Quantum microscopy of cells at the Heisenberg limit

Article

https://doi.org/10.1038/s41467-023-38191-4

Quantum microscopy of cells at the

Heisenberg limit

Zhe He

1,2

, Yide Zhang

1,2

, Xin Tong

1,2

,LeiLi

& Lihong V. Wang

Entangled biphoton sources exhibit non

classical characteristics and have been

applied to imaging techniques such as

ghost imaging, quantum holography,

and quantum optical coherence tomography. The development of wide-

fi

eld

quantum imaging to date has been hindere

dbylowspatialresolutions,speeds,

and contrast-to-noise ratios (CNRs). H

ere, we present quantum microscopy by

coincidence (QMC) with balanced pa

thlengths, which enables super-

resolution imaging at the Heisenberg limit with substantially higher speeds

and CNRs than existing wide-

fi

eld quantum imaging methods. QMC bene

fi

from a con

fi

guration with balanced pathleng

ths, where a pair of entangled

photons traversing symmetric paths wit

h balanced optical pathlengths in two

arms behave like a single photon with half the wavelength, leading to a two-

fold resolution improvem

ent. Concurrently, QMC resists stray light up to 155

times stronger than classical signals. The low intensity and entanglement

features of biphotons in QMC promis

e nondestructive bioimaging. QMC

advances quantum imaging to the microscopic level with signi

fi

cant

improvements in speed and CNR towar

d the bioimaging of cancer cells. We

experimentally and theoretically prove that the con

fi

guration with balanced

pathlengths illuminates an avenue fo

r quantum-enhanced coincidence ima-

ging at the Heisenberg limit.

Since the

fi

rst demonstration of entangled photon sources, the

biphoton state

–

has found extensive applications in quantum

computing

, quantum metrology

, and quantum information

.In

particular, the nonclassical behavior of biphotons motivates the search

for solutions that break classical limits, such as the uncertainty prin-

ciple or the diffraction limit

. The diffraction pattern of biphotons

has been demonstrated to be half as narrow as that of classical light

–

indicating the capability of biphoton imaging to achieve super reso-

lution beyond what is possible with classical light in diffraction-limited

linear imaging

A variety of approaches have been proposed for quantum imaging

using biphotons. Different nonlinear crystals, including

-barium

borate (BBO)

and periodically poled potassium titanyl phosphate

(PPKTP)

, were used for generating entangled photon pairs utilizing

the spontaneous parametric down-conversion (SPDC) effect

–

.In

addition, different types of detectors were employed for biphoton

detection. For example, single-photon avalanche diodes (SPADs) can

provide direct coincidence measurements based on the arrival times

of entangled photon pairs but do not have spatial resolution as they

are single-pixel detectors. Though SPAD-array cameras add spatial

resolution to single SPADs, they have a small number of pixels

–

Electron multiplying charge-coupled devices (EMCCDs) provide a

large number of resolvable pixels but are not capable of direct coin-

cidence measurements due to the low frame rate

–

. Thus far, two

methods have been developed and are frequently used to extract the

coincidence counts from an EMCCD camera

. However, these

methods typically require more than 2 × 10

frames to generate a single

coincidence image, which can take over 17 h, given the low frame rate.

Received: 15 August 2022

Accepted: 14 April 2023

Check for updates

Caltech Optical Imaging Laboratory, Andrew and Peggy Cherng Department of Medical Engineering, Department of Electrical Engineering, California

Institute of Technology, 1200 E. California Blvd., MC 138-78, Pasadena CA 91125, USA.

These authors contributed equally: Zhe He, Yide Zhang, Xin Tong.

e-mail:

LVW@caltech.edu

Nature Communications

| (2023) 14:2441

1234567890():,;

The development of wide-

fi

eld quantum imaging, therefore, is ham-

pered by the low acquisition rate. In comparison to classical wide-

fi

eld

imaging, quantum imaging bene

fi

ts from stray light resistance

enhancement of two-photon absorption

, and enhanced resolution

as a result of quantum correlation

. Despite these advantages,

EMCCD-based wide-

fi

eld quantum imaging with spatial resolution

fi

ne as 1.4

m has never been reported due to the use of low-

intensity light.

