Abstract
Entanglement between large numbers of quantum modes is the quintessential resource for future technologies such as the quantum internet. Conventionally, the generation of multimode entanglement in optics requires complex layouts of beamsplitters and phase shifters in order to transform the input modes into entangled modes. Here we report the highly versatile and efficient generation of various multimode entangled states with the ability to switch between different linear optics networks in real time. By defining our modes to be combinations of different spatial regions of one beam, we may use just one pair of multipixel detectors in order to measure multiple entangled modes. We programme virtual networks that are fully equivalent to the physical linear optics networks they are emulating. We present results for N=2 up to N=8 entangled modes here, including N=2, 3, 4 cluster states. Our approach introduces the highly sought after attributes of flexibility and scalability to multimode entanglement.
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Introduction
Multipartite entanglement is not only of fundamental scientific interest, it is also the key ingredient for quantum information technologies^{1,2,3,4}. In optics, several impressive demonstrations of multipartite entanglement have been shown recently including an 8photon cluster state^{5} and a 9mode state used for error correction^{6}. However, these schemes tend to employ one detection system per entangled mode/qubit, which introduces a lack of flexibility and is detrimental to its scalability. These optical setups are built to produce one set of outputs or to perform one given protocol; in order to change the output the optical hardware itself must be modified.
Currently, the wellestablished recipe for generating entanglement using continuous wave laser beams is to mix squeezed modes of light together at beamsplitters. It is possible to create Nmode entanglement given a network of N−1 beamsplitters with N input modes, even with less than N squeezed modes^{7}.
In our scheme, we copropagate all possible spatial modes of light within one beam. Entanglement between copropagating modes in one beam has been previously demonstrated with spatial modes^{8,9}, and also in the frequency domain^{10}. In the current work, we radically expand the idea of onebeam entanglement by introducing the notion of emulating linear optics networks, by programming virtual networks that mix together different spatial regions of the light beam. These softwarebased networks calculate the precise weighted combinations of the spatial regions required to emulate the physical networks. This is possible because the linear optical components in a typical network simply perform reversible operations, and can be represented by unitary matrices. It is worth stating explicitly that the entangled spatial modes that we produce are eventready and unconditional before the detection process. The realtime virtual networks allow us to match the detection basis to the desired spatial mode basis contained within the beam, analogous to shaping a reference local oscillator beam. We report here the ability to switch in real time between desired output states in a onebeam entanglement system using just one detection scheme. We programme virtual networks for 2, 3, 4, 5, 6, 7 and 8mode entangled states. As a further demonstration of the versatility of our setup, we produce linear 2, 3 and 4mode cluster states, which are highly entangled graph states garnering attention for their potential in quantum computing^{2}. The cluster state measurements demand more stringent squeezing requirements than noncluster inseparable states.
Results
Measuring spatial modes
By employing custommade multiphotodiodehomodynedetectors (MPHD) that each contain an array of eight photodiodes (see Fig. 1), we detect the light in eight spatial regions and assign individual electronic gains to each one. The linear combination of the eight gainadjusted photocurrents constitutes the measurement of one mode.
More generally, we can express the measurement process of a complete set of spatial modes in one beam by the following:
where is the set of N measured modes projected by the N×8 unitary matrix U acting on the eight homodynesubtracted photocurrent operators . is an N×8 matrix made up of the top N rows of U_{in}, the orthogonal 8×8 unitary matrix that recovers the important set of eight unmixed spatial modes that span the input basis (see Methods). Input modes are then mixed via U_{net}, which emulate linear optics networks, given by the N×N matrix:
where .
This allows us to uniquely define a mode â_{n} by the eight real numbers in the nth row of U, which we will label as the mode's gain vector G_{n}, such that â_{n} = G_{n}î. Therefore each spatial mode we measure, whether belonging to the input basis or an entangled mode basis, is defined by a unique pattern within the light beam. These spatial mode patterns, represented by Gaussian profiles modulated by respective electronic gains G_{n}, are shown visually in Fig. 2, while the detection stage of Fig. 1 shows how we implement this experimentally. The spatial modes are orthogonal to each other, spanning a basis so that the independent measurement of each mode is possible^{11,12}.
