Quantum process tomography of molecular dimers from two-dimensional electronic spectroscopy I: General theory and application to homodimers
Is it possible to infer the time evolving quantum state of a multichromophoric system from a sequence of two-dimensional electronic spectra (2D-ES) as a function of waiting time? Here we provide a positive answer for a tractable model system: a coupled dimer. After exhaustively enumerating the Liouville pathways associated to each peak in the 2D-ES, we argue that by judiciously combining the information from a series of experiments varying the polarization and frequency components of the pulses, detailed information at the amplitude level about the input and output quantum states at the waiting time can be obtained. This possibility yields a quantum process tomography (QPT) of the single-exciton manifold, which completely characterizes the open quantum system dynamics through the reconstruction of the process matrix. This is the first of a series of two articles. In this manuscript, we specialize our results to the case of a homodimer, where we prove that signals stemming from coherence to population transfer and viceversa vanish upon isotropic averaging, and therefore, only a partial QPT is possible in this case. However, this fact simplifies the spectra, and it follows that only two polarization controlled experiments (and no pulse-shaping requirements) suffice to yield the elements of the process matrix which survive under isotropic averaging. The angle between the two site transition dipole moments is self-consistently obtained from the 2D-ES. Model calculations are presented, as well as an error analysis in terms of the angle between the dipoles and peak overlap. In the second article accompanying this study, we numerically exemplify the theory for heterodimers and carry out a detailed error analysis for such case. This investigation provides an important benchmark for more complex proposals of quantum process tomography (QPT) via multidimensional spectroscopic experiments.
Multidimensional optical spectroscopies (MDOS) provide very powerful tools to study excited state dynamics of multichromophoric systems in condensed phases. These techniques distribute spectral features along several dimensions, uncluttering data which would otherwise appear obscured in linear spectroscopies and simultaneously yielding novel information on the dynamics of the probed system mukamel (). Possibilities in multidimensional techniques include decongesting spectral lineshapes, differentiating between homogeneous and inhomogeneous broadening mechanisms, providing unambiguous signatures about couplings between chromophores, and yielding signatures of coherent and incoherent processes involving excited states at the amplitude level minhaengbook (); minhaeng (). Although MDOS have been historically inspired by their NMR analogues, the timescales of the physical and chemical processes studied through MDOS are quite different from the ones in NMR mukamelacc (); jonas (); mukamel_accchemres (); harel (). The characteristic timescales of NMR are milliseconds, a resolution that does not allow for the observation of a wide variety of chemical dynamics in condensed phases ocurring in the orders of femto and picoseconds. On the other hand, femtosecond timescales can be easily accessed with ultrafast optical techniques. Examples of phenomena studied via MDOS are vast and include molecular reorientation processes and solvation dynamics prlliquids (); fayer-accchemres (), electron transfer moran-et (), vibrational coherences in organometallic complexes khalil:362 (); khaliljpc (); Baiz2009 () or halogens in rare gas matrices bihary (); guhr-bromine (), phonon dynamics in carbon nanotubes nanotube-uosaki (), protein unfolding kinetics tokmakoffprotein (), excitonic dynamics in light-harvesting systems engelchicago (); engelfleming (); scholes (); moran (); hauer_carotene (); kauffmann_accchemres () and organic polymers ppv (); collinischoles (), as well as many-body physics in semiconductor quantum wells Stone05292009 (); turner-nelson (); turnerjcp () and quantum dots Wong2010 ().
