ΜΗ-ΓΡΑΜΜΙΚΕΣ ΟΠΤΙΚΕΣ ΔΙΑΔΙΚΑΣΙΕΣ ΣΕ ΔΟΜΗΜΕΝΟ ΦΩΤΟΝΙΚΟ ΠΕΡΙΒΑΛΛΟΝ

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1 ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΤΩΝ ΥΛΙΚΩΝ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ ΜΗ-ΓΡΑΜΜΙΚΕΣ ΟΠΤΙΚΕΣ ΔΙΑΔΙΚΑΣΙΕΣ ΣΕ ΔΟΜΗΜΕΝΟ ΦΩΤΟΝΙΚΟ ΠΕΡΙΒΑΛΛΟΝ ΣΟΦΙΑ ΕΥΑΓΓΕΛΟΥ Τριμελής Συμβουλευτική Επιτροπή: ΑΝΑΠΛΗΡΩΤΗΣ ΚΑΘΗΓΗΤΗΣ ΕΜΜΑΝΟΥΗΛ ΠΑΣΠΑΛΑΚΗΣ (ΕΠΙΒΛΕΠΩΝ) ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΤΩΝ ΥΛΙΚΩΝ, ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΕΠΙΚΟΥΡΟΣ ΚΑΘΗΓΗΤΗΣ ΒΑΣΙΛΕΙΟΣ ΓΙΑΝΝΟΠΑΠΑΣ ΤΟΜΕΑΣ ΦΥΣΙΚΗΣ, ΣΧΟΛΗ ΕΦΑΡΜΟΣΜΕΝΩΝ ΜΑΘΗΜΑΤΙΚΩΝ ΚΑΙ ΦΥΣΙΚΩΝ ΕΠΙΣΤΗΜΩΝ, ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ PROFESSOR SIR PETER L. KNIGHT FRS DEPARTMENT OF PHYSICS, IMPERIAL COLLEGE LONDON ΠΑΤΡΑ, ΑΥΓΟΥΣΤΟΣ 2013

2 ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΤΩΝ ΥΛΙΚΩΝ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ ΜΗ-ΓΡΑΜΜΙΚΕΣ ΟΠΤΙΚΕΣ ΔΙΑΔΙΚΑΣΙΕΣ ΣΕ ΔΟΜΗΜΕΝΟ ΦΩΤΟΝΙΚΟ ΠΕΡΙΒΑΛΛΟΝ ΣΟΦΙΑ ΕΥΑΓΓΕΛΟΥ Τριμελής Συμβουλευτική Επιτροπή: ΑΝΑΠΛΗΡΩΤΗΣ ΚΑΘΗΓΗΤΗΣ ΕΜΜΑΝΟΥΗΛ ΠΑΣΠΑΛΑΚΗΣ (ΕΠΙΒΛΕΠΩΝ) ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΤΩΝ ΥΛΙΚΩΝ, ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΕΠΙΚΟΥΡΟΣ ΚΑΘΗΓΗΤΗΣ ΒΑΣΙΛΕΙΟΣ ΓΙΑΝΝΟΠΑΠΑΣ ΤΟΜΕΑΣ ΦΥΣΙΚΗΣ, ΣΧΟΛΗ ΕΦΑΡΜΟΣΜΕΝΩΝ ΜΑΘΗΜΑΤΙΚΩΝ ΚΑΙ ΦΥΣΙΚΩΝ ΕΠΙΣΤΗΜΩΝ, ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ PROFESSOR SIR PETER L. KNIGHT FRS DEPARTMENT OF PHYSICS, IMPERIAL COLLEGE LONDON ΠΑΤΡΑ, ΑΥΓΟΥΣΤΟΣ 2013 H παρούσα έρευνα έχει συγχρηματοδοτηθεί από την Ευρωπαϊκή Ένωση (Ευρωπαϊκό Κοινωνικό Ταμείο - ΕΚΤ) και από εθνικούς πόρους μέσω του Επιχειρησιακού Προγράμματος «Εκπαίδευση και Δια Βίου Μάθηση» του Εθνικού Στρατηγικού Πλαισίου Αναφοράς (ΕΣΠΑ) Ερευνητικό Χρηματοδοτούμενο Έργο: Ηράκλειτος ΙΙ. Επένδυση στην κοινωνία της γνώσης μέσω του Ευρωπαϊκού Κοινωνικού Ταμείου.

3 ΕΠΤΑΜΕΛΗΣ ΕΞΕΤΑΣΤΙΚΗ ΕΠΙΤΡΟΠΗ Νικόλαος Βάϊνος Καθηγητής, Τμήμα Επιστήμης των Υλικών, Πανεπιστήμιο Πατρών. Βασίλειος Γιαννόπαπας Επίκουρος Καθηγητής, Τομέας Φυσικής, Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών, Εθνικό Μετσόβιο Πολυτεχνείο (Μέλος Τριμελούς Συμβουλευτικής Επιτροπής). Sir Peter Knight Professor, Department of Physics, Imperial College London (Μέλος Τριμελούς Συμβουλευτικής Επιτροπής). Σωτήριος Μπασκούτας Αναπληρωτής Καθηγητής, Τμήμα Επιστήμης των Υλικών, Πανεπιστήμιο Πατρών. Εμμανουήλ Πασπαλάκης Αναπληρωτής Καθηγητής, Τμήμα Επιστήμης των Υλικών, Πανεπιστήμιο Πατρών (Επιβλέπων). Ανδρέας Τερζής Αναπληρωτής Καθηγητής, Τμήμα Φυσικής, Πανεπιστήμιο Πατρών. Δημήτριος Φωτεινός Καθηγητής, Τμήμα Επιστήμης των Υλικών, Πανεπιστήμιο Πατρών.

4 Αφιερωμένο στον αδελφό μου, Χρήστο

5 ΕΥΧΑΡΙΣΤΙΕΣ Υπάρχουν αρκετοί άνθρωποι που με βοήθησαν κατά τη διάρκεια της εκπόνησης αυτής της διδακτορικής διατριβής. Πρώτα από όλους, θα ήθελα να εκφράσω τις βαθιές ευχαριστίες μου στον επιβλέποντα μου Αναπληρωτή Καθηγητή Εμμανουήλ Πασπαλάκη για την καθοδήγησή, την στήριξη και την εμψύχωση που μου προσέφερε καθ όλη τη διάρκεια της διδακτορικής έρευνας, καθώς επίσης και για το ότι υπήρξε μια συνεχής πηγή νέων ιδεών που οδήγησαν την έρευνα της συγκεκριμένης διατριβής. Υπήρξα ιδιαιτέρως τυχερή καθώς μου έδωσε την ευκαιρία να εργαστώ σε μια συναρπαστική περιοχή της έρευνας και μου προσέφερε επιπλέον την απαραίτητη βοήθεια σε διάφορες περιπτώσεις. Έπειτα, θα ήθελα να ευχαριστήσω το μέλος της συμβουλευτικής επιτροπής του διδακτορικού, Επίκουρο Καθηγητή Βασίλειο Γιαννόπαπα, για την συνεργασία σε όλη τη διάρκεια της διδακτορικής έρευνας, καθώς και για την εκπόνηση των ηλεκτρομαγνητικών υπολογισμών των ρυθμών της αυθόρμητης εκπομπής κοντά σε μια πλασμονική νανοδομή, οι οποίοι συντέλεσαν στη διεξαγωγή των αποτελεσμάτων των κεφαλαίων 4-7. Επιπρόσθετα, εκτιμάται ιδιαιτέρως η συνεργασία με τον Αναπληρωτή Καθηγητή Ανδρέα Τερζή, στην ανάλυση των αποτελεσμάτων των κεφαλαίων 2, 3 και 7. Ευχαριστώ, επίσης, τον Δρ. Σπυρίδωνα Κοσιώνη για τη συμβολή του στη διεξαγωγή των αποτελεσμάτων του κεφαλαίου 2. Θα ήθελα, βέβαια, να ευχαριστήσω και το τρίτο μέλος της συμβουλευτικής επιτροπής του διδακτορικού, Καθηγητή Sir Peter L. Knight FRS, για την εμψύχωση της προόδου της διδακτορικής διατριβής. Επίσης, ευχαριστώ τα μέλη της εξεταστικής επιτροπής, Καθηγητή Δημήτριο Φωτεινό, Καθηγητή Νικόλαο Βάϊνο και Αναπληρωτή Καθηγητή Σωτήριο Μπασκούτα για τον χρόνο που διέθεσαν για την ανάγνωση της διδακτορικής διατριβής και τις πολύτιμες υποδείξεις τους. Τέλος, την βαθύτερη ευγνωμοσύνη μου

6 εκφράζω προς το ερευνητικό πρόγραμμα «Ηράκλειτος ΙΙ» που με στήριξε οικονομικά κατά τη διάρκεια της διδακτορικής έρευνας και μου έδωσε τη δυνατότητα να αφοσιωθώ αποκλειστικά στην έρευνα. Σε ένα πιο προσωπικό επίπεδο, θα ήθελα να ευχαριστήσω βαθύτατα την οικογένειά μου, η οποία στηρίζει κάθε βήμα και κάθε μου απόφαση. Σε εκείνους χρωστάω τα πάντα. Τέλος, θα ήθελα να πω ένα μεγάλο ευχαριστώ σε όσους φίλους μου στάθηκαν τόσα χρόνια και με βοήθησαν ο καθένας με τον δικό του τρόπο. H παρούσα έρευνα έχει συγχρηματοδοτηθεί από την Ευρωπαϊκή Ένωση (Ευρωπαϊκό Κοινωνικό Ταμείο - ΕΚΤ) και από εθνικούς πόρους μέσω του Επιχειρησιακού Προγράμματος «Εκπαίδευση και Δια Βίου Μάθηση» του Εθνικού Στρατηγικού Πλαισίου Αναφοράς (ΕΣΠΑ) Ερευνητικό Χρηματοδοτούμενο Έργο: Ηράκλειτος ΙΙ. Επένδυση στην κοινωνία της γνώσης μέσω του Ευρωπαϊκού Κοινωνικού Ταμείου.

7 ΔΗΜΟΣΙΕΥΣΕΙΣ Μέρος της διδακτορικής διατριβής έχει δημοσιευτεί στις παρακάτω εργασίες: 1. S. Evangelou, V. Yannopapas, and E. Paspalakis, Modifying free space spontaneous emission near a plasmonic nanostructure, Physical Review A 83, (2011). 2. S. Evangelou, V. Yannopapas, and E. Paspalakis, Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics, Physical Review A 83, (2011). 3. S. Evangelou, V. Yannopapas, and E. Paspalakis, Transparency and slow light in a four-level quantum system near a plasmonic nanostructure, Physical Review A 86, (2012). 4. E. Paspalakis, S. Evangelou, and A. F. Terzis, Control of excitonic population inversion in a coupled semiconductor quantum dot - metal nanoparticle system, Physical Review B 87, (2013). 5. S. Evangelou, V. Yannopapas, and E. Paspalakis, Transient properties of transparency of a quantum system near a plasmonic nanostructure, Optics Communications (special issue on Energy efficient nanophotonics: engineered light-matter interactions in sub-wavelength structures), in press, (2013). 6. S. Evangelou, V. Yannopapas, and E. Paspalakis, Modification of Kerr nonlinearity in a four-level quantum system near a plasmonic nanostructure, submitted, (2013).

8 7. E. Paspalakis, S. Evangelou, V. Yannopapas, and A. F. Terzis, Phasedependent optical effects in a four-level quantum system near a plasmonic nanostructure, submitted, (2013). 8. E. Paspalakis, S. Evangelou, S. G. Kosionis, and A. F. Terzis, Four-wave mixing in a coupled semiconductor quantum dot - metal nanoparticle system, submitted, (2013).

