The Effective Hamiltonian Method in the Thermodynamics of Two Resonantly Interacting Quantum Oscillators

Abstract

We investigate the classical problem of two resonantly interacting oscillators each of which is coupled to “its own” heat bath based on the effective Hamiltonian method and the quantum stochastic differential equation (in contrast to the well-known “global” and “local” approaches). We show that in the second order of the algebraic perturbation theory, each of the oscillators turns out to be also coupled to the “foreign” heat bath. We calculate the steady-state heat flows and prove that there is no heat flow from the cold heat bath to the hot one, as evidenced by some results of the local approach.

Technique for Calculating the Terms of the Effective Hamiltonian

It is convenient to write the interaction operators with the broadband fields as

$$\begin{gathered} {{V}_{c}}(t) = {{\gamma }_{c}}\sum\limits_{\forall \omega \ne 0}^{} {(c{{e}^{{ - i{{\omega }_{c}}t}}} + {{c}^{\dag }}{{e}^{{i{{\omega }_{c}}t}}}){{a}_{\omega }}{{e}^{{ - i\omega t}}},} \\ {{a}_{{ - \omega }}} = a_{\omega }^{\dag },\quad \omega > 0, \\ \end{gathered} $$

$$\begin{gathered} {{V}_{r}}(t) = {{\gamma }_{r}}\sum\limits_{\forall \omega \ne 0}^{} {(r{{e}^{{ - i{{\omega }_{r}}t}}} + {{r}^{\dag }}{{e}^{{i{{\omega }_{r}}t}}}){{a}_{\omega }}{{e}^{{ - i\omega t}}},} \\ {{b}_{{ - \omega }}} = b_{\omega }^{\dag },\quad \omega > 0. \\ \end{gathered} $$

The terms changing slowly with time are then

$$V_{c}^{'}(t) = {{\gamma }_{c}}\sum\limits_{\omega \in ({{\omega }_{c}})}^{} {(ca_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{c}})t}}} + {{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}})} ,$$

$$V_{r}^{'}(t) = {{\gamma }_{r}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {(rb_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{r}})t}}} + {{r}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}})} .$$

They define the operators of the first order in coupling constants.

The rapidly changing terms can be represented as

$$V_{c}^{{''}}(t) = {{\gamma }_{c}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {c{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}} + {{\gamma }_{c}}\sum\limits_{\forall \omega \notin ({{\omega }_{c}})}^{} {{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}},} $$

$$V_{r}^{{''}}(t) = {{\gamma }_{r}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{r}})}^{} {r{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}} + {{\gamma }_{r}}\sum\limits_{\forall \omega \notin ({{\omega }_{r}})}^{} {{{r}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}},} $$

where (ω_c) is the region with sizes |ω – ω_c| ≪ ω_c near ω = ω_c, but no less than the reciprocal rate of relaxation via this channel of coupling with the broadband field. The regions of the continuous spectrum (–ω_c), (ω_r), and (–ω_r) are defined similarly.

Identifying the rapidly and slowly changing terms in the interaction operator between the oscillators causes no difficulties at all in the case where the oscillator frequencies are equal, ω_c = ω_r:

$$V_{{c - r}}^{'}(t) = g(c{{r}^{\dag }} + {{c}^{\dag }}r),$$

$$V_{{c - r}}^{{''}}(t) = g({{c}^{\dag }}{{r}^{\dag }}{{e}^{{i({{\omega }_{c}} + {{\omega }_{r}})t}}} + cr{{e}^{{ - i({{\omega }_{c}} + {{\omega }_{r}})t}}}).$$

