Methods and systems for determining photon number statistics of a light source

Abstract

Individual intensity values are determined for each light pulse in a sequence of pulses emitted by a light source, perhaps a pulsed laser diode or light-emitting diode, and incident on a single detect

Description

FIELD Embodiments described herein relate to methods and systems for determining photon number statistics of a light source. BACKGROUND Quantum key distribution (QKD) is a technology for generating perfectly random quantum keys at two remote nodes, which can be used for data encryption to ensure secure communications. The basic operating principle of QKD relies on encoding and measuring quantum states, followed by discussion between the two nodes over an authenticated classical channel, which enables them to detect the presence of an eavesdropper. A key requirement for the security of QKD is that the light source used to transmit the states has a Poissonian photon number distribution. In many cases, the light source is assumed to have a Poissonian photon number distribution without this assumption being checked. However, deviations from this assumption can violate the security of QKD and open up a potential loophole that an eavesdropper could exploit. In cases where the assumption is checked, photon correlation measurements on the light source are performed. The photon correlations are used to bound the photon number statistics of the light source. The higher the order of correlation measured, the tighter these bounds can be made and the more accurate the result. The technology required to perform these checks is cumbersome and complicated to operate. In more detail, photon correlations are usually measured with either threshold single photon detectors or photon number resolving detectors. Photon number resolving detectors are not normally found outside research laboratories, whilst photon correlation measurements with single photon detectors take a long time to perform as they scale as Tn where T is the time period and n is the order of correlation. Such measurements also require the use of n separate detectors. As an example, in order to obtain a reasonable idea of the photon number distribution for a QKD system, it is necessary to measure photon correlations up to order n = 4, meaning that 4 single detectors will be required as well as the associated 4 channel correlation electronics. Using such a set-up, it may take many hours if not days to acquire the data with suitable precision. BRIEF DESCRIPTION OF DRAWINGS Embodiments of the invention will now be described by way of example with reference to the accompanying drawings in which: Figure 1 shows a system according to an embodiment; Figure 2 shows a schematic of pulses as measured on an oscilloscope in an embodiment; Figure 3 shows a matrix of intensity values compiled from measurements of intensity as carried out in an embodiment; Figure 4 shows a comparison of intensity correlation values for different laser bias currents according to an embodiment; and Figure 5 shows a comparison between the upper and lower bounds of a photon number distribution derived according to an embodiment, and a Poissonian photon number distribution. DETAILED DESCRIPTION According to a first embodiment, there is provided a computer-implemented method for determining photon number statistics of a light source, the method comprising: receiving intensity data for a sequence of light pulses emitted by a light source and incident on a single light detector; determining, based on the received intensity data, an individual intensity value for each light pulse in the sequence; determining, based on the intensity values, an average intensity value for the light pulses; determining intensity correlations for two or more orders n, wherein for each order n, determining the intensity correlation comprises comparing (i) an average of values obtained when each individual intensity value is raised to the power n and (ii) the average intensity value raised to the power n; and deriving photon number statistics for the light source based on the intensity correlations for the two or more orders n. The photon number statistics may define the probability that a given pulse of light emitted by the light source will comprise a particular number of photons 5 The method may comprise determining whether the light source is behaving as a Poissonian light source based on a comparison of the intensity correlations for the different orders. Determining whether the light source is behaving as a Poissonian light source may comprise determining whether the intensity correlations for the different 10 orders are each within a threshold distance of one another. The method may comprise determining photon number distribution bounds based on the intensity correlations determined for each order n. Determining the photon number distribution bounds may comprise determining, for each order n, a degree of variation 15 in the values obtained when each individual intensity value is raised to the power n. LA / The method may further comprise determining, based on the derived photon number CM statistics, whether the light source meets a criterion for use in a quantum key T” distribution system. 20 CO According to a second embodiment, there is provided a method for setting the power of a pulsed light source, the method comprising: adjusting the power of the light source to obtain two or more different output powers; 25 for each output power: illuminating a detector with a sequence of light pulses emitted by the light source; determining an individual intensity value for each detected light pulse; determining, based on the intensity values, an average intensity value 30 for the light pulses incident on the detector; determining intensity correlations for two or more orders n, wherein for each order n, determining the intensity correlation comprises comparing (i) an average of values obtained when each individual intensity value is raised to the power n and (ii) the average intensity value raised to the power n; and 35 selecting an output power based on a comparison of the intensity correlations for the different orders. The method may comprise determining, based on the comparison of the intensity correlations for the different orders, an output power at which the light source is behaving as a Poissonian light source. Determining whether the light source is behaving as a Poissonian light source may comprise determining whether the intensity correlations for the different orders are each within a threshold distance of one another other. The output power may be chosen as being the minimum output power for which the light source is determined to be behaving as a Poissonian light source. According to a third embodiment, there is provided a computer readable medium comprising computer executable code that when executed by a computer will cause the computer to carry out a method of the first embodiment. According to a fourth embodiment, there is provided a system comprising: one or more computer processors; and a computer readable medium according to the third embodiment. The system may be a quantum key distribution system. The system may comprise a light source, wherein the light source is a pulsed laser diode or pulsed light emitting diode. Embodiments described herein comprise a device for measuring intensity correlations which can be used to place bounds on the photon number distribution of a light source. An example embodiment is shown in Figure 1. Here, light from the light source 101 is directed on a broadband optical photodiode 103. The photodiode is connected to a broadband oscilloscope 105, which is itself connected to a computer 107. The light source may be one of a number of different types of light source, including a pulsed light emitting diode or pulsed laser diode, for example. In some embodiments, the light source may be one used in a QKD system. For example, the light source may form part of the transmitter in a BB84 QKD system or T12 QKD system. The light source may be that of the transmitter in a differential phase shift QKD system or a Continuous One-Way Protocol QKD system. The light source is configured to output pulses of light with a width at half maximum of t and a repetition period T. The photodiode is selected to have a bandwidth B where 1 B » -. The photodiode converts the optical pulses into electrical pulses, which are in turn passed onto the oscilloscope, this having a similarly high bandwidth as the 150 photodiode. The oscilloscope measures contiguous trains of the optical pulses. The pulse trains are stored in the oscilloscope memory where they are collected by the computer for processing. The computer processes the pulse trains by integrating the areas under each pulse, providing a single intensity value I for each pulse. 155 Figure 1 shows an example of an electrical pulse train having m pulses, as measured on the oscilloscope. The intensity values {1^, I2, / 3,..., Im] are stored as a listfile in the computer memory. Using these intensity values, it is possible to calculate intensity correlations G(n) as follows: 160 In a first step, the set of values {1^ I2, / 3,..., Im} as stored in the computer memory are raised to different powers, up to a value n, where n is the highest order under consideration. By doing so, the computer generates respective sets of values {if, / f, ■■■ Im}, Ui, II I3 - I?n} -, VL I2, I3 - Im}, allowing a matrix of values to be generated as shown in Figure 3. 165 The mean value { / ") in each set of values can be obtained, as well as the standard deviation an for the set. Thus, {In) will define the mean of the values { / ”, 12, I3 ... / «}, whilst an will define the standard deviation of those values. 170 The intensity correlations of order n, G(n) are now calculated using the following formula: where It = {4. / 2i, I3,... Im] denotes the measured intensities of the pulses i.e. column 1 of the matrix shown in Figure 2. Using this approach, it is possible to determine 175 intensity correlations up to arbitrary order n quickly and efficiently. For example, in the case of a 1GHz pulsed light source, a series of 106 pulses can be obtained in microseconds with a single detector only, allowing intensity correlations of up to order n = 10 to be computed with reasonable confidence. In contrast, if using a conventional technique of beamsplitters and single photon detectors, a total of ten detectors would 180 be required and the time required would be of the order of many months when using single photon detectors with unity detector efficiency. When the efficiency of the single photon detectors is at 25%, such as for single photon detectors based on avalanche photodiodes, the time required elongates to hundreds of thousands of years. With knowledge of the intensity correlations G(n), it is possible to determine whether or not the light source is behaving as a Poissonian light source. In the event that the light source is acting as a Poissonian light source, the values G(n) should coincide around 1. Thus, by measuring the values G(n), it is possible to identify an optimal operating power for the light source. By way of example, Figure 4 shows the output of a QKD transmitter light source for different levels of laser bias current. For each current value, the intensity correlations G(n) are determined and plotted for the range n = 1 to 9, together with a least square fit using a simple laser model with only one free fitting parameter. As the laser bias current is increased, the trends in the data are clearly reproduced qualitatively by the model, thereby validating the intensity correlator functions well. It can be seen that as the current increases, the difference in the values G(n) becomes smaller, eventually coinciding around 1 at an operating current of 11.9mA. In some embodiments, having computed the intensity correlations G(n), the values G(n) may be further used to calculate the photon number distribution bounds. Determining the photon number distribution bounds may be desirable in the case of a QKD system, for example, where knowledge of the photon distribution bounds can aid in calculating the secure bit rate for the QKD system. The photon number distribution bounds may be defined by reference to the combined standard deviation for each order G(n). The value for each intensity correlation G(n) can be calculated as: where is the standard deviation in the measured intensities of the pulses { / 1; 12, / 3,..., Im}. Using these values for aG(n), the photon number distribution bounds may be defined as: G& = G(2) - yac(2); G& = G(2) + yoG& G& = G(3) -yac&- G® = G(3) + ycG& 6(10) — G(10) — yOg(io); 6(10) — G(10) + Y^gClo) Here, the underbars and overbars indicate lower and upper bounds respectively. The term y represents the quantile that gives a probability 2e that the true value of 6(n) lies outside the confidence interval [G(n) - yaG(ny, G(n) + assuming each value G(n) has a normal probability distribution. For a given order n, the total probability that the true value of G(n) lies outside the confidence interval will depend on the total number of constraints. In the case of a QKD system, care must be taken to ensure that the overall probability is consistent with the final global security parameter of the system. The global security parameter for a QKD system may have a value of the order of 1010, equating to a key failure probability of less than once in 30,000 years, for example. The quantile y