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.

set_define_default_ops(t::Bool)

Set the preference for whether to define default operators (a, adag, f, fdag, σx, σy, σz, σp, σm) upon import. Default is true.

Note that changing this preference requires restarting the Julia session to take effect.

set_quindices_type(t::String)

Set the preference for the underlying type used to store indices. t must be either "Vector" (default) or "NTuple{N}" (where N is an integer).

• "Vector": Uses Vector{QuIndex}. Allows arbitrary number of indices but is slower due to heap allocation.
• "NTuple{N}": Uses NTuple{N,QuIndex}. Faster (stack allocation) but limits the number of indices per operator to a maximum of N.

Note that changing this preference requires restarting the Julia session to take effect.

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.