# Quotient sets
in nonabelian groups

Vsevolod F. Lev<EMAIL_ADDRESS>Department of Mathematics, The
University of Haifa at Oranim, Tivon 36006, Israel

###### Abstract.

We show that for a finite, nonempty subset $A$ of a group, the quotient set
$A^{-1}A:=\\{a_{1}^{-1}a_{2}\colon a_{1},a_{2}\in A\\}$ has size
$|A^{-1}A|\geq\frac{5}{3}\,|A|$, unless $A$ is densely contained in a coset,
or in a union of two cosets of a finite subgroup.

###### Key words and phrases:

Small doubling, Sumset, Kneser’s theorem

###### 2020 Mathematics Subject Classification:

Primary 11B75

##  1\. Introduction: Background and the Main Result

One of the cornerstones of the additive combinatorics is Kneser’s theorem
[Kn53, Kn55] relating the size of a sumset in an abelian group to the sizes of
the set summands. As shown by Olson [O84], a straightforward, simple-minded
analogue of Kneser’s theorem for nonabelian groups fails to hold. Many partial
extensions of the theorem in the nonabelian settings are known, however; see,
for instance, [F73, O84, O86, SW, T13, Z94, H13]. Particularly relevant in our
present context are the papers by Freiman [F73], Olson [O86], and Hamidoune
[H13].

In [F73], Freiman classified finite, nonempty subsets $A$ of a group with the
product set $A^{2}:=\\{a_{1}a_{2}\colon a_{1},a_{2}\in A\\}$ satisfying
$|A|^{2}<\frac{8}{5}\,|A|$.

Olson has extended Freiman’s result onto products with distinct set factors;
namely, as shown in [O84, Theorem 1], if $A$ and $B$ are finite, nonempty
subsets of a group, then “normally” the product set $AB:=\\{ab\colon a\in
A,b\in B\\}$ has size $|AB|\geq|A|+\frac{1}{2}\,|B|$.

Improving the _doubling coefficients_ $\frac{8}{5}$ and $\frac{1}{2}$ in the
Freiman-Olson estimates is a fascinating, mostly open, problem.

Addressing the case where $B=A^{-1}$, Hamidoune [H13] has established some
properties of the quotient set $A^{-1}A:=\\{a_{1}^{-1}a_{2}\colon
a_{1},a_{2}\in A\\}$ assuming that $|A^{-1}A|<\frac{5}{3}\,|A|$.

In this note, under the same assumption $|A^{-1}A|<\frac{5}{3}\,|A|$, we
completely determine the structure of the set $A$ itself, with an if-and-only-
if-type classification.

For a subgroup $H$ of a group $G$, let $N(H)$ denote the normalizer of $H$ in
$G$.

######  Theorem 1.

Let $A$ be a finite subset of a group $G$. Then $|A^{-1}A|<\frac{5}{3}\,|A|$
if and only if one of the following holds:

  * (i)

there is a finite subgroup $H\leq G$ such that $A$ is contained in a left
$H$-coset and $|A|>\frac{3}{5}\,|H|$;

  * (ii)

there is a finite subgroup $H\leq G$ and elements $a,b\in G$ with
$(a^{-1}b)^{2}\notin H$ and $a^{-1}b\in N(H)$ such that $A\subseteq aH\cup bH$
and $|A|>\frac{9}{5}\,|H|$.

Moreover, in the case (i) we have $A^{-1}A=H$, while in the case (ii) the set
$A^{-1}A$ is a disjoint union of $H$ and two double $H$-cosets of size $|H|$
each.

The coefficient $\frac{5}{3}$ corresponds to a structure threshold: say, if
$H$ is a finite subgroup, and $g\in N(H)$ is an element with $g^{i}\notin H$
for $i\in[1,4]$, then the set $A:=g^{-1}H\cup H\cup Hg$ satisfies
$|A^{-1}A|=\frac{5}{3}\,|A|$, while $A$ does not have the structure described
in the theorem.

