Non-Gaussian quantum states of a multimode light field

TL;DR

This paper demonstrates the generation of non-Gaussian multimode quantum states by photon subtraction, revealing non-Gaussianity spread among entangled modes, which advances quantum network applications.

Contribution

First experimental realization of non-Gaussian multimode quantum states through photon subtraction from Gaussian states.

Findings

Negativity of the Wigner function observed

Non-Gaussianity spreads among entangled modes

Potential applications in quantum networks

Abstract

Even though Gaussian quantum states of multimode light are promising quantum resources due to their scalability, non-Gaussianity is indispensable for quantum technologies, in particular to reach quantum computational advantage. However, embodying non-Gaussianity in a multimode Gaussian state remains a challenge as controllable non-Gaussian operations are hard to implement in a multimode scenario. Here, we report the first generation of non-Gaussian quantum states of a multimode light field by subtracting a photon in a desired mode from multimode Gaussian states, and observe negativity of the Wigner function. For entangled Gaussian states, we observe that photon subtraction makes non-Gaussianity spread among various entangled modes. In addition to applications in quantum technologies, our results shed new light on non-Gaussian multimode entanglement with particular emphasis on quantum networks.

Tables

Table 1. Extended Data Table 1: Purities of the Wigner functions in Fig.  2 (a). For comparison, the purity of a Wigner function by the ideal photon subtraction is provided in square brackets, which agrees well with the experimental result. Low purity in a photon-subtracted mode is attributed to a non-ideal input state  Ourjoumtsev:2006jn . No optical loss is corrected in the calculation. Errors noted in parentheses are 1 s. d.

			Subtraction mode
		Input	HG0	HG1	HG2
Measurement mode	HG0	0.91(1)	0.45(0) [0.53(2)]	0.88(1)	0.87(1)
	HG1	0.90(1)	0.82(1)	0.47(0) [0.49(1)]	0.86(1)
	HG2	0.91(1)	0.87(1)	0.77(1)	0.49(0) [0.50(2)]

Equations

$${\hat{\rho }}^{-}\propto \hat{A}\hat{\rho }{\hat{A}}^{†},\text{ where}\hat{A}=\sum _{k=0}{c}_k{\hat{a}}_k.$$

$$K_{\mathrm{ex}}=\frac{\frac{1}{S}{\sum }_{s=1}^S{x}_s^4}{{(\frac{1}{S}{\sum }_{s=1}^S{x}_s^2)}^2}-3$$

$$U^{\textrm{(LC)}}=\left({\begin{array}[]{*{20}{r}}{-0.344i}&{-0.421i}&{0.531i}&{0.650i}\\ {0.344}&{-0.765}&{-0.531}&{0.119}\\ {-0.765i}&{-0.344i}&{-0.119i}&{-0.531i}\\ {0.421}&{-0.344}&{0.650}&{-0.531}\end{array}}\right).$$

$$U^{\textrm{(SC)}}=\left({\begin{array}[]{*{20}{r}}{-0.316}&{0.632}&{0.707}&{0.000}\\ {0.632i}&{\,0.316i}&{0.000}&{-0.707i}\\ {-0.316}&{0.632}&{-0.707}&{0.000}\\ {0.632i}&{0.316i}&{0.000}&{0.707i}\end{array}}\right).$$

$${\hat{\rho }}^{-}=\mathcal{R}[\hat{\rho }]/\text{tr}[\mathcal{R}[\hat{\rho }]];\mathcal{R}[\hat{\rho }]=w_0\hat{\rho }+(1-w_0)\mathcal{S}[\hat{\rho }],$$

$$\mathcal{S}[\hat{\rho }]=\frac{N{p}_0-1}{N-1}\hat{A}\hat{\rho }{\hat{A}}^{†}+\frac{1-p_0}{N-1}\sum _{k=0}^{N-1}{\hat{a}}_k\hat{\rho }{\hat{a}}_k^{†},$$

$$W(x_0,p_0)={\int }_{{\mathbb{R}}^{2(N-1)}}d{x}_{1}d{p}_1...d{x}_{N-1}d{p}_{N-1}{W}^{(N)}.$$

