Abstract
Polarization is one of the fundamental properties of light, providing numerous applications in science and technology. While ‘dynamically unpolarized’ singlephoton sources are demanded for various quantum applications, such sources have never been explored. Here we demonstrate dynamically unpolarized singlephoton emission from a single [111]oriented nitrogen vacancy centre in diamond, in which the singlephoton stream is unpolarized, exhibiting intrinsic randomness with vanishing polarization correlation between time adjacent photons. These properties not only allow true random number generation, but may also enable fundamental tests in quantum physics.
Introduction
In classical optics, the polarization state of light is represented by Stokes parameters or the Poincaré sphere^{1}. Pure (linear, circular and elliptical) polarization states correspond to the surface of the sphere while unpolarized light is at the center of the sphere, unbiased relative to any pure polarization state. In quantum optics, polarization of a single photon is expressed by a twolevel system, i.e., a qubit^{2}, and the polarization state is represented by a density matrix or a Stokes vector. The unpolarized state of a single photon corresponds to the completely mixed state of a qubit. This means that the unpolarized state of a single photon is a statistical mixture of any two orthogonal polarization bases. If we measure the polarization of a single photon in a certain basis, the measurement outcome will be either of the two orthogonal polarizations with even probability, regardless of the choice of basis. The unpolarized mixed state of a single photon is crucial to explore fundamental problems for mixed states such as testing errordisturbance relations of measurements^{3,4} and exploring the nature of mixed states themselves^{5,6}, as well as to realize genuine random number generators^{7,8,9}. For these applications, in addition to the statical statistics predicted by the density matrix, we must examine the dynamical statistics of the measurement outcomes. In particular, to ensure true randomness it is essential to show that the polarizations between photons are not correlated with each other. However, to our knowledge, such dynamically unpolarized single photon sources have not been explored to date.
Negatively charged nitrogenvacancy (NV) centres have received broad attention as singlephoton sources over the last few decades^{10,11,12,13}. These properties have allowed NV centres to be used in applications ranging from singlephoton sources for quantum cryptography^{14,15} to tests of the foundations of quantum physics^{16,17,18}. Negatively charged NV centres consist of a substitutional nitrogen atom and an adjacent vacancy with an excess electron in diamond (Fig. 1(a)). When NV centres are nonresonantly excited from the ground state (^{3}A) to the excited state (^{3}E) by a 532 nm laser, the photoluminescence spectrum consists of the zerophonon line at 637 nm and broad phonon sidebands (600–850 nm) and the lifetime of ^{3}E is 11.6 ns^{10}. The excited states ^{3}E are composed of an orbital doublet of E_{x} which emits horizontally polarized photons and E_{y} which emits vertically polarized photons. Electric dipole transitions are allowed for dipoles associated with E_{x}, E_{y} in the plane perpendicular to the NV symmetry axis and perpendicular to each other due to the C_{3v} symmetry of the structure of a NV centre^{19,20} (Fig. 1(a)). In [111]oriented NV centres, which we used in this study, the orientation of these two dipoles are parallel to the diamond surface^{21,22,23,24,25}.
The degeneracy of the orbital doublet ^{3}E is lifted due to coupling to localized vibrational modes (the JahnTeller effect), thereby causing orbital mixing between E_{x} and E_{y} within the lifetime of ^{3}E^{13,24,26,27,28}. The polarization visibility of emitted photons from a [111]oriented NV centre around 4 K is below 60% for both the zerophonon line under nonresonant excitation^{24} and the phononsidebands under resonant excitation^{23}. This is likely due to the static JahnTeller effect^{13}. From around 4 K to 50 K, the polarization visibility decreases to 0% (i.e., unpolarized) exhibiting T^{5} dependence with thermal equilibration of the orbitals^{24}. This T^{5} dependence is clear evidence of the dynamic JahnTeller effect. This temperature dependence implies that single photons emitted from a [111]oriented NV centre are also unpolarized at room temperature under nonresonant excitation (Fig. 1(b)). Furthermore, as in thermal noise generators^{29}, it is reasonably expected that truly random selection takes place when photons are emitted from either the E_{x} or E_{y} orbital, leading to dynamically unpolarized single photon emission without polarization correlation between time adjacent single photons. However, as mentioned above, polarization correlation between single photons, including those emitted from a [111]oriented NV centre, has until now never been examined.
