Time-bin encoding Along Satellite-Ground Channels

Giuseppe Vallone,1 Daniele Dequal,1 Marco Tomasin,1 Francesco Vedovato,1

Matteo Schiavon,1 Vincenza Luceri,2 Giuseppe Bianco,3 and Paolo Villoresi11Dipartimento di Ingegneria dell’Informazione, Universita degli Studi di Padova, Padova, Italy.

2e-GEOS spa, Matera, Italy3Matera Laser Ranging Observatory, Agenzia Spaziale Italiana, Matera, Italy

Introduction - Time-bin encoding is an extensively usedtechnique to encode a qubit in Quantum Key Distribution(QKD) along optical fibers [1–4] and more in general forQuantum Information applications, such as fundamental testsof Quantum Mechanics [5–8], and for increasing the dimen-sion of the Hilbert space in which information can be en-coded [9]. Time-bin encoding exploits single photons passingthrough an unbalanced Mach-Zender interferometer (MZ).The difference in path length must be longer than the coher-ence length of the photon to avoid first order interference andto have the possibility to distinguish the taken path. If thephoton takes the short path, it is said to be in the state |S〉; ifit takes the long path, it is said to be in the state |L〉. If thephoton has a non-zero probability to take either path, then itis in a coherent superposition of the two states:

|ψ〉 = α|S〉+ β|L〉 . (1)

In typical QKD experiments, the qubits are encoded in thephase between the two paths with fixed amplitudes, typicallyat 50%, obtaining

|ψ〉 =|S〉+ eiφ|L〉√

2. (2)

Measurement in the |S〉, |L〉 basis is done by measuring thetime of arrival of the photon. Measurement in other basescan be achieved by using a second MZ interferometer beforethe photon detection, with the same unbalancement of the MZused on the generation stage.

Despite its success in fibers QKD (in particular in the“plug&play” systems [1]), time-bin encoding was never im-plemented in long-distance free-space QKD: indeed, turbu-lence effects on the wavefront are believed to make interfer-ence hardly measurable. Here we demonstrate that time-bininterference at the single photon level can be observed alongfree-space channels and in particular along satellite-groundchannels. To this purpose, we used a scheme similar to the“plug&play” systems: a coherent superposition between twowavepackets is generated on ground, sent on space and re-flected by a rapidly moving satellite at very large distancewith a total path length up to 5000 km. The beam returningon ground is at the single photon level and we measured theinterference between the two time-bins. We will demonstratethat the varying relative velocity of the satellite with respect tothe ground introduces a modulation in the interference patternwhich can be predicted by special relativistic calculations, asexplained below. Our results attest the viability of time-binencoding for Quantum Communications in Space.

PPLN

PMT

0 100 200 300 400 500Time (s)

-6

-4

-2

0

2

4

6

Ra

dia

l Ve

locity

(km

/s)

Long

Short

FIG. 1. Scheme of the experiment and satellite radial velocity. Inthe top panel we show the measured radial velocity of the Beacon-Csatellite ranging from−6 km/s to +6 km/s as a function of time dur-ing a single passage. In the bottom panel we show the unbalancedMZI with the two 4f -systems used for the generation of quantum su-perposition and the measurement of the interference. Light and darkgreen lines respectively represent the beams outgoing to and ingoingfrom the telescope. In the inset, we show the expected detection pat-tern: the number of counts Nc in the central peak varies according tothe kinematic phase ϕ imposed by the satellite. Right photo showsMLRO with the laser ranging beam and the Beacon-C satellite (notto scale). The phase ϕ(t) depends on the satellite radial velocity asdescribed in the text.

