# Disentangling the Impacts of Language and Channel Variability on Speech
Separation Networks

###### Abstract

Because the performance of speech separation is excellent for speech in which
two speakers completely overlap, research attention has been shifted to
dealing with more realistic scenarios. However, domain mismatch between
training/test situations due to factors, such as speaker, content, channel,
and environment, remains a severe problem for speech separation. Speaker and
environment mismatches have been studied in the existing literature.
Nevertheless, there are few studies on speech content and channel mismatches.
Moreover, the impacts of language and channel in these studies are mostly
tangled. In this study, we create several datasets for various experiments.
The results show that the impacts of different languages are small enough to
be ignored compared to the impacts of different channels. In our experiments,
training on data recorded by Android phones leads to the best
generalizability. Moreover, we provide a new solution for channel mismatch by
evaluating projection, where the channel similarity can be measured and used
to effectively select additional training data to improve the performance of
in-the-wild test data.

Index Terms: speech separation, domain mismatch

##  1 Introduction

The cocktail party problem [1] refers to the perception of each speech source
in a noisy social environment, and has been extensively studied for many
years. As the necessary pre-processing for downstream tasks, such as speaker
diarization [2] and automatic speech recognition [3], many efforts have been
made in speech separation [4] to address this problem.

Nowadays, most of the studies on speech separation are conducted on the
WSJ0-2mix dataset [5]. WSJ0-2mix was artificially synthesized from
WSJ0111https://catalog.ldc.upenn.edu/LDC93S6A, an English speech corpus. In
WSJ0-2mix, all mixed utterances are full overlaps of clean speech of two
speakers. A typical architecture among the popular methods today is the time-
domain audio separation network (TasNet) [6]. Many works based on TasNet have
achieved extraordinary performance [7, 8, 9, 10, 11]. However, WSJ0-2mix
unrealistically sets a limit on the fixed number of speakers (addressed by
WSJ0-nmix), the fully overlapping ratio, and the quiet scenario. To push the
field towards more practical and practicable scenarios, many efforts have been
made.

Some research focused on the agnosticism of the number of speakers in
utterances. In [12, 13], the model extracts the speakers recursively. In [14],
the authors used a counter head to determine the number of speakers. Still
others tried to extract the target speakers by some sort of information, such
as visual features [15], speech recognition information [16], and anchor audio
clips [17]. In addition, some expanded the studies by evaluating different
ratios of overlapping speech. For example, a sparsely overlapping version of
WSJ0-2mix was introduced in [18], and the study found the challenges of Deep
Clustering [5] on such mixtures. In order to simulate the actual meeting,
LibriCSS [19], a test set containing a lower overlapping ratio and longer
speech utterances, was made suitable for research focusing on downstream
speech recognition tasks [20, 21].

Despite the broad aspects of these previous works, domain mismatch between
training/test situations remains challenging for the application in actual
scenarios, since all of the datasets are synthesized artificially from audios
recorded in anechoic chambers. Domain mismatch can be attributed to four
factors: speaker, content, channel, and environment. With regards to speaker
mismatch, speakers in the test sets of all datasets are unseen in training
sets. However, the performance does not have an apparent drop, thus
demonstrating the speaker generalizability of the models. As for the
environment mismatch, we refer to the reverberation and noises we may
encounter in the reality. To deal with the issue, two datasets have been
presented: WHAM! [22] and WHAMR! [23], which are the noisy and reverberant
extensions of WSJ0-2mix, respectively. Furthermore, the authors of [24]
presented a realistic dataset that is recorded by multiple speakers in the
same reverberant chamber. In [25], the authors presented a new task for
separating speech, music, and sound effects.

In particular, content and channel mismatch are rarely investigated. Content
mismatch focuses on the contents spoken by speakers. In [26] and [27], they
believed that the larger the vocabulary presented in the training set, the
better generalizability the model possesses. Moreover, content mismatch is not
restricted by the vocabulary. It can also be extended to different languages
containing various phonemes. In [28], the GlobalPhoneMS2 dataset consisting of
22 spoken languages was presented, and the authors showed that when trained on
a multilingual dataset, the networks can improve their performance on unseen
languages as well as nearly all seen languages. However, the dataset can only
entangle the languages with the recording characteristics, which means that we
can not determine which aspect dominates the performance. Regarding channel
mismatch, it focuses on the type of microphones used in the recording. The
authors of [29] considered that near-field data is easier to separate than
far-field data, although both are recorded in the same environment. In the
pandemic era, virtual meetings become prevalent, and a wider variety of
microphones are used for recording. Therefore, channel mismatch should be
investigated in more depth to meet demand.

To delve into different aspects, we leverage several corpora and synthesize
them to create datasets of mixtures with similar statistics to WSJ0-2mix. Our
experiments are mainly divided into two categories. First, we focus on the
impact of language on some representative separation models. Second, we
control the language factors and compare them across different channels.
Details and source codes are open-sourced222https://github.com/Sinica-
SLAM/COSPRO-mix.

##  2 Datasets for Speech Separation

All the datasets used in this study (except WSJ0-2mix, which is widely used in
speech separation papers) are introduced in this section, including which task
the dataset is used for and how the dataset is used. Among these datasets,
COSPRO [30] and TAT [31] will be discussed in more detail, as these two
datasets were used as training sets in our experiments and were not used in
other speech separation papers.

###  2.1 COSPRO

Mandarin Continuous Speech Prosody Corpora (COSPRO) [30] was designed,
collected, and annotated at the Institute of Linguistics, Academia Sinica. As
the name suggests, the main feature of this dataset is continuous or fluent
speech prosody and its influence on speech production and perception. COSPRO
includes a total of approximately 130 hours of read speech and only 0.3 hours
of spontaneous narration in Mandarin Chinese. There are nine sets of speech
corpora, summarized in Table 1. Each corpus was designed to bring out
different prosodic features involved in fluent speech.

###  2.2 COSPRO-2mix

Before mixing the utterances in COSPRO, we first removed files that were
inappropriate, such as utterances containing only a few characters or carrier
sentences that differ a lot from actual conversations. Because most utterances
are much longer than those in WSJ0, we used a pre-trained speech recognizer to
segment the utterances. To this end, forced alignment was performed on all
utterances with word-level transcriptions using the Kaldi toolkit [32]. A
time-marked conversation file (CTM) containing time-aligned word
transcriptions was obtained for each utterance. Then, we trimmed each
utterance into multiple consecutive segments based on its CTM, subject to the
limit of the maximum segment length (approximately 10 seconds). Every cut
point must be at a word boundary.

