Abstract
Controlling the evolution of photonic quantum states is crucial for most quantum information processing and metrology tasks. Due to its importance, many mechanisms of quantum state evolution have been tested in detail and are well understood; however, the fundamental phase anomaly of evolving waves, called the Gouy phase, has had a limited number of studies in the context of elementary quantum states of light, especially in the case of photon number states. Here we outline a simple method for calculating the quantum state evolution upon propagation and demonstrate experimentally how this quantum Gouy phase affects twophoton quantum states. Our results show that the increased phase sensitivity of multiphoton states also extends to this fundamental phase anomaly and has to be taken into account to fully understand the state evolution. We further demonstrate how the Gouy phase can be used as a tool for manipulating quantum states of any bosonic system in future quantum technologies, outline a possible application in quantumenhanced sensing, and dispel a common misconception attributing the increased phase sensitivity of multiphoton quantum states solely to an effective de Broglie wavelength.
Main
The wave dynamics dictating the evolution of quantum states is of utmost importance in both fundamental studies of quantum systems and quantum technological applications. For photons, the evolution of their spatial structure has been the key in a plethora of promising techniques for quantum communication^{1,2}, information processing^{3,4}, simulation^{5} and metrology^{6}. One particular feature of a converging wave travelling through its focus is the acquisition of an additional phase shift when compared with a collimated beam or a plane wave travelling the same distance. This effect, which is known as the Gouy phase, was first observed and described by Gouy more than a century ago^{7,8}. Although the phenomenon is well established and can be described through methods in physical optics^{9,10}, the Gouy phase continues to be the topic of studies discussing its underlying physical origin by linking it to properties such as the geometry of the focus, geometric phases and the uncertainty principle^{9,11,12,13,14,15,16,17,18}. In addition to the continued interest aiming at providing an intuition for the phenomenon, this phase anomaly is often harnessed to realize tools in optics^{19,20,21,22}.
Despite the Gouy phase being a general wave phenomenon, studies investigating its role in quantum state evolution have been limited to a few matter wave studies^{23,24,25,26,27} and spatially separated photon pairs^{28,29}. Although these demonstrations use (locally) single quantum systems and thus observe the effect known for classical light waves, more complex quantum states consisting of multiple identical quantum systems (that is, bosonic systems with multiple excitations) have not been studied before. We term the specific phase acquired by such quantum states the quantum Gouy phase.
In general, any phase accrued by a mode of a photonic quantum system leads to a photonnumber dependent phase for the quantum state. This means that whereas a single photon or a classical field would acquire a phase ϕ upon propagation, when Nphotons occupy the same mode \((\leftN\right\rangle )\), the quantum state is left with N times the same phase, that is, \(\exp (iN\phi )\leftN\right\rangle\)^{30}. This increased phase sensitivity of photon number states is utilized in socalled N00N states, which have garnered popularity due to their potential to push the sensitivity of measurements to what is considered the absolute physical limit^{31}. N00N states can be compactly expressed for two orthogonal modes p and \({p}^{\prime}\) as
Hence, the enhancement in measurement sensitivity is enabled by the phase difference between the two components being N times the phase difference between the underlying modes. More importantly, using such a N00N state configuration allows for the study of the speedup of the quantum Gouy phase compared with the classical case.
In the present work we describe theoretically how an Nphoton number state evolves upon propagation and verify experimentally the speedup of the quantum Gouy phase with twophoton N00N states through interference in the transverse structure of a biphoton. We further show that the quantum Gouy phase speedup can be applied to superresolving longitudinal displacement measurements using the quantum Fisher information (QFI) formalism and solidifying its link to the uncertainty interpretation of the Gouy phase^{12}. Finally, we show that our results for Nphoton states cannot be simulated by classical light with a λ/N wavelength, demonstrating that the oftenused effective de Broglie wavelength approach for multiphoton states, although useful in specific cases^{32,33,34}, is not always accurate. As such, our work brings the fundamental wave feature of the Gouy phase to the quantum domain, thereby opening the path to its utilization in quantum technological applications through its unique quantum state manipulation properties.