When quantum entanglement is applied for resolution beyond the

classical limit

entangled photons may improve spatial resolution by

times, corresponding to the Heisenberg limit

.Usingabiphoton

NOON state in quantum lithography

enhances resolution by two fold,

and using an SPDC source also achieves the Heisenberg limit

.Both

methods utilize co-propagating biphotons to enhance resolution by a

factor of 2

. Moreover, recent studies indicate that resolution at the

Heisenberg limit could be realized even without requiring both entan-

gled photons to pass through the imaging object

Here, inspired by the wide-

fi

eld imaging method introduced in

refs.

, we develop quantum microsc

opy by coincidence (abbre-

viated as QMC in our work) with balanced pathlengths using an EMCCD

camera. Previous studies could not be used for practical microscopy for

the following reasons. First, they

did not have the option for high-

resolution imaging due to the small numerical aperture (NA) of the

imaging system. Second, they had a low imaging speed, requiring a

large number of frames for coincidence measurements. In our techni-

que, we improve the spatial resolution and speed by introducing a high-

NA microscopy design and a more ef

fi

cient algorithm. QMC relies on

the nonclassical properties of biphotons for super-resolution micro-

scopy with up to 5 times higher speeds, 2.6 times higher CNRs, and 10

times more resistance to stray light than existing wide-

fi

eld quantum

imaging techniques

. In contrast to the quantum imaging techniques

at the standard quantum limit

, QMC improves resolution by a factor

of 2 at the Heisenberg limit

. We demonstrate QMC for imaging cancer

cells with a 1.4

mresolutionanda100×50

fi

eld of view (FOV).

The combination of the improved s

peed, enhanced CNR, more robust

stray light resistance, super resolution, and low-intensity illumination

empowers QMC toward bioimaging.

Results

Experimental setup

The experimental setup of QMC is presented in Fig.

(see

“

Methods

”

for details). The

fi

rst prism separates the signal and idler photons into

two arms, and the optical pathlengths of the separated arms are

balanced (see

“

Discussion

”

). The arm associated with the object is

considered by itself the classical part for imaging, which can be

regarded as a wide-

fi

eld microscope. Therefore, an image acquired by

this arm alone is deemed a classical image.

Despite being used to image distinct types of objects, the previous

methods

(Supplementary Fig. 1) and QMC are based on similar

theories. However, the previous works demonstrated only macro-

scopic imaging because the two arms share the same lenses with a

small NA and a large

fi

eld of view (FOV). In comparison, QMC evenly

splits the beam at the source Fourier plane into the signal and idler

arms using a right-angle prism, which allows integrating high-NA

objectives in each arm. As shown in Fig.

, the two arms are built

symmetrically to ensure balanced optical pathlengths and magni

fi

ca-

tion ratios, which are the key conditions for super resolution at the

Heisenberg limit (see

“

Discussion

”

Estimation of coincidence

As shown in Fig.

