Quantum-randomized polarization of laser pulses derived from zero-point diamond motion

Douglas J. Little; Ondrej Kitzler; Seyed Abedi; Akael Alias; Alexei Gilchrist; Richard P. Mildren

doi:10.1364/OE.410287

1. Introduction

Randomness is an important resource [1,2], for applications in cryptography [3–5], computer networks [6], simulation [7], and even in certain algorithms (e.g. on primality testing [8,9]). Randomness utilized for these purposes is generally harvested from physical systems [10], then used to seed deterministic pseudo-random number generators [1,11,12]. Random number generators are evolving toward quantum systems (see the review [13] and references therein), due to the in-principle certifiability of the randomness and immunity from undetectable outside manipulation of the physical output [14,15]. Security-critical applications such as digital finance, mobile-based networks, cloud computing and storage, communications and gaming are envisaged to depend in future on cryptographic keys created and distributed using quantum-based protocols.

To date, most sources of quantum-randomness rely on the observation of single particles (typically photons). The prototypical photonic system is the observed reflection or transmission of single photons incident on a 50% reflector; which count as “heads" and “tails" in the quantum equivalent of a coin flip [16]. Photon counting statistics, time-of-flight measurements and spontaneous nonlinear processes have also been demonstrated as sources of extractable randomness [17–19]. Single-photon schemes though tend to be limited by detector speed, detector efficiency, and the notorious “dead time" that follows a detection event.

Generating macroscopic quantum states, defined as large particle-number systems with macroscopically-distinct observable outcomes [20], mitigates some of these shortcomings. Devices consisting of optical amplifying media can be engineered to “clone" photons spontaneously arising from the quantum vacuum, allowing the quantum-randomness to persist even at classical scales with billions of photons. Such states have been generated in inversion lasers [21], optical parametric oscillators [22], and Raman lasers [23,24], where conventional photodetectors extracted randomness from conventional laser pulses.

While observable macroscopic quantum-randomness has been demonstrated in the power and phase of laser pulses, it has not been demonstrated in the polarization state to date. Random polarizations are important in many quantum-information protocols, including some key distribution (QKD) schemes [25,26], where randomness from an external source is required to modulate the polarization of an optical field. Generating randomness in the polarization state directly would remove the need for modulation, potentially enabling new QKD system designs and reducing the cost and complexity of existing QKD schemes. Macroscopic quantum-random laser sources are also novel candidates for public randomness beacons [27,28], used to “grow" cryptographic keys [29].

In this paper, we report a simple, compact, robust scheme that produces Raman-laser pulses with randomly-oriented linear polarizations. We show that the unique, triply-degenerate Raman mode structure present in diamond (and other crystals of $O_h$ symmetry) is critical to the emergence of this behavioral regime, which arises when a $[110]$-propagating pump is polarized along the $[1\bar {1}0]$ crystal axis. At this crystal orientation an isotropic symmetry emerges in the Raman gain, which is then spontaneously broken by the zero-point motion of the diamond lattice, giving rise to randomly-orientated, linearly polarized laser outputs.

These quantum-random polarization states are passively-generated, meaning they do not require any active modulation to produce; they emerge spontaneously as a result of stimulated Raman scattering (SRS). We further demonstrate that polarization measurements can be performed with standard fast photo-detectors, and that these measurements are consistent with an independent, identical distribution, consistent with the hypothesized quantum origin of the randomness. The nature of SRS allows these pulses to be generated on-demand, and at any wavelength within the transmission window of the diamond cavity where a suitable pump source exists. The distribution of the polarizations is also continuously tunable by rotating the pump-polarization, giving users the freedom to optimize the polarization distribution for their chosen application.

2. Theory

SRS in diamond is driven by the 1332 cm$^{-1}$ vibration which comprises three degenerate, $F_{2g}$-symmetric Raman modes [30,31]. Thermal excitation of these vibrations is negligible by virtue of diamond’s high (around 2000 K) Debye temperature [32]. For multiple Raman modes oscillating at the same frequency, the quantum mechanical equations of motion for stimulated Raman scattering under canonical quantization generalize to [33],

(1a)$$\frac{\partial}{\partial \tau} \hat{Q}^\dagger_n(z,\tau) = -\Gamma \hat{Q}^\dagger_n(z,\tau) + \hat{F}^\dagger_n(z,\tau) + i\boldsymbol{\kappa}_n\mathbf{E}_p^*(\tau)\mathbf{\hat{E}}_S(z,\tau)$$