Full text

**Advanced quantum technologies require scalable and controllable quantum resources Andersen:2015dp ; Biamonte:2017ic . Gaussian states of multimode light such as squeezed states and cluster states are scalable quantum systems Yokoyama:2013jp ; Roslund:2014cb ; Chen:2014jx , which can be generated on demand. However, non-Gaussian features are indispensable in many quantum protocols, especially to reach a quantum computation advantage Mari:2012ep . Embodying non-Gaussianity in a multimode quantum state remains a challenge as non-Gaussian operations generally cannot maintain coherence among multiple modes Averchenko:2016gv . Here, we generate non-Gaussian quantum states of a multimode light field, and observe negativity of the Wigner function in adjustable modes. For this purpose, starting from the deterministic generation of Gaussian entangled states, we use sum-frequency generation to remove a single photon in a computer-controlled coherent superposition of optical modes. We reveal the induced non-Gaussian features and observe how they spread among the entangled modes, depending on the mode in which the photon is subtracted. The resulting non-Gaussian multimode quantum states will have broad applications for universal quantum computing Lloyd:1999vz ; Menicucci:2006ir , entanglement distillation Eisert:2002ft , and a nonlocality test Plick:2018wc . **

Our starting point is a squeezed vacuum state of light, a basic quantum resource for continuous-variable quantum technologies such as quantum-enhanced sensing Aasi:2013jb , deterministic quantum state teleportation Takeda:2013hn , and measurement-based quantum computing Menicucci:2006ir . Recent technological advances have extended the generation and control of squeezed vacuum from a single mode to multiple modes, which enables a determistic generation of large-scale multipartite entangled states Yokoyama:2013jp ; Roslund:2014cb ; Chen:2014jx . However, such quantum states are intrinsically Gaussian states, which always exhibit Gaussian statistics in electric field quadrature measurements. These states have limitations on quantum applications, e.g., universal quantum computing Lloyd:1999vz ; Menicucci:2006ir and entanglement distillation Eisert:2002ft . In particular, the ability to produce a non-Gaussian quantum state is essential to reach quantum advantages Mari:2012ep , which is connected to exotic quantum features of non-Gaussian quantum states. The most profound example thereof is contextuality, which goes hand in hand with negative values of the Wigner function Spekkens:2008kc . Another example is multimode entanglement, which can have conceptually different properties in non-Gaussian states as compared with their Gaussian counterparts Valido:2014iv .

The hybrid approach, which combines continuous-variable and discrete-variable quantum information processing, provides a solution Andersen:2015dp . Subtracting/adding a discrete number of photons Wenger:2004cw or coupling with a discrete-level quantum system Vlastakis:2013vi can generate non-Gaussian states such as a local or non-local superposition of coherent states Ourjoumtsev:2006jn ; Sychev:2017fq ; Ourjoumtsev:2009jh and hybrid entanglement Jeong:2014bl ; Morin:2014ip . Hitherto, this approach has only been successfully applied to a single- or two-mode quantum state, and the extension to highly multimode quantum states remains challenging due to the arduous task of maintaining coherence among multiple modes. For example, the conventional method of photon subtraction for a single-mode quantum state is based on a simple beam splitter Wenger:2004cw ; when applied to a multimode quantum state, however, the method results in the generation of a mixed quantum state Averchenko:2016gv .

To exploit the full potential of the large-scale entangled states available in the continuous-variable quantum information processing Yokoyama:2013jp ; Roslund:2014cb ; Chen:2014jx , it is essential that the hybrid approach is made compatible with multimode quantum states, e.g., via photon subtraction operating in multiple modes coherently. The concept of our experiment is illustrated in Fig. 1(a). If we call $\hat{\rho}$ the density operator of an input multimode quantum state, the output state $\hat{\rho}^{-}$ by a photon subtraction operator $\hat{A}$ becomes

[TABLE]

$c_{k}$ are complex numbers normalized as $\sum\left|{c_{k}}\right|^{2}=1$, and $\hat{a}_{k}$ is the annihilation operator for mode $k$. Note that $\hat{A}$ is, in general, a coherent superposition of annihilation operators in multiple modes. The ability to experimentally control both the $c_{k}$ coefficient and the multimode resource $\hat{\rho}$ is the key to tailor non-Gaussian multimode states and to achieve non-Gaussian entanglement for building non-Gaussian quantum networks Walschaers:2017bx ; Walschaers:2018wl .