Description of the experiment - In our scheme, a co-herent state |Ψout〉 in two temporal modes is generated atthe ground station with an unbalanced Mach-Zehnder inter-ferometer (MZI), sketched in the lower panel of Fig. 1.The ground station is the Matera Laser Ranging Observa-tory (MLRO) of the Italian Space Agency in Matera, Italy,that is equipped with a 1.5 m telescope designed for precisesatellite tracking and which acted as ground quantum-hub forthe first demonstrations of Space Quantum Communication(QC) [10, 11]. The delay ∆t ' 3.4 ns between the twotime-bins corresponds to a length difference between the twoarms of ` = c∆t ' 1 m (c is the speed of light in vacuum)and it is much longer than the coherence time τc ≈ 83 psof each wavepacket. Using the MLRO telescope, the state|Ψout〉 is directed towards satellites in a Low Earth Orbit(LEO) equipped with cube-corner retroreflectors (CCR): weselected three LEO satellites – Beacon-C, Stella and Ajisai –for our experiment. Thanks to the CCR properties, the stateis automatically redirected toward the ground station, where

2

it is injected into the same MZI used in the uplink. Afterthe reflection by the satellite and the downlink attenuation,the state collected by the telescope has a mean photon num-ber µ < 2 · 10−3 and it can be written as a time-bin qubit|Ψr〉 = (1/

√2)(|S〉 − eiϕ(t)|L〉).

The relative phase ϕ(t) is determined by the satellite in-stantaneous radial velocity with respect to ground, vr(t). In-deed, at a given instant t, the satellite motion determines ashift δr(t) of the reflector radial position, during the separa-tion ∆t between the two wavepackets. This shift can be esti-mated at the first order as δr(t) ≈ vr(t)∆t, and its value mayreach a few tens of micrometers for the satellites here used.Therefore, the satellite motion imposes during reflection theadditional kinematic phase ϕ(t) ≈ 2δr(t)(2π/λ) between thewavepackets |L〉 and |S〉, where λ is the pulse wavelength invacuum. By the relativistic calculation detailed in [12], theexact phase imposed by the satellite motion can be evaluatedas

ϕ(t) =2β(t)

1 + β(t)

2πc

λ∆t (3)

with β(t) = vr(t)c . We note that the first order approximation

of eq. (3) gives the phase ϕ(t) ≈ 4πvr(t)∆t/λ above de-scribed. For a typical LEO satellite passage the radial velocitymay range from −6 to +6 km/s (as shown in in the top panelof Fig. 1 for the selected passage of the Beacon-C satellite),corresponding to a phase range from −400rad to +400rad.

A single MZI for state generation and detection intrinsi-cally ensures the same unbalance of the arms and avoids ac-tive stabilization, necessary otherwise with two independentinterferometers. Two 4f -systems realizing an optical relayequal to the arm length difference were placed in the longarm of the MZI. The relay is required to match the interfer-ing beam wavefronts that are distorted by the passage throughatmospheric turbulence: otherwise, the latter may cause dis-tinguishability between the two paths, washing out the inter-ference.

The MZI at the receiver is able to reveal the interference be-tween the two returning wavepackets. At the MZI outputs weexpect detection times that follow the well known three-peakprofile (see Fig. 1): the first peak represents the pulse |S〉 tak-ing again the short arm, while the third represents the delayedpulse |L〉 taking again the long arm. In the central peak we ex-pect indistinguishably between two alternative possibilities:the |S〉 pulse taking the long arm and the |L〉 pulse taking theshort arm in the path along the MZI toward the detector. Thesignature of interference at the single photon level is then ob-tained when the counts in the central peak differ from the sumof the counts registered in the lateral peaks. To measure theinterference we used a single photon detector (PMT) placed atthe available port of the MZI, as shown in Fig. 1. For a mov-ing retroreflector, the probability Pc of detecting the photon inthe central peak is given by

Pc(t) =1− V(t) cosϕ(t)

2, V(t) = e

−(

∆tτc

√2πβ(t)

1+β(t)

)2

. (4)

-5 -4 -3 -2 -1 0 1 2 3 4 5

∆ = tmeas − tref (ns)

0

200

400

600

800 C No interference

0

20

40

60

Counts

B Destructive

interference

0

20

40

60 A Constructive

interference

FIG. 2. Constructive and destructive single photon interference(Beacon-C satellite, 11.07.2015 h 1.33 CEST). (A) Histogram of sin-gle photon detections as a function of time ∆ = tmeas − tref real-ized by selecting only the returns characterized by ϕ mod 2π ∈[4π/5, 6π/5] that lead to constructive interference. Solid line showsthe fit. (B) Histogram of single photon detections realized by select-ing only the returns characterized by ϕ mod 2π ∈ [−π/5, π/5].(C) Histogram of single photon detections without any selection onthe phase. As expected, interference is completed washed out. In allpanels, dotted red lines represent the expected counts in case of nointerference.

