Abstract
Path entanglement is a key resource for quantum metrology. Using pathentangled states, the standard quantum limit can be beaten and the Heisenberg limit can be achieved. However, the preparation and detection of such states scales unfavourably with the number of photons. Here we introduce sequential path entanglement, in which photons are distributed across distinct time bins with arbitrary separation, as a resource for quantum metrology. We demonstrate a scheme for converting polarization GreenbergerHorneZeilinger entanglement into sequential path entanglement. We observe the same enhanced phase resolution expected for conventional path entanglement, independent of the delay between consecutive photons. Sequential path entanglement can be prepared comparably easily from polarization entanglement, can be detected without using photonnumberresolving detectors and enables novel applications.
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Introduction
Quantum metrology, which harnesses the collective nature of entangled particles, is an emerging technology that promises to revolutionise our way of measuring and to radically alter the limits for parameter estimation^{1,2,3,4}. The most common example, which is also the one with the most technological applications, is the estimation of the relative optical phase between two arms of an interferometer: this problem is directly related to the realization of atomic clocks^{5}, the investigation of biological samples^{6} and the detection of gravitational waves^{7,8}. Generating light in an entangled state delocalised over the two arms can achieve a better scaling of the precision with the number of photons than is possible with any classical optical technique. Indeed, for a classical coherent state with average photon number N, the precision in phase estimation will mostly be limited by the Poissonian fluctuations of the photon number, the socalled shot noise, which contributes a scaling proportional to on the error of the phase estimate (this is the ‘standard quantum limit’). In the quantum domain N photons can be prepared in an entangled superposition of all photons occupying one arm and of all photons occupying the other arm. Since all photons travel together, phase is accumulated more rapidly, which allows for a phase estimate with enhanced precision whose error scales as 1/N, the Heisenberg limit^{9}.
The key elements for realising such an advantage are the ability to produce Nphoton pathentangled states and the capability of resolving the number of photons at both output ports of the interferometer. However, scaling up from the fewphoton level has proven to be technically challenging, since the best photon sources available produce multiphoton states with unwanted components, which degrades the quantum advantage^{10,11,12,13,14,15,16}. This motivates research into less technically demanding schemes that might retain the enhanced precision of conventional pathentangled states. Here we propose breaking up Nphoton entangled states so that the entangled photons are distributed over N time bins, with only one photon interacting with the sample at a time. This type of sequential pathentangled (SPE) state has a number of advantages. First, as we will show, SPE states can be produced directly from GreenbergerHorneZeilinger (GHZ) states^{17}; the current state of the art for producing GHZ states^{18,19} is more advanced than for N00N states^{15,16}, so that larger SPE states can be synthesised than N00N states. Second, detecting photons one at a time can be done with conventional avalanche photodetectors without requiring photonnumberresolving detection. Finally, SPE states sample a phase over a range of times, which enables the extraction of Fourier components of a dynamical phase.
A legitimate concern with a proposal to break up and separately detect an entangled state in this way is that the metrological power of a collective measurement on the entangled state is lost. However it is known that strategies based on collective measurements offer no advantage for quantum metrology^{1} and this is why SPE states can be just as powerful for metrology as conventional pathentanglement. In this paper we demonstrate the key features of this proposal by recording the interference produced by twophoton entangled states as a time delay between the photons is introduced. The results show that phase superresolution is maintained independent of the delay, confirming the power of SPE states as a resource for quantum metrology. We generate our SPE states by feeding a polarizationentangled photon pair into a modified MachZehnder interferometer^{20} (MZI). More generally, we establish that higher order SPE states could be deterministically produced from an initial resource of GHZ polarizationentangled states by means of optical switching.
Results
Sequential path entanglement
The archetypal pathentangled state is the N00N state^{3,21}, N0::0N〉 = N〉_{a}0〉_{b} + e^{iNϕ}0〉_{a}N〉_{b} (ignoring normalisation here and below), in which N photons are superposed between all propagating in one arm, a, of an MZI and all propagating in the other arm b (see Fig. 1 (a)), which accrues a phase rotation Nϕ, where ϕ is the relative phase between the two arms of the MZI. The advantage of this pathentangled state for metrology can be understood using the concept of the photonic de Broglie wavelength^{22}. The N photons are treated as an effective single entity and thus have an effective de Broglie wavelength λ/N, with λ the singlephoton wavelength.