We remark that the ostensible lack of symmetry in the statement of the theorem
clears off once we notice that any left coset is a right coset of a conjugate
subgroup, and vice versa.

With the exception of Section 5, the rest of the paper is devoted to the proof
of Theorem 1. In the next section we formally introduce the notation used and
gather some basic facts needed for the proof. In Section 3 we show that
conditions (i) and (ii) of Theorem 1 are sufficient for
$|A^{-1}A|<\frac{5}{3}\,|A|$ to hold, and also that they imply the last
assertion of the theorem (concerning the structure of the quotient set).
Additionally, in Section 3 we prove a lemma that will be used in the course of
the proof of necessity in Section 4. Finally, in the concluding Section 5 we
state and briefly discuss a conjectural extension of Theorem 1 onto the sets
$A$ satisfying $|A^{-1}A|<2|A|$.

##  2\. Preliminaries: notation and tools

For subsets $A$ and $B$ of a group, we denote by $A^{-1}$ the set of inverses
of the elements of $A$, and by $AB$ the product set:

$A^{-1}:=\\{a^{-1}\colon a\in A\\}\quad\text{and}\quad AB:=\\{ab\colon a\in
A,b\in B\\}.$

Thus, for instance, $A^{-1}A=\\{a^{-1}b\colon a,b\in A\\}$.

The subgroup generated by $A$ is denoted by $\langle A\rangle$, and the
identity element of the group by $1$. A left (right) coset of a subgroup $H$
is a set of the form $gH$ ($Hg$), where $g$ is an element of the group. A
_double_ $H$-coset if a set of the form $HgH$.

The following lemma summarizes the basic properties of double cosets.

######  Lemma 1.

If $H$ is a subgroup of a group $G$, then $G$ is a disjoint union of double
$H$-cosets. A set $S\subseteq G$ is a union of double $H$-cosets if and only
if it is stable under both left and right multiplication by $H$; that is, if
and only if $HS=SH=S$; alternatively, if and only if there is a set
$T\subseteq G$ such that $S=HTH$. For $a,b\in G$, we have $HaH=HbH$ if and
only if there exist $h_{1},h_{2}\in H$ such that $b=h_{1}ah_{2}$.

The _normalizer_ of a subgroup $H$, denoted $N(H)$, is the subgroup consisting
of all those group elements $g$ satisfying $gH=Hg$.

Given a finite subset $A$ of a group, and a group element $g$, by $r(g)$ we
denote the number of representations of $g$ in the form $g=a^{-1}b$ with
$a,b\in A$. Clearly, $r$ is supported on the quotient set $A^{-1}A$, and
$r(g^{-1})=r(g)\leq r(1)=|A|$ for any element $g$. Moreover, $r(g)=|A|$ if and
only if $Ag=A$. Therefore, the number of elements $g\in G$ satisfying $Ag=A$
is the size of the maximal subgroup $H$ such that $A$ is a union of left
$H$-cosets.

For a real $x$, the largest integer not exceeding $x$ and the smallest integer
not smaller than $x$ are denoted $\left\lfloor x\right\rfloor$ and
$\left\lceil x\right\rceil$, respectively.

######  Lemma 2.

For any finite subgroup $H$ and any group elements $a$ and $b$, either
$aH=Hb$, or $|aH\cap Hb|\leq\frac{1}{2}\,|H|$.

###### Proof.

Assuming that $aH\cap Hb$ is nonempty, fix an element $g\in aH\cap Hb$. Then
$a\in gH$ and $b\in Hg$, whence $|aH\cap Hb|=|gH\cap Hg|=|H\cap g^{-1}Hg|$.
The result follows since the intersection in the right-hand side is a subgroup
of $H$. ∎

######  Lemma 3.

Suppose that $H$ is a finite subgroup, and $a,b$ are elements of a group. For
$aH=Hb$ to hold, it is necessary and sufficient that $a,b\in N(H)$ and
$aH=bH$.