(1b)$$\frac{\partial}{\partial z} \mathbf{\hat{E}}_S(z,\tau) = -iC\sum_n\boldsymbol{\kappa}_n^* \mathbf{E}_p(\tau)\hat{Q}^\dagger_n(z,\tau).$$

Here $\tau = t - z/v$ is the time coordinate in the pump-pulse reference frame, where $v$ is the velocity of the pump and Stokes fields (i.e. no material dispersion). The direction of $z$ relative to the crystal structure is depicted in Fig. 1. $\hat {Q}^\dagger _n$ are the collective atomic operators for the $n$-th vibrational mode, $\hat {F}^\dagger _n$ are the corresponding Langevin operators that represent the zero-point motion of the crystal lattice [33,34], $\mathbf {\hat {E}}_S$ is the (total) Stokes field operator and $\mathbf {E}_p = \mathbf {\hat {e}}_pE_p$ is the (classical) pump field that is assumed to be undepleted, having no dependence on $z$. To include the polarization response of the medium, the material polarizabilities, $\boldsymbol {\kappa }_n$ are represented as tensors in this formulation. Finally, $\Gamma$ is the damping rate and $C = 2\pi N\hbar \omega _Sv/c^2$, where $N$ is the atomic density.

The resultant Stokes field is (see Supplement 1 for derivation)

(2a)$$\begin{aligned} &\hat{\mathbf{E}}_S(z,\tau) = iCdE_p(\tau)\int_0^\tau\int_0^z H(z,z',\tau,\tau')\times\\ &\left[\mathbf{\hat{x}}\tfrac{1}{\sqrt{2}}\left(\hat{F}_2^\dagger(z',\tau') + \hat{F}_3^\dagger(z',\tau')\right) + \mathbf{\hat{y}}\hat{F}_1^\dagger(z',\tau')\right]dz'd\tau', \end{aligned}$$

where $d$ is the magnitude of the polarizability tensor elements in the crystal basis, $\mathbf {\hat {x}}$ and $\mathbf {\hat {y}}$ are unit vectors in the rotated crystal basis, and

(2b)$$H(z,z',\tau,\tau') = e^{-\Gamma(\tau-\tau')}I_0\left(\sqrt{4(z-z')a(\tau,\tau')}\right),$$

where $I_0$ denotes the $0$-th order modified Bessel function of the first kind [33].

Fig. 1. Diagram illustrating the propagation direction $\mathbf {\hat {k}}_p$, and polarization $\mathbf {\hat {e}}_p$ with respect to the face-centered cubic axes of the diamond, and the rotated basis $\{x,y,z\}$. Positions of in-plane carbon atoms (filled circles) are depicted for reference.

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At low pump powers $I_0 = 1$ and $H = \exp (-\Gamma (\tau -\tau '))$ and so Eq. (2a) describes random perturbations that rapidly become uncorrelated with larger $\tau -\tau '$. This is the spontaneous Raman scattering regime, where $Cd$ can be inferred as the linear Raman scattering efficiency. In this regime individual Stokes photons tend to exhibit different (uncorrelated) polarizations.

At higher pump powers, the $I_0$ in Eq. (2b) acts as an amplification term due to its exponential-like character. This is the SRS regime, characterized by the persistent dependence of $H$ on $z-z'$. This dependence signifies that the Stokes photons become strongly correlated as the Stokes field amplifies, which is why Stokes pulses in this regime tend to exhibit a single, coherent polarization state rather than an ensemble of polarization states as in spontaneous Raman scattering.

In general, the polarization of a Stokes pulse aligns to the axis of highest Raman gain [35]. However, if the Raman gain is independent of polarization, then it falls to the random Langevin operators to spontaneously break the isotropic symmetry, leading to a Stokes pulse that is coherent with a randomized polarization orientation. To demonstrate this, the Stokes field is first projected onto an arbitrary vector in the $xy$-plane,

(3)$$\begin{aligned} \hat{\mathbf{e}}_S.\hat{\mathbf{E}}_S&(z,\tau) = iCdE_p(\tau)\int_0^\tau\int_0^z H(z,z',\tau,\tau')\times\\ &\left[\mathbf{\hat{x}}\tfrac{1}{\sqrt{2}}\cos\psi\left(\hat{F}_2^\dagger + \hat{F}_3^\dagger\right) + \mathbf{\hat{y}}\sin\psi \hat{F}_1^\dagger\right]dz'd\tau', \end{aligned}$$

where $\mathbf {\hat {e}}_S = (\cos \psi ,\sin \psi ,0)$ and $\psi$ is the angle with respect to the $x$-axis. The ensemble (pulse-to-pulse) average intensity of this field component is