The scheme that is most commonly used to produce approximate N00N states is shown in Fig. 1 (a). Two Fock states N/2〉 are made to interfere on the first beam splitter (BS) of an MZI. For N = 2, HongOuMandel (HOM) interference^{23} directly generates the required photonbunched 20::02〉 state^{16,24}. Unfortunately, for N > 2, generating a genuine N00N state by HOM interference in this way no longer works: the resulting state is instead a superposition of N0::0N〉 with an undesired contribution Ω(N)〉, which decreases the associated measurement precision^{13,25,26}. For instance, Nagata et al. prepared a 4photon N00N state with Ω(4)〉 ∝ 22〉, by interfering 2photon Fock states^{13}. States of this form are known as HollandBurnett^{27} states, and, while they are interesting for lossy phase estimation^{28}, they cannot achieve the Heisenberg limit. It has been proposed that Ω(N)〉 can be suppressed by using feedforward^{29}, however, this proposal has not been implemented yet.
Multiphoton SPE states can be generated without suffering from the complications described above that make N00N states difficult to produce for N > 2. We begin with a polarizationentangled GHZ state^{18,19} of N photons,
in which H〉_{i} (V〉_{i}) denotes a horizontally (vertically) polarized photon in spatial mode i = 1, 2, …, N. The conversion of this GHZ state into an SPE state is schematically shown in Fig. 1 (b). Each photon from the GHZ state is delayed with respect to the next and then all are transferred into a single spatial mode. In our experiments we combine the photons probabilistically, using a beam splitter, but in general this step could be done deterministically by means of a fast optical switch, such as a Pockels cell^{30}. Finally, path entanglement is created by directing the photons one by one onto a polarizing beam splitter followed by a halfwave plate at one output mode, which applies the operation H〉 → 1〉_{a}0〉_{b}; V〉 → 0〉_{a}1〉_{b}. This recipe allows the deterministic conversion of GHZ entanglement into SPE states within the MZI of the form
where now each temporal mode t_{j} contains a single pathentangled photon. Each of the N photons in mode b picks up the interferometer phase ϕ and so this state also exhibits superresolution, like the N00N state (see Methods). However, the time separation of the photons means that photonnumber resolving detectors are not required to identify the Nphoton events containing the enhanced metrological signal. In principle, ‘onoff’ detectors such as commerciallyavailable avalanche photodetectors can be used, since there is only a single particle in each time bin^{31}. Additionally, if the MZI phase is varying in time — as would be typical in pumpprobe type experiments — the Fourier components of the phase evolution can be accessed efficiently via the transmission statistics of SPE states by tuning the time delay T = t_{j}_{+1} − t_{j} into resonance with a component of interest (see Methods).
Experimental setup
Our experiment is shown schematically in Fig. 2. It contains two parts: the photon source (Fig. 2 (a)) which produces polarizationentangled photon pairs and the modified MZI (Fig. 2 (b)). In the experiment, as shown in Fig. 2 (a), typeII spontaneous parametric down conversion^{32} (SPDC) is employed to generate a polarizationentangled state (see Methods). A translation stage with μm precision adds an adjustable delay ΔL_{1} into path P_{1} and then P_{1} and P_{2} are combined at a 50:50 beam splitter BS1 and the resulting polarizationentangled state is coupled into an optical fibre in mode P_{4} for delivery to the MZI. We note that BS1 passively combines the photons into the same spatial mode in a probabilistic way, which allows us to study several delay settings, including ΔL_{1} = 0. But for a sufficiently large delay , where ξ is the single photon coherence length, the photons are wellseparated and could be deterministically combined using a fast optical switch, allowing the efficient conversion of higherphotonnumber GHZ states into SPE states. The state emerging from the fibre is converted into an SPE state of the form (2) at a polarizing beam splitter and a nmprecision piezoelectric actuator mounted on a μmprecision translation stage is used to control the optical path length difference ΔL_{2} within the modified MZI.
