Non-Hermiticity and conservation of orthogonal relation in dielectric microcavity
Non-Hermitian properties of open quantum systems and their applications have attracted much attention in recent years. While most of the studies focus on the characteristic nature of non-Hermitian systems, one important point has been overlooked: A non-Hermitian system can be a subsystem of a Hermitian system as one can clearly see in Feshbach projective operator (FPO) formalism. In this case, the orthogonality of the eigenvectors of the total (Hermitian) system must be sustained, despite the eigenvectors of the subsystem (non-Hermitian) satisfy the bi-orthogonal condition. Therefore, one can predict that there must exist some remarkable processes that relate the non-Hermitian subsystem and the rest part, and ultimately preserve the Hermiticity of the total system. In this paper, we study such processes in open elliptical microcavities. The inner part of the cavity is a non-Hermitian system, and the outer part is the coupled bath in FPO formalism. We investigate the correlation between the inner- and the outer-part behaviors associated with the avoided resonance crossings (ARCs), and analyze the results in terms of the Lamb shift. The ARC structures we examined depend on a trade-off between the relative difference of self-energies (simply known as Lamb shifts in atomic physics) and collective Lamb shifts. While the collective Lamb shift is maximized in the region of the center of ARC, but the relative difference of self-energies is minimized, and this naturally induce a crossing of imaginary part of eigenvalues. These results come from the conservation of the orthogonality in the total Hermitian quantum system.
pacs:42.50.-p, 42.55.Sa, 42.50.Nn, 13.40.Hq, 05.45.Mt
Hermicity of physical observables is one of the basic principles in quantum mechanics. For given Hermitian operator, all of its eigenvalues are real, and its eigenvectors corresponding to different eigenvalues are orthogonal to each other. On the other hand, a non-Hermitian system, which is related to openness has complex eigenvalues and its eigenvectors satisfy the bi-orthogonal relation. Recently, various non-Hermitian systems and their properties have been extensively studied theoretically as well as experimentally M11 (); F58 (); R09 (); CK09 (); R10 (); PRSB00 (); CW15 (), especially in the fields of avoided resonance crossings (ARCs) RLK09 (); W06 (); SGLX14 (); SGWC13 (); WH06 (); RPPS00 (); BP99 (); B96 (), exceptional points (EPs) K66 (); SKM+16 (); CK17 (), -symmetric Hamiltonian systems BB98 (); GS09 (); XC17 (), phase rigidity BRS07 (), and bi-orthogonal relations C06 (); B14 (); CS03 (); L09 (). Since the Hermitian system and the non-Hermitian system has quite different properties to each other, they are often considered being distinct and independent from each other.
For an open system, one can consider a total system composed of the open system and a bath interacting with the system. In other words, the total system, which is Hermitian, is decomposed into two orthogonal subspaces; one is a non-Hermitian (sub-)system, and the other is a bath coupled to it. This decomposition is known as Feshbach projective operator (FPO) formalism F58 (). It was first introduced by Feshbach in 1958 for describing a decay process in nuclear physics, and has been extended to other systems such as quantum dots R09 (), microwave cavities PRSB00 (), and dielectric microcavities PKJ16 (); PKJ16+ (). In studies employing FPO formalism, a focus is usually made on the non-Hermitian part rather than the coupled bath. However, it should be noted that the Hermitian properties of the total system is maintained despite of the existence of a non-Hermitian (sub-)system. There must occur some correlations in the system-bath interaction that preserve the Hermiticity of the total system.
In this paper, we study such correlations for two-dimensional dielectric microcavities. Even though the dielectric microcavities are classical systems, they can be a good platform to study the wave-mechanical properties of quantum mechanics due to an isomorphic nature of wave equations between optics and quantum mechanics RBM12+ (); DMB16+ (). The total system consists of the inner-part of the cavity (non-Hermitian) and the outer-part of the cavity (bath). We investigate the correlations between the inner- and the outer-parts of the wavefunctions in the context of the avoided resonance crossing (ARC), in which the resonances of the non-Hermitian part are strongly interacting and thus undergoing dramatic changes. The ARC in an open system is a natural extension of the avoided level crossing (ALC) LL65 () occuring in a closed system. Moreover, we adopt ellipses as the boundary shape of the microcavities in order to see the manifestation of the openness nature clearly; For a closed elliptic billiard, since there are no internal interactions causing the real-valued energy repulsion H10 (); T89 (), no ALC occurs (the elliptic billiard may have the long-range avoided crossings known as Demkov-type in some cases KK17 (), but we only consider Landau-Zener-type avoided crossing here). On the other hand, in an open elliptic system, there can occur ARCs W06 ().
