Video library: A. M. Chebotarev, T. V. Tlyachev, A. E. Teretenkov, V. V. Belokurov, On the normal form of multimode squeezings

VIDEO LIBRARY


International conference "QP 34 – Quantum Probability and Related Topics" September 17, 2013 12:00, Moscow, Steklov Mathematical Institute of RAS

On the normal form of multimode squeezings A. M. Chebotarev, T. V. Tlyachev, A. E. Teretenkov, V. V. Belokurov M. V. Lomonosov Moscow State University
Abstract: We describe the solution of algebraic equations for the coefficients of the normal factorization $$ U_t=e^{i\widehat H t}=e^{s_t}e^{-\frac{1}{2}(a^\dagger,R_ta^\dagger)-(g_t,a^\dagger)}\,e^{(a^\dagger,C_t a)}\,e^{\frac{1}{2} (a,\overline\rho_t a)+(\overline f_t,a)} $$ of the unitary group $U_t$ with Hamiltonian $$ \widehat H= \frac{i}{2}((a^\dagger,Aa^\dagger)-(a,\overline A a))+(a^\dagger,B a)+i(a^\dagger,h)-i(a,\overline h) $$ in terms of the matrices $\Phi_t$, $\Psi_t$ which define the canonical transformation of the creation-annihilation operators. Such a decomposition defines the normal symbol of squeezing and the inner products of squeezed states which are necessary for constructing a basis in a linear hull generated by a finite set of squeezed states. A new class of solvable quantum problems is related to Hamiltonians with $A$ and $B$ such that $[A\overline A,B]=0$ and $\operatorname{rank}A\overline A\ge \operatorname{rank}B$. In this case, the solution is expressed in terms of the eigenvalues of the Hermitian matrix $A\overline A-B^2$. Language: English