Input basis
We create two amplitude squeezed modes using optical parametric amplifiers (OPAs). The first mode is converted to a flip mode (FM) by phase delaying half its beam by half a wavelength, π (see inset of Fig. 1). The FM is overlapped in quadrature with the Gaussian mode (GM) output of the second OPA upon reflection of its output coupler^{13}. These two squeezed modes are the first two modes of what we refer to as the input basis; â_{1} and â_{2}. Six copropagating vacua modes are measured by calculating G_{n} vectors that are orthogonal to both â_{1} and â_{2}. These vacua modes (labelled â_{3}...â_{8}) complete the input mode basis (see middle row of Fig. 2). Measuring these modes amounts to matching the detection basis by following equation (1) and setting U_{net}=I (see Methods).
Entangled mode bases
Programming a virtual network amounts to calculating the precise expression for U_{net}. The unitaries we have access to in programming the virtual networks are beamsplitters and π phase shifts. U_{net} is the concatenation of all of these unitaries that make up a linear optics network. The π phase shift is equivalent to multiplying â by −1. Note that arbitrary phase shifts are forbidden as each measurement naturally corresponds to detection at a fixed phase defined by a shared reference beam, the local oscillator. Importantly, optimal virtual networks are calculated allowing for optimization of beamsplitters owing to asymmetries in the squeezing levels of input modes.
The most intuitive virtual network we create is the 2mode EinsteinPodolskyRosen (EPR) state^{14} shown in Fig. 3a with two squeezed inputs. Here, we engineer spatial mode patterns that have no spatial overlap; the left half of the beam is entangled with the right half (see the N=2 panel of Fig. 2). Entangled modes belonging to other bases share spatial overlap but are nevertheless spatially orthogonal.
Spatial modes measured in an entangled mode basis are given a superscript N to distinguish them from modes in the input basis; and represent the two modes spanning the N=2mode EPR basis. See the Methods section for details on how we create virtual networks for each of the N=2, 3, 4, 5, 6, 7 and 8mode bases. In general, we construct networks pertaining to N modes by concatenating N−1 virtual beamsplitters with vacua on unused input ports (see Fig. 3b). This is a highly efficient approach to create multimode entanglement as the arduous tasks of mode matching and alignment are replaced with the ease of programming.
Cluster states
Attracting attention for their potential in oneway quantum computing schemes, cluster states are a type of highly entangled Gaussian graph state^{15,16}. They satisfy the quadrature relation . As infinite squeezing would require infinite energy and are thus unrealizable, one is limited to the production of approximate cluster states in the laboratory, and there have been demonstrations of up to 4mode continuousvariable cluster states thus far^{17,18}.
In order to measure cluster states in one beam, we must be able to access the correct quadratures of each entangled mode. Here we measure 2, 3 and 4mode linear cluster states; however, measuring arbitrary cluster shapes would require modifying the optical setup (see Methods).
Discussion
Each spatial mode is characterized by the continuousvariable quadrature operators and of the electric field operator. The and variance measurements of the eight modes in the input basis are shown in Fig. 4a,b. Here, 〈[Δx_{GM}]^{2}〉=〈[Δx_{1}]^{2}〉=−4.3±0.05 dB and 〈[Δp_{FM}]^{2}〉=〈[Δp_{2}]^{2}〉=−3.7±0.05 dB below the quantum noise limit, and the variances of the vacua are verified to equal quantum noise.
In order to verify entanglement between measured modes, we use the wellestablished van Loock–Furusawa inseparability criteria^{19}. For an Nmode entangled state, it is sufficient to satisfy N−1 inseparability inequalities:
with free parameters g_{i}, to be optimized for maximum inseparability. We have omitted the superscript N here for clarity, as the above holds for any mode basis. The subscripts n and here indicate the nth mode in the Nmode basis. Table 1 summarizes the measured degrees of inseparability for all N−1 inequalities in each Nmode basis, given in roman numerals. Table 2 summarizes the more stringent inseparability required for unweighted cluster states (all homodyne gains {g_{i}} are set to 1). The relevant inseparability inequalities satisfy the cluster state quadrature relationship written in the form of equation (4), and are written out explicitly in the Methods for reference.