After segmentation, we referred to how WSJ0-2mix was mixed to make a dataset
comparable to WSJ0-2mix. Our procedure for making COSPRO-2mix is as follows.
First, we selected 81 speakers (34 males and 47 females) from the training
set, each with more than 400 seconds of speech. In COSPRO, there is a dataset
consisting of more speakers, but some of them with fewer utterances. To avoid
overfitting in training, we only selected speakers with longer speaking times.
For the evaluation set, seven males and seven females were selected. Then, we
followed the mixing procedure of WSJ0-2mix. For each mixed utterance in
WSJ0-2mix, we constructed a corresponding mixed utterance based on the same
gender combination and volume ratio. In this way, COSPRO-2mix and WSJ0-2mix
have the same number of mixed utterances and similar statistics. COSPRO-2mix
was used in language-related experiments.

Table 1: Information about the nine corpora in COSPRO. No. | Feature |  # Speakers (M / F) |  Length (hours)
---|---|---|---
01 | phonetically-balanced | 3 / 3 | 18.63
02 | multiple-speaker | 40 / 50 | 19.48
03 | intonation-balanced | 2 / 3 | 31.16
04 | stress-pattern balanced | 1 / 1 | 0.80
05 | lexically-balanced | 1 / 1 | 35.83
06 | focus-balanced prosody groups | 1 / 1 | 7.50
07 | multiple text-type/speaking-style | 1 / 1 | 1.53
08 | prosody-unit balanced | 1 / 1 | 15.20
09 | comparable spontaneous/read | 1 / 1 | 0.70

###  2.3 English Across Taiwan (EAT) & EAT-2mix

The English Across Taiwan (EAT)
corpus333http://www.aclclp.org.tw/doc/eat_brief_e.pdf contains microphone and
telephone recordings of college students in Taiwan. We selected English
utterances and English/Mandarin code-mixing utterances from the microphone
recordings to avoid the channel mismatch as much as possible and created an
English test set ($\textnormal{EAT-2mix}_{\textnormal{Eng}}$) and an
English/Mandarin code-mixing test set
($\textnormal{EAT-2mix}_{\textnormal{E\&M}}$) in the same way as COSPRO-2mix.
Both test sets contain the same speakers and similar utterance lengths. With
this dataset, we can compare performance within the same channel, which means
that the performance gap is mainly due to differences in content. The two test
sets were used in the first part of our experiment.

###  2.4 Taiwanese Across Taiwan (TAT)

The Taiwanese Across Taiwan (TAT) corpus [31] was originally intended for
Taiwanese speech recognition. Taiwanese, also known as Southern Min, is a
common dialect in Taiwan and belongs to the same language family as Mandarin.
The 200 speakers contributing to the corpus include men, women, and children
of all ages across Taiwan, with content ranging from written articles to
everyday conversations. Most notably, the corpus was recorded simultaneously
on six channels, including different microphones and smartphones. This feature
helps us investigate the impacts of different channels. The way of
synthesizing the datasets of mixed utterances is described below.

###  2.5 TAT-2mix

The six channels in TAT include: one close-talk (Audio-Technica AT2020), one
distant X-Y stereo microphone (ZOOM XYH-6 stereo microphone, containing left
and right channels), one lavalier (Superlux WO518+PS418D), iOS devices
(including iPhones, iPads, and iPods), and Android phones (produced by ASUS
and Samsung). According to these six channels, we created six datasets
respectively: $\textnormal{TAT-2mix}_{\textnormal{condenser}}$,
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-X}}$,
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-Y}}$,
$\textnormal{TAT-2mix}_{\textnormal{lavalier}}$,
$\textnormal{TAT-2mix}_{\textnormal{iOS}}$, and
$\textnormal{TAT-2mix}_{\textnormal{Android}}$.

Before mixing the utterances in TAT, we segmented the utterances using the
same method as in COSPRO-2mix to trim the noise in the front and tail of the
audio. Since the original length was already appropriate for speech separation
tasks, we did not cut the recording into shorter utterances.

After segmentation, the mixing procedure is similar to that of WSJ0-2mix. We
randomly selected 101 speakers (49 males and 52 females) for training and
validation. On the other hand, 18 speakers (11 males and 7 females) were
selected for testing. The number of speakers and the gender distribution are
exactly the same as WSJ0-2mix. Then, we followed the mixing procedure of
WSJ0-2mix to assure that TAT-2mix and WSJ0-2mix have the same number of mixed
utterances and similar statistics. All datasets are comprised of the same
mixtures of utterances but from different channels.

##  3 Speech Separation Models

In our experiments, we evaluated three representative speech separation
models, namely, Conv-TasNet [33], DPRNN [7], and DPT-Net [8]. These models
have been implemented in the Asteroid toolkit [34] so that it is easy to
compare the three models in the same coding environment. All three models are
based on TasNet consisting of an encoder, a separator, and a decoder. The
encoder is mainly a 1D convolutional layer, which transforms a waveform into
an embedding space. On the other hand, the decoder is a 1D transposed
convolutional layer, which transforms the embedding back to the waveform. The
main difference between the three models is the structure of the separator.

In Conv-TasNet, the separator is a fully-convolutional separation module,
which is composed of stacked 1D dilated convolutional blocks with increasing
dilation factors. The dilation factors increase exponentially to ensure that a
temporal context window is sufficiently large to take advantage of the long-
range dependency of the speech signal. In DPRNN, the separator splits the
output of the encoder into chunks with or without overlap, concatenates them
to form a 3D tensor, and then passes it to a dual-path BiLSTM module for
processing in two different dimensions: the chunk size and the number of
chunks. The two paths can help RNN see the information around and far away
from the current time frame to achieve better performance. As for DPT-Net, the
idea is similar to DPRNN. However, DPT-Net replaces the BiLSTM module with a
Transformer encoder, and replaces the first fully connected layer with an RNN
in the feed-forward network to learn the order information. It has shown the
best performance among these three models.

In our experiments, all model configurations followed the original
configuration implemented in Asteroid, except that we applied six dual-path
Transformer blocks in DPT-Net, which conforms to the configuration in the
original paper. We used the same loss function in terms of the scale-invariant
signal-to-distortion ratio (SI-SNR). $\text{SI-SNR}(\mathbf{x})$ is defined by

$10\log_{10}\frac{\langle\mathbf{\tilde{x}},\mathbf{\tilde{x}}\rangle}{\langle\mathbf{e},\mathbf{e}\rangle}\quad(\mathbf{\tilde{x}}=\frac{\langle\mathbf{x},\mathbf{s}\rangle}{\langle\mathbf{x},\mathbf{x}\rangle}\mathbf{x}\quad\textrm{and}\quad\mathbf{e}=\mathbf{\tilde{x}}-\mathbf{s}),$
(1)

where $\mathbf{x}$ denotes the output of the network and should ideally be
equal to the clean source $\mathbf{s}$. We only trained 100 epochs for each
model to speed up the progress of the experiments.