In our experiment, the controlled generation of non-Gaussian multimode quantum states is performed using quantum frequency combs as a resource. Figure 1(b) shows the experimental setup, whose details are presented in Methods. The optical modes in which we implement Eq. (1) are time-frequency modes Ansari:2018uj . The interest of these modes is that they are co-propagating in the same transverse mode, allowing for a large multimode quantum resource to keep its coherence and to access arbitrary superpositions of modes through elaborate techniques in ultrafast optics. We populate these modes with a highly multimode Gaussian state through a parametric down conversion process Roslund:2014cb . Tailoring the measurement mode basis allows for the generation of versatile multipartite entangled states Cai:2017cp . We combine this resource with a time-frequency mode-dependent photon subtractor to de-Gaussify such multimode Gaussian states.

More specifically, our multimode Gaussian resource $\hat{\rho}$ is a set of independent squeezed vacua whose eigen modes are conveniently approximated by Hermite-Gaussian modes HGk. In order to implement the concepts of Eq. (1), we associate these modes with the annihilation operators $\hat{a}_{k}$. Hence, it remains to control the $c_{k}$ coefficients for the photon subtraction. As shown in Fig. 1(b), this is implemented through a mode-selective sum-frequency generation between the Gaussian resource $\hat{\rho}$ and a gate beam Ra:2017ia . The non-linear interaction is designed such that pulse-shaping the gate allows for the control of the mode of photon subtraction Ansari:2018uj . Finally, detection of a single photon in the up-converted beam heralds the subtraction of a photon in the desired mode from the Gaussian resource.

In practice, in order to implement the operator $\hat{A}$, the mode of the gate should be set as $v_{g}=\sum_{k}(-1)^{k}c_{k}~{}$HGk Ra:2017ia , which is efficiently performed using a computer controlled pulse shaper. The intensity of the gate governs the efficiency of the operation, and hence the heralding probability. To characterize the generated non-Gaussian multimode quantum state, we employ a homodyne detection that can control the mode of measurement by pulse-shaping the local oscillator.

We first measure the input multimode squeezed vacua without photon subtraction (i.e., no gate field is applied). As expected, the measured state exhibits Gaussian distribution: see the input Wigner functions shown in the first column of Fig. 2(a). On the other hand, when a single photon is subtracted in HG0 (the second column), we observe that the Wigner function in HG0 becomes non-Gaussian while the Wigner functions in the other modes remain Gaussian. This result shows the mode-selective operation of the photon subtractor necessary for multimode quantum states. The non-Gaussian Wigner function in HG0 exhibits a negative value at the origin ($W_{0}$), as is required to achieve quantum advantages Mari:2012ep , and the negativity indicates a negative value in the entire multimode Wigner function (see Methods). When a photon is subtracted in HG1 (HG2), we similarly observe a non-Gaussian Wigner function only in HG1 (HG2). Compared with the photon subtraction in HG0, photon subtraction in the higher order modes results in a less non-Gaussian Wigner function. This is mainly because the input state in a higher order mode has a larger optical loss than HG0: we have found a high fidelity ($F$) between a non-Gaussian Wigner function obtained in experiment with the ideal Wigner function calculated by subtracting a photon from the corresponding input state.

Furthermore, the versatility of the experimental setup allows for the computer controlled subtraction of a photon in an arbitrary superposition of modes from a multimode quantum state. As an example, we subtract a photon in a superposition of HG0 and HG1 modes, HG${}_{0}-i$HG1 111The normalization constant is omitted for simplicity, and $-i$ is introduced to rotate the $x$-squeezed vacuum in HG1 to $p$-squeezed vacuum such that both HG0 and HG1 have a $p$-squeezed vacuum.. We now observe a non-Gaussian Wigner function in this mode (first row of Fig. 2(b)). On the other hand, in the orthogonal mode HG${}_{0}+i$HG1, a Gaussian Wigner function is obtained, and in a partially overlapping mode $i$HG1, an intermediate situation is obtained as expected, see Extended Data Fig. 2. When we subtract a photon in a superposition of three modes HG${}_{0}+i$HG1+HG2, we similarly observe a non-Gaussian Wigner function in the same superposed modes (second row of Fig. 2(b)).

