Multiphoton Quantum-State Engineering using Conditional Measurements

TL;DR

This paper presents a method to engineer multiphoton quantum states with tunable correlations using conditional measurements on two-mode squeezed vacuum states, advancing quantum optics and potential applications in entangled lasers.

Contribution

It introduces a novel photon subtraction technique to generate and control quantum-correlated multiphoton states with high-dimensional properties.

Findings

Generated multiphoton states with tunable correlations.

Demonstrated nearly Poissonian photon statistics.

Paved the way for entangled laser development.

Abstract

The quantum theory of electromagnetic radiation predicts characteristic statistical fluctuations for light sources as diverse as sunlight, laser radiation and molecule fluorescence. Indeed, these underlying statistical fluctuations of light are associated with the fundamental physical processes behind their generation. In this contribution, we demonstrate that the manipulation of the quantum electromagnetic fluctuations of a pair of vacuum states leads to a novel family of quantum-correlated multiphoton states with tunable mean photon numbers and degree of correlation. Our technique relies on the use of conditional measurements to engineer the excitation mode of the field through the simultaneous subtraction of photons from two-mode squeezed vacuum states. The experimental generation of multiphoton states with quantum correlations by means of photon subtraction unveils novel mechanisms to control fundamental properties of light. As a remarkable example, we demonstrate the engineering of a quantum correlated state of light, with nearly Poissonian photon statistics, that constitutes the first step towards the generation of entangled lasers. Our technique enables a robust protocol to prepare quantum states with multiple photons in high-dimensional spaces and, as such, it constitutes a novel platform for exploring quantum phenomena in mesoscopic systems.

Tables

Table 1. Table 1: Determinant of the second-order matrix of moments M 𝑀 M . Notice data with ( ∗ ) (*) indicate a theoretical value predicted by equations 3 and 4 using the fit values for z 𝑧 z , η 𝜂 \eta , ν 𝜈 \nu .

z	l=0	l=1	l=2	l=3
0.1	$-4.06\text{x}{10}^{-7}$	$-4.82\text{x}{10}^{-6}$	$-3.05\text{x}{10}^{-5\ast }$	$-8.79\text{x}{10}^{-5\ast }$
0.48	$-0.55\text{x}{10}^{-3}$	$-6.09\text{x}{10}^{-3}$	$-19.54\text{x}{10}^{-3}$	$-15.22\text{x}{10}^{-3}$
0.67	$-6.91\text{x}{10}^{-3}$	$-62.7\text{x}{10}^{-3}$	$-202.4\text{x}{10}^{-3}$	$-332.2\text{x}{10}^{-3}$

Equations

$$\begin{split}p(n,m&)=\frac{1}{n!m!}\sum_{j,k=l}^{\infty}\frac{B_{j}^{\left(l\right)}B_{k}^{\left(l\right)*}}{\left(j-l\right)!\left(k-l\right)!}\\ &\times\partial_{\alpha}^{j-l}\partial_{\alpha^{*}}^{k-l}\left\{\left(\eta\alpha^{*}\alpha+\nu\right)^{n}e^{-\left[\left(\eta-1\right)\alpha^{*}\alpha+\nu\right]}\right\}\Big{|}_{\alpha,\alpha^{*}=0}\\ &\times\partial_{\beta}^{j-l}\partial_{\beta^{*}}^{k-l}\left\{\left(\eta\beta^{*}\beta+\nu\right)^{m}e^{-\left[\left(\eta-1\right)\beta^{*}\beta+\nu\right]}\right\}\Big{|}_{\beta,\beta^{*}=0}.\end{split}$$

$$I=\frac{\sqrt{⟨({\hat{a}}_s^{†}{)}^2{\hat{a}}_s^2⟩⟨({\hat{a}}_i^{†}{)}^2{\hat{a}}_i^2⟩}}{⟨{\hat{a}}_s^{†}{\hat{a}}_s{\hat{a}}_i^{†}{\hat{a}}_i⟩}-1,$$

$$\begin{array}{r}⟨:{\hat{m}}_s^u\otimes {\hat{m}}_i^v:⟩=\sum _{m,n=0}^{\infty }p(m,n)m(m-1)...(m-u+1)\\ \times n(n-1)...(n-v+1).\end{array}$$

