Polish Academy of Sciences in partial fulfillment of the requirements for the degree of

Decoherence and entanglement decay of nitrogen-vacancy centers in diamond applied to spectroscopy of environmental noise

by

Damian Kwiatkowski B.Sc., University of Warsaw, 2014 M.Sc., University of Warsaw, 2015

A thesis submitted to the Institute of Physics

Polish Academy of Sciences in partial fulfillment of the requirements for the degree of

Doctor of Philosophy Institute of Physics

2020

The work was supported by funds of the Polish National Science Center (NCN), PhD Student Scholarship ETIUDA 6 No. 2018/28/T/ST3/00390

and the following grants: 2012/07/B/ST3/03616, 2015/19/B/ST3/03152.

Załącznik 2.

Warszawa, 17.04.2020 Damian Kwiatkowski

ul. Wileńska 6a/10 03-414 Warszawa PESEL: 91030207156

OŚWIADCZENIE autora rozprawy doktorskiej

Jako autor rozprawy doktorskiej pod tytułem:

Decoherence and entanglement decay of nitrogen-vacancy centers in diamond applied to spectroscopy of environmental noise

przygotowanej pod kierunkiem dr hab. Łukasza Cywińskiego, prof. IFPAN,

przedłożonej Radzie Naukowej Instytutu Fizyki Polskiej Akademii Nauk oświadczam, że:

1) rozprawa została napisana przeze mnie samodzielnie i nie zawiera treści uzyskanych w sposób niezgodny z obowiązującymi przepisami (ustawa z dnia 4 lutego 1994 r. o prawie autorskim i prawach pokrewnych – t.j. Dz. U. z 2006 r., Nr 90, poz. 631 z późn. zm.),

2) rozprawa nie była wcześniej przedmiotem procedur związanych z uzyskaniem stopnia naukowego w innym przewodzie doktorskim,

3) rozprawa doktorska w wersji cyfrowej umieszczona na załączonym nośniku danych jest tożsama z wersją wydrukowaną i przedłożoną w moim przewodzie doktorskim.

4) udzielam nieodpłatnie Instytutowi Fizyki Polskiej Akademii Nauk licencji niewyłącznej na umieszczenie pełnego tekstu ww. rozprawy doktorskiej w wersji elektronicznej na stronie

internetowej Instytutu w domenie ifpan.edu.pl i tym samym udostępnienie jej treści dla wszystkich

w sieci Internet.

v Kwiatkowski, Damian

Decoherence and entanglement decay of nitrogen-vacancy centers in diamond

applied to spectroscopy of environmental noise

Thesis directed by dr hab. Łukasz Cywiński, Prof. IF PAN

ABSTRACT

Nitrogen-vacancy center-based spin qubits in diamond offer a very special platform for inves- tigation of decoherence caused by their interaction with environment consisting of nuclear spins of

13

C, intrinsic to diamond. This environment is rather small - it consists of a few hundred spins, the minority of which is strongly coupled to the qubit, and the majority is weakly coupled. By tuning external magnetic field, one can control the nuclear spin dynamics that can be observed on the timescales of decoherence. In general, the whole environment is interacting due to presence of dipolar couplings between the nuclear spins. However, through a controlled expansion, in a fully quantum treatment - Cluster-Correlation Expansion based on a hierarchy of multi-spin correlations and timescales for formation of higher-order correlations - the corresponding many-body effects can be systematically introduced in calculation of the qubit decoherence.

Environment around the NV center is not large enough to be exactly described using a Gaus-

sian noise, which can be applied to environments consisting of many uncorrelated parts, each being

weakly coupled to the qubit. Timetrace of decoherence strongly depends on positions of the strongly

coupled spins, which can help to partially reconstruct the environment around the qubit. Moreover,

the state of the environment can be changed by generation of Dynamic Nuclear Polarization, by

appropriate manipulations on the qubit and we can see how the presence of this polarization affects

the decoherence. Here, we have shown how the non-classical effects (not amenable to description

by replacement of the environment by a source of classical noise) appear in a spin echo signal in

presence of polarized nuclei. We have also investigated the relation between the polarization and generation of qubit-environment entanglement and the process of appearance of states relevant for quantum Darwinism, so called Spectrum Broadcast Structures.

Analysis of two qubit decoherence allows for observations of spatial correlations in the environ- ment. Decoherence of two NV centers located a few nanometers from each other shows signatures of interaction with an environment common for both of the qubits. Analysing more precisely the deco- herence in a two-qubit system, one can also investigate how presence of one qubit affects decoherence of the other.

Results presented in this thesis show how the study of NV center and its nuclear environment

allows one to gain insight into the physics behind many nontrivial effects associated with deco-

herence: the quantum vs classical nature of environmental noise, generation of qubit-environment

entanglement, creation of Spectrum Broadcast Structures, correlated decoherence of multiple qubits

and back-action of qubit on the environment. The effects discussed here, can be easily tested ex-

perimentally.

vii Kwiatkowski, Damian

Decoherence and entanglement decay of nitrogen-vacancy centers in diamond

applied to spectroscopy of environmental noise

Praca doktorska pod kierunkiem dr hab. Łukasza Cywińskiego, Prof. IF PAN

STRESZCZENIE ROZPRAWY DOKTORSKIEJ

Kubity spinowe na bazie centrów azot-luka (NV) w diamencie mogą być wykorzystywane do badania dekoherencji wywołanej przez otoczenie składające się ze spinów jądrowych

C występują- cych w diamencie. Otoczenie to składa się z kilkuset spinów jądrowych, z których niewielka ilość jest silnie sprzężona do kubitu, a zasadnicza większość jest do niego słabo sprzężona. Poprzez zmianę zewnętrznego pola magnetycznego, można kontrolować dynamikę spinów jądrowych, obserwowalną na skali czasu dekoherencji. W ogólności, spiny otoczenia oddziałują ze sobą dipolowo. Niem- niej jednak możemy w kontrolowany sposób policzyć jego wpływ na kubit, korzystając z metody rozwinięcia CCE (ang. Cluster-Correlation Expansion), która opiera się na hierarchii skal czasu, w których istotne stają się korelacje wielospinowe o rosnącej liczbie spinów.