a, we develop a covariance algorithm to ef

fi

ciently

estimate the coincidence intensity of signal and idler photons using an

EMCCD camera. The signal and idler photons are detected by the left

and right regions of the camera, respectively. The calibration for the

EMCCD is shown in Supplementary Fig. 2. The total intensity

related to the coincidence intensity of biphotons

coin

and the intensity

of noise

noise

,where

and

represent the left and right regions,

respectively. In QMC, we use the mean value of coincidence intensity

coin

to estimate the intensity correlation

QMC

. Studies have demon-

strated that the distributions of entangled photon pairs in both regions

are symmetric about a center point due to their momentum antic-

orrelation in the far

fi

eld of the crystal

; therefore, the left and right

images can be inversely registered pixel by pixel according to the

symmetric center. The intensities of each pair of inversely registered

pixels in the left and right images are given by

coin

noise

coin

noise

The covariance between

and

is de

fi

ned by

cov

Þð

where

is the number of frames, and the subscript

refers to the frame

index. Eq. (

) can be simpli

fi

ed to

cov

coin

noise

while the

fi

rst two terms represent the variance of the signal, the

last two terms represent the covariance of the detection noise. Because

266 nm

CW laser

HWP

VWP

BBO

BPF

Source Fourier plane (

)

Object plane (

obj

)

Reference plane (

ref

)

Intermediate plane

Detection plane (

det

)

EMCCD

BPF

Fig. 1 | Experimental setup of QMC.

CW continuous wave, GL Glan-Laser polarizer,

HWP half-wave plate, VWP variable wave plate, BBO

-barium borate crystal, BPF

532 nm bandpass

fi

lter, PBS polarizing beam splitter, EMCCD electron multiplying

charge-coupled device camera.

= 50 mm,

= 180 mm,

= 9 mm,

= 300 mm,

and

= 200 mm. The source Fourier plane P

is set at the Fourier plane of the BBO

crystal.

Article

https://doi.org/10.1038/s41467-023-38191-4

Nature Communications

| (2023) 14:2441

the noise is primarily caused by the detector, which can be assumed to

be uncorrelated between the left an

d right regions, the covariance of

the detection noise approximately vanishes. In Supplementary Fig. 3, we

demonstrate that

noise

≪

coin

.Further,as

the coincidence intensity follows a Poisson distribution

,forwhichthe

variance equals the mean, we have

cov

coin

With enough frames, Eq. (

) directly estimates the expected

value of the coincidence intensity, which is not directly provided by

the existing algorithms

. Figure

b compares QMC with the

existing methods

in terms of the CNR versus the number of

frames. The CNR calculation work

fl

ow is shown in

“

Methods

”

and Supplementary Fig. 4. To achieve a CNR of 3, QMC requires

frames (10 ms per frame), which is approximately 40% and

20% of the required frames in refs.

, respectively. When

2×10

frames are used, QMC outperforms the methods given

in refs.

with 1.5 times and 2.6 times higher CNR, respectively

(see Supplementary Fig. 4). As expected, CNR increases with the

number of frames (Supplementary Fig. 5).

Resistance to stray light is a major advantage of quantum imaging.

Eq. (

) indicates that the covariance algorithm suppresses uncorre-

lated noise, such as stray light. While the methods introduced in

refs.

were reported to be effective in preventing stray light in

images using over 2 × 10

frames, their effectiveness is limited, espe-

cially when the frame number is less than 10

.Figure

cshowsthe

dependence of CNR on stray light intensity with 10

frames. The data

processing work

fl

ow for the stray light resistance in Fig.

cis

demonstrated in Supplementary Fig.

6. When the stray light intensity is

~12 times greater than the classical signal, the classical image is severely

disrupted because the CNR falls below unity; the methods in refs.

cannot maintain the CNR either. Nonetheless, with 10

frames, the CNR

of QMC remains higher than unity even when the stray light is ~120

times stronger than the classical signal. Figures

d,edisplaytheclas-

sical and QMC images of carbon

fi

bers in the presence of stray light

that is 8 times stronger than the class

ical signal. Whereas the classical

image is overwhelmed by the stray light (CNR = 0.92 ± 0.11), the QMC

′

Intensity

′

cov(

)≫cov(

′

)

Classical

Quantum

Fig. 2 | Coincidence measurement of QMC. a

The coincidence measurement relies

on the fact that the covariance between entangled photons in a sequence of frames

is much larger than the covariance between two random photons.

and

refer to

the left and right regions of the EMCCD, which are used to detect the signal and

idler photons, respectively.

and

are symmetric positions on the detector for

the signal and idler photons.

