonstration of an entanglement-enhanced
4 microscope, which is a confocal-type differential interference contrast microscope where an entan-
1 gled photon pair (N=2) source is used for illumination. An image of a Q shape carved in relief on
0
the glass surface is obtained with better visibility than with a classical light source. The signal to
2
noise ratio is 1.35 0.12 times betterthan that limited by theSQL.
±
n
a
J Quantum metrology involves using quantum mechan- glassplatesample,whereaQshapeiscarvedinreliefon
1 ics to realize more precise measurements than can be thesurfacewithaultra-thinstepof 17nm,isobtained
3 achieved classically [1]. The canonical example uses en- with better visibility than with a c∼lassical light source.
tanglementofN particlestomeasureaphasewithapre- The improvement of the SNR is 1.35 0.12, which is
] ±
h cision ∆φ = 1/N, known as the Heisenberg limit. Such consistent with the theoretical prediction of 1.35. We
p a measurement outperforms the ∆φ = 1/√N precision also confirm that the bias phase dependence of the SNR
-
limit possible with N unentangled particles—the stan- completelyagreeswiththetheorywithoutanyfittingpa-
t
n dardquantumlimit(SQL).Progresshasbeenmadewith rameter. We believe this experimental demonstration is
a
trapped ions [2–4] and atoms [5], while high-precision an important step towards entanglement-enhanced mi-
u
q optical phase measurements have many important ap- croscopy with ultimate sensitivity.
[ plications, including microscopy, gravity wave detection, Our entanglement-enhanced microscope is based on a
1 measurements of material properties, and medical and laserconfocalmicroscopecombinedwithadifferentialin-
v biological sensing. Recently, the SQL has been beaten terference contrast microscope (LCM-DIM)[19, 20]. An
5 with two photons [6–10] and four photons [11–13]. LCM-DIM can detect a very tiny difference between op-
7
Perhaps the natural next step is to demonstrate tical path lengths in a sample. The LCM-DIM works
0
entanglement-enhanced metrology[14–16]. Among the on the principle of a polarization interferometer (Fig.
8
. applications of optical phase measurement, microscopy 1a). In this example, the horizontal (H) and vertical
1
isessentialinbroadareasofsciencefromphysicstobiol- (V)polarizationcomponentsaredirectedtodifferentop-
0
ogy. Thedifferentialinterferencecontrastmicroscope[17] tical paths by a Nomarski prism. At the sample, the
4
1 (DIM) is widely used for the evaluation of opaque ma- two beams experience different phase shifts (∆φH and
: terials or the label-free sensing of biological tissues[18]. ∆φ ) depending on the local refractive index and the
v V
Forinstance,thegrowthoficecrystalshasrecentlybeen structure of the sample. After passing through the sam-
i
X
observed with a single molecular step resolution using ple, the two beams are combined into one beam by an-
r a laser confocal microscope combined with a DIM[19]. otherNomarskiprism. The difference in the phase shifts
a
Thedepthresolutionofsuchmeasurementsisdetermined canbe detected as a polarizationrotationatthe output,
by the signal to noise ratio (SNR) of the measurement, ∆φ=∆φ ∆φ .
V H
−
and the SNR is in principle limited by the SQL. In the We obtain differential interference contrast images for
advanced measurements using DIM, the intensity of the a sampleby scanning the relativepositionof the focused
probe light, focused onto a tiny area of 10−13 m2, is beams on the sample (Fig. 1c to 1e). When two beams
∼
tightly limited for a noninvasive measurement, and the probeahomogeneousregion,the outputintensityiscon-
limit of the SNR is becoming a critical issue. stant (Fig. 1c). At the boundary of the two regions, the
In this work, we demonstrated an entanglement- signal intensity increases or decreases, since the differ-
enhanced microscope, consisting of a confocal-type dif- ence in the phase shift ∆φ becomes non-zero (Fig. 1d).