##  4 Experiments

###  4.1 Impact of Language

We trained the representative models on WSJ0-2mix and COSPRO-2mix and
evaluated them on several datasets. The results are shown in Table 2, where
the columns refer to the training sets and the separation models; the rows
refer to the test sets. The metric used in this study is SI-SNRi (dB), the
improvement of SI-SNR (cf. Eq. 1), defined by

$\textrm{SI-SNRi}=\textrm{SI-SNR}(\mathbf{x})-\textrm{SI-SNR}(\mathbf{m}),$
(2)

where $\mathbf{m}$ and $\mathbf{x}$ denote the input mixture and output of the
network, respectively.

First, comparing each column in Table 2 under the two training sets, the
performance rankings are in line with past results: DPT-Net outperforms DPRNN,
and DPRNN outperforms Conv-TasNet. Then, we examine the performance in each
row. Looking at the first three rows (the group of WSJ0 & COSPRO), we can see
that the results of WSJ0-2mix and COSPRO-2mix are comparable, under both in-
domain (training/testing from the same dataset) and cross-domain
(training/testing from the different dataset) testing conditions. In
WSJ0$\times$COSPRO, each utterance is a mixture of a Mandarin utterance from
COSPRO and an English utterance from WSJ0. Due to different languages and
different recording conditions, WSJ0$\times$COSPRO is easier to separate than
WSJ0-2mix and COSPRO-2mix.

The results in the fourth and fifth rows of Table 2 (the group of EAT) show
the impact of language more explicitly since the only difference between the
two test sets is the language. We observe that the performance on EAT is much
worse than those on COSPRO-2mix and WSJ0-2mix, because the clean waveform
contains some microphone noise, which may degrade SI-SNR, and the recording
quality is not high, which causes many pop and explosive sounds. Compared to
the performance in other rows, it is obvious that the performance gap between
$\textnormal{EAT-2mix}_{\textnormal{E\&M}}$ and
$\textnormal{EAT-2mix}_{\textnormal{Eng}}$ is almost negligible. That is,
although language has an impact on speech separation, the impact is so small
to neglect, complying with the results in [35].

The results in the remaining rows of Table 2 (the group of TAT) reveal the
impact of channels since the only difference between these test sets is the
channel. There is a large difference between the performance of each row,
suggesting that channel variability seems to dominate the performance. We also
found that when the test set is more difficult, such as
$\textnormal{TAT-2mix}_{\textnormal{iOS}}$, the performance difference between
models is less pronounced. To investigate the impact of channel variability
and the relationship between channels, we performed the next set of
experiments.

Table 2: Performance (SI-SNRi in dB) of various separation models (ConvT: Conv-TasNet, DPR: DPRNN, DPT: DPT-Net) tested on various datasets and trained on WSJ0-2mix and COSPRO-2mix. WSJ0$\times$COSPRO is a test set, where each utterance is a mixture of a Mandarin utterance from COSPRO and an English utterance from WSJ0. Test Train | WSJ0-2mix | COSPRO-2mix
---|---|---
ConvT | DPR | DPT | ConvT | DPR | DPT
WSJ0-2mix | 15.84 | 16.65 | 17.11 | 12.49 | 13.15 | 13.81
COSRPO-2mix | 12.17 | 13.19 | 14.78 | 15.91 | 16.92 | 17.23
WSJ0$\times$COSPRO | 16.09 | 16.95 | 18.14 | 16.33 | 16.88 | 17.41
$\textnormal{EAT-2mix}_{\textnormal{E\&M}}$ | 3.02 | 3.93 | 4.77 | 2.68 | 3.16 | 3.20
$\textnormal{EAT-2mix}_{\textnormal{Eng}}$ | 2.84 | 3.90 | 4.46 | 2.42 | 2.83 | 2.89
$\textnormal{TAT-2mix}_{\textnormal{Android}}$ | 2.25 | 3.15 | 4.35 | 3.94 | 4.08 | 4.70
$\textnormal{TAT-2mix}_{\textnormal{iOS}}$ | -0.44 | 0.68 | 0.78 | 0.94 | 0.87 | 1.86
$\textnormal{TAT-2mix}_{\textnormal{condenser}}$ | 6.99 | 7.42 | 9.23 | 2.32 | 3.21 | 3.44
$\textnormal{TAT-2mix}_{\textnormal{lavalier}}$ | 5.39 | 6.55 | 7.57 | 5.91 | 3.75 | 5.96
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-X}}$ | 0.64 | 1.45 | 1.51 | -0.66 | 0.51 | 0.75
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-Y}}$ | -0.10 | 0.61 | 0.99 | -0.60 | -0.05 | 0.68

###  4.2 Impact of Channel Variability

In this set of experiments, we trained a DPT-Net on each TAT-2mix recorded by
a specific channel and evaluated on all TAT-2mix separately. The results are
shown in Table 3. The columns stand for the training sets, and the rows
present the test sets.

Overall, we can again see the severity of the impact of channel variability.
Although all datasets are composed of the same speech content and speakers,
the performance varies. In addition, the models perform best when the test
data is in-domain (training and testing are from the same channel), in line
with previous studies that emphasized the importance of in-domain training
data [26, 27, 36].

If we examine the SI-SNRi values in Table 3, condenser and lavalier are the
easiest channels, and XYH-6-X and XYH-6-Y are the hardest. The result is
reasonable because condenser is closest to the speaker (5–10cm) with a pop
filter, and lavalier is also very close to the speaker’s mouth. Android and
iOS are 15–20cm from the speaker. The distance between XYH-6 and the speaker
is about one meter. As reported in [29], speech separation is more difficult
for distant microphones.

By comparing the columns of XYH-6-X and XYH-6-Y, it is clear that the
performance between the left and right channels is almost equivalent, but
XYH-6-X has a slight advantage. When comparing the rows of XYH-6-X and
XYH-6-Y, XYH-6-Y is slightly harder than XYH-6-X. As expected, the left and
right channels are somewhat similar. As for the two smartphone systems,
whether comparing the rows of Android and iOS to see how other models perform
on the two test sets, or comparing the columns of Android and iOS to examine
the generalizability of the two models, the results show that the
characteristics of the iOS microphones and the Android microphones differ
considerably. However, for these two smartphone systems, their counterpart
model still performs the best among all cross-domain models, e.g., the iOS
model gives an SI-SNRi of 7.60 when tested on the Android test set, which is
better than the SI-SNRi values of other cross-domain models.

