Ultra large scale Continuous variable Clus Ter States Multiplexed in the Time Domain

I. INTRODUCTION

Section:

Quantum entanglement, which is often counterintuitive and was originally questioned by Einstein, Podolsky, and Rosen

(EPR),

¹ 1. A. Einstein, B. Podolsky, and N. Rosen, "Can quantum-mechanical description of physical reality be considered complete?," Phys. Rev. 47, 777 (1935). https://doi.org/10.1103/PhysRev.47.777 is now recognized as a versatile resource for quantum information protocols. ^2,3 2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000). 3. A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement (WILEY-VCH, 2011). In order to meet various requirements for various applications, an important technological development is to increase the number of available entangled qubits. In particular, measurement-based quantum computation (MBQC) requires the cluster type of large-scale entangled state, ^4,5 4. R. Raussendorf and H. J. Briegel, "A one-way quantum computer," Phys. Rev. Lett. 86, 5188 (2001). https://doi.org/10.1103/PhysRevLett.86.5188 5. R. Raussendorf, D. E. Browne, and H. J. Briegel, "Measurement-based quantum computation on cluster states," Phys. Rev. A 68, 022312 (2003). https://doi.org/10.1103/PhysRevA.68.022312 where individual components of the cluster state must be accessible by measuring devices. The number of entangled qubits with individual accessibility seems to be currently limited to a moderate scale. ^6–8 6. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O'Brien, "Quantum computer," Nature 464, 45 (2010). https://doi.org/10.1038/nature08812 7. T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, "14-qubit entanglement: Creation and coherence," Phys. Rev. Lett. 106, 130506 (2011). https://doi.org/10.1103/PhysRevLett.106.130506 8. X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, "Observation of eight-photon entanglement," Nat. Photonics 6, 225 (2012). https://doi.org/10.1038/nphoton.2011.354

Shifting to the continuous variable (CV) counterpart, ^9,10 9. J. Zhang and S. L. Braunstein, "Continuous-variable Gaussian analog of cluster states," Phys. Rev. A 73, 032318 (2006). https://doi.org/10.1103/PhysRevA.73.032318 10. N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster state," Phys. Rev. Lett. 97, 110501 (2006). https://doi.org/10.1103/PhysRevLett.97.110501 recently the number of fully inseparable qumodes of light fields drastically increased by introducing

multiplexing

either in the time domain ^11–13 11. N. C. Menicucci, X. Ma, and T. C. Ralph, "Arbitrarily large continuous-variable cluster states from a single quantum nondemolition gate," Phys. Rev. Lett. 104, 250503 (2010). https://doi.org/10.1103/PhysRevLett.104.250503 12. N. C. Menicucci, "Temporal-mode continuous-variable cluster states using linear optics," Phys. Rev. A 83, 062314 (2011). https://doi.org/10.1103/PhysRevA.83.062314 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 or in the frequency domain. ^14–17 14. N. C. Menicucci, S. T. Flammia, H. Zaidi, and O. Pfister, "Ultracompact generation of continuous-variable cluster states," Phys. Rev. A 76, 010302(R) (2007). https://doi.org/10.1103/PhysRevA.76.010302 15. N. C. Menicucci, S. T. Flammia, and O. Pfister, "One-way quantum computing in the optical frequency comb," Phys. Rev. Lett. 101, 130501 (2008). https://doi.org/10.1103/PhysRevLett.101.130501 16. M. Chen, N. C. Menicucci, and O. Pfister, "Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb," Phys. Rev. Lett. 112, 120505 (2014). https://doi.org/10.1103/PhysRevLett.112.120505 17. J. Roslund, R. M. de Araujo, S. Jiang, C. Fabre, and N. Treps, "Wavelength-multiplexed quantum networks with ultrafast frequency combs," Nat. Photonics 8, 109 (2014). https://doi.org/10.1038/nphoton.2013.340 The successes of the CV approach are based on the deterministic nature of field-quadrature squeezing. This is a strong advantage over the photonic qubit-based approach, where photons or photons pairs are generated with low probabilities and rare success events are postselected. ^8,18 8. X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, "Observation of eight-photon entanglement," Nat. Photonics 6, 225 (2012). https://doi.org/10.1038/nphoton.2011.354 18. P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, "Experimental one-way quantum computing," Nature 434, 169 (2005). https://doi.org/10.1038/nature03347 On the flip side, the disadvantage of the CV MBQC has been considered as finite squeezing in resource cluster states, which results in accumulation of noises during computation. ¹⁹ 19. M. Ohliger, K. Kieling, and J. Eisert, "Limitations of quantum computing with Gaussian cluster states," Phys. Rev. A 82, 042336 (2010). https://doi.org/10.1103/PhysRevA.82.042336 However, there has still been a possibility of a special encoding which can circumvent this problem, and indeed, a fault-tolerant threshold corresponding to about −20.5 dB of squeezing is recently proven by Menicucci ²⁰ 20. N. C. Menicucci, "Fault-tolerant measurement-based quantum computing with continuous-variable cluster states," Phys. Rev. Lett. 112, 120504 (2014). https://doi.org/10.1103/PhysRevLett.112.120504 for the Gottesman-Kitaev-Preskill (GKP) type of encoding. ²¹ 21. D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 Although the threshold of the squeezing levels is still unreached, it is good news that CV cluster states with finite squeezing can be a sufficient resource for quantum computation.