Traditionally, the spectroscopy of condensed phases is formulated as a response problem: The molecular system is perturbed with a sequence of short laser pulses and the coherent polarization response due to this set of perturbations (nonlinear polarization) is subsequently measured mukamel (). Informally, we can describe the exercise as ’kicking’ the quantum black box (molecular system) and ’listening to the whispers’ (measuring the response) due to the kicks, from which some properties of the box can be inferred. This description of spectroscopy is reminiscent to an idea stemming from the quantum optics and quantum information processing (QIP) communities, namely, quantum process tomography (QPT) nielsenchuangbook (); nielsenchuang (); dcqd (); resources (). Broadly speaking, QPT is a systematic procedure to characterize a quantum black box by sending a set of inputs, measuring their outputs, and analyzing the functional relationships between them. With the increasing effort of quantum engineering of gates and devices, QPT constitutes a cornerstone of QIP theory and experiment, as it provides a necessary check on the performance of the respective quantum black boxes. A natural question arises from the comparison of the two aforementioned concepts: Can the spectroscopy of condensed phases be formulated as a QPT? In a previous study yuenzhou (), we provided an affirmative answer to this question, at least for a molecular dimer. We showed that a set of two-color polarization controlled rephasing photon echo experiments is sufficient to reconstruct the density matrix elements associated with the dynamics of the single exciton manifold, and therefore, systematically characterize the excited state dynamics of the dimer, which can be regarded as the black box. For pedagogical reasons, we found it simpler and more convenient to concentrate our attention in the real time picture of the experiment, to make an explicit identification of the preparation, evolution, and detection steps of the QPT, with the coherence, waiting, and echo times, respectively. However, due to the widespread practice of displaying partially Fourier transformed data of the nonlinear optical polarization with respect to certain time intervals, it is worthwhile translating our results to the more visual two-dimensional electronic spectrum (2D-ES), and in fact, this is one of the main results of this work.
The present article is organized as follows: We begin in section I with a review of some relevant ideas of QPT and also introduce the process matrix as the main object to be reconstructed by means of QPT. In section II, relevant details on the dimer model system are presented. Section III describes the rephasing heterodyne photon echo experiment for the dimer and explains that the collected macroscopic polarization signal is a linear combination of elements of the process matrix at the waiting time . This implies that QPT can be performed by repeating several experiments with different pulse parameters in order to extract these elements. In section IV, the ideas of section III are mapped into the language of a 2D-ES, where each of the diagonal and cross peaks is associated with a set of elements of , and each of the axes of the spectrum can be associated with a preparation and a detection stage. Finally, in section V, these ideas are specialized to homodimer systems where, after isotropic averaging, only a partial QPT is possible, as some elements of are undetectable. Nevertheless, we note that the partial QPT is easily realized with current experimental capabilities, since it can be reconstructed with only two spectra resulting from different pulse polarization configurations for each given waiting time. The angle between the site dipoles is self-consistently obtained from these spectra, and an error analysis based on this angle as well as peak overlaps is carried out. Numerical calculations on a secular Redfield dissipation model are presented. A detailed analysis for heterodimers is carried out in the next article accompanying this investigation. Extensions of the procedure to account for inhomogeneous broadening, more sophisticated signal analysis, as well as bigger systems, are discuss at the end of this manuscript.
I Relevant concepts of Quantum Process Tomography and general definitions
Consider an arbitrary quantum system (quantum black box) interacting with an environment. We are interested in its evolution as a function of time in the form of a reduced density matrix . Very generally, this evolution is a linear transformation on the initial quantum state Choi1975285 (); sudarshan ():
is the central object of this article, and shall be called process matrix. Eq. (1) can be regarded as an integrated equation of motion for every 111We label time with instead of because the QPT we propose is identified with the waiting time of the PE experiment.. Eq. (1) can be expressed in terms of a basis for the Liouville space of the system:
For purposes of this article, we present two useful definitions. Consider the Liouville space of the system, and classify the vectors of in proper and improper density matrices. A state or a density matrix is proper if it satisfies all the conditions of a physical quantum state; namely, this is Hermitian, positive semidefinite, and has trace one. An improper state is any matrix that lives in the Liouville space but is not a proper density matrix. Clearly, any improper density matrix in the same Liouville space may be written as a unique linear combination of proper density matrices. In principle, Eq. (2), being a physical equation of motion, is restricted to the domain of proper density matrices . However, by linearity, its extension to any linear combination of proper states is well defined, so its validity for improper density matrices is not under question.