Furthermore, the model trained on
$\textnormal{TAT-2mix}_{\textnormal{Android}}$ achieves the highest average
performance on all test sets, indicating its robustness. One possible reason
is that Android contains smartphones from different companies, with different
frequency responses for their built-in microphones. It is also worth
mentioning that the model trained on
$\textnormal{TAT-2mix}_{\textnormal{condenser}}$ does not generalize well to
other channels. However, condenser microphones are the most popular devices
for recording corpora, which poses a problem in real-world scenarios. In the
next subsection, we proposed a solution that effectively selects additional
training data to improve performance on the in-the-wild test data.

Table 3: Cross-dataset evaluation (SI-SNRi in dB) of TAT-2mix on various channels. The model for evaluation is DPT-Net. Test Train |  And- roid | iOS |  cond- enser |  lav- alier |  XYH- 6-X |  XYH- 6-Y
---|---|---|---|---|---|---
Android | 10.89 | 7.60 | 5.32 | 6.98 | 6.19 | 5.59
iOS | 8.42 | 10.69 | 1.33 | 3.97 | 4.10 | 4.07
condenser | 10.47 | 4.53 | 14.07 | 11.97 | 9.95 | 9.26
lavalier | 11.62 | 9.57 | 9.34 | 14.10 | 5.62 | 4.24
XYH-6-X | 5.87 | 3.10 | 6.83 | 3.10 | 8.82 | 8.03
XYH-6-Y | 4.65 | 2.51 | 5.44 | 1.86 | 7.34 | 8.15
Average | 8.65 | 6.33 | 7.06 | 7.00 | 7.00 | 6.56

###  4.3 Data Selection by Pairwise PCA

We performed pairwise principal component analysis (PCA) to visualize the
relationship between channels. First, we encoded the results of individual
test sets (i.e., the rows in Table 3) into row vectors $\\{\mathbf{v}_{i}\\}$.
Then, we constructed the covariance matrix $\Lambda$ by

$\Lambda=\frac{1}{2N}\sum_{i=1}^{N}\sum_{j=1}^{N}\left(\mathbf{v}_{i}-\mathbf{v}_{j}\right)^{\intercal}\left(\mathbf{v}_{i}-\mathbf{v}_{j}\right),$
(3)

where $N$ denotes the number of vectors. We performed eigen-decomposition on
$\Lambda$ to find the two eigenvectors with the largest eigenvalues as the two
directions of the new feature space. Then, $\\{\mathbf{v}_{i}\\}$ was
projected onto the new 2D feature space.

Fig. 1 shows the scatterness of the projected points. From the figure, we can
see that XYH-6-X and XYH-6-Y are close. In order to evaluate the usefulness of
distance, we conducted two experiments. We found that if the training set
contains data close to the test set, the performance improves significantly.
For example, as shown by the solid arrows, when the training set contains
$\textnormal{TAT-2mix}_{\textnormal{condenser}}$ and
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-X}}$, the performance on
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-Y}}$ can be improved by at least 11%
compared to the performance of the model trained on
$\textnormal{TAT-2mix}_{\textnormal{XYH-6-X}}$ (8.17 vs 7.34). In contrast, if
both training sets are far away from the test data, even doubling the size of
the training data may not improve the performance much. For example, the
dotted arrows show that the performance on
$\textnormal{TAT-2mix}_{\textnormal{iOS}}$ is only improved by at most 7%
comparing the model trained on $\textnormal{TAT-2mix}_{\textnormal{XYH-6-X}}$
& $\textnormal{TAT-2mix}_{\textnormal{lavalier}}$ and the model trained on
$\textnormal{TAT-2mix}_{\textnormal{lavalier}}$ (4.23 vs 3.97).

The above results suggest that additional training data should be chosen
wisely to effectively enhance the robustness of the model. Therefore, we
propose the following solution. When encountering new data recorded by an
unknown channel, we can first use these six models to perform speech
separation on the data. Then, the test results are projected onto the 2D
space, so that we can observe the distances between the new dataset and other
points. Finally, we select existing datasets close to the new point and add
them to the training data to improve the performance of the model on the new
data.

Figure 1: 2D projection of the TAT-2mix test sets using pairwise PCA. D
denotes the distance between two points, and S denotes the SI-SNRi (dB) by the
model trained on the dataset at the arrow tail and tested on the dataset at
the arrow tip.

##  5 Conclusions and Future Work

In this paper, we have investigated the impact of language and channel
variability on speech separation by designing proportionate datasets and
comparing cross-domain performance. In the first set of experiments, we
created COSPRO-2mix and found that the impact of language is negligibly small
compared to channel variability. In the second set of experiments, we created
TAT-2mix from TAT recorded by different channels to observe the relationship
between the channels. Moreover, we proposed a solution to address channel
mismatch by evaluating projection and effectively selecting training data. In
the future, we hope to prove our method on in-the-wild test data and extend it
to domain generalization, where evaluation data are unknown.

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# Quantum Crooks fluctuation theorem and quantum Jarzynski equality in the
presence of a reservoir

H. T. Quan  Theoretical Division, MS B213, Los Alamos National Laboratory, Los
Alamos, NM, 87545, U.S.A.  H. Dong  Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing, 100190, P.R. China

###### Abstract

We consider the quantum mechanical generalization of Crooks Fluctuation
Theorem and Jarzynski Equality for an open quantum system. The explicit
expression for microscopic work for an arbitrary prescribed protocol is
obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum
Jarzynski Equality and their classical counterparts are clarified. Numerical
simulations based on a two-level toy model are used to demonstrate the
validity of the quantum version of the two theorems beyond linear response
theory regime.

###### pacs:

05.70.Ln, 05.40.-a

##  I Introduction:

Nonequilibrium thermodynamics has been an intriguing research subject for more
than one hundred years nonequilibrium . Yet our understanding about
nonequilibrium thermodynamic phenomena, especially about those far-from-
equilibrium regime (beyond the linear response regime), remains very limited.
In the past fifteen years, there are several significant breakthroughs in this
field, such as Evans-Searls Fluctuation Theorem evans , Jarzynski Equality
(JE) JE , and Crooks Fluctuation Theorem (Crooks FT) crooks . These new
theorems not only have important applications in nanotechnology and
biophysics, such as extracting equilibrium information from nonequilibrium
measurements, but also shed new light on some fundamental problems, such as
improving our understanding of how the thermodynamic reversibility arise from
the underlying time reversible dynamics.