The

multiplexing

allows many qumodes to be assigned to a single beam, which drastically reduces complexity and physical size of the optical system from the conventional implementation assigning a single qumode to a single optical beam. ^22–26 22. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, "Experimental generation of four-mode continuous-variable cluster states," Phys. Rev. A 78, 012301 (2008). https://doi.org/10.1103/PhysRevA.78.012301 23. R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. Yoshikawa, P. van Loock, and A. Furusawa, "Demonstration of unconditional one-way quantum computations for continuous variables," Phys. Rev. Lett. 106, 240504 (2011). https://doi.org/10.1103/PhysRevLett.106.240504 24. R. Ukai, S. Yokoyama, J. Yoshikawa, P. van Loock, and A. Furusawa, "Demonstration of a controlled-phase gate for continuous-variable one-way quantum computation," Phys. Rev. Lett. 107, 250501 (2011). https://doi.org/10.1103/PhysRevLett.107.250501 25. Y. Miwa, R. Ukai, J. Yoshikawa, R. Filip, P. van Loock, and A. Furusawa, "Demonstration of cluster-state shaping and quantum erasure for continuous variables," Phys. Rev. A 82, 032305 (2010). https://doi.org/10.1103/PhysRevA.82.032305 26. X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, "Gate sequence for continuous variable one-way quantum computation," Nat. Commun. 4, 2828 (2013). https://doi.org/10.1038/ncomms3828 Although both of the time-domain

multiplexing

and the frequency-domain

multiplexing

would be important developmental directions, the time-domain

multiplexing

is advantageous because of time translation symmetry. With the time-domain

multiplexing,

a cluster state of qumodes can be an arbitrarily long chain in the longitudinal direction by extending the operating time of the cluster-state generator.

In spite of the in-principle unlimitation of the time-domain-multiplexing, the previous experimental demonstration of a dual-rail CV cluster state was limited in the number of qumodes, ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 due to a technical reason as follows. The optical setup is a large

Mach-Zehnder interferometer

as depicted in Fig. 1 , including a long optical delay line for a shift of qumodes on a rail. The mathematical derivation of the cluster state with this setup is contained in Sec. S1 of the supplementary material. In order to operate this cluster-state generator, the relative phases at all interference points must be properly locked. For this purpose, modulated bright beams are injected into the optical paths of the cluster state as phase probes, and their classical interference signals are exploited as error signals for the feedback control. However, the modulated bright beams are noisy, which obscures objective quantum-level correlation signals of the cluster state. The noises were circumvented in the previous demonstration by chopping the bright beams and by detecting the cluster state when the bright beams are absent. Due to the above previous situation, the optical system was uncontrollable during the cluster-state generation. Phase drifts gradually accumulated and quantum correlations gradually degraded at the rear part of the cluster state. Finally, the full-inseparability criterion ²⁷ 27. P. van Loock and A. Furusawa, "Detecting genuine multipartite continuous-variable entanglement," Phys. Rev. A 67, 052315 (2003). https://doi.org/10.1103/PhysRevA.67.052315 became unsatisfied after the creation of around 16 000 qumodes within 1.3 ms. ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287

FIG. 1. Schematic of the optical setup to generate a dual-rail CV cluster state multiplexed in the time domain. Accompanied is a graph image of the quantum state at each stage, where qumodes are represented by nodes and their correlations are represented by edges. Edge colors in (iv) distinguish positive and negative correlations, as explained in Sec. S1 of the supplementary material. ¹² 12. N. C. Menicucci, "Temporal-mode continuous-variable cluster states using linear optics," Phys. Rev. A 83, 062314 (2011). https://doi.org/10.1103/PhysRevA.83.062314