9 Συγγραφέας: Σοφία Ευαγγέλου Επιβλέπων Καθηγητής: Εμμανουήλ Πασπαλάκης ΠΕΡΙΛΗΨΗ Μια σχετικά νέα περιοχή έντονης ερευνητικής δραστηριότητας ασχολείται με τη μελέτη των οπτικών ιδιοτήτων κβαντικών συστημάτων (ατόμων/μορίων και ημιαγώγιμων κβαντικών τελειών) συζευγμένων με πλασμονικές (μεταλλικές) νανοδομές. Τα ισχυρά πεδία και ο έντονος περιορισμός του φωτός που σχετίζονται με τους πλασμονικούς συντονισμούς οδηγούν σε ισχυρή αλληλεπίδραση μεταξύ των ηλεκτρομαγνητικών πεδίων και των κβαντικών συστημάτων κοντά σε πλασμονικές νανοδομές. Επιπλέον, χρησιμοποιώντας τα κβαντικά συστήματα μπορεί να επιτευχθεί εξωτερικός έλεγχος των οπτικών ιδιοτήτων της υβριδικής φωτονικής δομής. Στη διδακτορική διατριβή μελετάται θεωρητικά και υπολογιστικά η οπτική απόκριση συμπλεγμάτων κβαντικών συστημάτων με μεταλλικές νανοδομές, δίνοντας έμφαση σε μη-γραμμικές και κβαντικές οπτικές διαδικασίες. Στα συστήματα αυτά τα επιφανειακά πλασμόνια των μεταλλικών νανοδομών επηρεάζουν σημαντικά, τόσο το ηλεκτρομαγνητικό πεδίο που αλληλεπιδρούν τα κβαντικά συστήματα, όσο και το ρυθμό αυθόρμητης εκπομπής των κβαντικών συστημάτων. Μελετάμε απλές και πολύπλοκες μεταλλικές νανοδομές, όπως μια μεταλλική νανοσφαίρα και μια διδιάστατη διάταξη διηλεκτρικών νανοσφαιρών επικαλυμμένων με μέταλλο (μεταλλικοί νανοφλοιοί). Τα κβαντικά συστήματα είναι άτομα/μόρια και κυρίως ημιαγώγιμες κβαντικές τελείες και περιγράφονται από συστήματα δύο, τριών και τεσσάρων ενεργειακών επιπέδων. Δείχνουμε ότι, φαινόμενα όπως δημιουργία κβαντικής συμβολής στην αυθόρμητη εκπομπή, σύμφωνη ελεγχόμενη αναστροφή πληθυσμού, οπτική διαφάνεια και κέρδος χωρίς αναστροφή πληθυσμού, δημιουργία

10 αργού φωτός, τροποποιημένη οπτική μη-γραμμικότητα Kerr και μίξη τεσσάρων κυμάτων, όπως και φαινόμενα ελέγχου μέσω φάσης, εμφανίζονται στα κβαντικά συστήματα και τροποποιούνται σημαντικά λόγω της ύπαρξης της μεταλλικής νανοδομής.

11 ΣΥΝΟΨΗ ΤΩΝ ΑΠΟΤΕΛΕΣΜΑΤΩΝ ΤΗΣ ΔΙΔΑΚΤΟΡΙΚΗΣ ΔΙΑΤΡΙΒΗΣ Στο κεφάλαιο 1 κάνουμε μια μικρή εισαγωγή στα επιφανειακά πλασμόνια και περιγράφουμε τα θέματα που μελετώνται στην παρούσα διδακτορική διατριβή. Στη συνέχεια, στο κεφάλαιο 2 μελετάμε το φαινόμενο μίξης τεσσάρων κυμάτων σε ένα σύστημα μιας κβαντικής τελείας που βρίσκεται δίπλα σε μια πλασμονική νανοσφαίρα και αλληλεπιδρά σε κοντινό συντονισμό με ένα ισχυρό πεδίο σύζευξης και ένα ασθενές πεδίο ιχνηθέτη. Δείχνουμε ότι η ύπαρξη της πλασμονικής νανοδομής μπορεί να οδηγήσει σε σημαντική αλλαγή του φάσματος μίξης τεσσάρων κυμάτων και ειδικότερα σε ενίσχυση αλλά και σε συμπίεση του φάσματος μίξης τεσσάρων κυμάτων, ανάλογα με την απόσταση που έχουν η κβαντική τελεία και η μεταλλική νανοσφαίρα. Επίσης, βρίσκουμε ότι επιτυγχάνεται έλεγχος της μορφής του φάσματος τεσσάρων κυμάτων μέσω της έντασης και συχνότητας του πεδίου σύζευξης. Επίπλέον, περιγράφουμε την αλλαγή του φάσματος μίξης τεσσάρων κυμάτων με την απόσταση, με την συμπεριφορά της ενεργού συχνότητας Rabi που οδηγεί στη δημιουργία «πλασμονικών μετα-συντονισμών». Η συμπεριφορά της ενεργού συχνότητας Rabi εξηγείται μέσω αναλυτικών λύσεων των μη-γραμμικών εξισώσεων του πίνακα πυκνότητας που περιγράφουν τη δυναμική του συστήματος. Στο κεφάλαιο 3 μελετάμε τη δυνατότητα δημιουργίας ελεγχόμενης αναστροφής εξιτονικού πληθυσμού σε μια κβαντική τελεία που περιγράφεται από ένα σύστημα δύο ενεργειακών επιπέδων, στο ίδιο σύστημα συζευγμένης κβαντικής τελείας με μια μεταλλική νανοσφαίρα που αλληλεπιδρά με ένα παλμικό ηλεκτρομαγνητικό πεδίο. Δείχνουμε ότι η βασική μεθοδολογία που χρησιμοποιείται για τη δημιουργία ελεγχόμενης αναστροφής πληθυσμού σε κβαντικές τελείες απουσία

12 της μεταλλικής νανοσφαίρας, η εφαρμογή π-παλμών, μπορεί να επηρεαστεί σημαντικά από την αλληλεπίδραση εξιτονίων και πλασμονίων για μικρές αποστάσεις μεταξύ της κβαντικής τελείας και της μεταλλικής νανοσφαίρας, με αποτέλεσμα να μη δημιουργείται σημαντική αναστροφή πληθυσμού στις αποστάσεις αυτές. Επιπλέον, αναλύουμε την επίδραση της διάρκειας του παλμού στην απόδοση της μεθόδου και δείχνουμε ότι υψηλής απόδοσης αναστροφή πληθυσμού επιτυγχάνεται για μικρής διάρκειας ηλεκτρομαγνητικούς παλμούς. Τέλος, μέθοδοι που οδηγούν σε υψηλής απόδοσης ελεγχόμενη αναστροφή εξιτονικού πληθυσμού για όλες τις αποστάσεις μεταξύ της κβαντικής τελείας και της μεταλλικής νανοσφαίρας παρουσιάζονται μέσω αναλυτικών λύσεων των μη-γραμμικών εξισώσεων του πίνακα πυκνότητας για το υπό μελέτη σύστημα. Οι λύσεις αυτές χρησιμοποιούνται και για την εξήγηση της συμπεριφοράς του συστήματος για ηλεκτρομαγνητικούς παλμούς μικρής διάρκειας. Στο κεφάλαιο 4 μελετάμε τη δυναμική πληθυσμού λόγω αυθόρμητης εκπομπής σε ένα κβαντικό σύστημα τριών επιπέδων τύπου V με εκφυλισμένες διεγερμένες καταστάσεις που τοποθετείται κοντά σε μια πλασμονική νανοδομή. Η πλασμονική νανοδομή αποτελείται από μια διδιάστατη διάταξη διηλεκτρικών νανοσφαιρών επικαλυμμένων με μέταλλο και οδηγεί σε σημαντική διαφορά στο ρυθμό αυθόρμητης εκπομπής από το κβαντικό σύστημα σε ορθογώνιες κατευθύνσεις. Παρουσιάζουμε τη δυναμική πληθυσμού στις διάφορες ενεργειακές καταταστάσεις, μέσω αναλυτικών λύσεων του πίνακα πυκνότητας, και μελετάμε ιδιαίτερα την επίδραση των διαφορετικών αρχικών συνθηκών στη δυναμική πληθυσμού. Δείχνουμε ότι στην περίπτωση που μόνο μια διεγερμένη κατάσταση είναι αρχικά κατειλημμένη, τότε με την πάροδο του χρόνου και η δεύτερη διεγερμένη κατάσταση αποκτά πληθυσμό, και καταλήγουν και οι δύο καταστάσεις, μετά από κάποιο χρόνο, με τον

13 ίδιο πληθυσμό που αποσβένει αργά. Στην περίπτωση που οι διεγερμένες καταστάσεις έχουν αρχικά τον ίδιο πληθυσμό, η δυναμική του συστήματος εξαρτάται σημαντικά από την κβαντική συμφωνία της αρχικής υπέρθεσης, και οδηγεί είτε σε πολύ αργή απόσβεση (για αντισυμμετρική υπέρθεση), είτε σε πολύ γρήγορη απόσβεση (για συμμετρική υπέρθεση), είτε σε συνδυασμό πολύ γρήγορης και πολύ αργής απόσβεσης (για ασύμφωνη μίξη). Στο κεφάλαιο 5 μελετάμε την επίδραση της παραπάνω πλασμονικής νανοδομής (η πλασμονική νανοδομή αποτελείται από μια διδιάστατη διάταξη διηλεκτρικών νανοσφαιρών επικαλυμμένων με μέταλλο) στο φάσμα αυθόρμητης εκπομπής ενός κβαντικού συστήματος τεσσάρων επιπέδων τύπου διπλού V, όπου μια μετάβαση τύπου V γίνεται στο κενό ενώ η άλλη μετάβαση τύπου V γίνεται σε συχνότητες που επηρεάζονται από τα επιφανειακά πλασμόνια της νανοδομής. Δείχνουμε ότι το φάσμα της αυθόρμητης εκπομπής στο κενό επηρεάζεται σημαντικά από την ύπαρξη της πλασμονικής νανοδομής και μελετάμε την επίδραση των αρχικών συνθηκών του κβαντικού συστήματος και της απόστασης από τη νανοδομή στο φάσμα της αυθόρμητης εκπομπής στο κενό. Στο κεφάλαιο 6 παρουσιάζουμε τη μελέτη της οπτικής απόκρισης ενός κβαντικού συστήματος τεσσάρων επιπέδων που βρίσκεται δίπλα σε μια πλασμονική νανοδομή και αλληλεπιδρά με ένα γραμμικά πολωμένο ηλεκτρομαγνητικό πεδίο. Η πλασμονική νανοδομή είναι παρόμοια με αυτή που χρησιμοποιήθηκε στα κεφάλαια 4 και 5 (διδιάστατη διάταξη μεταλλικών νανοφλοιών), ενώ το κβαντικό σύστημα είναι τεσσάρων επιπέδων τύπου διπλού V. Δείχνουμε ότι η ύπαρξη της πλασμονικής νανοδομής μπορεί να οδηγήσει σε οπτική διαφάνεια, σε δημιουργία αργού φωτός, σε σημαντική αλλαγή στην χρονική εξάρτηση της απορρόφησης με δημιουργία

14 μεταβατικού κέρδους χωρίς αναστροφή πληθυσμού και σε σημαντικά τροποποιημένη οπτική μη-γραμμικότητα Kerr. Επιπλέον, βρίσκουμε ότι τα φαινόμενα αυτά έχουν σημαντική εξάρτηση από την απόσταση του κβαντικού συστήματος από την πλασμονική νανοδομή. Δείχνουμε ότι μεγαλύτερες αποστάσεις μεταξύ κβαντικού συστήματος και πλασμονικής νανοδομής, μέσα στα πλαίσια βέβαια που η πλασμονική νανοδομή επηρεάζει σημαντικά το κβαντικό σύστημα, οδηγούν σε καλύτερα αποτελέσματα, δηλαδή σε πιο έντονη μείωση της απορρόφησης (καλύτερη διαφάνεια), μικρότερες ταχύτητες ομάδας, εντονότερο μεταβατικό κέρδος χωρίς αναστροφή πληθυσμού και μεγαλύτερο συντελεστή οπτικής μη-γραμμικότητας Kerr. Επίσης, παρουσιάζουμε πλήθος αναλυτικών και αριθμητικών αποτελεσμάτων και ποσοτικοποιούμε τα φαινόμενα αυτά με ιδιαίτερη έμφαση στην εξάρτηση τους από τις διάφορες παραμέτρους του συστήματος, όπως ο ρυθμός αυθόρμητης εκπομπής στις κβαντικές μεταβάσεις που δεν επηρεάζονται από την πλασμονική νανοδομή. Στο κεφάλαιο 7 παρουσιάζουμε τη μελέτη της οπτικής απόκρισης ενός κβαντικού συστήματος τεσσάρων επιπέδων τύπου V, ίδιο με αυτό που χρησιμοποιήθηκε στο κεφάλαιο 6, που βρίσκεται δίπλα στην ίδια πλασμονική νανοδομή όπως στα κεφάλαια 4, 5 και 6, και αλληλεπιδρά με δύο ορθογώνια κυκλικά πολωμένα ηλεκτρομαγνητικά πεδία ίδιας συχνότητας, όπου η ένταση τους και η φάση τους μπορεί να αλλάζουν ανεξάρτητα. Δείχνουμε ότι στο συγκεκριμένο σύστημα η απορρόφηση και διασπορά του ενός πεδίου μπορεί να επηρεαστεί σημαντικά, και τελικά να ελεγχθεί, από την ύπαρξη του άλλου πεδίου και ιδιαίτερα από τη σχετική τους ένταση και από τη διαφορά φάσης μεταξύ τους. Έτσι, το σύστημα αυτό εμφανίζει φαινόμενα ελέγχου φάσης και πλάτους πεδίου. Επιπλέον, φαινόμενα όπως πλήρης οπτική διαφάνεια, κέρδος χωρίς αναστροφή πληθυσμού και μηδενική