Since the seminal work by Jarzynski and Crooks a dozen of years ago, the
studies of nonequilibrium thermodynamics in small system attract numerous
attention followup , and the validity and universality of these two theorems
in classical systems has been extensively studied not only by numerical
studies classicalnumerical , but also by experimental exploration
classicalexperiment  in single RNA molecules, and for both deterministic and
stochastic processes. For quantum systems, possible quantum extension of
Crooks FT and JE have also been reported quantumJE . Nevertheless, we notice
that almost all of these reports about quantum extension of Crooks FT focus on
isolated quantum systems isolatedcrooks , and the explicit expression of
microscopic work, and their distributions in the presence of a heat bath are
not extensively studied. In addition, the relationship between classical and
quantum Crooks FT is not addressed adequately so far. As a result, the
experimental studies of quantum Crooks FT and JE are not explored (an
exception is the experimental scheme of quantum JE of isolated system based on
trapped ions schmidt ).

Figure 1: (Color Online) Trajectories of a quantum system in a nonequilibrium
process. Similar to Ref. crooks  each step (from $t_{n}$ to $t_{n+1}$) is
divided into two substeps: the controlling substep of time $\tau_{Q}^{n}$, in
which the energy spectrum (black solid line) of the system change with time,
and the relaxation substep of time $\tau_{R}^{n}$ in which the energy spectrum
(black dashed line) remains unchanged. In the controlling substep (solid line)
work is done, but there is no heat exchange; While in the relaxation substep,
there is heat exchange between the system and the heat bath, but there is no
work done. Blue trajectory corresponds to fast controlling protocol, during
which there are usually interstate excitations in the controlling substep. Red
trajectory corresponds to slow (quantum adiabatic) controlling protocol, and
the system remains in its instantaneous eigenstate in the controlling substep.
Red trajectory is the counterpart of classical case.

In this paper, we will give a detailed proof of the validity of quantum Crooks
FT and quantum JE for an open quantum system based on the explicit expression
of microscopic work and their corresponding probability distributions for an
arbitrary prescribed controlling protocol. We also clarify the relation
between quantum Crooks FT, quantum JE and their classical counterparts. In the
last part of the paper, the studies based on a two-level system are given as
an illustration to demonstrate our central idea.

##  II Notations and assumptions:

Crooks FT crooks  is firstly derived in classical systems in a microscopically
reversible Markovian stochastic process. In the proof of a classical Crooks
FT, a key technique is to separate work steps from heat steps. In the
following discussion of quantum extension of Crooks FT and JE, we will employ
the same technique as that used in Ref. crooks  to separate the controlling
process into two substeps: controlling substep and relaxation substep (see
Fig. 1). The controlling substep proceeds so quickly in comparison with the
thermalization process of the system that we can ignore the influence of the
heat bath during the controlling substep. So there is only work done in the
controlling substep. In the relaxation substep, on the other hand, there is
only heat exchange.

Having clarified the main strategy (separating work substep from heat
substep), let us come to the details of the notations and assumptions. We
employ the same notations and assumptions as that in Ref. crooks  to prove the
quantum Crooks FT. In Ref. crooks  the author assumes discrete time and
discrete phase space. Here, the discrete energy spectrum in a quantum system
in place of the discrete phase space of a classical system arises naturally.
We also assume discrete time $t_{0}$, $t_{1}$, $t_{2}$, $t_{3}$, $\cdots$,
$t_{N}$ for the quantum system (see Fig. 1). The parameter $\lambda(t)$ is
controlled according to an arbitrary prescribed protocol
$\lambda(t_{0})=\lambda_{A}$, $\lambda(t_{1})=\lambda_{1}$,
$\lambda(t_{2})=\lambda_{2}$, $\cdots$, $\lambda(t_{N})=\lambda_{B}$, where
$A$ and $B$ depict the initial and final points of the process. Every step
$t_{n}\rightarrow t_{n+1}$ is seperated into controlling substep of time
$\tau_{Q}^{i}$ and relaxation substep of time time $\tau_{R}^{i}$,
$t_{i+1}=t_{i}+\tau_{Q}^{i}+\tau_{R}^{i}$ (see Fig. 1). If we use
$\left|i_{n},\lambda_{m}\right\rangle$ and $E(i_{n},\lambda_{m})$ to depict
the $i_{n}$-th instantaneous eigenstate and eigenenergy of the system
Hamiltonian $H(\lambda_{m})$, we can rewrite the trajectory $A\rightarrow B$
of Ref. crooks  in the following way

$\begin{split}\left|i_{0},\lambda_{0}\right\rangle\rightarrow\left|i_{0},\lambda_{1}\right\rangle\underrightarrow{\lambda_{1}}\left|i_{1},\lambda_{1}\right\rangle\rightarrow\left|i_{1},\lambda_{2}\right\rangle\underrightarrow{\lambda_{2}}\left|i_{2},\lambda_{2}\right\rangle\\\
\rightarrow\cdots\rightarrow\left|i_{N-1},\lambda_{N-1}\right\rangle\rightarrow\left|i_{N-1},\lambda_{N}\right\rangle\underrightarrow{\lambda_{N}}\left|i_{N},\lambda_{N}\right\rangle.\end{split}$
(1)

In the classical case, the system remains in its $i_{n}$-th state of the
discrete phase space during the controlling substep. Analogously, in quantum
systems, this process corresponds to the quantum adiabatic regime, i.e., the
system remains in its $i_{n}$-th eigenstate of the instantaneous Hamiltonian
when we control the parameter $\lambda(t)$ of the Hamiltonian $H[\lambda(t)]$
so slowly that the quantum adiabatic conditions are satisfied, and the above
trajectories (1) can be achieved (red trajectory of Fig. 1). However, if we
control the parameter of the Hamiltonian very quickly in the controlling
substep, and then the quantum adiabatic conditions are not satisfied, the
trajectory $A\rightarrow B$ in general should be written as (see blue
trajectory of Fig. 1)

$\begin{split}\left|i_{0},\lambda_{0}\right\rangle\rightarrow\left|i_{0}^{\prime},\lambda_{1}\right\rangle\underrightarrow{\lambda_{1}}\left|i_{1},\lambda_{1}\right\rangle\rightarrow\left|i_{1}^{\prime},\lambda_{2}\right\rangle\underrightarrow{\lambda_{2}}\left|i_{2},\lambda_{2}\right\rangle\\\
\rightarrow\cdots\rightarrow\left|i_{N-1},\lambda_{N}\right\rangle\rightarrow\left|i_{N-1}^{\prime},\lambda_{N}\right\rangle\underrightarrow{\lambda_{N}}\left|i_{N},\lambda_{N}\right\rangle.\end{split}$
(2)