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Here we demonstrate a new time-domain-multiplexing experiment, where a dual-rail CV cluster state with full inseparability is deterministically generated without limitation in the number of qumodes. The previous limitation explained above is removed by continuing the feedback control during the generation of a cluster state. Instead, the noises of the modulated bright beams are eliminated electrically after

homodyne

detections. The resulting effective squeezing levels are −4.3 dB, which are not as high as −5.0 dB for the first thousands of qumodes in the previous demonstration, ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 but in contrast the initial squeezing levels continue without degradation. The squeezing levels are sufficiently higher than −3.0 dB of the full-inseparability criterion. ^13,27 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 27. P. van Loock and A. Furusawa, "Detecting genuine multipartite continuous-variable entanglement," Phys. Rev. A 67, 052315 (2003). https://doi.org/10.1103/PhysRevA.67.052315 In principle, there is no limitation in the number of fully inseparable qumodes. We test the full inseparability of about 1.2 × 10⁶ qumodes, which are generated within 100 ms. The reason why we stop at about 1.2 × 10⁶ qumodes is simply the data size for verification. In order to estimate the effective squeezing levels, we repeatedly acquire

homodyne

signals of the 1.2 × 10⁶ qumodes, by which the data size reaches 100 GB.

In Sec. II, we summarize theories regarding quantum correlations in the dual-rail CV cluster state, full inseparability criterion, and longitudinal mode functions. In Sec. III, we describe experimental methods. In Sec. IV, we show experimental results and make discussions. In Sec. V, we summarize our results. For self-containedness, in Sec. S1 of the supplementary material, we describe derivation of the quantum correlations in the dual-rail CV cluster state for an ideal case, and in Sec. S2 of the supplementary material, we describe derivation of inequalities for full inseparability.

II. THEORY

Section:

A. Quantum correlations in the cluster state

The procedure of producing the dual-rail CV cluster state is as follows. First, we produce a series of equal-time two-mode squeezed states (ordinary

EPR

states) in two beams [(ii) of Fig. 1 ]. This can be done by combining two single-mode squeezed states at a balanced beamsplitter. The time-slot width of the

multiplexing

is denoted by T. Next, we add delay to half-parts of the

EPR

states by the time-slot width T, by unbalancing the optical path lengths [(iii) of Fig. 1 ]. Finally, we combine the staggered

EPR

states by another balanced beamsplitter [(iv) of Fig. 1 ]. The resulting extended

EPR

(ExEPR) state is equivalent to the dual-rail CV cluster state up to local phase redefinitions. ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287

In Sec. S1 of the supplementary material, the above procedure is mathematically followed for the ideal, infinitely squeezed case. A convenient way of representing the state, utilized in Sec. S1 of the supplementary material, is to specify the state by its nullifiers ²⁸ 28. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, "Quantum computing with continuous-variable clusters," Phys. Rev. A 79, 062318 (2009). https://doi.org/10.1103/PhysRevA.79.062318 in the form of a linear combination of single-mode quadrature operators ${\hat{x}}_{\mathrm{\Lambda },k}$ and ${\hat{p}}_{\mathrm{\Lambda },k}$ . Here, Λ ∈ {A, B} denotes the spatial mode index, and k ∈ ℕ denotes the temporal mode index, as depicted in (iv) of Fig. 1 . Quadrature operators obey a commutation relation similar to that of position and momentum operators, $[{\hat{x}}_{\mathrm{\Lambda },k},{\hat{p}}_{{\mathrm{\Lambda }}^{\prime },k^{\prime }}]=i\mathit{ħ}{\delta }_{\mathrm{\Lambda }{\mathrm{\Lambda }}^{\prime }}{\delta }_{k{k}^{\prime }}$ , where δ _qq′ is the Kronecker delta. Sec. S1 of the supplementary material, is summarized in the following set of nullifiers:

$${\displaystyle {\hat{X}}_k≔{\hat{x}}_{A,k}+{\hat{x}}_{B,k}+{\hat{x}}_{A,k+1}-{\hat{x}}_{B,k+1},}$$

(1a)

$${\displaystyle {\hat{P}}_k≔{\hat{p}}_{A,k}+{\hat{p}}_{B,k}-{\hat{p}}_{A,k+1}+{\hat{p}}_{B,k+1}.}$$

(1b)

The ideal ExEPR state is specified by the set of these nullifiers via the relations ${\hat{X}}_k|\text{ExEPR}〉=0$ and ${\hat{P}}_k|\text{ExEPR}〉=0$ for all k. These nullifiers also express the quantum correlations, which are experimentally observable. Note that all the nullifiers are mutually commutable, thus giving a simultaneous eigenstate.

In reality, the resource squeezing levels are finite. Therefore, when the nullifiers ${\hat{X}}_k$ and ${\hat{P}}_k$ are measured with respect to an experimental ExEPR state, there are residual noise variances,

$${\displaystyle 〈{\hat{X}}_k^2〉=2\mathit{ħ}{e}^{-2r_{x,k}},〈{\hat{P}}_k^2〉=2\mathit{ħ}{e}^{-2r_{p,k}}.}$$

(2)

Here, the bracket $〈\hat{O}〉$ represents the mean value of an observable $\hat{O}$ . The squeezing parameters r _x,k and r _p,k correspond to those of the first and second single-mode squeezing sources in (i) of Fig. 1 , respectively, when the subsequent transformations and measurements are ideal, which is apparent from calculations in Sec. S1 of the supplementary material.