The meaning of the process matrix is easy to grasp: Conditional on the initially state being prepared at , is the value of the entry of the quantum state after time , , i.e. . Therefore, denotes a state to state transfer amplitude. Note that is an improper density matrix if (coherences on their own are not valid quantum states). However, improper states are not necessarily unphysical as one expects at a first glance. Most of our intuition for nonlinear spectroscopies in the perturbative regime stems from the consideration of how a perturbative amplitude created at a certain entry of the total (proper) density matrix is transferred to other entries due to free evolution, as time progresses mukamel (). It is not the evolution of the total density matrix (which to leading order is unperturbed, mostly in its ground state, and not yielding a time-dependent dipole) what is effectively monitored in the phase-matched signal, but the evolution of an effective density matrix, such as , which can be improper. Terms such as transfer from population to population, coherence to coherence, population to coherence, and coherence to population are all ubiquitous in the jargon of MDOS. However, the monitoring of the latter is often ambiguous, incomplete, and in most cases, qualitative. Obtaining quantitative information about these events amounts to finding each of the elements of .
The transformation in Eq. (2) is limited by two classes of restrictions for the process matrix associated with Hermiticity and trace preservation:
We derive these conditions in Appendix A, but their content is intuitive: If is a proper density matrix, remains as a valid quantum state as evolves if these two requirements are preserved. In particular, elements of the form , which denote population transfers, are purely real as one expects, whereas the other elements are in general complex. As a comment to our previous discussion, by linearity, these conditions must also be satisfied even if is improper (notice that does not depend on ).
Equations (1) and (2) are remarkable because they guarantee that, in principle, if is known, the quantum black box described by is perfectly understood, as it predicts by linearity the evolution of an arbitrary initial state of . Although describes an open quantum system, details about the environment evolution need not be included explicitly, but only in an averaged sense in the elements of . We shall operationally define QPT as any procedure to reconstruct . A possible QPT is the following: (a) Prepare a linearly independent set of states that spans ; (b) for each of the prepared states, wait for a free evolution time and determine the density matrix at that time. Any protocol for determining a density matrix for a system is called Quantum State Tomography (QST) statetomography (); walmsley (); holography (); cinaprl (); cinapolyatomic (). In essence, QPT can be carried out for any system if both a selective preparation of initial states and QST can be achieved. Variants of this methodology exist although all of them operate within the same spirit nielsenchuangbook (); nielsenchuang (); dcqd (); resources (). QPT has been successfully implemented in a wide variety of experimental scenarios, including nuclear magnetic resonance cory (); childs (); veeman (), ion traps blatttraps (), single photons whitephotons (); steinbergphotons (), solid state qubits solidstate (), optical lattices opticallattices (), and Josephson junctions natphysqpt (). In this article, we show how to perform QPT for a model coupled heterodimer using two-color polarization controlled heterodyne photon-echo experiments, extending the domain of application of QPT to systems of chemical and biophysical interest.
Ii Model system: Coupled dimer
where and are creation and anhilation operators for a single Frenkel exciton in the site , are the first and second site energies, and is the coupling between the chromophores.
The standard diagonalization of this Hamiltonian, which is effectively a two-level system for the single-exciton manifold, follows from defining some convenient parameters: The average of the site energies , the difference , and the mixing angle . By introducing the operators:
the Hamiltonian in Equation (5) can be readily written as:
where the eigenvalues and of the single excitons are:
Denoting as the molecular ground state or the excitonic vaccuum, and are the excitons at each site, whereas , are the delocalized excitons. The biexcitonic state, expressed by , also plays a role in our study, as it is resonantly accesed through excited state absorption (ESA) after several pulses. Notice that the Hamiltonian does not contain two-body operators, and therefore does not account for exciton-exciton binding or repulsion terms, so the energy level of the biexciton is just the sum of the two exciton energies, minhaengbook (). Defining , the following relations hold:
Since we are concerned with the interaction of the chromophores with electromagnetic radiation, we make some remarks on the geometry of the transition dipoles (see Fig. 1). Let . Assume that the transition dipole moments from the ground to the single excitons in the site basis are and , respectively. It follows that the dipole moments for are located in the same plane, but in general have different magnitudes and directions:
In this model, we shall consider . As enumerated in our model, dipole mediated transitions only couple the ground state to the single excitons, and the single excitons to the biexciton.
