Abstract
Among the applications of optical phase measurement, the differential interference contrast microscope is widely used for the evaluation of opaque materials or biological tissues. However, the signaltonoise ratio for a given light intensity is limited by the standard quantum limit, which is critical for measurements where the probe light intensity is limited to avoid damaging the sample. The standard quantum limit can only be beaten by using N quantum correlated particles, with an improvement factor of √N. Here we report the demonstration of an entanglementenhanced microscope, which is a confocaltype differential interference contrast microscope where an entangled photon pair (N=2) source is used for illumination. An image of a Q shape carved in relief on the glass surface is obtained with better visibility than with a classical light source. The signaltonoise ratio is 1.35±0.12 times better than that limited by the standard quantum limit.
Introduction
Quantum metrology involves using quantum mechanics to realize more precise measurements than those that can be achieved classically^{1}. The canonical example uses entanglement of N particles to measure a phase with a precision Δφ=1/N, known as the Heisenberg limit. Such a measurement outperforms the precision limit possible with N unentangled particles—the standard quantum limit (SQL). Progress has been made with trapped ions^{2,3,4} and atoms^{5}, whereas highprecision optical phase measurements have many important applications, including microscopy, gravity wave detection, measurements of material properties, and medical and biological sensing. Recently, the SQL has been beaten with two photons^{6,7,8,9,10} and four photons^{11,12,13}.
Perhaps the next natural step is to demonstrate entanglementenhanced metrology^{14,15,16}. Among the applications of optical phase measurement, microscopy is essential in broad areas of science from physics to biology. The differential interference contrast microscope (DIM)^{17} is widely used for the evaluation of opaque materials or the labelfree sensing of biological tissues^{18}. For instance, the growth of ice crystals has recently been observed with a single molecular step resolution using a laser confocal microscope (LCM) combined with a DIM^{19}. The depth resolution of such measurements is determined by the signaltonoise ratio (SNR) of the measurement, and the SNR is in principle limited by the SQL. In the advanced measurements using DIM, the intensity of the probe light, focused onto a tiny area of ~ 10^{−13} m^{2}, is tightly limited for a noninvasive measurement, and the limit of the SNR is becoming a critical issue.
In this work, we demonstrate an entanglementenhanced microscope, consisting of a confocaltype differential interference contrast microscope equipped with an entangled photon source as a probe light source, with an SNR of 1.35 times better than that of the SQL. We use an entangled twophoton source with a high fidelity of 98%, resulting in a high twophoton interference visibility in the confocal microscope setup of 95.2%. An image of a glass plate sample, where a Q shape is carved in relief on the surface with a ultrathin step of ~ 17 nm, is obtained with better visibility than with a classical light source. The improvement of the SNR is 1.35±0.12, which is consistent with the theoretical prediction of 1.35. We also confirm that the bias phase dependence of the SNR completely agrees with the theory without any fitting parameter. We believe this experimental demonstration is an important step towards entanglementenhanced microscopy with ultimate sensitivity.
Results
The limit of differential interference contrast microscope
Our entanglementenhanced microscope is based on a laser confocal microscope combined with a differential interference contrast microscope (LCM–DIM)^{19,20}. An LCM–DIM can detect a very tiny difference between optical path lengths in a sample. The LCM–DIM works on the principle of a polarization interferometer (Fig. 1a). In this example, the horizontal (H) and vertical (V) polarization components are directed to different optical paths by a Nomarski prism. At the sample, the two beams experience different phase shifts (Δφ_{H} and Δφ_{V}) depending on the local refractive index and the structure of the sample. After passing through the sample, the two beams are combined into one beam by another Nomarski prism. The difference in the phase shifts can be detected as a polarization rotation at the output, Δφ=Δφ_{V}−Δφ_{H}.
We obtain differential interference contrast images for a sample by scanning the relative position of the focused beams on the sample (Fig. 1c–e). When two beams probe a homogeneous region, the output intensity is constant (Fig. 1c). At the boundary of the two regions, the signal intensity increases or decreases, as the difference in the phase shift Δφ becomes nonzero (Fig. 1d). The signal intensity returns to the original level after the boundary (Fig. 1e). The smallest detectable change in the phase shift is limited by the SNR, which is the ratio of the change in the signal intensity, C(φ), and the fluctuation of the uniform background level, ΔC, at a bias level of Φ_{0}. As is discussed in detail later, it is known that the SNR is limited by socalled ‘shot noise' or the SQL, when classical light sources such as lasers or lamps are used. That is, for a limited number of input photons (N), the SNR is limited by . This SNR limits the height resolution of the LCM–DIM when used to observe elementary steps at the surface of ice crystals^{19} or the difference in refractive indexes inside a sample. Thus, improving the SNR beyond the SQL is a revolutionary advance in microscopy.
Entanglementenhanced microscope
We propose to use multiphoton quantum interference to beat this standard quantum limit (Fig. 1b). Instead of a classical light, we use an entangled photon state (), socalled ‘NOON’ state, which is a quantum superposition of the states ‘N photons in the H polarization mode’ and ‘N photons in the V polarization mode’. The phase difference between these two states is NΔφ after passing through the sample, which is N times larger than the classical case (N=1). At the output, the result of the multiphoton interference (the parity of the photon number in the output) is measured by a pair of photon number discriminating detectors^{21,22}.