To find the position ΔL_{2} = 0 we first maximise the visibility of singlephoton interference in the MZI (see Fig. 2 (d)). For these measurements BS1 is replaced by a polarizer oriented at 45°, so that photons emitted from the source into path P_{1} are delocalised at the polarizing beam splitter, producing a fringe pattern in the singles count rates recorded in the paths P_{7}, P_{8}, as ΔL_{2} is scanned (photons in path P_{2} are simply dumped). The full width at half maximum of the pattern provides a measurement of the singlephoton coherence length ξ ≈ 130 μm, consistent with the result deduced from the measured HOM interference dip shown in Fig. 2 (c) (see Methods). We then set ΔL_{2} ≈ 0 and scan the piezoelectric actuator in steps of 10 nm to observe the fringe spacing, which matches the single photon de Broglie wavelength (simply the optical wavelength) λ ≈ 810 nm (see Fig. 3 (a)).
Next, we verify phase superresolution with a standard N00N state by setting ΔL_{1} = 0, so that both photons emitted from the source are temporally overlapped in the MZI. As shown in Fig. 2 (e), the coincidence count rate at detectors D_{1} & D_{2} exhibits an interference pattern with a broader envelope than the single photon interference pattern. This is due to spectral entanglement of the photon pairs emitted by our source, which introduces temporal correlations that are limited only by the coherence time of the downconversion pump laser, extending significantly beyond the singlephoton coherence time (see Methods for details). We comment that to our knowledge this is the first time the extended interference pattern of a 20::02〉 state generated with a spectrally anticorrelated photon pair source has been fully mapped out^{24}. The finegrained scan of the region around ΔL_{2} = 0 in Fig. 3 (b) shows the expected halfperiod fringes associated with the reduced de Broglie wavelength of the 20::02〉 state.
Independence of the separation between photons
To verify that SPE states have the same reduced de Broglie wavelength as the N00N state, independent of the delay between successive photons entering the MZI, we carried out experiments for ΔL_{1} = 200 μm and 10,000 μm. The setting ΔL_{1} = 200 μm > ξ is larger than the singlephoton coherence length but still less than the coherence length of the pump laser ξ_{pump} ≈ 300 μm, while ΔL_{1} = 10,000 μm is significantly larger than ξ_{pump}. Fig. 4 shows the interference patterns in the coincidence rates for detectors D_{1} and D_{2} produced by scanning ΔL_{2}. For each of the scans, the periodicity is always about 405 nm, half of the singlephoton wavelength λ = 810 nm, which confirms that SPE states retain the phase superresolution of a conventional N00N state, even when consecutive photons are separated well beyond their individual coherence lengths. It is clear that the coincidence rates fall as ΔL_{1} is increased, but this simply represents a reduction in the SPE preparation efficiency. This is because for ΔL_{1} < ξ, HOM interference at BS1 boosts the probability that photon pairs are coupled into our MZI, whereas for there is no longer HOM interference. In addition, the fibre coupling efficiency drops for large ΔL_{1}. In particular, the setting ΔL_{1} = 10,000 μm exceeds the Rayleigh range defined by our fibre, causing a significant drop in the coincidence rate for this measurement. However, the peak visibilities of the patterns in parts (b)–(f) of Fig. 4 remain high throughout the scan, at 0.98 ± 0.01, 0.95 ± 0.03 and 0.98 ± 0.03, respectively. From this we conclude that the quality of the biphoton interference is independent of ΔL_{1} and the reduced photonic de Broglie wavelength is independent of the distance between photons in an SPE state.
Discussion
We have introduced SPE states as an alternative resource for quantumenhanced optical interferometry. Conventional N00N states are difficult to produce and photonnumberresolving detectors^{33,34,35} are required to extract a metrological signal^{36}. While multiplexed onoff detectors can be realized easily, they only count photon number approximately and are therefore not suitable for metrological applications. By contrast, SPE states can be generated deterministically from GHZ states, which are easier to produce than N00N states. Photonnumberresolving detectors are also not needed, since photons can be detected sequentially^{1}. In practice, the performance of photodetectors is critical to achieving a metrological advantage with N00N states^{28} and this remains the case for SPE states also. However, realistic quantum interferometry could be realised using recently developed highly efficient, fast, detectors^{37}.