Probing the quantum Gouy phase
An interferometric measurement scheme can be used to observe the quantum Gouy phase of Nphotons. We chose to use the transversespatial modes of paraxial light beams as the different arms of the interferometric scheme, where one mode acts as the required reference arm. More specifically, we used Laguerre–Gaussian (LG) modes, which are a family of orthogonal solutions to the paraxial wave equation in cylindrical coordinates^{35}. In the case of a classical monochromatic field, the Gouy phase of these modes evolves as^{35}
where z is the propagation distance, k is the wavenumber, ℓ is an integer giving the number of orbital angular momentum quanta per photon, p is a positive integer defining the radial transverse structure of the field, w_{0} is the beam waist defining the transverse extent of the beam at its focus, and z_{0} gives the position of the beam focus along the optical axis. As the Gouy phase depends on the mode order S = 2p + ∣ℓ∣ + 1, its anomalous phase behaviour can be observed through the change of the transverse structure during propagation when the light is in a superposition of spatial modes of different mode orders^{36}. For radial modes, which are LG modes with ℓ = 0, this change results in a varying intensity along the optical axis (as can be seen in Fig. 1a); thus, to probe the quantum Gouy phase and distinguish it from its classical counterpart, we study the superposition of a Gaussian reference mode (p = 0) and different higherorder radial modes in both the classical domain and the aforementioned quantum setting, that is, a N00N state superposition. By measuring the change in intensity and twophoton detection rate, respectively, observed in a singlemode fibre (SMF) scanned through the focus, we are able to directly observe the speedup of the quantum Gouy phase.
Theoretical evolution upon propagation
In our measurement scheme, we expect the propagation to result in a photon numberdependent Gouy phase when the state \({\leftN\right\rangle }_{p}\) is translated through a focus. To verify these expectations theoretically, we start with N photons occupying a monochromatic paraxial mode at a position z = 0, with a complex field structure u_{ℓp}(ρ, 0). To translate the mode along the optical axis, we apply the translation operator \({\mathrm{e}}^{{{{\rm{i}}}}{\hat{P}}_{z}z/\hslash }\) to the mode in the angular spectrum representation, in which the quantized mode of light can be expressed as
where F_{ℓp}(κ, 0) represents the normalized complex amplitude of the plane wave mode with transverse wave vector κ, and \({\hat{a}}^{{\dagger} }({{{\mathbf{ \upkappa}}}})\) is the corresponding operator density ^{37,38}. After applying the translation operator, the mode takes the form
which is identical to the initial mode being propagated by z using the angular spectrum method (ASM)^{9,39}. We thus see that the quantized mode evolves identically to a classical light field, that is, the propagated LG mode has an identical spatial structure u_{ℓp}(ρ, z) that only differs by the propagationrelated changes to the wavefront curvature and beam radius. Due to the beam evolving according to the ASM, we can extract the Gouy phase evolution by defining a new mode \({\hat{b}}_{\ell p}^{{\dagger} }(z)\)—which has the structure of the field after translation—without the accumulated Gouy phase, that is, \({u}_{\ell p}({{{\mathbf{ \uprho}}}},z){\mathrm{e}}^{{{{\rm{i}}}}kz}{\mathrm{e}}^{{{{\rm{i}}}}{{{\Phi }}}_{\mathrm{G}}(z)}\). Using this new mode, we can express the mode after propagation as a single mode with a phase
We can then simply state the Gouy phase evolution of an Nphoton Fock state as
which explicitly contains the photon number dependent Gouy phase evolution. See Supplementary Section 1 for a detailed derivation.