The terms in equation (4) measure the degree of correlations between any two modes in a given basis. For the modes to be inseparable, each of these correlation variances (correlations in the x quadrature and anticorrelations in the p quadrature) must be in the quantum regime, that is below the normalized quantum noise of two units of vacua. Figure 4c,d shows this to be the case in our experimental measurements. Although we are limited here to eight modes owing to our detector, this scheme is scalable to higher numbers of mode entanglement even without increasing the number of squeezing resources, as shown in the simulation traces of Fig. 5. As we increase the simulated number of modes in the basis up to 30, the degree of inseparability approaches the classical bound of 1 owing to the vacuum noise penalty for each additional unsqueezed mode input. Entanglement is shown to hold here, however, even with current squeezing levels. Importantly, there is no loss incurred during the transformation of the squeezed input modes into a set of entangled modes, as can be seen by the agreement of the theoretical predictions and the experimental values of Fig. 5a. This equates to perfect mode matching at every virtual beamsplitter. Figure 5b explores how inseparability scales with different squeezing levels. Measuring a larger number of inseparable modes experimentally requires only an increase in the number of photodiodes in the MPHD, and importantly no modification of the optical setup. Note that this is not true for cluster states, and the number of squeezed inputs must be increased accordingly.
For the special case of N=2, optimal EPR entanglement^{20} is measured to be 0.58±0.01. Optimizing for the beamsplitter reflectivity^{21,22}, we find that owing to the slight asymmetry between input squeezing levels, the optimal beamsplitter ratio here is not 50%, but rather 48.8%, leading to a very slight improvement over the symmetric network. Each unique beamsplitter reflectivity changes the mapping of U_{net} such that formally the beam of light contains an infinite number of mode bases. The versatility of our scheme comes from being able to match the detection basis to a network that has been optimized for an arbitrary set of inputs.
The entanglement demonstrated in the current work allows for such protocols as quantum teleportation^{7,23,24}. To perform complex protocols such as oneway measurementbased quantum computations^{2,25,26}, we need to increase the degrees of freedom in our detection scheme as follows. First, we need to introduce the ability to measure each mode in an arbitrary phase quadrature at the MPHD. This may be achieved by manipulating the phases between the copropagating entangled modes. By introducing a unitary modeshaping device such as that explained in references^{27,28}, such access to individual phases becomes possible. These papers describe a process of manipulating both the amplitudes and phases of spatial modes in a lossless fashion via an adaptive process utilizing a network of deformable mirrors and lenses.
Second, in order to perform teleportationbased computations, we require the ability to feedforward the measurement results of arbitrary cluster nodes to remaining cluster nodes. This may be realised by combining a specifically shaped displacement beam containing the necessary phase space displacements with the beam containing the copropagating modes. By shaping the displacement beam to spatially match a specific mode to be displaced, it will interfere only with the desired mode owing to the orthogonal nature of the spatial mode basis. These modifications are feasible with existing technologies^{29}.
Emulating linear optics networks by mixing copropagating spatial modes is a highly efficient method for generating multimode entanglement. Otherwise arduous and potentially lossy tasks such as mode matching during the construction of a linear optics network are performed effortlessly and losslessly via softwarecontrolled combinations of the spatial modes. We have shown that although correlations weaken if more squeezing resources are not added, (noncluster) entangled modes scale here as the number of orthogonal modes measurable within the beam. The maximum number of measurable modes corresponds directly to the number of photodiodes in each pair of the multipixel detectors. We have demonstrated this by measuring N=2, 3, 4, 5, 6, 7 and 8mode entanglement within one beam, including up to 4mode cluster states, switching between them in real time. The ability to perform a wide range of protocols and optimize networks for asymmetry using just one optical setup offers versatility to future networks that will utilize entanglement as a resource.
Methods
Experimental setup
We use a dualwavelength continuouswave Nd:YAG laser at 1,064 nm and 532 nm. The OPAs each contain a periodically poled KTP crystal in a bowtie cavity. The squeezed beams are almost identical in purity, with squeezing levels of ∼−6 dB and antisqueezing of 8.5 dB. The beam containing the eight spatially orthogonal modes (see main text) is made highly elliptical in order to be measured by the MPHD, which has a linear array of eight photodiodes. The photodiode array used is a Hamamatsu InGaAs PIN photodiode array (G7150), which actually has 16 photodiodes. However, we choose to use only eight of these in the present experiment. The filling factor for the array is 90%, meaning that 10% of the light does not hit an active surface. The quantum efficiency for the photodiodes is 80%.