Many experiments have shown that two photons arriving simultaneously make a substantial difference in second harmonic generation^{38}, Kerrlike interactions^{39} and Rydberg atomic gases^{40}. This detrimental effect makes phase estimation modeling complicated. Furthermore, quantum metrology is often motivated by the need to probe refractive index variations in samples with extreme sensitivity to optical radiation^{41}. N00N states can excite 2photon or multiphoton transitions in a sample that scatter light out of the interferometer and such transitions can ionise quantum systems and damage biological molecules. The influence of multiple photons will become stronger if more photons are present at one time, as could be the case with highN N00N states, which are the ultimate goal of this avenue of research. By contrast even largeN SPE states can excite only singlephoton transitions because the constituent photons arrive one at a time.
It is also notable that SPE states can be used to extract the harmonic spectrum of a timevarying phase: if the consecutive time bins in an SPE state are separated by a constant delay T, the phase is sampled periodically, producing a signal proportional to the amplitude of the Fourier component of the dynamical phase at a frequency f = 1/T (see Methods).
We experimentally confirmed that 2photon SPE states posses the same phase superresolution as 2N00N states. Unlike schemes for achieving superresolution via postselection on classical light^{42}, this superresolution is of quantum origin and SPE states therefore represent a convenient and powerful platform for quantumenhanced phase sensing via optical interferometry.
Methods
Preparation of polarizationentangled photonic source
As shown in Fig. 2 a CW semiconductor laser produces a blue beam (power 34.5 mW, waist 100 μm, central wavelength 405 nm) that pumps typeII degenerate SPDC in a 2 mm βbariumborate (BBO) crystal, producing pairs of photons with central wavelength 810 nm^{32}. The downconverted signal and idler photons have different propagation velocities and travel along different paths inside the crystal due to its birefringence. The resulting walkoff effects are compensated by a combination of a halfwave plate and a 1 mm BBO crystal in each arm. Photons are then coupled into single mode fibres and we prepare a polarizationentangled state . The state is further transformed to be by placing a halfwave plate with 45° in mode P_{2}. The singles count rates measured directly in each of P_{1}, P_{2} are about 80,000 per second and the coincidence rate is about 20,000 pairs per second. The visibilities for the polarization correlations are about 98% in the H〉/V〉 basis and 93% for the  + 45°〉/ − 45°〉 basis. This shows that we have produced a highquality polarizationentangled photonic source.
Success rate for combination of photon pairs at BS1
When ΔL_{1} ≥ ξ, the photons do not overlap at BS1 and each photon has the same probability to be transmitted or reflected. Thus, the photon state in P_{4} can be a singlephoton state H〉 or V〉 with probability 0.25, respectively, or the maximally polarizationentangled state with probability 0.25 (the rest of the time, P_{4} contains no photons). When ΔL_{1} < ξ, HOM interference suppresses the single photon probabilities, boosting the probability for coupling both photons into P_{4} up to 1/2 for ΔL_{1} = 0. HOM interference with a visibility of 94.5 ± 0.4% is shown in Fig. 2 (c), obtained by scanning ΔL_{1} and measuring the coincidence rate at two output modes P_{3} and P_{4} (Fig. 2 (a)). The high visibility certifies good spatial and temporal overlap for the entangled photon pairs emitted by our source. A Gaussian fit to the HOM dip in Fig. 2 (c) yields the singlephoton coherence length ξ ≈ 126 μm.
Single and biphoton coherence time
Here we briefly outline the origin of the differing widths of the single and biphoton interference patterns shown in parts Fig. 2 (d) and (e). We consider that our downconversion source creates photon pairs in the entangled state
where ζ is the joint spectral amplitude for emitting signal and idler photons with frequencies ω + ω_{0}, ω′ + ω_{0}, with ω_{0} the centre frequency of the downconversion. ζ generally extends over a broad bandwidth set by the phasematching conditions in the downconversion crystal, but energy conservation enforces tight frequency anticorrelations, with a correlation width equal to the pump laser bandwidth.