Next, we employ the Lamb shift to understand the findings in the above investigations. The Lamb shift formally describes the openness nature of a quantum system in the system-bath coupling LR47 (); WKG04 (); N10 (). Originally, this concept was known as a small difference in energy levels of a hydrogen atom in quantum electrodynamics, caused by the vacuum fluctuations LR47 (). However, it is recently found that there are two types of the Lamb shift. One is, so-called, the self-energy and the other is the collective Lamb shift SS10 (); RWSS12 (); R13 (). The self-energy is simply known as the Lamb shift in atomic physics. It is an energy-level shift arising from individual interaction of energy-level with its bath. On the other hand, the collective Lamb shift is an energy-level shift due to the interaction of energy-levels with each other via the bath. Our former works considered only self-energy in circular and elliptic dielectric microcavity PKJ16 (); PKJ16+ (). However, in this study, we consider both collective Lamb shift and self-energy, and show that a interplay between collective Lamb shift and relative difference of self-energies determines the essential features of ARC. Moreover, it will be shown that this interplay comes from the orthogonality of wavefunctions for the total Hermitian Hamiltonian.
Ii Correlation of the system wavefunction with that of the bath
Any wavefunctions of a Hermitian Hamiltonian corresponding to the total Hilbert space can be decomposed into two orthogonal subspaces, an open quantum system and a bath part, by using the Feshbach projective operator formalism. Let be a total (Hermitian) Hamiltonian with real eigenvalues , and represents the corresponding eigenvector of for the total system . Then, for each , we have
where is an eigenvector of non-Hermitian Hamiltonian in open quantum system and is a resonance tail in bath omitting the homogeneous solution (or the plane-wave term) (See details in Appendix A). In the case of dielectric microcavity, the is an eigenmode corresponding to the inner-part of the cavity and is a emission pattern corresponding to the outer-part of the cavity PKJ16 (), respectively. Then the inner product between two eigenvectors of the total Hamiltonian will be given by
which directly follows from the fact that , , i.e., it must be always orthogonal. Thus, and : It comes from that the subspaces and are mutually orthogonal. Note that the LHS of Eq. (2) is non-trivial, since the eigenvectors of the non-Hermitian Hamiltonian are bi-orthogonal, i.e., . (See also details in Appendix B.) The Eq. (2) means that when the inner product of eigenvectors of , , increases, that of resonance tail, , must vary in a way to cancel out for preserving orthogonal relation in total Hilbert space. In this way, the inner-part of the microcavity and the outer-part of it have an intimate correlation.
ii.1 Inner-part behaviors of microcavity
There are two kinds of the real-valued energies in the eigenvalue trajectories represented by grey solid and brown solid curves in Fig. 1a. The grey solid curves are the real-valued energies of eigenvalue trajectories for the elliptic billiard belonging to integrable system, whereas the brown solid curves are those for the dielectric microcavity or open quantum system.
The upper thick and solid grey curve shows an eigenvalue trajectory with radial quantum number of and angular quantum number of varying from circular to elliptic billiard depending on the eccentricity , which is defined by , where and are the major and minor axes of the ellipse, respectively. The upper thick and solid brown curve shows that of an elliptic dielectric microcavity with refractive index for InGaAsP. On the other hand, the lower thick solid grey and brown curves also show the eigenvalue trajectories with different quantum number such that and . They start from the eccentricity (circle) to (ellipse) showing real-valued energies crossing for the billiard and avoided crossing for microcavity around . Notice those facts in Fig. 1b, i.e., the extended version of a green solid box in Fig. 1a.