′

is a random position in the right region and

different from

. Inset, intensities (

) at the three positions in different frames.

CNRs of QMC and the wide-

fi

eld quantum imaging methods in refs.

using

different numbers of frames. Data are plotted as means ± standard errors of the

means (

= 10).

CNRs of QMC and the wide-

fi

eld quantum imaging methods in

refs.

with 10

frames in the presence of stray light with different intensities.

Data are plotted as means ± standard errors of the means (

= 10). The mean

number of photons incident on the EMCCD is 0.49 per pixel per frame. Classical (

)

and QMC (

)imagesofcarbon

fi

bers in the presence of stray light with an intensity

of 8

, acquired using 2 × 10

frames. Scale bars, 20

Article

https://doi.org/10.1038/s41467-023-38191-4

Nature Communications

| (2023) 14:2441

image eliminates the stray light nearly completely by extracting the

coincidence intensity (CNR = 8.03 ± 1.22). In fact, with 2 × 10

frames,

QMC effectively suppresses stray light that is ~155 times stronger than

the classical signal (Supplementary Fig. 7). The stray light resistance

reaches its limit when the accidental coincidence caused by the stray

light equals the true coincidence. The covariance algorithm proves to

be the most effective at

fi

nding true coincidences of entangled pho-

tons and eliminating uncorrelated noise, providing the highest CNR

under the same stray light intensity.

Quanti

fi

cation of super resolution at the Heisenberg limit

We next quantify the enhanced spatial resolution of QMC. Figure

shows a simpli

fi

ed schematic of QMC. Figure

b demonstrates the

classical image of group 7 of a US Air Force (USAF) resolution

target that includes stripes of varying widths (from 2.76 to 3.91

m),

which approximate the highest resolution of our classical imaging

setup. In Fig.

c, the QMC image shows a higher resolution than the

classical image. We evaluate the resolution enhancement by deter-

mining the full width at half maximum (FWHM) of the line spread

functions (LSFs) near the focal point (see

“

Methods

”

for details).

Figure

d shows that the highest resolutions of classical imaging

and QMC are 2.9

m and 1.4

m, respectively, indicating that QMC

improves the spatial resolution of classical imaging by approximately

a factor of 2. The LSFs at different axial

coordinates are shown

in Fig.

QMC demonstrates a two-fold enhancement in spatial resolution

over classical imaging due to the fact that the equivalent wavelength of

the biphoton is half the wavelength of the SPDC photon. As shown in

Fig.

,and

are the coordinates of the source Fourier plane,

the object plane, and the detection plane in the signal arm, respec-

tively.

,and

are the corresponding coordinates in the idler

arm. The quantized

fi

eld operators

ðÞ

and

ðÞ

for the signal and idler

arms are shown in Supplementary Note 1. The intensity correlation

between the signal and idler photons for a given pixel pair is given by

the second-order correlation function:

QMC

ðÞ

where

∣

is the wavefunction of a photon pair emitted from the source

Fourier plane at

and

∣

and

are the entangled wavevectors of the signal and idler

photons from the source Fourier plane. Denoting

and

the

BBO

ℎ

μm

-10

μm

-20

μm

-30

μm

-40

μm

Classical

Quantum

Fig. 3 | Spatial resolution of QMC. a

Simple schematic of QMC. BBO,

-barium

borate crystal. EMCCD, electron multiplying charge-coupled device camera.

and

are the coordinates of the source Fourier plane, the object plane, and the

detection plane, respectively.