###### Proof.

If $aH=Hb$, then $b\in aH$ whence $a^{-1}b\in H$; equivalently, $b^{-1}a\in
H$, or $a\in bH$. As a result, $aH=bH$, and, consequently, $Hb=bH$, so that
$b\in N(H)$. In a similar way we get $a\in N(H)$. The opposite direction is
trivial: if $a,b\in N(H)$ and $aH=bH$, then $aH=bH=Hb$. ∎

######  Lemma 4.

Suppose that $H$ is a subgroup, and $g\notin H$ is an element of a group. For
the union $H\cup gH$ to be a subgroup, it is necessary and sufficient that
$g^{2}\in H$.

###### Proof.

If $H\cup gH$ is a subgroup, then $g^{-1}\in H\cup gH$ whence, indeed,
$g^{-1}\in gH$, and then $g^{2}\in H$. Conversely, if $g^{2}\in H$, then
$H\cup gH$ is easily seen to be closed under the “skew multiplication”
$(a,b)\mapsto a^{-1}b$. ∎

######  Lemma 5.

Suppose that $H$ is a finite subgroup of a group $G$. For an element $g\in G$,
the double $H$-coset $HgH$ has size $|HgH|=|H|$ if and only if $g\in N(H)$.

###### Proof.

Write $S=HgH$. Then $gH\subseteq S$ and $Hg\subseteq S$. Hence, $|S|=|H|$ if
and only if $gH=Hg$; that is, if and only if $g\in N(H)$. ∎

We will use the box principle in the following form.

######  Lemma 6.

Suppose that $A$ is a finite, nonempty subset of a group $G$. If
$g_{1},g_{2}\in G$ are group elements with $r(g_{1})+r(g_{2})>|A|$, then
$g_{1}^{-1}g_{2}\in A^{-1}A$.

###### Proof.

For $i\in\\{1,2\\}$, let $A_{i}$ be the set of all those elements $a\in A$ the
inverse of which appears as the first factor in some representation
$g_{i}=a^{-1}b$ with $b\in A$. Thus, $|A_{i}|=r(g_{i})$, and from
$r(g_{1})+r(g_{2})>|A|$ and $A_{1},A_{2}\subseteq A$ it follows that $A_{1}$
and $A_{2}$ have a common element; that is, there are $a,b_{1},b_{2}\in A$
such that $g_{1}=a^{-1}b_{1}$ and $g_{2}=a^{-1}b_{2}$. Consequently,
$g_{1}^{-1}g_{2}=b_{1}^{-1}aa^{-1}b_{2}=b_{1}^{-1}b_{2}\in A^{-1}A$. ∎

We need the following result of Kemperman and Wehn.

######  Theorem 2 (Kemperman-Wehn).

If $A$ and $B$ are finite, nonempty subsets of a group, then
$|AB|\geq|A|+|B|-r(g)$ for any element $g\in AB$.

Quoting from [O84],

> “Theorem 2 goes back to results of L. Moser and P. Scherk in the case of
> abelian groups, and was proved for nonabelian groups by J. H. B. Kemperman
> and (independently) D. F. Wehn. For proof see Kemperman’s paper [K56].”

##  3\. Proof of Theorem 1: sufficiency

If $G$ is a group, $H$ is a finite subgroup of $G$, and $A$ is a subset of $G$
contained in an $H$-coset and satisfying $|A|>\frac{3}{5}\,|H|$, then
$A^{-1}A=H$ by the box principle, whence $|A^{-1}A|<\frac{5}{3}\,|A|$. Thus,
condition (i) of the theorem is sufficient for $A$ to satisfy
$|A^{-1}A|<\frac{5}{3}\,|A|$, and it also implies the corresponding part of
the last assertion of the theorem. We now prove a similar result for condition
(ii).