Experiment
We first prepared laser light in a superposition of the Gaussian reference mode and one of the higherorder radial modes. The structuring of the laser beam was performed with a single hologram on a spatial light modulator (SLM) by using a holographic method commonly known as mode carving^{40}. After structuring, the beam was imaged one focal distance away from a 75 mm lens, which performs an optical Fourier transform on the transverse structure while focusing^{39}. As the transverse structure and its Fourier transform are identical for LG modes, the beam structure at the focus was identical to the structure carved at the SLM, up to a phase factor of π between the superposed LG modes, which needed to be accounted for with odd values of the radial index^{19,20}. To measure the Gouy phaseinduced change in the interference along the optical axis, we placed an SMF at the focus and moved it longitudinally using a stage with a computer controlled piezo actuator. The laser source was a continuouswave diode laser operating at 810 nm and the SLM used for structuring the light was wavefront corrected using the method described in ref. ^{41}. Furthermore, to get the generated modes as close as possible to the correct transverse structure at the desired beam radius, we employed an additional Gaussian correction in the mode carving that minimized any effect of the initial Gaussian beam structure in the carved mode (see Supplementary Section 5).
For a classical field, we can extract the theoretically expected measurement results simply by calculating the overlap of the Gaussian eigenmode of the SMF and the normalized transverse structure of the scalar field \({u}_{{{{\rm{total}}}}}({{{\mathbf{ \uprho}}}},z)\!=\!\frac{1}{\sqrt{2}}({u}_{0p}({{{\mathbf{ \uprho}}}},z)\!\!{\mathrm{e}}^{{{{\rm{i}}}}\theta }{u}_{0{p}^{\prime}}({{{\mathbf{ \uprho}}}},z)).\) Thus, for laser light, the amount of laser power coupled into the fibre is proportional to
where A_{j}(z) refer to the overlap between the normalized radial mode j, at a distance z from its focus, and the normalized Gaussian eigenmode of the fibre. To see the Gouy phase dependence of the detection probability, the above equation can then be stated as
where the term ϕ(z) is an extra phase contribution from the curvature of the wavefront acquired upon propagation. However, as the wavefront curvature is very small near the optical axis, the only substantial contribution to the phase of the overlaps A_{j}(z) comes from the Gouy phase difference ΔΦ_{G}(z). Thus, scanning the fibre through the focus results in a signal that oscillates as \(\cos \left(2({p}^{\prime}p)\arctan \left(\frac{2(z{z}_{0})}{k{w}_{0}^{2}}\right)\right)\) underneath some envelope function defined by the zdependence of the overlap functions.
For the measurements, we kept the reference mode (that is, a Gaussian mode with radial index p = 0) fixed and varied the index \({p}^{\prime}\) of the probe mode between 1 and 4, which lead to four unique measurement scenarios with differing Gouyphase contributions. The measured data can be found on the top row of Fig. 2. The measurements follow the probability introduced above very well, which we verified by fitting curves that match equation (7) to the data. In each fit, we fixed the mode field diameter of our fibre to the 5 μm specified by the manufacturer and only had four fitting parameters: an overall scaling factor of the function, the beam waist w_{0}, focal position z_{0} and the zindependent phase offset θ. The average adjusted R^{2} value of the fits was 0.986, meaning that the data correspond well with the theoretical model.
After first verifying the method’s viability using a laser and showing the effect of the Gouy phase on a classical interference pattern along the optical axis, we extended the measurement scheme to observe the quantum Gouy phase. Following the same general idea, we now generated different twophoton N00N states between a reference Gaussian mode (p = 0) and higherorder radial modes, and studied the twophoton interference pattern along the optical axis. To prepare such a N00N state, we first generated photon pairs through spontaneous parametric downconversion (see Supplementary Section 5 for more information) and then shaped each of the two photons individually into a welldefined superposition of the wanted radial modes using two holograms performing two different mode carvings. Once each of the photons was structured, we directed the photons into the same beam path using a beamsplitter. As demonstrated in ref. ^{6}, once in the same beam path, indistinguishable photons bunch into the desired spatial mode N00N state given in equation (1). A simplified sketch of the twophoton experimental setup can be seen in Fig. 3.