$$M^{(2,2)}=\left(\begin{array}{ccc}⟨:{\hat{m}}_s^0{\hat{m}}_i^0:⟩& ⟨:{\hat{m}}_s^1{\hat{m}}_i^0:⟩& ⟨:{\hat{m}}_s^0{\hat{m}}_i^1:⟩\\ ⟨:{\hat{m}}_s^1{\hat{m}}_i^0:⟩& ⟨:{\hat{m}}_s^2{\hat{m}}_i^0:⟩& ⟨:{\hat{m}}_s^1{\hat{m}}_i^1:⟩\\ ⟨:{\hat{m}}_s^0{\hat{m}}_i^1:⟩& ⟨:{\hat{m}}_s^1{\hat{m}}_i^1:⟩& ⟨:{\hat{m}}_s^0{\hat{m}}_i^2:⟩\end{array}\right).$$

Full text

††preprint: APS/123-QED

The identification of the photon as a fundamental carrier of information has triggered a wide variety of research that aims to improve the state of the art of photonic technologies brien2009 . Along these lines, the field of quantum photonics has focused on exploiting the quantum properties of light to dramatically improve the performance of protocols for communication, metrology, imaging, and information processing giovannetti2011 ; aspuru2012 ; willner2015 . Consequently, the successful implementation of functional quantum photonic technologies hinges on our ability to generate, manipulate, and measure complex multiphoton states dellanno2006 ; pan2012 ; hadfield2009 . However, the generation of high-dimensional entangled states comprising a large number of photons is nowadays one of the most challenging tasks in quantum optics huber2010 ; hiesmayr2016 .

Over the past few years, physicists and engineers have demonstrated the utilization of multiple degrees of freedom of single photons to perform information processing tasks for a wide range of applications. In this regard, multiple bits of information have successfully been encoded in a single photon by preparing complex superpositions in time, frequency, position, transverse momentum, angular position, orbital angular momentum and path xie2015 ; dixon2012 ; khan2007 ; kues2017 ; dada2011 ; maga2016 . The complexity of such superpositions has led to important improvements in the performance and tolerance of cryptographic protocols, in the estimation of small physical parameters, and in the development of novel schemes for information processing brien2009 ; giovannetti2011 ; aspuru2012 ; willner2015 . Furthermore, it is now recognized that such protocols can be further improved by incorporating a high number of photons, correlated and entangled wen2007-1 ; wen2007-2 ; tsang2009 ; wen2010 . Interestingly, the generation of a large number of entangled photons has been demonstrated through the use of stimulated parametric down-conversion eisenberg2004 . Furthermore, the preparation of three indistinguishable particles and their interference signatures have been studied in Ref. menssen2017 . In 2016, four-photon entanglement in the basis of orbital angular momentum was reported hiesmayr2016 . More recently, boson sampling with up to five particles was demonstrated by means of multiplexing of single-photon sources wang2016 .

Regardless of the enormous effort directed thus far at improving the efficiency of multiphoton sources eisaman2011 ; kwiat1995 , the challenges involved in the generation and manipulation of entanglement and correlations among the photons impose practical limitations to realistic quantum technologies aspuru2012 . In 1997, Dakna and coworkers introduced the concept of photon subtraction and the possibility of performing quantum state engineering at a new fundamental level in which the quantum vacuum and the excitation mode of the fields are manipulated dakna1997 . This seminal work triggered a wide variety of fundamental and applied research, including new fundamental tests of quantum mechanics, engineering of macroscopic states of light, and the development of novel schemes for quantum metrology ourjoumtsev2006 ; parigi2007 ; hashemi2017 . In particular, regarding quantum-state engineering, Takahashi $et$ $al.$ takahashi2010 demonstrated distillation of entanglement from a single-mode squeezed state and four years later Kurochkin and colleagues demonstrated entanglement distillation by subtracting photons from standard two-mode squeezed vacuum states (TMSVS) kurochkin2014 . More recently, in the same context, Carranza and Gerry predicted the possibility of generating mesoscopic states of light with tunable average photon numbers by subtracting an equal number of photons from each mode of TMSVS carranza2012 .