Then, a natural question is what levels of squeezing are required for the entanglement. Here we resort to the van Loock-Furusawa criterion for full inseparability. ^13,27 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 27. P. van Loock and A. Furusawa, "Detecting genuine multipartite continuous-variable entanglement," Phys. Rev. A 67, 052315 (2003). https://doi.org/10.1103/PhysRevA.67.052315 A sufficient condition for full inseparability is

$${\displaystyle 〈{\hat{X}}_k^2〉<\mathit{ħ},〈{\hat{P}}_k^2〉<\mathit{ħ},\text{for all}k.}$$

(3)

This condition corresponds to −3.0 dB squeezing of each nullifier. The derivation is contained in Sec. S2 of the supplementary material.

B. Longitudinal mode functions

In our demonstration, we use continuous-wave (CW) local oscillators (LOs) for

homodyne

detections. Therefore, the

homodyne

detection system continuously outputs real values ${\hat{x}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ or ${\hat{p}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ with Λ ∈ {A, B}, which correspond to instantaneous quadrature operators ${\hat{x}}_{\mathrm{\Lambda }}\left(t\right)$ or ${\hat{p}}_{\mathrm{\Lambda }}\left(t\right)$ but filtered by the detection system with finite bandwidth. There is flexibility in how we define the actual qumodes via weight functions when integrating these continuous signals. However, we must be careful about the orthogonality of qumodes, because the longitudinal modes are deformed by the introduced electric filters, as explained below.

The quadrature operator ${\hat{x}}_{\mathrm{\Lambda },k}$ of each qumode, specified by a longitudinal mode function f_k (t) = f ₀(t − kT) which is preferred to be contained in the k-th time slot kT ≤ t ≤ (k + 1) T, is connected to the instantaneous quadrature operators ${\hat{x}}_{\mathrm{\Lambda }}\left(t\right)$ via the relation

$${\displaystyle {\hat{x}}_{\mathrm{\Lambda },k}=\int f_k\left(t\right){\hat{x}}_{\mathrm{\Lambda }}\left(t\right)dt=\int f_0\left(t\right){\hat{x}}_{\mathrm{\Lambda }}(t+kT)dt.}$$

(4)

However, what we can directly choose in the experiment is not the optical longitudinal mode function f_k (t) but a computational weight function g_k (t), with which we perform weighted integration,

$${\displaystyle {\hat{x}}_{\mathrm{\Lambda },k}=\int g_k\left(t\right){\hat{x}}_{\mathrm{\Lambda }}^{det}\left(t\right)dt=\int g_0\left(t\right){\hat{x}}_{\mathrm{\Lambda }}^{det}(t+kT)dt.}$$

(5)

This integration is totally a postprocessing after acquisition of full-time quadrature values ${\hat{x}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ . The situation is the same for the conjugate quadrature operator ${\hat{p}}_{\mathrm{\Lambda },k}$ . Note that the integration is actually replaced by a summation in accordance with the sampling rate of

data acquisition.

If the response of the

homodyne

detection system is sufficiently flat, we can identify g_k (t) with f_k (t). However, this is not the case in this demonstration, in contrast to the previous demonstration, ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 because of the electric filters applied to the

homodyne

signals for the reasons explained in Sec. I and in more detail in Sec. III B. They are connected as g_k (t) ∝ (e*f_k )(t) via the impulse response function e(t) of the detection system including the electric filters, where * denotes convolution. Even if we choose the weight function g_k (t) so that they are nonzero only in the time slot kT ≤ t ≤ (k + 1) T, by which the weight functions themselves are orthonormal ∫g_k (t) g _k′(t)dt = δ _kk′, the orthonormality of actual

optical modes

∫f_k (t) f _k′(t)dt = δ _kk′ is not guaranteed in general. Note that the orthonormality of the mode functions {f_k (t)} _k∈ℕ is equivalent to the orthonormality of the qumodes via the relation

$${\displaystyle [{\hat{x}}_{\mathrm{\Lambda },k},{\hat{p}}_{\mathrm{\Lambda },k^{\prime }}]=\iint f_k\left(t\right)f_{k^{\prime }}\left(t^{\prime }\right)[{\hat{x}}_{\mathrm{\Lambda }}\left(t\right),{\hat{p}}_{\mathrm{\Lambda }}\left(t^{\prime }\right)]dtd{t}^{\prime }=i\mathit{ħ}\iint f_k\left(t\right)f_{k^{\prime }}\left(t^{\prime }\right)\delta (t-t^{\prime })dtd{t}^{\prime }=i\mathit{ħ}\int f_k\left(t\right)f_{k^{\prime }}\left(t\right)dt,}$$

(6)

where δ(t) is the Dirac delta function.

