The name "Loschmidt echo" is used in quantum physics for the quantity $$ M(t) \equiv \left| \langle\psi_0| e^{i H t/\hbar} e^{-i H_0 t/\hbar} |\psi_0\rangle \right|^2 $$ where $$ H = H_0 + \Sigma. $$ The idea is that $|\psi_0\rangle$ is the state of some quantum system at time zero and $\Sigma$ is a small perturbation; the system evolves under $H_0$ for time $t$ and we would like to discover the degree to which the result overlaps with evolution under $H$. One can interpret $M(t)$ also as if the system first evolved under $H_0$ and then under $-H$ for a further time $t$ (or backwards in time under $H$), and one wishes to know the degree to which $H$ 'undoes' the effect of $H_0$.

The name "Loschmidt" is, as I understand it, that of Johann Josef Loschmidt (15 March 1821 – 8 July 1895) who debated with Boltzmann about reversibility and entropy etc. in classical physics, so one can see why $M(t)$ might have got to be named after him. However in view of the fact that $M(t)$ as defined above is a thoroughly quantum-mechanical quantity I am not sure if the name is apt, and I wonder whether we might introduce the name Peres since Asher Peres used the quantity in his Stability of quantum motion in chaotic and regular systems, Phys. Rev. A 30, 1610 (1984). Anyway my question here is if anyone can elucidate the origins of the name "Loschmidt echo", and comments on the feasibility or appropriateness of introducing another name would also be welcome.

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    $\begingroup$ As you are no doubt aware many scientific concepts are named after someone other than the discoverer. $\endgroup$
    – mdewey
    Commented Dec 30, 2023 at 15:07
  • $\begingroup$ It is clearly named after the Loschmidt paradox, even if quantum. Any other response would be speculation. $\endgroup$
    – Mauricio
    Commented Dec 30, 2023 at 20:54
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    $\begingroup$ Just to say a response need not be speculation: maybe a responder can point to a first paper to use this name, or something like that. $\endgroup$ Commented Dec 30, 2023 at 20:56
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    $\begingroup$ Purely opinion, but I don't think replacing one person's name with another person's name would be helpful. The term "Loschimidt echo" is informative - on hearing it one immediately guesses that it will have something to do with a failure of time reversibility - whereas "Peres echo" doesn't tell you anything unless you're already familiar with the specific work. So if you really wanted to change the name I'd suggest trying to come up with something more descriptive rather than just assigning another person's name. (But "Loschimidt echo" is already pretty good in that respect, IMHO.) $\endgroup$
    – N. Virgo
    Commented Dec 31, 2023 at 3:18
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    $\begingroup$ Peres did not call it "echo" but "overlap, and the early uses of "echo" name a physical effect rather than a quantity that measures it. So if you'd like to use Peres's name, "Peres's overlap" would be more fitting. But "Loschmidt echo" seems to be too entrenched by now for the name change to succeed. $\endgroup$
    – Conifold
    Commented Dec 31, 2023 at 8:44

2 Answers 2


Here I present what I could find about the coinage of the term "Loschmidt echo" (mostly based of two reviews 1 and 2 and sources therein).

Erwin Hahn demonstrated and defined spin echos in the 1950, per 3. As soon as this happened the discussion of a "Loschmidt paradox" being realizable in the experiments was discussed. See 4, which is from 1957, "Loschmidt-Ehrenfest paradox" of 5 from 1965, and 1970 "Loschmidt demon" from 6. Ref 5 also cites 7 for having first made this association in 1959 between spin echoes and thermodynamic paradoxes.

However the earliest mention of "Loschmidt echo" I have found online is from Hahn himself in a lecture from 1978, see 8:

The path taken in precession phase may be in accord with a mirror sequence of pulses, in which case the Loschmidt echo scheme is realized; or the path may be cyclic, which does not retrace the forward evolution of phase, but nevertheless returns to the initial point by a different return route.

While this article is not cited, review 2 cites 9 to say:

Early on Erwin Hahn recognized that his procedure, which he viewed as a change in the sign of the system Hamiltonian, provided a quantum implementation of the Loschmidt proposal.

Hahn in Ref. 9 (1984) does not use the term Loschmidt echo but his realization of the Loschdmidt paradox is explained in more detail.