Standard sources of TMSVS are based on the process of spontaneous-parametric-down-conversion (SPDC) and, under ideal conditions, they produce infinite entangled superpositions of photons as described by the expression $\left|z\right>=\sqrt{1-\left|z\right|^{2}}\sum_{n=0}^{\infty}z^{n}\left|n\right>_{s}\left|n\right>_{i}$, where $n$ denotes the number of photons in the signal ($s$) and idler ($i$) modes, and $z$ represents the so-called squeezing parameter, which depends on the material properties. Under realistic conditions, however, SPDC sources operate at very low brightness and the probability of generating more than one photon per mode is extremely low. This is in fact the main limitation for the efficient generation of photon-subtracted TMSVS based on standard SPDC sources. In this work, we circumvent this challenge by utilizing a bright SPDC source harder2016 in combination with photon-number-resolving (PNR) detectors lita2008 , to demonstrate the generation of a novel family of correlated photon-subtracted TMSVS with a broad range of mean photon numbers and degrees of correlation. This is achieved by engineering quantum statistical fluctuations of light via conditional measurements. Moreover, our observations reveal that the generated states are composed of two multiphoton wavepackets which are highly correlated and exhibit nearly Poissonian statistics. Remarkably, this achievement constitutes an important step towards the development of entangled laser-like systems. Since this family of states naturally inhabits high-dimensional Hilbert spaces, we foresee their applicability to enrich protocols relying on multiphoton interference such as high-sensitivity interferometry, quantum imaging, and sub-Rayleigh lithography wen2007-1 ; wen2007-2 ; tsang2009 ; wen2010 .

I Generation of photon-subtracted TMSVS

Photon-subtracted TMSVS containing the same number of photons in each mode exhibit the highest degree of correlation carranza2012 ; barnett1989 . Consequently, we restrict ourselves to the case of symmetric photon-subtracted TMSVS. Mathematically, the subtraction of $l$-photons from ideal TMSVS is obtained by applying $l$-times the bosonic annihilation operators $\hat{a}_{s,i}$ to the TMSVS $\left|z\right>$ yielding $\left|z,-l\right>=(n_{s}n_{i})^{-1/2}\left|z,-l\right>=\hat{a}_{s}^{l}\hat{a}_{i}^{l}\left|z\right>=\sum_{j=l}^{\infty}B_{j}^{\left(l\right)}\left|j-l\right>_{s}\left|j-l\right>_{i}$, where $B_{j}^{\left(l\right)}=\mathcal{N}^{-1/2}\left[j!/\left(j-l\right)!\right]z^{j}$, and the normalization constant $\mathcal{N}=\sum_{n=0}^{\infty}\left|z\right|^{2j}\left[j!/\left(j-l\right)!\right]^{2}$. Note that by construction photon-subtracted TMSVS constitute a family of entangled states characterized by the correlation function $B_{j}^{\left(l\right)}$, which depends on the number of subtracted photons $l$ and the squeezing parameter $z$. The above expression is constructed for perfect TMSVS, close to unity beam splitter transmission and independent of losses. Below, we incorporate the optical losses due to the beam splitter transmission into a single efficiency value, a good approximation to our experiment. This implies that the degree of nonclassical correlations in photon-subtracted TMSVS can be readily controlled by tuning these parameters.

For our experiments we utilize a Ti:Sapphire laser and a spectral filter, composed of two gratings and a 4$f$ optical system, to produce 1 ps optical pulses that are used to pump a periodically poled potassium titanyl phosphate (ppKTP) waveguide with a length $L$ of $8$ mm, a nominal poling period of 46 $\mu$m and a width of 2 $\mu$m, see Fig. (1 a). This configuration allows us to produce bright TMSVS at a wavelength of 1570 nm by means of type-II parametric down-conversion process harder2016 . For this bright SPDC source we have the squeezing parameter given as $z=\text{tanh}\left(r\right)$, with $r=\chi_{\text{eff}}\omega_{p}L\sqrt{I_{p}}/\left(2n_{0}c\right)$, where $n_{0}$ is the refractive index and $\chi_{\text{eff}}$ is the effective nonlinearity of the waveguide, $\omega_{p}$ is the frequency and $I_{p}$ the intensity of the pump field.