As is well known from twopath interferometry with Nphoton states, entanglement can increase the sensitivity of a phase measurement by a factor of . In the entanglementenhanced microscope, we apply this effect to achieve an SNR that is times higher than that of the LCM–DIM. If the average number of Nphoton 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_{N} cos(Nφ+NΦ_{0}))/2 is the probability of detecting an odd (or even) number of photons in a specific output polarization (see Methods section). For a small positive phase shift of , the phase dependence of the signal is given by the slope of C(φ) at the bias phase Φ_{0}, lim_{Δ}_{φ→0}C(Δφ)−C(0)=∂C(φ)/∂φ_{φ=0} × Δφ, and the SNR is given by the ratio of the slope and the statistical noise of the detection. If the emission of the Nphoton states is statistically independent, the statistical noise is given by . The SNR is then given by
where ξ is the normalized overlap region between the two beams at the sample plane (see Methods section). By maximizing the function of cos(NΦ_{0}), we can find the maximum sensitivity (SNR_{max}) as follows:
at a bias phase of
For the ideal case of V_{N}=1, the maximum sensitivity . As the SNR for a classical microscope using kN photons , an entanglementenhanced microscope can improve the SNR by a factor of compared with a classical microscope for the same photon number.
Experimental setup
We demonstrate an entanglementenhanced microscope (Fig. 2a) using a twophoton NOON state (N=2). First, a polarizationentangled state of photons is generated from two beta barium borate crystals^{23,24} and is then delivered to the microscope setup via a singlemode fibre. The polarizationentangled state is then converted to a twophoton NOON state using a calcite crystal^{25} and focused by an objective lens. From the result of quantum state tomography (Fig. 2b), the fidelity of the state is 98% and the entanglement concurrence is 0.979, which ensures that the produced state is almost maximally entangled. The entangled photons pass through two neighbouring spots at the sample plane (Fig. 2a inset). Then, after passing through the collimating lens, the two paths are merged by a polarization beam splitter and the result of the twophoton interference is detected by a pair of singlephoton counters and a coincidence counter. The sample is scanned by a motorized stage to obtain an image. The beam diameters at the sample plane and the distance between the centre of the beams are all 45 μm (ξ=0.046).
Figure 2c,d shows the singlephoton and twophoton interference fringes using a classical light source and the NOON source, respectively. The fringe period of Fig. 2d is half that of Fig. 2c, which is a typical feature of NOON state interference. The visibilities of these fringes, V_{c}=97.1±0.4% (Fig. 2c) and V_{q}=95.2±0.6% (Fig. 2d), suggests the high quality of the classical and quantum interferences.
Obtained image beating the standard quantum limit
Figure 3 shows the main result of this experiment. We used a glass plate sample (BK7) on whose surface a Q shape is carved in relief with an ultrathin step of ~ 17 nm using optical lithography (Fig. 3a,b). Figure 3c,d shows the twodimensional (2D) scan images of the sample using entangled photons and single photons, respectively. The step of the Qshaped relief is clearly seen in Fig. 3c, whereas it is obscure in Fig. 3d. Note that for both images we set the bias phases to almost their optimum values given in equation 3 and the average total number of photons (N × k) contributed to these data are set to 920 per position assuming the unity detection efficiency.
For more detailed analysis, the crosssection of the images (coincidence count rate per single count rate at each position) are shown in Fig. 3e,f. The solid lines are theoretical fits to the data where the height and position of the step and the background level are used as free parameters. For Fig. 3e, the signal (the height of the peak of the fitting curve from the background level) is 20.21±1.13, and the noise (the standard deviation of each experimental counts from the background level of the fitting curve) is 9.48 (Fig. 3g). Thus, the SNR is 2.13±0.12. Similarly, the signal, the noise and the SNR are 17.7±1.22, 11.25 (Fig. 3h) and 1.58±0.11 for Fig. 3f, where classical light source (single photons) is used. The improvement in SNR is thus 1.35±0.12, which is consistent with the theoretical prediction of 1.35 (equation 2). The estimated height of the step was 17.0±0.9 nm (quantum) and 16.6±1.1 nm (classical), and is consistent with the estimated value of 17.3 nm from the atomic force microscope image in Fig. 3b.
As shown in equation 1, the SNR depends on the bias phase. Finally, we test the theoretical prediction of the bias phase dependence given by equation 1 in actual experiments. Figure 4 shows the bias phase dependence of the SNR for the twophoton NOON source (Fig. 4a) and classical light source (Fig. 4b). The solid curve is the theoretical prediction calculated by equation (1), where we used the observed visibilities of the fringes in Fig 2c,d for V_{N}. The theoretical curves are in good agreement with the experimental results.