15 απορρόφηση με μη-μηδενική διασπορά μπορεί να επιτευχθούν στο συγκεκριμένο σύστημα. Τα φαινόμενα που παρουσιάζονται υποστηρίζονται από εκτεταμένα αναλυτικά αποτελέσματα. Τέλος, στο κεφάλαιο 8, που αποτελεί και το τελευταίο κεφάλαιο της διδακτορικής διατριβής, παρουσιάζουμε μια σύνοψη των βασικών αποτελεσμάτων που έχουν εξαχθεί από αυτή τη μελέτη. Επιπλέον, σχεδιαγραφούμε και βασικές επεκτάσεις των αποτελεσμάτων της παρούσας διδακτορικής διατριβής, που μπορεί να γίνουν η βάση για περαιτέρω μελέτες στα συστήματα αυτά. Ακολουθεί η πλήρης διδακτορική διατριβή στην Αγγλική γλώσσα.

16 Nonlinear Optical Processes in Structured Photonic Environment A dissertation submitted by Sofia Evangelou to The Department of Materials Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Patras Patras, Greece August 2013

17 ii Members of the Examination Committee Associate Professor Emmanuel Paspalakis, Department of Materials Science, University of Patras, Greece (Thesis Advisor) Assistant Professor Vassilios Yannopapas, Department of Physics, National Technical University of Athens, Greece (Member of the Advisory Committee) Professor Sir Peter L. Knight FRS, Department of Physics, Imperial College, United Kingdom (Member of the Advisory Committee) Professor Demetrios J. Photinos, Department of Materials Science, University of Patras, Greece Professor Nikolaos A. Vainos, Department of Materials Science, University of Patras, Greece Associate Professor Sotirios Baskoutas, Department of Materials Science, University of Patras, Greece Associate Professor Andreas F. Terzis, Department of Physics, University of Patras, Greece

18 Author Sofia Evangelou Thesis advisor Emmanuel Paspalakis Nonlinear Optical Processes in Structured Photonic Environment Abstract A relatively new area of active research involves the study of the optical properties of quantum systems (atoms/molecules and semiconductor quantum dots) coupled to plasmonic (metallic) nanostructures. The large fields and the strong light confinement associated with the plasmonic resonances enable strong interaction between the electromagnetic field and quantum systems near plasmonic nanostructures. In addition, using the quantum system one may achieve external control of the optical properties of the hybrid photonic structure. In this thesis we analyze both theoretically and computationally the optical response of hybrid nanosystems comprised of quantum emitters and plasmonic nanostructures. We put emphasis on the study of nonlinear and quantum optical processes. In these systems the spontaneous decay rate and the electromagnetic field that interacts with the quantum emitter is significantly modified by the surface plasmons of the plasmonic nanostructures. We study cases of both simple and more involved plasmonic nanostructures. An example of a simple plasmonic nanostructure considered in this thesis is a metallic nanosphere, while a more involved one is a two-dimensional array of metal-coated dielectric nanospheres. The quantum systems are atoms/molecules and especially semiconductor quantum dots and are described by two-level, three-level or four-level systems. We find that several coherent optical phenomena that happen in the quantum systems can be strongly iii

19 Abstract iv influenced by the presence of the plasmonic nanostructure. Specifically, we show that effects such as quantum interference in spontaneous emission, controlled population inversion, optical transparency and gain without inversion, slow light, enhanced nonlinear optical Kerr effect and four-wave mixing as well as phase-dependent absorption and dispersion profiles can be created and modified.

20 Contents Title Page Members of the Examination Committee Abstract Table of Contents Citations to Published and Submitted Work Acknowledgments Dedication i ii iii v vii viii x 1 Outline of the thesis Introduction Structure of the thesis Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system Introduction Theoretical methodology Numerical results Conclusions Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system Introduction Theoretical model Analytical and numerical results Conclusions Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics Introduction Theory Results v

21 Contents vi 4.4 Conclusions Modification of free-space spontaneous emission near a plasmonic nanostructure Introduction Theoretical model and calculation of the spontaneous emission spectrum Form of the free-space spontaneous emission spectrum Conclusions Coherent optical phenomena in a four-level quantum system near a plasmonic nanostructure Introduction Density matrix equations Optical transparency and slow light near a plasmonic nanostructure Transient properties of transparency near a plasmonic nanostructure Modification of Kerr nonlinearity near a plasmonic nanostructure Conclusions Phase-dependent optical phenomena in a four-level quantum system near a plasmonic nanostructure Introduction Theoretical model and calculation of the linear susceptibility Phase-dependent effects in the absorption and dispersion spectra Conclusions Concluding remarks and outlook 119 Bibliography 122 A Electric field inside the semiconductor quantum dot 130

22 Citations to Published and Submitted Work Parts of this PhD thesis have appeared in the following papers: 1. S. Evangelou, V. Yannopapas, and E. Paspalakis, Modifying free space spontaneous emission near a plasmonic nanostructure, Physical Review A 83, (2011). 2. S. Evangelou, V. Yannopapas, and E. Paspalakis, Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics, Physical Review A 83, (2011). 3. S. Evangelou, V. Yannopapas, and E. Paspalakis, Transparency and slow light in a four-level quantum system near a plasmonic nanostructure, Physical Review A 86, (2012). 4. E. Paspalakis, S. Evangelou, and A. F. Terzis, Control of excitonic population inversion in a coupled semiconductor quantum dot - metal nanoparticle system, Physical Review B 87, (2013). 5. S. Evangelou, V. Yannopapas, and E. Paspalakis, Transient properties of transparency of a quantum system near a plasmonic nanostructure, Optics Communications (special issue on Energy efficient nanophotonics: engineered lightmatter interactions in sub-wavelength structures), in press, (2013). 6. S. Evangelou, V. Yannopapas, and E. Paspalakis, Modification of Kerr nonlinearity in a four-level quantum system near a plasmonic nanostructure, submitted, (2013). 7. E. Paspalakis, S. Evangelou, V. Yannopapas, and A. F. Terzis, Phase-dependent optical effects in a four-level quantum system near a plasmonic nanostructure, submitted, (2013). 8. E. Paspalakis, S. Evangelou, S. G. Kosionis, and A. F. Terzis, Four-wave mixing in a coupled semiconductor quantum dot - metal nanoparticle system, submitted, (2013). vii

23 Acknowledgments There are several people that helped me during the period in which this thesis work was carried out. First of all, I would like to deeply thank my supervisor Associate Professor Emmanuel Paspalakis for his guidance, support and encouragement during the entire length of my PhD research, as well as for being a continuous source of new ideas that led to the research of this thesis. I am very lucky that he gave me the opportunity to work in a very exciting research area and for providing the necessary help in several occasions. Then, I would like to thank the member of the PhD advisory committee Assistant Professor Vassilios Yannopapas for the collaboration in all this time and for providing the electromagnetic calculations for the decay rates near a plasmonic nanostructure that made the results of chapters 4-7 possible. In addition, the collaboration with Associate Professor Andreas F. Terzis in the analysis of the results of chapters 2, 3 and 7 is deeply appreciated. I also thank Dr. Spyridon G. Kosionis for his contribution in the results of chapter 2. Moreover, I would also like to thank the third member of the PhD advisory committee, Professor Sir Peter L. Knight FRS, for encouraging the progress of this thesis. Furthermore, the members of the examination committee Professor Demetrios J. Photinos, Professor Nikolaos A. Vainos, and Associate Professor Sotirios Baskoutas are acknowledged for reading the thesis and for their valuable suggestions. Finally, my deepest gratitude goes to research project Heracleitus II that financially supported me during the period of this thesis and gave me the opportunity to be fully committed in research. On a more personal level, I would like to deeply thank my family that supports me in every step I take in life and every decision I make. I owe everything to them. viii

24 Acknowledgments ix Finally, I would like to express a big thank you to those friends that have stood by me all these years and have helped me in their own special way. This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund.

25 Dedicated to my brother, Christos Aφιερωµένo στoν αδελφó µoυ, Xρήστo x

26 Chapter 1 Outline of the thesis 1.1 Introduction In recent years, the study of the particular way in which the metals respond to light, has emerged considerable interest in the area of science termed plasmonics [1, 2, 3]. Plasmonics is an interdisciplinary research area that mainly involves physics, chemistry, materials science and electrical engineering. The specific properties of plasmonic structures are determined by both their design and the materials that compose them. Several applications of plasmonic structures have been proposed, such as for example, in telecommunications, energy conversion and storage, high-density data storage, biological microscopy, medicine, photodetectors and sensors. The interaction of metals with electromagnetic radiation is mainly dictated by their free conduction electrons. Most metals posses a negative dielectric constant in the optical regime, which leads, for example, to a very high reflectivity. At optical frequencies the metal s free-electron gas produces surface and volume charge-density 1

27 Chapter 1: Outline of the thesis 2 oscillations, called plasmons, with distinct resonance frequencies. The electronic surface charge-density oscillations associated with surface plasmons at the interface between a metal and a dielectric can give rise to strongly enhanced optical near fields, which are spatially confined near the metal interface. In addition, if the electron gas is confined in three dimensions, as in the case of a nanoparticle, the overall displacement of the electrons with respect to the positively charge lattice leads to a restoring force, which in turn gives rise to specific particle-plasmon resonances depending on the geometry of the nanoparticle. At optical frequencies, a noble metal is characterized by a dielectric function ε m = ε 1 + iε 2, with ε 1 and ε 2 being its real and imaginary parts, respectively. In this case ε 1 < 0 and ε 1 is typically larger than ε 2. The opposite occurs for metals in the microwave or infrared parts of the spectrum. Plasmonics can be defined as the area of research that deals with the interaction of light with metals in the regime that ε 1 > ε 2 [1]. In addition, a surface plasmon may be defined as a fundamental electromagnetic mode of an interface between a material with negative dielectric constant (usually a metal) and a material with a positive dielectric constant having a well-defined frequency and which involves electronic surface charge-density oscillations [2]. For an introduction to surface plasmons see chapter 12 of Ref. [1] and for a detailed description see Ref. [2]. For recent progress in plasmonics see Ref. [3]. In parallel, significant research work has been done in the study of coherent control and quantum coherence and interference phenomena that occur under the interaction of coherent electromagnetic fields (laser fields) with multi-level quantum systems [4, 5]. Some of these effects are Rabi oscillations, controlled population transfer,

28 Chapter 1: Outline of the thesis 3 electromagnetically induced transparency, gain without inversion, slow light, and enhanced nonlinear optical processes. Initially, these effects were studied in atomic and molecular systems, but in the coming years, and especially in the past decade, significant amount of experimental and theoretical work has been done in semiconductor nanostructures, such as semiconductor quantum dots. The coherently driven quantum systems are considered to be active photonic structures and are expected to have significant applications in several areas of photonics and in quantum computing [6]. A relatively new area of active research, termed quantum plasmonics [7], involves the study of the optical properties of complex photonic structures that incorporate both plasmonic nanostructures and quantum emitters, such as atoms, molecules and semiconductor quantum dots. These complex (hybrid) structures are active photonic structures that are expected to have significantly enhanced optical response, in comparison to their constituents, if the properties of the two parts (plasmonic and quantum) are properly combined. The large fields and the strong light confinement associated with the plasmonic resonances enable strong interaction between the electromagnetic field and quantum emitters near plasmonic nanostructures. In addition, using the quantum emitter one may achieve external control of the optical and dynamical properties of the hybrid photonic structure. Examples of the phenomena already studied include the modification of spontaneous emission [8, 9, 10, 11], the alteration of resonance fluorescence [12, 13, 14], the creation of quantum interference effects in spontaneous emission [15, 16, 17] and Fano effects in energy absorption [18, 19, 20, 21, 22], as well as the creation and enhancement of resonance energy transfer [23, 24, 25] between distant quantum systems. In addition, the enhance-

29 Chapter 1: Outline of the thesis 4 ment of several nonlinear optical phenomena, such as up-conversion processes [26], Kerr nonlinearity [27], nonlinear optical rectification [28], second harmonic generation [29], difference-frequency generation [30], four-wave mixing [31], gain without inversion [32, 33], and intrinsic optical bistability [34, 35] has been studied in coupled quantum and plasmonic systems. The aim of this thesis is to analyze both theoretically and computationally the optical response of hybrid nanosystems comprised of quantum emitters and plasmonic nanostructures. We put emphasis on the study of nonlinear and quantum optical processes. In these systems the spontaneous emission rate and the electromagnetic field that interacts with the quantum emitter is significantly modified by the surface plasmons of the plasmonic nanostructures. and more involved plasmonic nanostructures. We consider cases of both simple An example of a simple plasmonic nanostructure presented in this thesis is a metallic nanosphere, while a more involved one is a two-dimensional array of metal-coated dielectric nanospheres. The quantum systems are atoms/molecules and especially semiconductor quantum dots and are described by two-level, three-level or four-level systems. We find that several coherent optical phenomena that happen in the quantum systems can be strongly influenced by the presence of the plasmonic nanostructure. Specifically, we show that effects such as quantum interference in spontaneous emission, controlled population inversion, optical transparency and gain without inversion, slow light, enhanced nonlinear optical Kerr effect and four-wave mixing as well as phase-dependent absorption and dispersion profiles can be created and modified.