The filter is a combination of both high-pass and low-pass filters, and the parameters are described in Sec. III B. A filter delays signals in general. Therefore, when the response e(t) is deconvoluted from the weight function g_k (t), the

optical mode

function f_k (t) is stretched in the forward direction, which may ruin the mode independence to some extent. Especially, the effect of a

high-pass filter

is not negligible in our setup, which is expected to produce a long tail in the forward direction. In order to suppress the non-orthogonality of the actual

optical modes,

a possible method is to use a weight function localized to the latter part of the time slot. This is a workable method, but is not sufficient in our case, as will be discussed by using autocorrelations of shot noises in Sec. IV B.

Therefore, here we use a weight function which has both positive and negative parts. By adjusting the balance between the positive part and the negative part, the long tail produced by the

high pass filter

is canceled. The actual un-normalized weight functions utilized in the experimental data analysis are shown in Fig. 2 . The un-normalized weight functions are the following functions:

$${\displaystyle g_k\left(t\right)=\{\begin{array}{ll}exp[-{\gamma }^2{(t-t_k)}^2]\hfill & \hfill \\ \times (t-t_k+t_c),\hfill & \text{if}2|t-t_k|\le T_w,\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}}$$

(7)

discretized at intervals of 10 ns. The width of the weight function window T_w is 120 ns, and the envelope fall time 1/γ is 40 ns. The balance between the positive part and the negative part is adjusted through an optimization parameter t_c , which is determined to be 2 ns. The center of the weight function is t_k = t ₀ + kT, where k ∈ ℕ is the temporal index as usual. The mode interval T is 160 ns, and t ₀ = 95 ns in Fig. 2 . Note that we do not have to care about the normalization of the weight functions in the actual data analysis, because the obtained quadrature values are normalized by using corresponding shot noises acquired in the same conditions.

FIG. 2. First three weight functions g ₀(t), g ₁(t), and g ₂(t), marked by squares, diamonds, and circles, respectively. They are discretized at intervals of 10 ns, in accordance with the sampling rate 100 MHz of

data acquisition.

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On the other side of the coin, the wavy weight functions lead to increased contribution of higher frequency components, compared to the previous demonstration. ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 However, the squeezing level degrades at higher frequencies with our squeezing sources, due to a finite bandwidth of optical cavities. The resulting correlation levels of −4.3 dB are not as high as the previous −5.0 dB for this reason.

III. EXPERIMENTAL METHODS

Section:

A. Optical setup

Figure 3 shows the experimental setup. The light source is a continuous-wave (CW) Ti:sapphire laser (SolsTiS, M Squared Lasers) operating at the wavelength of 860 nm and the power of about 2 W. About 1 W of the laser power is sent to a second harmonic generation (SHG) cavity. It is a bow-tie shaped cavity with a round-trip length of 500 mm, containing a KNbO₃ (KN) crystal as a nonlinear optical medium. The generated CW second harmonic light is used as pump beams for two

optical parametric oscillators (OPOs)

operating below threshold. The power of the pump beam for each

OPO

is about 150 mW. Each

OPO

is a bow-tie shaped cavity with a round-trip length of 230 mm, containing a periodically poled KTiOPO₄ (PPKTP) crystal (Raicol Crystals) as a nonlinear optical medium. It generates a squeezed light beam, which is treated as a stream of single-mode squeezed vacuum states. The bandwidth of the squeezing basically corresponds to the bandwidth of the

OPO

cavity, whose half-width at half-maximum is 17 MHz.

FIG. 3. Experimental setup. ISO, optical isolator; PM, phase modulation; HD,

homodyne

detector; Osc, oscilloscope; Ctrl, feedback controller.

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The two squeezed light beams from the

OPOs

are sent to an asymmetric

Mach-Zehnder interferometer,

composed of two balanced beamsplitters and an optical delay line. The delay line is in one of the two arms of the

Mach-Zehnder interferometer,

and implemented with an optical fiber. The length of the optical fiber is about 30 m, from which the time slot width T of the

multiplexing

is about 160 ns. Both ends of the fiber are anti-reflection coated, and the coupling efficiency is maximized by using special fiber aligners (FA1000S, First Mechanical Design). Total amount of losses regarding the optical fiber, including the coupling inefficiency and the propagation losses, was about 11%.

In order to characterize the resulting ExEPR state, the two output beams are each subject to a balanced

homodyne

detection. The bandwidth of the

homodyne

detectors without the electric filter is more than 30 MHz. The power of a CW local oscillator (LO) beam for each

homodyne

detection is 10 mW. The transverse mode of the LO beams is cleaned and stabilized by a cavity. The interference visibility between the ExEPR beams and the LO beams were about 97% on average. The quantum efficiency of the

homodyne

photodiodes is about 99%.

