Time-bin configuration

The time-bin configuration (English: time-bin-implementation, TBI) is a special, experimental setup for the implementation of the time-bin coding .

General

A possible setup of the Time-Bin configuration with two fiber optic MZI. At the beginning of the interferometer chain there is a laser diode (LD), at the end there are two detectors of the avalanche photodiode type (APD).

The time-bin coding is a way to make a qubit on a photon to encode . The simplest setup for performing the encoding consists of a Mach-Zehnder interferometer (MZI) through which a single photon is passed. If there is a phase-shifting element within the interferometer , the emerging photon has an encoded qubit of the form: ${\displaystyle \varphi }$

${\displaystyle |\psi \rangle ={|0\rangle +e^{-i\cdot \varphi }\cdot |1\rangle \over {\sqrt {2}}}}$

A measurement within the bases now available takes place via the arrival time of the photon. Measurement with other, additional bases can be achieved by letting the photon pass through a second MZI. If both interferometers are identical (except ), one speaks of a time-bin configuration. ${\displaystyle \left\{{\left|{\left.0\right\rangle ,\left|{\left.1\right\rangle }\right.}\right.}\right\}}$ ${\displaystyle \varphi }$

The TBI is the quantum mechanical extension of the delay line interferometer and thus an optical DPSK converter.

requirements

The photon radiated into the interferometer has the equation of state:

${\displaystyle |\psi \rangle =A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle }$

It is specified here that the state represents the case “photon along the short path” and the state represents the case “photon along the long path”. The complex-valued power at the detector can be calculated using: ${\displaystyle |0\rangle }$ ${\displaystyle |1\rangle }$

${\displaystyle P(\psi _{\bullet })=|\langle \psi ||\psi _{\bullet }\rangle |^{2}=|Q(\psi _{\bullet })|^{2}}$

With:

${\displaystyle A_{1}^{2}+A_{2}^{2}=1}$

The complete system response of the TBI can be calculated from the real and immediate components of . ${\displaystyle \Re }$ ${\displaystyle \Im }$ ${\displaystyle P(\psi _{\bullet })}$

Conventions

Detector signals derived from the conventions for from path 1, from path 4 and the central impulse from paths 2 and 3. Depending on and , there is constructive (red) or destructive (blue) interference for , but not for, the satellites. ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$ ${\displaystyle Z}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \varphi _{2}}$ ${\displaystyle Z}$

The photon has four possibilities (paths) to get to one of the two detectors at the end of the TBI:

• Path 1, index K: via the short arms of the two MZI

• Path 2, index M: over the short arm of the first MZI and over the long arm of the second MZI

• Path 3, index M: over the long arm of the first MZI and over the short arm of the second MZI

• Path 4, index L: over the long arms of the two MZI

Since path 2 and 3 are identical (except ), these paths have the same index. ${\displaystyle \varphi }$

The signal at the detector consists of a leading satellite , the central central pulse and the lagging second satellite . ${\displaystyle S_{1}}$ ${\displaystyle Z}$ ${\displaystyle S_{2}}$

State functions

The following status functions can be derived from the conventions:

• General

${\displaystyle Q(\psi _{\bullet })=(A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )\cdot \psi _{\bullet }}$

• Short path - S ₁

${\displaystyle Q(\psi _{K})=(A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot |0\rangle )}$

• Middle path - Z

${\displaystyle Q(\psi _{M})=2\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot e^{-j\cdot \varphi _{1}}\cdot |1\rangle )\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot e^{-j\cdot \varphi _{2}}\cdot |1\rangle )}$

• Long Path - S ₂

${\displaystyle Q(\psi _{L})=(A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )\cdot (A_{1}\cdot e^{-j\cdot (\varphi _{1}+\varphi _{2})}\cdot |1\rangle +A_{2}\cdot e^{-j\cdot (\varphi _{1}+\varphi _{2})}\cdot |1\rangle )}$

Configurations

Single particle interference

The photon interferes with itself, represented by:

${\displaystyle Q(\psi _{\gamma })(A_{1},A_{2},|0\rangle ,|1\rangle )=(A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )}$

${\displaystyle \Rightarrow }$

${\displaystyle {\begin{matrix}\Re (Q(\psi _{\gamma }))=1\quad &\quad \Im (Q(\psi _{\gamma }))=0\end{matrix}}}$

