# Broadband Optical Two-Dimensional Coherent Spectroscopy of a Rubidium Atomic
Vapor

Jieli Yan  Department of Physics, Florida International University, Miami, FL
33199, USA  Stephen Revesz  Department of Physics, Florida International
University, Miami, FL 33199, USA  Danfu Liang  Department of Physics, Florida
International University, Miami, FL 33199, USA  Hebin Li<EMAIL_ADDRESS>Department of Physics, Florida International University, Miami, FL 33199, USA

###### Abstract

Optical two-dimensional coherent spectroscopy (2DCS) has become a powerful
tool for studying energy level structure, dynamics, and coupling in many
systems including atomic ensembles. Various types of two-dimensional (2D)
spectra, including the so-called single-quantum, zero-quantum, and double-
quantum 2D spectra, of both D lines (D1 and D2 transitions) of potassium (K)
atoms have been reported previously. For rubidium (Rb), a major difference is
that the D-lines are about 15 nm apart as opposed to only about 3 nm for K.
Simultaneously exciting both D-lines of Rb atoms requires a broader laser
bandwidth for the experiment. Here, we report a broadband optical 2DCS
experiment in an Rb atomic vapor. A complete set of single-quantum, zero-
quantum, and double-quantum 2D spectra including both D-lines of Rb atoms were
obtained. The experimental spectra were reproduced by simulated 2D spectra
based on the perturbation solutions to the optical Bloch equations. This work
in Rb atoms complements previous 2DCS studies of K and Rb with a narrower
bandwidth that covers two D-lines of K or only a single D-line of Rb. The
broadband excitation enables the capability to perform double-quantum and
multi-quantum 2DCS of both D-lines of Rb to study many-body interactions and
correlations in comparison with K atoms.

##  I Introduction

Optical two-dimensional coherent spectroscopy (2DCS) [1, 2] is an optical
analog to two-dimensional nuclear magnetic resonance (NMR) spectroscopy [3].
The idea of implementing 2DCS in the optical regime was proposed by Tanimura
and Mukamel in 1993 and was realized experimentally first by using infrared
ultrafast pulses [4, 5]. Since then, various approaches [6, 7, 8, 9, 10, 11,
12, 13, 14, 15] have been developed in the near-infrared and visible range. By
unfolding a potentially congested 1D spectrum onto a 2D plane and correlating
the dynamics in two different frequency dimensions, optical 2DCS excels in
studying optical response of complex systems. The technique has become a
powerful spectroscopic tool to study energy level structure, dynamics, and
coupling in various systems, such as structural information in proteins [5],
dynamics of hydrogen bond in water [16], energy transfer processes in
photosynthesis [17, 18, 19], many-body interactions and correlations in atomic
ensembles [20, 21, 22, 23, 24, 25, 26, 27], ultrafast dynamics and couplings
of excitons in semiconductor quantum wells [28, 29, 30, 31, 32, 33, 34],
quantum dots [35, 36, 37, 38], 2D materials [39, 40], and perovskites [41, 42,
43, 44, 45, 46].

Potassium (K) and rubidium (Rb) atomic vapors were initially used as model
systems to validate optical 2DCS techniques. The energy level structures and
other parameters are well characterized and the D-lines can be excited by a
typical Ti:sapphire femtosecond oscillator, making K and Rb ideal samples to
test optical 2DCS implementations. Several early approaches of optical 2DCS
were first demonstrated in Rb [47, 13] and K [48] vapors. Atomic vapors were
also used to develop new advances such as optical 3D coherent spectroscopy
[49] and frequency-comb based optical 2DCS [50, 51], and quantitative analyses
such as pulse propagation effects [52, 53] and line shape analysis [54] in 2D
spectra.

Despite being considered initially as a model system, later optical 2DCS
measurements of atomic vapors provided interesting insights into many-body
interaction and correlation in atoms. Double-quantum 2DCS revealed two-atom
collective resonances due to dipole-dipole interaction in both K [20] and Rb
[21, 22] vapors. The method provides an extremely sensitive and background-
free detection to probe interatomic dipole-dipole interaction even in a dilute
vapor with a density as low as $4.81\times 10^{8}$ cm-3, corresponding to a
mean interatomic separation of 15.8 $\mu$m [24]. The technique was also
extended to multi-quantum 2DCS to probe multi-atom Dicke states with up to
eight atoms [23, 25]. Optical 2DCS can potentially be a useful tool to study
many-body physics in cold atoms. For K vapor, a complete set of single-
quantum, zero-quantum, and double-quantum 2D spectra involving both D1 and D2
lines have been reported [48, 20]. However, previous 2DCS studies of Rb vapor
measured only single-quantum 2D spectra [47, 13] of both D-lines or double-
quantum 2D spectra of an individual D-line [21, 22]. Compared to K, a major
difference in Rb is that the D-lines are 15 nm apart, requiring a broader
laser bandwidth to cover both D-lines simultaneously.

Here, we implemented broadband optical 2DCS in an Rb atomic vapor and obtained
a complete set of single-quantum, zero-quantum, and double-quantum 2D spectra
of both D-lines of the Rb atom. The single-quantum and zero-quantum 2D spectra
show the coherent coupling between the two D-line transitions, while the
double-quantum 2D spectrum reveals the collective resonances involving both
$5P_{1/2}$ and $5P_{3/2}$ states. We also present a theoretical treatment
based on the perturbative solution to the optical Bloch equation (OBE). The
simulated 2D spectra agree well with the experimental spectra. The rest of the
paper is organized as the following. Section II describes the experimental
setup of a collinear 2DCS implementation. Section III introduces the
theoretical approach based on the OBE. Section IV describes the experimental
and theoretical details of single-quantum and zero-quantum 2D spectra. Section
V the experimental and theoretical details of double-quantum 2D spectra.
Section VI is a conclusion.

##  II Experimental setup

The optical 2DCS experiment was performed in a collinear setup based on
acousto-optic modulators (AOM) [11]. As shown in Fig. 1, the primary apparatus
consists of an interferometer of two Mach-Zehnder interferometers. This nested
interferometer splits an input fs laser pulse into four pulses (A, B, C, and
D) and then combines them in one beam with three time delays between the
pulses. The delays are controlled by three translation stages. Each pulse goes
through an AOM and is phase modulated at a slightly different frequency
$\Omega_{i}$ ($i=$A, B, C, and D). The first-order diffraction from the AOM
output is used for the experiment while the zeroth-order output is terminated.
Meanwhile, the output of a continuous-wave (CW) laser (an external cavity
diode laser) is injected into the interferometer and propagates through the
same optical path and the AOMs as the fs laser pulses. The fs laser and CW
laser beams are offset such that they can be separated after exiting the
interferometer at beamsplitter 6 (BS6). The CW laser signal is detected by a
photodetector (PD1). The CW signal carries the AOM modulation frequencies and
their beating frequencies, which can be used as the reference for lock-in
detection. The CW laser beam also monitors the optical path fluctuations in
real time. The fs pulse sequence is incident on the window of an Rb atomic
vapor cell which is placed in an oven for temperature control. The resulting
fluorescence signal is collected and directed to a photodetector (PD2) by a
pair of lenses.

Figure 1: Schematic of the collinear optical 2DCS experimental setup. BS:
Beamsplitter. AOM: Acousto-optic modulator. PD: Photo detector.

The fourth-order nonlinear fluorescence signal can be measured by lock-in
detection using a proper combination of CW laser beating frequencies as the
reference. The nonlinear signal associated with different pulse sequences and
phase-matching conditions can be selectively detected by referencing to the
proper mixing frequency. For instance, for the rephasing single-quantum signal
in the phase-matched direction
$\mathbf{k}_{S}=-\mathbf{k}_{A}+\mathbf{k}_{B}+\mathbf{k}_{C}-\mathbf{k}_{D}$,
where $\mathbf{k}_{A,B,C,S}$ are the wave vectors for pulses A, B, C, and the
signal, the reference frequency for lock-in detection should be
$\Omega_{S1}=-\Omega_{A}+\Omega_{B}+\Omega_{C}-\Omega_{D}$. To obtain this
reference frequency, the CW laser signal is processed by a digital wave mixer
so that the beating frequencies $\Omega_{AB}=\Omega_{A}-\Omega_{B}$ and
$\Omega_{CD}=\Omega_{C}-\Omega_{D}$ are extracted by filtering and
subsequently mixed to generate $\Omega_{S1}=\Omega_{CD}-\Omega_{AB}$. The
rephasing single-quantum signal can be measured by lock-in detection
referencing to $\Omega_{S1}$. Other types of 2DCS signal can be obtained
similarly by using different mixing frequencies as the lock-in reference. The
time delay of each pulse can be controlled independently. There are three time
periods, $\tau$ between the first and second pulse, $T$ between the second and
third pulse, and $t$ between the third and fourth pulse. During the
experiment, the signal is recorded while scanning two or three time delays.
For instance, if both $\tau$ and $t$ are scanned, the signal can be
represented in the time domain as $S(\tau,t)$ with a fixed $T$. A 2D spectrum
$S(\omega_{\tau},\omega_{t})$ is generated by 2D Fourier-transforming the
time-domain signal $S(\tau,t)$ into the frequency domain. Depending on the
time ordering of the excitation pulses and which time delays are scanned, the
experiment can generate single-quantum, zero-quantum, and double-quantum 2D
spectra. The experimental detail and interpretation of these 2D spectra are
described in the following sections.