Iii Photon-echo experiment as Quantum Process Tomography
Consider a four-wave mixing experiment where an ensemble of identical dimers interacts with a series of three ultrashort laser pulses. The perturbation due to these pulses is given by:
where is the intensity of the electric field, which is assumed to be weak, is the dipole operator, denote the polarization 222Hereafter, we use the word polarization in two different ways: To denote (a) the orientation of oscillations of the electric field and (b) the density of electric dipole moments in a material. The meaning should be clear by the context. , time center, wavector, and carrier frequency of the pulse, and is the position of the center of mass of the molecule. is the slowly varying in time pulse envelope, which we choose as a Gaussian with width , or full-width half-maximum , . The pulses are sent to the sample in a non-collinear fashion to the sample, generating a time-dependent dipole in each of the molecules. Since the characteristic size of a molecule is much smaller than the wavelength of the radiation, , each molecule only experiences a potential that changes in time but is uniform in space, in consistency with the dipole approximation. Nonetheless, the spatial dependence of the pulses is still important, as the phases are imprinted to molecules located across different positions in the sample. The size of the sample is much larger than , so there is a considerable spacial modulation of the polarization due to these phases. Denoting the time-dependent state of the molecule at position by , a perturbative treatment allows us to decompose the density matrix into Fourier components:
where and are integer numbers. Notice that equals to a linear combination of the wavevectors associated with each pulse. As expected, the action of a pulse on each molecule attaches a spatial phase to its quantum state, so the total phase accumulated by it equals for each combination of perturbations. Each improper density matrix corresponds to one of these phases, and can be calculated by keeping track of the actions of the pulses in the bra and the ket of the system using double-sided Feynman diagrams. Eq. (12) implies that the optical polarization induced on the molecule can also be Fourier decomposed into different components seidner:3998 (); pullerits-prb (); gelin:164112 (): , where denotes the dipole operator of the molecule located at . The experimental setting we describe is analogous to the one of an array of dipole antennas which are spatially phased in a grating with respect to each other and oscillate in time. Classical electromagnetism predicts that the induced macroscopic polarization of this array emits radiation which is precisely concentrated along the vectors . This condition, which reflects conservation of momentum of the fields, is known as phase-matching echoes (). A fourth pulse of the same wavevector as one of the , known as the local oscillator, is allowed to interfere with the radiation along that direction. By varying the phases of this fourth field, two heterodyne detections can be carried out to extract the real and imaginary components of respectively, where is the polarization of the local oscillator mukamel ().
In this article, we are interested in the signal along , the so called photon-echo (PE) direction echoes-prl (). The frequency components of the pulses lie within the optical regime, so they can induce the transitions enumerated in the previous section. Traditionally, in the MDOS literature, the intervals between the time centers of the pulses are called coherence , waiting , and echo times, respectively. Here, is the time of detection of the signal chinesejournal (). We shall only consider rephasing photon-echo signals, where , where the inhomogeneous broadening is rephased cheng (). Due to these explicit interval dependences, the collected signal can be expressed as . As explained in our previous study yuenzhou (), we may regard the PE experiment as a QPT of the single-exciton manifold dynamics of the dimer as a function of . In fact, the polarization signal may be expressed as a linear combination of elements of the process matrix :
The coefficients for are frequency amplitudes of the laser pulse which is centered at , evaluated at the transition energy :
is the propagator of the optical coherences in the coherence and echo times, which, has been taken to be the product of a coherent oscillatory term beating at a frequency and an exponential decay with dephasing rate . This propagator is defined only for via the step function . The frequencies of the coherences in the coherence and echo intervals have opposite signs, reflecting the rephasing character of the signal. In optical PE experiments, it is customary to assume that the free-induction decay characterized by the evolution of optical coherences in the coherence and echo times is well characterized, and given by expressions of the form (17). The reason is that the characteristic energetic scales of the vibrational degrees of freedom are much lower than the optical gap, so the only nonunitary dynamics they induce in the optical coherence is, to a good approximation, restricted to pure dephasing which can be inferred from the polarization signal 333We anticipate that deviations from , if they were to happen, would most likely occur for short times , where the non-Markovian behavior of the bath will be stronger. This could consist of a non-secular transfer of optical coherences, for instance: . However, as we mention in section IV, the polarization is collected for many and points, and subsequently Fourier transformed along these dimensions. After processing the signal in this way, the errors due to the short time deviations will presumably be negligible, and the lineshape will be dominated by the functional dependence. We note that these problems are not alien to our protocol, but are generic concerns of any QPT with respect to errors of in the preparation and measurement stages.. The dynamics in the waiting time is more complex, consisting of small frequencies due to excitonic superpositions which are strongly influenced by the bath. It is the latter interval where QPT will prove most useful.
