State-independent quantum state tomography by photon-number-resolving measurements

Rajveer Nehra; Aye Win; Miller Eaton; Reihaneh Shahrokhshahi; Niranjan Sridhar; Thomas Gerrits; Adriana Lita; Sae Woo Nam; Olivier Pfister

doi:10.1364/OPTICA.6.001356

1. INTRODUCTION

Single and multiphoton sources prepared in Fock states are of fundamental importance: not only do they enable experiments that epitomize the wave-particle “duality” of quantum mechanics, they also can only be described by quantum theory due to the nonpositivity of their Wigner quasi-probability distribution [1,2].

Eugene Wigner originally defined the continuous phase-space quasi-probability distribution function to study quantum corrections to classical statistical systems [3]. For a quantum state of density operator $\hat{ρ}$ , the Wigner function is given by

W (q, p) = \frac{1}{π} \int_{- \infty}^{\infty} e^{2 i p y} ⟨ q - y | \hat{ρ} | q + y ⟩ d y,

where

q

and

p

are the respective eigenvalues of the position and momentum operators or, in our case, of the amplitude quadrature,

\hat{Q} = (\hat{a} + {\hat{a}}^{†}) / \sqrt{2}

, and phase quadrature,

\hat{P} = i ({\hat{a}}^{†} - \hat{a}) / \sqrt{2}

, of the quantized electromagnetic field,

\hat{a}

, and

{\hat{a}}^{†}

being the photon annihilation and creation operators, respectively. The Wigner function is uniquely defined and normalized over phase space. It provides a complete description of the quantum state. As is also the case for a classical joint probability distribution, the marginal distributions of the Wigner function are the probability density distributions of the quadratures

\int_{- \infty}^{\infty} d p W (q, p) = {| ψ (q) |}^{2},

\int_{- \infty}^{\infty} d q W (q, p) = {| ψ (p) |}^{2} .

However, unlike a classical joint distribution, the quantum Wigner distribution

W (q, p)

can be nonpositive (hence non-Gaussian for pure states [4]), e.g. for Fock states with

n > 0

. This acquires major significance in the context of quantum information and quantum computing over continuous variables (CVQC) [5,6] as it is well known that all-Gaussian (gates and states) CV quantum information suffers from no-go theorems for Bell inequality violation [7], entanglement distillation [8], and quantum error correction [9]. However, none of these no-go theorems apply to CVQC when including non-Gaussian states or gates [10–12]. Non-Gaussian resources are therefore essential to CVQC and can be implemented, for example, by Fock-state generation or detection [13]. It is therefore important to be able to characterize Fock states fully and efficiently, possibly in real time. One standard method of state tomography is Wigner function reconstruction. Quantum state tomography in phase space [14] can be performed by reconstructing the Wigner function from the measurement statistics of the generalized quadrature

\hat{Q} \cos ϕ + \hat{P} \sin ϕ

, measured by balanced homodyne detection (BHD) where phase

ϕ

is the tomographic angle. This was first done for heralded single-photon states in 2001 [15] and has recently improved [16].

An issue with BHD-based tomography is that the reconstruction process is computationally intensive, using the inverse Radon transform, or maximum likelihood algorithms [17]. A more direct approach to reconstruct the Wigner function was proposed by Wallentowitz and Vogel [18] and by Banaszek and Wodkiewicz [19]. It is based on the following expression of the Wigner function [20]:

W_{\hat{ρ}} (α) = \frac{1}{π} Tr [\hat{ρ} \hat{D} (α) {(- 1)}^{\hat{N}} {\hat{D}}^{†} (α)],

where

α = (q + i p) / \sqrt{2}

,

\hat{D} (α) = \exp (α {\hat{a}}^{†} - α^{*} \hat{a})

is the displacement operator, and

\hat{N} = {\hat{a}}^{†} \hat{a}

is the number operator. Equation (4) reveals that the Wigner function at a particular phase-space point

α

is the expectation value of the displaced parity operator

\hat{D} (α) {(- 1)}^{\hat{N}} {\hat{D}}^{†} (α)

over

\hat{ρ}

or, equivalently, the expectation value of the parity operator

{(- 1)}^{\hat{N}}

over the displaced density operator

{\hat{D}}^{†} (α) \hat{ρ} \hat{D} (α)

. This provides a direct measurement method, given that one has access to photon-number-resolving (PNR) measurements. In particular, the value of the Wigner function at the origin is the expectation value of the photon number parity operator

W (0) = \frac{1}{π} \sum_{n}^{\infty} {(- 1)}^{n} ρ_{n n} .