To calculate the Nphoton coincidence probability, we project the radial mode N00N state \(\left{{\Psi }}(z)\right\rangle\) onto the state where all of the photons have been coupled successfully into the SMF P = ∣〈Ψ(z)∣N〉_{SMF}∣^{2}. Assuming that we produce perfectly balanced N00N states of radial modes with a phase offset θ, the Nphoton detection probability can be reduced to the form
As before, we can express this coincidence probability as
which is similar to the detection probability of the classical field, leading to an oscillating interference underneath some envelope function. However, in the above equation we see the photon numberdependent scaling for both the frequency of the oscillation as well as the envelope term. Note that a probability curve with half the amplitude but the same shape can also be observed for photon pairs prepared similarly without bunching. Thus, to verify that we generate radial mode N00N states in our experiment, we prepared the two photons in the corresponding radial mode superpositions and showed that the probability of coupling both of the photons into the SMF roughly doubles when the photons are made indistinguishable in time, which is a clear signature of bunching (see Supplementary Fig. 2 for the measured data). See Supplementary Sections 2 and 3 for detailed derivations of the detection probabilities.
For the N00N state measurements, we used the same set of radial modes in superposition with the reference Gaussian mode leading to the data shown on the bottom row of Fig. 2. As before, the data follow very well the theoretically expected curves, verifying the abovepresented equations and their described behaviours. Fits of equation (8) to the data—with the same parameters as in the classical case—resulted in an average adjusted R^{2} value of 0.951. The slight imperfections in the data can all be accounted for by imperfections in the alignment, imaging, the SMF eigenmode, spatial mode generation and errors in the stage position. Aside from the errors in the stage positions, all of these can be effectively categorized as contaminations of our state space by modes not included in the theoretical analysis. Hence, our results demonstrate that the quantum Gouy phase leads to a speed up in the accumulated phase upon propagation and also modulates the underlying envelope function. As we will discuss next, both features shed new light on the fundamental understanding of the Gouy phase, as well as hint at quantum enhanced metrology applications.
Quantum Fisher information
As the quantum Gouy phase evolves faster with a larger number of photons, one application could be supersensitive measurements of longitudinal displacement. This prospect can be investigated by calculating the QFI achieved through translation, which is of the form^{31,42,43,44}
When calculating this variance for the radial mode N00N state \(\left{{\Psi }}(z)\right\rangle\), we get the QFI
where Δ^{2}k_{z}∣_{i} and \({\left\langle {k}_{z}\right\rangle }_{i}\) are the variance and average of k_{z} for mode i, respectively, calculated using the angular spectrum of the corresponding mode. It is worth noting that the QFI does not depend on z, as the angular spectrum of a mode only acquires a phase structure upon translation. From equation (10), we can see that the second term of the QFI has Heisenberg scaling. As we show in Supplementary Section 4, this term relates to the Gouy phase difference between modes p and \({p}^{\prime}\). Hence, radial mode N00N states along with their quantum Gouy phase properties should be able to enhance the sensitivity of longitudinal displacement measurements. However, although these states provide benefits such as intrinsic interferometric stability when translating the mode along z, the spatial extent of the modes change, making it challenging to devise a real measurement capable of saturating the QFI at any z. The form of equation (10) also shows that it could be possible to engineer different spatially structured quantum states to measure different physical parameters. Due to the form of the QFI, the key feature that needs to be optimized in such state engineering should be maximizing variance of a specific momentum of the quantum state. For example, this would mean maximizing the variance in orbital angular momentum for rotation sensing^{6} or linear momentum for sensing the longitudinal position (see equation (9)). See Supplementary Section 4 for derivations of the QFI and the Fisher information calculated for the projection used in our experiment.
