# Speech Enhancement for Wake-Up-Word detection in Voice Assistants

###### Abstract

Keyword spotting and in particular Wake-Up-Word (WUW) detection is a very
important task for voice assistants. A very common issue of voice assistants
is that they get easily activated by background noise like music, TV or
background speech that accidentally triggers the device. In this paper, we
propose a Speech Enhancement (SE) model adapted to the task of WUW detection
that aims at increasing the recognition rate and reducing the false alarms in
the presence of these types of noises. The SE model is a fully-convolutional
denoising auto-encoder at waveform level and is trained using a log-Mel
Spectrogram and waveform reconstruction losses together with the BCE loss of a
simple WUW classification network. A new database has been purposely prepared
for the task of recognizing the WUW in challenging conditions containing
negative samples that are very phonetically similar to the keyword. The
database is extended with public databases and an exhaustive data augmentation
to simulate different noises and environments. The results obtained by
concatenating the SE with a simple and state-of-the-art WUW detectors show
that the SE does not have a negative impact on the recognition rate in quiet
environments while increasing the performance in the presence of noise,
especially when the SE and WUW detector are trained jointly end-to-end.

Index Terms: keyword spotting, speech enhancement, wake-up-word, deep
learning, convolutional neural network

##  1 Introduction

Voice interaction with devices is becoming ubiquitous. Most of them use a
mechanism to avoid the excessive usage of resources, a trigger word detector.
This ensures the efficient use of resources, using a Speech-To-Text tool only
when needed and with the consequent start of a conversation. It is key to only
start this conversation when the user is addressing the device, otherwise the
user experience is notoriously degraded. Thus, the wake-up-word detection
system must be robust enough to avoid wake-ups with TV, music, speech and
sounds that do not contain the key phrase.

A common approach to reduce the impact of this type of noise in the system is
the adoption of speech enhancement algorithms. Speech enhancement consists of
the task of improving the perceptual intelligibility and quality of speech by
removing background noise [1]. Its main application is in the field of mobile
and internet communications [2] and related to hearing aids [3], but SE has
also been applied successfully to automatic speech recognition systems [4, 5,
6].

Traditional SE methods involved a characterization step of the noise spectrum
which is then used to try reduce the noise from the regenerated speech signal.
Examples of these approaches are spectral subtraction [3], Wiener filtering
[7] and subspace algorithms [8]. One of the main drawbacks of the classical
approaches is that they are not very robust against non-stationary noises or
other type of noises that can mask speech, like background speech. In the last
years, Deep Learning approaches have been widely applied to SE at the waveform
level [9, 10] and spectral level [6, 11]. In the first case, a common
architecture falls within the encoder-decoder paradigm. In [12], authors
proposed a fully convolutional generative adversarial network architecture
structured as an auto-encoder with U-Net like skip-connections. Other recent
work [13] proposes a similar architecture at the waveform level that includes
a LSTM between the encoder and the decoder and it is trained directly with a
regression loss combined with a spectrogram domain loss.

Inspired by these recent models, we propose a similar SE auto-encoder
architecture in the time domain that is optimized not only by minimizing
waveform and Mel-spectrogram regression losses, but also includes a task-
dependent classification loss provided by a simple WUW classifier acting as a
Quality-Net [14]. This last term serves as a task-dependent objective quality
measure that trains the model to enhance important speech features that might
be degraded otherwise.

##  2 Speech Enhancement

Speech enhancement is interesting for triggering phrase detection since it
tries to remove noise that could trigger the device, and at the same time
improves speech quality and intelligibility for a better detection. In this
case, we try to tackle the most common noisy environments where voice
assistants are used: TV, music, background conversations, office noise and
living room noise. Some of these types of background noise, such as TV and
background conversations, are the most likely to trigger the voice assistant
and are also the most challenging to remove.

Figure 1: End-to-end SE model at waveform level concatenated with a
classifier. Log-Mel Spectrogram and waveform reconstruction losses of the SE
model can be used together with the BCE loss of the classifier as a linear
combination to train the model.

###  2.1 Model

Our model has a fully-convolutional denoising auto-encoder architecture with
skip connections (Fig. 1), which has proven to be very effective in SE tasks
[12], working end-to-end at waveform level. In training, we input a noisy
audio $\bm{x}\in\mathbb{R}^{T}$, comprised of clean speech signal
$\bm{y}\in\mathbb{R}^{T}$ and background noise $\bm{n}\in\mathbb{R}^{T}$ so
that $\bm{x}=\lambda\bm{y}+(1-\lambda)\bm{n}$.

The encoder compresses the input signal and expands the number of channels. It
is composed of six convolutional blocks (ConvBlock1D), each consisting of a
convolutional layer, followed by an instance normalization and a rectified
linear unit (ReLU). Kernel size $K=4$ and stride $S=2$ are used, except in the
first layer where $K=7$ and $S=1$. The compressed signal goes through an
intermediate stage where the shape is preserved, consisting of three residual
blocks (ResBlock1D), each formed by two ConvBlock1D with $K=3$ and $S=1$ where
a skip connection is added from the input of the residual block to its output.
The last stage of the SE model is the decoder, where the original shape of the
raw audio is recovered at the output. Its architecture follows the inverse
structure of the encoder, where deconvolutional blocks (DeconvBlock1D) replace
the convolutional layers of the ConvBlock1D with transposed convolutional
layers. Skip connections from the encoder blocks to the decoder blocks are
also used to ensure low-level detail when reconstructing the waveform.

We use a regression loss function (L1 loss) at raw waveform level together
with another L1 loss over the log-Mel Spectrogram as proposed in [15] to
reconstruct a ”cleaned” signal $\hat{\bm{y}}$ at the output. Finally, we
include the classification loss (BCE Loss) when training the SE model jointly
with the classifier or concatenating a pretrained classifier at its output.
Thus, we also try to optimize the SE model to the specific task of WUW
classification. Our final loss function is defined as a linear combination of
the three losses:

$L_{T}=\alpha L_{raw}(\bm{y},\hat{\bm{y}})+\beta
L_{spec}(S(\bm{y}),S(\hat{\bm{y}}))+\gamma L_{BCE}$ (1)

where $\alpha$, $\beta$ and $\gamma$ are hyperparamters weighting each loss
term and $S(\cdot)$ denotes the log-Mel Spectrogram of the signal, which is
computed using 512 FFT bins, a window of 20 ms with 10 ms of shift and 40
filters in the Mel Scale.

##  3 Methodology

###  3.1 Databases

The database used for conducting the experiments here presented consists of
WUW samples labeled as positive, and other non-WUW samples labeled as
negative. Since the chosen keyword is ”OK Aura”, which triggers Telefónica’s
home assistant, Aura, positive samples are drawn from company’s in-house
databases. Some of the negative samples have been also recorded in such
databases, but we also add speech and other types of acoustic events from
external data sets, so the models gain robustness with further data.
Information about all data used is detailed in this section.

####  3.1.1 OK Aura Database

In a first round, around 4300 WUW samples from 360 speakers have been
recorded, resulting in 2.8 hours of audio. Furthermore, office ambient noise
has been recorded as well, with the aim of having samples for noise data
augmentation. The second data collection round has been done in order to study
and improve some sensitive cases where WUW modules typically underperform. For
instance, such dataset contains rich metadata about positive and negative
utterances, like room distance, speech accent, emotion, age or gender.
Furthermore, the negative utterances contain phonetically similar words to ”OK
Aura”, since these are the most ambiguous to recognize for a classifier.
Detailed information about data acquisition is explained in the following
subsection.

####  3.1.2 Data acquisition

A web-based Jotform form has been designed for data collection. Such form is
open and actually still receiving input data from volunteers, so readers are
also invited to contribute to the
dataset111https://form.jotform.com/201694606537056. Until the date of this
work, 1096 samples from 80 speakers have been recorded, which consists of 1.2
hours of audio. Volunteers are asked to pronounce various scripted utterances
at a close distance and also at two meters from the device mic. The similarity
levels are the following:

  1. 1.

Exact WUW, in an isolated manner: OK Aura.

  2. 2.

Exact WUW, in a context: Perfecto, voy a mirar qué dan hoy. OK Aura.

  3. 3.

Contains ”Aura”: Hay un aura de paz y tranquilidad.

  4. 4.

Contains ”OK”: OK, a ver qué ponen en la tele.

  5. 5.

Contains similar word units to ”Aura”: Hola Laura.

  6. 6.

Contains similar word units to ”OK”: Prefiero el hockey al baloncesto.

  7. 7.

Contains similar word units to ”OK Aura”: Porque Laura, ¿qué te pareció la
película?

####  3.1.3 External data

General negative examples have been randomly chosen from the publicly
available Spanish Common Voice corpus [16] that currently holds over 300 hours
of validated audio. However, we keep a 10:1 ratio between negative and
positive samples, since such ratio proves to yield good results in [17], thus
avoiding bigger ratios that lead to increasing computational times. Therefore,
we have used a Common Voice partition consisting of 55h for training, 7h for
development and 7h for testing.

Background noises were selected from various public datasets according to
different use case scenarios. Living room background noise (HOME-LIVINGB) from
the QUT-NOISE Database [18], TV audios from the IberSpeech-RTVE Challenge
[19], and music222https://freemusicarchive.org/ and
conversations333http://www.podcastsinspanish.org/ from free libraries.