ferential interference contrast microscope equipped with Thesignalintensityreturnstotheoriginallevelafterthe
anentangledphotonsourceas a probe lightsource,with boundary (Fig. 1e). The smallest detectable change in
an SNR of 1.35 times better than that of the SQL. We the phase shift is limited by the SNR, which is the ratio
useanentangledtwo-photonsourcewithahighfidelityof of the change in the signal intensity, C(φ), and the fluc-
98%,resultinginahightwo-photoninterferencevisibility tuation of the uniform background level, ∆C, at a bias
intheconfocalmicroscopesetupof95.2%. Animageofa level of Φ . As is discussed in detail later, it is known
0
2
As is well known from two-path interferometry with
N photon states, entanglement can increase the sensi-
tivity of a phase measurement by a factor of √N. In
the entanglement-enhanced microscope, we apply this
effect to achieve an SNR that is √N higher than that
of the LCM-DIM. If the average number of N-photon
states that pass through the microscope during a given
time interval is k, then the average number of detection
events in the output is given by C(φ) = kP(φ), where
P(φ) = (1 V cos(Nφ+NΦ ))/2 is the probability of
N 0
−
detecting an odd (or even) number of photons in a spe-
cificoutputpolarization(seemethodssection). Forsmall
phase shifts of ∆φ 1/N, the phase dependence of the
≪
signalis givenby the slope ofC(φ) atthe biasphase Φ ,
0
lim C(∆φ) C(0)= ∂C(φ)/∂φ ∆φ, and the
∆φ→0 − | |φ=0×
SNR is givenby the ratio of the slope and the statistical
noise of the detection. If the emission of the N-photon
states is statistically independent, the statistical noise is
given by ∆C = kP(0). The SNR is then given by
φ=0
|
p
k sin(NΦ )
0
SNR=(1 ξ) NV | | ∆φ, (1)
N
− r2 1 VN cos(NΦ0) ×
−
whereξ isthenormalizedpoverlapregionbetweenthetwo
beams at the sample plane (see methods section). By
maximizing the function of cos(NΨ ), we can find the
0
maximum sensitivity (SNR ) as follows:
max
SNR =(1 ξ)√kN 1 1 V2 ∆φ, (2)
FIG. 1: Illustration of (a) LCM-DIM and (b) the max − − − N ×
r
entanglement-enhanced microscope. The red and blue lines q
indicate horizontally and vertically polarized light. (c), (d) at a bias phase of
and(e)Thechangeinthesignalwhilethesampleisscanned.
cos(NΦ )= 1 1 V2 /V . (3)
0 − − N N
(cid:18) q (cid:19)
that the SNR is limited by so-called ‘shot noise’ or the For the ideal case of V = 1, the maximum sensitivity
N
SQL,when‘classical’lightsourcessuchaslasersorlamps SNR = √kN ∆φ. Since the SNR for a classical
max
×
are used. That is, for a limited number of input photons microscopeusingkN photons SNR =√kN ∆φ,an
max
(N), the SNR is limited by √N. This SNR limits the entanglement-enhancedmicroscopecanimprove×theSNR
heightresolutionofthe LCM-DIMwhen usedto observe byafactorof√N comparedtoaclassicalmicroscopefor
elementary steps at the surface of ice crystals[19] or the the same photon number.
difference in refractive indexes inside a sample. Thus, We demonstrate an entanglement-enhanced micro-
improving the SNR beyond the SQL is a revolutionary scope (Fig. 2a) using a two-photon NOON state (N =
advance in microscopy. 2). First, a polarization entangled state of photons
We propose to use multi-photon quantum interfer- (2;0 + 0;2 )/√2isgeneratedfromtwobetabar-
HV HV
| i | i
ence to beat this standard quantum limit (Fig. 1b). ium borate (BBO) crystals [23, 24] and is then delivered
Instead of a classical light, we use an entangled pho- to the microscope setup via a single-mode fiber. The
ton state ((N;0 + 0;N )/√2), so-called ‘NOON’ polarization entangled state is then converted to a two-
HV HV
| i | i
state, which is a quantum superposition of the states photonNOONstate(2 0 + 0 2 )/√2usingacal-
a b a b
| i | i | i | i
‘N photons in the H polarization mode’ and ‘N pho- cite crystal[25] and focused by an objective lens. From
tons in the V polarization mode’. The phase difference the result of quantum state tomography (Fig. 2b), the
between these two states is N∆φ after passing through fidelity of the state is 98 % and the entanglement con-
the sample, which is N times larger than the classical currence is 0.979,which ensures that the produced state
case (N = 1). At the output, the result of the multi- is almost maximally entangled. The entangled photons
photon interference (the parity of the photon number in pass through two neighboring spots at the sample plane
the output) is measuredby a pair ofphoton number dis- (Fig. 2a inset). Then, after passing through the colli-
criminating detectors (PNDs) [21, 22]. mating lens, the two paths are merged by a polarization
3
FIG. 2: (a) Experimental setup for a two-photon
entanglement-enhanced microscope. A 405-nm diode laser FIG.3: (a)Atomicforce microscope (AFM)image ofaglass
(line width < 0.02 nm) was used for the pump beam. A platesample(BK7)onwhosesurfacea‘Q’shapeiscarvedin
sharpcut filterwith acutoffwavelength below715 nmanda relief with an ultra-thin step using optical lithography. (b)
band pass filter with 4-nm bandwidth were used. The beam ThesectionoftheAFMimageofthesample,whichisthearea
displacement at the calcite crystal was 4 mm. The blue and outlined in red in (a). The height of the step is estimated to
red lines indicate the optical paths for the horizontally and be17.3nmfromthisdata. (c)Theimageofthesampleusing
verticallypolarized beams. Theinsetshowsanillustration of an entanglement-enhanced microscope where two photon en-
the sample with the trajectory of the two beams. (b) Single tangledstateisusedtoilluminatethesample. (d)Theimage
photon interference fringes of a classical microscope [cps] af- of the sample using single photons (a classical light source).