Hence, the PNR detection statistics of a quantum system of density operator

\hat{ρ}

yield a direct determination of the Wigner function at the origin. In order to recover the Wigner function at all points, one can simply displace

\hat{ρ}

by a raster scan of complex number

α

. This can be done by interference at a highly unbalanced beam splitter [21] of transmission to reflection coefficient ratio

t / r ≪ 1

, as depicted in Fig. 1.

Fig. 1. Implementation of a displacement by a beam splitter. The initial coherent state amplitude is $β$ with $| β | ≫ 1$ , so that we can have $t ≪ 1$ in order to preserve the purity of the quantum signal $\hat{ρ}$ , while still retaining a large enough value of $| α | = t | β |$ , as needed for the raster scan of the Wigner function in phase space.

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This technique is commonplace in quantum optics and was used, for example, to implement Bob’s CV unitary in the first unconditional quantum teleportation experiment [22]. In all rigor, the resulting Wigner function is a more general one, the $s$ -ordered Wigner function, $W (s α; s)$ , which tends toward $W (α)$ when $s = - t / r \to 0$ [23]. This method was implemented for quantum state tomography of phonon Fock states of a vibrating ion [24], as well as microwave photon states in cavity QED [25,26]. For quantum states of light, it has been experimentally realized for the positive Wigner functions of vacuum and coherent states, as well as phase-diffused coherent-state mixtures, initially detecting no more than one photon [27] and subsequently detecting several photons [28,29]. The nonpositive Wigner function of a single-photon state was confirmed using PNR measurements by time-multiplexing non-PNR, low efficiency avalanche photodiodes, albeit with the use of a priori knowledge of the input state in order to deconvolve the effect of losses [30]. Our work is the first demonstration of state-independent photon-counting quantum state tomography of a nonpositive Wigner function. The only assumption made here is that the initial quantum state consist of low photon numbers to avoid the saturation limit of the detector, which is less than five photons per microsecond for the superconducting transition-ddge sensor (TES) used in our experiment. Since no other prior knowledge is assumed about the state to be measured, this technique is equally applicable to any arbitrary quantum state with low photon flux. We directly observe negativity of the Wigner function with no correction for detector inefficiency.

2. EXPERIMENTAL SETUP AND METHODS

A. Setup Description

The experiment, depicted in Fig. 2, built upon our previous demonstration of coherent-state tomography [29] with the addition of the heralded single-photon source: a type II (YZY) quasi-phase-matched periodically poled ${KTiOPO}_{4}$ (PPKTP) crystal, of period 450 μm, was used in a doubly resonant optical parametric oscillator (OPO), as detailed in Supplement 1 [31]. The OPO was pumped by a stable frequency-doubled 532 nm Nd:YAG nonplanar ring oscillator laser (1 kHz FWHM). The two-mirror OPO cavity was one-ended, with a finesse of 300, a free spectral range of 1.5 GHz, and an FWHM of 5 MHz. One mirror’s inside facet was 99.995% reflective for the signal and idler fields near 1064 nm and 98% transmissive for the pump field at 532 nm (the outside mirror facet was uncoated); the other mirror’s inside facet was 98% reflective at 1064 nm and 99.95% reflective at 532 nm (its outside facet was antireflection-coated at 1064 nm). The cavity was near-concentric with a super-Invar structure, the mirrors’ radius of curvature being 5 cm and the mirrors $≃ 10 cm$ apart. The filter cavity (FC) was made of two 5 cm-curvature, 99% reflective mirrors placed $≃ 0.5 mm$ apart.