####  3.1.4 Data processing

All the audio samples are monoaural signals stored in Waveform Audio File
Format (WAV) with a sampling rate of 16kHz. The speech data that has been
collected was processed with a Speech Activity Detection (SAD) module
producing timestamps where speech occurs. For this purpose the tool from
pyannote.audio [20] has been used, which has been trained with the AMI corpus
[21]. This helped us to only use the valid speech segments of the audios we
collected.

As features to train the models we mainly used two: Mel-Frequency Cesptral
Coefficients (MFCCs) and log-Mel Spectrogram. The MFCCs were constructed first
filtering the audio with a band pass filter (20Hz to 8kHz) and then,
extracting the first thirteen coefficients with 100 milliseconds of windows
size and frame shifting of 50 milliseconds. The procedure to extract the log-
Mel Spectrogram ($S(\cdot)$) is detailed in section 2.1.

Train, development and test partitions are split ensuring that neither speaker
nor background noise is repeated between partitions, trying to maintain a
80-10-10 proportion, respectively. Total data, containing internal and
external datasets, consists of 50.737 non-WUW samples and 4.651 WUW samples.

###  3.2 Data augmentation

Several Room Impulse Responses (RIR) were created based on the Image Source
Method (ISM) [22], for a room of dimensions $(L_{x},L_{y},L_{z})$ where $2\leq
L_{x}\leq 4.5,2\leq L_{y}\leq 5.5,2.5\leq L_{z}\leq 4$ meters, with microphone
and source randomly located at any $(x,y)$ point within a height of $0.5\leq
z\leq 2$ meters. Every TV and music original recordings were convolved with
different RIRs to simulate the signal picked up by the microphone of the
device in the room.

Adding background noise to clean speech signals is the main data augmentation
technique used in the training stage. We use background noises of different
scenarios (TV, music, background conversations, office noise and living room
noise) and a wide range of SNRs to improve the performance of the models
against noisy environments. In each epoch, we create different noisy samples
by randomly selecting a sample of background noise for each speech event and
combining them with a randomly chosen SNR in a specified range.

###  3.3 Wake-Up Word Detection Models

With the aim of assessing the quality of the trained SE models, we use several
trigger word detection classifier models, reporting the impact of the SE
module at WUW classification performance. The WUW classifiers used here are a
LeNet, a well-known standard classifier, easy to optimize [23]; `Res15`,
`Res15-narrow` and `Res8` based on a reimplementation by Tang and Lin [24] of
Sainath and Parada’s Convolutional Neural Networks (CNNs) for keyword spotting
[25], using residual learning techniques with dilated convolutions [26]; a
`SGRU` and `SGRU2`, two Recurrent Neural Network (RNNs) models, based on the
open source tool named Mycroft Precise[27], which is a lightweight wake-up-
word detection tool implemented in TensorFlow. These are two bigger variations
that we have implemented in PyTorch. We also use a `CNN-FAT2019`, a CNN
architecture adapted from a kernel [28] in Kaggle’s FAT 2019 competition [29],
which has shown good performance in tasks like audio tagging or detection of
gender, identity and speech events from pulse signal [30].

Figure 2: Macro F1-score box plot for different SNR ranges. Classifiers
trained with low noise ($[5,30]$ dB SNR). Figure 3: Macro F1-score box plot
for different SNR ranges. Classifiers trained with a very wide range of noise
($[-10,50]$ dB SNR).

###  3.4 Training

Speech signals and background noises are combined randomly following the
procedure explained in 3.2 with a given SNR range. The SE model is trained to
cover a wide SNR range of $[-10,50]$ dBs, whereas WUW models are trained to
cover two scenarios: a classifier trained with the same SNR range as the SE
model, and a classifier less aware of noise with a narrower SNR range of
$[5,30]$ dBs. This way, it is possible to study the impact of the SE model
regarding if the classifier has been trained with more or less noise.

Data imbalance is addressed balancing the classes in each batch using a
weighted sampler. We use a fixed window length of 1.5 seconds based on the
annotated timestamps for our collected database, and random cuts for the rest
of the Common Voice samples.

All the models are trained with early stopping based on the validation loss
with 10 epochs of patience. We use the Adam optimizer with a learning rate of
0.001 and a batch size of 50. Loss (1) allows to train the models in multiple
ways and we define different SE models and classifiers based on the loss
function used:

  1. a)

Classifier: we remove the auto-encoder from the architecture (Fig. 1) and
train any of the classifiers using the noisy audio as input: $\alpha=\beta=0$
and $\gamma=1$

  2. b)

SE model (SimpleSE): we remove the classifier from the architecture and
optimize the auto-encoder based on the reconstruction losses only:
$\alpha=\beta=1$ and $\gamma=0$

  3. c)

SE model + frozen classifier (FrozenSE): operations of the classifier are
dropped from the backward graph for gradient calculation, optimizing only the
SE model for a given pretrained classifier (LeNet). $\alpha=\beta=\gamma=1$

  4. d)

SE model + classifier (JointSE): auto-encoder and LeNet are trained jointly
using the three losses: $\alpha=\beta=\gamma=1$

###  3.5 Tests

All the models take as input windows of 1.5 seconds of audio, to ensure that
common WUW utterances are fully within it, since the average ”OK Aura” is
about 0.8 seconds long. Therefore, we perform an atomic test evaluating if a
single window contains the WUW or not. Both negative and positive samples are
assigned a background noise sample with which they are combined with a random
SNR between certain ranges, as described in section 3.4.

Given the output scores of the models, the threshold to decide if a sample
test is a WUW or not is chosen as the one yielding the biggest difference
between true and false positive rates, based on Youden’s J statistic [31].
Once the threshold is decided, macro F1-score is computed in order to balance
WUW/non-WUW proportions in the results. We average such scores across all WUW
classifiers described in section 3.3, for every SNR range.

##  4 Results

Figure 2 illustrates the improvement of the WUW detection in noisy scenarios
by concatenating our FrozenSE model with all WUW classifiers described in
section 3.3 trained with low noise ($[5,30]$ dB SNR), which we could find in
simple voice assistant systems. Applying SE in quiet scenarios maintains
fairly good results, and improves them in lower SNR ranges.

If we train the classifiers with more data augmentation ($[-10,50]$ dB SNR),
the baseline classifier results for noisier scenarios improve. Results using
the FrozenSE do not decrease but the improvement in ranges of severe noise is
not as large as in Figure 2, see Figure 3.

In section 3.4 we have defined the parameters of the loss function (1) to
train a classifier (case a)), and different approaches to train the SE model,
either standalone (b), c)) or in conjunction with the classifier (d)). In
Figure 4 we can see how JointSE performs better than all the other cases in
almost every SNR range. From 40 dB to 10 dB of SNR, the results are very
similar for the 4 models. In contrast, in the noisiest ranges we can see how
the classifier without SE model is the worst performer, followed by the
SimpleSE case where only the waveform and spectral reconstruction losses are
used. We found that the FrozenSE case, which includes the classification loss
in the training stage, improves the results for the wake-up-word detection
task. However, the best results are obtained with the JointSE case where the
SE model + LeNet are trained jointly using all three losses.

Figure 4: Comparison of different training methods for the SE models and LeNet
classifier, in terms of the macro F1-Score for different SNR ranges. All
models trained in the range of $[-10,50]dB$ SNR.

We compared the WUW detection results of our JointSE with other SOTA SE models
(SEGAN [12] and Denoiser [13]), followed by a classifier (data augmented
LeNet) in different noise scenarios. In Table 1, it can be observed how when
training the models together with the task loss, the results in our setup are
better than with other more powerful but more general SE models, since there
is no mismatch between the SE and classifier in the end-to-end and it is also
more adapted to common home noises. JointSE improves the detection over the no
SE model case, especially in scenarios with background conversations, loud
office noise or loud TV, see Table 2.

Table 1: Macro F1-score enhancing the noisy audios with SOTA SE models and
using a LeNet as a classifier.

SNR $[dB]$ |  | No SE | SEGAN | Denoiser | JointSE
---|---|---|---|---|---
$[20,10]$ | Clean | $0.980$ | $0.964$ | $0.980$ | $0.990$
$[10,0]$ | Noisy | $0.969$ | $0.940$ | $0.955$ | $0.972$
$[0,-10]$ | Very noisy | $0.869$ | $0.798$ | $0.851$ | $0.902$

Table 2: Macro F1-score percentage difference between JointSE and LeNet
without any SE module, for different background noise types. Positive values
mean that the JointSE score is bigger than the single LeNet’s.

SNR $[dB]$ |  | Music | TV | Office | Living Room | Conversations
---|---|---|---|---|---|---
$[20,10]$ | Clean | $1.0$ | $-0.9$ | $1.4$ | $0.4$ | $2.3$
$[10,0]$ | Noisy | $0.0$ | $-1.2$ | $0.8$ | $0.4$ | $1.9$
$[0,-10]$ | Very noisy | $0.5$ | $3.9$ | $11.2$ | $3.1$ | $3.8$

##  5 Conclusions

In this paper we proposed a SE model adapted to the task of WUW in voice
assistants for the home environment. The SE model is a fully-convolutional
denoising auto-encoder at waveform level and it is trained using a log-Mel
Spectrogram and waveform regression losses together with a task-dependent WUW
classification loss. Results show that for clean and slightly noisy
conditions, SE in general does not bring a substantial improvement over a
classifier trained with proper data augmentation, but in the case of very
noisy conditions SE does improve the performance, especially when the SE and
WUW detector are trained jointly end-to-end.