The main difference of the above two kinds of trajectories (1) and (2) is that
after the controlling substep the system may not be in the same eigenstate as
that before the controlling, i.e., $i_{n}\neq i^{\prime}_{n}$. The internal
excitation
$\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle$
is due to randomness caused by quantum non-adiabatic transition and has no
classical counterpart. Actually this difference of trajectories (1) and (2)
highlights the main difference between the quantum and classical Crooks FT.
For a quantum system, the microscopic work done in every controlling substep
is equal to the difference of the energy before and after the controlling
substep: $W_{n}=E(i_{n}^{\prime},\lambda_{n+1})-E(i_{n},\lambda_{n})$, and the
heat exchanged with the heat bath is equal to the difference of the energy of
the system before and after the relaxation substep
$Q_{n}=E(i_{n},\lambda_{n})-E(i_{n-1}^{\prime},\lambda_{n})$. For the
trajectory (2) as a whole, we must make $2N$ times quantum measurements to
confirm the microscopic work done and heat exchanged with the heat bath.
Similar to the classical case, the total work $W$ performed on the system, and
the total heat $Q$ exchanged with the heat bath are given by the summation of
work and heat in every step,
$W=\sum_{n=0}^{N-1}\left[E(i_{n}^{\prime},\lambda_{n+1})-E(i_{n},\lambda_{n})\right],Q=\sum_{n=0}^{N}\left[E(i_{n},\lambda_{n})-E(i_{n-1}^{\prime},\lambda_{n})\right]$,
and the total change in energy is $\Delta
E=Q+W=E(i_{N},\lambda_{N})-E(i_{0},\lambda_{0})$. Note that the work and heat
depend on the trajectory, but the energy change depends only on the initial
and final energy, and does not depend on the trajectory.

Similar to the classical case crooks  we assume the trajectory (2) to be
Markovian, and the forward process starts from the thermal equilibrium
distribution $P(\left|i_{0},\lambda_{0}\right\rangle)=e^{-\beta
E(i_{0},\lambda_{0})}/(\sum_{i}e^{-\beta E(i,\lambda_{0})})$. The joint
probability for a given trajectory (2) can be expressed as

$\begin{split}P_{F}(A\rightarrow
B)=&P(\left|i_{0},\lambda_{0}\right\rangle)\prod_{n=0}^{N-1}P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)\\\
&\times
P_{F}(\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle\rightarrow\left|i_{n+1},\lambda_{n+1}\right\rangle).\end{split}$
(3)

It can be seen that the above probability (3) of a trajectory for a quantum
case is different from the classical case crooks  by the extra term
$P(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)$
arising from randomness due to quantum non-adiabatic transition. When the
quantum adiabatic conditions are satisfied,
$P(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=\delta_{i_{n},i_{n}^{\prime}}$,
we regain the probability of a trajectory in classical systems crooks . We
will see later that the quantum Crooks FT and quantum JE in the quantum
adiabatic regime are the counterpart of classical Crooks FT and classical JE.

To prove the quantum Crooks FT, we also need to consider the time-reversed
trajectory reverse  of the original trajectory (2). The time-reversed
trajectory corresponding to the forward time trajectory $A\leftarrow B$ in Eq.
(2) can be written as

$\begin{split}\Theta\left|i_{0},\lambda_{0}\right\rangle\leftarrow\Theta\left|i_{0}^{\prime},\lambda_{1}\right\rangle\underleftarrow{\lambda_{1}}\Theta\left|i_{1},\lambda_{1}\right\rangle\leftarrow\Theta\left|i_{1}^{\prime},\lambda_{2}\right\rangle\underleftarrow{\lambda_{2}}\\\
\cdots\leftarrow\Theta\left|i_{N-1},\lambda_{N}\right\rangle\leftarrow\Theta\left|i_{N-1}^{\prime},\lambda_{N}\right\rangle\underleftarrow{\lambda_{N}}\Theta\left|i_{N},\lambda_{N}\right\rangle\end{split}$
(4)

where
$\Theta\left|i_{n},\lambda_{n}\right\rangle=\left|i_{n},\lambda_{n}\right\rangle^{\ast}$
is the microscopic state in the time-reversed trajectory sakurai . The
sequence in which states are visited is reversed, as is the order in which
$\lambda$ is changed. The work done $W$, the heat exchange $Q$ with the heat
bath, the change of the internal energy $\Delta E$, and the change of free
energy $\Delta F$ for the reversed time direction are the negative value of
that of the forward time trajectory. The joint probability for time reversed
trajectory $A\leftarrow B$ can be expressed as

$\begin{split}P_{R}(A\leftarrow
B)=&\prod_{n=0}^{N-1}P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)\\\
&\times
P_{R}(\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle\leftarrow\Theta\left|i_{n+1},\lambda_{n+1}\right\rangle)\\\
&\times P(\Theta\left|i_{N},\lambda_{N}\right\rangle),\end{split}$ (5)

where $P(\Theta\left|i_{N},\lambda_{N}\right\rangle)=e^{-\beta
E(i_{N},\lambda_{N})}/\sum_{i}e^{-\beta E(i,\lambda_{N})})$ is the initial
thermal distribution for the time-reversed trajectory. Also there is en extra
term
$P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)$
arising due to the randomness caused by quantum non-adiabatic transition in
comparison with the classical case.

##  III Proof of quantum crooks FT and quantum JE

As we have mentioned before, in a trajectory every step consists of two
substeps, the controlling substep (not necessarily to be quantum adiabatic)
and the relaxation substep. The relaxation substeps are assumed to be
microscopically reversible, and therefore obey the detailed balance crooks ;
chandler  for all fixed value of the external control parameter $\lambda$

$\frac{P_{F}(\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\rightarrow\left|i_{n},\lambda_{n}\right\rangle)}{P_{R}(\Theta\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n},\lambda_{n}\right\rangle)}=\frac{e^{-\beta
E(i_{n},\lambda_{n})}}{e^{-\beta E(i_{n-1}^{\prime},\lambda_{n})}}.$ (6)

To compare the ratio of the probabilities of forward (3) and time-reversed (5)
trajectories, we also need to know the ratio of the probabilities in the
controlling substep. In the following we will focus on the study of
controlling substep and its time reversal. As we mentioned before, during the
controlling substep, the system can be regarded as an isolated quantum system
and the evolution is completely determined by a time-dependent Hamiltonian
$H[\lambda(t)]$. For example, when the controlling parameter $\lambda$ is
changed from $\lambda_{n}$ to $\lambda_{n+1}$, the probability of the
transition from a microscopic state $\left|i_{n},\lambda_{n}\right\rangle$ to
another microscopic state $\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle$
can be expressed as