Fig. 2. Experimental setup. The red dotted lines denote the locking beam paths for the on/off PDH servo loops of the OPO and the FC. The displacement operation is contained in the black dashed-dotted box at the top. BP, Brewster prism; DM, dichroic mirror; EOM, electro-optic modulator; FI, Faraday isolator; FR, Faraday rotator; HWP, half-wave plate; IF, interference filter; LO, local oscillator input to the displacement field; PBS, polarizing beam splitter; PD, photodiode; POL, polarizer; PZT, piezoelectric transducer.

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The OPO mode was aligned and mode-matched to all parts of the experiment (FC, TES fibers) by using a seed beam that was injected into the OPO through its highly (99.995%) reflecting mirror and exited through its output coupler. The seed beam was carefully mode-matched to the OPO so as to be a pure ${TEM}_{00}$ mode before being sent to the rest of the setup. It was also used for interference visibility optimization with the displacement field. A significant contribution to photon loss is any mode mismatch between the OPO and the FC, which must also be locked on resonance simultaneously, as detailed in the next section. By careful mode-matching of a seed OPO beam to the FC, we achieved 83% transmission of the OPO mode through the FC.

B. Stabilization Procedure

Both the OPO and the FC cavities were Pound–Drever–Hall (PDH)-locked [32] to a reference laser beam provided by the undoubled output of the pump laser. This was achieved by way of an “on/off” locking system, effected by a system of computer-controlled diaphragm shutters. In the “on” locking phase, the input to the single-photon sensitive PNR detectors was closed, and the reference laser was unblocked and sent into both the OPO and the FC (dotted lines in Fig. 2), whose PDH lock loops were closed for a few seconds. Because of its super-Invar structure, the OPO drift was low and the PDH loops could then be open, in the “off” phase, with their correction signals held constant. The shutter of the reference laser was closed and the paths between the OPO and the PNR detectors were open for data acquisition, for as long as 3 s; see Fig. 3. This procedure allowed us to lock the OPO to its doubly resonant, frequency-degenerate mode at $ω_{s} = ω_{i} = ω_{p} / 2$ . This was essential, as the displacement field, also provided by the undoubled output of the pump laser, had to be at the same frequency as the OPO’s quantum signal beam and phase-coherent with it. Note that finding this frequency-degenerate, doubly resonant OPO mode is nontrivial since the double resonance condition

ω_{s} = ω_{i},

\Leftrightarrow \frac{m_{s}}{L + n_{s} (T) ℓ} = \frac{m_{i}}{L + n_{i} (T) ℓ},

features two different indices of refraction

n_{s} \neq n_{i}

(

L

is the cavity length in air only,

ℓ

the crystal length, and

m_{s, i} \in N

the mode numbers). It is, however, possible to temperature-tune the OPO crystal to achieve stable frequency degeneracy [33–35]. This required temperature control of the PPKTP crystal to the level a few millidegrees, around 27.810°C, using a commercial temperature controller.

Fig. 3. On/off cycles of alternated active locking and data acquisition. Data collection begins for a period of 800 ms while the auxiliary locking beam is blocked, followed by a period of 200 ms where the auxiliary locking (broken red line in the experiment schematic) is enabled for active locking, and the signal channel is blocked. This process occurs cyclically during data collection to prevent excess photon flux from damaging the TES while ensuring a stable OPO cavity mode.

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C. PNR Detection

Our PNR detection system comprises two TESs, consisting of tungsten chips in a cryostat, coupled through standard telecom fiber. A detailed description of the TES system can be found in Refs. [29,36]. The TESs are cooled using an adiabatic demagnetization fridge at a stable temperature of 100 mK, at the bottom edge of the steep superconducting transition slope (resistance versus temperature). When one photon is detected, its energy is absorbed by the tungsten chip, yielding a sharp increase in its resistance that is detected by a superconducting quantum interference device over a rise time on the order of 100 ns. The heat is then dissipated through a weak thermal link, over a time on the order of 1 μs. During this time, the TES is still active (as opposed to, say, nanowire detectors or avalanche photodiodes). Due to the finiteness of its superconducting transition slope, the TES can resolve up to five photons. The absolute maximum photon flux sustainable by the TES without the tungsten driven into the normal conductive regime is therefore 5 photons/μs in the continuous-wave regime, i.e., a power of 1 pW. The OPO’s average power was kept at 100 fW by setting the pump power to 200 μW (the OPO threshold was 200 mW). We observed that the background counts were negligible when the TES signal was suppressed by rotating the pump’s linear polarization by 90°, thereby completely phase-mismatching the nonlinear interaction in PPKTP.