(4)$$I(\psi) \propto \langle( \hat{\mathbf{e}}_S.\hat{\mathbf{E}}_S)(\hat{\mathbf{e}}_S.\hat{\mathbf{E}}_S^\dagger)\rangle.$$

Expanding the RHS of Eq. (4) yields,

(5)$$\begin{aligned} \propto \int_0^\tau \int_0^z \int_0^\tau \int_0^z H(z,z',\tau,\tau')&H(z,z^{\prime\prime},\tau,\tau^{\prime\prime})\times\\ \bigg[\tfrac{1}{2}\cos^2\psi\left(\langle \hat{F}_2^\dagger \hat{F}_2\rangle + \langle F_2^\dagger F_3 \rangle + \langle \hat{F}_3^\dagger \hat{F}_2 \rangle + \langle \hat{F}_3^\dagger \hat{F}_3 \rangle \right) &+ \sin^2\psi \langle \hat{F}_1^\dagger \hat{F}_1 \rangle \bigg]dz^{\prime\prime}d\tau^{\prime\prime}dz'd\tau', \end{aligned}$$

where the $\hat {F}^\dagger$ operators are functions of $z',\tau '$, and the $\hat {F}$ operators are functions of $z'',\tau ''$. Given that the correlation properties of the Langevin operators are

(6)$$\langle \hat{F}_i^\dagger(z',\tau') \hat{F}_j(z^{\prime\prime},\tau^{\prime\prime})\rangle = \frac{2\Gamma}{\rho}\delta_{ij}\delta(z'-z^{\prime\prime})\delta(\tau'-\tau^{\prime\prime}),$$

where $\delta _{ij}$ is the Kronecker delta and $\rho$ is the linear density of atoms in $z$. Equation (5) becomes

(7)$$I(\psi) \propto \int_0^\tau \int_0^z \frac{2\Gamma}{\rho} H(z,z',\tau,\tau')^2 dz' d\tau',$$

which is independent of $\psi$. Therefore, when passing the Stokes pulses through a linear polarizer, the power statistics are independent of the orientation of the polarizer. This implies that the (linear) output Stokes polarizations in this crystal/pump configuration are random, uniformly-distributed over $\psi$.

3. Experiment

To realize quantum-random polarizations experimentally, a frequency-doubled Q-switched Nd:YAG laser operating at 532 nm was used to pump a diamond in a cavity (Fig. 2). The pump pulses had a duration of 10 ns and a repetition rate of 1 kHz. The diamond was an 8 mm long, CVD-grown ultra-low-birefringence ($\Delta n < 10^{-6}$) diamond (Element 6) oriented so that the pump propagated along the $[110]$ direction, and pump polarization oriented in the $[1\bar {1}0]$ direction.

Fig. 2. Schematic of the system used to realize coherent Stokes pulses with random polarizations. Control of the pump power and polarization was through a linear polarizer in-between two half-wave plates. Stokes polarizations were calculated by measuring at the relative energy of orthogonally-polarized pulse components. One polarization component was time-delayed, allowing measurement of both components on a single detector. The attenuator was inserted to balance the loss between polarization components.

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The cavity input coupler was high-transmitting at the pump wavelength and high-reflecting at the first Stokes wavelength (573 nm). The output coupler was high-reflecting at the pump wavelength and 40% transmitting at the Stokes wavelength. The input coupler was a flat mirror, while the curvature of the output coupler was 4 m, yielding collimated pump and Stokes beams. The cavity itself was 30 mm long, with the diamond positioned in the center. Strictly, the cavity is a deviation from the theory in Section 2., however since the isotropic symmetry of the system is not affected, the theory remains valid for describing the polarization behaviour of the Stokes field. A pump pulse energy of 130 $\mu$J with a mode diameter of 300 $\mu$m was chosen to operate the laser close to threshold, complying with the requirement of an undepeleted pump.