######  Proposition 1.

Let $H$ be a finite subgroup of a group $G$, and suppose that $A\subseteq
aH\cup bH$ where $a,b\in G$ are elements with $a^{-1}b\in N(H)$ and
$(a^{-1}b)^{2}\notin H$. If $|A|>\frac{9}{5}\,|H|$, then
$|A^{-1}A|<\frac{5}{3}\,|A|$; moreover, in this case $A^{-1}A$ is a disjoint
union of $H$ and two double $H$-cosets of size $|H|$ each.

###### Proof.

The assumption $|A|>\frac{9}{5}\,|H|$ implies that the cosets $aH$ and $bH$
are disjoint. We write $A=aX\cup bY$ with $X,Y\subseteq H$ and notice that

$A^{-1}A=(X^{-1}a^{-1}\cup Y^{-1}b^{-1})(aX\cup bY)\\\
=((X^{-1}X)\cup(Y^{-1}Y))\cup(X^{-1}a^{-1}bY)\cup(Y^{-1}b^{-1}aX).$ (1)

Since $|X|+|Y|=|A|>|H|$, we have either $|X|>\frac{1}{2}\,|H|$, or
$|Y|>\frac{1}{2}\,|H|$; accordingly, by the box principle, either $X^{-1}X=H$,
or $Y^{-1}Y=H$. Thus, $(X^{-1}X)\cup(Y^{-1}Y)=H$. Furthermore, since
$a^{-1}b\in N(H)$, we have $a^{-1}bY\subseteq a^{-1}bH=Ha^{-1}b$;
consequently, there is a subset $Y^{\prime}\subseteq H$ such that
$a^{-1}bY=Y^{\prime}a^{-1}b$, and then
$X^{-1}a^{-1}bY=X^{-1}Y^{\prime}a^{-1}b=Ha^{-1}b$, as $X^{-1}Y^{\prime}=H$ in
view of $|X^{-1}|+|Y^{\prime}|=|X|+|Y|>|H|$. Therefore $|X^{-1}a^{-1}bY|=|H|$.
Taking the inverses, we get $|Y^{-1}b^{-1}aX|=|H|$. Hence, $|A^{-1}A|\leq
3|H|<\frac{5}{3}\,|A|$. Next, $X^{-1}a^{-1}bY=Ha^{-1}b=a^{-1}bH$ shows that
$X^{-1}a^{-1}bY$ is a double $H$-coset, and so is its inverse
$Y^{-1}b^{-1}aX=Hb^{-1}a=b^{-1}aH$. Finally, the two double $H$-cosets are
disjoint from $H$ and from each other thanks to the assumption
$(a^{-1}b)^{2}\notin H$. ∎

Next, we prove a lemma that provides a simple criterion for a given set to
satisfy conditions (i) and (ii) of Theorem 1; this lemma will be used in the
proof of necessity in the next section.

######  Lemma 7.

Let $H$ be a finite subgroup of a group $G$, and suppose that $A\subseteq
aH\cup bH$ where $a,b\in G$. If $|A|>\frac{9}{5}\,|H|$ and $|A^{-1}A|\leq
3|H|$, then $A$ satisfies either condition (i), or condition (ii) of Theorem
1, according to whether $(a^{-1}b)^{2}\in H$ or $(a^{-1}b)^{2}\notin H$.

###### Proof.

As in the proof of Proposition 1, we write $A=aX\cup bY$ with $X,Y\subseteq
H$, and use (1). If $a^{-1}b\notin N(H)$, then $Ha^{-1}b\neq a^{-1}bH$ by the
definition of the normalizer subgroup; hence,

$|X^{-1}a^{-1}b\cap a^{-1}bY|\leq|Ha^{-1}b\cap a^{-1}bH|\leq\frac{1}{2}\,|H|$

by Lemma 4. Without loss of generality, we assume $a,b\in A$ whence $1\in
X\cap Y$. Consequently, both $X^{-1}a^{-1}b$ and $a^{-1}bY$ lie in
$X^{-1}a^{-1}bY$, and we conclude that

$|X^{-1}a^{-1}bY|\geq|X^{-1}a^{-1}b\cup a^{-1}bY|=|X|+|Y|-|X^{-1}a^{-1}b\cap
a^{-1}bY|>|H|.$

