# First occurrence of the Bloch sphere in the scientific literature

The Bloch sphere is a geometric representation of a single qubit. I am having trouble figuring out when it came into common usage in the scientific literature. The wikipedia article, as at the time of writing this question, references a journal article by Bloch (Bloch, Felix (Oct 1946). "Nuclear induction". Phys. Rev. 70 (7–8): 460–474) but the paper contains neither an image of a Bloch sphere, nor (as far as a cursory read suggests) a description of one.

I realise that the Poincaré sphere is mathematically the same object, but I am interested in when it was first explicitly used to represent a two state quantum system, and specifically when a picture of it first occurred in a scientific publication to that end.

## 1 Answer

"Qubits" were only named by Schumacher in 1995, and even early ideas about "quantum computing" do not appear until 1960-s. "Bloch sphere" refers to representing the pure states of a 2D quantum system (2 refers to complex dimension). What comes to be called "the Bloch sphere" is spelled out by Feynman, Vernon and Hellwarth in Geometrical Representation of the Schrödinger Equation for Solving Maser Problems (1957), and they do have a picture on p. 52 of "oscillator diagram in rotating coordinates", at least:

"A simple, rigorous geometrical representation for the Schrödinger equation is developed to describe the behavior of an ensemble of two quantum‐level, noninteracting systems which are under the influence of a perturbation. In this case the Schrödinger equation may be written, after a suitable transformation, in the form of the real three‐dimensional vector equation $$dr/dt=ω×r$$, where the components of the vector r uniquely determine ψ of a given system and the components of ω represent the perturbation. When magnetic interaction with a spin ½ system is under consideration, "r" space reduces to physical space. By analogy the techniques developed for analyzing the magnetic resonance precession model can be adapted for use in any two‐level problems. The quantum‐mechanical behavior of the state of a system under various different conditions is easily visualized by simply observing how r varies under the action of different types of ω.

On the subject of the origins they only say:"The extensive and explicit use of rotating coordinate procedures, as was introduced by Bloch, Ramsey, Rabi, and Schwinger for special kinds of magnetic transitions, is generally applicable in dealing with the $$r$$ space". The name "Bloch sphere", with a picture, appears, for example in Stroud et al., Superradiant Effects in Systems of Two-Level Atoms (1972), p. 1103, and by then they call it "the familiar Bloch-vector form". "Rotating coordinate procedures" (inspired by Majorana) appear e.g. in Bloch's paper with Rabi Atoms in Variable Magnetic Fields (1945). There are no pictures:

"As an essential feature in the derivation of his formulae, Majorana has shown that the problem of a system with arbitrary angular momentum can be reduced to the consideration of $$j$$ representative points on the unit sphere, each representing the direction of an angular momentum with value $$\frac12$$... Majorana's method, while remarkable in its elegance, has the disadvantage of somewhat obscuring the physical significance of the representative systems with spin $$\frac12$$. It is clear that a simple intuitive understanding of the procedure and of the essential formulae will be very useful to many. In this paper we shall arrive at such an understanding by the application of the familiar vector model where the total spin operator $$M$$ is treated as a sum of $$2j$$ spin operators each representing a system with angular momentum $$\frac12$$ and with the same "$$g$$" value as the total system. Mathematically this will be expressed by a different representation of the system in which the variables of the Schroedinger wave function $$\psi$$ are the spin variables of the constituent systems with spin $$\frac12$$ and which we shall call the "composite" representation in contrast to the usual representation which uses only a single spin variable, referring to the total system of spin $$j$$."

But the mathematics of "rotating coordinate procedures" appears already in Bloch's PhD thesis Quantum Mechanics of the Electrons in Crystal Lattices (1929) written under Heisenberg, or, for that matter, Poincare's optical paper from 1892, but outside the quantum mechanical context.