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# Physical realization of coupled Hilbert-space mirrors for quantum-state
engineering

E. S. Kyoseva  Department of Physics, Sofia University, James Bourchier 5
blvd., 1164 Sofia, Bulgaria  N. V. Vitanov  Department of Physics, Sofia
University, James Bourchier 5 blvd., 1164 Sofia, Bulgaria  Institute of Solid
State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussée 72, 1784
Sofia, Bulgaria  B. W. Shore  Fachbereich Physik, Technische Universität
Kaiserslautern, 67653 Kaiserslautern, Germany  Permanent address: 618
Escondido Circle, Livermore, CA 94550, USA

###### Abstract

Manipulation of superpositions of discrete quantum states has a mathematical
counterpart in the motion of a unit-length statevector in an $N$-dimensional
Hilbert space. Any such statevector motion can be regarded as a succession of
two-dimensional rotations. But the desired statevector change can also be
treated as a succession of reflections, the generalization of Householder
transformations. In multidimensional Hilbert space such reflection sequences
offer more efficient procedures for statevector manipulation than do sequences
of rotations. We here show how such reflections can be designed for a system
with two degenerate levels – a generalization of the traditional two-state
atom – that allows the construction of propagators for angular momentum
states. We use the Morris-Shore transformation to express the propagator in
terms of Morris-Shore basis states and Cayley-Klein parameters, which allows
us to connect properties of laser pulses to Hilbert-space motion. Under
suitable conditions on the couplings and the common detuning, the propagators
within each set of degenerate states represent products of generalized
Householder reflections, with orthogonal vectors. We propose physical
realizations of this novel geometrical object with resonant, near-resonant and
far-off-resonant laser pulses. We give several examples of implementations in
real atoms or molecules.

Householder reflection, quantum state engineering, coherent excitation,
degenerate levels

###### pacs:

03.67.Lx; 32.80.Bx; 33.80.Be

##  I Introduction

Manipulation of discrete quantum states has long held interest, most recently
for application to quantum information processing QI . In the simplest
realizations one deals with a nondegenerate two-state system (a “two-level
atom”AE ; Shore ) and through pulsed resonant coherent excitation produces a
specified superposition of the two states, starting from a single state.
Following the availability of laser light sources, much attention centered on
producing complete excitation, i.e. complete population transfer from the
ground state to an excited state Shore . More recently interest has shifted to
the production of specific superpositions of the two states, and to producing
transitions between superposition states. In the context of quantum
information a two-state system serves as a qubit; more general tasks involve
multiple quantum states (a qunit), possibly driven by pulsed nonresonant
light.

The goal of quantum-state manipulation is to realign the statevector $\Psi(t)$
from some given Hilbert-space direction $\Psi(t_{i})$ at an initial time
$t_{i}$ to some prescribed state $\Psi(t_{f})$ at a final time $t_{f}$, each
of these statevectors being defined by the set of complex-valued probability
amplitudes $C_{n}(t)$ associated with a set of quantum states
$|\psi_{n}\rangle$. Expressed in matrix form, the goal is to obtain a
propagator matrix $\mathsf{U}(t_{f},t_{i})$ that transforms the vector
$\mathbf{C}(t)$ of components $C_{n}(t)$,

$\mathbf{C}(t_{f})=\mathsf{U}(t_{f},t_{i})\mathbf{C}(t_{i}).$ (1)

As noted in the following section, there exists an extensive literature
describing analytic solutions to the two-state system for a variety of pulsed-
excitation scenarios. There exist also numerous analytic solutions to
multistate systems.

Here we consider, as a particular generalization of the two-state system, a
two-level system involving two sets of degenerate sublevels: a less energetic
set of $N$ states $|\psi_{n}\rangle\ (n=1,2,\ldots,N)$ with common energy
$E_{N}$ (the _lower_ set), and a more energetic set of $M$ states
$|\varphi_{m}\rangle\ (m=1,2,\ldots,M)$ that share energy $E_{M}$ (the _upper_
set). Thus we consider a Hilbert space of dimension $N+M$. For definiteness we
here assume that $N\geqq M$. We assume that interactions produced by laser
fields can induce direct transitions between lower sublevels and upper
sublevels, but not directly within either manifold of states (i.e. we allow
electric-dipole transitions only).

Because the statevector has unit length at all times, any such change can be
regarded as a rotation, and any allowable motion in several dimensions can be
decomposed into a succession of two-dimensional rotations, analogous to the
three Euler angles that define an arbitrary rotation in Euclidean space.
However, length-preserving changes in multi-dimensional space can also be
produced by a succession of reflections. When expressed in matrix form these
are generalizations of the Householder reflections used in matrix calculus
Householder . As has been shown previously, such unitary operations can be
implemented very efficiently using coherent excitation techniques Kyoseva ;
Ivanov ; as subsequently noted Ivanov2 , the use of quantum Householder
reflections (QHR) permits efficient quantum-state engineering of transitions
between arbitrary superpositions.

The QHR implementation proposed earlier Kyoseva ; Ivanov  requires a
particular multistate linkage pattern, in which a set of $N$ low-lying
degenerate states all link, via radiative interaction, with a _single_ upper
state – a generalization of the tripod linkage termed an _N-pod_. Such a
linkage pattern occurs with a lower level having angular momentum $J=1$ (three
sublevels) excited to an upper level having $J=0$ (a single sublevel) but the
pattern is difficult to realize for more than 3 lower states. It is therefore
desirable to extend the QHR technique to more general linkage patterns.

The present manuscript describes a procedure that allows implementation of the
QHR when there are multiple states in the upper set. Such an extension makes
possible the application of QHR to arbitrary angular momentum states, as occur
with free atoms and molecules. As we show, the propagator for such situations
is not a single QHR, but a product of QHRs, moreover with orthogonal vectors.
The expression for the resulting propagator has a clear geometric
interpretation as the effect of a succession of reflections, i.e. _coupled
mirrors_.

The key to this extension of the QHR is a transformation of the underlying
basis states, the so-called Morris-Shore (MS) transformation MS ; Vitanov MW ;
other MS . This replaces the original system, with its multiple linkages
between states, by a set of independent nondegenerate two-state systems,
thereby allowing us to utilize the considerable literature of analytic
solutions to two-state systems and produce analytic solutions for degenerate
multistate systems Kyoseva .

In this paper we present the solution of the degenerate two-level problem in a
simple closed form involving sums of projectors of MS states. These
expressions are useful for deriving analytical solutions, generalizations of
known two-state solutions, for systems having degenerate levels. For certain
conditions, wherein the transition probabilities between states from different
sets vanish, the propagator for states of the lower set is given by a product
of QHRs with orthogonal vectors, each vector being a bright state; a similar
property applies to the upper set. We discuss properties of this novel
geometric object, the coupled mirrors. To illustrate the procedure we develop
a useful explicit analytic formalism for two upper states and present some
examples.

##  II The degenerate two-level system

###  II.1 Two non-degenerate states

We consider the controlled alteration of a multistate system, expressed as a
redirection of the statevector, induced by a set of laser pulses. The simplest
example of coherent excitation occurs when there are just two nondegenerate
states, indexed 1 and 2. The time-dependent Schrödinger equation prescribes
the changes in Hilbert space as (here and henceforth we set $\hbar=1$, thereby
making no distinction between energy and frequency units)

$i\frac{d}{dt}\mathbf{C}(t)=\mathsf{H}(t)\mathbf{C}(t),$ (2)

where the column vector $\mathbf{C}(t)$ has the complex-valued probability
amplitudes $C_{n}(t)$ as elements, $\mathbf{C}(t)=[C_{1}(t),C_{2}(t)]^{T}$.
Commonly one neglects in the Hamiltonian $\mathsf{H}(t)$ elements that vary
rapidly compared with the characteristic response times of the system, the so-
called rotating-wave approximation (RWA) Shore . Then the Hamiltonian matrix,
with suitable choice of energy zero-point, reads

$\mathsf{H}(t)=\left[\begin{array}[]{cc}0&\frac{{}_{1}}{{}^{2}}\Omega(t)\\\
\frac{{}_{1}}{{}^{2}}\Omega(t)^{\ast}&\Delta(t)\end{array}\right].$ (3)

Here the slowly-varying function $\Omega(t)$, the Rabi frequency, quantifies
the coupling between the two states. For a laser-driven electric-dipole
transition in an atom or a molecule, $\Omega(t)$ is proportional to the
transition dipole moment and the electric-field envelope. The detuning
$\Delta(t)$ measures the frequency offset of the carrier laser frequency
$\omega$ from the Bohr transition frequency $\omega_{0}$,
$\Delta=\omega_{0}-\omega$. A time dependence in the detuning can be
introduced by both $\omega$ (due to chirping femto ) and $\omega_{0}$ (due to
Stark and Zeeman shifts).

When the transition is exactly resonant the associated detuning vanishes,
$\Delta=0$. Then the response depends only upon the initial conditions and
upon the time-integrated Rabi frequency (the temporal pulse area),

$A(t)=\int_{0}^{t}dt^{\prime}\,|\Omega(t^{\prime})|.$ (4)

At all times the probability amplitudes are expressible as trigonometric
functions of $A(t)$; when $A(t)$ is some even-integer multiple of $\pi$ (e.g.,
a $2\pi$__ pulse), the probabilities repeat their initial values. Thus the
response, for resonant excitation, does not depend on any details of the pulse
shape. Of particular use are the $\pi$ pulses, which produce complete
population inversion between the two states, and half-$\pi$ pulses that create
an equal coherent superposition of the two states. Resonant pulses having
precise temporal area have had wide application, most notably in nuclear
magnetic resonance NMR  and coherent atomic excitation Shore . They are now a
common tool in quantum information processing QI .