,and

are the corresponding coordinates in

the signal arm, and

,and

are the coordinates in the idler arm.

and

are the PSFs from

and from

and

are the PSFs from

and from

is the amplitude

transmission coef

fi

cient of the object. Classical (

)andQMC(

) images of group 7

(2.76 ~ 3.91

m) of a USAF 1951 resolution target. Scale bars, 20

m. All the images

are normalized by their maximum and minimum intensities (see

“

Methods

”

Spatial resolution of classical imaging and QMC versus the axial

coordinate from

the classical focal point. The highest spatial resolutions for classical imaging and

QMC are 2.9

mand1.4

m, respectively. Data are plotted as means ± standard

errors of the means (

=14).

Normalized lateral LSFs of classical imaging and QMC

versus the lateral coordinate at different

positions.

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Nature Communications

| (2023) 14:2441

position and wavevector of the pump light, Eq. (

)demonstratesa

spatially entangled state with

and

denotes the probability amplitude of the state

∣

.The

momentum correlation width of the entangled photons is demon-

strated in Supplementary Fig. 8.

As derived in Supplementary Note 1, the QMC image can be

described by

ðÞ

QMC

ðÞ

QMC

ρ

ðÞ

∣

ρ

ðÞ

∣

QMC

;

ρ

;

ρ

ρ

where

(

ρ

) is the amplitude transmission coef

fi

cient of the object, and

ρ

is the 2D coordinates on the object plane.

QMC

;

ρ

is the dis-

tribution of squared intensity with a wavelength of

/2 on the object

plane.

(

;

ρ

ρ

) is the point spread function (PSF) from

ρ

on the

object plane (P

obj

)to

ρ

on the detection plane (P

det

) for the light with

a wavelength of

/2.

is the magni

fi

cation ratio from the object to the

detector. Similarly, the classical counterpart is given by

ρ

ðÞ

∣

ρ

ðÞ

∣

;

ρ

ðÞ

∣

;

ρ

ρ

ðÞ

∣

where

;

ρ

ðÞ

is the intensity distribution of wide-

fi

eld illumination

with a wavelength of

on the object plane.

;

ρ

ρ

ðÞ

denotes the

point spread function (PSF) from

ρ

on the object plane (P

obj

)to

ρ

thedetectionplane(P

det

) for the light with a wavelength of

In contrast to linear classical imaging, QMC is based on a pure

quantum effect to achieve the Heisenberg limit. Unlike the quantum

super-resolution methods requiring both signal and idler photons to

pass through the object

, the idler photons in our experiment do not

traverse the object. Indicated by Eqs. (S5) and (S6), while calculating

the QMC image, the point spread functions

and

Fig.

a are multiplied for each photon pair instead of being multiplied

classically as

. This concept has been theoretically proven for

quantum imaging based on a simpli

fi

ed model

,which,however,

cannot be applied to the complex setup in this work (see Fig.

a).

For example, as shown in Fig.

b, c, the spatial distributions of the

wide-

fi

eld illumination for the classical imaging and QMC are different,

which was not considered in ref.

but can be explained by

different wide-

fi

eld illumination functions

QMC

;

ρ

and

;

ρ

ðÞ

Eqs. (

The classical resolution in our experiment is limited by the

effective NA of the objectives, which may be lower than the nominal

NA of the objectives (NA = 0.4) due to under

fi

lling.

Imaging cells by QMC

In Fig.

, we demonstrate classical (Fig.

a) and QMC (Fig.

b) imaging

of cancer cells. Figure

c shows the normalized intensities between the

arrows in Fig.

a, b. QMC clearly distinguishes the cell structures that

cannot be resolved in its classical counterpart. Note that the lumpy

features in both images are due to imperfect sample preparation.

Whereas the images in Fig.

were averaged over 2 × 10

frames (10 ms

per frame) to achieve a high CNR, the images of the cells shown in

Supplementary Fig. 9 were acquired using fewer (10

)frames.