Virtual networks
For evennumbered mode bases (N=2, 4, 6 and 8), the method for creating the virtual network is as follows. The two squeezed modes a_{1} and a_{2} are combined on a half reflecting beamsplitter (HBS). As the output of this HBS is an EPR state, we choose to call this the EBS. The EBS outputs are symmetrically combined with N−2 vacua, as in Fig. 1 of the main text. The beamsplitters (BSs) are then given by where n is the number of BSs between the EBS and the BS in question. For N=4 and N=6 and N=8, mode output 2 is swapped with mode output N−1. For N=8, an additional swap of output modes 4 and 5 is made. For oddnumbered mode bases (N=3, 5 and 7), the method is the same with the following modifications. The EBS has its reflectivity changed to r = 1/2 − 1/(2N) (see, for example, refs 30,31 for more details on N=3). The vacua are mixed using beamsplitters as above, with one output arm having one less vacuum input. π phase shifts are applied to all BS outputs on the left of the EBS except for the one left output exiting the last BS. Mode outputs 1 and N−1 are swapped, and the network for N=7 has an additional swap between output modes 3 and 4. The homodyne gains g_{i} are optimized using a genetic algorithm in order to maximally satisfy the van Loock–Furusawa inequalities. These gains g_{i} scale the contributions of the quadrature variances and are independent from calculations regarding U_{net}. Here, optimal homodyne gains are calculated using two measures: minimizing the mean of the N−1 inequalities; and minimizing the variance of the set of inequalities. A tradeoff between the two measures is needed, and preference is given to minimizing the mean of the inequalities.
Spatial mode bases
The input matrix is defined as follows:
with . Each row of U_{in} represents the eight electronic gains that match the detection basis to the input modes. For example, the top row containing all ones recovers the standard GM, and the second row recovers the phaseflipped FM. By setting U_{net}=I, we can label each row of U_{in} as . Formally, O_{N,(8−N)})U_{in}, where O_{N,(8−N)} is a zero matrix of size N by (8−N).
The linear optics network for the ideal and symmetric 2mode EPR basis is simply a HBS:
From equation (4) we get:
from which we see that indeed and share no part of the detected light. We show this ideal EPR basis in Fig. 2 in order to emphasize the spatial separation. Note that the factor has been omitted from the scale in Fig. 2 for clarity.
The optimized network uses a beamsplitter reflectivity of 48.8%, and produces the following output modes:
Measuring cluster states
The difference between a 2mode cluster state and a 2mode EPR state is a Fourier transform on one mode. The Fourier transform is a rotation of π/2 degrees:
Therefore, we get . It follows that the homodyne measurements we perform in the 2mode cluster basis {p_{1}−x_{2}, p_{2}−x_{1}}_{cluster} and the 2mode EPR basis {x_{1}−x_{2}, p_{1}+p_{2}}_{EPR} are equivalent. Therefore, we may perform local Fourier transforms so long as we can match the homodyne detection basis for individual modes. It is important to note that this convenient basis change will not always be possible for different clusters. However, by shaping the local oscillator we may have access to arbitrary cluster states within the one beam. This was out of scope for the current experiment.
The criteria for verifying the measurements of the various cluster states are given below^{15,17} with the results summarized in Table 2:
Additional information
How to cite this article: Armstrong, S. et al. Programmable multimode quantum networks. Nat. Commun. 3:1026 doi: 10.1038/ncomms2033 (2012).
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Acknowledgements
This research was conducted by the Australian Research Council Centre of Excellence for QuantumAtom Optics (project number CE0348178), in collaboration with the network 'Highdimensional entangled systems' (HIDEAS FP7ICT221906) funded by the European Union, as well as the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project number CE110001027). We thank Akira Furusawa and the Furusawa group for discussions. S.A. is grateful for funding from the AustraliaAsia Prime Minister`s Award. B.H. acknowledges funding from the Alexander von Humboldt foundation.
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S.A., J.F.M., J.J., N.T. and H.A.B. designed the experiment. J.J. designed and built both of the OPAs. S.A., J.F.M. and J.J. constructed and performed the experiment. B.H. designed and built the multiphotodiode homodyne detectors, taught S.A. how to code the digital locking system and designed the software filters used in data analysis. P.K.L. and H.A.B. supervised the experiment. S.A. calculated and optimized the virtual networks, performed the data analysis and wrote the manuscript. J.F.M. wrote the data acquisition code and provided support in the data analysis.
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Armstrong, S., Morizur, JF., Janousek, J. et al. Programmable multimode quantum networks. Nat Commun 3, 1026 (2012). https://doi.org/10.1038/ncomms2033
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DOI: https://doi.org/10.1038/ncomms2033
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