Otoczenie wokół centrum NV nie jest wystarczająco duże, aby można było je dokładnie opisać

używając szumu o gaussowskiej statystyce. Przebieg dekoherencji zależy bowiem silnie od położeń

spinów znajdujących się blisko centrum NV. Z drugiej strony, ten fakt może być wykorzystany

do częściowej rekonstrukcji otoczenia wokół kubitu. Co więcej, dynamiczna polaryzacja jądrowa

pozwala w pewnym stopniu zmienić stan otoczenia i następnie zaobserwować jak ta zmiana wpłynęła

na dekoherencję. W tej pracy pokazałem jak nieklasyczne efekty (takie, które nie da się opisać

poprzez zastąpienie otoczenia spinowego źródłem klasycznego szumu) pojawiają się w sygnale echa

spinowego w obecności polaryzacji jąder. Dodatkowo zbadałem zależność między polaryzacją a

generację splątania między kubitem a otoczeniem, jak również proces powstawania tzw. stanów

rozgłaszających spektrum (ang. Spectrum Broadcast Structures), co ma związek z kwantowym Darwinizmem.

Analiza dekoherencji dwóch kubitów pozwala dodatkowo na obserwację przestrzennych ko- relacji w otoczeniu. W dekoherencji dwóch centrów NV, zlokalizowanych kilka nanometrów od siebie, można zauważyć cechy oddziaływania z otoczeniem wspólnym dla obu tych kubitów. Do- datkowo w układzie dwukubitowym można również zbadać wpływ jednego kubitu na dekoherencję drugiego.

Wyniki przedstawione w tej rozprawie doktorskiej pokazują jak analiza centrum NV i otoczenia złożonego ze spinów jądrowych, pozwala badać efekty takie jak: natura przejścia kwantowo-klasycznego dla szumu otoczenia, generacja splątania pomiędzy kubitem a otoczeniem, formacja stanów rozgłasza- jących spektrum, skorelowana dekoherencja wielu kubitów, jak i wpływ kubitu na stan otoczenia.

Rozważania przedstawione w tej pracy mogą być również przebadane eksperymentalnie.

ix List of Publications

(1) Piotr Szańkowski, Guy Ramon, Jan Krzywda, Damian Kwiatkowski, Łukasz Cywiński, Environmental noise spectroscopy with qubits subjected to dynamical decoupling,

Journal of Physics: Condensed Matter 29, 333001 (2017) (2) Damian Kwiatkowski, Łukasz Cywiński,

Decoherence of two entangled spin qubits coupled to an interacting sparse nuclear spin bath, Physical Review B 98, 155202 (2018)

(3) Katarzyna Roszak, Damian Kwiatkowski, Łukasz Cywiński,

How to detect qubit-environment entanglement generated during qubit dephasing, Physical Review A 100, 022318 (2019)

(4) Damian Kwiatkowski, Piotr Szańkowski, Łukasz Cywiński,

Influence of nuclear spin polarization on the spin-echo signal of an NV-center qubit, arXiv:1909.06438 (accepted for publication in Phys. Rev. B)

Beyond the scope of this thesis

(5) Małgorzata Strzałka, Damian Kwiatkowski, Łukasz Cywiński, Katarzyna Roszak,

Qubit-environment Negativity versus Fidelity of conditional environmental states for an NV- center spin qubit interacting with a nuclear environment,

arXiv:1911.08867 (submitted to Phys. Rev. A)

(6) Ashley J. Ruiter, Krzysztof Belczyński, Stuart A. Sim, Ivo R. Seitenzahl, Damian Kwiatkowski, The effect of helium accretion efficiency on rates of Type Ia supernovae: double-detonations in accreting binaries,

Monthly Notices of the Royal Astronomical Society: Letters 440, L101-L105, (2014)

Acknowledgements

At the very beginning, I would like to express my sincere gratitude to my supervisor dr hab.

Łukasz Cywiński, prof. IFPAN for his guidance through the course of my study. Without his patience, commitment and support, this work would never come to being.

♠

I would like to thank my collaborators: dr Katarzyna Roszak, dr Jarosław Korbicz for sharing their insight to theory of decoherence and its consequences and dr Friedemann Reinhard for extensive support in understanding and testing the experimental capabilities and limitations on quantum sensing using NV centers.

♠

I will always be grateful for sharing inexhaustible passion everyday with my fellow, Jan Krzywda.

♠

I would like to express my thanks to my family and friends for all the unconditional support.

♠

I cannot begin to express my deepest gratitude to my love, Magdalena Sieniawska, who was always

there for me. I would never reach this moment without her boundless support and belief in the

purpose of my effort.

Contents

Chapter

1 Introduction 1

1.1 Who let the cat out? . . . . 1

1.1.1 A ball or a dice? . . . . 2

1.1.2 A faulty machinery . . . . 5

1.1.3 Rotating a qubit . . . . 5

1.1.4 Two qubits enter . . . . 6

1.1.5 Qubits: computer or sensor? . . . . 8

2 Nitrogen-vacancy in diamond 9 2.1 Energy level structure . . . . 9

2.2 Interaction with magnetic field . . . . 10

2.3 Optical initialization and readout . . . . 11

3 Poking the world with a qubit 14 3.1 You spin me right round...: Rabi oscillations . . . . 14

3.2 Hamiltonian of the NV center and its environment . . . . 16

4 Fade To Black: decoherence 21 4.1 Initial state of the qubit and the environment . . . . 21

4.2 Pure-dephasing Hamiltonian . . . . 22

4.3 One . . . . 23

4.4 Non-interacting nuclear bath . . . . 24

5 Lost in the Echo 27 5.1 Non-interacting nuclear bath . . . . 28

6 Many-body effects observable through decoherence 31 6.1 A practical implementation of Cluster-Correlation Expansion . . . . 32