terthedarkcountsofthedetectors(100cps)subtracted. (c) (e) and (f) are 1D finescan data for thearea outlined in red
Two-photoninterferencefringesofanentanglement-enhanced in(c)and(d)forthesamephotonnumberof920. Themea-
microscope [counts per 5 s]. While varying the phase by ro- surement was made at a bias phase of 0.41 (e) and 0.66 (f),
tating the phase plate (PP), we counted the detected events where optimal bias phase are 0.38 and 0.67 respectively.
intheoutput. Theclassical fringewasmeasuredbyinserting
apolarizertransmittingthediagonallypolarizedphotonsand
counting single-photon detection events. imately17nmusingopticallithography(Fig. 3). Figures
3c and 3d show the 2D scan images of the sample using
entangled photons and single photons, respectively. The
beam splitter and the result of the two photon interfer- step of the Q-shaped relief is clearly seen in Fig. 3c,
ence is detected by a pair of single-photon counters and while it is obscure in Fig. 3d. Note that for both images
a coincidence counter. The sample is scanned by a mo- we set the bias phases to almost their optimum values
torized stage to obtain an image. The beam diameters given by Eq. (3) and the average total number of pho-
at the sample plane and the distance between the center tons(N k)contributedtothese dataaresetto920per
×
of the beams are all 45 µm (ξ =0.046). position assuming the unity detection efficiency.
Figures 2c and 2d show the single-photon and two- Formoredetailedanalysis,the crosssectionofthe im-
photon interference fringes using a classical light source ages (coincidence count rate/single count rate at each
and the NOON source respectively. The fringe period position) are shown in Figs. 3e and 3f. The solid lines
of Fig. 2d is half that of Fig. 2c, which is a typical are theoretical fits to the data where the height and po-
feature of NOON state interference. The visibilities of sition of the step and the background level are used as
thesefringes,Vc =97.1 0.4%(Fig. 2c)andVq =95.2 free parameters. For Fig. 3e, the signal (the height of
± ±
0.6% (Fig. 2d), suggests the high quality of the classical the peak of the fitting curve from the background level)
and quantum interferences. is 20.21 1.13, and the noise ( the standard deviation
±
Figure3showsthemainresultofthisexperiment. We of each experimental counts from the background level
used a glass plate sample (BK7) on whose surface a Q of the fitting curve ) is 9.48 (Fig. 3g). Thus, the SNR
shapeiscarvedinreliefwithanultra-thinstepofapprox- is 2.13 0.12. Similarly, the signal, the noise, and the
±
4
entanglement-enhanced microscope, which is a confocal-
type differential interference contrast microscope
equipped with an entangled photon source as a probe
light source, with an SNR 1.35 0.12 times better
±
than the SQL. Imaging of a glass plate sample with an
ultra-thin step of 17 nm under a low photon number
∼
condition shows the viability of the entanglement-
enhanced microscope for light sensitive samples. To
test the performance of the entanglement-enhanced mi-
FIG. 4: Dependence of the SNR on the bias phase using (a) croscope, we used modest-efficiency detectors, however,
an entanglement-enhanced microscope (N = 2) and (b) a recently developed high-efficiency number-resolving
classical microscope (N =1). The SNR was calculated from photon detectors would markedly improve detection
the experimental data similar to Fig. 3E and 3F taken for
efficiency[38, 39]. We believe this experimental demon-
different bias phases. The total photon number Nk = 1150
stration is an important step towards entanglement-
for (a) and 1299 for (b). The solid line shows thetheoretical
enhanced microscopy with ultimate sensitivity, using
curveusing Eq. (1).