This flexibility of the setup allows us to extend photon subtraction to entangled input states. We first investigate an Einstein-Podolsky-Rosen (EPR) entangled state, which exhibits quantum correlations between two superposed modes: EPR${}_{0}=$HG0+HG1 and EPR${}_{1}=$HG${}_{0}-$HG1 (see Methods). The last two rows in Fig. 2(b) show the experimentally obtained Wigner functions. Without photon subtraction, the reduced quantum state in each of EPR0 and EPR1 is a thermal state as expected. When a photon is subtracted in EPR0, the introduced non-Gaussian characteristic appears in the other mode (EPR1) with almost no effect on the mode in which the photon is actually subtracted, and vice versa. In striking contrast with the aforementioned separable input state, the effect of photon subtraction on an entangled state is not localized but is transferred to another mode.

To open genuine perspectives for applications in quantum technologies, we show the scalability of our approach to larger multimode entangled states. We consider a four-mode linear cluster (LC) state and a four-mode square cluster (SC) state (see Methods), where we denote the four modes of the linear one as LCk ($k=0,1,2,3$) and the square one as SCk ($k=0,1,2,3$). Due to the large dimension of the state, full tomography becomes impractical. Thus, we quantify the non-Gaussianity in each mode by evaluating the phase-averaged excess kurtosis (to compare with the kurtosis of 3 by a Gaussian distribution). This is obtained from the quadrature measurement outcomes $x_{1},x_{2},...,x_{S}$ where the measurement phase is randomized:

[TABLE]

For a quantum state with zero mean ($\langle\hat{a}\rangle=\langle\hat{a}^{\dagger}\rangle=0$), which corresponds to all the quantum states in our experiment, a Gaussian state always exhibits $K_{\rm{ex}}\geq 0$, thus $K_{\rm{ex}}<0$ indicates a non-Gaussian state. Figure 3 shows excess kurtosis of the generated states. For the linear cluster state, excess kurtosis in each mode is initially close to zero. When a photon is subtracted in LC3, it is LC2 in which excess kurtosis becomes highly negative (i.e. non Gaussian) while those in the other modes remain close to 0. For photon subtraction in LC2, only LC3 exhibits a significant negativity. We can concentrate more non-Gaussianity in LC3 by subtracting a photon in a superposition of four modes, as shown in the last figure of Fig. 3(a). For the square cluster state, excess kurtosis without photon subtraction is close to zero in each mode. Photon subtraction in SC0, however, does not affect the distributions in its nearby modes (SC1 and SC3) much, but it introduces non-Gaussianity mostly in SC2 which is two steps away from the mode of photon subtraction.

We have generated non-Gaussian quantum states of a multimode light by subtracting a photon from multimode Gaussian states. The selectivity and the controllability of the mode(s) for the photon subtraction make it possible to extend the non-Gaussianity of a quantum state to the multimode regime, which has been a main obstacle for scalable quantum information processing Andersen:2015dp ; Biamonte:2017ic . The availability of non-Gaussian multimode states will stimulate fundamental studies on multipartite entanglement Valido:2014iv and multimode quantumness Hudson:1974eh by going beyond the Gaussian realm, as well as applications in quantum computing Lloyd:1999vz ; Menicucci:2006ir and quantum communication Eisert:2002ft ; Plick:2018wc . In particular, the observed nontrivial interplay between photon subtraction and cluster states, confirming recent theoretical predictions Walschaers:2018wl , provides new insights into the fields of quantum networks Kimble:2008if and quantum transport Walschaers:2016bm .