The first-order transformation generators are found from the relations

$$\frac{{d{{S}^{{(1,0,0)}}}(t)}}{{dt}} = - {{\hbar }^{{ - 1}}}V_{{c - r}}^{{''}}(t),$$

$$\frac{{d{{S}^{{(0,1,0)}}}(t)}}{{dt}} = - {{\hbar }^{{ - 1}}}V_{c}^{{''}}(t),$$

$$\frac{{d{{S}^{{(0,0,1)}}}(t)}}{{dt}} = - {{\hbar }^{{ - 1}}}V_{r}^{{''}}(t).$$

They are fast functions of time. Under the assumption of adiabatic interaction switching, the first-order transformation generators are defined by the expressions

$${{S}^{{(1,0,0)}}}(t) = cr\frac{{g{{e}^{{ - i({{\omega }_{c}} + {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}} - {{c}^{\dag }}{{r}^{\dag }}\frac{{g{{e}^{{i({{\omega }_{c}} + {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}},$$

$${{S}^{{(0,1,0)}}}(t)\, = \,{{\gamma }_{c}}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}}}{{i\hbar (\omega + {{\omega }_{c}})}}} \, + \,{{\gamma }_{c}}\sum\limits_{\omega \notin ({{\omega }_{c}})}^{} {\frac{{{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}}}{{i\hbar (\omega - {{\omega }_{c}})}}} ,$$

$${{S}^{{(0,0,1)}}}(t)\, = \,{{\gamma }_{r}}\sum\limits_{\omega \notin ( - {{\omega }_{r}})}^{} {\frac{{r{{b}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{r}})t}}}}}{{i\hbar (\omega + {{\omega }_{r}})}}} \, + \,{{\gamma }_{r}}\sum\limits_{\omega \notin ({{\omega }_{r}})}^{} {\frac{{{{r}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}}}{{i\hbar (\omega - {{\omega }_{r}})}}} .$$

Then,

$$\begin{gathered} \text{[}{{S}^{{(1,0,0)}}}(t),V_{c}^{{''}}(t)]' = \frac{{g{{\gamma }_{c}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}} \\ \times \sum\limits_{\omega \in ({{\omega }_{r}})}^{} {ra_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{r}})t}}}} + \frac{{g{{\gamma }_{c}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {{{r}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}} , \\ \end{gathered} $$

$$\begin{gathered} \text{[}{{S}^{{(0,1,0)}}}(t),V_{{c - r}}^{{''}}(t)]' \\ = {{\gamma }_{c}}g\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {\frac{{a_{\omega }^{\dag }r{{e}^{{i(\omega - {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}}} + {{\gamma }_{c}}g\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {\frac{{{{a}_{\omega }}{{r}^{\dag }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}}} . \\ \end{gathered} $$

As a result, the term that describes the coupling of the single-model oscillator with frequency ω_r to the foreign heat bath appears in the effective Hamiltonian:

$$\begin{gathered} {{{\tilde {H}}}^{{(1,1,0)}}}(t) = - \frac{{g{{\gamma }_{c}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {{{r}^{\dag }}c{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}} \\ - \frac{{g{{\gamma }_{c}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {ra_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{r}})t}}}} . \\ \end{gathered} $$

Here, we set ω_c = ω_r ≈ ω.

Similarly, it is easy to obtain

$$\begin{gathered} {{{\tilde {H}}}^{{(1,0,1)}}}(t) = - \frac{{g{{\gamma }_{r}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{c}})}^{} {{{c}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}} \\ - \frac{{g{{\gamma }_{r}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{c}})}^{} {bc_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{c}})t}}}} . \\ \end{gathered} $$

The operator ${{\tilde {H}}^{{(0,1,1)}}}$(t) describes the second-order coupling between the heat baths that does not affect the open system and, therefore, below it is disregarded.