Results and Discussion
Static polarization measurements
First, we evaluated static polarization properties of emitted photons by measuring polarization angular dependencies, and applying quantum state tomography and quantum process tomography. The excitation by a 532 nm laser and the collection of the phonon sideband emission from single NV centres was carried out by a standard confocal microscopy method at room temperature (See Methods). The detection part of the experimental setup is shown in Fig. 1(c). We used type IIa (111) diamond synthesized by a high pressure high temperature (HPHT) method. In order to search for a [111]oriented NV centre among four possible 〈111〉 orientations and investigate polarization states of the photons emitted from the NV centre, we measured detection polarization angular dependencies of the fluorescence intensity in both linear and circular polarization bases. In the linear polarization basis measurements, we measured the polarizer angle dependence of the fluorescence intensity by rotating polarizer1 (purple rectangle shown in Fig. 1(c)) for excitation laser polarizations set to horizontal (H), vertical (V), diagonal (D), antidiagonal (A), right circular (R) and left circular (L). These data are shown in Fig. 2(a). We observed almost no angular dependence, with an average visibility of 6.0% for an excitation power of 1 mW (0.26P_{sat}, where P_{sat} is the saturation excitation power). All measurements conducted in this paper were made for the same [111]oriented NV centre. Furthermore, we confirmed similar unpolarized properties for several other [111]oriented NV centres. In the circular polarization basis measurements, we measured the polarizer angle dependence of the fluorescence intensity by rotating polarizer1 followed by a λ/4 waveplate (green rectangle shown in Fig. 1(c)) at 45° for excitation laser polarizations of H, V, D, A, R and L as shown in Fig. 2(b). Again, we found only minimal angular dependence, with an average visibility of 1.3% for an excitation power of 1 mW.
Quantum state tomography
We reconstructed a density matrix ρ_{exp} for the polarization of the photons emitted from the NV centre from the data shown in Fig. 2(a,b) by a maximum likelihood quantum state tomography method^{30}. The density matrix is shown in Fig. 2(c). The density matrix shows that the polarization state is nearly the completely mixed state which is given by
in either the R, L basis or the H, V basis, where I is the twodimensional identity matrix. The Stokes vector calculated from the density matrix is (S_{1}/S_{0}, S_{2}/S_{0}, S_{3}/S_{0}) = (0.057, −0.003, −0.004). This vector is located near to the origin of the Poincaré sphere, indicating an almost completely unpolarized state. We also estimated the average fidelity . Therefore, the single photons emitted from the measured single [111]oriented NV centre are statistically statically unpolarized.
Quantum process tomography
Next, in order to find a key quantum operator describing the excitationemission process of the NV centre, we evaluated the χ matrix by quantum process tomography^{31,32} (See Methods). The input state of the process is assumed to be the polarization state of the 532 nm excitation laser and the output state of the process is assumed to be single photons emitted from the NV centre. The ideal χ matrix χ_{ideal} for the process, which outputs the completely mixed state ρ_{mixed out} for any input ρ_{in}, is diagonal with all entries equal to 1/4 because of the identity relation ρ_{mixed out} = (Iρ_{in}I + Xρ_{in}X + Yρ_{in}Y + Zρ_{in}Z)/4 = I/2, where X, Y and Z are the Pauli matrices, which form the basis of the χ matrix. χ_{exp}, which is the χ matrix estimated from the data shown in Fig. 2(a,b), is shown in Fig. 2(d). The result demonstrates that the process outputs a nearly completely mixed polarization state of emitted single photons for any input excitation polarization state. We also estimated the process fidelity^{33} = 0.999.
Dynamical polarization correlation measurements
Finally, in order to demonstrate the dynamical nature of the randomness of the emitted photon polarization, we measured the polarization correlation between time adjacent emitted single photons by measuring the modified secondorder correlation function g^{(2)}(τ) using the setup shown in Fig. 1(c). The basic setup was a standard Hanbury BrownTwiss type setup in which the photon stream was divided by the insertion of a nonpolarizing beamsplitter (NPBS), and detected at the outputs by avalanche photodiodes (APD1 and APD2). For the polarization correlation measurements, polarizer2, 3 (thick blue rectangles shown in Fig. 1(c)) were introduced between the NPBS and the APDs. We measured the modified secondorder correlation function at weak excitation (0.20 mW = 0.05P_{sat}), where ij denotes the polarizer settings HH, VV, HV and VH and the first (second) character denotes the orientation of polarizer2 (polarizer3). This means that if the polarizer settings are HV, APD1 detects only H polarization photons and APD2 detects only V polarization photons and g^{(2)}(τ) is measured for these photons. If there is polarization correlation, for example time adjacent photons tend to have same H polarization, HH, the temporal profile of would give a narrower antibunching dip (i.e. a smaller decay time) than that of others, and . Thereby we can investigate dynamic polarization correlation using this method. The data we measured are shown in Fig. 3(a). The data show that the four are identical to each other within experimental accuracy. The average of over the four polarizer settings was found to be . Note that the measured value of g^{(2)}(0) < 0.5 (antibunching) indicates strong evidence of single photons for all polarizer settings. These data indicate that time adjacent emitted single photons have essentially no polarization correlation. We also measured under strong excitation conditions (8.8 mW = 2.3P_{sat}) as shown in Fig. 3(b). In addition to the antibunching dip with , bunching occurred around the antibunching dip due to optical cycling in the threelevel system^{10}. Again, we found that the four were identical to each other within experimental accuracy.