7 Pulse: Dynamical Decoupling 34 7.1 Spin echo revisited . . . . 34

7.2 Dynamical Decoupling . . . . 37

7.3 Dynamical Decoupling: classical case . . . . 41

7.3.1 Comparison of quantum and classical result . . . . 45

7.4 What comes next . . . . 45

8 Echo of a spin interacting with a polarized nuclear environment 47 8.1 How to polarize nuclear spins in the bath? . . . . 47

8.1.1 Thermal polarization . . . . 47

8.1.2 Dynamic Nuclear Polarization: Hartmann-Hahn conditions for cross polariza- tion . . . . 47

8.2 A polarized society . . . . 50

8.2.1 Down with the weakness . . . . 51

8.3 Dynamical Decoupling of a quantum Gaussian bath . . . . 52

8.3.1 Relation to linear response theory . . . . 54

8.3.2 Application to spin echo of non-interacting nuclear bath . . . . 56

8.4 Results for an NV center in diamond in a spin echo sequence . . . . 56

8.5 Experimental implications of this result . . . . 60

xiii

9 Qubit-Environment Entanglement 63

9.1 Experimental protocol for QEE detection . . . . 64

9.2 Outlook . . . . 69

10 Aerials: Spectrum Broadcast Structures 70 10.1 „The Moon only exists when I look at it”, says Albert E. . . . 70

10.2 Spectrum Broadcast Structures and the emergence of objectivity . . . . 73

10.3 Dynamics of a partially reduced density matrix: NV center in diamond . . . . 75

10.3.1 Timescale of SBS formation for weakly coupled nuclei . . . . 77

10.3.2 Nuclear bath polarized by DNP: numerical results . . . . 78

10.4 Reaching Nirvana . . . . 82

11 Decoherence of two qubit states: introduction 85 11.1 Motivation . . . . 85

11.2 Pure dephasing in case of two qubits . . . . 87

11.3 Two-qubit spin echo . . . . 89

11.4 Decoherence in low magnetic fields . . . . 90

11.5 Decoherence in high magnetic fields . . . . 91

12 Decoherence of entangled pair of qubits 92 12.1 Entangled pair of qubits interacting with a common magnetic noise of Gaussian statistics . . . . 92

12.2 How do we observe the common environment? . . . . 94

12.3 Adaptation of Cluster-Correlation Expansion to observation of correlations . . . . 94

12.4 Nuclear pair contribution to correlation function . . . . 96

12.5 Results for entanglement dynamics under two-qubit echo . . . . 98

12.5.1 From decoherence-free subspace of |Ψ

i to realistic distances between the

qubits . . . . 99

12.6 Intermediate interqubit distances . . . 102 12.7 Large interqubit distances . . . 104 12.8 Discussion and Outlook . . . 110

13 Comfortably Numb: second qubit as a witness 111

13.1 Correlated pair dynamics . . . 112 13.2 Decoherence in low magnetic fields . . . 114 13.3 Numerical results . . . 115 13.4 Picture on the timescales relevant to quantum computer gate operation times . . . . 116 13.5 A take home message . . . 119

14 Outro: summary of results 123

Bibliography 129

Chapter 1

Introduction

At the moment when this thesis is written, technology is fighting for the smallest and the most accurate detectors and sensors. Heading there, the quantum effects emerge more and more visibly. But what happens if we start touching quantum mechanics and exploit it for sensing? What properties, that we will use or observe, are really quantum?

A university course of Quantum Mechanics 1.01. typically works with wavefunctions, position and momentum representation, finally the last lecture ends up somewhere between hydrogen atom and description of a quantum harmonic oscillator. But the very beginning is the Schrödinger’s cat...

Now for the reader it might now come in handy to look at the row of their bookshelf and pick a random book about quantum open systems. The scientific content of this thesis will begin the same.

Let us roast the Schrödinger’s cat.

1.1 Who let the cat out?

Even Google’s Artificial Brain has learnt to recognize cats. There is no doubt, why this

poor pet has been chosen for a particularly atrocious gedankenexperiment. Namely, Schrödinger

proposed a box containing an unstable atom, whose decay would cause that a connected hammer

would break a glass bottle containing a deadly poison. Next part is quite simple, as should you find

a cat, there is no doubt that a long-distance attraction between cats and boxes would work. Once

the cat enters the box, it is sealed and further life of the innocent pet depends on the whim of an

atom.

Once this odd experiment starts, we may think of the state of the atom as a superposition of decayed and not decayed. But also the "life"-state of the cat is correlated with state of an atom, so we should be able to write that the whole system state is:

|Ψ

i = 1

√ 2



 

        

+

⊗      

+ +

+

⊗     

+ 

 

 (1.1)

Assuming that cat’s "life"-states are mutually orthogonal and also "poison"-states can be considered as such, this state represents entanglement. Now, only when we decided to measure the cat’s state (I believe there shouldn’t be any problem with choice of the basis), we would determine whether it deceased or not.

Truth is, such quantum states of macroscopic objects are extremely fragile. Any sort of interaction with the external world, like particles coming from the cosmos or even merger of two black holes, producing a gravitational wave, would cause that at some point, the quantum state described as a vector in Hilbert space is gone. Then, in principle the state of a cat at the end of this experiment is equivalent to a situation when its fate can be decided with a dice.

To observe the decay of quantum mechanical description of a system, we need to extend the formalism with density matrices.

1.1.1 A ball or a dice?

For the moment, let us leave the cat in the box and think about the experimenter’s life, in the real world where pure quantum states are sensitive to literally anything. If so, there is also a very high chance that the machinery, which produces pure states, is faulty. How to describe a situation that the machine can prepare one pure state with a certain probability, but also a set of undesired states can be produced?

At first let us consider the two level system, a qubit. In fact, even when we will discuss a

system of N interacting environmental species, these also will be qubits by definition.