The polarization signal yields a linear combination of elements weighted by the probability amplitude to prepare a state with the first two pulses and detect with the third pulse and the fourth heterodyning pulse. These probability amplitudes can be controlled by manipulating the polarization of the pulses and the frequency amplitudes for the resonant transitions . In essence, state preparation and QST are implicit in the coherence and echo times (see yuenzhou ()). In a different context, Gelin and Kosov had previously hinted at a similar idea by identifying these times as “doorway” and “window” intervals Gelin2008177 (). By conducting several experiments varying these control knobs and collecting the signal from each of these settings, a system of linear equations can be established whereby the elements of can be inverted, and therefore QPT is achieved. This statement is correct provided that besides the free-induction decay rates , the parameters , , , , , and are all known or can be obtained self-consistently during the experiment. We will elaborate on these points for the case of a homodimer in section V.
Notice that Eqs. (13), (14), and (15) monitor all the 12 real valued paramenters involving for , so that they allow for the QPT of the single exciton manifold, which is an effective quantum bit (qubit) system. However, we have also kept track of the elements , that is, the possibility of amplitude leakage errors from the single-exciton dynamics channel to . It is known that whereas the excitonic dynamics occurs in femtosecond timescales, exciton recombination happens in the order of nanoseconds. Therefore, these decay channels could be potentially ignored in many experimental systems. We shall keep them in our theoretical analysis as they do not increase the complexity of the problem by much, although in situations where this could be problematic, we could accordingly disregard them.
Iv QPT from 2D spectrum of PE
As mentioned, QPT can be carried out from data resulting from a series of experiments varying colors and polarizations of the pulses. The necessary information can in principle be obtained by collecting a single point for a fixed pair of and points for each of the experiments. Often, however, the PE signal is collected across many points, and conveniently processed into a 2D correlation spectrum in the conjugate frequency variables and :
which still evolves in the coordinate444The factor of arises due to the phase shift relating the macroscopic polarization and the detected signal corresponding to the emitted electric field.. By performing the integrals of Eq. (18) using Eqs. (13), (14), and (15), we obtain:
The spectrum consists of a sum of four resonances at , which correspond to the frequencies of the optical coherences at the coherence and echo times. These resonances are modulated by lineshape functions of the form,
that correspond to the one-sided Fourier transform of the propagator along each and axis555The fully dispersive and absorptive lineshapes only show up after including the non-rephasing signal in the 2D-ES. See gallagher () for more information on this issue.. The peaks are centered about and have a width parameter . The difference in signs for the Fourier transform in Eq. (18) guarantees that all the resonances appear in the first quadrant of both frequency axes. The expressions for the amplitudes , associated with peaks centered at , are given by 666These expressions were already displayed in procedia () without the background in this article.:
Typically, the probed samples are in solution, so the molecules in the ensemble are isotropically distributed. The isotropic average for a tetradic is given by thirunamachandran ():
where and are the polarizations of the pulses in the lab and the molecular frame, respectively. The isotropic average consists of a sum of molecular frame products weighted by the isotropically invariant tensor .
Since the information in Eqs. (13), (14), (15) is in principle contained in Eqs. (13), (14), and (15), several conclusions from our previous study are immediately transferable: The elements of contained in Eqs. (22), (23), (24), and (25) can be all be extracted by repeating a number of experiments with different polarization configurations for the fields and two different waveforms for the pulses. Under different motivations, theoretical proposals for manipulating 2D-ES using pulse-shaping capabilities have been previously reported voronine:044508 (); abrchemphys (). An extensive study of this possibility for a heterodimer will be presented in the second article accompanying this study.