${\displaystyle \Rightarrow }$

${\displaystyle P(\psi _{\gamma })=1}$

Principle case

• S ₁

${\displaystyle P(\psi _{K})(A_{1},A_{2},|0\rangle ,|1\rangle ,\varphi _{1},\varphi _{2})=(A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )^{2}\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot |0\rangle )^{2}}$

• S ₂

${\displaystyle P(\psi _{L})(A_{1},A_{2},|0\rangle ,|1\rangle ,\varphi _{1},\varphi _{2})=(A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )^{2}+(A_{1}\cdot |1\rangle +A_{2}\cdot |1\rangle )^{2}}$

• Z with two EOM

${\displaystyle P(\psi _{M})(A_{1},A_{2},|0\rangle ,|1\rangle ,\varphi _{1},\varphi _{2})=4\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )^{2}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =A_{1}\cdot A_{1}\cdot |0\rangle \cdot |0\rangle +A_{1}\cdot A_{2}\cdot |0\rangle \cdot |1\rangle \cdot (\cos \varphi _{1}+\cos \varphi _{2})+A_{2}\cdot A_{2}\cdot |1\rangle \cdot |1\rangle \cdot \cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =A_{1}\cdot A_{2}\cdot |0\rangle \cdot |1\rangle \cdot (\sin \varphi _{1}+\sin \varphi _{2})+A_{2}\cdot A_{2}\cdot |1\rangle \cdot |1\rangle \cdot \sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(A_{1},A_{2},|0\rangle ,|1\rangle ,\varphi _{1},\varphi _{2}=0)=4\cdot (A_{1}\cdot |0\rangle +A_{2}\cdot |1\rangle )^{4}\cdot (A_{1}\cdot A_{1}\cdot |0\rangle \cdot |0\rangle +2\cdot A_{1}\cdot A_{2}\cdot |0\rangle \cdot |1\rangle \cdot \cos \varphi _{1}+A_{2}\cdot A_{2}\cdot |1\rangle \cdot |1\rangle )}$

General time bin configuration

• S ₁

${\displaystyle P(\psi _{K})(A_{1},A_{2},\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot {(A_{1}+A_{2})^{4}}}$

• S ₂

${\displaystyle P(\psi _{L})(A_{1},A_{2},\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot {(A_{1}+A_{2})^{4}}}$

• Z with two EOM

${\displaystyle P(\psi _{M})(A_{1},A_{2},\varphi _{1},\varphi _{2})={\frac {1}{2}}\cdot (A_{1}\cdot A_{2})^{2}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =A_{1}\cdot A_{1}+A_{1}\cdot A_{2}\cdot (\cos \varphi _{1}+\cos \varphi _{2})+A_{2}\cdot A_{2}\cdot \cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =A_{1}\cdot A_{2}\cdot (\sin \varphi _{1}+\sin \varphi _{2})+A_{2}\cdot A_{2}\cdot \sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(A_{1},A_{2},\varphi _{1},\varphi _{2}=0)={\frac {1}{2}}\cdot (A_{1}+A_{2})^{4}\cdot (A_{1}\cdot A_{1}+2\cdot A_{1}\cdot A_{2}\cdot \cos \varphi _{1}+A_{2}\cdot A_{2})}$

The power of the central impulse of an ideal TBI as a function of and . ${\displaystyle P}$ ${\displaystyle Z}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \varphi _{2}}$

Ideal time bin configuration

• S ₁

${\displaystyle P(\psi _{K})(\varphi _{1},\varphi _{2})=1}$

• S ₂

${\displaystyle P(\psi _{L})(\varphi _{1},\varphi _{2})=1}$

• Z with two EOM

${\displaystyle P(\psi _{M})(\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =1+\cos \varphi _{1}+\cos \varphi _{2}+\cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =\sin \varphi _{1}+\sin \varphi _{2}+\sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(\varphi _{1},\varphi _{2}=0)=4\cdot \cos ^{2}{\frac {\varphi _{1}}{2}}}$

Real time bin configuration with the imbalance ${\displaystyle \eta }$

In the practical operation of an experimental setup for the TBE in time-bin configuration, asymmetries at the beam splitters (imbalance ) and / or attenuation at the necessary splices (imperfection ) occur. These losses lead to the calculation basis of a real TBI. ${\displaystyle 0\leq \eta \leq 2}$ ${\displaystyle 0\leq \chi \leq 2}$