3. EXPERIMENTAL RESULTS

A. Heralding Ratio

The heralding ratio determines the quality of the single-photon source. It is the probability of seeing one photon in the OPO signal (heralded) beam with no displacement field, provided one photon was detected in the filtered idler (heralding) beam. The pump power was kept low enough so as to suppress two-photon events in the OPO signal in the absence of a displacement field. Results are displayed in Table 1. We can see that the heralded channel has many more counts than the heralding channel, as expected, since the latter is filtered by the FC and the interference filter (IF). The heralding efficiency was

η_{h} = \frac{N_{c}}{N_{i}} = 0.58 \pm 0.02,

and can also be considered the overall detection efficiency of the heralded channel, i.e., of the quantum signal.

Table 1. Experimentally Measured Number of Single-Photon Counts on Both Channels^a

View Table

B. Quantum Tomography of a Single-Photon State

We now turn to the state tomography results. Figure 4 shows the reconstructed Wigner function along with a fit with the following Wigner function of the single-photon state mixed with vacuum [31],

W (q, p) = η W_{| 1 ⟩ ⟨ 1 |} (q, p) + (1 - η) W_{| 0 ⟩ ⟨ 0 |} (q, p),

where we took the overall detection efficiency of the signal,

η

, as a fit parameter [31]. The Wigner function is plotted for experimentally measured values of

| α |

, where phase-space coordinates are

(q, p) = (\sqrt{2} | α | \cos ϕ, \sqrt{2} | α | \sin ϕ)

, where

ϕ

is the tomographic angle. We can clearly see the negativity around the origin of the phase space,

W (0, 0) = - 0.035 \pm 0.005 .

Errors in the displacement amplitudes were considered to be negligible due to the long-term amplitude stability of the laser producing the displacement field and the high accuracy of the calibration as mentioned in the “displacement calibration” section of Supplement 1 [31]. The Wigner function error bars (

1 σ

) at zero displacement were obtained from the statistics of multiple data sets with the displacement field blocked. At nonzero displacement, in order to speed up the measurement process and minimize experimental drifts, we decided to use the statistics of the measurement results at 10 different phases for the same displacement amplitude. This procedure yields a conservative estimate of the Wigner function error bars (

1 σ

), in the particular case of a single-photon Fock state, because it assumes that the measured Wigner function has the required cylindrical symmetry about the origin of phase space. The results are plotted in Fig. 5. Note that the fact that Wigner function is not significantly altered by this averaging—in fact, both the two-dimensional (2D) fit in Fig. 4 and the one-dimensional (1D) fit in Fig. 5 yield

η = 0.57 (3)

—speak to the high quality of the phase-space rotational symmetry of our data. One can notice that the fit residuals are reasonably small around the origin of phase space but grow larger in the outskirts of the function, near our maximum displacement values. These correspond to larger detected photon numbers on the TES, for which photon pileups during the TES’ cooling time make data analysis more arduous [29].

Fig. 4. Top, reconstructed Wigner function; black points, reconstructed values from raw data; solid surface, least-square Wigner-function fit, Eq. (9). Bottom, fit residuals.

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Fig. 5. Phase-averaged Wigner function. The Wigner function fit yielded $η = 0.57 (3)$ , which is consistent with the heralding efficiency 0.58(2). Error bars are discussed in the text.

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4. CONCLUSION

We have demonstrated state-independent photon-counting quantum state tomography with PNR measurements using a superconducting TES system and evidenced clear negativity in the single-photon Fock Wigner function with no correction for photon loss. This work has been limited by two factors: when working with continuous-wave detection, photon fluxes become overwhelming to the TES when $| α | \to 1$ . Moreover, photon pileups, in particular during the TES cooling time, greatly complicate data analysis [29]. In the future, we will multiplex several TES channels in order to access larger displacement amplitudes, i.e., larger regions of phase space. This will also reduce the photon pileup effect. Finally, owing to the intrinsic simplicity of photon-counting quantum tomography, we believe it is possible to herald and visualize Fock state Wigner functions in real time for quantum information applications.