In the current work, the fs laser pulse is provided by a Ti:sapphire
oscillator (Coherent Vitara). The output pulse spectrum has a bandwidth (FWHM)
of 67.45 nm with a central wavelength of 810.89 nm. The repetition rate is 80
MHz. The fs laser power at the sample is 18 mW in total. The fs pulses have
sufficient bandwidth to simultaneously excite two D-lines of Rb atoms to
obtain 2D spectra of both D-lines.

##  III Theoretical approach

Using the density matrix formalism, the light-matter interaction in this
experiment can be described by the equation of motion [55]

$\dot{\rho}_{ij}=-\frac{i}{\hbar}\sum_{k}(H_{ik}\rho_{kj}-\rho_{ik}H_{kj})-\Gamma_{ij}\rho_{ij},$
(1)

where $\rho_{ij}$ are the density matrix elements. The Hamiltonian has matrix
elements $H_{ij}=\hbar\omega_{i}\delta_{ij}-\mu_{ij}E(t)$, where
$\hbar\omega_{i}$ is the energy of state $|i\rangle$, $E(t)$ is the electric
field, $\mu_{ij}$ ($i\neq j$) is the dipole moment of the transition between
$|i\rangle$ and $|j\rangle$, and $\delta_{ij}$ is the Kronecker delta
function. The relaxation operator $\Gamma$ has matrix elements
$\Gamma_{ij}=\frac{1}{2}(\gamma_{i}+\gamma_{j})+\gamma_{ij}^{ph}$, where
$\gamma_{i}$ and $\gamma_{j}$ are the population decay rates for states
$|i\rangle$ and $|j\rangle$, respectively, and $\gamma_{ij}^{ph}$ is the pure
coherence dephasing rate ($\gamma_{ij}^{ph}=0$ for $i=j$).

The equations of motion represented by Eq. (1) include a series of coupled
differential equations. To solve these equations perturbatively, we substitute
$E$ with $\lambda E$ and expand the density matrix elements as

$\rho_{ij}=\rho_{ij}^{(0)}+\lambda\rho_{ij}^{(1)}+\lambda^{2}\rho_{ij}^{(2)}+\lambda^{3}\rho_{ij}^{(3)}+\cdots,$
(2)

where $\lambda$ is a constant to track the order. Plug them into Eq. (1) and
collect coefficients of $\lambda^{n}$ ($n=0,1,2,\cdots$), we can find the
perturbative solutions for each order. In general, the $n$-th order solution
of density matrix element is related to the $(n-1)$-th order solution. Under
the excitation of a field
$\textbf{E}(t)=\hat{E}_{n}e^{i\mathbf{k}_{n}\cdot\mathbf{r}-i\omega_{n}t}+c.c.$,
the $(n-1)$-th order density matrix element $\rho_{ik}^{(n-1)}$ evolves to the
$n$-th order density matrix element $\rho_{jk}^{(n)}$ or $\rho_{il}^{(n)}$.
There are four possibilities that can be calculated by the following integrals

$\displaystyle\mathrm{(a)}\ \ \ \rho_{jk}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{i\mu_{ij}}{2\hbar}e^{i\mathbf{k}_{n}\cdot\mathbf{r}}\int^{t}_{-\infty}\hat{E}_{n}(t^{\prime})e^{-i\omega_{n}t^{\prime}}$
(3) $\displaystyle
e^{-i\Omega_{jk}(t-t^{\prime})}\rho^{(n-1)}_{ik}(t^{\prime})dt^{\prime},$
$\displaystyle\mathrm{(b)}\ \ \ \rho_{jk}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{i\mu_{ij}}{2\hbar}e^{-i\mathbf{k}_{n}\cdot\mathbf{r}}\int^{t}_{-\infty}\hat{E}^{*}_{n}(t^{\prime})e^{i\omega_{n}t^{\prime}}$
(4) $\displaystyle
e^{-i\Omega_{jk}(t-t^{\prime})}\rho^{(n-1)}_{ik}(t^{\prime})dt^{\prime},$
$\displaystyle\mathrm{(c)}\ \ \ \rho_{il}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{i\mu_{kl}}{2\hbar}e^{-i\mathbf{k}_{n}\cdot\mathbf{r}}\int^{t}_{-\infty}\hat{E}^{*}_{n}(t^{\prime})e^{i\omega_{n}t^{\prime}}$
(5) $\displaystyle
e^{-i\Omega_{il}(t-t^{\prime})}\rho^{(n-1)}_{ik}(t^{\prime})dt^{\prime},$
$\displaystyle\mathrm{(d)}\ \ \ \rho_{il}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{i\mu_{kl}}{2\hbar}e^{i\mathbf{k}_{n}\cdot\mathbf{r}}\int^{t}_{-\infty}\hat{E}_{n}(t^{\prime})e^{-i\omega_{n}t^{\prime}}$
(6) $\displaystyle
e^{-i\Omega_{il}(t-t^{\prime})}\rho^{(n-1)}_{ik}(t^{\prime})dt^{\prime},$

where $\Omega_{ij}=\omega_{i}-\omega_{j}-i\Gamma_{ij}$. A convenient way to
track the time evolution of density matrix elements in the perturbation
calculation is to use double-sided Feynman diagram. The interaction with a
field is described by the vertex of an arrow with a vertical line. These four
integrals can be represented by four vertices, as shown in Fig. 2(a-d),
respectively. An arrow represents a field that changes one index of the
density matrix element. A photon is absorbed (emitted) if the arrow points
towards (away from) the vertical lines. An arrow pointing to the right
indicates that the field is $E(t)$, while an arrow pointing to the left means
that the field is conjugated, $E^{*}(t)$.

Figure 2: Four different types of vertices in double-sided Feynman diagrams.

A doubled-side Feynman diagram representing an excitation quantum pathway may
include multiple orders of these four vertices. The corresponding nonlinear
signal can be calculated order by order by using the integrals in Eqs. (3-6).
In an experiment, the nonlinear signal usually includes contributions from
multiple excitation pathways represented by multiple double-sided Feynman
diagrams. The contributions from individual diagrams can be calculated
separately and added together to obtain the overall nonlinear signal.

##  IV Single-quantum and zero-quantum 2D spectra

Single-quantum and zero-quantum rephasing 2D spectra were obtained by using
the pulse sequence shown in Fig. 3(a), in which pulses A and D are considered
conjugated. The Rb atom is considered a three-level $V$ system with the
relevant energy levels shown in Fig. 3(b). A fourth-order nonlinear signal is
generated in the sample by the excitation of the four-pulse sequence. Briefly,
the first pulse, A, creates coherence between the ground and excited states.
The second pulse, B, converts the coherence to a population in either the
ground state or the excited states, depending on the relative phase between
the first pulses. For a three-level $V$ system, the first two pulses also
create a Raman-like coherence between the two excited states. The third pulse,
C, converts the population and the “Raman” coherence to a third-order
coherence between the ground and excited states. The fourth pulse, D, converts
the third-order coherence to a population in the excited states which emits
fluorescence as the fourth-order nonlinear signal. The dynamics measured
during the first time delay $\tau$ reveal the time evolution of the coherence
between the ground and excited states and the corresponding coherence
dephasing time $T_{2}$. During the second time delay $T$, the dynamics include
the population decay term and the “Raman” term. The population term leads to a
spectral signal at $\omega_{T}=0$ which can reveal the population decay time
$T_{1}$, while the “Raman” term results in a signal at
$\omega_{T}=\pm\omega_{e_{2}e_{1}}$. The process consists of eight specific
excitation quantum pathways represented by the double-sided Feynman diagrams
shown in Fig. 3(c).

Pulse D can be seen as a local oscillator for heterodyne detection of the
third-order coherence within the sample itself. The resulting signal is
fluorescence emitted by the fourth-order population. To selectively detect the
fourth-order fluorescence signal that is a specific response to the pulse
sequence in Fig. 3(a), the output of PD2 in Fig. 1 is demodulated by a lock-in
amplifier referenced to the mixing signal
$\Omega_{S1}=-\Omega_{A}+\Omega_{B}+\Omega_{C}-\Omega_{D}$ generated from the
CW laser signal. The AOM modulation frequencies used in this experiment are
$\Omega_{A}$ = 80 MHz, $\Omega_{B}$ = 80.0173 MHz, $\Omega_{C}$ = 80.104 MHz,
and $\Omega_{D}$ = 80.107 MHz so the reference frequency is $\Omega_{S1}=14.3$
kHz.