Other, possibly relevant, second-order terms are

$$\begin{gathered} {{{\tilde {H}}}^{{(2,0,0)}}}(t) = - \frac{i}{2}[{{S}^{{(1,0,0)}}}(t),V_{{c - r}}^{{''}}(t)]' \\ = - \frac{{{{g}^{2}}}}{{2\hbar {{\omega }_{c}}}}({{c}^{\dag }}c + {{r}^{\dag }}r + 1), \\ \end{gathered} $$

$${{\tilde {H}}^{{(0,2,0)}}}(t) = - \frac{i}{2}[{{S}^{{(0,1,0)}}}(t),V_{c}^{{''}}(t)]',$$

$$\begin{gathered} \text{[}{{S}^{{(0,1,0)}}}(t),V_{c}^{{''}}(t)]' \\ = \left[ {{{\gamma }_{c}}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}}}{{i\hbar (\omega + {{\omega }_{c}})}}} } \right. + {{\gamma }_{c}}\sum\limits_{\omega \notin ({{\omega }_{c}})}^{} {\frac{{{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}}}{{i\hbar (\omega - {{\omega }_{c}})}}} , \\ \left. {{{\gamma }_{c}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}} + {{\gamma }_{c}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}} } \right] \\ = \left[ {{{\gamma }_{c}}\sum\limits_{\omega \notin ({{\omega }_{c}})}^{} {\frac{{{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}}}{{i\hbar (\omega - {{\omega }_{c}})}},{{\gamma }_{c}}\sum\limits_{\forall \omega ' \notin ( - {{\omega }_{c}})}^{} {c{{a}_{{\omega '}}}{{e}^{{ - i(\omega ' + {{\omega }_{c}})t}}}} } } \right] \\ + \left[ {{{\gamma }_{c}}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}}}{{i\hbar (\omega + {{\omega }_{c}})}},{{\gamma }_{c}}\sum\limits_{\forall \omega ' \notin ({{\omega }_{c}})}^{} {{{c}^{\dag }}{{a}_{{\omega '}}}{{e}^{{ - i(\omega ' - {{\omega }_{c}})t}}}} } } \right]. \\ \end{gathered} $$

As a result, we have the contributions ${{\tilde {H}}^{{(0,2,0)}}}$(t) and ${{\tilde {H}}^{{(0,0,2)}}}$(t) to the effective Hamiltonian:

$$\begin{gathered} {{{\tilde {H}}}^{{(0,2,0)}}}(t) = - \gamma _{c}^{2}({{c}^{\dag }}c + 1)\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {\frac{1}{{\hbar (\omega + {{\omega }_{c}})}}} \\ - \gamma _{c}^{2}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{a_{\omega }^{\dag }{{a}_{{\omega '}}}{{e}^{{i\omega t}}}{{e}^{{ - i\omega 't}}}}}{{2\hbar (\omega + {{\omega }_{c}})}} - \gamma _{c}^{2}\sum\limits_{\substack{ \omega ' \notin ( - {{\omega }_{c}}) \\ \forall \omega ' \notin ( - {{\omega }_{c}})} } ^{} {\frac{{a_{{\omega '}}^{\dag }{{a}_{\omega }}{{e}^{{i(\omega ' - \omega )t}}}}}{{2\hbar (\omega + {{\omega }_{c}})}},} } \\ \end{gathered} $$

while the operator ${{\tilde {H}}^{{(0,0,2)}}}$(t) is derived from ${{\tilde {H}}^{{(0,2,0)}}}$(t) by the simultaneous substitutions of the subscripts and the like operators c → r and broadband field operators a_ω → b_ω. The operators ${{\tilde {H}}^{{(0,2,0)}}}$(t) and ${{\tilde {H}}^{{(0,0,2)}}}$(t) can be represented by quantum counting processes similarly to [30, 31, 33]. These processes obey a different algebra of Ito differentials [33] that generalizes the well-known Hudson–Parthasarathy algebra [34]. If we use the results [30, 31, 33] of an analysis of their role, along with Wiener processes, then it can be stated that the terms ${{\tilde {H}}^{{(0,2,0)}}}$(t) and ${{\tilde {H}}^{{(0,0,2)}}}$(t) in the statement of the problem considered here should be neglected.