Interface

Avac(A::QuExpr), vacA(A::QuExpr)

Simplify operator by assuming it is applied to the vacuum from the left or right, respectively. To be precise, Avac(A) returns $A'$ such that $A'|0⟩ = A|0⟩$, while vacA(A) does the same for $⟨0|A$.

BosonCreate(name,inds): represent bosonic creation operator $name_{inds}$

BosonDestroy(name,inds): represent bosonic annihilation operator $name_{inds}$

CorrPerm_isless(a,b)

isless for Tuples of Tuples representing products of Corr (see above) of a permutation of operators. E.g., ((1,3),(2,)) represents <A1 A3>C <A2>C. It is assumed that the total number of operators in a and b is equal, i.e., that sum(length.(a)) == sum(length.(b)).

CorrTup_isless(a,b)

isless for Tuples of integers that represent Corr of sorted operators (with n representing An such that n<m == An<Am). (n,m,...) ≡ Corr(AnAm...). Defined in such a way that the same order is obtained as with BaseOperator objects

FermionCreate(name,inds): represent fermionic creation operator $name_{inds}$

FermionDestroy(name,inds): represent fermionic annihilation operator $name_{inds}$

TLSCreate(name,inds): represent TLS creation operator $name_{inds}$

TLSDestroy(name,inds): represent TLS annihilation operator $name_{inds}$

TLSx(name,inds): represent TLS x operator $name_{inds}$

TLSy(name,inds): represent TLS y operator $name_{inds}$

TLSz(name,inds): represent TLS z operator $name_{inds}$

boson_ops(name::Symbol): return function for creating bosonic annihilation and creation operators with name name (i.e., wrappers of BosonDestroy and BosonCreate)

expval(A::QuExpr): replace expression A by its (formal) correlator ⟨A⟩c.

expval(A::QuExpr): replace expression A by its (formal) expectation value ⟨A⟩.

expval2corrs_inds(N::Int)

for N operators, create an array of tuples of tuples that represents the terms in a sum of products of correlators. Each tuple corresponds to a sum term, see CorrTup_isless and CorrPerm_isless for details of the format. The returned array and terms are sorted such that if the N operators are sorted, the represented expression is also sorted with the conventions of the QuantumAlgebra package. This allows to directly return a normal-ordered form.

expval_as_corrs(expr::QuExpr)

Take an expression expr=A B C + D E... and write its expectation value in terms of correlations $⟨A⟩_c, ⟨B⟩_c, ⟨AB⟩_c, ⟨ABC⟩_c, \ldots$. Note that $⟨A⟩_c = ⟨A⟩$.

E.g., expval_as_corrs(a'(:n)*a(:n)) returns $⟨a^\dagger_n a_n⟩_c + ⟨a^\dagger_n⟩_c ⟨a_n⟩_c$ (which is equal to $⟨a^\dagger_n a_n⟩$), while expval_as_corrs(a'(:n)*a(:m)*a(:n)) returns $\langle a_{n}^\dagger a_{m} a_{n} \rangle_{c} + \langle a_{n}^\dagger \rangle_{c} \langle a_{m} \rangle_{c} \langle a_{n} \rangle_{c} + \langle a_{n}^\dagger \rangle_{c} \langle a_{m} a_{n} \rangle_{c} + \langle a_{m} \rangle_{c} \langle a_{n}^\dagger a_{n} \rangle_{c} + \langle a_{n} \rangle_{c} \langle a_{n}^\dagger a_{m} \rangle_{c}$.

See also: expval, corr

extindices(A) return externally visible indices of an expression

fermion_ops(name::Symbol): return function for creating fermionic annihilation and creation operators with name name (i.e., wrappers of FermionDestroy and FermionCreate)

heisenberg_eom(A,H,Ls=()) calculates $\dot{A} = i [H,A] + \sum_i (L_i^† A L_i - ½ \{L_i^† L_i, A\})$, where Ls = (L_1,L_2,...) is an iterable of Lindblad operators.

symmetric_index_nums(A) return sequence of numbers of exchange-symmetric indices

tlspm_ops(name::Symbol): return functions for creating jump operators for a two-level system with name name. The output of these functions depends on setting of use_σpm.

tlsxyz_ops(name::Symbol): return functions for creating Pauli operators for a two-level system with name name. The output of these functions depends on setting of use_σpm.

Avac(A::QuExpr), vacA(A::QuExpr)

Simplify operator by assuming it is applied to the vacuum from the left or right, respectively. To be precise, Avac(A) returns $A'$ such that $A'|0⟩ = A|0⟩$, while vacA(A) does the same for $⟨0|A$.

vacExpVal(A::QuExpr,S::QuExpr=1)

Calculate the vacuum expectation value $⟨0|S^\dagger A S|0⟩$, i.e., the expectation value $⟨ψ|A|ψ⟩$ for the state defined by $|ψ⟩= S|0⟩$`.

∑(ind,A::QuExpr): return (formal) sum of expression A over index ind.

@anticommuting_fermion_group name1 name2 ...: define a group of mutually anticommuting fermionic operators

@boson_ops name: define function $name for creating bosonic annihilation operators with name name (also defines deprecated $(name)dag, use $(name)' instead)

@fermion_ops name: define function $name for creating fermionic annihilation operators with name name (also defines deprecated $(name)dag, use $(name)' instead)

@tlspm_ops name: define functions $(name)m and $(name)p creating jump operators for a two-level system with name name.

@tlsxyz_ops name: define functions $(name)x, $(name)y, and $(name)z creating Pauli operators for a two-level system with name name.