Figure 3: (a) Time ordering of excitation pulses for performing zero-quantum
and single-quantum 2DCS. (b) Relevant energy levels of Rb atoms. (c) Double-
sided Feynman diagrams representing possible excitation quantum pathways in
zero-quantum and single-quantum 2DCS using the pulse sequence in (a).

Single-quantum rephasing 2D spectra can be obtained by scanning time delays
$\tau$ and $t$, while fixing the second time delay $T$, and 2D Fourier-
transforming the signal into the frequency domain. A typical single-quantum 2D
spectrum is shown in Fig. 4 (a), where the spectral amplitude is plotted with
the maximum normalized to 1. The $x$ and $y$ axes are the emission frequency
$\omega_{t}$ and the absorption frequency $\omega_{\tau}$ corresponding to the
time delays $t$ and $\tau$, respectively. The absorption frequency has a
negative sign because the excitation pulse A is conjugated. The dotted
diagonal line indicates that the absorption and emission frequencies have the
same absolute value ($|\omega_{\tau}|=\omega_{t}$). Peak SA (SB) on the
diagonal line is due to the D1 (D2) transition, absorbing and emitting at the
same frequency. In contrast, the off-diagonal peak SC (SD) absorbs at the D1
(D2) frequency and emits at the D2 (D1) frequency, revealing the coupling
between the D1 and D2 transitions. Each peak has contributions from two of the
double-sided Feynman diagrams in Fig. 3(c) which are grouped and labeled
accordingly. For peaks SA and SB, the two diagrams represent similar
processes. For peaks SC and SD, the two diagrams describe two different
dynamics, ground state depletion and “Raman” coherence, during the second time
delay $T$. The process involving “Raman” coherence can be isolated in a zero-
quantum 2D spectrum.

Figure 4: (a) Red curve shows the laser spectrum between 375.107 THZ and
386.23 THz. Note that the laser spectrum is much broader than the spectra
range shown here. The projection of the single-quantum spectra onto the
emission axis is shown as the blue curve. The D1 and D2 transitions are shown
respectively. (b) Experimental measured single-quantum 2D spectrum of Rubidium
vapor at 160 ∘C. (c) Simulated single-quantum 2D spectrum matched to
experimental data.

Zero-quantum 2D spectra can be obtained by scanning time delays $T$ and $t$,
while fixing the first time delay $\tau$, and 2D Fourier-transforming the
signal into the frequency domain. A typical zero-quantum 2D spectrum is shown
in Fig. 5 (a), where the amplitude is plotted with the maximum normalized to
1. The $x$ axis is the emission frequency $\omega_{t}$. The $y$ axis is the
mixing frequency $\omega_{T}$ corresponding to the second time delay $T$.
There are four peaks in the spectrum. Peaks ZA and ZB are located on the
$\omega_{T}=0$ line. They each are contributed by the three pathways involving
a population decay during the second time delay $T$, as labeled in Fig. 3(c)
accordingly. Peak ZC (ZD) has a mixing frequency of $\omega_{T}=7.12$
($\omega_{T}=-7.12$) THz, the frequency difference between the $5^{2}P_{1/2}$
and $5^{2}P_{3/2}$ states.

The experimental 2D spectra can be reproduced in simulation based on the
theoretical approach described in Section III. The contribution from each
pathway in Fig. 3(c) needs to be calculated. As an example, assuming the
excitation pulses are Delta pulses, the signal from the fourth diagram (SD2
shown in Fig. 3-b) can be calculated by using the integrals in Eqs. (3-6) as:

$\rho^{(4)}_{e_{1}e_{1}}(\tau,T,t)=S_{0}\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}e^{-\Gamma_{e_{1}e_{1}}\tau^{{}^{\prime}}}e^{(-i\omega_{e_{1}g}-\Gamma_{e_{1}g})t}e^{(-i\omega_{e_{1}e_{2}}-\Gamma_{e_{1}e_{2}})T}e^{(-i\omega_{ge_{2}}-\Gamma_{ge_{2}})\tau}\Theta(\tau^{{}^{\prime}})\Theta(t)\Theta(T)\Theta(\tau),$
(7)

where

$S_{0}=-\frac{E_{A}E_{B}E_{C}E_{D}}{16\hbar^{4}}\rho_{gg}^{(0)}e^{-i(\mathbf{k}_{D}-\mathbf{k}_{C}-\mathbf{k}_{B}+\mathbf{k}_{A})\cdot\mathbf{r}}.$
(8)

Here $\tau^{\prime}$ is the fluorescence emission time, $E_{i}$ ($i=A,B,C,D$)
is the electric field amplitude for each pulse, $\rho_{gg}^{(0)}$ is the
initial population in the ground state, $\mathbf{k}_{i}$ is the wave vector,
and $\Theta$ is Heaviside step function. Taking the 2D Fourier Transform
(2DFT) of Eq. (7) gives the fourth-order frequency-domain signal with
$\tau^{\prime}=0$ as the following:

$\displaystyle S^{(4)}_{e_{1}e_{1}}(\omega_{\tau},\omega_{T},\omega_{t})$
$\displaystyle=$ $\displaystyle
S_{0}\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}\bigg{[}\frac{1}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\times\frac{1}{\omega_{T}-\omega_{e_{1}e_{2}}+i\Gamma_{e_{1}e_{2}}}\times\frac{1}{\omega_{\tau}-\omega_{ge_{2}}+i\Gamma_{ge_{2}}}\bigg{]}$
(9)

Repeating the above derivation for each pathway shown in Fig. 3(c) and summing
each contribution gives the overall nonlinear signal in the frequency domain.
Evaluating the 2DFT of the time-domain expression at $T=0$ gives the single-
quantum 2D frequency-domain spectrum solution shown in Eq.
(LABEL:eq:S1Signal), while evaluating at $\tau=0$ gives the zero-quantum 2D
frequency-domain solution shown in Eq. (LABEL:eq:S0Signal). Simulations for
each spectra are shown in Fig. 4(c) and Fig. 5(b), respectively.

$\displaystyle S^{(4)}_{1Q}(\omega_{\tau},\omega_{t})$ $\displaystyle=$
$\displaystyle\frac{2S_{0}\mu_{e_{1}g}^{4}}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\times\frac{1}{\omega_{\tau}-\omega_{ge_{1}}+i\Gamma_{ge_{1}}}$
$\displaystyle+$
$\displaystyle\frac{2S_{0}\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\times\frac{1}{\omega_{\tau}-\omega_{ge_{2}}+i\Gamma_{ge_{2}}}$
$\displaystyle+$
$\displaystyle\frac{2S_{0}\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\times\frac{1}{\omega_{\tau}-\omega_{ge_{1}}+i\Gamma_{ge_{1}}}$
$\displaystyle+$
$\displaystyle\frac{2S_{0}\mu_{e_{2}g}^{4}}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\times\frac{1}{\omega_{\tau}-\omega_{ge_{2}}+i\Gamma_{ge_{2}}}$
$\displaystyle S^{(4)}_{0Q}(\omega_{T},\omega_{t})$ $\displaystyle=$
$\displaystyle\frac{\mu_{e_{1}g}^{4}+\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\times\frac{S_{0}}{\omega_{T}-\omega_{gg}+i\Gamma_{e_{1}e_{2}}}$
$\displaystyle+$
$\displaystyle\frac{\mu_{e_{1}g}^{4}}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\times\frac{S_{0}}{\omega_{T}-\omega_{e_{1}e_{1}}+i\Gamma_{e_{1}e_{2}}}$
$\displaystyle+$
$\displaystyle\frac{\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\times\frac{S_{0}}{\omega_{T}-\omega_{e_{1}e_{2}}+i\Gamma_{e_{1}e_{2}}}$
$\displaystyle+$
$\displaystyle\frac{\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\times\frac{S_{0}}{\omega_{T}-\omega_{e_{2}e_{1}}+i\Gamma_{e_{2}e_{1}}}$
$\displaystyle+$
$\displaystyle\frac{\mu_{e_{1}g}^{4}+\mu_{e_{1}g}^{2}\mu_{e_{2}g}^{2}}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\times\frac{S_{0}}{\omega_{T}-\omega_{gg}+i\Gamma_{e_{1}e_{2}}}$
$\displaystyle+$
$\displaystyle\frac{\mu_{e_{1}g}^{4}}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\times\frac{S_{0}}{\omega_{T}-\omega_{e_{2}e_{2}}+i\Gamma_{e_{1}e_{2}}}$

To conduct the single-quantum and zero-quantum simulations, the center
frequencies for each resonance and their associated relaxation linewidths were
extracted from the experimental spectra by fitting Lorentzian curves. Emission
center frequencies $\omega_{e_{1}g}=2\pi$ $\times$ 377.210 THz and
$\omega_{e_{2}g}=2\pi\times 384.245$ THz were extracted for resonances SA and
SB shown in Fig. 4. The relaxation linewidths $\Gamma_{e_{1}g}$ = $2\pi$
$\times$ 0.2446 THz and $\Gamma_{e_{2}g}=2\pi\times 0.1450$ THz were extracted
by slicing along the diagonal of SA and SB respectively. Slicing along the
diagonal through SC and SD extracted parameters $\Gamma_{e_{1}e_{2}}$ = 2$\pi$
$\times$ 0.1446 THz and $\Gamma_{e_{2}e_{1}}$ = 2$\pi$ $\times$ 0.0716 THz
respectively. The zero-quantum simulated spectra utilized the same parameters
as the single-quantum simulation. The transition dipole moments were
$\mu_{e_{1}g}=2.537\times 10^{-29}$ C$\cdot$m and $\mu_{e_{2}g}=3.584\times
10^{-29}$ C$\cdot$m.