Usual g^{(2)}(τ) data for emitted singe photons from a single NV centre are fitted by the following function which is derived by analysis of the threelevel system,
where τ_{a} is the decay time of antibunching, τ_{b} is the decay time of bunching, c is the coefficient of bunching and ρ is signaltobackground ratio^{10,34}. For our cases, the data were fitted by Eq. (2). Decay times τ_{a} of antibunching for the four polarizer settings (estimated by fitting for the data shown in Fig. 3(a,b)) are shown in Fig. 3(c,d), respectively. These figures show that the decay times for the four polarizer settings are equal within the estimated standard error of the fitted values. These results are consistent with the expectation of uncorrelated decay times and we therefore conclude that the emitted single photons are dynamically unpolarized for weak and strong excitation cases.
Next, we evaluated the polarization correlation between same and different polarizations using a polarization correlation function C(τ) defined by
C(τ) is a quantitative measure of the dynamical polarization correlation. In particular, if measured photons have exactly the same (different) polarization, i.e., HH or VV (HV or VH), C(τ) is equal to +1 (−1) for any τ. On the other hand, if the measured photons have no polarization, C(τ) is equal to 0. C(τ) values for our data under weak and strong excitation condition are shown in Fig. 3(e,f), respectively. The data show that C(τ) is close to 0 for all τ that we measured; C(τ) averaged over all τ is 0.0008, 0.0022 and the standard deviation is 0.0206, 0.0109 for weak and strong excitation, respectively. Almost all of the 90% confidence intervals for C(τ) values (shaded area in Fig. 3(e,f)) include 0. Note that the slight peak and dip seen around τ = 0 in Fig. 3(e,f) respectively are due to fluctuations in the background which become prominent due to the vanishing correlation signal at τ = 0. We have therefore demonstrated essentially no polarization correlation between time adjacent emitted single photons for both weak and strong excitation conditions.
Conclusion
We have presented the first demonstration of unpolarized single photons emitted from a [111]oriented NV centre in diamond in both a statical and dynamical sense. The new ‘dynamically unpolarized’ property, i.e. the property of no polarization correlation between time adjacent single photons, constitutes a new criterion of randomness for unpolarized single photons which is useful for true random number generation^{7,8,9} as well as for tests of fundamental quantum mechanics pertaining to mixed states^{3,4,5,6}.
Methods
Confocal microscopy
A continuouswave 532 nm laser was used for nonresonant excitation of the NV centre. The laser was focused with a 1.4 numerical aperture (NA) oil immersion objective lens, which also collected the photons emitted from a single NV centre. Using bandpass filters, photons emitted from the NV centre ranging from 650 nm to 800 nm were measured. The measurements of the secondorder correlation function for the filtered photons were performed using a modified Hanbury BrownTwiss type setup as explained in the main text.
Quantum process tomography
Quantum process tomography is a method for estimating the χ matrix, which finds quantum operators describing a given quantum process^{31,32}. The process for the one qubit case, having input state ρ_{in} and output state ρ_{out} can be written as
where σ_{m,n} ∈ {I, X, Y, Z}. Here, I is the twodimensional identity matrix and X, Y and Z are the Pauli matrices, which form the basis of the χ matrix.
For our case, accurately speaking, the polarization state of this laser is a classical polarization rather than a quantum state. However the excitationemission process is a linear process which means that a singlephoton in the excitation laser is absorbed by a NV center and subsequently a singlephoton is emitted from the NV center. In this case the polarization state of the absorbed single photon is identical to that of the excitation laser. This allows us to evaluate correlations arising from the process of excitation followed by emission of a single photon. We followed the concrete procedure of quantum process tomography given in ref. 32.
Additional Information
How to cite this article: Abe, N. et al. Dynamically unpolarized singlephoton source in diamond with intrinsic randomness. Sci. Rep. 7, 46722; doi: 10.1038/srep46722 (2017).
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Acknowledgements
This research was supported in part by JSPS KAKENHI (no. 22244035) and MIC SCOPE (no. 121806010). N.A. thanks Tohoku University Division for Interdisciplinary Advanced Research and Education for the financial support. This research has been partly carried out at the Fundamental Technology Center, Research Institute of Electrical Communication, Tohoku University. We used the VESTA ver.3.3.2 software to draw the atomic structure in Fig. 1(a).
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N.A. carried out the measurements, analysis and wrote the paper. Y.M. and M.S. contributed insights on the experimental technique and the interpretation of the results. K.E. supervised all aspects of the project. All authors discussed the results and commented on the manuscript.
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Abe, N., Mitsumori, Y., Sadgrove, M. et al. Dynamically unpolarized singlephoton source in diamond with intrinsic randomness. Sci Rep 7, 46722 (2017). https://doi.org/10.1038/srep46722
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DOI: https://doi.org/10.1038/srep46722
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