B. Modulations, filters, and

data acquisition

In order to probe the optical phases and thereby to lock the relative phases of all optical interferences by feedback control, modulated bright probe beams are utilized. At each interference point, there is a corresponding photo-detector, and demodulation of the detector signal gives an error signal, which is fed back to a piezoelectrically actuated mirror, as depicted in Fig. 3 . First, a phase modulation at 32 MHz is commonly added by an electro-optic modulator (EOM) just after the Ti:sapphire laser output. This modulation is utilized for the Pound-Drever-Hall lock ²⁹ 29. E. D. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," Am. J. Phys. 69, 79 (2001). https://doi.org/10.1119/1.1286663 of the optical cavities including the

OPOs.

Next, two probe beams are phase modulated by piezoelectric transducers (PZTs) at 231 kHz and 326 kHz, respectively, and then injected into the individual

OPOs

from the back side of a highly reflective mirror composing of the

OPO.

The power of the probe beams is adjusted to about 2.5 μW at the output of the

OPOs.

These lower-frequency modulations are utilized for the lock of the following three. First, the phase-sensitive parametric amplification at each

OPO

is locked to the point minimizing the probe beam power, by which the phase of the squeezed light is associated with that of the probe beam. Second, the two interferences of the

Mach-Zehnder interferometer

are locked, where the beat signal of the two modulations is exploited by demodulating at the difference frequency of 95 kHz. Third, the LO phases are locked so that the

homodyne

detectors measure either the objective ${\hat{x}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ or ${\hat{p}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ quadrature values. The measurement basis is selected via the choice of the demodulation frequency between 231 kHz ad 326 kHz. These modulations of bright probe beams become unwanted noises in the

homodyne

detections, which wipe out the objective quantum correlations and thus must be removed somehow.

As stated in Sec. I, these noises were eliminated at the level of the optical system in the previous work, ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 employing periodic switching. The probe beams were alternately turned on and off by using a pair of acousto-optic modulators (AOMs), and the feedback control of the optical system was active when the probe beams were on, while the optical system was held and the ExEPR state was tested when the probe beams were off. In that demonstration, uncontrollable drifts of the optical system were negligible for first several thousands of qumodes, but gradually degraded the correlation in the ExEPR state, finally spoiling the full inseparability after creation of about 16 000 qumodes. In contrast, here we employ the method of eliminating the noises electrically after the

homodyne

detections, by using frequency filters. In this demonstration, the switching AOMs are always on, and the feedback control of the optical system continues during creation of the ExEPR state.

The electric filter applied to each

homodyne

signal in this demonstration is a serial combination of a 5th-order

high pass filter

with the cutoff frequency of 1.5 MHz, a 3rd-order inverse Chebyshev low-pass filter which has a notch characteristic at 32 MHz, and a 3rd-order low-pass filter with the cutoff frequency of 40 MHz for the purpose of anti-aliasing. After the electric filtering and appropriate low-noise amplification, the

homodyne

signals are digitized by an 8-bit oscilloscope (DPO7054, Tektronix) with the sampling rate of 100 MHz. The

data acquisition

frame width is 100 ms, and we acquire 1500 frames of data for each measurement quadrature in order to determine the residual noise variance of each nullifier with acceptable precision. In the same condition, we also acquire optical shot noises for references, which correspond to vacuum fluctuations. The shot noises are obtained by simply blocking the optical beam paths of the ExEPR dual rail.

Figure 4 shows the power

spectra

from the detection system with and without the electric filters, obtained from the fast Fourier transform (FFT) of oscilloscope data. We can see that the signals around the modulation frequencies are largely attenuated by the filters.

FIG. 4. Power

spectra

with and without the electric filter. (i) Optical shot noise without the electric filter. (ii) Optical shot noise with the electric filter. (iii) Dark noise without the electric filter. (iv) Dark noise with the electric filter. (v) Noise floor of the oscilloscope. The dark noise is the electric noise of the detection system when the LO beam is absent. 2048 data points are used for each fast Fourier transform, and the resulting power

spectrum

is averaged over 6000 times.