I Methods

Experimental details. Non-Gaussian multimode quantum states are generated by several nonlinear interactions on femtosecond pulses, as described in Fig. 1. The fundamental light source is a Ti:Sapphire laser, which produces a train of pulses (duration: 90 fs, central wavelength: 795 nm) at a repetition rate of 76 MHz. The laser beam is split into three beams: one is used for generating a multimode Gaussian state, another for photon subtraction, and the third for the homodyne detection.

The first beam is up-converted to a femtosecond pulse having 397.5-nm central wavelength in NC1 (0.2-mm-thick BiB3O6) which is used as a pump for a parametric-down-conversion process in NC2 (2-mm-thick BiB3O6) inside a cavity, the SPOPO. The length of the SPOPO is locked to the length of the Ti-Sapphire laser via the Pound-Drever-Hall method, such that the train of pump pulses is synchronized with the down-converted pulses which circulate inside the SPOPO. Transmittance of the output coupler of the SPOPO is 50%. The light coming out through the output coupler is a multimode Gaussian state, containing roughly twelve squeezed vacua in orthogonal time-frequency modes Roslund:2014cb ; Cai:2017cp . Among the twelve modes, we focus on the first four dominating modes, whose covariance matrix is given in Extended Data Fig. 3.

For photon subtraction, we perform sum-frequency interaction between the multimode Gaussian state and the second beam from the Ti-Sapphire laser (the gate; 1 mW power) inside NC3 (2.5-mm-thick BiB3O6). Detection of a single photon generated by the sum-frequency interaction heralds photon subtraction from the multimode squeezed vacua, where the time-frequency mode of the gate determines the photon subtraction mode Ra:2017ia . To engineer the time-frequency mode of the gate, we employ a homemade pulse shaper whose core element is a spatial light modulator, having a spectral resolution of 0.2 nm. Conversion efficiency of the nonlinear interaction is 0.1 %, and we have typically 110 Hz of heralding rate with background noise of 6 Hz.

The last beam is used as the local oscillator (LO) of the homodyne detection to measure the generated quantum state. The measurement mode is the mode of the LO, which is engineered by another pulse shaper having a spectral resolution of 0.2 nm. For each event of photon subtraction, photocurrent difference between the two PDs is sampled every 2 ns during a 2-$\mu s$ time window, and one quadrature outcome is obtained by calculating the dot product between the samples and the double-sided-decaying-shape temporal mode of the SPOPO  NeergaardNielsen:2006hl . To reconstruct a Wigner function in Fig. 2, we collect 20,000 $\sim$ 30,000 quadrature outcomes. In the case of no photon subtraction, we monitor the variance of the quadrature outcomes. Phase dependance of the quadrature squeezing of the multimode Gaussian state provides the phase information of the LO relative to this multimode light.

Preparation of entangled states. We prepare an entangled state by choosing a specific basis of modes in which quantum correlations among desired modes emerge Roslund:2014cb ; Cai:2017cp . In the HG mode basis, even-order (odd-order) modes exhibit $p$-quadrature ($x$-quadrature) squeezed vacuum. To prepare an EPR entangled state, we use a basis of $\textrm{EPR}_{0}=\frac{1}{\sqrt{2}}\left(\textrm{HG}_{0}+\textrm{HG}_{1}\right)$ and $\textrm{EPR}_{1}=\frac{1}{\sqrt{2}}\left(\textrm{HG}_{0}-\textrm{HG}_{1}\right)$. We have obtained $\langle\Delta^{2}(\hat{x}^{\textrm{EPR}}_{0}-\hat{x}^{\textrm{EPR}}_{1})\rangle+\langle\Delta^{2}(\hat{p}^{\textrm{EPR}}_{0}+\hat{p}^{\textrm{EPR}}_{1})\rangle=2.51(6)<4$ (the Duan entanglement criterion Duan:2000fw ; Simon:2000fd ) and $\langle\Delta^{2}\hat{x}^{\textrm{EPR}}_{1|0}\rangle\langle\Delta^{2}\hat{p}^{\textrm{EPR}}_{1|0}\rangle=0.71(4)<1$ (the EPR criterion Bowen:2003kk ). To prepare a linear cluster state, we use a basis of LCk ($k=0,1,2,3$) which is obtained by applying a unitary matrix $U^{\textrm{(LC)}}$ to the basis of HGk ($k=0,1,2,3$), where