3 A pure state of two-level system can be described as follows:

|Ψi = c

|0i + c

|1i , c

, c

∈ C (1.2)

with |c

|

+ |c

|

= 1. One way of satisfying this normalization condition is a trigonometric function of a single angle, i.e.

sin

θ + cos

θ = 1 (1.3)

Since c

and c

are complex, we can write:

c

= cos θ (1.4)

c

= sin θe

(1.5)

Why not having both numbers described as complex? Well, simply because in quantum mechanics the global phase is not important.

Once we constructed the general pure state of a two-level system, let us write what is a density matrix for this entity. For now, it will be quite artificial, but its use will become more clear, once we introduce the idea of faulty machinery or generally open system.

ˆ

ρ

= |Ψi hΨ| =

=



 

h↑| h↓|

|↑i

cos

cos

sin

e

|↓i

cos

sin

e

sin



  =



 

1

2

(1 + cos θ)

(sin θ cos φ − i sin θ sin φ)

1

2

(sin θ cos φ + i sin θ sin φ)

(1 − cos θ)



  =

= 1

2 (I + sin θ cos φ ˆ σ

+ sin θ sin φ ˆ σ

+ cos θ ˆ σ

), (1.6)

where σ ˆ

, with i ∈ {x, y, z}, correspond to Pauli matrices. Finally, we obtained one of the most

useful representations of two level systems. Notice that each factor standing next to a Pauli matrix

of a given index, actually corresponds to transformation from cartesian to spherical coordinates. In

other words, a two-level system state can be represented on a so-called Bloch sphere. Normalization

of a state is satisfied by trace of the density matrix, here, a term multiplying identity matrix (times the length of the matrix), as a consequence of the fact that all Pauli matrices are traceless:

Tr(ˆ ρ) = 1 (1.7)

Because we want to represent the state on a sphere, it is useful to have a formula for length of the Bloch vector. And again, it is somewhat artificial, because we know the result at once, from the construction above. Length of the Bloch vector, will be defined as:

|~r

|

= Tr ˆ ρ

(1.8)

Anyway, when we use the identities for Pauli matrices:

ˆ

σ

σ ˆ

= i 

σ ˆ

, i 6= j (1.9) ˆ

σ

σ ˆ

= I (1.10)

we get the expected |~r

|

= n

+ n

+ n

=1, where n

is the coefficient multiplying the respective Pauli matrix.

We learned that the trace of the density matrix is equal to unity. Once we diagonalize the matrix, it can be represented as:

ˆ

ρ = |ψ

i hψ

| . (1.11)

5 1.1.2 A faulty machinery

Now imagine that, an experimentalist has a setup which generates two mutually orthogonal pure states ( |Ψ

i , |Ψ

i) with some probability. An ensemble of such outputs can be written as:

ˆ

ρ = p |Ψ

i hΨ

| + (1 − p) |Ψ

i hΨ

| (1.12) Now, when we decide to check the length of the Bloch vector, we find that:

ˆ

ρ

= p

|Ψ

i hΨ

|Ψ

ihΨ

| + p(1 − p)(|Ψ

i hΨ

|Ψ

ihΨ

| + |Ψ

i hΨ

|Ψ

ihΨ

) + (1 − p)

|Ψ

i hΨ

| (1.13) Tr ˆ ρ

= p

+ (1 − p

) = 1 − 2p(1 − p) ≤ 1 (1.14)

which illustrates that such faulty machine in fact generates states which enter the inside of a Bloch ball. Those are called mixed states, because they correspond to a mixture of pure states created in a classical lottery.

A general mixture of pure states as such, can be described as a following sum:

ˆ ρ =

X

p

|ψ

i hψ

| (1.15)

where each coefficient p

is non-negative and their sum from definition of density matrix is equal to one. From such a description we can also see that each of these p

can be thought of as probability to draw state |ψ

i from the faulty machinery.

1.1.3 Rotating a qubit

As we know now the description of the two-level system, now we should construct a corre- sponding evolution operator. Let us write the operator as an infinite sum:

U = exp ˆ



−iˆn · ˆσ Ωt 2



= 1 − i(ˆn · ˆσ) Ωt 2 + 1

2

 Ωt 2



(ˆ n · ˆσ)(ˆn · ˆσ)

| {z }

1

− i

3!

 Ωt 2



(ˆ n · ˆσ) + ..., (1.16) And in the end, we obtain the expansion of cosine multiplied by identity and sine, by a vector of Pauli matrices:

U = ˆ 1 cos Ωt

2 − i(ˆn · ˆσ) sin Ωt

2 (1.17)

1.1.4 Two qubits enter

Prior sections showed almost an entire description of single qubit. To start anywhere with multiqubit registers needed for applications of quantum mechanics to real problems, let us start considering two qubits. A general superposition of two-qubit pointer states can be represented as follows:

|Ψ

i = c

|00i + c

|01i + c

|10i + c

|11i (1.18) This state is shared by two entities. Experimentally, we may, for example, have access only to one of the qubits. How do we see if two qubits are correlated?

To answer this question, we need to define separable states:

|Ψ

i = |Ψ

i ⊗ |Ψ

i (1.19)

If we want to extend the definition of separability to mixed states we would write that:

ˆ

ρ

= X

w

ρ ˆ

⊗ ˆ ρ

, (1.20)

where w

≥ 0, P

i

w

= 1 (see e.g. [41, 43, 114]).

