T1 Time

The energy relaxation time: how long a qubit can stay in the excited state before decaying to ground.

T1 (also called the relaxation time or longitudinal relaxation time) measures how long a qubit maintains its energy before spontaneously releasing it to the environment.

Definition

If a qubit is prepared in $|1⟩$, the probability of finding it still in $|1⟩$ after time $t$:

$$P_1(t)=e^{-t/T_1}$$

T1 is the time constant for exponential decay to the ground state.

Physical Picture

A qubit in $|1⟩$ has higher energy than $|0⟩$. Over time:

• Energy leaks to the environment (as photons, phonons, etc.)
• Qubit relaxes: $|1⟩\to |0⟩$
• This is irreversible (energy is gone)

Typical Values

Causes of T1 Decay

Measurement

Inversion Recovery

1. Prepare $|1⟩$ (with X gate)
2. Wait time $t$
3. Measure
4. Repeat for various $t$
5. Fit exponential to get T1

|1⟩ ──[wait t]── Measure → P(1) = e^{-t/T1}

Why T1 Matters

Information Loss

A qubit in superposition will have its $|1⟩$ component decay: $$\alpha |0⟩+\beta |1⟩\stackrel{T_1}{\to }\alpha |0⟩+\beta e^{-t/2T_1}|1⟩+\dots$$

Circuit Depth Limit

Must complete computation before T1 decay destroys information.

Fundamental Bound on T2

T1 limits T2: $T_2\le 2T_1$

You can’t have coherence (T2) without energy (T1).

Improving T1

Relation to T2

• T1: Energy relaxation (longitudinal)
• T2: Total coherence (includes dephasing)
• $T_2\le 2T_1$ always
• $T_2\ll T_1$ when dephasing dominates

See also: T2 Time, Decoherence, Superconducting Qubit