Funding

National Science Foundation (PHY-1708023, PHY-1521083).

Acknowledgment

The authors thank Rafael Alexander, Carlos Andreas González Arciniegas, Xu Yi, Avi Pe’er, Chun-Hung Chang, Jacob Higgins, Chaitali Joshi, and Xuan Zhu for helpful discussions. We would also like to thank Scott Glancy and Arik Avagyan for valuable comments and feedback during the revision process.

See Supplement 1 for supporting content.

REFERENCES AND NOTES

1. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

2. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

3. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]

4. R. L. Hudson, “When is the Wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974). [CrossRef]

5. S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999). [CrossRef]

6. S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficient classical simulation of continuous variable quantum information processes,” Phys. Rev. Lett. 88, 097904 (2002). [CrossRef]

7. J. S. Bell, “EPR correlations and EPW distributions,” in Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987), Chap. 21, pp. 196–200.

8. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002). [CrossRef]

9. J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for Gaussian quantum error correction,” Phys. Rev. Lett. 102, 120501 (2009). [CrossRef]

10. D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310 (2001). [CrossRef]

11. N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006). [CrossRef]

12. N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014). [CrossRef]

13. S. Ghose and B. C. Sanders, “Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps,” J. Mod. Opt. 54, 855–869 (2007). [CrossRef]

14. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993). [CrossRef]

15. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001). [CrossRef]

16. O. Morin, V. D’Auria, C. Fabre, and J. Laurat, “High-fidelity single-photon source based on a Type II optical parametric oscillator,” Opt. Lett. 37, 3738–3740 (2012). [CrossRef]

17. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009). [CrossRef]

18. S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A 53, 4528–4533 (1996). [CrossRef]

19. K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Phys. Rev. Lett. 76, 4344 (1996). [CrossRef]

20. A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977). [CrossRef]

21. M. G. A. Paris, “Displacement operator by beam splitter,” Phys. Lett. A 217, 78–80 (1996). [CrossRef]

22. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998). [CrossRef]

23. K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969). [CrossRef]

24. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281–4285 (1996). [CrossRef]

25. P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the Wigner function of a one-photon Fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002). [CrossRef]

26. B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science 342, 607–610 (2013). [CrossRef]

27. K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674–677 (1999). [CrossRef]

28. M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444–1446 (2009). [CrossRef]

29. N. Sridhar, R. Shahrokhshahi, A. J. Miller, B. Calkins, T. Gerrits, A. Lita, S. W. Nam, and O. Pfister, “Direct measurement of the Wigner function by photon-number-resolving detection,” J. Opt. Soc. Am. B 31, B34–B40 (2014). [CrossRef]

30. K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative Wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105, 253603 (2010). [CrossRef]

31. See supplemental document.

32. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983). [CrossRef]

33. S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262 (2003). [CrossRef]

34. S. Feng and O. Pfister, “Quantum interference of ultrastable twin optical beams,” Phys. Rev. Lett. 92, 203601 (2004). [CrossRef]

35. S. Feng and O. Pfister, “Realization of an ultrastable twin-beam source for continuous-variable entanglement of bright beams,” Proc. SPIE 5161, 109–115 (2004). [CrossRef]

36. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008). [CrossRef]

	$N_{s}$	$N_{i}$	$N_{c}$
Single-photon events	$54, 320 \pm 90$	$1556 \pm 30$	$903 \pm 17$

State-independent quantum state tomography by photon-number-resolving measurements

Abstract

1. INTRODUCTION

2. EXPERIMENTAL SETUP AND METHODS

A. Setup Description

B. Stabilization Procedure

C. PNR Detection

3. EXPERIMENTAL RESULTS

A. Heralding Ratio

B. Quantum Tomography of a Single-Photon State

4. CONCLUSION

Funding

Acknowledgment

REFERENCES AND NOTES

Supplementary Material (1)

Cited By

Figures (5)

Tables (1)

Equations (10)

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