In order to access the information about one of the qubits, we should define a partial trace:

ρ

= T r

(ˆ ρ

) = X

i

D ν

   ˆ ρ

  ν

E , (1.21)

where   ν

E

- represent all states of qubit A in a given basis, e.g. {|↑

i , |↓

i}. The result of this operation is called reduced density matrix. In other words, removal of one of the subsystems degree of freedom is happening when calculating a sum of expectation values over all basis states of that subsystem. For two qubit states, we can distinguish four special pure states:

|ψi = 1

√ 2 ( |01i ± |10i) (1.22)

|φi = 1

√ 2 ( |00i ± |11i) (1.23)

7 which are called Bell states. One can observe, already on the level of vectors of states that they are not separable. In fact these are maximally entangled. Let us try to find out what it means. Their corresponding density matrices are as follows:

ˆ ρ

= 1

2



 

 

 

 



1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1



 

 

 

 



(1.24)

ˆ ρ

= 1

2



 

 

 

 



1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1



 

 

 

 



(1.25)

Let us calculate the reduced density matrices for those states:

ˆ

ρ

= ˆ ρ

= 1

2 1 (1.26)

ˆ

ρ

6= ˆ ρ

⊗ ˆ ρ

(1.27)

And the reduced states are maximally mixed, as they correspond to normalized identity matrix.

Quite naturally, when we try to rotate identity to another basis, we will obtain unity again. There- fore, this state is also fully classical, with equal probabilities of obtaining any result in any measure- ment basis. All the quantum information is then exclusively encoded onto two-body correlations.

Entanglement as a term has much sense here, because when extracting information encoded on such entangled state, we need to operate on two qubits simultaneously. Otherwise, we are lost without any intelligible response from the system.

A general criterion to distinguish separable states was introduced by Peres and Horodecki

in 1996 [84, 41]. If we take density matrix and transpose it only on the subspace of one of the

subsystems (partial transposition) and then calculate its eigenvalues, we would find that whenever

it has any negative eigenvalues, the state is entangled. From this, one can define negativity, which

is a sum of negative eigenvalues and can be used as an entanglement measure.

But for bipartite pure state, the criterion is naturally much simpler. When we calculate the reduced density matrices for each subspace and both of them are pure, the state of the whole system is separable.

1.1.5 Qubits: computer or sensor?

In current endeavor to exploit the quantum information there are three leaders: quantum computing and communication, as well as the quantum sensing . The former two have many faces. Theoretically, by creating superpositions, resonantly operating on all qubits and storing some information, like Fourier series coefficients, we could "easily" calculate the Fourier transform of a given function. The problem currently left to solve by experimentalists, lies between quantum computing theory and openness of a realistic register. First of all, finding a space of parameters to operate a qubit realized in a given physical system is already a major challenge. Nevertheless, more and more experimental groups report quantum control over different realizations of qubits in solid state systems (e.g. [18, 29, 55, 85]). Nevertheless, the basic units of a quantum computer, still remain extremely vulnerable to any perturbation and, well, a full isolation from the external world is highly unlikely.

What if this is actually a great thing? The high vulnerability of a spin qubit can be actually

exploited. Recent experimental works show a spin qubit (or an ensemble of spin qubits) attached

to an STM tip [15], inside a chip [11] on which a sample can be placed. But we can go even further

into the microscopic nature of the environment around a qubit. This will be a starting point for

quantum sensing of a quantum neighbourhood. The lion’s share of experiments, tending to observe

such signals, has been realized using a nitrogen-vacancy center in diamond and now I would like

to show what is the structure of this qubit and how can we initialize pure quantum states of this

entity, really flawlessly.

Chapter 2

Nitrogen-vacancy in diamond

2.1 Energy level structure

Nitrogen-vacancy center consists of a nitrogen atom and adjacent vacancy, which substitute carbon in the lattice. In fact the electronic structure of the complex has been found to be inter- connected only with three close neighbour carbon atoms [35, 28, 69, 57, 62, 72, 44, 16, 26, 27, 19]

, each sharing an electron with the complex, as the nitrogen atom is a source of a free pair of

electrons). It is a point defect of C

point-group symmetry. This means it is spatially invariant

against identity, two

π rotations about the axis between nitrogen and vacancy and three vertical

reflection planes. As an effect, atomic orbitals forming a molecule hybridize into a group of four

sp

molecular orbitals, which are occupied by five electrons from the molecule and additional sixth

electron from the lattice. Total electronic spin number of these is S = 1. To be more precise, this

last electron is needed for NV

defect, which is more stable than neutral NV

[112, 4], and therefore

in further investigations, we will focus only on the former. Now, in order to resolve the electronic

structure of the defect, one needs to add contributions to its Hamiltonian, systematically. First, we

need to take a Hamiltonian for Coulomb field of nuclei and electrons in the lattice, interacting with

electrons in the defect. Taking a linear combination of atomic orbitals of the molecule and using a

six-electron model of the defect, one can compute the energy level structure [20] (Fig.2 in Doherty,

New J Phys 13 2011). Now the a

level of a nitrogen atom mixes with a

of carbon to create

a

and a

levels, with the latter falling into the valence band of diamond. In this case, it has been

found that a

level remains filled in both ground and first excited state, whereas the first ground

state is occupied as follows: a

e

and the excited state as: a

e

. Introduction of Coulomb repulsion between electrons in the center, further split both of theses electronic energy levels. Ground state is split into one lower triplet -

A

and two singlet levels-

E and

A

of higher energy and the excited state into one triplet

E and higher singlet

E

. [20, 19](Fig. 3 in Doherty New J. Phys 2012).

Spin-spin and spin-orbit coupling values in this system are on the order of MHz-GHz [5, 102, 103] and when we compare them with splitting between singlet and triplet states (10

− 10

GHz), we can justify their treatment as first order perturbation to the prior solution and their values have been determined by observation of splitting change by application of electric fields and strain at low temperatures. Finally, it has been found that the fine structure splitting between m

= 0 and m

= ±1, a most fundamental property of the ground-state NV center, corresponds to electronic spin-spin interactions [19]:

∆

= 3µ

g

µ

8π he

(r

), e

(r

) | 1 − 3 cos(θ

)

|r

|

( |e

(r

), e

(r

) i − |e

(r

), e

(r

) i) = 2.87GHz, (2.1) where r

and r

are coordinates of electrons occupying e

and e

energy levels and θ

is the angle between those, with respect to the nitrogen nucleus.