[TABLE]

To check correlations among different modes, we use the four nullifiers associated with a linear cluster state vanLoock:2007ky ; Cai:2017cp , $\hat{\delta}_{k}^{\textrm{(LC)}}=\hat{x}_{k}-\sum_{l}V_{kl}\hat{p}_{l}$ ($V_{kl}$ is the adjacency matrix defining the topology of a cluster state, where $V_{kl}=1$ if $k$ and $l$ are connected, and 0 otherwise). They all exhibit a variance less than the vacuum fluctuation: $\langle\Delta^{2}\hat{\delta}_{k}^{\textrm{{(LC)}}}\rangle/\langle\Delta^{2}\hat{\delta}_{k}^{\textrm{(LC)}}\rangle_{\textrm{vacuum}}=$ 0.75(2), 0.67(2), 0.68(2), and 0.64(2) for $k=$ 0, 1, 2, and 3, respectively. Similarly, we prepare a square cluster state in the basis of SCk ($k=0,1,2,3$), which is obtained by applying a unitary matrix $U^{\textrm{(SC)}}$ to the HG basis, where

[TABLE]

Each of the four nullifiers $\hat{\delta}_{k}^{\textrm{(SC)}}$ associated with a square cluster state exhibits a variance less than the vacuum fluctuation: $\langle\Delta^{2}\hat{\delta}_{k}^{\textrm{{(SC)}}}\rangle/\langle\Delta^{2}\hat{\delta}_{k}^{\textrm{(SC)}}\rangle_{\textrm{vacuum}}=$ 0.72(2), 0.77(2), 0.61(2), and 0.75(2) for $k=$ 0, 1, 2, and 3, respectively.

Theoretical model. To calculate $\hat{\rho}^{-}$, we model a single-photon subtractor that takes into account experimental imperfections Ra:2017ia :

[TABLE]

where $w_{0}$ is the weight of background noise (e.g. dark counts) of the SPD which does not alter the input state. $\mathcal{S}[\hat{\rho}]$ is the actual photon subtraction from an input state:

[TABLE]

where $N$ is the number of modes. $\hat{A}$ is the desired photon subtraction in Eq. (1), whose weight is $p_{0}$; the remaining weight, $1-p_{0}$, corresponds to the photon subtraction in the incoherent mixture of the other modes with equal contribution. To consider the experimental conditions, we use $w_{0}=0.0094$, $p_{0}=0.95$, $N=4$, and $\hat{\rho}$ in Extended Data Fig. 3. $K_{\textrm{ex}}$ of $\hat{\rho}^{-}$ in a specific mode is then obtained by following the method presented in Refs. Walschaers:2017bx ; Walschaers:2018wl , which is based on calculating multimode correlation functions.

Negativity of a multimode Wigner function. A negative value in a single-mode Wigner function, which is reduced from a multimode Wigner function, is sufficient to show a negative value in the multimode Wigner function. The proof is straightforward by considering the contraposition of the statement: a non-negative multimode Wigner function always leads to a non-negative Wigner function when reduced to a single mode. Consider a reduced phase space $\mathbb{R}^{2}$ of an arbitrary single mode defining quadratures $(x_{0},p_{0})$ and the entire phase space $\mathbb{R}^{2N}$ of $N$ modes defining quadratures $(x_{0},p_{0},...,x_{N-1},p_{N-1})$. The Wigner function in the $N$ modes is assumed to be non negative: $W^{(N)}(x_{0},p_{0},...,x_{N-1},p_{N-1})\geq 0$. Then, the Wigner function in the reduced phase space $W(x_{0},p_{0})$ can be obtained by integrating the multimode Wigner function over the phase space of $(x_{1},p_{1},...,x_{N-1},p_{N-1})$:

[TABLE]

As $W^{(N)}\geq 0$, the reduced Wigner function $W(x_{0},p_{0})$ cannot have a negative value.