Figure 5: (a) Experimental and (b) simulated zero-quantum 2D spectra of a Rb
vapor. The spectral amplitude is plotted with the maximum normalized to 1.

##  V Double-quantum 2D spectra

Double-quantum 2D spectra were acquired by using the pulse sequence shown in
Fig. 6(a), where the conjugated pulses A and D arrive after pulses B and C.
The first two pulses can excite double-quantum coherence between the ground
state and doubly excited states. The relevant energy levels, as shown in Fig.
6(b), include the ground state $|g\rangle$, two singly excited states
$|e_{1}\rangle=|5^{2}P_{1/2}\rangle$ and $|e_{2}\rangle=|5^{2}P_{3/2}\rangle$,
and three doubly excited states which are two-atom states
$|d_{1}\rangle=|5^{2}P_{1/2},5^{2}P_{1/2}\rangle$,
$|d_{2}\rangle=|5^{2}P_{1/2},5^{2}P_{3/2}\rangle$, and
$|d_{3}\rangle=|5^{2}P_{3/2},5^{2}P_{3/2}\rangle$. For convenience, the
frequencies of $|e_{1}\rangle$ and $|e_{2}\rangle$ are labeled D1 and D2,
respectively. The frequencies of the doubly excited states $|d_{1}\rangle$,
$|d_{2}\rangle$, and $|d_{3}\rangle$ are 2D1, D${}_{1}+$D2, and 2D2,
respectively. Using this pulse sequence, the excitation process is different
from the single-quantum excitation. The first pulse, B, creates single-quantum
coherence between the ground state and the singly excited states. The second
pulse, C, converts the single-quantum coherence to double-quantum coherence
between the ground state and the doubly excited states. The third pulse, A,
converts the double-quantum coherence back to single-quantum coherence. The
fourth pulse, D, converts the single-quantum coherence to a population in the
singly or doubly excited states, which emits a fluorescence signal. To detect
the double-quantum signal, the fluorescence is measured by PD2 and its output
is demodulated by a lock-in amplifier referenced to the mixing signal
$\Omega_{S2}=\Omega_{B}+\Omega_{C}-\Omega_{A}-\Omega_{D}$.

Figure 6: (a) Time ordering of excitation pulses for performing double-quantum
2DCS. (b) Relevant energy levels of Rb atoms including single-atom
($e_{1}$,$e_{2}$) and two-atom states ($d_{1}$,$d_{2}$,$d_{3}$). Two-atom
interactions cause an energy shift ($\Delta_{\omega}$) to the two-atom states
shown as the green dashed lines. No interactions would be described by the
solid black lines. Shown are two example transitions between states
g$\rightarrow d_{1}$ with energy $\omega_{d_{1}g}$ and another transition
between states $e_{2}\rightarrow d_{2}$ with energy $\omega_{d_{2}e_{2}}$. (c)
Double-sided Feynman diagrams representing possible excitation quantum
pathways in double-quantum 2DCS using the pulse sequence in (a).

For the energy levels in Fig. 6(b), this excitation process includes 18
pathways represented by the double-sided Feynman diagrams shown in Fig. 6(c).
In all pathways, a double-quantum coherence between the ground state and one
of the doubly excited states evolves during the second time delay $T$. The
dynamics of double-quantum coherence can be measured by scanning $T$ in
double-quantum 2DCS. For two independent, non-interacting Rb atoms, the
contributions from all 18 pathways cancel out, leading to a vanishing double-
quantum signal. However, the cancellation is not complete if there is the
interaction between the Rb atoms that breaks the symmetry, resulting in a non-
zero double-quantum signal. Double-quantum 2DCS has been proven to be
extremely sensitive detection to dipole-dipole interactions and collective
resonances in dilute K and Rb atomic vapors [20, 21, 22, 24, 25, 27].

The double-quantum signal was measured as time delays $T$ and $t$ are scanned
while time delay $\tau$ is fixed. Fourier-transforming the signal into the
frequency domain generates double-quantum 2D spectra. A typical double-quantum
2D spectrum is shown in Fig. 7(a), where the spectral amplitude is plotted
with the maximum normalized to 1. The $x$ and $y$ axes are the emission
frequency $\omega_{t}$ and the double-quantum frequency $\omega_{T}$,
corresponding to the time delays $t$ and $T$. There are four peaks in the
double-quantum 2D spectrum. The double-quantum frequencies of these peaks
match the frequencies of the two-atom doubly excited states 2D1, D${}_{1}+$D2,
and 2D2, since the double-quantum coherence oscillates at these frequencies
during time delay $T$. The emission frequencies of these peaks are D1 and D2.
Each peak has contributions from multiple pathways represented by double-sided
Feynman diagrams labeled accordingly in Fig. 6(c). The double-quantum 2D
spectrum reveals the two-atom collective resonances and dipole-dipole
interactions involving both $5^{2}P_{1/2}$ and $5^{2}P_{3/2}$ states. For Rb
atoms, there is also a single-atom doubly excited state $5^{2}D$ at
$2\pi\times 770.5$ THz. The double-quantum signal associated with $5^{2}D$ has
been observed before [21] as two off-diagonal peaks with the double-quantum
frequency at $\omega_{T}=2\pi\times 770.5$ THz and the emission frequency at
$\omega_{t}=2\pi\times 384.2$ and $2\pi\times 386.3$ THz. However, in the
current experiment, the laser spectrum was centered at $\sim$810 nm and does
not have sufficient intensity at the wavelength required for exciting
$5^{2}D$. The double-quantum signal associated with $5^{2}D$ was not observed
here.

Figure 7: (a) Experimental and (b) simulated double-quantum 2D spectra of a Rb
vapor. The spectral amplitude is plotted with the maximum normalized to 1.

The experimental frequency-domain double-quantum 2D spectrum can be reproduced
by calculating contributions from all pathways in Fig. 6(c).

$\displaystyle S^{(4)}_{DA}(\omega_{T},\omega_{t})$
$\displaystyle=\frac{S_{0}(\mu_{e_{1}g}\mu_{d_{2}e_{1}}+\mu_{e_{2}g}\mu_{d_{2}e_{2}})}{\omega_{T}-\omega_{d_{2}g}+i\Gamma_{d_{2}g}}$
$\displaystyle\times\bigg{(}\frac{\mu_{e_{2}g}\mu_{d_{2}e_{2}}}{\omega_{t}-\omega_{d_{2}e_{2}}+i\Gamma_{d_{2}e_{2}}}-\frac{\mu_{d_{2}e_{1}}\mu_{ge_{1}}}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\bigg{)}$
$\displaystyle S^{(4)}_{DB}(\omega_{T},\omega_{t})$
$\displaystyle=\frac{S_{0}(\mu_{e_{1}g}\mu_{d_{2}e_{1}}+\mu_{e_{2}g}\mu_{d_{2}e_{2}})}{\omega_{T}-\omega_{d_{2}g}+i\Gamma_{d_{2}g}}$
$\displaystyle\times\bigg{(}\frac{\mu_{e_{1}g}\mu_{d_{2}e_{1}}}{\omega_{t}-\omega_{d_{2}e_{1}}+i\Gamma_{d_{2}e_{1}}}-\frac{\mu_{d_{2}e_{2}}\mu_{e_{2}g}}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\bigg{)}$
$\displaystyle S^{(4)}_{DC}(\omega_{T},\omega_{t})$
$\displaystyle=\frac{S_{0}\mu^{2}_{e_{2}g}\mu^{2}_{d_{3}e_{2}}}{\omega_{T}-\omega_{d_{3}g}+i\Gamma_{d_{3}g}}$
$\displaystyle\times\bigg{(}\frac{1}{\omega_{t}-\omega_{d_{3}e_{2}}+i\Gamma_{d_{3}e_{2}}}-\frac{1}{\omega_{t}-\omega_{e_{2}g}+i\Gamma_{e_{2}g}}\bigg{)}$
$\displaystyle S^{(4)}_{DD}(\omega_{T},\omega_{t})$
$\displaystyle=\frac{S_{0}\mu^{2}_{e_{1}g}\mu^{2}_{d_{1}e_{1}}}{\omega_{T}-\omega_{d_{1}g}+i\Gamma_{d_{1}g}}$
$\displaystyle\times\bigg{(}\frac{1}{\omega_{t}-\omega_{d_{1}e_{1}}+i\Gamma_{d_{1}e_{1}}}-\frac{1}{\omega_{t}-\omega_{e_{1}g}+i\Gamma_{e_{1}g}}\bigg{)}$