Discussion

The balanced pathlengths require symmetry in the optical paths of the

signal and idler photons from the source Fourier plane to the detection

planes, such that the paired photons are correlated in positions and

momentums concurrently, and the phases of the paired photons can

be combined. This requirement, however, cannot be satis

fi

ed through

classical sources because two unentangled photons can only be cor-

related in either position or momentum in accordance with the

uncertainty principle

. As a consequence of the path symmetry, all

entangled photon pairs should appear at positions symmetric about

the same center within the source Fourier plane, the object plane, and

the detection plane. We mirrored the signal arm setup onto the

idler arm to maintain the path symmetry as precisely as possible.

In particular, the photon pairs on the symmetric positions on

the source Fourier plane propagate symmetrically due to the SPDC

phase matching, and they propagate through the identical pairs of free-

space 4

systems to reach the object plane and the reference plane,

respectively. The sign

al photon on the object plane can be scattered by

the object, leading to different wavevectors between the signal and idler

photons. In Fig.

a, in the paraxial approximation, photons traversing

the same position on the object plane would arrive at the same position

on the detection plane, indicating

identical optical pathlengths

according to Fermat

’

s principle. Though the scattering effect appears to

disrupt the path symmetry, the pathlength symmetry is maintained

because the conjugation between the object and detection planes bal-

ances the optical pathlengths of a scattered signal photon and the

related idler photon. Therefore, we can utilize the con

fi

guration of the

balanced pathlengths to describe t

he biphoton propagation from the

sourceFourierplanetothedetectionplane(seeEq.(S5)).

We have attempted to provide the most practical setup and

algorithm for quantum bioimaging with a spatial resolution down to

1.4

m. However, the current implementation of QMC is not intended

to compete with state-of-the-art classical microscopy techniques in

termsofCNRsbecauseofthelowSPDCef

fi

ciency of the BBO crystal.

For example, to achieve a CNR of 3, QMC requires the acquisition of

about 10

frames over 17 min, whereas classical imaging may only need

a single frame captured in less than a second. With more powerful

quantum sources in the future

, QMC could demonstrate quantum

advantages over state-of-the-art classical imaging. Furthermore, com-

pared with classical methods, such as SHG microscopy

, that reject

background noise through spectral

fi

ltering, QMC eliminates both

temporally and spatially uncorrelated background noise through

coincidence detection.

In conclusion, we have demonstrated quantum microscopy of

cancer cells at the Heisenberg limit. QMC is advantageous over exist-

ing wide-

fi

eld quantum imaging methods due to the 1.4

m resolution,

Classical

Quantum

Fig. 4 | Imaging of cancer cells with QMC.

Classical (

)andQMC(

)imagesoftwo

HeLa cells. Scale bars, 20

Normalized classical and QMC intensities between

the arrows in (

)and(

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Nature Communications

| (2023) 14:2441

up to 5 times higher speed, 2.6 times higher CNR, and 10 times more

robustness to stray light. With low-intensity illumination, we have

demonstrated that QMC is suitable for nondestructive bioimaging at a

cellular level, revealing details that cannot be resolved by its classical

counterpart. Finally, while the resolution of classical imaging can be

improved in various ways

,thecon

fi

guration used in QMC can

further improve the resolution by halving the wavelength, thus push-

ing the boundary of classical super-resolution imaging techniques with

quantum enhancement.