• S ₁

${\displaystyle P(\psi _{K})(\eta ,\varphi _{1},\varphi _{2})=1}$

• S ₂

${\displaystyle P(\psi _{L})(\eta ,\varphi _{1},\varphi _{2})=1}$

• Z with two EOM

${\displaystyle P(\psi _{M})(\eta ,\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =\eta ^{2}+\eta \cdot (2-\eta )\cdot (\cos \varphi _{1}+\cos \varphi _{2})+(2-\eta )^{2}\cdot \cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =\eta \cdot (2-\eta )\cdot (\sin \varphi _{1}+\sin \varphi _{2})+(2-\eta )^{2}\cdot \sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(\eta ,\varphi _{1},\varphi _{2}=0)=\eta ^{2}+2\cdot \eta \cdot (2-\eta )\cdot \cos \varphi _{1}+(2-\eta )^{2}}$

Real time bin configuration with the imperfection ${\displaystyle \chi }$

• S ₁

${\displaystyle P(\psi _{K})(\chi ,\varphi _{1},\varphi _{2})=\chi ^{2}}$

• S ₂

${\displaystyle P(\psi _{L})(\chi ,\varphi _{1},\varphi _{2})=(2-\chi )^{2}}$

• Z with two EOM

${\displaystyle P(\psi _{M})(\chi ,\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =\chi ^{2}+\chi \cdot (2-\chi )\cdot (\cos \varphi _{1}+\cos \varphi _{2})+(2-\chi )^{2}\cdot \cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =\chi \cdot (2-\chi )\cdot (\sin \varphi _{1}+\sin \varphi _{2})+(2-\chi )^{2}\cdot \sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(\chi ,\varphi _{1},\varphi _{2}=0)=\chi ^{2}+2\cdot \chi \cdot (2-\chi )\cdot \cos \varphi _{1}+(2-\chi )^{2}}$

Real time bin configuration with uncoupled imbalance and imperfection ${\displaystyle \eta ,\chi }$

• S ₁

${\displaystyle P(\psi _{K})(\eta ,\chi ,\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot \chi ^{2}\cdot (\eta \cdot \chi +(2-\eta )\cdot (2-\chi ))^{2}}$

• S ₂

${\displaystyle P(\psi _{L})(\eta ,\chi ,\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot (2-\chi )^{2}\cdot (\eta \cdot \chi +(2-\eta )\cdot (2-\chi ))^{2}}$

• Z with two EOM

${\displaystyle P(\psi _{M})(\eta ,\chi ,\varphi _{1},\varphi _{2})={\frac {1}{16}}\cdot (\eta \cdot \chi +(2-\eta )\cdot (2-\chi ))^{2}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =\eta ^{2}\cdot \chi ^{2}+\eta \cdot (2-\eta )\cdot \chi \cdot (2-\chi )\cdot (\cos \varphi _{1}+\cos \varphi _{2})+(2-\eta )^{2}\cdot (2-\chi )^{2}\cdot \cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =\eta \cdot (2-\eta )\cdot \chi \cdot (2-\chi )\cdot (\sin \varphi _{1}+\sin \varphi _{2})+(2-\eta )^{2}\cdot (2-\chi )^{2}\cdot \sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(\eta ,\chi ,\varphi _{1},\varphi _{2}=0)={\frac {1}{16}}\cdot (\eta \cdot \chi +(2-\eta )\cdot (2-\chi ))^{4}\cdot (\eta ^{2}\cdot \chi ^{2}+2\cdot \eta \cdot (2-\eta )\cdot \chi \cdot (2-\chi )\cdot \cos \varphi _{1}+(2-\eta )^{2}\cdot (2-\chi )^{2})}$

The power of the central impulse of a real TBI as a function of and . ${\displaystyle P}$ ${\displaystyle Z}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \kappa }$

Real time bin configuration with coupled imbalance and imperfection ${\displaystyle \kappa }$

With and as well . ${\displaystyle 0\leq \kappa \leq 2}$ ${\displaystyle \eta =\kappa }$ ${\displaystyle \chi =2-\kappa }$

• S ₁

${\displaystyle P(\psi _{K})(\kappa ,\varphi _{1},\varphi _{2})=\kappa ^{2}\cdot (2-\kappa )^{4}}$

• S ₂

${\displaystyle P(\psi _{L})(\kappa ,\varphi _{1},\varphi _{2})=\kappa ^{4}\cdot (2-\kappa )^{2}}$