There are intensity plots of wavefunctions corresponding to the eigenvectors of elliptic billiards, in Fig. 1c for blue dots (from A1 to H1). All of them are entirely localized on the interior regions of the elliptic boundary, and have well-defined quantum numbers by an elliptic and hyperbolic coordinate. In Fig. 1d, intensity plots of wavefunctions for red dots (from A2 to H2) are presented. They do exit on the total space not inside elliptic boundary and do not have well-defined quantum number around ARC region. However, To focus on the properties of non-Hermitian Hamiltonian, we deal only inner parts of microcavity in this section. The interior wavefunctions of blue and red dots are quite similar to each other away from ARC region (self-engergy), while those of blue and red dots are quite different around ARC region (collective Lamb shift). The wavefunctions B1, C1, D1, F1, G1, and H1 retain their shape as stable bouncing-ball-type modes whereas the wavefunctions for C2 and G2 show pronounced deformation from bouncing-ball-type modes and [B2 H2, F2 D2] undergo mode exchanges due to coherent superpositions caused by the strong-interaction. The wavefunctions C2 and G2 are symmetric and anti-symmetric superpositions with B2 and F2 known as bowtie- and -type modes. These kinds of behaviors can be explained by three-type of Hamiltonians in Fig. 2.
ii.2 Outer-part behaviors of microcavity
We plot Husimi probability distributions and resonance tails of each resonance modes in Fig. 3a which corresponds to the resonance modes in Fig. 1d with the eccentricity , , and . It is well-known that the Husimi distributions below critical lines (’s) determine the shape of the resonance tails (i.e., the emission pattern) LYMLA07 (); SLY+07 (); CCSN00 (); SSWS+10 (). This fact is well-described in Fig. 3a, i.e., the shaded grey regions in each figures between critical lines () and the resonance tails provide our claims. Therefore, it is legitimate to compare the Husimi distributions below critical lines instead of resonance tails themselves which are extending to infinity.
More precisely, we can depict a Bhattacharyya distance B43 (); MMPZ08 () between two decay channels (or resonance tails) as a function of the eccentricity in Fig. 3b. The in general measures a similarity between two probability distributions and . Any given probability distributions, it is defined as
where the factor (Bhattacharyya coefficient) is given by Together with , we measure the degree of sharing of common decay channels by comparing the Husimi probability distributions below the critical lines. The extremal point of is lying at where the bouncing-ball-type modes (black circles) are mixed at bowtie-type and -type modes. It shows that the two mixed modes (i.e., bowtie- and -type mode; green circle in Fig. 3b) more resemble each other than the two bouncing ball-type modes, and simultaneously their imaginary part of the complex eigenvalues are crossing. Two quality factors , which are formally defined by (from the equation (16) in the Methods section), for red dashed curves in Fig. 1b, is shown Fig. 3c. These quality factors are obtained from inset c and they have same order of magnitude.
ii.3 Correlation of the conservation with Lamb shift and avoided crossing
The correlation in conservation of the orthogonal relation for dielectric microcavity is shown in Fig. 4 as a trade-off between the wavefunctions’ overlap for inner-part of the microcavity and the Bhattacharyya distance for outer-part of the cavity. The overlap in Fig. 4a is obtained from as a function of the eccentricity . It shows a parabolic curve whose maximum value is near at the center of the avoided resonance crossing () where two bouncing-ball-type modes are mixed at bowtie- and -type modes, respectively and decrease as deviating from the returning to bouncing-ball-type modes. An orange star line in bottom in Fig. 4a clearly shows a bi-orthogonal relation:. Instead of the inner product between resonance tails at infinity , we use the Bhattacharyya distance in Fig. 4b to quantify the similarity between two resonance tails and it shows a symmetric curve with respect to the overlap curve in Fig. 4a. This symmetric shape between the overlap curve and the Bhattacharyya distance shows certain correlation between inner-part of and outer-part of the microcavity as a result of conservation of the orthogonal relation for the total Hilbert space. In this paper, we insist that this correlation can be used to understand the structure of the Lamb shift and avoided crossing relating to the non-Hermitian Hamiltonian itself.