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IV. RESULTS AND DISCUSSION

Section:

A. Power

spectra

of

homodyne

signals

Since now the ExEPR state is stabilized by the continuous feedback, first we discuss the power

spectra

of the

homodyne

signals. Figure 5 shows the power

spectra

of the quadrature signals ${\hat{x}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ and ${\hat{p}}_{\mathrm{\Lambda }}^{det}\left(t\right)$ of the output ExEPR beams, together with that of the optical shot noise ${\hat{x}}_{\mathrm{\Lambda }}^{\text{vac,det}}\left(t\right)$ for references, for each of the two output ports Λ ∈ {A, B}. We can see that the output of the ExEPR state generator is, in the frequency domain, oscillating between $\hat{x}$ -squeezed states and $\hat{p}$ -squeezed states, and that the squeezed quadratures are opposite between the two output ports. This interesting point was not discussed in the previous work. ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287

FIG. 5. Power

spectra

of

homodyne

detection signals. (a) HD-A. (b) HD-B. (i) $\hat{x}$ quadrature signals of the ExEPR output. (ii) $\hat{p}$ quadrature signals of the ExEPR output. (iii) Optical shot noises for reference. 2048 data points are used for each fast Fourier transform, and the resulting power

spectrum

is averaged over 6000 times.

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By using the schematic in Fig. 1 , we discuss how this periodic behavior in the frequency domain is understood. The situation is that two squeezed beams, one is $\hat{x}$ -squeezed and the other is $\hat{p}$ -squeezed, enter a

Mach-Zehnder interferometer

with asymmetric lengths of arms. When we focus on a single sideband frequency component, the unbalance of the arm lengths acts as just a phase shift. The amount of the phase shift is θ = 2πfT, proportional to the sideband frequency f and the difference between the propagation times of the two arms T. Then, the two-input two-output

Mach-Zehnder interferometer

as a whole works as a single beamsplitter with a reflectivity R = cosθ dependent on the phase shift θ. At the phases of R = 1, the $\hat{x}$ -squeezed beam fully goes to one output port and the $\hat{p}$ -squeezed beam the other output port, and at the phases of R = 0, the opposite occurs. Between the above two cases, the beamsplitter combines the two squeezed beams and makes entanglement between the output ports. These cases periodically appear in the frequency domain because the phase shift θ is proportional to the sideband frequency f. The period of the oscillation corresponds to the propagation time difference T.

B. Orthogonality of qumodes

We apply the weighted integration to the above

homodyne

signals by using weight functions described in Sec. II B. Before that, we check the orthogonality of the qumodes by applying the weighted integration to the optical shot noises,

$${\displaystyle {\hat{x}}_{\mathrm{\Lambda },k}^{\text{vac}}=\int g_k\left(t\right){\hat{x}}_{\mathrm{\Lambda }}^{\text{vac,det}}\left(t\right)dt=\int f_k\left(t\right){\hat{x}}_{\mathrm{\Lambda }}^{\text{vac}}\left(t\right)dt.}$$

(8)

An unfiltered shot noise ${\hat{x}}_{\mathrm{\Lambda }}^{\text{vac}}\left(t\right)$ is a white noise whose autocovariance is represented by a Dirac delta function. Therefore, if the integrated shot noise values of a temporal mode ${\hat{x}}_{\mathrm{\Lambda },k}^{\text{vac}}$ have some correlation with those of neighboring temporal modes ${\hat{x}}_{\mathrm{\Lambda },k+m}^{\text{vac}}$ , m ≠ 0, the correlation is due to the overlap between the

optical mode

functions f_k (t) and f _k+m (t), via the relation

$${\displaystyle 〈{\hat{x}}_{\mathrm{\Lambda },k}^{\text{vac}}{\hat{x}}_{\mathrm{\Lambda },k+m}^{\text{vac}}〉=\iint f_k\left(t\right)f_{k+m}\left(t^{\prime }\right)〈{\hat{x}}_{\mathrm{\Lambda }}^{\text{vac}}\left(t\right){\hat{x}}_{\mathrm{\Lambda }}^{\text{vac}}\left(t^{\prime }\right)〉dtd{t}^{\prime }=\frac{\mathit{ħ}}{2}\iint f_k\left(t\right)f_{k+m}\left(t^{\prime }\right)\delta (t-t^{\prime })dtd{t}^{\prime }=\frac{\mathit{ħ}}{2}\int f_k\left(t\right)f_{k+m}\left(t\right)dt.}$$

(9)

Figure 6 shows the correlation of the integrated shot noise values among neighboring qumodes,

$${\displaystyle C_{\mathrm{\Lambda }}\left(m\right)≔\frac{〈{\hat{x}}_{\mathrm{\Lambda },k}^{\text{vac}}{\hat{x}}_{\mathrm{\Lambda },k+m}^{\text{vac}}〉}{〈{\left({\hat{x}}_{\mathrm{\Lambda },k}^{\text{vac}}\right)}^2〉},}$$

(10)

for |m| ≤ 5. As we can see with trace (i), the correlation is almost canceled, |C _Λ(m)| < 0.01 for m ≠ 0, when the weight functions g_k (t) in Eq. (7) and Fig. 2 is used. For comparison, trace (ii) shows the correlation when only the positive parts of the weight functions $g_k^{+}\left(t\right)\propto max(g_k\left(t\right),0)$ are used. In spite of the spacing between neighboring weight functions, a negative correlation C _Λ(±1) ≈ − 0.13 is observed, which is mainly due to the

high-pass filter.