30 Chapter 1: Outline of the thesis Structure of the thesis We begin our study in chapter 2 with an analysis of four-wave mixing in a coupled semiconductor quantum dot - spherical metal nanoparticle structure which interacts with a weak probe field of varying frequency and a strong pump field of fixed frequency. Depending on the values of the pump field intensity and frequency, we find that there is a critical distance that changes the form of the spectrum. Above this distance the four-wave mixing spectrum shows an ordinary three-peaked form and the effect of controlling its magnitude by changing the interparticle distance can be obtained. Below this critical distance the four-wave mixing spectrum becomes single-peaked, and as the interparticle distance decreases the spectrum is strongly suppressed. The behavior of the system is explained using the effective Rabi frequency that creates plasmonic metaresonances in the hybrid structure. Furthermore, the behavior of the effective Rabi frequency is explained via an analytical solution of the nonlinear density matrix equations. In chapter 3 we study the potential for controlled population inversion in the same coupled semiconductor quantum dot - spherical metal nanoparticle structure as in chapter 2 which interacts with a pulsed electromagnetic field. We show that the widely used method of population inversion by a π-pulse can be modified for small interparticle distances. This modification depends strongly on the pulse duration. We also present analytical solutions of the nonlinear density matrix equations, for specific pulse envelope, which lead to efficient excitonic population inversion in the quantum dot for several distances between the semiconductor quantum dot and the metal nanoparticle.

31 Chapter 1: Outline of the thesis 6 Chapter 4 deals with the effects of quantum interference in spontaneous decay on the population dynamics of a three-level V-type quantum system next to a plasmonic nanostructure. It has been recently shown that the placement of a V-type quantum emitter in the proximity of metallic nanostructures can create dynamics similar to that of quantum interference in spontaneous emission. Here, we continue this work and present results on the population dynamics of a three-level V-type quantum emitter for various initial conditions, in the presence of a two-dimensional array of metal-coated dielectric nanospheres. In chapter 5 we present results on the effects of the presence of a plasmonic nanostructure on free-space spontaneous emission in a four-level quantum system, where one V-type transition is influenced by the interaction with surface plasmons and the other V-type transition occurs in free-space. The plasmonic nanostructure that we consider is a two-dimensional array of metal-coated dielectric nanospheres. We show that the spectrum of spontaneous emission in the free-space modes is strongly influenced by the presence of the plasmonic nanostructure, and explore the dependence of the spectrum on different initial conditions of the quantum system and on the distance from the nanostructure. Chapter 6 explores the form of the χ (1) and χ (3) susceptibilities of a four-level double V-type quantum system near a two-dimensional array of metal-coated dielectric nanospheres. In the quantum system under study one V-type transition is influenced by the interaction with surface plasmons while the other V-type transition interacts with free-space vacuum, similar to the system studied in chapter 5. The structure also interacts with a linearly polarized laser field, which couples the lowest state

32 Chapter 1: Outline of the thesis 7 with the upper states in the free-space transitions. Phenomena such as optical transparency, slow light, enhanced Kerr nonlinearity and transient gain without inversion are presented. We find that these effects are strongly influenced by the distance between the quantum system and the plasmonic nanostructure as well as by free-space spontaneous decay. The last chapter with research results is Chapter 7 analyzes the linear absorption and dispersion properties of a four-level double-v-type quantum system near a plasmonic nanostructure, in the case that the system interacts with two orthogonal circularly polarized laser fields with the same frequency, and different phases and electric field amplitudes. We find that the presence of the plasmonic nanostructure leads to strong modification of the absorption and dispersion spectra for one of the laser fields, in the presence of the other, and show that one can use the phase difference and the relative electric field amplitudes of the two laser fields for efficient control of the optical properties of the system. Effects such as complete optical transparency, zero absorption with non-zero dispersion and gain without inversion are obtained. Finally, in the closing chapter, chapter 8, we briefly conclude our findings and discuss future research directions which could build on the results presented in this thesis.

33 Chapter 2 Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 2.1 Introduction The study of the optical properties of complex photonic structures that involve the interaction of excitons (confined electron-hole pairs) from semiconductor quantum dots and surface plasmons from metallic nanostructures has attracted significant attention recently. This is a relatively new area of active research in nanophotonics which falls within quantum plasmonics [7]. In this research area, several interesting experimental results have been presented [36, 37, 38, 39, 40, 41] that have stimulated the research for theoretical studies. A structure that has attracted particular attention in the theoretical studies is a hybrid nanocrystal complex composed of a 8

34 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 9 semiconductor quantum dot and a spherical metal nanoparticle [18, 19, 20, 21, 22, 31, 32, 33, 34, 35]. The optical response of this structure interacting with a weak probe field of varying frequency and a strong pump field of fixed frequency has already been considered in some works [31, 32, 42, 43, 44]. In particular, Lu and Zhu [43] investigated the probe absorption and dispersion of the probe field in the presence of the pump field and showed that a hole induced by coherent population oscillation appears at the absorption spectrum of the probe field when excitons and plasmons interact. They also pointed out that the system may exhibit slow light and that the slow light effect is greatly modified by the distance between the quantum dot and the metal nanoparticle due to the coupling of excitons and plasmons. In the same system, Sadeghi [32, 44] studied the generation of tunable gain without inversion in semiconductor quantum dots using plasmonic effects. He showed that when such a system is exposed to a laser field and the interparticle distance is reduced the initial impact of plasmons is enhancement of the ac-stark shift in the quantum dot. Moreover, when this distance reaches a critical value, a creation of plasmonic metaresonances occurs [45] and an abrupt formation of a significant amount of gain without inversion in the quantum dot appears. Another nonlinear optical phenomenon that has been studied in the same complex nanostructure is the four-wave mixing [42]. Initially, Lu and Zhu [42] showed that significantly enhanced four-wave mixing can occur due to exciton-plasmon interaction, and that the spectrum of the real and imaginary part of the relevant χ (3) susceptibility is significantly dependent on the distance between the quantum dot and the metal

35 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 10 nanoparticle. Quite recently, Li et al. [31] predicted induced bistable behavior of the real and imaginary part of the χ (3) susceptibility of the four-wave mixing effect. They gave specific emphasis to the behavior of the relevant χ (3) susceptibility in the bistability region and showed that the bistability region can be tuned by adjusting the size of metal nanoparticle, the interparticle distance and the intensity of the pump field. In this chapter we also address the problem of the four-wave mixing in a coupled semiconductor quantum dot - spherical metal nanoparticle structure and give emphasis to the behavior of the four-wave mixing spectrum with the interparticle distance. We find that there is a critical interparticle distance that strongly modifies the four-wave mixing spectrum. This critical distance is the one that creates plasmonic metaresonances in the hybrid structure [32, 45]. Above this critical distance the four-wave mixing spectrum shows an ordinary three-peaked form [46, 47, 48, 49] and the effect of controlling its magnitude by changing the interparticle distance can be obtained. Below this critical distance the four-wave mixing spectrum becomes single-peaked, and as the interparticle distance decreases the spectrum is strongly suppressed. The chapter is organized as follows: in the next section we present the density matrix equations for the several nonlinear optical processes for the interaction of the semiconductor quantum dot system with the pump and probe electromagnetic fields in the presence of the metal nanoparticle. Then, in section 2.3 we present results obtained from numerical solutions of the density matrix equations and study systematically the dependence of the four-wave mixing spectrum on interparticle distance,

36 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 11 pump field intensity and frequency. Moreover, an explanation of the behavior of the four-wave mixing spectrum is given in terms of an effective Rabi frequency. Finally, in section 2.4 we conclude our findings. 2.2 Theoretical methodology The hybrid structure that we study is composed of a spherical metal nanoparticle of radius α and a small spherical semiconductor quantum dot of radius β, in an environment with real dielectric constant ε env, see Fig The center-to-center distance between the two particles is denoted by R. We also assume that β α and consider that R > α. The quantum dot is characterized by a two-level system, with 1 being the ground state and 2 being the single exciton state. The biexciton state is omitted here. This is an approximation that has been done in all the theoretical papers in this area. For CdSe-based quantum dots, that are of interest here, the biexciton binding energy is large (for example values up to 50 mev have been reported by Sewall et al. [50]) and its value increases as the quantum dot size decreases [50]. Therefore, we do not expect the biexciton state to have significant influence in the system studied here. The energy difference between the two states is hω 0. This system interacts with two linearly polarized oscillating electromagnetic fields, a pump and a probe field, with total electric field E(t) = ẑ [E a cos(ω a t) + E b cos(ω b t)], (2.1) that excites the interband transition between the two energy levels of the semiconductor quantum dot. Here, ẑ is the polarization unit vector (along the z direction),

37 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 12 Figure 2.1: (Color online) The coupled system consists of a semiconductor quantum dot of radius β and a spherical metallic nanoparticle of radius α. The centers of the two particles are separated by distance R. E a (E b ) is the electric field amplitude of the pump (probe) field, and ω a (ω b ) is the angular frequency of the pump (probe) applied field. The dielectric constant of the semiconductor quantum dot is represented by ε S, while we treat the metal nanoparticle as a classical dielectric particle with dielectric function ε m (ω). The electromagnetic field also excites plasmons, on the surface of the metal nanoparticle. These plasmonic excitations provide a strong continuous spectral response. Such surface plasmons influence the excitons and induce electromagnetic interactions between excitons and plasmons [18, 19]. The Hamiltonian of the system, in the dipole approximation, takes the form H = hω µe SQD (t) ( ), (2.2) where µ represents the dipole moment of the semiconductor quantum dot corresponding to the single exciton transition, and E SQD represents the electric field inside the semiconductor quantum dot. In the dipole approximation the total electric field in-

38 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 13 side the quantum dot consists of two parts, the first part being the applied external fields and the second part corresponding to the induced fields produced by the polarization of the metal nanoparticle (treated as a classical metallic nanosphere). A calculation of E SQD is presented in Appendix A that makes clear the different terms that appear in it. For the present case, E SQD, in the quasi-static approximation, is explicitly written as [18, 19, 20]: E SQD (t) = h µ n=a,b [ ] Ωn 2 e iωnt + G n ρ 21 (t) + Ω n 2 eiωnt + G nρ 12 (t), (2.3) with ρ nm (t) being the density matrix elements. We also defined the Rabi frequencies of the pump (Ω a ) and probe (Ω b ) fields as [18, 19, 20] Ω n = and the parameter G n as [18, 20] µe n hε effs ( 1 + s aγ n α 3 R 3 ), (2.4) Here, ε effs = 2ε env+ε S 3ε env, γ n = εm(ωn) εenv ε m(ω n)+2ε env G n = 1 s 2 aγ n α 3 µ 2. (2.5) 4πε env hε 2 effsr6 with n = a, b, and s a = 2 as the applied field is taken parallel to the interparticle axis of the system (the interparticle axis is the z-axis). The Rabi frequency has two terms, the first is related to the direct coupling of the quantum dot to the applied field, and the second term is related to the electric field from the metal nanoparticle that is induced by the external applied field. In other words, the external applied field induces a dipole moment on the metal nanoparticle, which then acts as a single dipole producing the dipolar interaction with the quantum dot showing the well-known 1/R 3 dependence with the distance between the two particles. In addition, the parameter G n arises due to the dipole-dipole interactions