Taking the inverses, we get $|Y^{-1}b^{-1}aX|>|H|$, and then from (1) we
obtain $|A^{-1}A|>3|H|$, contradicting the assumptions. Thus, $a^{-1}b\in
N(H)$. By Lemma 3, the set $F:=H\cup(a^{-1}b)H$ is a subgroup if and only if
$(a^{-1}b)^{2}\in H$. In this case $A\subseteq aF$ and
$|A|>\frac{9}{5}\,|H|=\frac{9}{10}\,|F|>\frac{3}{5}\,|F|$ so that $A$
satisfies condition (i). Finally, if $(a^{-1}b)^{2}\notin H$, then $A$
satisfies condition (ii). ∎

##  4\. Proof of Theorem 1: necessity

Let $A\subseteq G$ be a finite subset with $|A^{-1}A|<\frac{5}{3}\,|A|$, and
suppose that the assertion is true for all sets ${\mathcal{A}}\subseteq G$
satisfying either $|{\mathcal{A}}^{-1}{\mathcal{A}}|<|A^{-1}A|$, or
$|{\mathcal{A}}^{-1}{\mathcal{A}}|=|A^{-1}A|$ and $|{\mathcal{A}}|>|A|$. We
show that $A$ is contained ether in a coset, or in a union of two cosets, as
specified in the conditions (i) and (ii) of the theorem.

We write $Q:=A^{-1}A$; thus, $|A|>\frac{3}{5}\,|Q|$.

Recall, that for an element $g\in G$, we have denoted by $r(g)$ the number of
representations $g=a^{-1}b$ with $a,b\in A$. By Theorem 2,

$r(g)\geq 2|A|-|Q|,\quad g\in Q.$

Let $Q^{+}:=\\{g\in Q\colon r(g)>|Q|-|A|\\}$. We notice that $Q^{+}$ is
nonempty as, for instance, it contains the identity element. Also, $Q^{+}$ is
stable under inversion.

For any $g\in Q$ and $g_{0}\in Q^{+}$ we have

$r(g)+r(g_{0})>(2|A|-|Q|)+(|Q|-|A|)=|A|.$

Hence, $g_{0}^{-1}g\in Q$ by Lemma 6, implying $g\in g_{0}Q$. It follows that
$g_{0}Q=Q$ for any $g_{0}\in Q^{+}$. Therefore, $Q^{+}Q=Q$ and, considering
the inverses, $QQ^{+}=Q$. Letting $F:=\langle Q^{+}\rangle$, we furthermore
conclude that $QF=FQ=Q$. As a result,

$(AF)^{-1}(AF)=FA^{-1}AF=FQF=FQ=Q=A^{-1}A.$ (2)

From these equalities and by the choice of $A$, either $AF=A$, or there is a
finite subgroup $H\leq G$ such that one of the following holds:

  * –

$AF$ is contained in a left $H$-coset, $|AF|>\frac{3}{5}\,|H|$, and
$(AF)^{-1}(AF)=H$;

  * –

$AF$ meets exactly two left $H$-cosets, $|AF|>\frac{9}{5}\,|H|$, and
$(AF)^{-1}(AF)$ is a disjoint union of $H$ and two double $H$-cosets of size
$|H|$ each.

In the first case, recalling (2) we get $Q=(AF)^{-1}(AF)=H$; as a result, $A$
is contained in a single left $H$-coset, and
$|A|>\frac{3}{5}\,|Q|=\frac{3}{5}\,|H|$; thus, $A$ satisfies condition (i).

In the second case, with (2) in mind, $|Q|=|(AF)^{-1}(AF)|=3|H|$ showing that
$|A|>\frac{3}{5}\,|Q|=\frac{9}{5}\,|H|$. By Lemma 7, the set $A$ satisfies
condition (i) or condition (ii).