A variety of nonresonant ($\Delta\neq 0$) pulses also lead to exact analytic
expressions for the probability amplitudes $C_{1}(t)$ and $C_{2}(t)$. Amongst
the soluble two-state models are the models of Rabi Rabi , Landau-Zener LZ ,
Rosen-Zener RZ , Allen-Eberly AE ; Hioe , Bambini-Berman BB , Demkov-Kunike DK
, Demkov Demkov , Nikitin Nikitin , and Carroll-Hioe CH . Methods for
approximate solutions are also available, such as the perturbation theory, the
adiabatic approximation STIRAP , the Magnus approximation Mag54 , and the
Dykhne-Davis-Pechukas approximation DDP . The latter, in particular, is a very
useful tool for deriving very accurate approximations in various cases of
interest, e.g., transform-limited Vasilev-Gaussian  and chirped Gaussian
pulses Vasilev-Gaussian-chirped . The two-state dynamics acquires interesting
new features when the pulsed field possesses some symmetries symmetry , or
when it is a sequence of identical pulses (pulse train) train .

The two-state atom provides a very basic model of coherent excitation, widely
used because of the relative simplicity with which it can be treated
mathematically and fabricated experimentally. However, with this mastery has
come interest in more general systems, involving more than two discrete
quantum states. It is with a class of these that the present paper deals.
Specifically, we describe a straightforward technique that allows efficient
redirection of the statevector for such a system by means of laser pulses, and
the use of the many exact and approximate solutions available for two-state
systems.

###  II.2 Multiple states as two degenerate levels

An important extension is the linear chain of $N$ states, each state being
linked only to its nearest neighbor. The simplest of this is the three-state
chain, involving two independent pulses, the mathematics of which has been
extensively reported three state , as have lengthier chains chains . Here we
consider an extension in which each of the states of the two-state system is
replaced by a degenerate set of states that together form an energy _level_.
Such situations occur commonly when one deals with angular-momentum
eigenstates, as happens with the electronic structure of atoms in free space.
For a given angular momentum $J$ there are $2J+1$ magnetic sublevels that, in
the absence of an external electric or magnetic field, all have the same
energy.

We shall consider the possibility that there be no nonzero elements of the
Hamiltonian matrix linking any states of the same energy; nonzero couplings
occur only between states from the lower set and the states in the upper set.
We make no restriction on the number of states that connect with any single
state. This generalizes the usual situation of angular momentum states excited
by electric-dipole radiation, when any state can link to no more than three
other states (the selection rules $\Delta m=-1,0,+1$). Figure 1 illustrates
the linkage pattern we consider.

Figure 1: (Color online) Schematic linkage pattern for multistate system
consisting of two coupled sets of degenerate levels.

We shall assume that we can control the magnitude and the phase of each
interaction (e.g. each electric field envelope), and that _every Rabi
frequency has the same time dependence_ , which we write as

$\Omega_{mn}(t)=2V_{mn}f(t),$ (5)

where $V_{mn}$ is a complex-valued constant and $f(t)$ a real-valued pulse-
shaped function of time bounded by unity. We assume that the duration of the
pulse is shorter than any decoherence time, so that the dynamics is governed
by the Schrödinger equation. Such is the situation for excitations of free
atoms and molecules by picosecond or femtosecond pulses femto .

We shall also assume that each transition has exactly the same detuning
$\Delta$. In the simplest situations, those associated with angular-momentum
states, three distinct fields can be distinguished by their polarization,
while sharing a common carrier frequency. By utilizing three independent
polarization directions of a laser pulse we can ensure three linkages with
common time dependence. More general linkage patterns, still within the model
described here, are also possible other MS .

##  III Exact analytical solution

###  III.1 The RWA Hamiltonian

In the RWA the Schrödinger equation provides a prescription for the time
dependence of the probability amplitudes $C_{n}(t)$. Written in matrix form it
is

$i\frac{d}{dt}\mathbf{C}(t)=\mathsf{H}(t)\mathbf{C}(t),$ (6)

where the column vector $\mathbf{C}(t)$ has $C_{n}(t)$ as elements. We shall
arrange these elements with those of the lower set first, followed by the
upper set. Then the RWA Hamiltonian has the block structure

$\mathsf{H}(t)=\begin{bmatrix}\mathsf{O}&\mathsf{V}f(t)\\\
\mathsf{V}^{\dagger}f(t)&\mathsf{D}(t)\end{bmatrix}.$ (7)

Here $\mathsf{O}$ denotes the $N$-dimensional square null matrix; the zeros
signify that the lower states do not interact with each other and that they
all have the same energy, which we take as the zero-point of our energy scale.
The matrix $\mathsf{V}$ is an $(N\times M)$-dimensional matrix whose elements
$V_{nm}$ are the magnitudes of the couplings between the lower and the upper
states; $\mathsf{V}^{{\dagger}}$ is its hermitian conjugate. Lastly,
$\mathsf{D}(t)$ is an $M$-dimensional square diagonal matrix whose elements
are all equal to the shared detuning $\Delta(t)$,

$\mathsf{D}(t)=\Delta(t)\mathsf{I},$ (8)

where $\mathsf{I}$ is the unit matrix. The diagonal nature of $\mathsf{D}$
indicates the absence of interaction of the upper states amongst themselves.
We allow the detuning $\Delta(t)$ to vary with time, bearing in mind use of
known analytic solutions to the two-state model with frequency-swept detuning.
It proves useful to write the matrix of interactions $\mathsf{V}$ as a row
vector of $N$-dimensional column vectors
$\left|V_{n}\right\rangle=\left[V_{1n},V_{2n},\ldots,V_{Nn}\right]^{T}$,

$\mathsf{V}=\begin{bmatrix}V_{11}&V_{12}&\cdots&V_{1M}\\\
V_{21}&V_{22}&\cdots&V_{2M}\\\ \cdots&\cdots&\ddots&\vdots\\\
V_{N1}&V_{N2}&\cdots&V_{NM}\end{bmatrix}=[\left|V_{1}\right\rangle,\left|V_{2}\right\rangle,\ldots,\left|V_{M}\right\rangle].$
(9)

###  III.2 The Morris-Shore transformation

A significant property of the linkage pattern shown in Fig. 1 is that a
transformation of Hilbert-space coordinates – the Morris-Shore (MS)
transformation – reduces the dynamics to that of a set of independent two-
state systems, together with decoupled states MS . When $N>M$, as is the case
with Fig. 1, the $N_{0}=N-M$ additional states are a part of the lower-level
manifold. They have no connection with excited states, and hence they cannot
produce excitation followed by fluorescence; they are termed _dark states_
dark state , by contrast to the states that can produce excitation (and thence
fluorescence), the _bright states_. The number of bright states is the lesser
of $N$ and $M$, in this case $M$. Figure 2 shows the new linkage pattern in
the MS basis.

Figure 2: (Color online) Linkages of Fig. 1 transformed by the Morris-Shore
transformation into a set of $M$ independent nondegenerate two-state systems,
with couplings $\lambda_{n}$, and a set of $N_{0}=N-M$ decoupled (dark)
states.

Because, by assumption, all elements of the interaction matrix share a common
time dependence, the required MS transformation is achieved by a constant
unitary transformation

$\mathsf{S}=\left[\begin{array}[]{cc}\mathsf{A}&\mathsf{O}\\\
\mathsf{O}&\mathsf{B}\end{array}\right].$ (10)

Here $\mathsf{A}$ is a unitary $N$-dimensional square matrix
($\mathsf{AA}^{\dagger}=\mathsf{A}^{\dagger}\mathsf{A}=\mathsf{I}$) which
transforms the lower set of states, and $\mathsf{B}$ is a unitary
$M$-dimensional square matrix
($\mathsf{B}\mathsf{B}^{\dagger}=\mathsf{B}^{\dagger}\mathsf{B}=\mathsf{I}$)
which transforms the upper set. This transformation casts the dynamics into
the MS basis, with new MS bright lower states
$\\{|\alpha_{m}\rangle\\}_{m=1}^{M}$, dark lower states
$\\{|\gamma_{k}\rangle\\}_{k=1}^{N_{0}}$, and upper states
$\\{|\beta_{m}\rangle\\}_{m=1}^{M}$. The transformed MS Hamiltonian has the
form

$\widetilde{\mathsf{H}}(t)=\mathsf{S}\mathsf{H}(t)\mathsf{S}^{\dagger}=\left[\begin{array}[]{cc}\mathsf{O}&\widetilde{\mathsf{V}}f(t)\\\
\widetilde{\mathsf{V}}^{\dagger}f(t)&\mathsf{D}(t)\end{array}\right],$ (11)

where

$\widetilde{\mathsf{V}}=\mathsf{A}\mathsf{V}\mathsf{B}^{\dagger}.$ (12)