The simulated double-quantum spectra used parameters extracted from the
experimental spectra with fitted Lorentzian curves. The double-quantum center
frequencies extracted from the experimental spectra were
$\omega_{d_{1}g}=2\pi\times$ 754.213 THz, $\omega_{d_{2}g}=2\pi\times$ 761.358
THz, and $\omega_{d_{3}g}=2\pi\times$ 768.509 THz. The emission center
frequencies ($\omega_{e_{1}g}$, $\omega_{e_{2}g}$) were identical to those
extracted from the single-quantum spectra. The remaining frequency terms are

$\displaystyle\omega_{d_{3}e_{2}}$
$\displaystyle=\omega_{e_{2}g}-\Delta_{\omega}$ (16)
$\displaystyle\omega_{d_{2}e_{2}}$
$\displaystyle=\omega_{e_{1}g}-\Delta_{\omega}$ (17)
$\displaystyle\omega_{d_{2}e_{1}}$
$\displaystyle=\omega_{e_{2}g}-\Delta_{\omega}$ (18)
$\displaystyle\omega_{d_{1}e_{1}}$
$\displaystyle=\omega_{e_{1}g}-\Delta_{\omega}$ (19)

where $\Delta_{\omega}=2\pi\times 100$ MHz. The terms $\omega_{e_{1}g}$ and
$\omega_{e_{2}g}$ are the same used for the single-quantum spectra simulation.
The remaining relaxation terms are

$\displaystyle\Gamma_{d_{3}e_{2}}=\Gamma_{e_{2}g}+\Delta_{\Gamma}$ (20)
$\displaystyle\Gamma_{d_{2}e_{2}}=\Gamma_{e_{1}g}+\Delta_{\Gamma}$ (21)
$\displaystyle\Gamma_{d_{2}e_{1}}=\Gamma_{e_{2}g}+\Delta_{\Gamma}$ (22)
$\displaystyle\Gamma_{d_{1}e_{1}}=\Gamma_{e_{1}g}+\Delta_{\Gamma}$ (23)

where $\Delta_{\Gamma}=2\pi\times 15$ GHz. The terms $\Gamma_{e_{1}g}$ and
$\Gamma_{e_{2}g}$ were the values extracted from the experimental single-
quantum spectra.

The energy splitting $\Delta_{\omega}$ and relaxation shift $\Delta_{\Gamma}$
are brought about by the interaction of individual atoms. A case where
$\Delta_{\omega}$ = $\Delta_{\Gamma}$ = 0 would suggest no interactions
between individual atoms. This would lead to a double-quantum spectra with
S${}^{(4)}_{DD}$ = S${}^{(4)}_{DC}$ = 0 and S${}^{(4)}_{DA}$ =
S${}^{(4)}_{DB}\neq$ 0 which is not reflected by the spectra in Fig. 7.

##  VI Conclusion

In conclusion, we have implemented a broadband collinear optical 2DCS
experiment on Rb atoms and obtained a complete set of single-quantum, zero-
quantum, and double-quantum 2D spectra including both D-line transitions of
Rb. The single-quantum 2D spectrum shows the coherent coupling between two
D-line transitions. The zero-quantum 2D spectrum reveals the coherence between
two excited states. The double-quantum 2D spectrum is a result of the dipole-
dipole interaction and collective resonance between two atoms. Simulated 2D
spectra based on the perturbative solutions to the OBEs agree well with the
experimental spectra. The measurements in Rb atoms complement previous 2DCS
studies of K and Rb with a narrower bandwidth that covers two D-lines of K or
only a single D-line of Rb. The broadband excitation enables the possibility
of double-quantum and multi-quantum 2DCS of both D-lines of Rb to study many-
body interactions and correlations in comparison with K atoms.

###### Acknowledgements.

This work was partially supported by NSF under Grants PHY-1707364 and
CHE-2003785.

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# Quantum tunneling from Schwarzschild black hole in non-commutative gauge
theory of gravity

Abdellah Touati*  X<EMAIL_ADDRESS>Department of Physics,
Faculty of Sciences of Matter, University of Batna-1, Batna 05000, Algeria
Slimane Zaim<EMAIL_ADDRESS>Department of Physics, Faculty of
Sciences of Matter, University of Batna-1, Batna 05000, Algeria

###### Abstract

In this Letter, we present the first study of the Hawking radiation as a
tunneling process in the non-commutative (NC) gauge theory of gravity. First,
we reconstruct the non-commutative Schwarzschild black hole (SBH) in the gauge
theory of gravity, using the Seiberg-Witten (SW) map and the star product.
Then, we calculate the emission spectrum of the outgoing massless particles
using the quantum tunneling mechanism. In first case, we calculate the
tunneling rate of massless particles across the event horizon of the NC SBH
with lower frequency, where the obtained results show a pure thermal
radiation, where we find that the Hawking temperature is the same in the two
approaches; the classical thermodynamics and the quantum tunneling approach
which indicate the equivalence between these two in the NC spacetime. However,
in the case of masseless particles emission with large frequency we take into
account the conservation of energy.The tunneling rate deviates from the pure
thermal radiation, but it’s consistent with an underlying unitary quantum
theory, then we find it related to the change in the black hole entropy, where
this relation show a logarithmic correction to the entropy in this geometry.
Moreover, we obtain the NC density number of the particles emission and
finally, we discuss the results.

Quantum tunneling process, Non-commutative gauge theory, Schwarzschild black
hole

##  I Introduction

Since the first paper that connected quantum process and gravity in 1975 by
S.Hawking [1], this paper is considered the first bridge to the quantum theory
of gravity. In that paper Hawking demonstrated that a black hole can emit pure
thermal radiation as a black body and thus evaporate [1, 2], and their
calculation based on quantum field theory in curved spacetime near the event
horizon. This phenomena is known as Hawking radiation, and that allowed us the
study black hole thermodynamics, which became an essential tool to investigate
and understand the quantum effect near the event horizon and the black hole
physics [2, 3, 4, 5, 6, 7, 8, 9]. Recently, a new method was initiated by
Kraus and Wilezek [10] and developed by Parikh and Wilezek [11, 12, 13], in
which describes the Hawking radiation as a tunneling process based on a semi-
classical approach. This method became useful to study Hawking radiation for
many black holes with static spherical symmetry [14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28].

In this Letter, we aim to extend this method to calculate tunneling rate from
deformed SBH in the NC gauge theory, where this theory is motivated by the
string theory [29]. The main idea of NC theory is the quantization of gravity
by the quantization of spacetime itself, and that is done by imposing a
commutation relation between the coordinates themselves:

$[x^{\mu},x^{\nu}]=i\Theta^{\mu\nu},$ (1)

where $\Theta^{\mu\nu}$ is an anti-symmetric real matrix. The above
commutation relation modified the ordinary product of two arbitrary functions
$f(x)$ and $g(x)$ defined over this spacetime, which became defined by a new
product called the star product (Moyal product) ”$*$” given by:

$(f*g)(x)=f(x)e^{\frac{i}{2}\Theta^{\mu\nu}\overleftarrow{\partial_{\mu}}\overrightarrow{\partial_{\nu}}}g(x).$
(2)

It is worth to mention that some paper studied the tunneling process from non-
commutative black hole using the ”coordinate coherent states approach” [30,
31, 32, 33, 34, 35, 36]. Here, we use the gauge theory of gravity, the star
product and the SW maps [29] to investigate the tunneling process in the NC
gauge theory.

In this present work, we obtain th NC SBH in gauge theory of gravity. The
corrections to the tunneling rate from NC SBH are obtained in two cases, lower
frequency in the case of conservation energy, where the first one shows a pure
thermal radiation and is consistent with the thermodynamic process, and the
second case the results deviate from the thermal radiation but it’s consistent
with an underlying unitary theory. Then we investigate the Hawking temperature
in this geometry, and we obtain a logarithmic correction to the NC black hole
entropy. Finally we obtain the NC correction to the density number of
particles creation, and our result shows that, the non-commutativity plays the
same role as the mass, where increasing the strength of the gravitational
field of the deformed SBH and leads to decreasing in the density number of the
particle tunneling from this black hole .