As mentioned in Appendix A for the details of non-Hermitian (effective) Hamiltonian, let us now write as matrix form with respect to the eigenbasis of explicitly to understand the Lamb shift and avoided crossing in Fig. 1, under the continuous change the parameter , as
with (Note that is Hermitian). The first matrix represents Hamiltonian of a closed quantum billiard system and second one does system-bath interaction. Since is a Hamiltonian of a closed quantum billiard system, it is a Hermitian matrix and has real eigenvalues (See Fig. 2). Especially, if the billiard systems are one of the integrable systems H10 (); T89 (), there are no interactions in it so that the off-diagonal term (internal coupling) must be vanished giving . For convenience, we omit the parameter in the case of the meaning of the variable is obvious. When the eccentricity is equal to a crossing point, i.e., , the eigenvalue of are degenerated with , but not eigenvectors. This explains the process of thick grey curves in Fig. 1. On the other hand, the second matrix is quite different from that of first one. It is a Hamiltonian due to system-bath coupling with complex entries. When the Hamiltonian represents time reversal system, it becomes symmetric so that is same to L09 (). Therefore the final form of matrix in our case under the parameter is given by
where . Then, there are primarily three types of interactions depending on BP99 (); RPPS00 (); B96 (). First, ’s are pure real leading to the repulsion of real parts and the crossing of imaginary parts in the complex energy. Second, are pure imaginary leading to a repulsion of imaginary parts and a crossing of real parts. Third, are the complex number resulting in the repulsion of both parts simultaneously. Since we here deal with only first case (strong coupling), we must set to be . We can straightforwardly diagonalize the Eq. (5) above, and then we obtain a following equation satisfying
where . Note that the the diagonal eigenvalues denote the eigenvalues of the system’s Hamiltonian . Here, we assume that is negligible, i.e., . This is a reasonable assumption that the real part of the complex energy is repulsion, but the imaginary part is crossing with RPPS00 (); BP99 (); B96 () and consider . Then, the repulsion of real part near the center of ARC () given by:
The first term of the square root is corresponding to the relative difference of self-energy, and the second term is corresponding to the collective Lamb shift, respectively. It is important to note that the physical meaning of the relation, , is the relative difference of with interaction of the system-bath at each energy levels, instead of just difference of unperturbed eigenvalues, . In our previous work PKJ16+ (), we confirmed that the crossings of self-energies of resonances, i.e., take place in the region where the has extremal point with same order of quality factors of the modes. Therefore, the relative difference of the self-energies as a first part of the square root have zero values near , and so it is becoming larger as far from . At the same time, at extremal point, i.e., the most similar decay channel leads to the crossing of imaginary parts (). Now we focus on a relation between the wavefunctions’ overlap and the (i.e., the collective Lamb shift). The overlap is maximized near at the center of ARC () due to a mixing of wavefunctions by collective Lamb shift. Thus this fact reveals that (the off-diagonal component) is also maximized near at . It indicates that these off-diagonal terms—an interaction via the bath—become prominent in the regime of overlapping resonance R09 (); R01 (); ER13 (). From these results, we can conclude that the first term of the square root is minimized leading a crossing of imaginary part when the second term of the square root is maximized, and there exits a trade-off relation between those two terms for the conservation of orthogonal relation in the total Hilbert space.
We have investigated a specific correlation between inner-part of and outer-part of a dielectric microcavity as a consequence in conservation law of orthogonal relations for the total (Hermitian) Hilbert space. This correlation is non-trivial, because the non-Hermitian system with bi-orthogonal condition is always a subpart of the total Hilbert space. That is, the Hermitian total space can be decomposed into two orthogonal subspaces, one is a non-Hermitian system as an inner-part of the microcavity and the other is a resonance tail as an outer-part of the cavity lying on the bath. Since the inner product of wavefunctions in the non-Hermitian with different eigenvalues is non-zero in general, so that of resonance tails in bath must vary to remove this non-zero value for preserving the orthogonal relation in the total Hilbert space.
