The QKD scheme described in chapter 1, encodes quantum information on photon polarisations using two non-orthogonal bases. An alternative encoding approach uses the phase difference between successive photon pulses interfering at a beam splitter at the end of a Mach-Zehnder interferometer. This method of encoding information is known as Differential Phase Shift (DPS) QKD35.
One of the key components in a DPS-QKD systems is a Mach-Zehnder interferometer. To illustrate the operation principles of DPS-QKD, it is important to explain how a Mach-Zehnder interferometer work for large photon number pulses. In this type of interferometers, the input laser beam is split in a 50:50 beam splitter, resulting in two beams, one can be called the modulated beam (marked yellow) and the other can be called the reference beam (marked red), as shown in Figure 2.12.
Figure 2.12: A block diagram of a typical DPS-QKD system with two pulses, coloured red and yellow, taking the different path in the Mach-Zehnder interferometers. The laser is input from the bottom left as shown by the blue arrow and split by a beam splitter. One arm contains a phase modulator which causes phase shifting one arm relative to the other. Constructive and destructive phase shifting can be recorded by detectors 0 and 1 respectively.
Letβs define π1 and π2 as the total path length for the modulated and reference beams respectively. When the light passes through a glass beam splitter it exhibits a phase shift equivalent to 2ππ‘/π, where π‘ is the thickness of the beam splitter. For detector 0, the modulated beam takes a phase shift of π from the
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first beam splitter, a π phase shift from the top mirror, a 2ππ‘/π from the last beam splitter, and the path length of 2ππ1/π, before reaching detector 0 . This gives a total of
2π + 2π (π1+ π‘
π )
The reference beam takes a phase shift of 2ππ‘/π as it goes through the first beam splitter, a π phase shift as it gets reflected by the bottom mirror and another π phase shift as it gets reflected by the second beam splitter toward detector 0. Hence the phase difference between the two beams π, which is caused by the phase modulator is:
2π + 2π (π1+ π‘ π ) β 2π β 2π ( π2+ π‘ π ) = π 2π (π1β π2 π ) = π
In a similar fashion, the phase difference between the two arms for detector 1 is:
2π + 2π (π1+ 2π‘ π ) β π β 2π ( π2+ 2π‘ π ) = π π + 2π (π1β π2 π ) = π + π
Hence, it is clear to note that when the phase modulator causes a zero phase shift, there is a constructive interference on the path to detector 0, and a destructive interference to detector 1. By varying π, this condition can be changed, hence changing the probability of arrival at either detector.
A simplified example of a DPS-QKD system is shown inFigure 2., where Alice generates a train of single photon pulses and passes it through a Mach-Zehnder interferometer. In the interferometer, Alice splits the signal through a beam splitter, and applies a random phase modulation of 0, Ο2, 3Ο2, or Ο on one arm and recombines the two arms at the junction of a second beam splitter. She later sends the signal to Bob through a quantum channel such as a fibre optic or free space. At Bobβs side, he performs a similar Mach-Zehnder interferometry process, where he splits the signal into two arms, performs a random 0
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or Ο2 phase-shift through one arm and recombines the two arms in a beam splitter. There are three detection time slots at which Bob can measure. The first is when two consecutive pulses take the long paths on Aliceβs and Bobβs interferometers. The second is when the two pulses both take the short paths of the interferometers. Both of these events arenβt of use in this protocol. The only used events are when a pulse takes the long and short arm of Alice and Bob sides while the other takes the short and long arms of Alice and Bob respectively, as shown in Figure 2.. If the interferometers on both sides are matched, as when Alice and Bob use a similar basis, a 0 or Ο phase shift takes place between two successive photon pulses, causing trigger events in either detector 0 or 1, as shown in Figure 2.. In a real implementation, each single photon pulse is accompanied by one proceeding bright pulse for timing purposes. The bright photon pulse will appear as two pulses on Bobβs detectors as it goes through the two long and two short paths in Alice and Bob sides.
Figure 2.13: A block diagram of a typical DPS-QKD system with two pulses, coloured red and yellow, taking the different path in the Mach-Zehnder interferometers.
After receiving the qubits, Bob announces publicly the time slots at which a photon was detected by one of his detectors, but does not reveal which detector detected it. From Aliceβs modulation data, she can know which detector in Bobβs end recorded the event. By designating detection events recorded by detector 0 and 1 as key bit values 0 and 1, respectively, they can share an identical bit string.
In an ideal system where there are no errors in the detection events, the sifted key created is unconditionally secure. However, in practice, as discussed in chapter 1, all QKD systems have a baseline error rate, hence error correction and privacy amplification procedures are essential in a practical DPS-QKD system.
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Figure 2.14: An example of a click event observed on detector 0 due to a 0 degrees phase shift representing a 0 bit. This occurs when two consecutive pulses take opposite and alternate arms on Alice and Bobβs ends. The figure also shows timing pulses measured by the detector.
After error correction, when the error threshold is low enough to allow carrying on with the protocol, to nullify any information that may have been obtained by an eavesdropper, Eve, privacy amplification is performed36. This allows making Eveβs information of the final shared key extremely small. In
privacy amplification, Alice and Bob apply hash algorithms that uses the error free shared key as an input and outputs a new shorter key.
Bright timing pulses
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