##  II Non-commutative Schwarzschild black hole

We review briefly the formalism of the NC gauge theory of gravity [37, 38, 39,
40], which is a generalization of the classical theory based of the de Sitter
groupe as local symmetry [41, 42]. Where we use the tetrad formalism and both
the star $\ast-$ product and the SW map to construct a NC gauge theory for a
Schwarzschild metric. According to the article [37]. The NC corrections to the
commutative tetrad fields $\hat{e}^{a}_{\mu}(x,\Theta)$ are obtained using the
perturbation form for the SW map, which is described as a development in the
power of $\Theta$ up to the second-order

$\hat{e}^{a}_{\mu}(x,\Theta)=e^{a}_{\mu}(x)-i\Theta^{\nu\rho}e^{a}_{\mu\nu\rho}(x)+\Theta^{\nu\rho}\Theta^{\lambda\tau}e^{a}_{\mu\nu\rho\lambda\tau}(x)+\mathcal{O}(\Theta^{3})$
(3)

where

$e^{a}_{\mu\nu\rho}=\frac{1}{4}[\omega^{ac}_{\nu}\partial_{\rho}e^{d}_{\mu}+(\partial_{\rho}\omega^{ac}_{\mu}+R^{ac}_{\rho\mu})e^{d}_{\nu}]\eta_{cd}$
(4)

and

$\displaystyle\hat{e}_{\mu}^{a}$
$\displaystyle=\frac{\Theta^{\nu\rho}\Theta^{\lambda\tau}}{32}\left[2\\{R_{\tau\nu},R_{\mu\rho}\\}^{ab}e^{c}_{\lambda}-\omega^{ab}_{\lambda}(D_{\rho}R_{\tau\nu}^{cd}+\partial_{\rho}R_{\tau\nu}^{cd})e^{m}_{\nu}\eta_{dm}-\\{\omega_{\nu},(D_{\rho}R_{\tau\nu}+\partial_{\rho}R_{\tau\nu})\\}^{ab}e^{c}_{\lambda}-\partial_{\tau}\\{\omega_{\nu},(\partial_{\rho}\omega_{\mu}+R_{\rho\mu})\\}^{ab}e^{c}_{\lambda}\right.$
$\displaystyle\left.-\omega^{ab}_{\lambda}\left(\omega^{cd}_{\nu}\partial_{\rho}e^{m}_{\mu}+\left(\partial_{\rho}\omega_{\mu}^{cd}+R_{\rho\mu}^{cd}\right)e^{m}_{\nu}\right)\eta_{dm}+2\partial_{\nu}\omega_{\lambda}^{ab}\partial_{\rho}\partial_{\tau}e^{c}_{\mu}-2\partial_{\rho}\left(\partial_{\tau}\omega_{\mu}^{ab}+R_{\tau\mu}^{ab}\right)\partial_{\nu}e^{c}_{\lambda}-\\{\omega_{\nu},(\partial_{\rho}\omega_{\lambda}+R_{\rho\lambda})\\}^{ab}\partial_{\tau}e^{c}_{\mu}\right.$
$\displaystyle\left.-\left(\partial_{\tau}\omega_{\mu}+R_{\tau\mu}\right)\left(\omega^{cd}_{\nu}\partial_{\rho}e^{m}_{\lambda}+\left((\partial_{\rho}\omega_{\lambda}+R_{\rho\lambda})\right)e^{m}_{\nu}\right)\eta_{dm}\right]\eta_{cb}+\mathcal{O}\left(\Theta^{3}\right),$
(5)

where $\hat{e}_{a}^{\mu}$ and $\omega^{ab}_{\mu}$ are the commutative tetrad
field and spin connection, and

$\displaystyle\\{\alpha,\beta\\}^{ab}$
$\displaystyle=\left(\alpha^{ac}\beta^{db}+\beta^{ac}\alpha^{db}\right)\eta_{cd},$
(6) $\displaystyle[\alpha,\beta]^{ab}$
$\displaystyle=\left(\alpha^{ac}\beta^{db}-\beta^{ac}\alpha^{db}\right)\eta_{cd}$
(7) $\displaystyle D_{\mu}R_{\rho\sigma}^{ab}$
$\displaystyle=\partial_{\mu}R^{ab}_{\rho\sigma}+\left(\omega_{\mu}^{ac}R^{db}_{\rho\sigma}+\omega_{\mu}^{bc}R^{da}_{\rho\sigma}\right)$
(8)

where the $\hat{e}_{a}^{\mu}$ is the inverse of the vierbein
$\hat{e}_{\mu}^{a}$ defined as

$\hat{e}_{\mu}^{b}\hat{e}_{a}^{\mu}=\delta_{a}^{b},\quad\hat{e}_{\mu}^{a}\hat{e}_{a}^{\nu}=\delta_{\mu}^{\nu}\,.$
(9)

For the NC metric $\hat{g}_{\mu\nu}$, we use the expression that found in Ref.
[38]

$\hat{g}_{\mu\nu}=\frac{1}{2}(\hat{e}_{\mu}^{b}\ast\hat{e}_{\nu
b}+\hat{e}_{\nu}^{b}\ast\hat{e}_{\mu b})\,.$ (10)

To compute the deformed metric $\hat{g}_{\mu\nu}$, we choose the following NC
anti-symmetric matrix $\Theta^{\mu\nu}$

$\Theta^{\mu\nu}=\left(\begin{matrix}0&0&0&0\\\ 0&0&\Theta&0\\\
0&-\Theta&0&0\\\ 0&0&0&0\end{matrix}\right),\qquad\mu,\nu=0,1,2,3$ (11)

We follow the same steps outlined in Ref. [43], we choose the following
general tetrads field

$\displaystyle\underline{e}_{\mu}^{0}$
$\displaystyle=\left(\begin{array}[]{cccc}\left(1-\frac{2m}{r}\right)^{\frac{1}{2}},&0,&0,&0\end{array}\right),$
(13) $\displaystyle\underline{e}_{\mu}^{1}$
$\displaystyle=\left(\begin{array}[]{cccc}0,&\left(1-\frac{2m}{r}\right)^{-\frac{1}{2}}sin\theta
cos\phi,&rcos\theta cos\phi,&-rsin\theta sin\phi\end{array}\right),$ (15)
$\displaystyle\underline{e}_{\mu}^{2}$
$\displaystyle=\left(\begin{array}[]{cccc}0,&\left(1-\frac{2m}{r}\right)^{-\frac{1}{2}}sin\theta
sin\phi,&rcos\theta sin\phi,&rsin\theta cos\phi\end{array}\right),$ (17)
$\displaystyle\underline{e}_{\mu}^{3}$
$\displaystyle=\left(\begin{array}[]{cccc}0,&\left(1-\frac{2m}{r}\right)^{-\frac{1}{2}}cos\theta,&-rsin\theta,&0\end{array}\right).$
(19)

We calculate the components of the deformed metric $\hat{g}_{\mu\nu}$ for the
Schwarzschild black hole using Eqs. (5) together with Eq. (19), and we use of
the definition (10), we obtain the non-zero components of the NC metric
$\hat{g}_{\mu\nu}$ in the leading order on $\Theta$

$\displaystyle-\hat{g}_{00}=$
$\displaystyle\left(1-\frac{2m}{r}\right)+\Theta^{2}\left\\{\frac{m\left(88m^{2}+mr\left(-77+15\sqrt{1-\frac{2m}{r}}\right)-8r^{2}\left(-2+\sqrt{1-\frac{2m}{r}}\right)\right)}{16r^{4}(-2m+r)}\right\\}+\mathcal{O}(\Theta^{4}),$
(20) $\displaystyle\hat{g}_{11}=$
$\displaystyle\left(1-\frac{2m}{r}\right)^{-1}+\Theta^{2}\left\\{\frac{m\left(12m^{2}+mr\left(-14+\sqrt{1-\frac{2m}{r}}\right)-r^{2}\left(5+\sqrt{1-\frac{2m}{r}}\right)\right)}{8r^{2}(2m-r)^{3}}\right\\}+\mathcal{O}(\Theta^{4}),$
(21) $\displaystyle\hat{g}_{22}=$ $\displaystyle
r^{2}+\Theta^{2}\left\\{\frac{-18m^{2}\left(-6+\sqrt{1-\frac{2m}{r}}\right)r+m\left(-57+29\sqrt{1-\frac{2m}{r}}\right)r^{2}2\left(5-3\sqrt{1-\frac{2m}{r}}\right)r^{3}-68m^{3}}{16r(-2m+r)^{2}}\right\\}+\mathcal{O}(\Theta^{4}),$
(22) $\displaystyle\hat{g}_{33}=$ $\displaystyle
r^{2}sin^{2}(\theta)+\Theta^{2}\left\\{\frac{m^{2}\left(50-6\sqrt{1-\frac{2m}{r}}\right)r+m\left(-43+23\sqrt{1-\frac{2m}{r}}\right)r^{2}+2\left(5-3\sqrt{1-\frac{2m}{r}}\right)r^{3}}{32r(r-2m)^{2}}\right.$
$\displaystyle\left.+\frac{-8m^{3}+(2m-r)\left(4m^{2}+3m(-1+\sqrt{1-\frac{2m}{r}})r-2(1+\sqrt{1-\frac{2m}{r}})r^{2}\right)cos(2\theta)}{32r(r-2m)^{2}}\right\\}+\mathcal{O}(\Theta^{4}).$
(23)