Finally, we applied this correlation to the Lamb shift and avoided crossing relating to the non-Hermitian Hamiltonian. As a result, we confirm that, on the contrary of the typical Hamiltonian model, our case not only the off-diagonal-interaction terms but also the diagonal-interaction terms contributes together to avoided crossings under the given non-Hermitian Hamiltonian. The off-diagonal-interaction—the collective Lamb shift—which is related to the inner product of inner-part of the cavity is maximized at the center of ARC, when it measured by the overlap of wavefunctions. But, the diagonal-interaction—the relative difference of self-energies—which is related to the outer-part of the microcavity is minimized (also leading to the crossing of imaginary part), when it measured by Bhattacharyya distance near at the center of ARC, respectively. The trade-off between these two terms determines the structure of the avoided crossing such that while the collective Lamb shift is dominant near at the center of ARC (), the relative difference of self-energies is becoming more dominant as growing apart from, and it also explains the origin of the crossing of imaginary part near at the center of the ARC.
We are grateful to Kyungwon An and Ha-Rim Kim for valuable comments. This work was supported by a grant from Samsung Science and Technology Foundation under Project No. SSTF-BA1502-05. We thank Korea Institute for Advanced Study for providing computing resources (KIAS Center for Advanced Computation Linux Cluster) for this work. J.K. acknowledges financial support by the KIST Institutional Program (Project No. 2E26680-16-P025). H.J. and K.J. also acknowledge financial supports by the National Research Foundation of Korea (NRF) through a grant funded by the Korea government (MSIP) (Grant No. 2010-0018295) and by the KIST Institutional Program (Project No. 2E26680-16-P025). In addition, K.J. acknowledges financial support by the National Research Foundation of Korea (NRF) through a grant funded by the Korean government (Ministry of Science and ICT) (NRF-2017R1E1A1A03070510 and NRF-2017R1A5A1015626).
Appendix A. Derivation of the non-Hermitian Hamiltonian and Lamb shift
In this Appendix, we recapitulate the non-Hermitian quantum mechanics for an elliptic dielectric microcavity. First of all, let us consider a time-independent Schrödinger equation with its total Hilbert space composed of two subsystems as follows:
where is a total (Hermitian) Hamiltonian with (real energy) eigenvalues , and represents the corresponding eigenvector of on a given (total) system . For convenience, each subspaces and denote a closed quantum system and a bath (or an environment), respectively.
The first subspace corresponds to a discrete state of closed quantum system , and the second one is a continuous-scattering state of the bath such that projection operators, and , satisfy and . Here, is a projection operator onto the closed quantum system whereas is a projection onto the bath. The operator is an identity operator defined on the total space . With these projection operators, we can define useful matrices such as , , , and .
The total Hamiltonian in equation (8) can be represented by a matrix
where and denote the Hamiltonians of the closed quantum system and the bath with eigenvalues and eigenvectors such that and , respectively. Here, and are interaction Hamiltonians between the system and bath, respectively. The total eigenvector on is also given by
Notice that and , and typically we have used usual addition ’’ notation instead of the direct sum ’’. (e.g. See the formalism as in Ref. R09 ().) By using these definitions, the Schrödinger equation of equation (8) can be rewritten F58 (); R09 () in the form of
Also note that the states restricted on and after the projections are given by R09 ()
where —in general, it is called by homogeneous solution or plane wave—is the eigenvector of , and is an out going Green function in the subspace . The equation (12) means that the eigenvector localized in subsystem can be obtained through the incoming vector penetrating into the subsystem via coupling term and propagating by effective Green function .
Then, by using the total Hamiltonian in equation (9), we can formally define the effective non-Hermitian Hamiltonian as
where means the (Cauchy) principal value depending on each decay channels.
This non-Hermitian Hamiltonian has generally complex eigenvalues under the Feshbach projection-operator (FPO) formalism: (See Ref. PKJ16 () and the references therein.)
and its complex energies (eigenvalues) of are given by (for each )
Now we consider the Lamb shift. The Lamb shift, which is a small energy difference between a closed and an open system due to the system-bath interaction, can be also obtained by the effective non-Hermitian Hamiltonian in equation (14) PKJ16 (); PKJ16+ (); R13 (). That is, it is the difference between the eigenvalues of and the real part of the eigenvalues of :
In general, while itself has real eigenvalues, the effective non-Hermitian Hamiltonian has complex energy eigenvalues. Thus we consider a complex matrix as followings:
If we consider the case of a two-dimensional vector space, the non-Hermitian Hamiltonian with respect to a given eigenbasis of , i.e., can be represented by toy-Hamiltonian-model as follows ():
For more additional works, above equation can be reordered as (Note that and below indicate diagonal and off-diagonal matrix of the effective Hamiltonian, respectively.)
where each components have following forms (actually this division of conforms our main suggestion)
where , thus we have the real value and the imaginary value for any complex number .