For the single photon interference measurement (Fig. 2 (d)) we dump photons P_{2}, leaving the photons in P_{1} in the mixed state where The MZI implements the transformation
at the two output ports P_{7}, P_{8} (postselected on no photon in the other port) and the single photon count probability is , where p(ω) = f(ω, ω) is the marginal spectrum. This exhibits fringes with period λ = 2πω_{0}/c and visibility V given by a Fourier transform, , so that the coherence length ξ is determined by the inverse of the bandwidth of p(ω). Note that the fringe patterns for the two output ports are π out of phase, as seen in Fig. 3 (a).
For the measurement of the two photon N00N state interference (Fig. 2 (e)), the MZI produces the state
at each of its two output ports (postselected on no photons in the other port). The coincidence probability at detectors D_{1}, D_{2} is then , which produces an interference pattern with period λ/2 and visibility given by V(ΔL_{2}) = g(ΔL_{2}/c, ΔL_{2}/c), where is the twodimensional Fourier transform of the joint spectral intensity ζ(ω, ω′)^{2}. The tight frequency anticorrelations between the photons produce strong temporal correlations such that the width of g(ΔL_{2}/c, ΔL_{2}/c) is proportional to the coherence length ξ_{pump} of the pump laser.
Superresolution with SPE states
To convert N00N states into SPE states we simply introduce a delay ΔL_{1} between photons, which for the biphoton case corresponds to the transformation . However, as described above, the biphoton interference visibility depends only on the joint spectral intensity, which is unaffected by this phase rotation. This is why our SPE state retains phase superresolution for all values of ΔL_{1}. For higher photon number states with N > 2 the visibility is determined by higher order correlation functions, but the coincidence rate is always invariant under phase rotations of the joint spectrum, so that the metrological power of SPE states is independent of the delay between photons.
Extracting dynamical phase information with SPE metrology
An interesting feature of SPE states is that the photons sample the MZI phase over a set of discrete times t_{j}, meaning that the metrological signal is of the from V cos^{2}(ϕ), where V is a visibility and where . Substituting in the Fourier relation , where is the spectrum associated with the timevarying phase (note that here ω is the Fourier conjugate of t and not an optical frequency), we obtain
where we assumed a constant interphoton delay t_{j} = t_{0} + (j − 1)T (in our experiment we had T = ΔL_{1}/c). The function Δ comprises a series of spectral peaks of height , located at the frequencies ω_{k} = 2πk/T; k = …, −2, −1, 0, 1, 2, ...., each with width δω = 4π/NT. In the limit that N is large, we may approximate Δ as a series of Dirac delta functions, so that the SPE signal can be written as
We would like to extract information about the Fourier components of the phase; as an example we consider the quantity , where τ = π/ω_{1} = T/2, calculated from two measurements with different values of the initial arrival time t_{0}. The zerofrequency component is removed from D_{τ} and we have
where we have used the fact that (since ϕ is real) and where we also assumed that the phase variation is sufficiently narrowband that , which will hold for a wide class of smooth phase dynamics with compact temporal support of duration . This shows that just two measurements with SPE states provide access to the Fourier component of a dynamical phase at a frequency f = ω_{1}/2π = 1/T fixed by the delay T between the probe photons. We note that the accuracy of this procedure is ultimately limited by the timing jitter of the nonclassical light sources that produce the photons in the SPE state, but parametric photon sources have demonstrated extremely low timing jitter^{43}.
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Acknowledgements
XMJ is grateful for insightful discussions with Animesh Datta and Yoshihisa Yamamoto. This research was supported by the Chinese Academy of Sciences, the National Natural Science Foundation of China (No.11004183) and the Royal Society, EU IP QESSENCE(248095), EPSRC (EP/J000051/1 and EP/H03031X/1) and AFOSR EOARD (FA86550913020). XMJ is supported by a EU MarieCurie Fellowship (PIIFGA2011300820).
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X.M.J. and I.A.W. conceived the project and contributed to the design of the experiment. X.M.J. and C.Z.P. performed the experiment. M.B., J.N. contributed to the theoretical analysis. X.M.J., Y.D., M.B., J.N. analysed the data and wrote the paper.
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Jin, XM., Peng, CZ., Deng, Y. et al. Sequential Path Entanglement for Quantum Metrology. Sci Rep 3, 1779 (2013). https://doi.org/10.1038/srep01779
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DOI: https://doi.org/10.1038/srep01779
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