Momentum uncertainty
In addition to showing the potential for Heisenberg scaling, there is an interesting connection between the QFI and the uncertainty interpretation of the Gouy phase that fundamentally links the potential change in the spread of the transverse momentum to the evolution of the Gouy phase^{12}. Feng and Winful also noted that a larger momentum spread of higherorder modes results in a bigger Gouy phase shift^{12}. As the Gouy phase is increased by the photon number N, which is accompanied by a photon numberdependent momentum spread, as can be seen in equation (10), our results make a further connection between the quantum Gouy phase and its uncertainty interpretation. Similarly to ref. ^{12}, one can further link this behaviour to a tighter spatial confinement of the photons which can be made visible, for example by measuring the spatial extent of the Nphoton state as shown in Fig. 1c).
The de Broglie wavelength of light
Finally, our results show that the behaviour of a twophoton N00N state cannot be replicated simply by switching to a classical field with half the wavelength. The difference is clear if we note that the Gouy phase has a nonlinear dependence on the wavenumber, which means that simply ascribing an effective de Broglie wavelength λ/N to the Nphoton state does not produce the correct quantum Gouy phase. This is in contrast to the phase accrued by a nonconverging field upon propagation as well as arguments discussed in such a context^{32,33,34}. To investigate this fundamental difference in more detail, in Fig. 4 we plotted the measured data for two radial mode N00N states, along with overlap curves calculated for classical 405 nm modes with two different mode orders and waists. From these comparisons we see that the effect is not reproduced by a simple switching of the wavelength or doubling of the mode order.
Based on the comparison in Fig. 4 and equation (6) the only exact description of the Nphoton Fock state evolution seems to be that it evolves as the underlying mode, taken to the power of N. Although doubling the mode order and halving the wavelength seems to replicate quite well the shown twophoton behaviour. As the state evolves as the mode taken to power N, this evolution of the Nphoton quantum state results in a more rapid phase change and tighter confinement of the Nphoton. Both of these features have been taken advantage of in different studies and experiments. Either in the form of N00Nstate superresolution measurements^{30,45} or in increasing the confinement^{46}.
Conclusion
In summary, we have verified theoretically and experimentally that the increased phase sensitivity of multiphoton quantum states also extends to the fundamental phase anomaly of converging waves called the Gouy phase. We have shown through singlepath interferometric measurements along the optical axis that twophoton N00N states experience twice the Gouy phase when travelling through a focus. As the Gouy phase is a fundamental feature of converging waves, our results should apply broadly to quantum states of any bosonic system. Moreover, as the Gouy phase is an important factor in systems such as optical cavities^{46,47}, and a powerful tool in various applications such as mode sorters and mode converters^{19,20,21}, our results can be widely utilized in applications in quantum optics and quantum information science. In addition to providing a tool for quantum state manipulation, we showed that our results allow Heisenberglimited scaling in measurements of the longitudinal displacement and, as such, might inspire new superresolution measurement schemes.
Aside from these possible technological applications, we have linked the speedup of the Gouy phase in the quantum domain to an increased spread in the momentum of an Nphoton state. Hence, our results show that the uncertainty interpretation of the phase anomaly^{12} holds true in the quantum domain. Finally, due to the nonlinear relation between the Gouy phase and the wavenumber, our results unambiguously demonstrate that an Nphoton state cannot be rigorously modelled by using a classical field with a wavelength λ/N. However, our results suggest that an additional Nfold increase in the mode order can approximate the effect of the quantum Gouy phase when the beam Rayleigh lengths are matched. This hints at a possible link between an Nphoton state and the Nth harmonic of a classical field, which introduces an increase of the mode order and decrease of the beam waist, in addition to doubling the frequency. Thus, our study not only outlines possible applications using the quantum features of spatially structured photons, it also sheds new light on the fundamental understanding of the Gouy phase, a property intrinsic to all systems described by converging or diverging waves.