provides a means for bounding confidence in the values determined for 6(n); this acknowledges that there may be an error in the measurement of values G(n) and that the values 6(n) can only be determined to a certain degree of precision. Typically for QKD systems, the quantile should be around 7 to 10, but it could be larger than this. By way of example, taking the starting point as a 7-quantile for a Normal distribution, this will provide a starting value of 2.56 x 10-12 for e. If one considers up to 6(4) for simplicity, there will be 6 constraints from the equations 6(2) <G(2) <G(2); 6(3) <G(3) <G(3); 6(4) <G(4) <G(4), an additional 6 constraints for the singlephoton yield, and a further constraint for the error rate. Taking these together, the total number of constraints will be 13, with a resultant total value of 13 x 2.56 x 10-12 = 3.33 x 10-11 for e. A check can then be made that the final total value e is much smaller than the global security parameter for the QKD system. The values 6(n) may be used as constraints in a linear optimisation for minimising the value of i.e. the probability that a given pulse will comprise a single photon only. In contrast to conventional methods in which it is assumed at the outset that the light source has a Poissonian photon number distribution, embodiments remove this assumption by replacing the Poissonian photon number distributions generated by the Poissonian formula with measured values derived from the intensity correlations G(n). Minimisation is used in order to find the lowest probability that a given pulse will comprise a single photon only, thereby assuming the most pessimistic situation in terms of security. The value of is minimised subject to: CO 1 = Pn n=0 G(l) G(2) 6(3) CO = P n=0 co = Pnn(n - l) / p2 n=Q CO = pnn(n — l)(n — 2) / / z3 n=0 and continuing for 6(4), 6(5) etc. up to 6(10): CO £(10) = ^pnn(n n=0 l)(n - 2)(n - 3)(n - 4)(n - 5)(n - 6)(n - 7)(n - 8)(n - 9) / p10 and where: 6(2) <G(2) <G(2) 6(3) <G(3) <G(3) 6(10) <G(10) <G(10) In the above, the values pn define the probability of emitting n photons per pulse, and p is the mean intensity of the light signal. The value p can be measured with high precision using an optical power meter, or by means of a calibrated power meter, whose output is fed into a feedback loop to guarantee the high stability of the measured value. For many applications, including QKD systems, the light source will be attenuated down to the single photon level, with the value p defining the mean intensity of the attenuated light signal. The above minimisation problem involves simple linear programming that can be carried out by the computer. In order to address the problem of infinite sums, the sums may be truncated at a particular value of n e.g. n = 200, ensuring that all photon number distributions that could be of interest are still covered. Figure 5 shows the result of using a photon flux of 0.4 photons per pulse and y =12 (giving a confidence interval 2e= 1 - 3.6 x 10A-33) As can be seen visually, there is no difference between the upper and lower bounds of the derived photon number distribution and a Poissonian photon number distribution. Whilst the intensity correlations G(n) may be determined prior to attenuating the light signal, it is reasonable to expect the intensity correlations G(n) to still hold for the attenuated light, as the correlations will be independent of light source intensity. Using this assumption, it will be possible to calculate the bounds on the single photon number distribution for intensities at the single photon level. In principle, this can be done for any order n, although in the embodiments described herein, a value of n = 10 is chosen as a practical limit. It can be seen that embodiments described herein enable accurate characterisation of the photon number distribution for light sources. Embodiments are fast to operate; depending on the light source, it may be possible to collect sufficient data to determine the intensity correlations within minutes or less. Moreover, embodiments described herein are simple to operate, requiring a single high bandwidth oscilloscope and photoreceiver only. Embodiments can be used directly with quantum key distribution systems to characterise their light source photon number distribution. By doing so, embodiments can help to improve the implementation security of quantum key distribution systems. In particular, in the event that a QKD system’s light source is characterised, the photon number bounds can also be calculated and inserted into the QKD system’s secure bit rate calculation. It will be appreciated, however, that the method of determining intensity correlations as described herein is not exclusively applicable to QKD systems and embodiments described herein may be applied to any system or circumstance in which it is desirable to confirm that a light source is operating in the Poissonian regime. Implementations of the subject matter and the operations described in this specification can be realized in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the subject matter described in this specification can be realized using one or more computer programs, i.e., one or more modules of computer program instructions, encoded on computer storage medium for execution by, or to control the operation of, data processing apparatus. Alternatively or in addition, the program instructions can be encoded on an artificially generated propagated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal that is generated to encode information for transmission to suitable receiver apparatus for execution by a data processing apparatus. A computer storage medium can be, or be included in, a computer-readable storage device, a computer-readable storage substrate, a random or serial access memory array or device, or a combination of one or more of them. Moreover, while a computer storage medium is not a propagated signal, a computer storage medium can be a source or destination of computer program instructions encoded in an artificially generated propagated signal. The computer storage medium can also be, or be included in, one or more separate physical components or media (e.g., multiple CDs, disks, or other storage devices). While certain embodiments have been described, these embodiments have been presented by way of example only and are not intended to limit the scope of the invention. Indeed, the novel methods, devices and systems described herein may be embodied in a variety of forms; furthermore, various omissions, substitutions and changes in the form of the methods and systems described herein may be made without departing from the spirit of the invention. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the invention.