Methods

Experimental setup

In the QMC system, a

-barium borate (BBO) crystal (5 × 5 × 0.5 mm

PABBO5050-266(I)-HA3, Newlight Photonics) was cut for type-I SPDC

at 266 nm wavelength. The pump was a 266 nm continuous-wave laser

(FQCW266-10-C, CryLaS) with an output power of 10 mW. A UV-coated

Glan-Laser polarizer (GLB10-UV, Thorlabs) and a half-wave plate

(WPH05M-266, Thorlabs) were used to adjust the polarization angle of

the pump laser beam. For imaging, the pump beam was adjusted to be

vertically polarized. The pump laser beam then passed through the

BBO crystal and generated a ring of SPDC photons with a half-opening

angle of 3°. A bandpass

fi

lter with a center wavelength of 532 nm and a

bandwidth of 2 nm (64-252, Edmund Optics) was used to block the

pump beam. The generated SPDC photon pairs propagated through an

=50mmlenstotheFourierplane,i.e.,thesourceFourierplane(P

and were spatially separated using a knife-edge right-angle prism

mirror (MRAK25-P01, Thorlabs). The separated signal and idler pho-

tons propagated to the object plane (P

obj

) and the reference plane

ref

), respectively, by two identical 4

imaging systems comprising

of an

= 180 mm lens and an

= 9 mm objective (LI-20X, 0.4 NA,

Newport). The sample was placed on the object plane. The object

plane, the reference planes, and the intermediate plane were conjugate

through the other two identical 4

imaging systems, which consist of

an identical set of

= 9 mm objectives and

= 180 mm lenses and

another right-angle prism mirror. Each objective was followed by an

HWP mounted on a motorized precision rotation mount (PRM1Z8,

Thorlabs). The intermediate plane and the detection plane (P

det

)ofan

EMCCD camera (iXon Ultra 888, Andor) were conjugated through a 4

system consisting of

= 300 mm and

= 200 mm lenses. Another BPF

was placed in front of the EMCCD camera to block unwanted stray

light. The EMCCD was operated at

−

65 °C, with a horizontal pixel shift

readout rate of 10 MHz, a vertical pixel shift speed of 1.13

μ

s, and an

electron multiplier (EM) gain of 1000. The whole setup was covered by

a light-shielding box.

Sample preparation

”

×2

”

(5.08 × 5.08 cm) positive 1951 USAF resolution target (58-

198, Edmund Optics) was used to quantify the spatial resolution and

DOF of our system. The carbon

fi

ber sample was prepared by ran-

domly distributing carbon

fi

bers with a diameter of 6

μ

m on top of a

glass slide. The

fi

bers were mixed with UV-curing optical adhesive

(NOA61, Thorlabs) and sealed with a cover glass. The optical adhesive

was then cured by illumination of UV light from a light-emitting diode

(LED). HeLa cells were placed on sterile glass slides and cultured in

DMEM supplemented with 10% fetal bovine serum and a penicillin-

streptomycin mixture (all from Invitrogen/Life Technologies) at

37 °C in a 5% CO

air atmosphere. When cells were 70% con

fl

uent on

the glass slides, we

fi

xed them with an ice-cold mixture of ethanol and

methanol (1:1 volume ratio). The glass slides placed in a 10-cm petri-

dish were covered by the organic solvents and then incubated in a

freezer (

−

20 °C) for 5

–

7 min. The organic solvents preserved the

cells by removing lipids, dehydrating tissue, and denaturing and

precipitating the proteins in the cells. After

fi

xation, the glass slides

were gently rinsed with phosphate-buffered saline to remove any

fi

xation agent.

Data acquisition and processing

A custom-written LabVIEW (National Instruments) program utilizing

the library from the Andor software development kit (SDK) was used to

control the EMCCD for data acquisition. The imaging data were saved

as 16-bit Flexible Image Transport System (FITS)

fi

les with each

fi

containing 1000 frames. The FITS

fi

les were imported into MATLAB

(MathWorks) and processed with custom-written scripts. The EMCCD

frames were extracted from the

fi

les and were used to calculate the

coincidence intensity using our QMC algorithm. The reconstructed

images were then interpolated based on a cubic spline using not-a-knot

end conditions for better visualization. The maximum pixel number of

the EMCCD camera is 1024 × 1024. We utilized an area of 100 × 50

pixels after binning of 2.