2.2 Interaction with magnetic field

Zero-field term due to symmetries in the NV center in diamond, splits only m

= 0 and m

= ±1 in the ground and excited electronic states. However in order to remove the degeneracy between energy levels of non-zero spin number, we need for the considered complex to interact with external magnetic field B:

H ˆ

= µ

~ X

i

(g

s

+ l

) · B + 1 2m

c

h s

× ∇ ˆ V

(r

) i

· (B × r

), (2.2)

where g

is the electron Landé factor, l is the electron orbital magnetic moment p - electron mo-

mentum, ˆ V

is the Coulomb interaction between electrons forming the center, lattice electrons and

nuclei in the center m

- mass of the electron and c - speed of light.

11 In fact, this full form of magnetic Hamiltonian can be represented as: ˆ H

=

~

S · g · B, with g being the effective g-factor tensor: g = g

(ˆ x + ˆ y) + g

z. Measurements of these terms ˆ [35, 66, 22] show a slight deviation from free electron g-factor ( ∼ 10

) as well as a little anisotropy between perpendicular and parallel terms (g

− g

∼ 10

). These shifts most likely correspond to the relativistic term (containing gradient of Coulomb potential) which includes g-factors coming from A

levels.

Figure 2.1: Orbital and fine structure of electronic ground and first excited states in nitrogen- vacancy in diamond

2.3 Optical initialization and readout

Optical transition between ground

A

and excited

E energy levels corresponds to 637 nm

laser wavelength and is in general spin conserving, due to weak spin-orbit interaction [5]. Due to

phonon sideband of the excited state, absorption/fluorescence is also possible for lower wavelengths

(for fluorescence - higher), which resembles Stokes and anti-Stokes shifts for transitions [60]. When

looking at the energy level structure, singlet states originating from ground state of the complex lie between

E and

A

and in fact there are two mechanisms of relaxation from

E excited state - radiative decay into triplets, which conserves spin or transition through forbidden transition to

1

A

via intersystem crossing (ISC), which does not conserve spin. Nevertheless probability of such transition is non-zero only for m

= ±1 states of the excited triplet state [90, 106]. Then the electron relaxates to

E and then another ISC to

A

happens. Probability of a transition from

E to m

= 0 level of the ground state is 1.1-2 times higher than to other spin states of

A

. In other words, intersystem crossings, when optical pumping is applied, effectively depopulate the m

= ±1 levels after a few pump photon absorption cycles. This subtle difference between ISC transitions is a starting point for spin polarization of an NV center. Considering the timescales needed for polarization, light should polarize the NV center into |0i state within few hundreds of nanoseconds [33, 25], however, the experimental practice is to a laser pulse of a duration of ≈ 1 µs [98] in order to increase the polarization degree.

3

E

1

A

3

A

Δ=2.88 GHz Ω =2gμB

637 nm

Figure 2.2: Optical initialization of NV center. Solid arrows correspond to laser excitation and radiative decay. Dashed arrows show a non-radiative path of decay through singlet levels.

Spin states can be recognized by the timetrace of fluorescence or rather reduced intensity of

13

this fluorescence, when NV center is in m

= ±1 states. This is again due to intersystem crossings,

which reduces number of optical photons during a certain time window of collecting fluorescent

photons from NV, when passing through the non-radiative transfers, and generally causing a different

observed lifetime of excited state, i.e. for m

= 0 it is 12 ns and for m

= ±1 - 7.8 ms [77]. Measuring

this fluorescence contrast can therefore be understood as a destructive spin state measurement for

our further theoretical investigation. A measurement sequence described here, in fact, causes loss

of spin polarization of the NV center. In order to achieve a spin-projective measurement, one

needs to perform the experiment in low temperatures (T ∼1-10 K, e.g. [91]), where phonon-induced

spin-mixing, happening in excited state phonon sideband, [73, 102, 103] can be suppressed while

increasing the spin-conserving transition probabilities.

Poking the world with a qubit

3.1 You spin me right round...: Rabi oscillations

The construction of a physical spin qubit hidden inside the electronic structure of ground state of the nitrogen-vacancy center in diamond is complete. Before we enter deeper into the real world, let me leave you in a little suspense for now. Having constructed a self-Hamiltonian for the defect, we can start poking the qubit. Our playground will start with rotating a qubit state.

Using a procedure described above, we initialize the qubit in m

= 0 state, which from now on will correspond to the following vector of state:

|0i ≡  

A

, m

= 0 

(3.1) The remaining eigenstates of a free Hamiltonian for this spin-1 system, will be assigned as:

|±1i ≡  

A

, m

= ±1 

(3.2) Initial state |0i is an eigenstate of the free Hamiltonian and it will not rotate on the Bloch sphere, if we do not apply an oscillating magnetic field in transverse direction:

H ˆ

= ω

2 |1i h1| + ω

2 (cos ωt ˆ σ

+ sin ωt ˆ σ

), (3.3) where σ ˆ

= |0i h1| + |1i h0| and ˆσ

= i( − |0i h1| + |1i h0|).

In the previous chapter we learnt how to initialize a pure state |0i of an NV center. In order

to solve this time-dependent problem, we need to write the Hamiltonian and state in the rotation

frame with respect to frequency of the oscillating magnetic field ω:

15

H = ˆ ˆe U

H ˆ ˆ U

− ω

2 σ ˆ

= ω

2 σ ˆ

+ ω − ω

2 σ ˆ

, (3.4)

where ˆ U

= exp −i

|1i h1|

. Naturally the initial state in the rotating frame is still    e ψ(0) E

= U ˆ

|0i = |0i = |ψ(0)i.

Now a corresponding evolution operator (in rotation frame) is time-independent and can be expressed using Eq. (1.17) and a state (in the same frame) at time t is:

   e ψ(t) E

=

 cos Ωt

2 + i ω − ω

Ω sin Ωt 2



|0i − i ω

Ω sin Ωt

2 |1i (3.6)

with Ω

= (ω

− ω)

+ ω

.