Claims

1. A computer-implemented method for determining photon number statistics of a light source, the method comprising:5 receiving intensity data for a sequence of light pulses emitted by a light sourceand incident on a single light detector;determining, based on the received intensity data, an individual intensity value for each light pulse in the sequence;determining, based on the intensity values, an average intensity value for the10 light pulses;determining intensity correlations for two or more orders n, wherein for each order n, determining the intensity correlation comprises comparing (i) an average of values obtained when each individual intensity value is raised to the power n and (ii) the average intensity value raised to the power n; and15 deriving photon number statistics for the light source based on the intensitycorrelations for the two or more orders n.LDCXJ* 2. A computer-implemented method according to claim 1, wherein the photon numberT— statistics define the probability that a given pulse of light emitted by the light source will20 comprise a particular number of photons.CO1 3. A computer-implemented method according to claim 1, comprising determiningwhether the light source is behaving as a Poissonian light source based on a comparison of the intensity correlations for the different orders.

254. A computer-implemented method according to claim 3, wherein determining whether the light source is behaving as a Poissonian light source comprises determining whether the intensity correlations for the different orders are each within a threshold distance of one another.

5. A computer-implemented method according to claim 1, further comprising determining photon number distribution bounds based on the intensity correlations determined for each order n.35 6. A computer-implemented method according to claim 5, wherein determining thephoton number distribution bounds comprises determining, for each order n, a degreeof variation in the values obtained when each individual intensity value is raised to the power n.40 7. A computer-implemented method according to claim 1, further comprising:determining, based on the derived photon number statistics, whether the light source meets a criterion for use in a quantum key distribution system.

8. A method for setting the power of a pulsed light source, the method comprising:45 adjusting the power of the light source to obtain two or more different outputpowers;for each output power:illuminating a detector with a sequence of light pulses emitted by the light source;50 determining an individual intensity value for each detected lightpulse;determining, based on the intensity values, an average intensity LO value for the light pulses incident on the detector;CM determining intensity correlations for two or more orders n,T” 55 wherein for each order n, determining the intensity correlation comprisescomparing (i) an average of values obtained when each individualCO intensity value is raised to the power n and (ii) the average intensity1 value raised to the power n; andselecting an output power based on a comparison of the intensity correlations60 for the different orders.

9. A method according to claim 8, comprising determining, based on the comparison of the intensity correlations for the different orders, an output power at which the light source is behaving as a Poissonian light source.6510. A method according to claim 9, wherein determining whether the light source is behaving as a Poissonian light source comprises determining whether the intensity correlations for the different orders are each within a threshold distance of one another other.7011. A method according to claim 9, wherein the output power is chosen as being the minimum output power for which the light source is determined to be behaving as a Poissonian light source.75 12. A computer readable medium comprising computer executable code that whenexecuted by a computer will cause the computer to carry out a method according to claim 1.

13. A system comprising:80 one or more computer processors; anda computer readable medium according to claim 12.

14. A system according to claim 13, wherein the system is a quantum key distribution system.8515. A system according to claim 13, further comprising a light source, wherein the light source is a pulsed laser diode or pulsed light emitting diode.LDCM

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