a higher NOON state, a squeezed state[40, 41], and
other hybrid approaches[42] or adaptive estimation
SNR are 17.7 1.22,11.25(Fig. 3h), and 1.58 0.11 for schemes[43, 44].
± ±
Fig. 3f where classical light source (single photons) are
used. TheimprovementinSNRisthus1.35 0.12,which
±
is consistent with the theoretical prediction of 1.35 (Eq.
2). The estimated height of the step was 17.0 0.9 nm Acknowledgment
±
(quantum)and16.6 1.1nm(classical),andisconsistent
±
with the estimatedvalue of17.3nm fromAFM image in The authors thank Prof. Mitsuo Takeda, Prof. Yoko
Fig. 3b. Miyamoto, Prof. Hidekazu Tanaka, Prof. Masamichi
AsEq. (1)shows,theSNRdependsonthebiasphase. Yamanishi, Dr. Shouichi Sakakihara, and Dr. Kouichi
Finally, we test the theoretical prediction of the bias Okada. This work was supported in part by FIRST
phasedependencegivenbyEq. (1)inactualexperiments. of JSPS, Quantum Cybernetics of JSPS, a Grant-in-Aid
Figure 4 shows the bias phase dependence of the SNR fromJSPS,JST-CREST,SpecialCoordinationFundsfor
for the two-photonNOON source (Fig. 4a) and classical PromotingScienceandTechnology,ResearchFoundation
light source (Fig. 4b). The solid curve is the theoreti- for Opto-Science and Technology, and the GCOE pro-
cal prediction calculated by Eq. (1), where we used the gram.
observed visibilities of the fringes in Figs. 2c and 2d for
V . The theoretical curves are in good agreement with
N
the experimental results.
Note that the entanglement-enhanced microscope we
∗
reportedhereisdifferentfromthe‘entangled-photonmi- Electronic address: [email protected]
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6
APPENDIX with a variance of σ in the x-y plane (Fig. 5). After
passing through the sample, the two beams experience a
To perform the parity measurement used in (ref. phase shift (φ) in the gray region (x > 0). The state,
21,22), it is required to count both of the events where Ψ(x,y,φ) , after the sample is written as
| i
‘even’ and ‘odd’ number of photons are detected in the
output. However,experimentalimplementationsbecome 1
Ψ(x,y,φ) = ψ (x,y,φ) +eiNΦ0 ψ (x,y,φ) ,
much easier if it is sufficient for us to just count ‘odd’ | i √2 | H i | V i
(or‘even’)number photon-detectionevents. Inaddition, (cid:0) (cid:1)(4)
the distance between the two beams and the beam size where ψ (x,y,φ) and ψ (x,y,φ) representsthestates
H V
| i | i
may effect on the SNR. Here we consider these technical of N photons in the horizontal and vertical polarization
effects on SNR and derive Eq.(1). modes respectively, and Φ is a bias phase. We assume
0
To derive Eq. (1), we consider that the two beams at that the phase shift is described by a step function and
the sample are separated at a distance of α along the x- the N photons are in the same spatial modes. These
axis,andeachsetofN photonshasGaussiandistribution states are written as
∞ ∞ 1 N
ψ (x,y,φ) = e−iχ(x)Nφ f(x α/2,y) aˆ†(x,y) 0 dxdy
| H i Z−∞∞Z−∞∞ p − √N!(cid:16) H (cid:17) | i
= e−iχ(x)Nφ f(x α/2,y)N;0,x,y dxdy,
HV
− | i
Z−∞Z−∞
∞ ∞ p 1 N
ψ (x,y,φ) = e−iχ(x)Nφ f(x+α/2,y) aˆ†(x,y) 0 dxdy
| V i Z−∞∞Z−∞∞ p √N!(cid:16) V (cid:17) | i
= e−iχ(x)Nφ f(x+α/2,y)0;N,x,y dxdy, (5)
HV
| i
Z−∞Z−∞
p
where 0 is a vacuum state, aˆ†(x,y) and aˆ†(x,y) are After passingthroughthe second calcite crystal,the two
| i H V
the creation operators in H and V polarization modes beamsaredisplacedadistanceof α/2forHpolarization
−
at the position of (x,y) respectively, and χ(x) is a step and α/2 for V polarization along the x-axis, resulting in
function that χ(x) = 0 for x 0 and χ(x) = 1 for the two beams in the same spatial mode. The state can
≤
x > 0. f(x α/2,y) and f(x+α/2,y) represent the N then be written as
−
photonprobabilitydensitiesinthehorizontalandvertical
polarization modes written as
f(x,y)= 1 e−(x22+σ2y2); ∞ ∞ f(x,y) dxdy =1.