Image normalization

Denoting

as the image intensity, the normalized intensity is calcu-

lated by

norm

min

max

min

where

max

and

min

are the maximum and minimum values of

Contrast-to-noise ratio estimation

Denoting

and

as the intensities of the object of interest and the

background, respectively, the contrast-to-noise ratio (CNR) is calcu-

lated by

CNR =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where

and

are the mean values;

and

are the standard devia-

tions of

and

Measurements of resolution and depth of

fi

eld

To measure the resolution of our system, the line pro

fi

perpendicular to an edge in the USAF resolution target was extracted

and

fi

tted to an edge spread function (ESF) centered at

, i.e.,

ESF

ðÞ

erf

,where

and

are coef

fi

cients, and

the radius of the beam. A Gaussian line spread function (LSF) was

obtained by taking the derivative of the ESF, i.e., LSF

ðÞ

ESF

exp

ffiffiffiffi

. The resolution was estimated by the

FWHM of the LSF, i.e.,

R

ffiffiffiffiffiffiffi

ln2

Imaging with stray light

A 532 nm continuous-wave laser (MLL-III-532, CNI) was used to

introduce stray light to the detection plane. Transmitting through

a ground glass diffuser (DG20-1500, Thorlabs), the 532 nm laser

created speckle patterns on the detection plane, leading to sig-

fi

cantly reduced CNR in the raw EMCCD images. To evaluate how

robust the classical imaging and QMC were against the stray light, we

acquired images under different stray light intensities and calculated

their CNRs.

Statistics and reproducibility

Statistical analysis was performed using MATLAB (R2021a). Data are

presented as means ± standard errors of the means in all

fi

gure parts in

which error bars are shown. No statistical method was used to pre-

determine sample sizes. We determined sample sizes based on our

preliminary studies and on the criteria in the

fi

eld to experimentally

demonstrate the imaging system. All experiments except for that

shown in Fig.

were replicated at least twice. All attempts at replica-

tion were successful. Cell imaging in Fig.

with 2 × 10

frames was not

replicated because we repeated the measurement under the same

Article

https://doi.org/10.1038/s41467-023-38191-4

Nature Communications

| (2023) 14:2441

condition with 10

frames (Supplementary Fig. 9). Besides, we have

repeated experiments on other samples under the same condition

with 2 × 10

frames.

Reporting summary

Further information on research design is available in the Nature

Portfolio Reporting Summary linked to this article.

Data availability

Imaging data for the cell images generated in Fig.

are available in the

Github online at

http://github.com/ZheHE2022/Quantum-Microscopy-

of-Cells-at-the-Heisenberg-Limit

. All data used in this study are available

from the corresponding auth

or upon reasonable request.

Code availability

The code for the covariance algorithm is provided in the Supplemen-

tary Software and Github online at

http://github.com/ZheHE2022/

Quantum-Microscopy-of-Cells-at-the-Heisenberg-Limit

. All custom

codes used in this study are available from the corresponding author

upon reasonable request.

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Acknowledgements

We thank Dr. Qiyuan Song and Samuel A. Solomon for their assistance

with the experiment. We also thank Dr. Kelvin Titimbo Chaparro and

Siddik Suleyman Kahraman for the discussion. This project has been

made possible in part by grant number 2020-225832 from the Chan

Zuckerberg Initiative DAF, an advised fund of Silicon Valley Community

Foundation, and National Institutes of Health grants R35 CA220436

(Outstanding Investigator Award) and R01 EB028277.

Author contributions

Z.H., Y.Z., and X.T. built the imaging system, performed the experiments,

and analyzed the data. Z.H. developed the quantum imaging theory. Y.Z.

Article

https://doi.org/10.1038/s41467-023-38191-4

Nature Communications

| (2023) 14:2441

developed the data acquisition program. L.L. prepared the biological

samples. L.V.W. conceived the concept and supervised the project.

All authors contributed to writing the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information

The online version contains

supplementary material available at

https://doi.org/10.1038/s41467-023-38191-4

Correspondence

and requests for materials should be addressed to

Lihong V. Wang.

Peer review information

Nature Communications

thanks the anon-

ymous reviewers for their contribution to the peer review of this work.

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