Because the matrices
$\widetilde{\mathsf{V}}\widetilde{\mathsf{V}}^{\dagger}=\mathsf{A}\mathsf{V}\mathsf{V}^{\dagger}\mathsf{A}^{\dagger}$
and
$\widetilde{\mathsf{V}}^{\dagger}\widetilde{\mathsf{V}}=\mathsf{B}\mathsf{V}^{\dagger}\mathsf{V}\mathsf{B}^{\dagger}$
must be diagonal (possibly after removing null rows and rearanging the basis
states), the matrices $\mathsf{A}$ and $\mathsf{B}$ are defined by the
conditions that they diagonalize $\mathsf{V}\mathsf{V}^{\dagger}$ and
$\mathsf{V}^{\dagger}\mathsf{V}$, respectively.
$\mathsf{V}^{\dagger}\mathsf{V}$ has $M$ generally nonzero eigenvalues
$\lambda_{n}^{2}$ ($n=1,2,\ldots,M$). The $N$-dimensional matrix
$\mathsf{V}\mathsf{V}^{\dagger}$ has the same $M$ eigenvalues as
$\mathsf{V}^{\dagger}\mathsf{V}$ and additional $N_{0}=N-M$ zero eigenvalues.
From the vector form (9) of $\mathsf{V}$ we obtain

$\displaystyle\mathsf{VV}^{\dagger}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{M}\left|V_{n}\right\rangle\left\langle
V_{n}\right|,$ (13a) $\displaystyle\mathsf{V}^{\dagger}\mathsf{V}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}\left\langle
V_{1}|V_{1}\right\rangle&\left\langle
V_{1}|V_{2}\right\rangle&\cdots&\left\langle V_{1}|V_{M}\right\rangle\\\
\left\langle V_{2}|V_{1}\right\rangle&\left\langle
V_{2}|V_{2}\right\rangle&\cdots&\left\langle V_{2}|V_{M}\right\rangle\\\
\vdots&\vdots&\ddots&\vdots\\\ \left\langle
V_{M}|V_{1}\right\rangle&\left\langle
V_{M}|V_{2}\right\rangle&\cdots&\left\langle
V_{M}|V_{M}\right\rangle\end{array}\right].$ (13f)

$\mathsf{V}^{\dagger}\mathsf{V}$ is the Gram matrix Gantmacher  for the set of
vectors $|V_{n}\rangle$, ($n=1,2,\ldots,M$). Thus if all these vectors are
linearly independent then $\det\mathsf{V}^{\dagger}\mathsf{V}\neq 0$ and all
eigenvalues of $\mathsf{V}^{\dagger}\mathsf{V}$ are nonzero Gantmacher ;
however, this assumption of independence is unnecessary.

The MS Hamiltonian (11) has the explicit form

$\widetilde{\mathsf{H}}(t)=\left[\begin{array}[]{c|c}\mathsf{O}&\mathsf{O}\\\
\hline\cr\mathsf{O}&\begin{array}[]{cccccccc}0&0&\cdots&0&\lambda_{1}f(t)&0&\cdots&0\\\
0&0&\cdots&0&0&\lambda_{2}f(t)&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0&0&0&\cdots&\lambda_{M}f(t)\\\
\lambda_{1}f(t)&0&\cdots&0&\Delta&0&\cdots&0\\\
0&\lambda_{2}f(t)&\cdots&0&0&\Delta&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\lambda_{M}f(t)&0&0&\cdots&\Delta\end{array}\end{array}\right].$
(14)

The structure of $\widetilde{\mathsf{H}}(t)$ shows that in the MS basis the
dynamics decomposes into sets of $N_{0}$ decoupled single states and $M$
independent two-state systems
$|\alpha_{n}\rangle\leftrightarrow|\beta_{n}\rangle$ ($n=1,2,\ldots,M$), each
composed of a lower state $|\alpha_{n}\rangle$ and an upper state
$|\beta_{n}\rangle$, and driven by the two-state RWA Hamiltonians

$\widetilde{\mathsf{h}}_{n}=\left[\begin{array}[]{cc}0&\lambda_{n}f(t)\\\
\lambda_{n}f(t)&\Delta\end{array}\right],\quad(n=1,2,\ldots,M).$ (15)

Each of these two-state Hamiltonians has the same detuning $\Delta$, but they
differ in the couplings $\lambda_{n}$. Each of the new lower states
$|\alpha_{n}\rangle$ is an eigenstate of $\mathsf{V}\mathsf{V}^{\dagger}$ that
corresponds to a specific eigenvalue $\lambda_{n}^{2}$ of
$\mathsf{V}\mathsf{V}^{\dagger}$. Similarly the new upper state
$|\beta_{n}\rangle$ is the eigenstate of $\mathsf{V}^{\dagger}\mathsf{V}$,
corresponding to the same eigenvalue $\lambda_{n}^{2}$. The square root of
this common eigenvalue, $\lambda_{n}$, represents the coupling (half the Rabi
frequency) in the respective independent MS two-state system
$|\alpha_{n}\rangle\leftrightarrow|\beta_{n}\rangle$. The $N_{0}$ zero
eigenvalues of $\mathsf{V}\mathsf{V}^{\dagger}$ correspond to decoupled (dark)
states in the lower set (we assume thoughout that $N\geqq M$; therefore, dark
states, if any, are in the lower set). The dark states are decoupled from the
dynamical evolution because they are driven by one-dimensional null
Hamiltonians.

###  III.3 The propagator in the original basis

The eigenvectors of $\mathsf{V}\mathsf{V}^{\dagger}$ and
$\mathsf{V}^{\dagger}\mathsf{V}$ form the transformation matrices $\mathsf{A}$
and $\mathsf{B}$,

$\mathsf{A}=\left[\begin{array}[]{c}\langle\gamma_{1}|\\\ \vdots\\\
\langle\gamma_{N_{0}}|\\\ \langle\alpha_{1}|\\\ \vdots\\\
\langle\alpha_{M}|\end{array}\right],\quad\mathsf{B}=\left[\begin{array}[]{c}\langle\beta_{1}|\\\
\vdots\\\ \langle\beta_{M}|\end{array}\right].$ (16)

They obey the completeness relations

$\displaystyle\sum_{n=1}^{M}|\alpha_{n}\rangle\langle\alpha_{n}|+\sum_{k=1}^{N_{0}}|\gamma_{k}\rangle\langle\gamma_{k}|=\mathsf{I},$
(17a)
$\displaystyle\sum_{n=1}^{M}|\beta_{n}\rangle\langle\beta_{n}|=\mathsf{I}.$
(17b)

The propagator $\widetilde{\mathsf{u}}_{n}(t_{f},t_{i})$ for the independent
MS two-state system $|\alpha_{n}\rangle\leftrightarrow|\beta_{n}\rangle$
($n=1,2,\ldots,M$) is unitary and therefore can be parameterized in terms of
the complex-valued Cayley-Klein parameters $a_{n}$ and $b_{n}$,

$\widetilde{\mathsf{u}}_{n}(t_{f},t_{i})=\begin{bmatrix}a_{n}&b_{n}\\\
-b_{n}^{\ast}e^{-i\delta}&a_{n}^{\ast}e^{-i\delta}\end{bmatrix},$ (18)

where $\left|a_{n}\right|^{2}+\left|b_{n}\right|^{2}=1$ and
$\delta=\int_{t_{i}}^{t_{f}}\Delta(t^{\prime})dt^{\prime}$. In what follows we
express the desired control of statevector motion in terms of constraints on
these two-state Cayley-Klein parameters. The unimportant phase factor
$e^{-i\delta}$ originates from the chosen representation of the Hamiltonian
(7), which facilitates the application of the MS transformation. In the
interaction representation, where the diagonal elements are nullified and the
detunings appear in phase factors multiplying the couplings, the factor
$e^{-i\delta}$ disappears. (Note: This phase factor has been omitted in Eqs.
(17), (18) and (20) of Ref. Kyoseva , where it should have been associated
with $a^{\ast}$ and $b^{\ast}$; however, it did not affect any result there.)

By taking into account the MS propagators (18) for the two-state MS systems,
the ordering of the states (16), and the MS Hamiltonian (14), we write the
propagator of the full system in the MS basis as

$\widetilde{\mathsf{U}}=\left[\begin{array}[]{c|c}\mathsf{I}&\mathsf{O}\\\
\hline\cr\mathsf{O}&\begin{array}[]{cccccccc}a_{1}&0&\cdots&0&b_{1}&0&\cdots&0\\\
0&a_{2}&\cdots&0&0&b_{2}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&a_{M}&0&0&\cdots&b_{M}\\\
-b_{1}^{\ast}e^{-i\delta}&0&\cdots&0&a_{1}^{\ast}e^{-i\delta}&0&\cdots&0\\\
0&-b_{2}^{\ast}e^{-i\delta}&\cdots&0&0&a_{2}^{\ast}e^{-i\delta}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&-b_{M}^{\ast}e^{-i\delta}&0&0&\cdots&a_{M}^{\ast}e^{-i\delta}\end{array}\end{array}\right].$
(19)

It is straightforward to show that the propagator in the original basis
$\mathsf{U}=\mathsf{S}^{\dagger}\widetilde{\mathsf{U}}\mathsf{S}$ reads

$\mathsf{U}=\left[\begin{array}[]{cc}\mathsf{U}_{N}&\mathsf{U}_{NM}\\\
\mathsf{U}_{MN}&\mathsf{U}_{M}\end{array}\right],$ (20a) where
$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{M}a_{n}|\alpha_{n}\rangle\langle\alpha_{n}|+\sum_{k=1}^{N_{0}}|\gamma_{k}\rangle\langle\gamma_{k}|,$
(20b) $\displaystyle\mathsf{U}_{NM}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{M}b_{n}|\alpha_{n}\rangle\langle\beta_{n}|,$ (20c)
$\displaystyle\mathsf{U}_{MN}$ $\displaystyle=$
$\displaystyle-e^{-i\delta}\sum_{n=1}^{M}b_{n}^{\ast}|\beta_{n}\rangle\langle\alpha_{n}|,$
(20d) $\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\sum_{n=1}^{M}a_{n}^{\ast}|\beta_{n}\rangle\langle\beta_{n}|.$
(20e)

The propagator $\mathsf{U}_{N}$ connects states within the lower set,
$\mathsf{U}_{M}$ connects states within the upper set, and $\mathsf{U}_{NM}$
and $\mathsf{U}_{MN}$ mix states from the lower and upper sets. By using the
completeness relations (17a) and (17b) we find

$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\mathsf{I}+\sum_{n=1}^{M}(a_{n}-1)|\alpha_{n}\rangle\langle\alpha_{n}|,$
(21a) $\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\left[\mathsf{I}+\sum_{n=1}^{M}(a_{n}^{\ast}-1)|\beta_{n}\rangle\langle\beta_{n}|\right].$
(21b)

Hence the propagator $\mathsf{U}_{N}$ does not depend on the decoupled states
$|\gamma_{k}\rangle$ $(k=1,2,\ldots,N_{0})$. This has to be expected because,
owing to their degeneracy, the choice of the decoupled states is not unique:
any superposition of them is also a zero-eigenvalue eigenstate of
$\mathsf{V}\mathsf{V}^{\dagger}$. Because the dynamics in the original basis
must not depend on such a leeway of choice, the full propagator $\mathsf{U}$
must not depend on the decoupled states at all, and this is indeed the case.