To measure the polarization of the Stokes pulses, a polarizing beam splitter was used to separate them into two orthogonally-polarized components. Pulse traces were measured using a digitized (Picoscope 6404C) single fast photodiode, which provided pulse-polarization measurements using

(8)$$\theta = \begin{cases} \tan^{-1}\left(\left[\frac{E_h}{\eta E_v}\right]^{1/2}\right), & \textrm{if } E_h \leq \eta E_v\\ \frac{\pi}{2} - \tan^{-1}\left(\left[\frac{\eta E_v}{E_h}\right]^{1/2}\right), & \textrm{if } E_h > \eta E_v, \end{cases}$$

where $\theta$ is the polarization orientation with respect to the $[1\bar {1}0]$ direction, $E_h$ and $E_v$ are the integrated energies of the horizontally- and vertically-polarized pulse components, and $\eta$ is a correction factor included to more-precisely balance the loss experienced by each pulse component, in addition to the attenuator (Fig. 2). The value for $\eta$ was determined a priori by orienting the pump polarization to yield Stokes pulses with a known, deterministic orientation (35.3 degrees to the $[1\bar {1}0]$ direction for a $[111]$-oriented pump [35]).

Figure 3 depicts the measured histogram of Stokes polarizations, $p(\theta )$, as the pump polarization is rotated away from the $[1\bar {1}0]$ axis of the diamond. The sensitive dependence of $p(\theta )$ on the pump-polarization is consistent with a symmetry-based mechanism for the observed randomness. The collapse of the isotropic Raman gain symmetry as the pump polarization is rotated away from the $[1\bar {1}0]$ orientation is smooth, enabling the variance of the polarization distribution to be continuously tuned.

Fig. 3. Histogram of $\theta$ values for $10^5$ Stokes pulses with the pump oriented a) $0^\circ$, b) $1^\circ$, c) $2^\circ$, and d) $5^\circ$ degrees from the $[1\bar {1}0]$ diamond axis. The uncertainty in pump orientation was $\pm 0.25^\circ$. Bin widths were set to $1^\circ$.

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Figure 4 shows the same histogram as Fig. 3(a) and compares it with one where the basis had been rotated by $31^\circ$. Departure of the observed distributions from uniformity occurred because $\theta$ could not be measured for pulses with components below the noise floor, or exceeding the maximum range of the digitizer. This occurred most frequently for polarizations with one low-energy component (i.e. $\theta$ close to 0 or $90^\circ$) and was exacerbated by operating the laser close to threshold. The dashed curves in Fig. 4 indicate the expected measured distribution. To calculate this distribution, the shape parameter of the exponentially-distributed total Stokes energies was estimated using a standard maximum-likelihood approach. The fraction of Stokes pulses where both pulse components were within range of the digitizer, $f(\theta )$, was then calculated as a function of polarization orientation. Since the polarization distribution was assumed to be uniform, the expected measured distribution is just $f(\theta )$ normalized over the interval $0-90^\circ$.

Fig. 4. Histogram of measured $\theta$ for $10^5$ Stokes pulses obtained a) in the original basis and b) in the basis rotated by $31^\circ$. Bin widths were set to $1^\circ$. Curves indicate the expected distribution based on the digitizer limits.

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Rotating the measurement basis by $31^\circ$ using a quartz rotator yielded no shift in the center of $p(\theta )$ (Fig. 4(b)), verifying that the observed non-uniformity is not intrinsic to the Stokes pulses, but arises as a consequence $\theta$ not always being measurable. High-fidelity observations of pulse-to-pulse random linear polarizations is a unique measurement problem, and is a topic of future work.

The observed displacement of probability mass toward the center of the $0-90^\circ$ domain compared to the expected measured distribution is consistent with the presence of classical uncertainty in the measurement of $E_h$ and $E_v$. Other second order effects, such as incoherence between longitudinal Stokes modes and the pump polarization not being perfectly linear, may also contribute to this displacement.

Raw sequences of measured Stokes polarizations were found to be compliant with the NIST 800-90B standard [36], demonstrating them to be consistent with an independent, identically-distributed (iid) random output, indicating a lack of correlation between successive observations. The full test log from the NIST 800-90B test suite for this data set is included as supplementary material (see Supplement 1). As an additional iid check, data sets were subject to permutation entropy (PE) tests. PE testing involves encoding the original continuous-variable data and then comparing the Shannon entropy of the distribution with that expected under the iid assumption [37]. In its most primitive form ($D = 2$), step-increases/decreases between two data elements are encoded as 0’s and 1’s, but more sophisticated encodings are possible by comparing more than two elements ($D > 2$) [38]. Here, the tests were repeated for different embedding delays, $T$, which compared every second ($T = 2$), third ($T = 3$), etc. element, up to $T = 1,000$.