The simultaneous subtraction of $l$-photons is achieved by routing each of the signal and idler modes into a 90:10 fiber coupler. Each coupler subtracts photons by directing approximately 10$\%$ of the injected photons to a transition edge sensor (TES) that acts as a photon number resolving detector. The typical electrical output traces generated by a TES are shown in Fig. (1 b). The measured pulse energy due to the absorbed photons, shown in Fig. (1 c), clearly demonstrates the photon-number resolution of the TESs lita2008 . A series of Gaussian fits to the pulse-energy distributions are then used to estimate photon number probabilities. Crucially, the photon events detected by the central TESs permit conditional measurements on the external TESs that register the remaining 90$\%$ of the input photons. We then monitor the correlations between the photon-subtracted states.

In Fig. (2) we present the measured joint photon number distributions (top row) for the first generation of quantum-correlated photon-subtracted TMSVS inhabiting in a multi-dimensional space for different number of subtracted photons, (a) $l=0$, (b) $l=1$, (c) $l=2$, and (d) $l=3$. Importantly, we observe that photon subtraction generates a family of quantum states, whose marginal photon statistics are modified from thermal to nearly-Poissonian, with increasingly larger mean photon numbers, see Fig. (3). These effects become more evident for the case of $l=3$ and $z=0.66$, see Fig. (3 d). In that particular case, we clearly observe that the marginal photon number distributions for the signal and idler modes, strongly resemble the photon number distribution that characterizes a laser (red dashed curves in Fig. 3) with a mean photon number of $\left\langle n\right\rangle=2.05$ and not that of a thermal distribution as in conventional SPDC sources (black dotted curve in Fig.3). In other words, the generated light field is composed of two wavepackets exhibiting nearly Poissonian statistics. In that sense, the generated states share some similarities with a system of entangled lasers carranza2012 .

In order to characterize the generated states we use a theoretical model that assumes perfect TMSVs and photon detectors with non-unit detection efficiencies and contributions from dark counts. These are reasonable assumptions for sources of this kind as described in harder2016 ; sergey2017 . This models allows the computation of the realistic probability of simultaneously detecting $m$ photons in the signal and $n$ photons in the idler mode (see Supplementary Materials)

[TABLE]

Here, the parameters $\eta<1$ and $\nu>0$ represent the overall detector quantum efficiency, including the effect of all losses in the setup, and the dark counts, respectively sperling2012 . Our model considers losses only after beam splitters and assumes beam splitters with almost perfect transmissivity. Note that the complex parameters $\alpha$ and $\beta$ arise from the Glauber-Sudarshan $P$-function corresponding to the photon-subtracted TMSVS sudarshan1963 ; glauber1963 ; mehta1967 ; book_knight . It is worth remarking that the theoretical model used to obtain Eq. (1) can be readily extended to describe an asymmetric subtraction of photons from TMSVS, as well as photon subtraction from any two-mode quantum field (see Supplementary Materials).

Our theoretical results, shown in Fig. (2) (bottom row) are obtained by evaluating Eq. (1), with the best fit corresponding to an overall system efficiency of $16.25\%$. We compare this to our estimation of the overall efficiency of our setup. The source efficiency, beam splitter transmission and detection efficiency are estimated to be 90 $\%$, 90 $\%$ and 80 $\%$, leaving a fiber coupling efficiency of about 25 $\%$.

II Non-classicality test of photon-subtracted TMSVS

A natural metric to investigate the non-classicality of the photon-subtracted states is through applying the Cauchy-Schwarz inequality $\langle(\hat{a}_{s}^{\dagger})^{2}\hat{a}_{s}^{2}\rangle\langle(\hat{a}_{i}^{\dagger})^{2}\hat{a}_{i}^{2}\rangle\geq|\langle\hat{a}_{s}^{\dagger}\hat{a}_{s}\hat{a}_{i}^{\dagger}\hat{a}_{i}\rangle|^{2}$, which is fulfilled by any classical two-mode light field. That is, the inequality is obeyed if and only if the corresponding P-function, $P(\alpha,\beta)$, is a classical probability distribution. Accordingly, any violation of this inequality unequivocally indicates the nonclassical nature of any bimodal state, or equivalently that $P(\alpha,\beta)$ is non-classical Agarwal1988 . As a measure of this violation, we consider the Agarwal parameter

