Squeeze operator explained

In quantum physics, the squeeze operator for a single mode of the electromagnetic field is^[1]

\hat{S}(z) = \exp \left ( { 1 \over 2 } (z ^{*} \hat{a} ^{2} - z \hat{a} ^{\dagger} ) \right ) , z = r e ^{i\theta}

where the operators inside the exponential are the ladder operators. It is a unitary operator and therefore obeys

S(\zeta)S ^{\dagger} (\zeta) = S ^{\dagger} (\zeta)S(\zeta) = \hat 1 , where \hat 1 is the identity operator.

Its action on the annihilation and creation operators produces

\hat{S} ^{\dagger} (z)\hat{a}\hat{S}(z) = \hat{a}\cosh r - e ^{i\theta} \hat{a} ^{\dagger} \sinh r and \hat{S} ^{\dagger} (z)\hat{a} ^{\dagger} \hat{S}(z) = \hat{a} ^{\dagger} \cosh r - e ^{-i\theta} \hat{a} \sinh r

The squeeze operator is ubiquitous in quantum optics and can operate on any state. For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state.

The squeezing operator can also act on coherent states and produce squeezed coherent states. The squeezing operator does not commute with the displacement operator:

\hat{S}(z) \hat{D}(\alpha) ≠ \hat{D}(\alpha) \hat{S}(z),

nor does it commute with the ladder operators, so one must pay close attention to how the operators are used. There is, however, a simple braiding relation,

\hat{D}(\alpha)\hat{S}(z) = \hat{S}(z)\hat{S} ^{\dagger} (z)\hat{D}(\alpha)\hat{S}(z) = \hat{S}(z)\hat{D}(\gamma), where \gamma = \alpha\cosh r + \alpha ^{*} e ^{i\theta} \sinh r ^[2]

Application of both operators above on the vacuum produces displaced squeezed state:

\hat{D}(\alpha)\hat{S}(r)| 0 \rangle = |r, \alpha\rangle .

Or Squeezed coherent state:

\hat{S}(r)\hat{D}(\alpha)| 0 \rangle = |\alpha,r\rangle .

Derivation of action on creation operator

As mentioned above, the action of the squeeze operator

S(z) on the annihilation operator a can be written as $$S^\backslash dagger(z)\; a\; S(z)=\backslash cosh(|z|)a-\backslash frac$$\sinh(|z|)a^\dagger. To derive this equality, let us define the (skew-Hermitian) operator A\equiv (z a ^{\dagger} - z ^{*} a ^{2)/2} , so that S ^{\dagger} = e ^{A} .

The left hand side of the equality is thus

e ^{A} a e ^{-A} . We can now make use of the general equality $$e^A\; B\; e^=\backslash sum\_^\backslash infty\; \backslash frac[\backslash underbrace\{A,[A,\backslash dots,[A\}\_\{k\backslash ,\backslash text\{times\}\},B]\backslash dots]],$$ which holds true for any pair of operators A and B . To compute e ^{A} a e ^{-A} thus reduces to the problem of computing the repeated commutators between A and a .As can be readily verified, we have$$[A,a]=\backslash frac[z\; a^\{\backslash dagger\; 2\}-z^*\; a^2,a]=\; \backslash frac[a^\{\backslash dagger\; 2\},a]\; =\; -z\; a^\backslash dagger,$$$$[A,a^\backslash dagger]=\backslash frac[z\; a^\{\backslash dagger\; 2\}-z^*\; a^2,a^\backslash dagger]=\; -\backslash frac[a^\{2\},a^\backslash dagger]\; =\; -z^*\; a.$$Using these equalities, we obtain [\underbrace{A,[A, ... ,[A} _{k} ,a] ... ]] = \begin{cases} |z| ^{k} a, & for k even ,\\ - z|z| ^{k-1} a ^{\dagger,} & for k odd . \end{cases}

so that finally we get

e ^{A} a e ^{-A} = a \sum ^{inf ty} _{k =0} (|z| ^{2k})/(( 2 k)!) - a ^{\dagger} (z)/(|z|) \sum ^{inf ty} _{k =0} (|z| ^{2k+1})/(( 2 k +1 )!) = a\cosh|z| - a ^{\dagger} e ^{i\theta} \sinh|z|.

See also

• Squeezed coherent state

Notes and References

1. Book: Introductory quantum optics . Gerry, C.C. . Knight, P.L. . amp . 978-0-521-52735-4 . 2005 . Cambridge University Press . 182.
2. M. M. Nieto and D. Truax (1995), Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System. 1997 . 10.1002/prop.2190450204. quant-ph/9506025. Nieto . Michael Martin . Truax . D. Rodney . Fortschritte der Physik/Progress of Physics . 45 . 2 . 145–156 . 14213781 . Eqn (15). Note that in this reference, the definition of the squeeze operator (eqn. 12) differs by a minus sign inside the exponential, therefore the expression of \gamma is modified accordingly (\theta → \theta + \pi ).