The results of PE tests on the data in Fig. 4(a) are shown in Table 1. Under the iid null hypothesis, it is expected on average that 30 tests will have $p$-values less than 0.01 (23 observed), and 3 tests will have $p$-values less than 0.001 (3 observed). Thus the PE test results are consistent with Stokes polarization orientations being iid.

Table 1. Number of $p$-values out of 1,000 below thresholds of 0.1, 0.01 and 0.001, for embedding dimensions, $D$ of 3, 4 and 5, and the expected number assuming the iid null hypothesis is true. Each test for a given $D$ was calculated with a different embedding delay (up to $T = 1,000$)

View Table

4. Discussion

These results are commensurate with the hypothesized quantum origin of the observed randomness. The min-entropy of polarizations measured in the original basis (Fig. 4(a)) was calculated to be 6.7 bits, compared to 2.9 bits when the laser is operating in a deterministic configuration (pump oriented in the $[111]$ direction), thus the quantum entropy rate produced by this system is estimated to be 3.8 bits per pulse, or 3.8 kb/s given our 1 kHz pulse repetition-rate. This is a significant addition to the quantum entropy that Bustard et. al. were able to extract from SRS pulses; 3.4 and 6.0 bits per pulse for the power and phase respectively [23,24]. Fundamentally, the bit rate is limited by the phonon de-phasing time, which dictates how quickly the crystal can “reset" after each pulse, avoiding correlations between subsequent pulses. Bustard et. al. compute a maximum potential random bit-rate of the order of 1 Tb/s in diamond; incorporating random polarization onto such schemes would represent a substantial (around 50%) enhancement of this rate. Bit rates of the order of Mb/s are realistically achievable with current technology, which is principally limited by the availability of appropriate pump lasers.

Laser sources with macroscopic quantum-randomly-polarized outputs are viable substitutes for active polarization-modulation schemes used in quantum-key distribution, reducing system complexity while improving security by avoiding the need for external randomness sources. For communication schemes that utilise public randomness beacons to grow an existing shared private key, this system can be used to distribute entropy in native quantum states, rather than an encoded bit string. This would simplify the entropy-generating architecture and randomizing polarizations, rather than phase, mean that interferometric “read-out" schemes are not required. Such a randomness beacon would need to operate via direct optical interconnects, as opposed to being web-based, however this is advantageous from a security standpoint. Furthermore, the wavelength-flexibility of this system enables outputs at frequencies that are difficult for adversaries to replicate. With a tunable pump laser, frequency-shift keying can be employed for an added layer of security.

5. Conclusion

In conclusion, we have demonstrated that laser pulses with randomly-oriented linear polarizations are produced when the direction and polarization of the pump beam are oriented along the $[110]$ and $[1\bar {1}0]$ axes of diamond respectively. Experimental data were consistent with randomness that is quantum in origin, arising from the diamond’s zero-point motion spontaneously breaking the isotropic Raman gain symmetry; established by the unique, triply-degenerate $F_{2g}$ Raman mode structure of the $O_h$-symmetric lattice. This theoretical mechanism is supported by the observed filtered uniform distribution of polarization measurements, and established consistency of measurement sequences with an iid random output via the NIST 800-90B standard and permutation entropy tests. Certifying the quantum origin of this randomness remains a topic of future work [14].

These macroscopic quantum-random polarization states are relevant to applications where randomness natively resides in the polarization, such as in QKD, in addition to standard applications such as quantum-random number generation, simulation and machine learning where correlation-free randomness is a vital resource.

Funding

Australian Research Council (CE170100009, DP150102054, LP160101039); Air Force Research Laboratory (FA2386-18-1-4117).

Acknowledgments

The authors thank Mike Steel, Alex Sabella and Adam Bennett for valuable discussions.

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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	$D = 3$	$D = 4$	$D = 5$	Expected
$p < 0.1$	100	94	96	100
$p < 0.01$	5	12	6	10
$p < 0.001$	0	2	1	1

Quantum-randomized polarization of laser pulses derived from zero-point diamond motion

Abstract

1. Introduction

2. Theory

3. Experiment

4. Discussion

5. Conclusion

Funding

Acknowledgments

Disclosures

References

Supplementary Material (1)

Cited By

Figures (4)

Tables (1)

Equations (10)

Optics Express

Douglas J. Little	https://orcid.org/0000-0001-7333-9564
Ondrej Kitzler	https://orcid.org/0000-0002-0252-2297
Richard P. Mildren	https://orcid.org/0000-0002-1521-2423