We note that for a retroreflector at rest we expect Pc = 0.The above relation is obtained by time-of-flight calculationstogether with the Doppler effect that changes the angular fre-quency of the reflected pulses from ω0 ≡ 2πc

λ to 1−β1+βω0

(see [12]). The theoretical visibility V(t) is approximately 1since the β factor is upper bounded by 3 · 10−5 in all the ex-perimental studied cases, while ∆t/τc is of the order of 102.

Experimental results - The value of ϕ(t) originating fromthe satellite motion can be precisely predicted on the base ofthe sequence of measurements of the instantaneous distanceof the satellite, or range r, which is realized in parallel. Therange is measured by a strong Satellite Laser Ranging (SLR)signal at 10 Hz and energy per pulse of 100 mJ. Then, bymeasuring the range every 100 ms, the instantaneous satellitevelocity relative to the ground station vr(t) can be estimated,from which ϕ(t) can be derived by Eq. (3). Since vr(t) iscontinuously changing along the orbit as shown in Fig. 1 thevalue of ϕ(t) is varying accordingly.

By the synchronization technique described in detailin [12], we determined of the expected (tref ) and the mea-sured (tmeas) instant of arrival of each photon. In this way, thehistogram of the detections as a function of the temporal dif-ference ∆ = tmeas − tref can be obtained. In Fig. 2 we showsuch histograms corresponding to constructive and destructiveinterference in the case of satellite Beacon-C.

In particular, for the constructive interference, Fig. 2A,we selected the detections corresponding to ϕ (mod 2π) ∈[4π/5, 6π/5]. For the destructive interference, Fig. 2B, weselected a kinematic phase ϕ (mod 2π) ∈ [−π/5, π/5]. Thedetections in the central peak are respectively higher or lower

3

0 π/2 π 3π/2 2π

ϕ (mod 2π)

0

0.2

0.4

0.6

0.8

1

Pc

Ajisai

0

0.2

0.4

0.6

0.8

1

Pc

Stella

0

0.2

0.4

0.6

0.8

1

Pc

Beacon-C

FIG. 3. Experimental interference pattern. Experimental prob-abilities P (exp)

c as a function of the kinematic phase measured forthree different satellites. By fitting the data we estimate the visibili-ties Vexp = 67 ± 11% for Beacon-C, Vexp = 53 ± 13% for Stellaand Vexp = 38 ± 4% for Ajisai. Dashed lines correspond to thetheoretical value of Pc predicted by eq. (4).

than the sum of the two lateral peaks in the two cases. Wenote that the peak width is determined by the detector timingjitter which has standard deviation σ = 0.5 ns. These twohistograms clearly show the interference effect in the centralpeak. On the contrary, Fig. 2C is obtained by taking all thedata without any selection on ϕ. In this case, the interferenceis completely washed out. These results show that, in order toprove the interference effect, it is crucial to correctly predictthe kinematic phase ϕ imposed by the satellite motion.

By using the data of Fig. 3, we experimentally evaluate theprobability P (exp)

c as the ratio of the detections associated thecentral peak Nc to twice the sum N` of the detections associ-ated to the side peaks, namely

P (exp)c =

Nc2N`

. (5)

The values P (exp)c = 0.87±0.10 and P (exp)

c = 0.20±0.03 areobtained for constructive and destructive interference respec-tively. The values deviates with clear statistical evidence from0.5, which is the expected value in the case of no interference.