It is clear that, for $\Theta\rightarrow 0$, we obtain the commutative
Schwarzschild solution. The NC line element according to the above metric is
given by

$d\hat{s}^{2}=-\hat{g}_{00}(r,\Theta)dt^{2}+\hat{g}_{11}(r,\Theta)dr^{2}+\hat{g}_{22}(r,\Theta)d\theta^{2}+\hat{g}_{33}(r,\Theta)d\phi^{2},$
(24)

For such a black hole, we find that the event horizon in non-commutative
space-time is where the NC metric (20) satisfies this conditions
$\frac{1}{\hat{g}_{11}}=0$, and it’s solution gives us the NC event horizon of
the Schwarzschild black hole (SBH) [44]:

$r_{h}^{NC}=r_{h}\left[1+\frac{3}{8}\left(\frac{\Theta}{r_{h}}\right)^{2}\right]$
(25)

where $r_{h}=2m$ is the event horizon in the commutative case when $\Theta=0$.
The effect of non-commutativity is small, which is reasonable to expect since
at large distances it can be considered neglected.

##  III Quantum tunneling in NC spacetime

According to the authors in Refs. [11, 13, 12], in the description of the
tunneling process it is required to use stationary coordinates, unlike
Schwarzschild coordinates,that are not singular at the event horizon, and that
ensures the energy conservation in the derivative of the spectrum of
radiation. For that we write the NC SBH in the Painlevé-Gullstrand form [11,
13, 12, 28], the NC line element (24) for a radial became:

$\displaystyle d\hat{s}^{2}$
$\displaystyle=-\hat{g}_{00}(r,\Theta)dt^{2}+2\hat{h}(r,\Theta)dtdr+dr^{2}+\hat{g}_{22}(r,\Theta)d\theta^{2}$
$\displaystyle+\hat{g}_{33}(r,\Theta)d\phi^{2},$ (26)

where

$\hat{h}(r,\Theta)=\sqrt{\hat{g}_{00}(r,\Theta)(\hat{g}_{11}(r,\Theta)^{-1}-1)}$
(27)

In the semi-classical tunneling of particles, the tunneling rate is related to
the imaginary part of the action [11, 13, 12],

$\Gamma\sim e^{-Im\hat{S}},$ (28)

where $\hat{S}$ is tunneling action of particles in the NC curved spacetime,
which is defined by:

$\hat{S}=\int p_{\mu}dx^{\mu},$ (29)

where $p_{\mu}=\hat{g}_{\mu\nu}\frac{dx^{\nu}}{d\tau}$, is the conjugate
momentum and $\tau$ is the affine parameter. We are interested in the
tunneling of a massless particles with radial motion $L=0$ and in the
equatorial plan $\theta=\pi/2$. The imaginary part of the action is reduced to

$\displaystyle Im\hat{S}$
$\displaystyle=Im\int(p_{t}dt+p_{r}dr)=Im\int_{r_{i}}^{r_{f}}\int_{0}^{p_{r}}dp^{\prime}_{r}dr.$
(30)

As we see, the first term of integration is real, so it does not contribute in
the calculation of the imaginary part of the action. Using the Hamiltonian
equation, where the Hamiltonian of the system is given by
$H=m-\omega^{\prime}$:

$\dot{r}=\frac{dH}{dp_{r}}=\frac{d(m-\omega^{\prime})}{dp_{r}},$ (31)

where $\dot{r}=\frac{dr}{dt}$, put all of that in Eq. (30),

$\displaystyle
Im\hat{S}=Im\int_{m}^{m-\omega}\int_{r_{i}}^{r_{f}}\frac{d(m-\omega)}{\dot{r}}dr.$
(32)

We consider that the particle tunneling across the event horizon has a radial
motion ($L=0$) and in the plane of $\theta$, i.e., ($d\theta=0$). Therefore,
the radial null geodesic can be obtained using
$\hat{g}_{\mu\nu}U^{\mu}U^{\nu}=0$

$\displaystyle\frac{dr}{dt}=-\hat{g}_{01}+\sqrt{\hat{g}_{01}^{2}+\hat{g}_{00}}.$
(33)

We put the above equation into Eq. (32)

$\displaystyle
Im\hat{S}=Im\int_{m}^{m-\omega}\int_{r_{i}}^{r_{f}}\frac{d(m-\omega)}{-\hat{g}_{01}+\sqrt{\hat{g}_{01}^{2}+\hat{g}_{00}}}dr.$
(34)

where $\hat{g}_{01}=\hat{h}$ and $dH=d(m-\omega)=-d\omega$, after rearranging
our expression we find

$\displaystyle
Im\hat{S}=Im\int_{0}^{\omega}(-d\omega)\int_{r_{i}}^{r_{f}}\frac{dr}{\sqrt{\hat{g}_{00}\hat{g}_{11}}\left[1-\sqrt{1-\frac{1}{\hat{g}_{11}}}\right]}.$
(35)

This expression has a single pole at the NC event horizon $r=r^{NC}_{h}$. The
integral can be evaluated by deformed the contour around the pole (using the
residue theorem), and to make sure that the Boltzmann factor111In the
approximation of lower frequency $\omega\ll m$. is conserved $\Gamma\sim
e^{-2ImS}\sim e^{-\beta\omega}$ we follow the same steps to evaluate the
integral over $\omega$ as in Refs. [26, 27]

$\displaystyle Im\hat{S}$
$\displaystyle=\pi\int_{0}^{\omega}\left(4m+\frac{3\Theta^{2}}{2m}\right)d\omega=\pi\omega\left(4m+\frac{3\Theta^{2}}{2m}\right).$
(36)

The tunneling rate from the NC SBH is therefore:

$\displaystyle\hat{\Gamma}\sim
exp\left[-2\pi\omega\left(4m+\frac{3\Theta^{2}}{2m}\right)\right].$ (37)

In the limit $\Theta\rightarrow 0$, the commutative expression is
recovered[11, 13].

Figure 1: The behavior of tunneling rate from NC SBH as function of the
emitting particle energy $\omega$, with $m=1$.

In Fig. 1 we show the effect of the non-commutativity on the tunneling rate of
particle from the NC SBH. As we observe, the non-commutativity has a small
effect on the tunneling rate of emitting particles with frequency
$\omega\in[0.04,0.15]$, which decreases $\hat{\Gamma}$ with increasing in
$\Theta$, and for other values of $\omega$ or small $\Theta$ the effect of
non-commutativity becomes negligible.

###  III.1 NC correction to the Hawking temperature

It is worth to note that, in the Planck law, particles with frequency $\omega$
should be emitted at a $\Gamma\sim exp\left[-\omega/T\right]$ rate. Therefore,
the black hole temperature of our case (37)

$\displaystyle\hat{T}_{H}$
$\displaystyle=\frac{1}{2\pi\left(4m+\frac{3\Theta^{2}}{2m}\right)}=\frac{T_{H}}{\left(1+\frac{3\Theta^{2}}{2r_{h}^{2}}\right)}.$
(38)

where in the case of $\Theta=0$, the NC Hawking temperature is reduced to the
commutative one $T_{H}=\frac{1}{4\pi r_{h}}$.

Figure 2: The behavior of Hawking temperature as a function of the black hole
mass $r_{h}$.

In Fig. 2 we show the behavior of Hawking temperature as a function of the
black hole event horizon $r_{h}$. As we see the non-commutativity of spacetime
removes the divergence of the Hawking temperature, and leads the black hole to
reach a new maximum $\hat{T}^{max}_{H}\approx\frac{0.03249}{\Theta}$ at the
critical size $r_{h}^{crit}\approx 1.22475\,\Theta$ before turning to a zero
at $r_{h}=0$ at this point the black hole is completely evaporated.

For a massive black hole $\frac{\Theta}{m}\ll 1$, the above expression can be
developed at the second order in $\left(\Theta/m\right)$.

$\displaystyle\hat{T}_{H}=T_{H}\left(1-\frac{3\Theta^{2}}{8m^{2}}\right).$
(39)

It’s important to mention that, the above expression is equivalent to the one
obtained in our previous work using the surface gravity [44] (which is
analyzed in detail), and this result indicates that the non-commutativity
conserved the equivalence between the quantum tunneling process of evaporation
and the thermodynamic process.