$P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=|\left\langle
i_{n}^{\prime},\lambda_{n+1}\right|U\left|i_{n},\lambda_{n}\right\rangle|^{2}$
(7)

where $U=\mathrm{T}\exp\\{-i\int_{t_{0}}^{t_{1}}H[\lambda(t)]dt\\}$ is the
unitary matrix describing the evolution of the isolated quantum system in the
controlling substep, and $\mathrm{T}$ is the time-ordered operator. Similarly,
in the time-reversed trajectory the excitation probability from the
microscopic state $\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle$ to
another microscopic state $\Theta\left|i_{n},\lambda_{n}\right\rangle$ in the
time reversed trajectory can be expressed as theta

$\begin{split}P_{R}(\Theta&\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)\\\
&=|\left(\left\langle
i_{n},\lambda_{n}\right|\overleftarrow{\Theta}\right)\Theta
U\overleftarrow{\Theta}\left(\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle\right)|^{2},\end{split}$
(8)

where $\Theta
U\overleftarrow{\Theta}=\mathrm{T}\exp\\{-i\int_{t_{0}}^{t_{1}}H[\lambda(t_{0}+t_{1}-t)]dt\\}=(U^{\dagger})^{\ast}=U^{T}$
is the time-reversed unitary matrix. Because of the property of the time-
reversed transformation
$\Theta\left|i_{n},\lambda_{n}\right\rangle=\left|i_{n},\lambda_{n}\right\rangle^{\ast}$,
and the property of the Hermitian conjugate matrix,

$(\left\langle
i_{n},\lambda_{n}\right|)^{\ast}U^{T}(\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)^{\ast}\equiv\left\langle
i_{n}^{\prime},\lambda_{n+1}\right|U\left|i_{n},\lambda_{n}\right\rangle$ (9)

it is not difficult to prove that

$\frac{P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)}{P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)}\equiv
1.$ (10)

Based on the above two results (6), (10) and Eqs. (3) and (5), we reproduce
the Crooks FT for a quantum mechanical system

$\frac{P_{F}(A\rightarrow B)}{P_{R}(A\leftarrow B)}=e^{\beta(W-\Delta F)}.$
(11)

From Eq. (11) we group all those trajectories with the same amount of
microscopic work, and obtain

$\frac{P_{F}(W|_{a})}{P_{R}(-W|_{-a})}=e^{\beta(a-\Delta F)}.$ (12)

Eq. (12) is the Crooks FT. Similar to the derivation in Ref. crooks , we
obtain the JE for a quantum open system straightforwardly $\left\langle
e^{-\beta W}\right\rangle=e^{-\beta\Delta F}$ from $\int P_{R}(-W|_{-a})da=1$.
Here, we would like to emphasize that though quantum generalization of Crooks
FT and JE have been reported in some previous work, the explicit consideration
of the influence of the heat bath, i.e., the explicit expression of
microscopic work in the presence of a heat bath has not been reported before.
Also the relation between quantum and classical trajectories are not addressed
clearly. Hence our quantum mechanical extensions of Crooks FT and JE are
highly nontrivial.

##  IV Illustration of quantum Crooks FT and quantum JE in a two-level system

Figure 2: (Color Online) Microscopic work distribution $P_{F}(W)$ of forward
trajectories (solid lines), and the negative reverse work distribution
$P_{R}(-W)$ of their corresponding time-reversed trajectories (dashed lines).
The probabilities have been normalized. Here we fix $\Delta(t_{0})$ and
$\Delta(t_{N})$. Different distributions represent different controlling time
(the more steps, the longer control time). The controlling steps are chosen to
be $N=5$ (red $\bullet$), $N=10$ (blue $\bigtriangleup$), $N=15$ (green
$\square$), and $N=20$ (black $\bigcirc$). It can be seen that the work
distributions for both forward and reversed trajectories are not Gaussian.
Moreover, with the decrease of the controlling speed, the fluctuation of the
distributions decreases, and the difference between the work distribution of
the forward and time-reversed trajectories becomes less obvious. The
corresponding forward and negative reverse work distribution cross at
$W=\Delta F$, and this is a direct consequence of the quantum Crooks FT. The
free energy difference $\Delta F$ ia marked by the red vertical dash-dotted
line.

Having generalized the Crooks FT and JE to quantum systems in the presence of
a heat bath. In the following, we use the studies based on a two-level system
quan08  as an illustration to demonstrate our main idea. The Hamiltonian of
the two-level system is $H=\Delta(t)\left(\sigma_{z}+1\right)/2$, where
$\Delta(t)$ is the parameter of the Hamiltonian, and $\sigma_{z}$ is Pauli
matrix. The initial and final value of the parameter are
$\Delta_{A}=\Delta(t_{0})$ and $\Delta_{B}=\Delta(t_{N})$ respectively. The
controlling scheme is the same as that in Ref. quan08 : We divide the whole
process into N even steps. Hence the parameter in the $n$th step is
$\Delta(t_{n})=\Delta(t_{0})+n\Delta$, $n=1$, $2$, $\cdots$, $N$, where
$\Delta=(\Delta_{B}-\Delta_{A})/N$ is the change of the parameter in every
step. Every step consists of two substeps: the controlling substep, in which
we change the parameter from $\Delta(t_{n})$ to
$\Delta_{n+1}=\Delta(t_{n})+\Delta$, and the relaxation substep. For
simplicity, we consider the case where the system reaches thermal equilibrium
with the heat bath in every relaxation substep. Hence, the probability for the
forward and reverse relaxation substep can be expressed as
$P_{F}(\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\rightarrow\left|i_{n},\lambda_{n}\right\rangle)=e^{-\beta
E(i_{n},\lambda_{n})}/(\sum_{i}e^{-\beta E(i,\lambda_{n})})$, and
$P_{R}(\Theta\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n},\lambda_{n}\right\rangle)=e^{-\beta
E(i_{n-1}^{\prime},\lambda_{n})}/(\sum_{i}e^{-\beta E(i,\lambda_{n})})$. Also
we assume the quantum adiabatic conditions are satisfied in every controlling
substep. That is
$P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=\delta_{i_{n},i_{n}^{\prime}}$,
and
$P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=\delta_{i_{n},i_{n}^{\prime}}$.
Based on these assumptions, the microscopic work distribution for the forward
trajectories can be obtained quan08

$P_{F}(W|_{k\Delta})=P^{F}_{e}\prod_{l=0}^{N-k-1}\frac{e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+l\Delta)}}{e^{\beta(l+1)\Delta}-1},$
(13)

where

$P^{F}_{e}=\prod_{j=1}^{N}\frac{e^{-\beta[\Delta_{A}+(j-1)\Delta]}}{1+e^{-\beta[\Delta_{A}+(j-1)\Delta]}},k=0,1,2,\cdots,N.$
(14)