A more clear evidence of the role of ϕ(t) can be demon-strated by evaluating the experimental probabilities P (exp)

c asa function of ϕ. Fig. 3 shows P (exp)

c for ten different values ofthe kinematic phase ϕ and for the three different satellites. Byfitting the data by P (exp)

c = 12 (1 − Vexp cosϕ), we estimated

the experimental visibilities Vexp = 67± 11% for Beacon-C,Vexp = 53 ± 13% for Stella and Vexp = 38 ± 4% for Aji-sai. The data were collected at the following satellite distanceranges: from 1600 to 2500 km (Ajisai, 12.07.2015, h 3.42CEST), from 1100 to 1500 km (Stella, 12.07.2015, h 3.08CEST) and from 1200 to 1500 km (Beacon-C, 11.07.2015,h 1.33 CEST), giving two-way channel lengths ranging from2200 up to 5000 km. The interference patterns in Fig. 4

clearly demonstrate that the quantum superposition is pre-served along these thousand kilometer scale channels withrapidly moving retroreflectors. We attribute the different visi-bilities to residual vibrations of the unbalanced MZI betweenthe upgoing and downgoing pulses.

Conclusions - Interference between two temporal modesof single photons was observed along a path that includes arapidly moving retroreflector on a satellite and with lengthup to 5000 km. We have experimentally demonstrated thatthe relative motion of the satellite with respect to the groundinduces a varying phase that modulates the interference pat-tern. This varying phase is not present in the case of fixedterminals. The effect resulted from the measured interferencepattern during passages of three satellites, Beacon-C, Stellaand Ajisai, having different relative velocities and distancesfrom MLRO ground station. We have demonstrated that at-mospheric turbulence is not detrimental for time-bin encodingin long distance free-space propagation. Indeed, the two tem-poral modes separated by a few nanoseconds are identicallydistorted by the propagation in turbulent air, whose dynamicsis in the millisecond scale [13]: the key point here is the care-ful matching of the interfering wavefronts in the two arms, asshown in Fig. 1. The results here presented attest the feasi-bility of time-bin/phase encoding technique in the context ofSpace Quantum Communications.

Furthermore, the measurement of interference in Space is amilestone to investigate the interplay of Quantum Theory withGravitation. As recently proposed by M. Zych et al. [14, 15],single photon interference in Space is a witness of generalrelativistic effects: gravitational phase shift between a su-perposition of two photon wavepackets could be highlightedin the context of large distance quantum optics experiment.The interference patterns measured in the present experimentdemonstrate that a coherent superposition between two tem-poral modes holds in the photon propagation and its interfer-ence can be indeed observed over very long channels involv-ing moving terminals at fast relative velocity. We believe thatthe results here described attest the viability of the use of pho-ton temporal modes also for fundamental tests of Physics.

[1] A. Muller, et al., Applied Physics Letters 70, 793 (1997).[2] J. Brendel, et al., Phys. Rev. Lett. 82, 2594 (1999).[3] N. Gisin, et al., Reviews of Modern Physics 74, 145 (2002).[4] V. Scarani, et al., Rev. Mod. Phys. 81, 1301 (2009).[5] J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).[6] W. Tittel, et al., Physical Review Letters 81, 3563 (1998).[7] G. Lima, et al., Phys. Rev. A 81, 2 (2010).[8] G. Carvacho, et al., Phys. Rev. Lett. 115, 030503 (2015).[9] I. Ali-Khan, et al., Physical Review Letters 98, 060503 (2007).

[10] P. Villoresi, et al., New J. Phys. 10, 033038 (2008), 0803.1871.[11] G. Vallone, et al., Physical Review Letters 115, 040502 (2015).[12] G. Vallone, et al., [arxiv.org:1509.07855] (2015), 1509.07855.[13] I. Capraro, et al., Phys. Rev. Lett. 109, 200502 (2012).[14] M. Zych, et al., Nature Communications 2, 505 (2011).[15] D. Rideout, et al., Classical Quant. Grav. 29, 224011 (2012).