The estimation of the NC parameter $\Theta$ can be obtained by the use of the
back-reaction point which means the maximum of temperature, at this point the
thermal energy is given by222In the natural system of unity
($\hbar=k_{B}=c=1$). $E_{th}=\hat{T}^{max}_{H}\approx\frac{0.03249}{\Theta}$
at the point $r_{h}^{crit}=2m\approx 1.22475\,\Theta$, and the total mass of
the black hole333In this study we use the reduced Planck mass
$M_{Planck}=2.435\times 10^{18}Gev$
$M=\frac{1}{2}r_{h}^{crit}\frac{1}{G}\approx
0.612375\,\Theta\,M_{Planck}^{2}$. The NC parameter can be estimated as
follows:

$\Theta\approx 1.854\times 10^{-35}m\sim l_{Planck}.$ (40)

This result is in the Planck scale, which is a good according to the one
obtained using the gravitational wave experimental data [45] and our previous
work using the thermodynamics process of the NC SBH [44]. However, there are
some paper that obtained a bound on the NC parameter through the study of the
black holes thermodynamics, e.g, [46, 47, 48, 49], and it’s expected to be
$\sqrt{\theta}\sim 10^{-1}.l_{p}$. According to this result and our previous
works [43, 44, 50], with some references in literature as [46, 47, 48, 49,
45], confirming that the NC property of spacetime appears close to the Planck
scale.

###  III.2 Logarithmic corrections to the entropy in NC spacetime

In this subsection, we will extract the entropy correction from the tunneling
process in the case of large frequency $\omega$. For that, let’s use the
definition (34) and integrate the relation (36) over the quantity
$d(\hat{m}-\omega)$444Where $\hat{m}=r^{NC}_{h}/2$ denotes the NC correction
to the black hole mass, then the element of integration in the NC case becomes
$dH=d(m-\omega)\rightarrow
d\hat{H}=\left(1-\frac{3\Theta^{2}}{32(m-\omega)^{2}}\right)d(m-\omega)$ from
the initial stat $m$ to the final one $m-\omega$ (conservation of energy), and
according to the Refs. [11, 13] the mass of black hole in (36) becomes
$m-\omega$

$\displaystyle Im\hat{S}=$
$\displaystyle-\pi\int_{m}^{m-\omega}\left(4(m-\omega)+\frac{3\Theta^{2}}{2(m-\omega)}\right)d(m-\omega)$
$\displaystyle+\pi\int_{m}^{m-\omega}\left(\frac{3\Theta^{2}}{8(m-\omega)}\right)d(m-\omega)+\mathcal{O}(\Theta^{4}),$
$\displaystyle=$
$\displaystyle-\pi\left(2(m-\omega)^{2}+\frac{9\Theta^{2}}{16}log\left(4\pi(m-\omega)^{2}\right)-2(m)^{2}\right.$
$\displaystyle\left.-\frac{9\Theta^{2}}{16}log\left(4\pi m^{2}\right)\right).$
(41)

the usual relation between the tunneling rate and entropy is recovered [11,
13, 12]

$\displaystyle\hat{\Gamma}\sim e^{-2Im\hat{S}}=e^{\Delta\hat{S}_{BH}}.$ (42)

where $\Delta\hat{S}_{BH}=\hat{S}_{BH}(m-\omega)-\hat{S}_{BH}(m)$ is the
difference of Bekenstein-Hawking entropy of the NC SBH.

$\displaystyle\hat{S}_{BH}=4\pi m^{2}+\frac{9\pi\Theta^{2}}{8}log(4\pi
m^{2}).$ (43)

As we see in the NC spacetime the relation of black hole area and entropy is
not conserved, and this result can be written in a similar way to the one
obtained with the string theory and loop quantum gravity [22]:

$\displaystyle\hat{S}_{BH}=\frac{A}{4}+\alpha\,log(A).$ (44)

where $A=4\pi m^{2}$ is the commutative area of SBH, and $\alpha$ is a
constant, which is defined with different values for each theory. In the
Quantum Loop Gravity case, this constant has a negative value
$\alpha=-\frac{1}{2}$ [51, 52], and in the String theory the constant $\alpha$
is a 4-D central charge, which depends on the number of field and can take a
negative and a positive values [53]. However, in our work using the NC gauge
theory of gravity, $\alpha$ has always a positive value, which is related to
the NC parameter by $\alpha=\frac{9\pi\Theta^{2}}{8}$, using our result of NC
parameter values (40), this constant can be related to the Planck length
$\alpha\sim\frac{9\pi}{4}l_{Planck}^{2}$, and that indicates in NC gauge
theory the coefficient of logarithm correction to the entropy represents a
quantum area at the Planck level, which means that the spacetime is quantized
at the Planck scale via the non-commutativity. In other hand the NC correction
to the tunneling rate (42) can be written as [22]

$\displaystyle\hat{\Gamma}=e^{\Delta\hat{S}_{BH}}=\left(1-\frac{\omega}{m}\right)^{\frac{9\pi}{4}l_{Planck}^{2}}e^{-8\pi
m\omega\left(1-\frac{\omega}{2m}\right)}.$ (45)

The above expression shows an exponential term with correction factor, in
which this exponential denotes the usual commutative case with non-thermal
radiation [22], and correction factor in our case of NC gauge theory depends
on the Planck length, which indicates a quantum correction that appears. In
the semi-classical limit we set $l_{Planck}=0$, and our expression reduces to
the commutative one [22].

###  III.3 NC correction to the density number of particles emitted

According to the Ref. [54, 55, 56] the density number of particles emitted is
related to the tunneling rate [57, 28], so in our case the density number from
the tunneling rate (37) can be written as [54]

$\displaystyle\hat{n}=\frac{\hat{\Gamma}}{1-\hat{\Gamma}}=\frac{1}{e^{8\pi
m\omega\left(1+\frac{3\Theta^{2}}{8m^{2}}\right)}-1},$ (46)

the obtained expression is in the form of the Planck distribution of the black
body radiation and it’s related to the NC parameter of the deformed geometry,
and when $\Theta=0$ the density number of particle creation is reduced to the
commutative case [1]:

$\hat{n}=\frac{1}{e^{8\pi m\omega}-1}.$ (47)

Figure 3: The density number behavior of particles emitted $\hat{n}$ from a NC
SBH as a function of particle frequency $\omega$ (left panel) with $m=1$ and
different NC parameter $\Theta$. (middle panel) The density number of
particles emitted as function of the NC parameter $\Theta$ with $m=1$ and
different frequency $\omega$. (right panel) The commutative behavior
$\Theta=0$ of $\hat{n}$ as function of the black hole mass $m$, with different
frequency $\omega$.

The behavior of density number of particle emitted from the tunneling process
in the NC SBH as a function of particle frequency $\omega$ is shown in Fig. 3
(left panel). The main observation in this figure is the effect the non-
commutativity, where it’s effect decreases the density number of particles
emitted by the black hole as $\Theta$ increases, and that means the non-
commutativity plays a similar role as potential well in quantum mechanics. In
(middle panel) we show the effect of the non-commutativity, in which decreases
the density number of particles emitted, and this observation is equivalent to
the commutative case when we increase the black hole mass $m$ the density
number of particles emitted by the black hole decreasing (right panel), That
means the non-commutativity plays a similar role as the black hole mass which
means non-commutativity increases the gravitational field of the black hole
and that explains the decreasing in the density number of particles escaping
from the black hole.

##  IV Conclusions

Based on the NC gauge gravity, we construct the deformed SBH using a general
form of tetrad fields. Then, we extend the semi-classical tunneling approach
[11, 13, 12] to the NC SBH, where we are studying tWO cases. Firstly, we
consider the tunneling particles from NC SBH are massless with lower
frequency, and that leads to a pure thermal radiation of black hole as it is
expected from the Hawking radiation, and we find the equivalence between the
quantum tunneling process and the thermodynamic process in the NC spacetime.
The analysis of the thermal radiation shows the effect of non-commutativity on
the tunneling rate of massless particles from the NC SBH, which is found
playing a similar role as well potential in quantum mechanics. Then we show a
new behavior of Hawking temperature in which the black hole stop radiation
when his completely evaporated $r_{h}=0$, and the estimation of $\Theta$ show
that the non-commutativity of spacetime appears at Planck scale.

Secondly, when conservation of energy [13] is taken into account with large
frequency, the tunneling rate deviates from thermal radiation, but it’s
consistent with an underlying unitary quantum theory in which the tunneling
rate is related to the change of black hole entropy. Then, we show that the NC
correction to the entropy of NC SBH is a logarithmic correction, which is
consistent with other theory of quantum gravity as the String theory and
Quantum Loop Gravity, and we show that the coefficient of logarithmic
correction in the NC gauge theory of gravity is related directly to the Planck
length $l_{Planck}$ and describes a quantum surface at Planck level.

The NC correction to the density number of particles creation by the NC SBH
are obtained in the pure thermal radiation (at lower frequency approximation),
and the result confirms the effect of the non-commutativity in which decreases
the density number of particles.

###### Acknowledgements.

This work is supported by PRFU Research Project B00L02UN050120230003, Univ.
Batna 1, Algeria.

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