[TABLE]

which is negative if the inequality is violated Agarwal1988 . In Fig. (4) we present the experimental Agarwal parameter $I$ as a function of the squeezing parameter $z$ for the photon-subtracted states $\left|z,-l\right>$ with $l=0,1,2,$ and $3$. The theoretical curves are computed using Eq. (2) and the photon-number distributions obtained for our realistic experimental conditions through Eq. (1). For small values of squeezing $(z\rightarrow 0)$ we observe the greatest degree of violation of the inequality. Conversely, by increasing the degree of squeezing $(z\rightarrow 1)$ $I$ approaches zero. At this point it is worth emphasizing that a product of coherent states, which represents the classical-quantum boundary, gives an Agarwal parameter $I=0$. Hence, it is tempting to think that as we subtract an increasing number of photons and $(z\rightarrow 1)$, the state becomes a product of coherent states. However, as we show below, this is not the case, since at this particular regime the generated states become highly correlated.

To analyze the nature of the correlations between the generated photon wavepackets we explore the two-dimensional characteristic function $\sum_{m,n=0}^{\infty}p(m,n)x^{m}y^{n}$, where $p(m,n)$ is the probability of simultaneously detecting $m$ photons in the signal and $n$ photons in the idler modes. Then, by defining the operators $\hat{m}_{s(i)}=(\eta\hat{n}_{s(i)}+\nu)$ we obtain the joint factorial moments

[TABLE]

Using Eq. (3), it is possible to construct a matrix of moments of any order. However, it is sufficient to explore the behavior of the second order matrix of moments to demonstrate the nonclassical nature of the correlations between modes

[TABLE]

Crucially, the determinant of $M^{(2,2)}$ yields information about the mean values, the variances, and the covariance of the joint-photon statistics $\text{det}(M^{(2,2)})=\langle:(\Delta\hat{m}_{s})^{2}:\rangle\langle:(\Delta\hat{m}_{i})^{2}:\rangle-\langle:\Delta\hat{m}_{s}\Delta\hat{m}_{i}:\rangle^{2}$. For any two-mode classical field the covariance square $\langle:\Delta\hat{m}_{s}\Delta\hat{m}_{i}:\rangle^{2}$ is always greater than the product of the marginal variances $\langle:(\Delta\hat{m}_{s})^{2}:\rangle\langle:(\Delta\hat{m}_{i})^{2}:\rangle$ (see Supplementary Materials). Thus, in order to show the non-classicality of the correlations among the signal and idler modes, it is sufficient to show the contrary, that is, $\text{det}(M^{(2,2)})<0$.

In Table (1), we present the experimental $\text{det}(M^{(2,2)})$ for $l=0,1,2,3$ and for three different values of $z$. Quite remarkably, as we subtract more photons, the corresponding determinant becomes more negative. This implies that by subtracting a higher number of photons the covariance square becomes much larger than the product of marginal variances, thus indicating an increasingly larger degree of correlation between modes.

A further criterion to demonstrate the non-classicality of the correlations is to probe the non-negativity of $M^{(2,2)}$. That is, since the matrices of moments are non-negative (positive definite) for classical states, any violation of the non-negativity implies genuine non-classical second order correlations for the generated photon-subtracted TMSVS (see Supplementary Materials). In order to demonstrate the non-negativity of $M$, we use its normalized eigenvectors $\mathbf{V}$ and the corresponding eigenvalues $\lambda$. Since $\mathbf{V}^{\dagger}M^{(2,2)}\mathbf{V}=\lambda$, if all the eigenvalues of $M^{(2,2)}$ are non-negative, it holds that $M^{(2,2)}\geq 0$. On the contrary, if at least one eigenvalue is negative we have a violation of the non-negativity of $M^{(2,2)}$, that is, $M^{(2,2)}\ngeq 0$. Here, we have computed the minimal eigenvalues of $M^{(2,2)}$ and the results are shown in Fig. (4 b). Clearly, all the eigenvalues are negative and we can conclude that $M^{(2,2)}\ngeq 0$, as a result, the correlations between the idler and signal modes are nonclassical.