When we hit the resonance with the oscillation frequency of the transverse field, i.e. when ω = ω

, the state after time t, becomes:

|ψ(t)i = cos ω

t

2 |0i − i sin ω

t

2 |1i (3.7)

Therefore, there are two particularly interesting cases, when applying the resonant field for:

(1) t

=

xy

, the initial state is rotated to |ψ(t

) i = |1i, which can also be called a π pulse around x axis

(2) t

=

xy

, we obtain that |ψ(t

) i =

( |0i − i |1i) and this will correspond to a π/2 pulse around x axis.

A representation of evolution for initial state |ψ(t = 0)i = |0i on the Bloch sphere, starting with |0i is shown on Fig. 3.1 for resonant and off-resonant case.

∗ If we started the evolution from a general superposition |ψi = a |0i + b |1i, we would have to additionally take care of the fact that moving to a rotating frame also applies to rotation of this state, namely that:

|ψi = a |0i + b |1i −^U−^ˆ→⁰   ψeE

= a |0i + be^−itω|1i , (3.5)

x

y

|0i

|1i

x

y

|0i

|1i

Figure 3.1: Representation of Rabi nutations on a Bloch sphere for initial state of the qubit |ψ(0)i =

|↑i. On both panels, the green and orange arrow represent state of the qubit at t = 0 and t =

, respectively. Panel (a) corresponds to a resonant case, when ω = ω

and panel (b), when ω = 0.9 ω

.

3.2 Hamiltonian of the NV center and its environment

Control of a single qubit is very easy. In theory. Experimental reality, however, begs to differ.

In the incoming content, we will learn again that the qubit is extremely vulnerable to interaction with the external world, starting from its natural neighbourhood.

Diamond consists mostly of

C nuclei. However, the remaining carbon-13 isotope (of spin- 1/2) is the one to blame for qubit’s death. Diamond naturally contains 1.1% of this isotope. In principle, NV center also couples to phonons in diamond lattice, however the spin-orbit coupling for light elements is very weak and therefore we can state that spin and orbital degrees of freedom are decoupled from each other on the timescales of decoherence, leaving only nuclear bath as the observable environment.

Finally, having such a system and environment setting, we can divide the Hamiltonian into 4

17 fundamental parts:

H = ˆ ˆ H

+ ˆ H

+ ˆ H

+ ˆ H

(3.8)

First two parts correspond to self-Hamiltonian of NV center spin and nuclear spins response to external magnetic field:

H ˆ

= ∆

S ˆ

+ Ω ˆ S

, (3.9)

where ∆

is the zero-field splitting, whose origin was extensively elaborated in prior chapter and Ω is the Zeeman splitting of m

= ±1 levels. ˆ S is the electronic spin operator. Here already we assume that the magnetic field is chosen along the direction pointing from nitrogen to vacancy.

Nuclear self-Hamiltonian is also not particularly complicated:

H ˆ

= X

ω

I ˆ

, (3.10)

with ˆ I

= 1 ⊗ 1... ⊗ 1 ⊗ I ˆ

|{z}

α−th spin

⊗1...1.

Couplings between nuclei in the environment can be described by the dipolar interaction:

H ˆ

= X

α6=β

D

h I ˆ

· ˆ I



δ

− 3 

I ˆ

· n

  n

· ˆ I

i

, (3.11)

here α and β are indices of spins, i and j -directions in space, D

=

corresponds to a coupling tensor, where γ

is the gyromagnetic ratio of αth nucleus and r

is a distance between nuclei.

For magnetic fields equal or greater than B & 1 mT, i.e. those for which spin bath spectroscopy is performed experimentally, only Zeeman energy conserving terms in nuclear space give any relevant contribution to dynamics, and therefore we can apply a secular approximation for the dipolar couplings in the environment:

H ˆ

= X

α6=β

d



I ˆ

I ˆ

+ ˆ I

I ˆ

− 4 ˆ I

I ˆ



, (3.12)

where d

=

1 − 3 cos

θ

and θ

is the angle between displacement vector between a considered pair of nuclei and their quantization axis, which, in this setting, coincides with the quantization axis of an NV center.

Interaction between the NV center spin qubit with nuclei ( hyperfine interaction) can be divided into 2 contributions - Fermi contact and dipolar coupling:

H ˆ

= A

S ˆ

I ˆ

+ A

h S ˆ

· ˆ I



δ

− 3 

S ˆ

· n

 

n

· ˆ I

i

(3.13) The first one, Fermi contact coupling, is proportional to electronic wavefunction density at the location of a nucleus, i.e. A

=

|ψ

(r

) |

. Wavefunction of a deep defect is strongly localized. In case of NV center, the Fermi contact part gives measureable contribution to hyperfine coupling for nuclei that are at most 0.5 nm away from the defect [26].

Second part is the dipolar interaction, which is represented structurally by the same tensor as D

, except that we substitute for one of the gyromagnetic ratios the value for an electron and the displacement vector r

connects the vacancy and nucleus α locations.

Effectively, the Fermi contact interaction contains only contributions along the quantization axes, because the splitting between qubit levels is assumed to be still much larger than any A

, so that its transverse parts (proportional to ∼ ˆ S

, ˆ S

) can be dropped. For the same reasons, only dipolar interaction tensor terms proportional to ˆ S

give any contribution to decoherence. NV center possesses a zero-field splitting of 2.87 GHz as well as Zeeman term due to external magnetic field.