Eqs. (22), (23), (24), and (25) can also be derived by book-keeping the double-sided Feynman diagrams that oscillate at the particular frequencies for the coherence and waiting times in each of the four resonances (we refer the reader to Fig. 1). In analyzing the possible pathways in Liouville space, we make use of the rotating-wave approximation (RWA): Perturbations which are proportional to can excite the ket and de-excite the bra, whereas the ones proportional to can deexcite the ket and excite the bra. As an illustration, consider the signal , which arises from diagrams oscillating with frequency at the coherence time and at the echo time (Fig. 1(b)). The two states at the coherence time which can oscillate at are or , but the latter cannot be produced by a single action of the dipole operator on the initial ground state . Therefore, is the only possible state for the coherence interval, and is produced by acting the first pulse on the bra of the ground state: . Similar considerations imply that the state at the echo time must be or . Given these constraints, we are ready to enumerate the possible initial and final states for the waiting time interval which are compatible with these restrictions. By exciting the ket or deexciting the bra of with the second pulse, the following initial states for the quantum channel can be produced: . The final states can all give rise to or by exciting the ket or deexciting the bra with the third pulse. Therefore, in principle, there are possibilities for which can be detected in at . However, we assume that the state does not evolve to other states due to the bath:
This assumption is quite reasonable, as we are ignoring processes where phonons can induce optical excitations from . This condition is present in Eqs. (14), and (15) in the function terms and in Eqs. (22), (23), (24), and (25) in the “-1” terms, which correspond to . This leaves possibilities for .
To be more explicit, consider the pathways in that monitor the population to coherence process . These are displayed in Fig. (2). The pathway on the left involves represents an excited state absorption (ESA) from the single-exciton manifold, and is proportional to , an expression which can be immediately read out from the diagram: Each interaction with the field picks up a factor corresponding to the amplitude of the transition, which depends on the alignment of the corresponding dipole with the polarization of the pulse, as well as the frequency amplitude of the pulse at the given transition. A minus sign is included if the perturbation is on the bra. Similarly, the pathway on the right involves stimulated emission (SE) and is proportional to .
The rest of the diagrams for all the peaks can be systematically analyzed in the way described above. In general, the pathways we need to consider can be classified in ESA, SE, and ground state bleaching (GSB) processes. GSB processes are the ones that take at the end of the waiting time to a dipole active coherence involving an excited state. ESA pathways, which are proportional to dipole transitions involving the excited state, differ in sign from SE and GSB pathways, as can be easily seen by inspection.
Fig. (3) provides a mnemonic device to keep track of the Liouville pathways that each peak in the 2D electronic spectrum monitors, and therefore, also provides a scheme of the QPT protocol. The axis can be associated with a particular state preparation whereas the axis with a particular detection. Each peak reflects a nontrivial number of processes in Liouville space. As an illustration (see Fig. (4)), we consider the ideal case where the bath does not interact with the system, in which case, a very simple picture is recovered: The off-diagonal peaks beat at the coherence frequency and the diagonals remain static. This case can be easily derived from Eqs. (22), (23), (24), and (25) by substituting , that is, populations remain static whereas coherences beat at difference frequencies.
V The case of the homodimer
To gain insights into the described QPT protocol, we specialize the results above to a coupled homodimer. In the following subsections, we discuss, for this particular case, (A) the Hamiltonian and the transition dipole moments involved in the experiments, (B) properties of the spectroscopic signals under isotropic averaging, (C) stability of the numerical inversion, (D) analytical expressions of the elements of in terms of the peak amplitudes of the spectra, (E) a procedure to extract the angle between the dipoles, (F) a summary of the QPT procedure, and (G) a numerical example with a model system. A similar study focused on the heterodimer will be presented in the second part of this series.
v.1 Hamiltonian and transition dipole moments
which we have diagonalized with the symmetric and antisymmetric single exciton states given by:
The splitting between the two delocalized states is just . Using Eq. (10), the transition dipoles take the simple forms:
Interestingly, these expressions are independent of the coupling . Also, notice that and are perpendicular to and (see Fig. 5). Denoting the norm of each site dipole by
the following relationships follow: