Optical equivalence theorem

Key takeaways

Optical equivalence theorem belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Optical equivalence theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Optical equivalence theorem from memory before moving on to harder problems.

Reference excerpt

The optical equivalence theorem in quantum optics asserts an equivalence between the expectation value of an operator in Hilbert space and the expectation value of its associated function in the phase space formulation with respect to a quasiprobability distribution. The theorem was first reported by George Sudarshan in 1963 for normally ordered operators and generalized later that decade to any ordering. Let Ω be an ordering of the non-commutative creation and annihilation operators, and let g Ω ( a ^ , a ^ † ) {\displaystyle g_{\Omega }({\hat {a}},{\hat {a}}^{\dagger })} be an operator that is expressible as a power series in the creation and annihilation operators that satisfies the ordering Ω. Then the optical equivalence theorem is succinctly expressed as

Here, α is understood to be the eigenvalue of the annihilation operator on a coherent states and is replaced formally in the power series expansion of g. The left side of the above equation is an expectation value in the Hilbert space whereas the right hand side is an expectation value with respect to the quasiprobability distribution. We may write each of these explicitly for better clarity. Let ρ ^ {\displaystyle {\hat {\rho }}} be the density operator and Ω ¯ {\displaystyle {\bar {\Omega }}} be the ordering reciprocal to Ω. The quasiprobability distribution associated with Ω is given, then, at least formally, by

ρ ^ = 1 π ∫ f Ω ¯ ( α , α ∗ ) | α ⟩ ⟨ α | d 2 α . {\displaystyle {\hat {\rho }}={\frac {1}{\pi }}\int f_{\bar {\Omega }}(\alpha ,\alpha ^{*})|\alpha \rangle \langle \alpha |\,d^{2}\alpha .}

The above framed equation becomes

tr ⁡ ( ρ ^ ⋅ g Ω ( a ^ , a ^ † ) ) = ∫ f Ω ¯ ( α , α ∗ ) g Ω ( α , α ∗ ) d 2 α . {\displaystyle \operatorname {tr} ({\hat {\rho }}\cdot g_{\Omega }({\hat {a}},{\hat {a}}^{\dagger }))=\int f_{\bar {\Omega }}(\alpha ,\alpha ^{*})g_{\Omega }(\alpha ,\alpha ^{*})\,d^{2}\alpha .}

For example, let Ω be the normal order. This means that g can be written in a power series of the following form:

g N ( a ^ † , a ^ ) = ∑ n , m c n m a ^ † n a ^ m . {\displaystyle g_{N}({\hat {a}}^{\dagger },{\hat {a}})=\sum _{n,m}c_{nm}{\hat {a}}^{\dagger n}{\hat {a}}^{m}.}

The quasiprobability distribution associated with the normal order is the Glauber–Sudarshan P representation. In these terms, we arrive at

tr ⁡ ( ρ ^ ⋅ g N ( a ^ , a ^ † ) ) = ∫ P ( α ) g ( α , α ∗ ) d 2 α . {\displaystyle \operatorname {tr} ({\hat {\rho }}\cdot g_{N}({\hat {a}},{\hat {a}}^{\dagger }))=\int P(\alpha )g(\alpha ,\alpha ^{*})\,d^{2}\alpha .}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Optical equivalence theorem

Start with the simplest possible case. Write down what Optical equivalence theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Optical equivalence theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Optical equivalence theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Optical equivalence theorem

In research: Optical equivalence theorem appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Optical equivalence theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Optical equivalence theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum optics, Theorems in quantum mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Optical equivalence theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Optical equivalence theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Optical equivalence theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Optical equivalence theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Optical equivalence theorem in simple terms?

The optical equivalence theorem in quantum optics asserts an equivalence between the expectation value of an operator in Hilbert space and the expectation value of its associated function in the phase space formulation with respect to a quasiprobability distribution. The theorem was first reported…

Why does Optical equivalence theorem matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Optical equivalence theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Optical equivalence theorem.