39 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 14 between excitons and plasmons. This term originates from the induced dipole on the metal nanoparticle, which is now produced by the dipole induced by the applied field on the semiconductor quantum dot [18, 20]. G n is also called the self-interaction term of the quantum dot [20] and is responsible for the Förster energy transfer between the quantum dot and the metal nanoparticle [51]. We proceed with the density matrix equations. The general equation for the density matrix ρ(t) reads i h ρ = [H, ρ] + relaxation terms, (2.6) where the denotes time derivative. For the two-level system the density matrix is written as ρ(t) = ρ 11 (t) ρ 21 (t) ρ 12 (t) ρ 22 (t), (2.7) where ρ 21 (t) = ρ 12(t). We introduce the slowly varying quantities of the density matrix elements σ 21 (t) = ρ 21 (t)e iω at, σ 12 (t) = ρ 12 (t)e iωat, σ 11 (t) = ρ 11 (t), σ 22 (t) = ρ 22 (t) and obtain the density matrix equations, which describe the dynamics of the system, under the rotating wave approximation, as σ 21 (t) = (i a + 1 ) σ 21 (t) ig [σ 22 (t) σ 11 (t)] σ 21 (t) T2 i 2 σ 22 (t) σ 11 (t) = i ( Ω a + Ω b e iδt) σ 12 (t) ( Ωa + Ω b e iδt) [σ 22 (t) σ 11 (t)], (2.8) i ( Ω a + Ω be iδt) σ 21 (t) 4G I σ 21 (t)σ 12 (t) σ 22(t) σ 11 (t) + 1 T 1. (2.9)

40 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 15 In Eqs. (2.8) and (2.9), a = ω 0 ω a is the detuning of the pump field from resonance, δ = ω b ω a is the detuning between the two fields, and G I represents the imaginary part of the parameter G = G a + G b. Moreover, T 1 is the population relaxation time due to spontaneous emission and T 2 is the relaxation time due to dephasing processes of the semiconductor quantum dot. The relaxation times T 1 and T 2 are in principle influenced by the presence of the metal nanoparticle. Using the same approximations as above, we can calculate T 1 next to the metallic nanosphere as [52, 53] T 1 (ω n ) = T (c/ω n ) 3 Im(γ n )α 3 /R 6, (2.10) where T 10 is the population relaxation time in the absence of the metal nanoparticle and c is the speed of light in the vacuum. In addition, from the general form of T 2 we obtain [46] T 2 (ω n ) = 2T 1(ω n )T 20 T T 1 (ω n ), (2.11) where T 20 is relaxation time of the quantum dot due to pure dephasing processes in the absence of the metal nanoparticle. However, for the parameters that we use here the values of T 1 and T 2 do not practically change in the frequency region of interest, and therefore will be considered constant in this study, as in previous papers concerning similar systems [18, 19, 20, 21, 31, 32, 33, 34, 35]. We take that the pump field is a strong field and its interaction with the system will be treated to all orders while the probe field is a weak field and its interaction with the system will be treated to first order. This means that Ω b Ω a. We then expand σ 21 (t) as [46, 47, 48] σ 21 (t) = σ (ω a) 21 (t) + σ (ω b) 21 (t)e iδt + σ (2ω a ω b ) 21 (t)e iδt, (2.12)

41 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 16 and σ 22 (t) σ 11 (t) as [46, 47, 48] σ 22 (t) σ 11 (t) = w (0) (t) + w ( δ) (t)e iδt + w (δ) (t)e iδt. (2.13) Here, w (0) (t) is the population inversion and σ (ω a) 21 (t) is the off-diagonal term of the matrix element in the case that E b = 0. Also, w (δ) (t) = w ( δ) (t) as the population difference should be real. The last two terms describe the effect of population pulsation in the system [47]. The terms σ (ω b) 21 (t), σ (2ω a ω b ) 21 (t), w ( δ) (t), w (δ) (t) are considered to be small, in the sense that σ (ω b) 21 (t), σ (2ω a ω b ) 21 (t) σ (ω a) 21 (t) and w ( δ) (t), w (δ) (t) w (0) (t). We also define σ ( ωa) 12 (t) = σ (ωa) ( ω 21 (t), σ b ) 12 (t) = σ (ω b) 21 (t) and σ (ω b 2ω a) 12 (t) = σ (2ωa ω b) 21 (t). Following the above procedure we obtain the following differential equations for the density matrix elements: ẇ (0) (t) = i [ Ω a σ ( ω a) 12 (t) Ω aσ (ω a) 21 (t) ] 4G I σ ( ω a) 12 (t)σ (ω a) 21 (t) w(0) (t) + 1, (2.14) T 1 ẇ ( δ) (t) = (iδ + 1 ) w ( δ) (t) + iω a σ ( ω b) 12 (t) iω aσ (2ω a ω b ) 21 (t) T1 σ (ωa) 21 (t) = σ (ω b) 21 (t) = σ (2ω a ω b ) 21 (t) = iω bσ (ωa) [ ( ω 21 (t) 4G I σ b ) 12 (t)σ (ωa) 21 (t) + σ ( ωa) 12 (t)σ (2ωa ω b) 21 (t) ] (2.15), (i a + 1 ) T2 σ (ωa) 21 (t) i Ω a 2 w(0) (t) igσ (ωa) 21 (t)w (0) (t), (2.16) (i a iδ + 1 ) σ (ω b) 21 (t) i Ω a T2 2 w(δ) (t) igσ (ω a) 21 (t)w (δ) (t) igσ (ω b) 21 (t)w (0) (t) i Ω b 2 w(0) (t), (2.17) (i a + iδ + 1 T2 ) σ (2ω a ω b ) 21 (t) i Ω a 2 w( δ) (t) igσ (ω a) 21 (t)w ( δ) (t) igσ (2ω a ω b ) 21 (t)w (0) (t). (2.18) We will solve these equations for the description of the four-wave mixing process. The essential difference of Eqs. (2.14)-(2.18) and those describing two-level systems

42 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 17 without the plasmonic nanostructure [46, 47, 48] is due to the nonlinear terms, i.e. the terms containing the parameter G. As Eqs. (2.14)-(2.18) are nonlinear differential equations we will follow a numerical solution of them using a Runge-Kutta method. The four-wave mixing spectrum is given by [46, 47, 48, 49] F W M = σ (2ω a ω b ) 21 (t = T ) 2, (2.19) where T is the length of the applied electromagnetic fields that is taken long enough so as the system under study has reached steady state behavior. Eqs. (2.14) and (2.16) can be solved independently. Combining the steady state solutions of these two equations, we find that the steady state population inversion can be calculated as one of the roots of a third-order equation [22]: w (0) ss 3 + c2 w ss (0) 2 + c1 w ss (0) + c 0 = 0, (2.20) where c 2 = 1 + 2T 2 2 a G R 2T 2 G I T 2 2 G 2, (2.21) c 1 = T 1T 2 Ω a 2 + T a + 2T 2 2 a G R 2T 2 G I + 1 T 2 2 G 2, (2.22) c 0 = 1 + T a T 2 2 G 2, (2.23) with w (0) ss denoting the value of w (0) (t) in steady state and G R being the real part of the parameter G. The solutions for w (0) ss are (w ss (0) ) 1 = p + + p c 2 3, (2.24) (w ss (0) ) 2 = p + + p c 2 2 (w ss (0) ) 3 = p + + p c i i 2 (p + p ), (2.25) 3 2 (p + p ), (2.26)

43 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 18 where p ± = ( r ± S ) 1/3, (2.27) q = 3c 1 c r = 9c 1c 2 27c 0 2c , (2.28), (2.29) S = q 3 + r 2. (2.30) As w (0) ss represents the population inversion, we only account for the real roots that satisfy the condition 1 w (0) ss 1. These solutions will be used in the following section for the explanation of the behavior of the four-wave mixing spectrum. 2.3 Numerical results In all the calculations we take the quantum dot initially in the ground state, leading to w (0) (0) = 1 and the rest density matrix elements being zero. The parameters that we use are T 1 = 0.8 ns, T 2 = 0.3 ns, ε env = ε 0, α = 7.5 nm, µ = 0.65 e nm, hω 0 = 2.5 ev, ε S = 6ε 0, with ε 0 being the dielectric constant of the vacuum. These values correspond to colloidal quantum dots (typically CdSe-based quantum dots) and have been used in various studies [18, 19, 20, 22, 31, 32, 33, 34]. For ε m (ω) we use experimental values of gold [54]. We solve numerically, using a fourth-order Runge-Kutta method, Eqs. (2.14)-(2.18) for T = 100 ns to obtain the steady state behavior of the spectrum. Below, in all calculations the intensity of the probe field is taken four orders of magnitude smaller than the intensity of the pump field. In Fig. 2.2, we present the four-wave mixing spectrum for different interparticle distances when the pump field is at exact resonance with the quantum dot, a = 0.

44 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 19 8 a b FWM FWM Detuning, ns Detuning, ns 1 c d FWM Detuning, ns 1 FWM Detuning, ns 1 Figure 2.2: (Color online) The four-wave mixing spectra in the coupled system as a function of the detuning δ. The intensity of the pump field is I a = 10 2 W/cm 2 in (a) and (c) and I a = W/cm 2 in (b) and (d). The pump field excitation is at exact resonance, a = 0. In (a) solid curve R = 100 nm, dashed curve R = 20 nm, dotted curve R = 18 nm and dot-dashed curve R = 16.5 nm. In (b) solid curve R = 100 nm, dashed curve R = 20 nm, dotted curve R = 17 nm and dot-dashed curve R = 15 nm. In (c) solid curve R = 15 nm, dashed curve R = 14.5 nm and dotted curve R = 14 nm. In (d) solid curve R = 14 nm, dashed curve R = 13.5 nm and dotted curve R = 13 nm.

45 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system a 25 b FWM FWM Detuning, ns 1 c Detuning, ns 1 d FWM FWM Detuning, ns 1 Detuning, ns 1 Figure 2.3: (Color online) The four-wave mixing spectra in the coupled system as a function of the detuning δ. The intensity of the pump field is I a = 10 2 W/cm 2 in (a) and (c) and I a = W/cm 2 in (b) and (d). The pump field excitation is detuned by a = 5 ns 1 in (a) and (c) and a = 10 ns 1 in (b) and (d). In (a) solid curve R = 100 nm, dashed curve R = 20 nm, dotted curve R = 18 nm and dot-dashed curve R = 16 nm. In (b) solid curve R = 100 nm, dashed curve R = 20 nm, dotted curve R = 17 nm and dot-dashed curve R = 15 nm. In (c) solid curve R = 15 nm, dashed curve R = 14.5 nm and dotted curve R = 14 nm. In (d) solid curve R = 13.5 nm, dashed curve R = 13 nm and dotted curve R = 12.5 nm.

46 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 21 FWM a FWM b Detuning, ns Detuning, ns c 0.8 d FWM FWM Detuning, ns Detuning, ns 1 Figure 2.4: (Color online) The four-wave mixing spectra in the coupled system as a function of the detuning δ. The intensity of the pump field is I a = 10 2 W/cm 2 in (a) and (c) and I a = W/cm 2 in (b) and (d). The pump field excitation is detuned by a = 5 ns 1 in (a) and (c) and a = 10 ns 1 in (b) and (d). In (a) solid curve R = 100 nm, dashed curve R = 22 nm, dotted curve R = 20 nm and dot-dashed curve R = 18 nm. In (b) solid curve R = 100 nm, dashed curve R = 20 nm, dotted curve R = 18 nm and dot-dashed curve R = 16 nm. In (c) solid curve R = 16 nm, dashed curve R = 15 nm and dotted curve R = 14 nm. In (d) solid curve R = 14.5 nm, dashed curve R = 14 nm and dotted curve R = 13.5 nm.