III Conclusions

The possibility of controlling multiphoton states in high-dimensional Hilbert spaces paves the way towards novel quantum protocols in the mesoscopic regime, being of particular relevance for quantum simulators konrad2018 , quantum optical interferometry dowling2008 , and quantum networks ourjoumtsev2009 ; sanders2012 ; israel2018 . More importantly, the ability of conditioning the mean photon number of photon-subtracted states enables the possibility of using our setup to perform experiments where heralding of multiple photons is required, a task that remains challenging harder2016 . Along these lines, in this work we have demonstrated the first experimental generation of a novel family of quantum-correlated photon-subtracted states with tunable mean photon numbers and degree of quantum correlations. We certified the quantum nature of the correlations in the generated photon-subtracted states through the use of the Agarwal parameter. Furthermore, the control of the degree of quantum correlations by means of photon subtraction was quantified. Given the enormous interest in high-dimensional quantum photonics with multiple particles, we anticipate the use of our technique for testing fundamental multiphoton physics pan2012 , and for the development of novel schemes for quantum metrology and information processing brien2009 ; giovannetti2011 ; aspuru2012 ; boto2000 .

IV Acknowledgements

We acknowledge J. P. Dowling, S. Glancy and D. Reddy for helpful discussions. We thank N. Montaut for useful advices regarding the alignment of the waveguide source.

Contribution of NIST, an agency of the U.S. government, not subject to copyright

V Methods

Detectors fabrication The superconducting transition edge sensor (TES) is a highly-sensitive microcalorimeter that measures the amount of heat absorbed in the form of photons. By exploiting the sharp superconducting-to-normal resistive transition in a superconductor as an extremely sensitive thermometer, we can measure the change in resistance from absorbing one or more photons. Absorption of two photons causes a temperature rise that is ideally twice that of a single photon. As a result, the TES generates an output signal that is proportional to the cumulative energy in an absorption event. This proportional pulse-height enables the determination of the energy absorbed by the TES and consequently the direct conversion of sensor pulse-height into photon number.

We utilize Tungsten (W) as the superconductor material for the fabrication of the TESs used in this work. The device size is 20 $\times$ 20 $\mu$m2, with a superconducting transition temperature of $\sim$150 mK. The W film is embedded in an optical stack designed to maximize photon absorption at the target wavelength, in this case 1550 nm. The optical stack consists of a highly reflective bottom mirror (electron-beam evaporated Ag film $\sim$80 nm), a dielectric spacer (physical vapor deposition (PVD) grown layer of SiO2 $\sim$232 nm thick plus a layer of direct current (DC)-sputtered amorphous-Si (a-Si) $\sim$20 nm thick), the active detector layer (DC-sputtered high-purity W, 20 nm thick) and an anti-reflection (AR) coating (DC-sputtered a-Si $\sim$56 nm and a PVD-grown layer of SiO2 $\sim$183 nm). Our fiber-packaging scheme ensures sub-micron alignment of the device active area to the fiber core of SMF-28 standard telecom fiber. The optimized optical stack as well as the packaging scheme ensures detection efficiency $>$ 95$\%$ of photons transmitted through the fiber.

Detectors operation TES are required to operate at low temperatures, below their transition temperature. Therefore, elaborate cooling systems are required to operate these devices. We operate the TESs in a commercial dilution refrigerator at a temperature around 30 mK. To operate the TESs, we voltage-bias the TESs which allows stable operation within the transition region [M1]. The TES readout is typically accomplished by use of superconducting quantum interference devices (SQUIDs) [M2], which serve as low noise amplifiers of the current flowing through the TES. The TES/SQUID responses due to different optical energies, i.e. photon numbers, are further amplified by room-temperature electronics and digitized. To maximize the signal to noise ratio, we perform post-processing of the output waveforms. One fast, reliable method is optimum filtering using a Wiener filter [M3]. This method yields a projection of each waveform onto a known template, resulting in an outcome that is approximately proportional to the absorbed energy, hence the number of photons for a given absorption event.

[M1] Irwin, K.D., An application of electrothermal feedback for high resolution cryogenic particle detection. App. Phys. Lett., 66(15), 1998-2000 (1995).

[M2] Jaklevic, R., et al., Quantum Interference Effects in Josephson Tunneling. Phys. Rev. Lett., 12(7), 159-160 (1964).

[M3] Fixsen, D.J., et al., Pulse estimation in nonlinear detectors with nonstationary noise. Nucl. Instrum. Methods Phys. Res. A, 520(1–3), 555-558 (2004).