The Bloch Equations

The applet on this page uses the Bloch sphere to visualize the evolution of a quantum two-level system whose Hamiltonian is

              (         *    †)
H  = ˉhΔ σz + ˉh Ω(t)σ + Ω (t)σ

In this expression, σ is a “lowering” operator given by σ = (σx - y)2, and σ = (σx + y)2. The operators σx, σy and σz are given by the Pauli matrices. The constant Δ is entered in the “detuning” field below, and the “Rabi frequency” Ω(t) is an arbitrary complex function that is entered below also in the appropriate field.

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Supported functions and symbols: Do not worry about capitalization. The applet is not case sensitive.

+ -Exp(t)Sin(t)Csc(t)Sinh(t)Csch(t)Rect(t)
* /Ln(t)Cos(t)Sec(t)Cosh(t)Sech(t)Unit(t)
( )Sqrt(t)Arcsin(t)Arccsc(t)Arcsinh(t)Arccsch(t)
[ ]Arccos(t)Arcsec(t)Arccosh(t)Arcsech(t)
{ }Arctan(t)Arccot(t)Arctanh(t)Arccoth(t)

The fields labeled “Phi” and “Theta” are used to specify the initial state as a point on the surface of the Bloch sphere. The constant “π” can be entered as “Pi” in any field. The imaginary unit i can be entered as a plain lowercase “i”. Any algebraic expression involving these constants and ordinary functions such as trigonometric and hyperbolic functions is acceptable in the Rabi frequency field (only note that the “time” variable must be called “t”). There is also a function “rect(t)” that is defined as 1 for -12 t 12 and zero elsewhere, which can be used to represent “square” pulses.

In addition to the Hamiltonian evolution given by (1), phenomenological damping constants Gamma1 and Gamma2 (corresponding to 1∕T1 and 1∕T2, respectively) may also be specified. Note that one must always have Gamma2Gamma1/2, or else unphysical trajectories will result.

To be precise, if the expectation values of σx, σy, σz are written as x, y, z, corresponding to the coordinates in the Bloch sphere plot, the equations being integrated are

x˙ =   - Δy - γ x + Ω z                                  (2)
               2     i
y˙ =   Δx - γ2y + Ωrz                                    (3)
z˙ =   - γ1(z + 1)- Ωry - Ωix                             (4)
where Ωr and Ωi are the real and imaginary parts of Ω, respectively.

The equations (2)-(4) are called the Bloch equations. They were first introduced in the context of nuclear magnetic resonance (NMR), where they give the evolution of a spin (elementary magnetic moment) in a combined static magnetic field along the z axis and a radiofrequency field in the x-y plane. In this case the quantity Ω(t) is proportional to the slowly-varying complex amplitude of the rf field, and the Bloch vector represents the actual orientation of the spin in space (only perhaps in a rotating frame of reference).

The same equations can be used to describe a “two-level atom” with energy levels ∣+z 〉 and ∣- z〉 in a near-resonant laser field, and in this case Ω(t) is proportional to the slowly-varying complex amplitude of the laser field. For the two-level atom, the projection of the Bloch vector onto the z axis is a measure of the average energy (with the zero of energy chosen to be exactly halfway between the two levels), and its projection onto the x - y plane gives the amplitude and phase of the atom’s electric dipole moment.

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