###  III.4 Analytical solutions for degenerate levels

Equation (20a) expresses the dynamics of the degenerate two-level system in
terms of the dynamics of the $M$ independent non-degenerate two-state systems,
each with the same detuning $\Delta(t)$ and pulse shape $f(t)$ of the
couplings but with different coupling strengths $\lambda_{n}$. Therefore, Eq.
(20a) allows one to generalize any analytical non-degenerate two-state
solution to a pair of degenerate levels. Recently, such a generalization of
the Landau-Zener model to two degenerate levels has been presented Vasilev .
This generalization displays several interesting properties, for instance, not
all transition probabilities between degenerate states are defined for
infinite time duration. Other analytical solutions involving two degenerate
levels have been derived for five chainwise-coupled states in $M$ or $W$
linkage configurations Vitanov MW . We present below another interesting
aspect of the solution (20): its geometrical nature.

##  IV Quantum-state reflections

###  IV.1 Coupled reflections

Of particular significance is the special case when the Cayley-Klein
parameters $b_{n}$ are all equal to zero,

$b_{n}=0\quad(n=1,2,\ldots,M);$ (22)

then all transition probabilities in the MS basis, as well as these in the
original basis from the lower set to the upper set, vanish, i.e.
$\mathsf{U}_{NM}=\mathsf{U}_{MN}=0$. Equation (22) implies that
$\left|a_{n}\right|=1$, or

$a_{n}=e^{i\phi_{n}}\quad(n=1,2,\ldots,M),$ (23)

for all MS two-state systems, where $\phi_{n}$ are arbitrary phases.

After substituting Eqs. (22) and (23) in Eqs. (20), the propagator in the
original basis reads

$\displaystyle\mathsf{U}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\mathsf{U}_{N}&\mathsf{O}\\\
\mathsf{O}&\mathsf{U}_{M}\end{array}\right],$ (24c)
$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\mathsf{I}+\sum_{n=1}^{M}(e^{i\phi_{n}}-1)|\alpha_{n}\rangle\langle\alpha_{n}|,$
(24d) $\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\left[\mathsf{I}+\sum_{n=1}^{M}(e^{-i\phi_{n}}-1)|\beta_{n}\rangle\langle\beta_{n}|\right],$
(24e)

where the propagator $\mathsf{U}_{N}$ operates in the lower set of states and
$\mathsf{U}_{M}$ acts in the upper set. Taking into account that the coupled
lower states $|\alpha_{n}\rangle\ (n=1,2,\ldots,M)$ are orthonormal basis
vectors, i.e. $\langle\alpha_{n}|\alpha_{k}\rangle=\delta_{nk}$, as are the
upper MS states $|\beta_{n}\rangle$ $(n=1,2,\ldots,M)$, we rewrite
$\mathsf{U}_{N}$ and $\mathsf{U}_{M}$ as the products

$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{M}\mathsf{M}(\alpha_{n},\phi_{n}),$ (25a)
$\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\prod_{n=1}^{M}\mathsf{M}(\beta_{n},-\phi_{n}),$ (25b)

where

$\mathsf{M}(\nu;\phi)=\mathsf{I}+(e^{i\phi}-1)|\nu\rangle\langle\nu|.$ (26)

These individual matrices are, in fact, generalizations of the matrices used
to produce Householder reflection in matrix computations Householder . We
refer to them as _generalized quantum Householder reflection_ (QHR) operators
Kyoseva ; Ivanov ; Ivanov2 . The generalized QHR operator (26) is unitary,
$\mathsf{M}(\nu;\phi)^{-1}=\mathsf{M}(\nu;\phi)^{\dagger}=\mathsf{M}(\nu;-\phi)$,
and its determinant has unit magnitude, $\det\mathsf{M}(\nu;\phi)=e^{i\phi}$.
For $\phi=0$ the QHR operator reduces to the identity,
$\mathsf{M}(\nu;0)=\mathsf{I}$, while for $\phi=\pi$, the QHR operator reduces
to the standard reflection,

$\mathsf{M}(\nu;\pi)=\mathsf{I}-2|\nu\rangle\langle\nu|.$ (27)

The orthogonality of the QHR vectors $|\alpha_{n}\rangle$ in Eq. (25a)
automatically ensures the commutation of the QHRs,

$\left[\mathsf{M}(\alpha_{n},\phi_{n}),\mathsf{M}(\alpha_{m},\phi_{m})\right]=0.$
(28)

Therefore their ordering in the product in Eq. (25a) is unimportant. The same
argument applies to Eq. (25b).

The importance of QHR derives from the fact that any $N$-dimensional unitary
matrix can be decomposed into a set of at most $N$ generalized QHRs Ivanov .
The resulting ease with which pulse sequences can be designed to realize the
QHR therefore enables one to synthesize any desired unitary transformation of
a qunit state, for example a quantum Fourier transform Ivanov  or transition
between any two pure or mixed qunit states Ivanov2 .

###  IV.2 Special case: orthogonal interaction vectors

In the special case when the interaction vectors $|V_{n}\rangle$ are
_orthogonal_ , $\langle V_{m}|V_{n}\rangle=|V_{n}|^{2}\delta_{mn}$, the MS
eigenvalues and the MS states simplify greatly. Then the matrix
$\mathsf{V}^{\dagger}\mathsf{V}$ of Eq. (13f) becomes diagonal, and hence its
eigenvalues are $\lambda_{n}^{2}=\left|V_{n}\right|^{2}$ ($n=1,2,\ldots,M$).
Moreover, the eigenstates of $\mathsf{V}^{\dagger}\mathsf{V}$ – the MS states
in the upper set – coincide with the original states,

$|\beta_{n}\rangle\equiv|\varphi_{n}\rangle,\quad(n=1,2,\ldots,M).$ (29)

The coupled MS states in the lower set – the eigenstates of
$\mathsf{VV}^{\dagger}$ – are readily found from Eq. (13a),

$|\alpha_{n}\rangle=\frac{1}{\left|V_{n}\right|}\left|V_{n}\right\rangle\equiv|\hat{V}_{n}\rangle.$
(30)

The propagator $U_{N}$ in the lower set is a product of QHRs, with the
normalized interaction vectors $|\hat{V}_{n}\rangle$ serving as QHR vectors,
whereas the propagator $\mathsf{U}_{M}$ in the upper set is a phase gate,

$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{M}\mathsf{M}(\hat{V}_{n},\phi_{n}),$ (31a)
$\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\sum_{n=1}^{M}e^{-i\phi_{n}}|\varphi_{n}\rangle\langle\varphi_{n}|.$
(31b)

The advantage of having orthogonal interaction vectors $|V_{n}\rangle$ is that
they serve as QHR vectors. The implication is that in order to construct a
pre-selected coupled-QHR transformation (31a), the required couplings are
directly obtained from Eq. (30). Otherwise, in the general case of non-
orthogonal interaction vectors, a set of pre-selected QHR vectors
$|\alpha_{n}\rangle$ ($n=1,2,\ldots,M$), defined as the eigenvectors of
$\mathsf{VV}^{\dagger}$, demand numerical derivation of the required couplings
from Eq. (13a).