• Z with two EOM

${\displaystyle P(\psi _{M})(\kappa ,\varphi _{1},\varphi _{2})={\frac {1}{4}}\cdot \kappa ^{6}\cdot (2-\kappa )^{6}\cdot (\Im ^{2}+\Re ^{2})}$

With:

${\displaystyle \Re =1+\cos \varphi _{1}+\cos \varphi _{2}+\cos(\varphi _{1}+\varphi _{2})}$

${\displaystyle \Im =\sin \varphi _{1}+\sin \varphi _{2}+\sin(\varphi _{1}+\varphi _{2})}$

• Z with an EOM

${\displaystyle P(\psi _{M})(\kappa ,\varphi _{1},\varphi _{2}=0)=4\cdot \kappa ^{6}\cdot (2-\kappa )^{6}\cdot \cos ^{2}{\frac {\varphi _{1}}{2}}}$

Implementation

An alternative setup for performing the TBE . Two Michelson interferometers in TBI. The animation shows the development of the satellites and , as well as the central impulse up to the exit of the arrangement. ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$ ${\displaystyle Z}$

A great advantage of the TBI is the property that only one fiber or a free space beam path has to be established on the path between transmitter and receiver. This enables the temperature stabilization to be limited to the interferometer. The disadvantage of this is the required, difficult to realize mechanical and optical equality of the interferometers. Therefore, in addition to temperature stabilization, optical stabilization must also be implemented. A delay line can be used for coarse adjustment in the range , for path length differences around a fiber stretcher and for differences in the wavelength range, the electro-optical modulator (EOM). ${\displaystyle 10^{-3}}$ ${\displaystyle (mm)}$ ${\displaystyle 10^{-6}}$ ${\displaystyle (\mu m)}$ ${\displaystyle 10^{-9}}$ ${\displaystyle (nm)}$

Since a control system can not be set up with a single photon , another laser beam is coupled into the interferometer at a distance of the single photon wavelength. In addition to maintaining the optical stability, this also takes over the technologically necessary communication between transmitter and receiver.

For the construction and to show the feasibility, instead of an expensive single photon source (SPS), a strongly damped laser ("attenuated laser") can be used for the time being. The attenuation is increased with each successful experimental step in order to be able to replace this partial setup with a PLC later.

Implementation of a time bin configuration. For explanations of the identifiers, see text. The photo shows a medium level of expansion.

For testing the experimental setup and for the measurements at the output of the received interferometer, a quickly measurable, meaningful and reproducible variable is required, which enables an assessment of the functional quality of the TBI. Such a variable is the interference contrast (visibility) and / or the time-bin criterion .

Example of the components of an interferometer within a time-bin configuration:

1 = fiber pool for taking up long fibers

2 = Power supply carrier with the boards for +12 V, +5 V and the reference +2,500 V underneath space for the EOM

3 = Terminal support for holding the 50/50 coupler, the fiber dummy, the circuit board for voltage balancing

4 = space for the carrier of the high voltage power supply and the DAC

5 = Carrier for the fiber-wound piezo ring with the future controller board and the sensor (arrow points to a fiber fixator)

6 = Faraday mirror holder

7 = temperature sensor holder

literature

• Matthias Leifgen: Protocols and components for quantum key distribution . PDF accessed on March 20, 2018 (English)

• Björnstjerne Zindler: Construction of fiber-based interferometers for quantum cryptography . PDF accessed on March 20, 2018 (German) (2.363 MB)

Individual evidence

1. ^ Todd Pittman: It's a Good Time for Time-Bin Qubits. Physical Review, October 9, 2013, accessed March 26, 2018 . 
2. ↑ C.-A. Bunge: high bit rate optical transmission systems. HfT Leipzig, March 16, 2009, accessed December 7, 2018 . 
3. ↑ Matthias Leifgen: Chapter 7.1.1.1 Requirements for the interferometer . In: Protocols and Components for Quantum Key Distribution . 2016, doi : 10.18452 / 17473 .
4. ↑ Björnstjerne Zindler: Chapter 2.2.1 The Thermobox . In: Structure of fiber-based interferometers for quantum cryptography . 2011 ( nadirpoint.de [PDF]).
5. ↑ Björnstjerne Zindler: Chapter 3.1.1 The Visibility . In: Structure of fiber-based interferometers for quantum cryptography . 2011 ( nadirpoint.de [PDF]).