Discussion
Note that the entanglementenhanced microscope we reported here is different from the ‘entangled photon microscope’ theoretically proposed by Teich and Saleh^{26}, which is the combination of two photon fluorescence microscopy and the entangled photon source. In the proposal, the increase in twophoton absorption rate and the flexibility in the selection of target regions in the specimen were predicted. The application of entangled photon sources for imaging also includes quantum lithography^{27}, where the lateral resolution of the generated pattern is improved^{25,28}, and ghost imaging^{29}, where the spatial correlation of entangled photons is utilized. In this context, this work is the first application of entanglementenhanced optical phase measurement beyond the SQL for imaging including microscopy. Note also that the entanglement is indispensable to improve the SNR of the phase measurement beyond the SQL^{11,30}. This situation is different from the improvement in the contrast of the ghost imaging using strong thermal light^{31,32,33,34,35,36,37}.
In conclusion, we proposed and demonstrated an entanglementenhanced microscope, which is a confocaltype 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 ultrathin step of ~ 17 nm under a low photon number condition shows the viability of the entanglementenhanced microscope for lightsensitive samples. To test the performance of the entanglementenhanced microscope, we used modestefficiency detectors, however, recently developed highefficiency numberresolving photon detectors would markedly improve detection efficiency^{38,39}. We believe this experimental demonstration is an important step towards entanglementenhanced microscopy with ultimate sensitivity, using a higher NOON state, a squeezed state^{40,41} and other hybrid approaches^{42} or adaptive estimation schemes^{43,44}.
Methods
SNR of an entanglementenhanced microscope
To perform the parity measurement used by Gerry^{21} and Seshadreesan et al.^{22}, it is required to count both of the events where ‘even’ and ‘odd’ number of photons are detected in the output. However, experimental implementations become much easier if it is sufficient for us to just count ‘odd’ (or ‘even’) number photondetection events. In addition, the distance between the two beams and the beam size may have effect on the SNR. Here we consider these technical effects on SNR and derive equation (1).
To derive equation (1), we consider that the two beams at the sample are separated at a distance of α along the x axis, and each set of N photons has Gaussian distribution with a variance of σ in the x–y plane (Fig. 5). After passing through the sample, the two beams experience a phase shift (φ) in the grey region (x>0). The state, Ψ(x, y, φ), after the sample is written as
where ψ_{H}(x,y,φ) and ψ_{V}(x,y,φ) represents the states of N photons in the horizontal and vertical polarization modes, respectively, and Φ_{0} is a bias phase. We assume that the phase shift is described by a step function and the N photons are in the same spatial modes. These states are written as
where 0› is a vacuum state, and are the creation operators in H and V polarization modes at the position of (x,y), respectively, and χ(x) is a step function that χ(x)=0 for x ≤0 and χ(x)=1 for x>0. f(x−α/2, y) and f(x+α/2, y) represent the Nphoton probability densities in the horizontal and vertical polarization modes written as
After passing through the second calcite crystal, the two beams are displaced at a distance of −α/2 for H polarization and α/2 for V polarization along the x axis, resulting in two beams in the same spatial mode. The state can then be written as
Here, we assume that the state is projected onto the state where odd number of photons are in the minus diagonal polarization mode at the output. The measurement operator in the basis of plus (P) and minus (M) diagonal polarization is therefore written as
where
The probability of odd number photon detection can then be written as
where we denote the phaseindependent term of as ξ(α) which is the overlap integral between H and V polarized beams at the sample plane.
We now calculate the SNR of our microscope using the NOON state including the effect of the overlap between the two beams at the sample plane. If the emission of the Nphoton states is statistically independent, the statistical noise at the bias phase is given by
For a small positive phase shift of , the signal is
Considering the visibility of the interference fringe V_{N}, the SNR is given by
Thus one can confirm that counting odd (or even) number photondetection events can also achieve the phase super sensitivity. Note also that the dependence of SNR on the size and the distance between the two beams is simply given by (1−ξ). This means that it is reasonable to compare the SNR between an entanglementenhanced microscope and a classical microscope (N=1) for the same ξ.
Additional information
How to cite this article: Ono, T. et al. An entanglementenhanced microscope. Nat. Commun. 4:2426 doi: 10.1038/ncomms3426 (2013).
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Acknowledgements
We thank Dr. Shouichi Sakakihara for sample fabrication, Professor Hidekazu Tanaka and Dr. Koichi Okada for atomic force microscope measurements. We also thank Professor Mitsuo Takeda, Professor Yoko Miyamoto, Professor Gen Sazaki, Professor Holger F. Hofmann and Professor Masamichi Yamanishi for helpful discussion. This work was supported in part by FIRST of JSPS, Quantum Cybernetics of JSPS, a GrantinAid from JSPS, JSTCREST, Special Coordination Funds for Promoting Science and Technology, Research Foundation for OptoScience and Technology and the GCOE programme.
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Experiments, measurements and data analysis were performed by T.O. with assistance of R.O. and S.T. The project was planned by S.T. and supervised by R.O. and S.T. The manuscript was written by all the authors.
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Ono, T., Okamoto, R. & Takeuchi, S. An entanglementenhanced microscope. Nat Commun 4, 2426 (2013). https://doi.org/10.1038/ncomms3426
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DOI: https://doi.org/10.1038/ncomms3426
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