###  IV.3 Realizations

####  IV.3.1 Off-resonant hyperbolic-secant pulses

The condition (23) can be realized with the Rosen-Zener model RZ ; Kyoseva ,
which assumes constant detuning and hyperbolic-secant time dependence $f(t)$
for the couplings, with pulse duration $T$,

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle\mbox{sech}(t/T),$ (32a)
$\displaystyle\Delta(t)$ $\displaystyle=$ $\displaystyle\text{const}.$ (32b)
The independent MS two-state systems share the same detuning $\Delta$ and
pulse shape $f(t)$, but have different MS couplings $\lambda_{n}$. For the
Rosen-Zener model the Cayley-Klein parameters $a_{n}$ ($n=1,2,...,M$) read RZ
; Kyoseva  $a_{n}=\frac{\Gamma^{2}\left(\frac{1}{2}+\frac{1}{2}i\Delta
T\right)}{\Gamma\left(\frac{1}{2}+\lambda_{n}T+\frac{1}{2}i\Delta
T\right)\Gamma\left(\frac{1}{2}-\lambda_{n}T+\frac{1}{2}i\Delta T\right)},$
(33)

where $\Gamma(z)$ is Euler’s $\Gamma$-function. Using the reflection formula
$\Gamma(1/2+z)\Gamma(1/2-z)=\pi/\cos\pi z$, we find

$\left|a_{n}\right|^{2}=1-\frac{\sin^{2}(\pi\lambda_{n}T)}{\cosh^{2}\left(\frac{1}{2}\pi\Delta
T\right)}.$ (34)

It follows that the condition $\left|a_{n}\right|=1$ is satisfied when
$\lambda_{n}T=l$ $(l=0,1,2,\ldots)$. The phase $\phi_{n}$ of
$a_{n}=e^{i\phi_{n}}$ depends on the detuning $\Delta$, but not on the
corresponding coupling $\lambda_{n}$, and for an arbitrary integer $l$ we find
Kyoseva

$\phi_{n}=2\arg\prod_{k=0}^{l-1}\left[\Delta T+i\left(2k+1\right)\right].$
(35)

Hence the QHR phase $\phi_{n}$ can be produced by an appropriate choice of the
detuning $\Delta$. This result shows that even though the couplings for the MS
two-state systems are not the same, the phases $\phi_{n}$ of the Cayley-Klein
parameters $a_{n}$ coincide, $\phi_{n}\equiv\phi$. This feature is unique for
the sech pulse. For other non-resonant pulses, e.g. Gaussian Vasilev-Gaussian
, the phase $\phi_{n}$ would depend also on the coupling and therefore will be
generally different for each MS two-state system.

####  IV.3.2 Resonant pulses

For exact resonance ($\Delta=0$), the Cayley-Klein parameter $a_{n}$ reads
$a_{n}=\cos(A_{n}/2)$, where the pulse area is
$A_{n}=2\lambda_{n}\int_{-\infty}^{\infty}f(t)dt$. When the pulse area is
$A_{n}=2(2l+1)\pi$ ($l=0,1,2,\ldots$), the phase $\phi_{n}$ is equal to $\pi$;
hence we obtain a physical realization for the standard QHR (27). This result
is not resticted to the sech pulse (32a) but it is valid for any pulse shape
with such an area Kyoseva ; Ivanov . When the pulse areas are multiples of
$4\pi$, the phases $\phi_{n}$ vanish, $a_{n}=1$, and the corresponding QHRs
reduce to the identity. Resonant pulses therefore do not produce variable QHR
phases $\phi_{n}$, which can be used as free parameters.

####  IV.3.3 Far-off-resonant pulses

Far-off-resonant pulses provide the opportunity for easy adjustment of the
phase $\phi_{n}$, albeit only approximately. Then the condition (23) is
fulfilled automatically because of the smallness of the transition
probabilities in each MS system. Specifically, if the common detuning $\Delta$
exceeds sufficiently much the largest MS coupling $\lambda_{n}$, then all
transition probabilities will be negligibly small, $\left|b_{n}\right|\ll 1$.
By adiabatic elimination of each upper MS state one finds

$\phi_{n}\approx\frac{\lambda_{n}^{2}}{\Delta}\int_{-\infty}^{\infty}f^{2}(t)dt.$
(36)

For a sech pulse the integral is equal to $2T$, and for a Gaussian to
$\sqrt{\pi/2}$. Because each MS coupling $\lambda_{n}$ is generally different,
each phase $\phi_{n}$ will also be different. Any desired phase $\phi_{n}$, or
a set of such phases, can easily be produced by choosing the original
couplings $V_{mn}$, and hence the MS couplings $\lambda_{n}$, appropriately.

##  V Two degenerate upper states

Above, we described the dynamics of the degenerate two-level quantum system in
the general case when the lower and upper levels had arbitrary degeneracies,
$N$ and $M$ respectively. In this section, we will illustrate these results
with a specific example: when the upper set consists of just two degenerate
states, i.e. $M=2$. This case is insteresting because of the possible
implementations in different real physical systems, several examples of which
will be presented below. Moreover, this special case allows for an elegant
analytical treatment.

###  V.1 General case

We retain the notation for the lower states $|\psi_{n}\rangle$
$(n=1,2,...,N)$, and we denote the two upper states $|\varphi^{\prime}\rangle$
and $|\varphi^{\prime\prime}\rangle$. The interaction matrix (9) reads

$\mathsf{V}=\left[\begin{array}[]{cc}V_{1}^{\prime}&V_{1}^{\prime\prime}\\\
V_{2}^{\prime}&V_{2}^{\prime\prime}\\\ \vdots&\vdots\\\
V_{N}^{\prime}&V_{N}^{\prime\prime}\end{array}\right]\equiv\left[\left|V^{\prime}\right\rangle,\left|V^{\prime\prime}\right\rangle\right],$
(37)

where $\left|V^{\prime}\right\rangle$ and
$\left|V^{\prime\prime}\right\rangle$ are $N$-dimensional interaction vectors
comprising the couplings between the lower states and the corresponding upper
state. The product $\mathsf{V}^{{\dagger}}\mathsf{V}$ reads

$\mathsf{V}^{\dagger}\mathsf{V}=\left[\begin{array}[]{cc}\left|V^{\prime}\right|^{2}&\left\langle
V^{\prime}|V^{\prime\prime}\right\rangle\\\ \left\langle
V^{\prime\prime}|V^{\prime}\right\rangle&\left|V^{\prime\prime}\right|^{2}\end{array}\right].$
(38)

With the introduction of parameters $\theta$ and $\sigma$ through the
definitions

$\displaystyle\frac{2|\langle
V^{\prime}|V^{\prime\prime}\rangle|}{|V^{\prime\prime}|^{2}-|V^{\prime}|^{2}}$
$\displaystyle=$ $\displaystyle\tan 2\theta\quad(0<\theta<\pi/2),$ (39a)
$\displaystyle\arg\langle V^{\prime}|V^{\prime\prime}\rangle$ $\displaystyle=$
$\displaystyle\sigma,$ (39b)

we write the eigenvalues $\lambda_{m}$ and the associated eigenvectors
$|\beta_{m}\rangle$ ($m=1,2$) within the upper set as

$\displaystyle\lambda_{1,2}^{2}$ $\displaystyle=$
$\displaystyle\frac{|V^{\prime}|^{2}+|V^{\prime\prime}|^{2}}{2}\pm\frac{|V^{\prime}|^{2}-|V^{\prime\prime}|^{2}}{2\cos
2\theta},$ (40a) $\displaystyle|\beta_{1}\rangle$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\cos\theta\\\
-e^{-i\sigma}\sin\theta\end{array}\right],\qquad|\beta_{2}\rangle=\left[\begin{array}[]{c}e^{i\sigma}\sin\theta\\\
\cos\theta\end{array}\right].$ (40f) The next step is to find the MS states
$|\alpha_{1}\rangle$ and $|\alpha_{2}\rangle$ within the lower set of states.
They are the eigenstates of the $N$-dimensional matrix,
$\mathsf{V}\mathsf{V}^{{\dagger}}=|V^{\prime}\rangle\langle
V^{\prime}|+|V^{\prime\prime}\rangle\langle V^{\prime\prime}|,$ (41)

which correspond to the (nonzero) eigenvalues (40a). We construct them as
superpositions of the interaction vectors $|V^{\prime}\rangle$ and
$|V^{\prime\prime}\rangle$, and find after simple algebra

$\displaystyle|\alpha_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda_{1}}\left(\cos\theta\left|V^{\prime}\right\rangle-e^{-i\sigma}\sin\theta|V^{\prime\prime}\rangle\right),$
(42a) $\displaystyle|\alpha_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda_{2}}\left(e^{i\sigma}\sin\theta|V^{\prime}\rangle+\cos\theta|V^{\prime\prime}\rangle\right).$
(42b)

###  V.2 The propagators

After we have found the explicit form of the MS states (40f) and (42), we
obtain the exact form of the propagators $\mathsf{U}_{N}$ and
$\mathsf{U}_{M}$,

$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\mathsf{M}(\alpha_{1},\phi_{1})\mathsf{M}(\alpha_{2},\phi_{2})$
(43a) $\displaystyle=$
$\displaystyle\mathsf{I}+(e^{i\phi_{1}}-1)|\alpha_{1}\rangle\langle\alpha_{1}|+(e^{i\phi_{2}}-1)|\alpha_{2}\rangle\langle\alpha_{2}|,$
(43b) $\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\left[e^{-i\phi_{1}}|\beta_{1}\rangle\langle\beta_{1}|+e^{-i\phi_{2}}|\beta_{2}\rangle\langle\beta_{2}|\right],$
(43c)

with $\phi_{1}$ and $\phi_{2}$ being the phases of the Cayley-Klein parameters
for the MS two-state propagators (18).

It is easy to verify that the bright states (42a) and (42b) are eigenstates of
the propagator $\mathsf{U}_{N}$ with eigenvalues $e^{i\phi_{1}}$ and
$e^{i\phi_{2}}$, respectively. Physically this means that if the qunit starts
in one of these states it will end up in this same state, acquiring only a
phase factor. This occurs because of the conditions (22) and the independence
of the different MS two-state systems. If the QHR phases are equal,
$\phi_{1}=\phi_{2}$, then any superposition of $|\alpha_{1}\rangle$ and
$|\alpha_{2}\rangle$ is also an eigenvector of $\mathsf{U}_{N}$. The other
eigenstates of the propagator are all degenerate, with unit eigenvalue, and
they are orthogonal to $|\alpha_{1}\rangle$ and $|\alpha_{2}\rangle$. For a
qutrit ($N=3$) there is only one such eigenvector (up to an unimportant global
phase factor) and it is proportional to
$|\alpha_{1}\rangle\times|\alpha_{2}\rangle$. For higher-dimensional qunits,
any vector in a hyperplane orthogonal to $|\alpha_{1}\rangle$ and
$|\alpha_{2}\rangle$ is an eigenvector of $\mathsf{U}_{N}$.