This correlation is canceled by the negative parts of the weight functions. An insight regarding this cancellation is that the wavy weight functions make little use of the low-frequency components, which are affected by the

high-pass filter

and cause overlap between qumodes.

FIG. 6. Autocorrelation of a series of integrated shot noises. (a) HD-A. (b) HD-B. (i) When the weight functions in Eq. (7) and Fig. 2 are used. (ii) When only the positive part of the weight functions is used. Average is taken over 1.5 × 10⁶ times.

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Note that the fractions of a qumode invading neighboring qumodes correspond to square of the correlation coefficients |C _Λ(m)|², which are very small and thus experimentally negligible.

C. Full inseparability

Finally we check the inseparability condition. Figure 7 shows the residual variances of the nullifiers, $〈{\hat{X}}_k^2〉$ and $〈{\hat{P}}_k^2〉$ in Eq. (2), together with the variances of the same observables calculated from the shot noises for references. We can clearly see the variances well squeezed below the −3 dB bound for full inseparability, for up to k = 6 × 10⁵. Taking into account the two spatial mode indices A and B, the number of qumodes tested here is 1.2 × 10⁶. As mentioned in Sec. III B, the variance of each nullifier is averaged over 1500 times. The average and the standard deviation of the squeezing levels over the 6 × 10⁵ temporal indices are − 4.3 ± 0.2 dB for ${\hat{X}}_k$ and − 4.3 ± 0.2 dB for ${\hat{P}}_k$ . The worst squeezing level among the 6 × 10⁵ temporal indices is −3.6 dB for ${\hat{X}}_k$ and −3.5 dB for ${\hat{P}}_k$ . They will approach the average −4.3 dB if we could increase the number of data frames for the averaging.

FIG. 7. Residual noise variances of the nullifiers. (i) Variances of ${\hat{X}}_k$ . (ii) Variances of ${\hat{P}}_k$ . (iii) Corresponding reference shot noises. (iv) A sufficient condition of −3 dB bound for full inseparability. Horizontal axes are the temporal mode index k. Any correction of optical losses or detection noises is not applied.

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In contrast to the previous work, ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 the squeezing levels do not degrade during the generation of over-one-million qumodes. This is owing to the stabilized optical system by introducing the continuous feedback.

V. CONCLUSION

Section:

We have demonstrated the deterministic generation and verification of a fully inseparable dual-rail CV cluster state consisting of more than one million qumodes of light, by employing a time-domain

multiplexing

scheme. Compared to the previous work, ¹³ 13. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain," Nat. Photonics 7, 982 (2013). https://doi.org/10.1038/nphoton.2013.287 the cluster state generator does not degrade during operation, owing to the continuous feedback of the optical system. We can in principle further increase the number of qumodes, but we stopped at around one million qumodes because of the data size for verification. Time domain

multiplexing

is one of the key technologies of CV information processing. We note that our scheme is potentially upgraded to two-dimensional CV cluster states, where one dimension is finite and the other dimension is unlimited, by combining two optical delay lines. ¹² 12. N. C. Menicucci, "Temporal-mode continuous-variable cluster states using linear optics," Phys. Rev. A 83, 062314 (2011). https://doi.org/10.1103/PhysRevA.83.062314 Open questions from technological aspects are how to combine the cluster states with photon counters aiming at universal processing, ²⁸ 28. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, "Quantum computing with continuous-variable clusters," Phys. Rev. A 79, 062318 (2009). https://doi.org/10.1103/PhysRevA.79.062318 as well as how to increase the squeezing levels toward fault tolerance. We mention recent experiments of optical filters for quantum states, ³⁰ 30. S. Takeda, H. Benichi, T. Mizuta, N. Lee, J. Yoshikawa, and A. Furusawa, "Quantum mode filtering of non-Gaussian states for teleportation-based quantum information processing," Phys. Rev. A 85, 053824 (2012). https://doi.org/10.1103/PhysRevA.85.053824 as well as real-time integration of

homodyne

signals, ³¹ 31. H. Ogawa, H. Ohdan, K. Miyata, M. Taguchi, K. Makino, H. Yonezawa, J. Yoshikawa, and A. Furusawa, "Real-time quadrature measurement of a single-photon wave packet with continuous temporal-mode matching," Phys. Rev. Lett. 116, 233602 (2016). https://doi.org/10.1103/PhysRevLett.116.233602 which in the future may be combined with this entanglement

multiplexing

technologies.

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Source: https://aip.scitation.org/doi/10.1063/1.4962732

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