Bloch Sphere
single qubitSingle-qubit state space
Any pure single-qubit state can be written as |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, mapping to a point on the unit sphere (Bloch sphere) with polar angle θ ∈ [0,π] and azimuthal angle φ ∈ [0,2π]. The Bloch vector 𝐧 = (sin θ cos φ, sin θ sin φ, cos θ) represents the expectation values ⟨σx⟩, ⟨σy⟩, ⟨σz⟩.
Special states
|0⟩ (north pole): θ=0, Bloch vector = (0,0,+1)
|1⟩ (south pole): θ=π, Bloch vector = (0,0,−1)
|+⟩ = (|0⟩+|1⟩)/√2: θ=π/2, φ=0 → (+x equator)
|+i⟩ = (|0⟩+i|1⟩)/√2: θ=π/2, φ=π/2 → (+y equator)
Mixed state: ρ = I/2 → Bloch vector = origin (not on sphere)
Quantum Gates
unitary operationsUnitary transformations on the Bloch sphere
Single-qubit gates are 2×2 unitary matrices acting on |ψ⟩. Geometrically they are rotations of the Bloch sphere. Pauli gates rotate by π around x, y, or z axes. The Hadamard H swaps the |0⟩/|1⟩ basis with |±⟩. Phase gates (S, T) rotate around z.
Superposition & Interference
Mach-ZehnderMach-Zehnder Interferometer
A single photon enters a 50/50 beamsplitter (H gate), acquires a phase φ on one arm, then hits a second beamsplitter. The output probabilities are:
Output Probabilities
Measurement & Born Rule
projection, statisticsBorn rule
Measuring |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩ in the computational basis yields |0⟩ with probability P₀ = cos²(θ/2) and |1⟩ with probability P₁ = sin²(θ/2), regardless of φ. The post-measurement state collapses to the measured outcome.
Measurement histogram
Entanglement & Bell States
non-separabilityBell states — maximally entangled two-qubit states
A state is entangled if it cannot be written as |ψ_A⟩⊗|ψ_B⟩. The four Bell states form an orthonormal basis for the two-qubit Hilbert space and are maximally entangled: measuring one qubit instantly determines the other's outcome, regardless of separation — "spooky action at a distance" (Einstein, 1935).
|Φ⁺⟩
(|00⟩ + |11⟩)/√2
|Φ⁻⟩
(|00⟩ − |11⟩)/√2
|Ψ⁺⟩
(|01⟩ + |10⟩)/√2
|Ψ⁻⟩
(|01⟩ − |10⟩)/√2
|00⟩ (separable)
|0⟩⊗|0⟩ — not entangled
|++⟩ (separable)
(|0⟩+|1⟩)/√2 ⊗ same
Rydberg Atoms & Blockade
neutral-atom QCRydberg blockade mechanism
Exciting two nearby atoms to Rydberg states shifts one atom's resonance by V(R) = C₆/R⁶ due to van der Waals interactions. When V(R) ≫ ℏΩ_R, the double excitation |rr⟩ is off-resonant — this is the Rydberg blockade. Within the blockade radius R_b, only one atom can be excited at a time, enabling two-qubit entangling gates.
V(R) = C₆/R⁶ interaction potential
Two-Qubit Gates
entangling operationsEntangling gates
Two-qubit gates act on the 4D Hilbert space spanned by |00⟩, |01⟩, |10⟩, |11⟩. CNOT, CZ, and iSWAP can generate maximal entanglement from product states — they are universal when combined with single-qubit gates.
Rabi Oscillations
coherent driveDriven two-level system
A resonant laser drive with Rabi frequency Ω causes coherent oscillations between |0⟩ and |1⟩. Off-resonance (detuning Δ), the generalised Rabi frequency is Ω' = √(Ω²+Δ²) and the oscillation amplitude is reduced to (Ω/Ω')².
P₁(t) — population in |1⟩
Decoherence: T₁ & T₂
open quantum systemsBloch sphere collapse
Real qubits interact with their environment. T₁ (relaxation time) is how long the excited state population survives — the qubit decays from |1⟩ to |0⟩. T₂ (coherence/dephasing time) is how long superpositions survive. Always T₂ ≤ 2T₁. The off-diagonal density matrix element |ρ₀₁| decays as e^{−t/T₂}.
Population P₁(t) and Coherence |ρ₀₁(t)|
Laser Cooling Physics
Doppler, tweezersDoppler Cooling
A red-detuned laser (δ < 0) preferentially scatters photons from atoms moving toward the beam (Doppler blueshift → resonance). Each scatter kick opposes motion → viscous "optical molasses" force F = −αv. Minimum temperature:
T_D = ℏΓ / (2k_B)For Cs D₂ (Γ/2π = 5.234 MHz): T_D = 125 μK. Sub-Doppler cooling (gray molasses, PGC) reaches the photon recoil limit T_r ≈ 100s of nK.
Optical Tweezer Trapping
A tightly focused, red-detuned laser (e.g. 1064 nm for Cs) creates an AC Stark shift potential well at the intensity maximum. Trap depth U₀ = α·I_peak/(cε₀). Atoms trapped in the ground motional state ⟨n⟩ < 0.1 via resolved-sideband or EIT cooling.
ω_r/2π = √(4U₀/mw₀²) / (2π) → tens to hundreds of kHzRadial trap frequencies 50–200 kHz, axial 10–50 kHz for typical tweezer parameters. Arrays of 1000+ individually controlled tweezers are now routinely operated.
Cooling stages
Wigner Function
phase-space portraitPhase-space quasi-probability distribution
The Wigner function W(x,p) is the quantum analogue of a classical phase-space distribution. It can take negative values — a signature of non-classical states. Classical coherent states are Gaussians (W ≥ 0). Fock states |n⟩ show oscillating rings with n negative regions. Cat states exhibit interference fringes.
Reading the Wigner function
Green/positive: regions where the state is "more classical".
Red/negative: quantum interference — impossible for any classical distribution.
Coherent state: single Gaussian blob → minimal uncertainty.
Fock state |n⟩: concentric rings, central sign (−1)^n → strongly non-classical.
Cat state: two blobs + oscillating interference fringes between them.