Methods
Source
The photon pair source uses a 12mmlong, type0 periodically poled potassium titanyl phosphate nonlinear crystal that is pumped by a 133.5 mW continuouswave 405 nm freespace laser. The downconverted photons are filtered through a 3nmwide bandpass filter centred around 810 nm, and coupled into separate singlemode fibres. Before the singlemode fibres, one photon is sent through an adjustable delay line. The rate of photon pairs after the singlemode fibres is roughly 3 MHz (correcting for accidentals, efficiency and nonlinearity of detectors).
Spatial mode manipulation
The spatial structures of the photons were modulated with a Holoeye Pluto2 SLM. To independently shape each photon, a pair of amplitude and phase modulating holograms were displayed on the phaseonly SLM. The amplitude modulation was implemented using a method that spatially changes the efficiency of the holograms grating. The N00N states were created by structuring the photons in equal and orthogonal superpositions of the two modes in the N00N state. A detailed figure of the experimental system is shown in the Supplementary Fig. 2 and more experimental details are given in Supplementary Section 5. The Gaussian beam waist of the photon spatial modes was roughly 774 μm before they were focused down to the final SMF (Thorlabs 780HP FC/PC).
Detection
The singlemode fibre to which the final state of light was projected on was scanned around the focus using a translation stage with a computer controlled piezo actuator (Thorlabs PIA13). A coupling stage (xyzcontrol) and a mount with tip/tilt controls was placed on top of the translation stage to allow maximum control of the alignment of the fibre. As the manufacturer of the piezo actuator stated that the step size of the actuator might differ depending on the direction, the actuator was scanned in the same direction in all measurements. The typical step size provided by the manufacturer (20 nm per piezo step) was used in the data processing. To detect the photon pair, a fibre beamsplitter was used to probabilistically split the photons. Subsequently, two singlephoton avalanche photodiodes (Laser Components CountT) were used in combination with a coincidence counter (IDQ ID900) to postselect for twophoton detections occurring between the two detectors. The coincidence window used to determine coinciding detections was τ = 1 ns and the accidental coincident detections were calculated using the approximate formula R_{1}R_{2}τ, where R_{i} refer to single photon detection rates in the two detectors. For measuring the coupling efficiency of laser light, two power meters were used. The first one was placed in one output of a fibre beamsplitter, which split the light coming out of the laser into two outputs. The power recorded with this power meter was used to monitor the output power of the laser as a reference signal. The second output of the fibre beamsplitter was fed to the spatial mode manipulation setup. To record the classical signal, the second power meter was placed directly after the final SMF. The final data is calculated as the power in the second power meter divided by the power in the first one to eliminate the effects of possible laser power fluctuations from the data.
Data availability
Source Data are provided with this paper.
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Acknowledgements
We thank F. Bouchard and S. Prabhakar for fruitful discussions. We acknowledge the support of the Academy of Finland through the Competitive Funding to Strengthen University Research Profiles (decision no. 301820; grant no. 308596), and the Photonics Research and Innovation Flagship (PREIN—decision no. 320165). M.H. acknowledges support from the Doctoral School of Tampere University and the Magnus Ehrnrooth foundation through its graduate student scholarship. R.F. acknowledges support from the Academy of Finland through the Academy Research Fellowship (decision no. 332399).
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M.H. and R.F. conceived and designed the experiment. M.H. constructed and performed the experiment, and processed the data. R.F.B. and R.F. supervised and assisted at every stage of the study. M.H., R.F.B. and M.O. derived the theoretical framework. M.H., R.F.B. and R.F. wrote the manuscript. All of the authors edited and proofed the manusript.
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Nature Photonics thanks Xuemei Gu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Supplementary Figs. 1 and 2, derivations and experimental details.
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Unprocessed relative power data, unprocessed coincidence data, and unprocessed single photon rate data.
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Hiekkamäki, M., Barros, R.F., Ornigotti, M. et al. Observation of the quantum Gouy phase. Nat. Photon. 16, 828–833 (2022). https://doi.org/10.1038/s4156602201077w
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DOI: https://doi.org/10.1038/s4156602201077w
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