47 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 22 Results for two different pump field intensities are shown. From Figs. 2(a) and 2(b) we find that for both pump field intensities the spectrum is triple peaked, with one peak at δ = 0 and two symmetric peaks at positive and negative values of δ. These are called Rabi sidebands. The decrease of the interparticle distance increases the magnitude of the four-wave mixing spectrum. In addition, the positions of the Rabi sidebands change to larger or smaller detunings depending on the interparticle distance and the pump field intensity. These results are in agreement with that of refs. [31, 42], where the real and imaginary part of the susceptibility which are responsible for the four-wave mixing process were presented. However, for even smaller interparticle distances the results are markedly different, as can be seen in Figs. 2(c) and 2(d). There, the spectrum is single-peaked and as the interparticle distance decreases the spectrum is suppressed. The same general behavior is found in Figs. 2.3 and 2.4 that the pump field is detuned from resonance, for both positive (Fig. 2.3) and negative (Fig. 2.4) detunings. We note that the form of the spectrum and the actual distance for which the form of the spectrum changes from triple-peaked to singlepeaked depends on the pump field intensity and frequency, and for a certain value of the distance the spectrum can be either triple-peaked or single-peaked depending on the values of the pump field intensity and frequency. For the dependence on pump intensity see for example the case of R = 15 nm in Fig. 2(b) and 2(c). The explanation of the form of the four-wave mixing spectrum shown in Figs. 2.2 to 2.4 lies in the behavior of the effective Rabi frequency Ω eff = Ω a + 2Gσ (ω a) ss 21, (2.31) where σ (ω a) ss (ω 21 denotes the value of σ a ) 21 (t) in steady state [32, 45]. The effective Rabi

48 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 23 frequency Ω eff is defined in order to make the nonlinear density matrix equations (2.14)-(2.18) in steady state to be written in the same form as the (linear) density matrix equations of a regular two-level system [46], where the pump field Rabi frequency has been replaced by the effective Rabi frequency. Of course, the effective Rabi frequency is not a regular Rabi frequency as its value depends on the value of the density matrix element σ (ωa) ss 21 ; however, it can be used for understanding the behavior shown in Figs. 2.2 to 2.4, as can be seen from Figs. 2.5 and 2.6. From there we note that the value of the effective Rabi frequency changes strongly with interparticle distance. This change denotes two regions, the region of large value of Ω eff where the triple-peaked spectrum appears and the region of low value of Ω eff where the singlepeaked spectrum appears. This is an indication of plasmonic metaresonances in the system, a concept proposed and explored by Sadeghi [32, 44, 45]. We also find that the positions of the Rabi sidebands are roughly approximated by δ = ± 2 a + Ω eff 2. The larger values of Ω eff are obtained for positive detunings of the pump field and the lower values for negative detunings of the pump field. Also, for negative detunings of the pump field the change between the two regions of Ω eff is rather smooth, while for zero or positive detunings this change is rather abrupt. The behavior of Ω eff can be used for determining the critical distance that changes the form of the spectrum from triple-peaked to single-peaked as the distance decreases. For the case of zero or positive detunings the critical distance is the distance that the abrupt change in the value of Ω eff occurs. For example, for the case of Fig. 2.5 these values are approximately nm for a = 0 and nm for a = 5 ns 1. For the case of negative detunings the exact value of the critical distance is not uniquely

49 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 24 Re eff ns a Interparticle Distance, R nm b 8 Im eff ns eff ns Interparticle Distance, R nm c Interparticle Distance, R nm Figure 2.5: (Color online) The real (a), imaginary (b) and absolute value (c) of the effective Rabi frequency Ω eff as a function of the interparticle distance R. The intensity of the pump field is I a = 10 2 W/cm 2 and the pump field detuning is a = 0 (solid curve), a = 5 ns 1 (dashed curve) and a = 5 ns 1 (dotted curve).

50 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system a 20 Re eff ns Interparticle Distance, R nm b 15 Im eff ns Interparticle Distance, R nm c 25 eff ns Interparticle Distance, R nm Figure 2.6: (Color online) The real (a), imaginary (b) and absolute value (c) of the effective Rabi frequency Ω eff as a function of the interparticle distance R. The intensity of the pump field is I a = W/cm 2 and the pump field detuning is a = 0 (solid curve), a = 10 ns 1 (dashed curve) and a = 10 ns 1 (dotted curve).

51 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 26 defined as the change in Ω eff occurs smoothly, however a region of distances that this change occurs can still be determined. For example, for the case of Fig. 2.5 these values are approximately in the region 16.5 nm to 17.5 nm. In order to further stress that the pump field detuning can be used for the control of the form of the four-wave mixing spectrum, we present in Fig. 2.7 the spectrum for the same interparticle distance and pump field intensity, but for different values of the pump field detuning. This shows that the spectrum changes from single-peaked in Fig. 2.7(a) to triple-peaked in Fig. 2.7(b). The explanation of this is shown in Fig. 2.8, where it can be seen that the value of Ω eff changes from the region of low value to the region of large value with the increase of the detuning of the pump field for the specific value of interarticle distance. For the explanation of the behavior of Ω eff the relation between Ω eff and w (0) ss will be used. From Eqs. (2.14) and (2.16), with the use of Eq. (2.31), we obtain: Ω eff 2 = 1 + T a T 1 T 2 ( ) w ss (0). (2.32) The solutions of the third-order equation describing w (0) ss, Eqs. (2.24)-(2.26) strongly depend on the values of the parameters of the system, as can be seen by the expressions for the coefficients of the third-order algebraic equation, Eqs. (2.21)-(2.23). The nature of the roots is purely determined by the parameter S. We have one real root and two complex roots, with (w (0) ss ) 2 = (w (0) ss ) 3, when S 0 and three real roots (bistability case [31, 34, 35]) when S < 0. We start with the a = 0 case, and try to explain the observed behavior of the magnitude of Ω eff [solid curve in Fig. 2.5(c)] by understanding the behavior of w (0) ss. We find that for short distances between the quantum dot and the metal nanoparticle

52 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system a 1.0 FWM Detuning, ns 1 b FWM Detuning, ns 1 Figure 2.7: (Color online) The four-wave mixing spectra in the coupled system as a function of the detuning δ. The intensity of the pump field is I a = 10 2 W/cm 2 and the pump field detuning is a = 1 ns 1 in (a) solid curve, a = 4 ns 1 in (a) dashed curve, a = 7 ns 1 in (b) solid curve and a = 10 ns 1 in (b) dashed curve. The interparticle distance is R = 15.3 nm in every case.

53 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 28 a Re eff ns Interparticle Distance, R nm Im eff ns b Interparticle Distance, R nm c 15 eff ns Interparticle Distance, R nm Figure 2.8: (Color online) The real (a), imaginary (b) and absolute value (c) of the effective Rabi frequency Ω eff as a function of the interparticle distance R. The intensity of the pump field is I a = 10 2 W/cm 2 and the pump field detuning is a = 1 ns 1 (solid curve), a = 4 ns 1 (dashed curve), a = 7 ns 1 (dotted curve) and a = 10 ns 1 (dash-dotted curve).

54 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 29 the parameter S is positive and we have just one real and negative root, of rather large absolute value, for example at R = 13 nm, w (0) ss is As the interparticle distance increases the absolute value of the population inversion decreases, and at R = 14.5 nm and R = nm w (0) ss becomes 0.87 and 0.75, respectively. At R = nm, the parameter S becomes zero and then for a just higher value of the interparticle distance it becomes negative and we obtain three real roots ( 0.74, 0.09 and 0.08) for w (0) ss. In the region from R = nm until R = nm, the S-parameter is always negative and the three real roots are all negative. The largest one, (w (0) ss ) 1, decreases in absolute value as the distance increases. Also, the second root, (w (0) ss ) 2, increases in absolute value with the increase of the interparticle distance. In addition, the third root, (w (0) ss ) 3, has a rather small value and a weak R dependence, as at the two limits of the bistability region, i.e at R = nm and at nm the root is approximately 0.08 and 0.05, respectively. At distance R = nm the parameter S becomes zero. In this case we have one root of small value (approximately 0.05) and two equal roots (w (0) ss ) 2 = (w (0) ss ) 3, as p + = p. At an interparticle distance just above nm the S-parameter becomes positive again, and there is one real root, (w (0) ss ) 1, and two complex roots. At this distance we have a very abrupt change in w (0) ss as the two larger in absolute value real roots (for R = nm, the roots are 0.42, 0.42 and 0.05) become complex (having a rather small imaginary part) and the third one (now the only real one) has a very small value. A very similar behavior is found for positive detuning of the pump field. For the a =5 ns 1, the bistability region in which the parameter S is negative is [14.32 nm

55 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system nm]. Hence, the largest distance at which the S-parameter becomes zero is at nm and as in the previous case at this distance we have the occurrence of the abrupt change of the population inversion. Obviously, this is the distance at which we have the abrupt change of the magnitude of the effective Rabi frequency in the dotted curve of Fig. 2.5(c). For the case of negative detuning of the pump field studied in Fig. 2.5(c), there are no distances for which the S-parameter becomes zero and therefore there is not an abrupt change in the population inversion. However, we should mention that this is not a general trend for a < 0, as for small values of negative a, for example for a = 1 ns 1, we have found a bistability region and an abrupt change of Ω eff at a distance of nm (not shown here). Therefore, we conclude that the most important parameter in the explanation of the abrupt change in the magnitude of Ω eff is the S-parameter and especially the largest distance at which it becomes zero. Applying this condition for the largest intensity of the pump field (I a = W/cm 2 ), we find R = nm for a = 0 and R = nm for a = 10 ns 1. These values are in excellent agreement with the values suggested in Fig. 2.6(c). 2.4 Conclusions We have studied the four-wave mixing effect in a coupled semiconductor quantum dot - spherical metal nanoparticle structure giving emphasis to the behavior of the four-wave mixing spectrum with the interparticle distance. Depending on the values of the pump field intensity and frequency, we find that there is a critical distance that

56 Chapter 2: Four-wave mixing effects in a coupled semiconductor quantum dot - metal nanoparticle system 31 changes the form of the spectrum. Above this distance the four-wave mixing spectrum shows an ordinary three-peaked form and the effect of controlling its magnitude by changing the interparticle distance can be obtained. Below this critical distance the four-wave mixing spectrum becomes single-peaked, and as the interparticle distance decreases the spectrum is strongly suppressed. The behavior of the system is explained by the effective Rabi frequency that creates plasmonic metaresonances in the hybrid structure. Furthermore, the behavior of the effective Rabi frequency is explained via an analytical solution of the density matrix equations.