2πσ2
Z−∞Z−∞
(6)
1
Ψ′(x,y,φ) = ψ (x+α/2,y,φ) +eiNΦ0 ψ (x α/2,y,φ) . (7)
H V
| i √2 | i | − i
(cid:0) (cid:1)
Here,weassumethatthestateisprojectedontothestate erator in the basis of plus (P) and minus (M) diagonal
where odd number of photons are in the minus diagonal polarization is therefore written as
polarization mode at the output. The measurement op-
7
∞ ∞ ∞ N/2 n;2m 1,x,y n;2m 1,x,y dxdy if N is even
Πˆ = −∞ −∞ n=0 m=1 | − iPM PMh − | (8)
(R−∞∞R−∞∞ P∞n=0 Pm(N=+11)/2 |n;2m−1,x,yiPM PMhn;2m−1,x,y| dxdy if N is odd.
R R P P
where
1 n m
n;m,x,y = aˆ†(x,y) aˆ† (x,y) 0
| iPM √n!m! P M | i
(cid:16) (cid:17) (cid:16) (cid:17) n m
1 1 1
= aˆ†(x,y)+aˆ†(x,y) aˆ†(x,y) aˆ†(x,y) 0 .
√n!m! √2 H V √2 H − V | i
(cid:18) (cid:16) (cid:17)(cid:19) (cid:18) (cid:16) (cid:17)(cid:19)
(9)
The probability of odd number photon-detection can then be written
P(φ) = Ψ′(x,y,φ)Πˆ Ψ′(x,y,φ)
h | | i
1 1
= (1 cos(Nφ+NΦ ))(1 ξ(α))+ (1 cos(NΦ ))ξ(α), (10)
0 0
2 − − 2 −
where we denote the phase independent term of super sensitivity. Note also that the dependence of SNR
∞ −α/2f(x,y) dxdy + ∞ ∞ f(x,y) dxdy as ξ(α) on the size and the distance between the two beams is
∞ −∞ ∞ α/2
which is the overlapintegralbetween H and V polarized simply givenby (1 ξ). This means thatitis reasonable
R R R R −
to comparethe SNR betweenanentanglement-enhanced
beams at the sample plane.
WenowcalculatetheSNRofourmicroscopeusingthe microscope and a classical microscope (N = 1) for the
NOON state including the effect of the overlap between same ξ.
the two beams at the sample plane. If the emission of
the N-photonstatesis statisticallyindependent, the sta-
tistical noise at the bias phase ∆C(φ) = kP(0) as
φ=0
|
given by
p
1
∆C(0)=√k (1 cos(NΦ )) . (11)
0
2 −
r
For a small phase shift of ∆φ 1/N, the signal is
≪
k
∂(C(φ))/∂φ ∆φ=(1 ξ(α)) N sin(NΦ ) ∆φ.
| ||φ=0× − 2 | 0 |×
(12)
Considering the visibility of the interference fringe V ,
N
the SNR is given by
k sin(NΦ )
0
SNR=(1 ξ) NV | | ∆φ. (13)
N
− r2 1 VNcos(NΦ0)×
−
Thus one can confirm thapt counting odd (or even) num-
ber photon-detection events can also achieve the phase
8
FIG. 5: The schematic of the two probe beams. The beams
are in V polarization (blue) and H polarization (red) on the
sample,correspondingtoFigs1c-1e. Thegrayshadedregion
has thephase change of φ.