Similarly, the microscopic work distribution for the time-reversed trajectory
can be expressed as

$P_{R}(-W|_{-k\Delta})=P^{R}_{e}\prod_{l=0}^{N-k-1}\frac{e^{\beta\Delta}[e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+l\Delta)}]}{e^{\beta(l+1)\Delta}-1},$
(15)

where

$P^{R}_{e}=\prod_{j=1}^{N}\frac{e^{-\beta[\Delta_{B}-(j-1)\Delta]}}{1+e^{-\beta[\Delta_{B}-(j-1)\Delta]}},k=0,1,2,\cdots,N.$
(16)

We plot the above distributions (13) and (14) of microscopic work in Fig. 2.
Here the probability distribution in the excited state are
$P_{e}(\Delta_{A})=e^{-\beta\Delta_{A}}/(1+e^{-\beta\Delta_{A}})=1/3$, and
$P_{e}(\Delta_{B})=e^{-\beta\Delta_{B}}/(1+e^{-\beta\Delta_{B}})=1/5$. The
free energy difference is $\Delta
F_{AB}=\left[\ln(1+1/2)-\ln(1+1/4)\right]k_{B}T\approx 0.263\ln 2k_{B}T$. It
can be seen (see Fig. 2) that the corresponding forward and negative reverse
work distributions cross at $W=\Delta F$, no matter what the controlling
protocol is, and this result is a direct consequence of Crooks FT. It should
be pointed out that the work distributions (13) and (15) are non-Gaussian
quan08 . Hence, the processes discussed here are beyond the linear response
regime. Yet we will see both Crooks FT and JE holds. We also plot the
logarithm of the ratio of the forward and negative reverse work distribution
(See Fig. 3(a)). It can be seen that all data collapse onto the same straight
line. In addition, the slope of the line is equal to unit, and the line cross
the horizontal axis at $W=0.263\ln 2k_{B}T=\Delta F_{AB}$. Thus our numerical
simulation confirms the validity of quantum Crooks FT when the process is
beyond the linear response regime. We also plot the logarithm of the exponent
averaged work $\ln\left\langle e^{-\beta W}\right\rangle$ and averaged work
$\left\langle W\right\rangle$ of the forward process (see Fig. 3(b)) to test
the validity of quantum JE. It can be seen that the averaged work is greater
than the free energy difference $\left\langle W\right\rangle\geqslant\Delta
F$, while the logarithm of the exponent averaged work is identical to the
difference of the free energy $\ln\left\langle e^{-\beta
W}\right\rangle\equiv\Delta F\approx 0.1823k_{B}T$ no matter what the
controlling protocol is. Hence, Fig. 3(b) verifies quantum JE when the process
is beyond the linear response regime.

Figure 3: (Color Online) (a) The logarithm of the probabilities of forward and
time-reversed trajectories as a function of work. It can be seen that all data
of different work and different control protocols ($N=5$ (red $\bullet$),
$N=10$ (blue $\bigtriangleup$), $N=15$ (green $\square$), and $N=20$ (black
$\bigcirc$) ) collapse onto the same straight line. The slop of the line is
equal to unity, and the line cross the horizontal axes at $W=\Delta F$. Thus
the numerical result verifies the quantum Crooks FT
$\ln\left[P_{F}(W|_{a})/P_{R}(-W|_{-a})\right]=\beta(a-\Delta F)$. (b) The
averaged work VS. the logarithm of averaged exponent work for different
control protocols. It can be seen that the averaged work $\left\langle
W\right\rangle$ (red $\bigcirc$) is always greater than the difference of free
energy $\Delta F_{AB}$ and differ from one control protocol to another, while
the logarithm of the exponentially averaged work $\ln\left\langle\exp[-\beta
W]\right\rangle$ (blue $\square$) is always equivalent to the difference of
free energy irrespective of the control protocols. Thus the numerical result
verifies the JE $\ln{\left\langle\exp[-\beta W]\right\rangle}\equiv\Delta F$.

##  V Conclusion and remarks

In this paper, we explicitly consider the quantum Crooks FT and quantum JE in
the presence of an external heat bath. Our proof includes the proof of
classical Crooks FT as a special case. When the quantum adiabatic conditions
are satisfied, we reproduce the result of Crooks FT and JE for classical
systems. Our work indicates that in quantum systems, the probabilities (Eqs.
(3) and (5)) comes from the quantum non-adiabatic transition and statistical
mechanical randomness, while in classical system, the randomness only comes
from the later case. We use the two-level system as an illustration to
demonstrate the validity of quantum Crooks FT and quantum JE beyond the linear
response regime.

Before concluding the paper, we would like to mention the following points.
First, though the quantum non-adiabatic transition is introduced into the
controlling substep, this substep is time reversal symmetric. I. e., all the
time asymmetry is due the relaxation substep (statistical mechanical
randomness), rather than the controlling substep (quantum non-adiabatic
transition). This is the same as the classical case. Second, when we change
the Hamiltonian slowly, we reproduce the proof of Crooks for classical
systems. In this sense, we say that our proof includes the classical Crooks FT
and classical JE as a special case. Third, for classical system, the Crooks FT
and JE have been experimentally verified classicalexperiment . However, for a
quantum mechanical system, the experimental exploration on Crooks FT and JE
has not been reported (an exception is schmidt ). This perhaps is mainly due
to the fact that microscopic work in a quantum mechanical system is not a well
defined observable hanggi . There is no well defined pressure or force for a
quantum system quan0811 . Hence, we cannot follow the way that we do in
classical system to measure the force and make the integral of the force by
the extension. On the contrary, we will have to introduce quantum measurement
processes to confirm the initial and final energy of the system and then
calculate the microscopic work done from the difference of the initial and
final energy difference mukemal . Fourth, though the numerical simulations
consider only the special cases: 1) the system reach thermal equilibrium with
the heat bath in every relaxation substep, and 2) the quantum adiabatic
conditions are satisfied in every controlling substep, the quantum Crooks FT
and quantum JE are not constrained in these special cases. Finally, our
numerical simulations based on a two-level system can possibly be testified by
employing Josephson junction charge qubit chargequbit . Discussion about
employing Josephson Junction qubit to test the quantum Crooks FT and quantum
JE will be given later.

##  VI acknowledgments

HTQ thanks Wojciech H. Zurek, G. Crooks and Rishi Sharma for stimulating
discussions and gratefully acknowledges the support of the U.S. Department of
Energy through the LANL/LDRD Program for this work.

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