Ratio of Zeeman splittings of electron and nuclear spin levels, corresponds to a ratio of magnetic moments of these entities, which is on the order of 10

. Therefore, hyperfine interaction, even for the most extreme case of nearest neighbour nuclear spin, cannot cause the electron spin flip in a finite external magnetic field (Table 3.1 indicates numerical values for NV center and

C nuclei), leaving only ˆ S

proportional terms:

H ˆ

= A

S ˆ

I ˆ

+ A

h S ˆ

I ˆ



− 3 ˆ S

(ˆ z · n

) 

n

· ˆ I

i

, (3.14)

where only the dipolar interaction contains terms transverse to nuclear quantization axes. It is

now very important to notice that this type of interaction corresponds to a class of so called pure-

19 dephasing Hamiltonians. Simply, this coupling term commutes with the system Hamiltonian, there- fore, there is no energy exchange between the system and the environment. As a result, the qubit can only dephase. In case of high magnetic fields (for NV center that is B > 0.1 T [119]), we can also assume that nuclear Zeeman splitting excludes also transverse couplings for nuclei, so called isotropic approximation:

H ˆ

= A

S ˆ

I ˆ

+ A

(1 − 3 cos

θ

) ˆ S

I ˆ

, (3.15)

where θ

is the angle between vacancy to nucleus displacement vector and direction from nitrogen

to vacancy. This approximation will be justified in numerical simulations of qubit decoherence in

the latter chapters.

Physical constants

Electron gyromagnetic ratio, γ

GHz/T -28.02

13

C nucleus gyromagnetic ratio, γ

/2π MHz/T 10.71 magnetic permeability of vacuum, µ

/4π N/A 10

T

nm

µeV

1.6 ·10

Splittings

Zero-field splitting, ∆

µeV 11.95

GHz 2.89

Electron Zeeman splitting, Ω µeV 116 · B [T]

GHz 28.02 · B [T]

Nuclear Zeeman splitting, ω µeV 0.045 · B [T]

MHz 10.71 · B [T]

Couplings

Hyperfine dipolar coupling, A µeV 8.19 ·10

(r [nm])

kHz 1.956 (r[nm])

Nuclear dipolar coupling, b µeV 3.1 ·10

(r [nm])

Hz 7.45 (r [nm])

Couplings for nearest-neighbours (NN)

NN distance nm 0.154

Hyperfine dipolar coupling, A

µeV 1.636 ·10

MHz 3.927

µT

141

τ

=

N N

ns 251.6

Nuclear dipolar coupling, b

µeV 6.204 ·10

kHz 1.489

τ

=

N N

µs 672

Table 3.1: Table of physical constants and couplings, as well as the corresponding timescales, listed

specifically for NV center in diamond. Dipolar couplings do not contain the term depending on

angle between an entity and NV quantization axis, but the couplings for nearest-neighbours have

been calculated for maximum value of the term containing angular dependence. (

- magnetic field

created by the nearest neighbour nucleus, as seen by the NV center )

Chapter 4

Fade To Black: decoherence

4.1 Initial state of the qubit and the environment

The initial state of the qubit is a fully controlled parameter. Experiments of quantum sensing, typically start by initialization of an eigenstate of self-Hamiltonian for the qubit and subsequent rotation by π/2 angle around the x axis on the Bloch sphere, to create a superposition of those eigenstates:

|ψ(0)i = 1

√ 2 ( |m

i + |m

i) (4.1)

Choice of this initial state will be more strongly justified in the following sections.

Environment consisting of nuclear spins, in thermal equllibrium, can be expressed as:

ˆ

ρ

(0) = 1 Z exp 

−β ˆ H



= O

i=1

1 2



1 + tanh βω

2 σ



, (4.2)

where β =

BT

, simply corresponds to inverse of the temperature T (k

- Boltzmann constant) and ω

corresponds to Zeeman splitting of i-th nucleus, with σ

being Pauli-Z for this nucleus.

When we look at the energy scales corresponding to the Zeeman splittings of

C nuclei, con- sidering conditions typical for experiments in diamond, which are performed in room temperature, the term proportional to σ ˆ

is, in fact, negligible. As an effect, in the following sections we shall consider an initial state of nuclear bath as completely mixed, i.e. ρ ˆ

=

1

, where 1

is a 2

by 2

identity matrix. Otherwise, we will start from the initial state of an environment, which is separable, i.e.:

ˆ ρ

(0) =

O

ˆ

ρ

(0), (4.3)

where ρ ˆ

(0) is the initial density matrix of i-th nucleus.

4.2 Pure-dephasing Hamiltonian

Having constructed a Hamiltonian for the NV center spin qubit coupled to a bath of interacting nuclei, we can start poking the qubit to see how the environment is affecting its state. As it was mentioned above, we will consider a pure-dephasing Hamitonian. But what are the consequences of such choice?

Let us make a tiny step back and write a general form of such class of Hamiltonians for qubit interacting with a system:

H ˆ

= h

s ˆ

+ ˆ s

V + ˆ ˆ H

, (4.4)

where s ˆ

= m

|m

i hm

| + m

|m

i hm

| is an analogue of Pauli-z operator in a chosen two-level subspace of NV center spin states. The first term defines qubit splitting proportional to ˆ s

, but the second term also is proportional to this operator in qubit space. Last term contains splittings and couplings inside the environment. Now, consider an initial state of the qubit and the environment as:

ˆ

ρ

(0) = |ψi hψ| ⊗ ˆ ρ

(4.5)

where qubit (defined between m

and m

spin states) is initialized in a pure state: |ψi = a |m

i + b |m

i and for now we will not consider anything specific about the state of the environment.

The dephasing Hamiltonian from Eq. (4.4) can be rewritten as a set of qubit state dependent Hamiltonians for the environment:

H ˆ

= |m

i hm

| ⊗ ˆ H

+ |m

i hm

| ⊗ ˆ H

(4.6)

where ˆ H

and ˆ H

act only on the space of the environment. A corresponding evolution operator also can be divided into qubit conditional operators:

U ˆ

(T ) = |m

i hm

| ⊗ ˆ U

(T ) + |m

i hm

| ⊗ ˆ U

(T ), (4.7)

where ˆ U

(T ) = exp 

−i ˆ H

T 

and ˆ U

(T ) = exp 

−i ˆ H

T 

, with evolution time T .