Time-bin encoding (time-bin qubits)

Time-bin encoding represents a qubit by a single photon spread over two time slots separated by a fixed delay $\tau$, typically 0.1 to a few nanoseconds. The two basis states are $|e\rangle$, the photon in the early bin, and $|l\rangle$, the photon in the late bin, and a general qubit is the superposition

$$|\psi\rangle = \alpha\,|e\rangle + \beta\, e^{i\varphi}\,|l\rangle,$$

with $|\alpha|^2 + |\beta|^2 = 1$. The relative phase $\varphi$ is defined with respect to the phase of the laser or interferometer that prepared the state. Time bins are the usual qubit format in fiber quantum key distribution and in fiber-based entangled-photon experiments, because they survive transmission through kilometers of fiber whose birefringence drifts.

Preparation

The standard source sends a short laser pulse, much shorter than $\tau$, through an unbalanced Mach-Zehnder interferometer whose arms differ by a delay $\tau$. Each laser pulse leaves as a pair of pulses separated by $\tau$, with the splitting ratio setting $\alpha$ and $\beta$ and a phase shifter in one arm setting $\varphi$. Attenuated to well below one photon per pair, this gives a time-bin qubit. An intensity modulator can instead carve the two pulses out of a continuous-wave laser, and a phase modulator then sets $\varphi$ directly, which lets the state change from one clock cycle to the next as QKD requires.

In fiber, the delay corresponds to a length difference $\Delta L = c\tau/n_g$. With group index 1.468, a 1 ns separation needs

$$\Delta L = \frac{(3.00 \times 10^8)(10^{-9})}{1.468} \approx 0.20\ \text{m}.$$

Measurement with an unbalanced interferometer

Measuring in the early/late basis only requires recording the photon's arrival time, so the detector's timing jitter must be small compared with $\tau$; superconducting nanowire detectors, with jitter of tens of picoseconds, resolve bins well below a nanosecond.

Measuring the phase requires a second unbalanced interferometer with the same delay $\tau$. A photon in the analyzer can take the short or long arm, so it exits in one of three time slots:

• the first slot: early bin through the short arm, probability 1/4;
• the middle slot: early bin through the long arm or late bin through the short arm, probability 1/2;
• the last slot: late bin through the long arm, probability 1/4.

In the middle slot the two paths are indistinguishable and interfere, so which output port fires depends on $\varphi$ minus the analyzer phase $\theta$. Selecting middle-slot events gives a measurement in the superposition basis; the side slots reveal arrival time. With passive splitting, half of the detected photons fall in the side slots. The analyzer delay must match the preparation delay to well within the photon's coherence length, and the fringe visibility of the middle slot is the figure of merit for the whole link.

Phase stability

The difficulty lies in keeping the two interferometers' phases stable relative to each other. The optical path length of fiber changes by about 10⁻⁵ per kelvin (the thermo-optic coefficient of silica, 8.5 × 10⁻⁶ K⁻¹, plus thermal expansion), so a 0.20 m imbalance at 1550 nm shifts by

$$\frac{2\pi (0.20\ \text{m})(9.3 \times 10^{-6})}{1.55\ \mu\text{m}} \approx 7.7\ \text{rad/K},$$

about 1.2 fringes per kelvin. Holding the phase to 0.1 rad therefore needs temperature control at the 10 mK level, or active phase tracking with reference light. A Faraday-mirror (Michelson) analyzer removes the analyzer's polarization dependence but still needs this phase control, and temperature-controlled planar interferometers are the integrated alternative.

Why fiber links favor time bins

Both bins travel through the same fiber within nanoseconds of each other, so slow changes in fiber birefringence and polarization mode dispersion, which scramble polarization qubits, affect both bins almost identically and leave $\varphi$ unchanged. The cost is the interferometer stabilization and, with passive analyzers, the factor of two lost to the side slots.

Time-bin entanglement

Pumping spontaneous parametric down-conversion with two pump pulses separated by $\tau$ creates the entangled state $(|e\rangle|e\rangle + e^{i\varphi}|l\rangle|l\rangle)/\sqrt{2}$, and two analyzers, one per photon, measure two-photon interference in the coincidences. A continuous-wave pump with a long coherence time gives the related energy-time entanglement analyzed by a Franson interferometer. A two-photon fringe visibility above $1/\sqrt{2} \approx$ 70.7% violates the CHSH Bell inequality. Time bins are also used in photonic quantum computing schemes that process many qubits in sequence through one loop of delay lines, and more than two bins encode higher-dimensional states.

Common questions

What is the difference between time-bin and polarization encoding?

Polarization encoding uses horizontal and vertical (or other) polarization states and needs only waveplates and polarizers to measure, but fiber birefringence drifts and must be tracked. Time-bin encoding is insensitive to that drift in transit and needs stabilized unbalanced interferometers at each end.

How is the bin separation chosen?

It must exceed the pulse duration plus the detector timing jitter, so the bins do not overlap, and it sets the interferometer imbalance. Shorter separations allow higher clock rates and smaller, more stable interferometers; values from about 100 ps to a few ns cover most systems.

References: J. D. Franson, "Bell inequality for position and time," Phys. Rev. Lett. 62, 2205 (1989); J. Brendel, N. Gisin, W. Tittel and H. Zbinden, "Pulsed energy-time entangled twin-photon source for quantum communication," Phys. Rev. Lett. 82, 2594 (1999); N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 (2002).