57 Chapter 3 Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 3.1 Introduction In some of the studies involving a coupled nanostructure consisting of a semiconductor quantum dot and a metal nanoparticle [45, 55, 56, 57], the controlled dynamics of the populations of the ground and single exciton states of the quantum dot was analyzed, giving emphasis to the influence of the presence of the metal nanoparticle to the coherent population transfer from the ground to the exciton states. Specifically, it has been shown that the period of Rabi oscillations in the quantum dot, that is modeled as a two-level system, is significantly altered due to the presence of the plasmonic nanoparticle [55, 56]. In addition, Sadeghi showed that the Rabi oscillations 32

58 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 33 can be destroyed for certain interparticle distances due to the rise of plasmonic metaresonances [45]. More recently, Antón et al. [57] showed that if the quantum dot is modeled as a three-level V-type system, then selective excitonic population transfer can be achieved due to the presence of the plasmonic nanoparticle. Furthermore, methods for population transfer in a single hole spin state in a p-doped InAs/GaAs self-assembled quantum dot system coupled to a metallic nanoparticle, using both continuous wave and pulsed electromagnetic fields, have been recently proposed [58]. In this chapter we analyze the potential for controlled population inversion in a semiconductor quantum dot - metal nanoparticle hybrid system [59]. The hybrid complex that we study is comprised of a small quantum dot and a spherical metal nanoparticle. The quantum dot is described by a two-level system and the interaction of the system with an external electromagnetic field is modeled by the modified nonlinear density matrix equations that take into account the interaction between excitons and surface plasmons [18, 19, 20]. We consider the interaction with a pulsed electromagnetic field with hyperbolic secant envelope and allow for time-dependent frequency (chirp). Here, we first show that the widely used method of population inversion by a π-pulse can be modified for small interparticle distances due to the interaction between excitons and surface plasmons. This modification depends strongly on the pulse duration. We also present analytical solutions of the density matrix equations which lead to efficient excitonic population inversion in the quantum dot for several, even small, distances between the semiconductor quantum dot and the metal nanoparticle. We note that recently, several articles have considered the coherent manipulation

59 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 34 of quantum dot systems interacting with hyperbolic secant electromagnetic pulses both theoretically [60] and experimentally [61, 62]. In addition, analytical solutions of nonlinear optical Bloch equations which lead to high efficiency excitonic population inversion in quantum dots with local field effects have been presented for hyperbolic secant pulses [63]. The chapter is organized as follows: in the next section we present the density matrix equations for the interaction of the semiconductor quantum dot system with the electromagnetic pulse in the presence of the metal nanoparticle. Then, in section 3.3 we present results obtained from (approximate) analytical and numerical solutions of the density matrix equations, with emphasis given to cases that lead to large excitonic population inversion in the system. Finally, in section 3.4 we summarize our findings. 3.2 Theoretical model The hybrid structure that we study is composed of a spherical metal nanoparticle of radius α and a small spherical semiconductor quantum dot of radius β, in an environment with real dielectric constant ε env. The system is similar to that presented in Fig. 2.1 and we apply a two-level system approximation for the study of the quantum dot with 1 being the ground state and 2 being the single exciton state. This system interacts with a linearly polarized oscillating electromagnetic field with E(t) = ẑe 0 f(t) cos[ωt + ϕ(t)], (3.1)

60 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 35 that excites the interband transition between the two energy levels of the semiconductor quantum dot. Here ẑ is the polarization unit vector (along the z direction), E 0 is the electric field amplitude, f(t) is the dimensionless pulse envelope, ω is the angular frequency, and ϕ(t) is the time-dependent phase of the applied field. We assume that only these two levels contribute to the dynamics of the system and an oscillating dipole moment is induced. The dielectric constant of the semiconductor quantum dot is represented by ε S, while we treat the metal nanoparticle as a classical dielectric particle with dielectric function ε m (ω). The electromagnetic field also excites plasmons, on the surface of the metal nanoparticle. These plasmonic excitations provide a strong continuous spectral response. Such surface plasmons influence the excitons and induce electromagnetic interactions between excitons and plasmons [18, 19, 20]. This interaction is responsible for the coupling between the two particles and leads to Förster energy transfer [51]. The Hamiltonian of the system, in the dipole approximation, takes the form H = hω µe SQD (t) ( ), (3.2) where µ represents the dipole moment of the semiconductor quantum dot corresponding to the single exciton transition, and E SQD represents the electric field inside the semiconductor quantum dot. In the dipole approximation the total electric field inside the quantum dot consists of two parts, the first part being the applied external field and the second part corresponding to the induced field produced by the polarization of the metal nanoparticle (treated as a classical metallic nanosphere) [18, 19, 20]: E SQD (t) = h [ ( ) Ω(t) + Gσ(t) µ 2 e i[ωt+ϕ(t)]

61 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 36 + ( Ω ) ] (t) + G σ (t) e i[ωt+ϕ(t)]. (3.3) 2 Above, we introduced the slowly varying quantities σ(t) = ρ 21 (t)e i[ωt+ϕ(t)] and σ (t) = ρ 12 (t)e i[ωt+ϕ(t)], with ρ ij (t) being the density matrix elements. We also defined the time-dependent Rabi frequency Ω(t) as [18, 19, 20] and the parameter G as [18, 20] Here, ε effs Ω(t) = Ω 0 f(t), Ω 0 = µe ( s aγ 1 α 3 ), (3.4) hε effs R 3 = 2ε env+ε S 3ε env, γ 1 = εm(ω) εenv ε m(ω)+2ε env, and s a G = 1 s 2 aγ 1 α 3 µ 2. (3.5) 4πε env hε 2 effsr6 = 2 as the applied field is taken parallel to the interparticle axis of the system (the interparticle axis is the z-axis). The density matrix equations, which describe the dynamics of the system, under the rotating wave approximation, are given by [18, 19, 20]: σ(t) = 1 T 2 σ(t) + i Ω(t) 2 (t) + ig (t)σ(t) + iδσ(t) + i ϕ(t)σ(t), (3.6) (t) = iω (t)σ(t) iω(t)σ (t) + 4G I σ(t)σ (t) (t) 1 T 1. (3.7) In Eqs. (3.6) and (3.7), (t) = ρ 11 (t) ρ 22 (t) corresponds to the population difference between the ground and single exciton states, which is a real quantity, δ = ω ω 0 is the detuning of the applied field from resonance and G I represents the imaginary part of the parameter G(= G R + ig I ). Moreover, T 1 is the population relaxation time due to spontaneous emission and T 2 is the relaxation due to dephasing processes of the semiconductor quantum dot. The relaxation times T 1 and T 2 are in principle influenced by the presence of the metal nanoparticle [see Eqs. (2.10) and (2.11)].

62 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 37 However, we will consider them constant in this study similar to several previous works, as for the parameters that we use here the values of T 1 and T 2 do not practically change in the frequency region of interest. 3.3 Analytical and numerical results We take the quantum dot initially in the ground state [ (t = 0) = 1, σ(t = 0) = 0] and study for the moment the population dynamics of the quantum dot under short pulse excitation, so the relaxations times T 1 and T 2 are considered infinite, in order to analyze the regime of fully coherent excitation of the system. In the case that ϕ(t) = 0, and the electromagnetic field is at exact resonance with the quantum dot, δ = 0, then if we ignore the effect of the G-parameter, i.e. take G = 0, we can obtain the following analytical solution for (t) t (t) = cos[θ(t)], Θ(t) = Ω 0 f(t )dt, (3.8) 0 where Θ(t) is the time-dependent pulse area. At the end of the pulse, Θ(t) takes a constant value that is known as pulse area θ. Eq. (3.8) clearly shows how important pulse area can be. If θ is an odd multiple of π, then complete excitonic population inversion is succeeded at the end of the pulse, while if θ is an even multiple of π, then the population returns to the ground state at the end of the pulse. Below we assume that the system interacts with a pulse with hyperbolic secant envelope, with f(t) = sech[(t t 0 )/t p ], where t p characterizes the pulse width, and the pulse center t 0 is chosen such as the pulse is practically zero at t = 0 and t = 2t 0.

63 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 38 In this case θ = Θ(2t 0 ) = π Ω 0 t p. (3.9) Therefore, it is clear from Eqs. (3.4) and (3.9) that the pulse area depends on the interparticle distance, R. Therefore, if one chooses the electric field amplitude as E 0 = hε effs µt p in order to obtain a π-pulse in the absence of the metal nanoparticle, then the presence of the metal nanoparticle will change the dynamics of the quantum dot populations and complete excitonic population inversion will not occur. However, one may choose an R-dependent electric field amplitude as E 0 = hε effs µt p 1 + s a γ 1 a 3 R 3, (3.10) in order to obtain a π-pulse for every R. In this case the population dynamics may be still strongly influenced from the presence of the metal nanoparticle for small interparticle distances and excitonic population inversion can be small. This can be seen in Fig. 3.1 that shows the time evolution of (t) for the same parameters as in Chapter 2, i.e., T 1 = 0.8 ns, T 2 = 0.3 ns, ε env = ε 0, α = 7.5 nm, µ = 0.65 e nm, hω 0 = 2.5 ev, ε S = 6ε 0, with ε 0 being the dielectric constant of the vacuum. For ε m (ω) we use experimental values of gold [54]. The time evolution of (t) in Fig. 3.1 has been calculated from the numerical solution of Eqs. (3.6) and (3.7) taking G nonzero. The strong influence in the population dynamics in this case happens due to the nonzero values of G that increases for small interparticle distances. This plays an important role and makes the solution of Eq. (3.8) invalid. The above findings are not limited to hyperbolic secant pulses; they hold for any pulse envelope. Interestingly, the influence of the parameter G on the excitonic population dynamics depends strongly on the pulse duration, as can be seen from Fig.

64 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system t Time ps Figure 3.1: (Color online) The time evolution of (t) from the numerical solution of Eqs. (3.6) and (3.7) for R = 80 nm (solid curve), R = 14 nm (dotted curve), R = 13 nm (dashed curve), and R = 12 nm (dot-dashed curve), with t 0 = 22.5 ps and t p = 3 ps. The electric field amplitude E 0 is taken different in every distance according to Eq. (3.10), so it always gives π pulse area As the pulse duration becomes smaller the influence of the parameter G becomes weaker [see Fig. 3.2(a) and (b)], and for short pulses no influence is found [see Fig. 3.2(c)]. This means that a pulse with area π for every R may lead to significant excitonic population inversion for short pulses. In addition, as the pulse becomes shorter, the influence of the population decay and dephasing processes weakens and larger final populations are obtained. In order to explore further the dependence of the inversion in pulse area, we present in Fig. 3.3 the final value of (t), i.e., the value of (t) at t = 2t 0, as a function of the pulse area θ. This is obtained from the numerical solution of Eqs. (3.6) and (3.7) taking G nonzero. We find that for longer pulses, the pulse areas for maximum excitonic population inversion can be quite different than π and depend on the value of the interparticle distance. For example, for the case of Fig. 3.3(a),

65 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system a 0.5 t Time ps b t Time ps c t Time ps Figure 3.2: (Color online) The same as in Fig. 3.1 with (a) t 0 = 11.5 ps and t p = 1.5 ps, (b) t 0 = 6 ps and t p = 0.75 ps and (c) t 0 = 0.75 ps and t p = 0.1 ps. Solid curve: R = 80 nm, dotted curve: R = 14 nm, dashed curve: R = 13 nm, and dot-dashed curve: R = 12 nm.

66 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system a 1.0 b t t Θ Π Θ Π 1.0 c 1.0 d t t Θ Π Θ Π Figure 3.3: (Color online) The final value of (t) [ (2t 0 )] as a function of pulse area θ. The pulse area is measured in multiples of π. In (a) t 0 = 22.5 ps and t p = 3 ps, (b) t 0 = 11.5 ps and t p = 1.5 ps, (c) t 0 = 6 ps and t p = 0.75 ps, and (d) t 0 = 0.75 ps and t p = 0.1 ps. Solid curve: R = 80 nm, dotted curve: R = 14 nm, dashed curve: R = 13 nm, and dot-dashed curve: R = 12 nm.

67 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system a 0.5 t Time ps b t Time ps Figure 3.4: (Color online) The time evolution of (t) from the numerical solution of Eqs. (3.6) and (3.7) for R = 80 nm (solid curve), R = 14 nm (dotted curve), R = 13 nm (dashed curve), and R = 12 nm (dot-dashed curve), with t 0 = 22.5 ps and t p = 3 ps. The different curves coincide. In (a) Eq. (3.12) is fulfilled and in (b) Eq. (3.13) is fulfilled.