The next textual mention of the "Loschmidt echo" (as far as I can find online) does not appear until the 2001 work of Rodolfo Jalabert and Horace Patawski (10) who also refer to Ref. 9. The 2001 publication also clearly defines the formula that you mention. Worth highlighting Asher Peres who also introduced the same formula in 1984, without calling it Loschdmidt echo, in 11.

Reviews 1 and 2 suggest that the echo was named after the Loschmidt paradox. Review 2 was also written partially by Jalabert and Patawski, if their use of the term was inspired by some other earlier work they missed to indicate it. The term Loschmidt echo might have existed informally to refer a series of spin echo results until its formalization in 2001.

Disclaimer: I will comment very little on the "feasibility" of introducing an alternative name. This site is not the forum for that unless it is already a matter of historical name conflict. Also it clearly does not help with the vast family of spin echo names.


  1. Gorin, T., Prosen, T., Seligman, T. H., and Žnidarič, M. (2006). Dynamics of Loschmidt echoes and fidelity decay. Physics Reports, 435(2-5), 33-156. doi:10.1016/j.physrep.2006.09.003

  2. Scholarpedia's Loschmidt echo

  3. Hahn, E. L. (1950). Spin echoes. Phys. Rev., 80(4), 580. doi:10.1103/PhysRev.80.580

  4. Rothstein, Jerome. "Nuclear spin echo experiments and the foundations of statistical mechanics." Am. J. Phys. 25.8 (1957): 510-518. doi:10.1119/1.1934539

  5. Baur, M., Jordan, J. R., Jordan, P. C., and Mayer, J. E. (1965). Towards a theory of linear nonequilibrium statistical mechanics. Ann. Phys., 35(1), 96-163. doi:10.1016/0003-4916(65)90071-0

  6. Rhim, W-K., Alexander Pines, and John S. Waugh. "Violation of the spin-temperature hypothesis." Phys. Rev. Lett. 25.4 (1970): 218. doi:10.1103/PhysRevLett.25.218

  7. Blatt, J. M. (1959). An alternative approach to the ergodic problem. Pro. Theo. Phys., 22(6), 745-756. doi:10.1143/PTP.22.745

  8. Hahn, E., "Pulsed nuclear magnetic resonance in solids. A survey." Faraday Symposia of the Chemical Society. Vol. 13. Royal Society of Chemistry, 1978. doi:10.1039/FS9781300007

  9. Brewer, R.G., and Hahn, E. L. "Atomic Memory". Scientific American, Vol. 251(6), pp. 50-57 (1984). (Available in jstor)

  10. Jalabert, R. A., and Pastawski, H. M. (2001). Environment-independent decoherence rate in classically chaotic systems. Phys. Rev. Lett., 86(12), 2490. doi:10.1103/PhysRevLett.86.2490

  11. Peres, A. (1984). Stability of quantum motion in chaotic and regular systems. Phys. Rev. A, 30(4), 1610. doi:10.1103/PhysRevA.30.1610

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    $\begingroup$ It seems to me that all the early authors apply "echo" with various attachments to a phenomenon rather than to a numerical measure. On the other hand, Peres, who introduced it in 1984, did not call it "echo", let alone "Loschmidt echo", but rather "overlap". The shifted use of "Loschmidt echo" for the measure does not seem to appear until 2000-s, Jalabert-Pastawski is an early example. $\endgroup$
    – Conifold
    Commented Dec 31, 2023 at 8:40
  • $\begingroup$ @Conifold you are right, I have updated my answer to give more credit to the later $\endgroup$
    – Mauricio
    Commented Dec 31, 2023 at 11:41

@Mauricio quotes 1978 work mentioning Loschmidt echo. Looks like the use of the term in that work was ignored, as ref. 1 says: "At time 2t, one measures a maximum in the return probability, that we call Loschmidt echo (LE)", and ref. 2 quotes ref. 1 as the source of the term: "quantity measuring sensitivity to perturbations of the backward evolution is nowadays known as the Loschmidt echo after Ref. [7]" Furthermore, it does not look like the term in 1978 work was applied specifically to the expression of the OP.

As for the appropriateness of the term, it looks OK to me, but it's a matter of preference.

ref 1: Jalabert and Pastawski, Phys. Rev. Lett. 86 (2001) 2490-2493

ref 2: T. Gorin et al. / Physics Reports 435 (2006) 33 –156


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