In quantum physics, the squeeze operator for a single mode of the electromagnetic field is[1]
![{\displaystyle {\hat {S}}(z)=\exp \left({1 \over 2}(z^{*}{\hat {a}}^{2}-z{\hat {a}}^{\dagger 2})\right),\qquad z=r\,e^{i\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6c179a0d0d3818c86507ef6e3872800fbb8d87)
where the operators inside the exponential are the ladder operators. It is a unitary operator and therefore obeys
, where
is the identity operator.
Its action on the annihilation and creation operators produces
![{\displaystyle {\hat {S}}^{\dagger }(z){\hat {a}}{\hat {S}}(z)={\hat {a}}\cosh r-e^{i\theta }{\hat {a}}^{\dagger }\sinh r\qquad {\text{and}}\qquad {\hat {S}}^{\dagger }(z){\hat {a}}^{\dagger }{\hat {S}}(z)={\hat {a}}^{\dagger }\cosh r-e^{-i\theta }{\hat {a}}\sinh r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f344cbcb4f77c48153634b316d989bbba3f72e12)
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:
![{\displaystyle {\hat {S}}(z){\hat {D}}(\alpha )\neq {\hat {D}}(\alpha ){\hat {S}}(z),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f131346982b81ca892c38bb88eea21118e7fd1c)
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,
[2]
Application of both operators above on the vacuum produces displaced squeezed state:
.
Or Squeezed coherent state:
.
Derivation of action on creation operator
As mentioned above, the action of the squeeze operator
on the annihilation operator
can be written as
![{\displaystyle S^{\dagger }(z)aS(z)=\cosh(|z|)a-{\frac {z}{|z|}}\sinh(|z|)a^{\dagger }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a16d116dcc2915c50a74d7b737f463b05425a82a)
To derive this equality, let us define the (skew-Hermitian) operator
![{\displaystyle A\equiv (za^{\dagger 2}-z^{*}a^{2})/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3617e0f86527900c499da72058acce6e99bec3d7)
, so that
![{\displaystyle S^{\dagger }=e^{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1377b54e378061b1ac103bc59efee5063ce40467)
.
The left hand side of the equality is thus
. We can now make use of the general equality
![{\displaystyle e^{A}Be^{-A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}[\underbrace {A,[A,\dots ,[A} _{k\,{\text{times}}},B]\dots ]],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d1944da09e2049fd780aa580393cfc9a2af10e)
which holds true for any pair of operators
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
and
![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
. To compute
![{\displaystyle e^{A}ae^{-A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8083e0caf23a6ede1843d42221b450c45204484e)
thus reduces to the problem of computing the repeated commutators between
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
and
![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
. As can be readily verified, we have
![{\displaystyle [A,a]={\frac {1}{2}}[za^{\dagger 2}-z^{*}a^{2},a]={\frac {z}{2}}[a^{\dagger 2},a]=-za^{\dagger },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f62585473e2b387edf0ebde45d723db1030f641f)
![{\displaystyle [A,a^{\dagger }]={\frac {1}{2}}[za^{\dagger 2}-z^{*}a^{2},a^{\dagger }]=-{\frac {z^{*}}{2}}[a^{2},a^{\dagger }]=-z^{*}a.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed65b76372543b52560b3dec6418505d316586cb)
Using these equalities, we obtain
so that finally we get
See also
References
- ^ Gerry, C.C. & Knight, P.L. (2005). Introductory quantum optics. Cambridge University Press. p. 182. ISBN 978-0-521-52735-4.
- ^ M. M. Nieto and D. Truax (1995), Nieto, Michael Martin; Truax, D. Rodney (1997). "Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System". Fortschritte der Physik/Progress of Physics. 45 (2): 145–156. arXiv:quant-ph/9506025. doi:10.1002/prop.2190450204. S2CID 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
is modified accordingly (
).
Operators in physics
General | Space and time | - d'Alembertian
- Parity
- Time
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Particles | |
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Operators for operators | - Anti-symmetric operator
- Ladder operator
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Quantum | Fundamental | - Momentum
- Position
- Rotation
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Energy | - Total energy
- Hamiltonian
- Kinetic energy
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Angular momentum | |
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Electromagnetism | |
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Optics | |
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Particle physics | |
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