68 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 43 which is the longer pulse considered here, the pulse area for maximum inversion is about 1.075π for R = 14 nm, 1.12π for R = 13 nm, and 1.2π for R = 12 nm. The displayed dependence explains the results of Fig. 3.1, as one may see that a π pulse area leads to good final inversion for R = 14 nm, smaller final inversion for R = 13 nm and very small final inversion for R = 12 nm. As the pulse duration becomes shorter, then the pulse area that gives maximum inversion goes closer to π for the various interparticle distances. For the shortest case considered, that of Fig. 3.3(d), the maximum excitonic population inversion occurs for pulse area π (and odd multiples of π) independent of the value of the interparticle distance. Moreover, it will be interesting if solutions of (t) from Eqs. (3.6) and (3.7) are obtained, that lead to complete excitonic population inversion for non-zero G in the time regime that the relaxation times are omitted. This can be achieved for hyperbolic secant pulse envelope at exact resonance, δ = 0, in combination with ϕ(t) that has a tanh[(t t 0 )/t p ] form. If we require (t) = tanh[(t t 0 )/t p ], that leads to complete excitonic population inversion at t = 2t 0, and recall that dtanh(t)/dt = sech 2 (t), and dsech(t)/dt = sech(t)tanh(t), then by inspecting Eqs. (3.6) and (3.7) and using the normalization condition, we obtain that σ(t) has a sech[(t t 0 )/t p ] dependence. Using these facts, a direct substitution to Eqs. (3.6) and (3.7) gives that (t) = tanh[(t t 0 )/t p ] is obtained for ϕ(t) = Ktanh[(t t 0 )/t p ] and E 0 = hε effs [(G R K) 2 + ( G I + 1 t p ) 2 ] 1/2 µ 1 + s aγ 1 a 3 R 3. (3.11) Below we present two special cases of Eq. (3.11). In the first case ϕ(t) = 0 (K = 0)

69 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system 44 and E 0 = hε effs [G 2 R + ( ) ] 2 1/2 G I + 1 t p µ 1 + s aγ 1 a 3, (3.12) R 3 and in the second case K = G R ϕ(t) = G R tanh[(t t 0 )/t p ], E 0 = hε ( ) effs GI + 1 t p µ 1 + s aγ 1 a 3. (3.13) R 3 In the case that G = 0 Eq. (3.12) reduces to the π-pulse case, Eq. (3.10). To assess the validity of the analytical results in Fig. 3.4 we present the time evolution of (t) for both special cases [Fig. 3.4(a) is for Eq. (3.12) and Fig. 3.4(b) is for Eq. (3.13)] for the same material parameters as before and for different interparticle distances. The time evolution of (t) in Fig. 3.4 has been calculated from the numerical solution of Eqs. (3.6) and (3.7). The results are the same for the different interparticle distances, as the curves coincide. In both cases, the presence of relaxation processes leads to incomplete excitonic population inversion, however highefficiency inversion is accomplished (0.966 of the population is excited to the single exciton state) for all the values of the interparticle distance. Furthermore, we find that the pulse area values obtained from Eq. (3.12) are the same as those of Fig. 3.3 that lead to maximum excitonic population inversion for every case of interparticle distance and pulse duration. We can also use the analytical result of Eq. (3.12) in order to explain the behavior of the population inversion with the pulse duration displayed in Figs. 3.2 and 3.3. For a specific quantum dot and metallic nanoparticle and for a given interparticle distance R, G I and G R take a constant value. Then, smaller pulse duration leads to larger values for 1/t p, and for short enough pulses G 2 R + (G I + 1/t p ) 2 1/t p, so in this limit Eq. (3.12) approximates the π-pulse result of Eq. (3.10).

70 Chapter 3: Control of population dynamics in a coupled semiconductor quantum dot - metal nanoparticle system Conclusions In this chapter we have analyzed the potential for controlled population inversion in a coupled system comprised of a semiconductor quantum dot and a spherical metal nanoparticle. The quantum dot is described by a two-level system and for the description of the interaction of the system with an external electromagnetic field the modified nonlinear density matrix equations are used. The system interacts with a pulsed electromagnetic field with hyperbolic secant envelope. We first show that the widely used method of population inversion by a π-pulse can be modified for small interparticle distances due to the interaction between excitons and surface plasmons. This modification depends strongly on the pulse duration. We also present analytical solutions of the density matrix equations, using proper approximations, which lead to efficient excitonic population inversion in the quantum dot for several, even small, distances between the semiconductor quantum dot and the metal nanoparticle. The present results can be used in areas where controlled population transfer in a semiconductor quantum dot is needed, such as in quantum information processing and in the creation of ultrafast nanoswitches.

71 Chapter 4 Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 4.1 Introduction For several years it has been realized that the spontaneous emission of quantum emitters, such as atoms, molecules and quantum dots, can be strongly influenced by the presence of nanostructures, for reviews see, e.g., Refs. [64, 65]. An important effect in such systems is the significantly modified spontaneous decay rate in different emitter dipole moment directions, for example for orthogonal dipole directions. Agarwal [66] showed that this effect can be used for simulating quantum interfer- 46

72 Chapter 4: Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 47 ence effects in spontaneous emission [67, 68, 69, 70, 71, 72]. He proposed to place a three-level quantum emitter with orthogonal dipole moments within or near a structure which suppresses spontaneous emission for a specific dipole orientation. Applying this idea, it has been recently shown [15], by using a rigorous electromagnetic Green s tensor technique [73, 74, 75], that the placement of a three-level V-type quantum emitter in the proximity of simple, or complex, metallic nanostructures can boost the degree of quantum interference in spontaneous emission. Here, we continue this work and present results on the population dynamics of a three-level V-type quantum emitter for different initial conditions, in the presence of a two-dimensional array of metal-coated dielectric nanospheres [76]. We note that quite recently the effects of quantum interference in spontaneous emission has been studied in the population dynamics [16] and in the form of the spontaneous emission spectra [17, 77] of four-level quantum systems near plasmonic nanostructures. In addition, the study of spontaneous emission and resonance fluorescence of quantum emitters near various plasmonic structures has attracted significant attention recently [8, 9, 10, 11, 12, 13, 14]. We note that the effects of spontaneous decay and resonance fluorescence of quantum emitters near plasmonic structures have been studied since the 1970 s; for some earlier work see Refs. [52, 78, 79, 80, 81, 82, 83]. Moreover, several successful experiments have recently shown that the fluorescence near plasmonic nanostructures can be quite different than in free space, see for example Refs. [65, 84, 85, 86, 87, 88, 89, 90]. This chapter is organized as follows: in the next section we present the density matrix equations for the spontaneous decay dynamics of the quantum emitter in

73 Chapter 4: Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 48 the presence of a plasmonic nanostructure and explain the appearance of quantum interference effects in spontaneous decay. Then, in section 4.3 we present results for the population dynamics of the quantum emitter, obtained from analytical solutions of the density matrix equations, for different initial conditions of the quantum system. Finally, in section 4.4 we conclude our findings. 4.2 Theory The quantum system of interest is shown in Fig. 4.1(a). We consider a V-type system with two degenerate Zeeman sublevels for the upper states 2 and 3, and one lower state 1. The quantum system is located in vacuum at distance d from the surface of the plasmonic nanostructure. The dipole moment operator is taken as µ = µ( 2 1 ˆε ˆε + ) + H.c., where ˆε ± = (e z ± ie x )/ 2, describe the rightrotating (ˆε + ) and left-rotating (ˆε ) unit vectors and µ is taken to be real. Both excited levels 2 and 3 decay spontaneously to the lower level with decay rate 2γ. The spontaneous decay dynamics in the above system is studied by a density matrix approach. By considering solely spontaneous emission effects, the time-dependent density matrix equations describing the interaction of the atom with its environment, in the rotating-wave and Weisskopf-Wigner approximations, are given by [15, 91, 92, 93, 94] ρ 22 = 2γρ 22 κ(ρ 23 + ρ 32 ), (4.1) ρ 33 = 2γρ 33 κ(ρ 32 + ρ 23 ), (4.2) ρ 23 = 2γρ 23 κ(ρ 22 + ρ 33 ), (4.3)

74 Chapter 4: Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 49 metal 2 3 ω ω 1 silica S c S a d (a) (b) (c) (d) Figure 4.1: (Color online)(a) A V-type, three-level system. (b) Metallic nanoshell made from a silica core of radius S c and metal coating of thickness S S c. (c) Square lattice of metallic nanoshells (monolayer) with period a. (d) Side view of the monolayer, where d is the distance of the quantum emitter from the surface of a nanoshell. with ρ 11 + ρ 22 + ρ 33 = 1 and ρ nm = ρ mn. Here, κ describes the coupling coefficients between states 2 and 3 due to spontaneous emission in a modified anisotropic vacuum [66], and it is responsible for the effects of quantum interference. The values of γ and κ are obtained by [15, 91, 92, 93, 94, 95] γ = µ 0µ 2 ω 2 ˆε ImG(r, r; ω) ˆε + (4.4) h κ = µ 0µ 2 ω 2 ˆε + ImG(r, r; ω) ˆε +. (4.5) h Here, G(r, r ; ω) is the dyadic electromagnetic Green s tensor, where r refers to the position of the quantum emitter, and µ 0 is the permeability of vacuum. Also, ω is the resonance frequency between states 1 and 2 (or 3 ). For the quantization procedure that leads to the relation between the spontaneous decay rate and the dyadic electromagnetic Green s tensor, see Ref. [96] or the Appendix B of Ref. [1].

75 Chapter 4: Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 50 The electromagnetic Green s tensor obeys the equation G(r, r ; ω) ϵ(r, ω)ω2 c 2 G(r, r ; ω) = µ 0 Iδ(r r ), (4.6) where, ϵ(r, ω) is the spatially- and frequency-dependent dielectric function of the system, I = e x e x + e y e y + e z e z is the unit dyad (unit tensor), and c is the speed of light in the vacuum. From Eqs. (4.4) and (4.5) we obtain the values of γ and κ as [15, 91, 92, 93, 94, 95] γ = µ 0µ 2 ω 2 2 h Im[G (r, r; ω) + G (r, r; ω)] = 1 ( ) Γ + Γ, (4.7) 2 κ = µ 0µ 2 ω 2 2 h Im[G (r, r; ω) G (r, r; ω)] = 1 ( ) Γ Γ. (4.8) 2 Here, G (r, r; ω) = G zz (r, r; ω), G (r, r; ω) = G xx (r, r; ω) denote components of the electromagnetic Green s tensor where the symbol ( ) refers to a dipole oriented normal - along the z-axis (parallel - along the x-axis) to the surface of the plasmonic nanostructure. Finally, we define the spontaneous emission rates normal and parallel to the surface as Γ, = µ 0 µ 2 ω 2 Im[G, (r, r; ω)]/ h. The degree of quantum interference is defined as p = Γ Γ Γ + Γ. (4.9) For p = 1 we have maximum quantum interference in spontaneous emission [72]. This can be achieved by placing the emitter close to a structure that completely quenches Γ. We stress that when the emitter is placed in vacuum, Γ = Γ and κ = 0, so no quantum interference occurs in the system.

76 Chapter 4: Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 51 The corresponding electromagnetic Green s tensor which provides the corresponding spontaneous emission rates Γ and Γ is given by [15, 73, 75] G EE ii (r, r ; ω) = gii EE (r, r ; ω) i d 2 k 8π 2 SBZ g 1 c 2 K + g;z v gk ;i(r) exp( ik + g r)ê i (K + g ), (4.10) with and v gk ;i(r) = R g ;g(ω, k ) exp( ik g r)ê i(k g ) (4.11) g K ± g = (k + g, ±[q 2 ( k + g ) 2 ] 1/2 ). (4.12) The vectors g denote the reciprocal-lattice vectors corresponding to the 2D periodic lattice of the plane of scatterers and k is the reduced wavevector which lies within the surface Brillouin zone associated with the reciprocal lattice [74]. When q 2 = ω 2 /c 2 < (k + g) 2, K ± g defines an evanescent wave. The term gii EE(r, r ; ω) of Eq. (4.10) is the free-space Green s tensor and ê i (K ± g ) the polar unit vector normal to K ± g. R g ;g(ω, k ) is the reflection matrix which provides the sum (over g s) of reflected beams generated by the incidence of plane wave from the left of the plane of scatterers [74]. Also, in Eq. (4.10), the terms corresponding to s-polarized waves (those containing components with the azimuthial unit vector ê i (K ± g ) normal to K ± g ) have small contribution to the decay rates and have been, therefore, neglected. 4.3 Results The plasmonic nanostructure considered in this study is a two-dimensional array of touching metal-coated spherical dielectric (silica, SiO 2 ) nanoparticles, and is shown

77 Chapter 4: Quantum interference in spontaneous decay near a plasmonic nanostructure: Population dynamics 52 y x z metallic shell dielectric core (a) (b) Figure 4.2: (Color online)(a) A metal-coated dielectric nanosphere and (b) a twodimensional array of such spheres used in this work. in Figs. 4.1(b)-(d) and 4.2. Ordered arrays of metallic nanoshells can be fabricated via self-assembly [97, 98] and nanopatterning/nanolithographic [99, 100] techniques. The dielectric function of the shell, ϵ m (ω), is provided by a Drude-type electric permittivity given by ϵ m (ω) = 1 ω 2 p ω(ω + i/τ), (4.13) where ω p is the bulk plasma frequency and τ the relaxation time of the conductionband electrons of the metal. A typical value for the plasma frequency for gold is hω p = 8.99 ev. This also determines the length scale of the system as c/ω p 22 nm. The dielectric constant of silica is taken to be ϵ = 2.1. In the calculations we have taken τ 1 = 0.05ω p. The lattice constant of the square lattice is a = 2c/ω p and the sphere radius S = c/ω p with core radius S c = 0.7c/ω p. Fig. 4.3 shows the spontaneous decay rates near the plasmonic nanostructure described above. This is calculated with the method analyzed in Refs. [15, 73, 74,