In Refs. SS10 (); R13 (), the term ‘self-energy’ due to the diagonal components was known as ‘Lamb shift’ in atomic physics. On the other hand, the off-diagonal terms were known as ‘collective Lamb shift’. That is, the self-energy is an energy shift due to the individual energy-level interactions with the bath independently, while due to the energy-level interactions via the bath each other, is called the collective Lamb shift by Rotter R13 ().
Appendix B. Bi-orthogonality condition
For convenience, let us abbreviate as
where and satisfying non-Hermitian Hamiltonian with eigenvectors and eigenvalues in the form of
We also assume that the eigenvalues of are not degenerate. In addition to the eigenvector of , it will be convenient to invoke an eigenvector of the Hermitian adjoint matrix in the form of
The reason for invoking the additional states is since the eigenvector of are not orthogonal in general. This result can be conformed by noting the facts:
Therefore, the identity operators are given by
where we note that, in our case, the effective Hamiltonian has always full-rank. By using this relations, we will get the following results:
Due to the fact that the vector is an eigenvector of not of and , the numerators and are non-zero in general. As a result, the inner product are non-zero B14 (). So an analogous result can be found as
The eigenvectors with different eigenvalues belonging to non-Hermitian matrices are not orthogonal. Thus, it is natural and desired to establish an orthogonal conditions again. To do so, let us introduce ‘right’ and ‘left’ eigenvectors of the non-Hermitian Hamiltonian, which are different from each other in general. First of all, if we consider equation (15) again
then, by multiplying to the left-hand side with left eigenvectors , one obtains that
where the right and left eigenvectors are assumed to be orthonormal. Thus we can easily check that
which is corresponding to the Schrödinger equation for the left bra-vectors. If we apply a dagger operation to this vector, we get
In the case of Hermitan Hamiltonian operator, that is, , it directly lead to is real and . However, when is not the case of Hermitian Hamiltonian, i.e., , furthermore, if is symmetric,
then, this gives birth to the relation between the left and right eigenvectors
This is a typical physical system because time reversal condition is usually satisfied L09 (). In fact, by comparing the previous results, we can notice that
This bi-orthogonal relation is dominant as approaching the exceptional point, whereas nearly orthogonal () away from the exceptional point in non-Hermitian Hamiltonian R09 ().
Appendix C. Relation between Lamb shift and ARC
To find out clues of relation between Lamb shift and ARC, let us consider relative difference between a pair of Lamb shift and that of the real-valued energies in eigenvalue trajectories for microcavity depending on simultaneously. To do that, we define as follows:
Then corresponds to thick brown solid curves in Fig. 1a and do to thick grey solid curves in Fig. 1a, respectively. Their Lamb shift for each are shown by the inset of Fig. 5a. The violet solid curve is a which is a Lamb shift of upper adjacent thick grey and thick brown solid curves in Fig. 1a while magenta solid one is a which is a that of lower adjacent thick grey and thick brown solid curves, respectively. Then we can define the relative difference of Lamb shift depending on the eccentricity as
This one is depicted as a black solid curve in Fig. 5a. Similarly,
This one is a red solid curve depicted in Fig. 5a.
At the center of ARC (), we get
It means that the real-valued energies difference for open quantum system consists of two part, i.e., first one is a relative difference of the real-valued energies of the billiard and second one is that of the Lamb shift. The center of ARC exits around crossing point in general, but in order to get deeper understanding clearly, let us consider the circumstance when two points are almost coincident (); thus, . Then the difference of real-valued energies for open system is exact difference of Lamb shift: At the eccentricity ,
These are clearly shown in Fig. 5b. The crossing point () and the center of ARC () are same in so that is equal to with .
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