In the special case when the vectors $\left|V^{\prime}\right\rangle$ and
$\left|V^{\prime\prime}\right\rangle$ are orthogonal, $\left\langle
V^{\prime}|V^{\prime\prime}\right\rangle=0$, the expressions simplify. Then
$\theta=0$ and $\sigma=0$, and hence

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle|V^{\prime}|,\quad\lambda_{2}=|V^{\prime\prime}|,$ (44a)
$\displaystyle|\alpha_{1}\rangle$ $\displaystyle=$
$\displaystyle|\hat{V}^{\prime}\rangle,\quad|\alpha_{2}\rangle=|\hat{V}^{\prime\prime}\rangle,$
(44b) $\displaystyle|\beta_{1}\rangle$ $\displaystyle=$
$\displaystyle|\varphi_{1}\rangle,\quad|\beta_{2}\rangle=|\varphi_{2}\rangle.$
(44c)

Then the propagator $\mathsf{U}_{N}$ in the lower set is a product of QHRs, in
which the interaction vectors $|V^{\prime}\rangle$ and
$|V^{\prime\prime}\rangle$ serve as QHR vectors, while the propagator
$\mathsf{U}_{M}$ in the upper set is a phase gate,

$\displaystyle\mathsf{U}_{N}$ $\displaystyle=$
$\displaystyle\mathsf{M}(\hat{V}^{\prime},\phi_{1})\mathsf{M}(\hat{V}^{\prime\prime},\phi_{2}),$
(45a) $\displaystyle\mathsf{U}_{M}$ $\displaystyle=$ $\displaystyle
e^{-i\delta}\begin{bmatrix}e^{-i\phi_{1}}&0\\\ 0&e^{-i\phi_{2}}\end{bmatrix}.$
(45b)

###  V.3 Examples

Following are examples of linkage patterns amongst angular-momentum states,
which allow application of the QHR theory.

####  V.3.1 Two levels, $J=3/2\leftrightarrow J=1/2$

Figure 3: (Color online) Linkage pattern for the four degenerate magnetic
sublevels of $J=3/2$, the lower set, shown coupled by arbitrary polarization
of electric-dipole radiation to the two sublevels of $J=1/2$, the upper set.
The states are labeled by their magnetic quantum number.

Figure 3 shows linkage patterns possible with arbitrary polarization between
the four magnetic sublevels of $J=3/2$, the lower set, and the two of $J=1/2$,
the upper set. The interaction matrix has the elements (with the Clebsch-
Gordan coefficients included),

$\mathsf{V}=\frac{1}{\sqrt{6}}\begin{bmatrix}\sqrt{3}V_{+}&0\\\
-\sqrt{2}V_{0}&V_{+}\\\ V_{-}&-\sqrt{2}V_{0}\\\ 0&\sqrt{3}V_{-}\end{bmatrix},$
(46)

where the subscripts $+$, $-$ and 0 refer to right circular ($\sigma^{+}$),
left circular ($\sigma^{-}$) and linear ($\pi$) polarizations.

Figure 4: (Color online) Time evolution of the numerically calculated
populations of the magnetic sublevels of a $J=3/2$ level coherently coupled to
a $J=1/2$ level by three polarized ($\sigma^{+}$, $\sigma^{-}$ and $\pi$)
pulsed laser fields, with sech shape $f(t)=\text{sech}(t/T)$,
$V_{-}=V_{0}=V_{+}=8.5T^{-1}$ and detuning $\Delta=80T^{-1}$. Then
$\theta=\pi/4$, $\sigma=\pi$, $\lambda_{1}T=106.3$, $\lambda_{2}T=38.2$,
$\phi_{1}=2.65772$, $\phi_{2}=0.954776$. The arrows on the right indicate the
values derived by the QHR theory.

Figure 4 shows an example of time evolution of the populations of the magnetic
sublevels in the $J=3/2$ level, starting with all population in state
$|\psi_{-3/2}\rangle$. The conditions $a_{n}=e^{i\phi_{n}}\ (n=1,2)$ are
realized approximately, by using large detuning from the upper $J=1/2$ level.
In the end of the interaction, the numerically calculated populations are seen
to approach the values predicted by the QHR theory (the arrows on the right).

####  V.3.2 Two levels, $J=2\leftrightarrow J=1$

The linkage pattern for electric-dipole couplings between sublevels of $J=2$,
the lower set, and $J=1$, the upper set will, for polarization expressed as a
combination of left- and right-circular polarization, appear as two uncoupled
systems: three states form a $\Lambda$-linkage, while five form an M-linkage,
as shown in Fig. 5. The interaction matrix is

$\mathsf{V}=\frac{1}{\sqrt{10}}\begin{bmatrix}\sqrt{6}V_{+}&0\\\
V_{-}&V_{+}\\\ 0&\sqrt{6}V_{-}\end{bmatrix}.$ (47)

Then $\tan 2\theta=|V_{+}V_{-}|/[5(|V_{-}|^{2}-|V_{+}|^{2})]$ and $\sigma=\arg
V_{+}-\arg V_{-}$.

Figure 5: (Color online) Linkage pattern for five states in an $M$
configuration.

####  V.3.3 Three levels, $J=0\leftrightarrow J=1\leftrightarrow J=0$

The MS transformation can be applied not only to a pair of degenerate levels
but also to a ladder of degenerate levels, as long as there is only a single
detuning. Figure 6 illustrates an example in an angular momentum basis between
the magnetic sublevels of three levels with angular momenta $J=0,1,0$. The
interaction matrix has the form

$\mathsf{V}=\begin{bmatrix}V_{+}^{\prime}&V_{+}^{\prime\prime}\\\
V_{0}^{\prime}&V_{0}^{\prime\prime}\\\
V_{-}^{\prime}&V_{-}^{\prime\prime}.\end{bmatrix},$ (48)

and the formalism of this section applies. In this case all fields can be
changed independently.

Figure 6: (Color online) Linkage pattern for three-level ladder involving a
degenerate middle level. The two ends of the chain, magnetic sublevels with
$J=0$ have the same detuning from the three intermediate sublevels of $J=1$.
The various linkages are invoked by adjusting the direction of the
polarization with respect to the quantization axis. The sublevels with $J=1$
form the lower set, while those with $J=0$ form the upper set.

##  VI Discussion and conclusions

We have here extended the earlier work on QHR in Hilbert space to allow more
general linkage patterns between the quantum states, with particular attention
to degenerate sublevels that occur with angular momentum states. The extension
relies on the use of the Morris-Shore transformation to reduce the original
multi-linkage Hamiltonian to a set of independent two-state systems, thereby
allowing the utilization of various known two-state analytic solutions. Such
solutions, when expressed in terms of Cayley-Klein parameters, readily lead to
conditions upon the pulse areas and the time-varying detunings of the
excitation pulses.

We have found that the propagator within the lower (or upper) set of
degenerate states represents coupled quantum-state mirrors. The realization of
these coupled QHRs within the lower (or upper) set of states requires certain
conditions on the interaction parameters, specifically that all transition
probabiities between the lower and upper sets must vanish. We have proposed
three physical realizations of this condition: with resonant, near-resonant
(hyperbolic-secant), and far-off-resonant pulses. Resonant and near-resonant
pulses provide exactly zero transition probabilities, but offer less
flexibility in the QHR phases $\phi_{n}$; moreover they require a carefully
chosen pulse area in each MS two-state system, which leads to a number of
conditions on the interaction parameters. Far-off-resonant pulses fulfill the
zero-probability conditions only approximately, but offer much more
flexibility. Then the only restriction is for sufficiently large detuning,
without specific constraints on pulse areas, because the zero-probability
conditions are fulfilled simultaneously in all MS two-state systems.

For angular-momentum states, there are six independent interaction parameters:
three polarization amplitudes, two relative phases between different
polarizations, and the common detuning. Therefore the constructed QHR has six
free parameters (with the far-off-resonance realization). For a more general
linkage, the number of independent parameters can be much larger.

In a subsequent paper multiQHR , we shall describe the mathematical aspects of
this largely unknown operator of coupled reflections, which however, as we
have shown here, arises naturally in quantum systems. In particular, we shall
show how one can factorize, and therefore synthesize, an arbitrary U($N$)
propagator by such objects. The procedure is more efficient than a set of
rotations that would produce the same transformation.

We conclude by pointing out that the confinement of the statevector evolution
to the lower set of states, and the availability of simple and efficient tools
for its engineering, such as coupled QHRs, can be an essential ingredient for
decoherence-free quantum computing decoherence .

###### Acknowledgements.

Quantum information processing, and the needed manipulation of statevectors in
Hilbert space, has long interested Sir Peter Knight, in whose honor the
present issue of Journal of Modern Optics has been assembled. We are pleased
to offer the present article in his honor. This work has been supported by the
EU ToK project CAMEL (Grant No. MTKD-CT-2004-014427), the EU RTN project EMALI
(Grant No. MRTN-CT-2006-035369), and the Alexander von Humboldt Foundation.

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