Having ruled out the exceptional cases where $AF$ is contained in a single
coset, or in a union of two cosets, we proceed with the proof using the
additional assumption

$AF=A,\quad F=\langle Q^{+}\rangle.$ (3)

Thus, $A$ is a union of left $F$-cosets, and so is $Q=\cup_{a\in A}a^{-1}A$.
It follows that $F\subseteq Q$. Indeed, we have $F=Q^{+}$: here the inclusion
$Q^{+}\subseteq F$ is trivial, while $F\subseteq Q^{+}$ follows by observing
that if $g\in F$, then $r(g)=|A|>|Q|-|A|$ by (3), whence $g\in Q^{+}$.

As we have just observed, if $g\in F$, then $r(g)=|A|$. Conversely, if $g\in
Q$ is an element with $r(g)=|A|$, then $r(g)>|Q|-|A|$ showing that $g\in
Q^{+}=F$. As a bottom line, $r(g)=|A|$ if and only if $g\in F$.

If $g\in G$ is an element with $r(g)>|Q|-|A|$, then $g\in Q^{+}=F$, whence,
indeed, $r(g)=|A|$. Thus, we have

$2|A|-|Q|\leq r(g)\leq|Q|-|A|$

for all elements $g\in Q$ with $r(g)<|A|$. We remark that the first of the two
inequalities is just Theorem 2, but the second one is new and, in our view, is
quite amazing.

We write $k:=|A|/|F|$. If $k=1$, then $A$ is a single left $F$-coset;
therefore $A$ satisfies condition (i). If $k=2$, then $A$ is a union of two
left $F$-cosets and $|A^{-1}A|<\frac{5}{3}|A|<4|F|$; therefore, applying Lemma
7, we conclude that $A$ satisfies condition (ii). Suppose thus that $k\geq 3$.

Since $A$ and $Q$ are unions of left $F$-cosets, both $|A|$ and $|Q|$ are
divisible by $|F|$. From this observation and
$|Q|<\frac{5}{3}\,|A|=\frac{5}{3}\,k|F|$, we get

$|Q|\leq\left(\left\lceil\frac{5}{3}\,k\right\rceil-1\right)\,|F|.$ (4)

Furthermore, to any representation $g=a^{-1}b$ with $a,b\in A$ there
correspond $|F|$ representations $g=(fa)^{-1}(fb),\ f\in F$. (Notice that
these representations are “legal” in the sense that both $af$ and $bf$ lie in
$A$.) Therefore, also $r(g)$ is divisible by $|F|$, for any $g\in G$.
Moreover, since $g$ is constant on any left $F$-coset, for any given positive
integer $m$, the number of elements $g\in Q$ with $r(g)=m$ is divisible by
$|F|$;

Let $Q_{0}$ and $Q_{1}$ denote the sets of all those elements $g\in Q$ with
$r(g)\leq\frac{1}{2}\,|A|$ and with $r(g)>\frac{1}{2}\,|A|$, respectively. We
write $N_{0}:=|Q_{0}|$ and $N_{1}:=|Q_{1}|$ and define

$\sigma_{0}:=\sum_{g\in Q_{0}}r(g)\ \text{and}\ \sigma_{1}:=\sum_{g\in
Q_{1}}r(g);$

thus, $N_{0}+N_{1}=|Q|$, $\sigma_{0}+\sigma_{1}=|A|^{2}$, and
$N_{0},N_{1},\sigma_{0}$, and $\sigma_{1}$ are all divisible by $|F|$. The sum
$\sigma_{0}$ has $N_{0}$ terms, each of them divisible by $|F|$ and not
exceeding $\frac{1}{2}\,|A|=\frac{1}{2}\,k|F|$; therefore,
$\sigma_{0}\leq\left\lfloor\frac{1}{2}\,k\right\rfloor|F|N_{0}$. The sum
$\sigma_{1}$ has $N_{1}$ terms, of them $|F|$ are equal to $|A|$, and each of
the remaining $N_{1}-|F|$ terms does not exceed $|Q|-|A|$. Therefore,

$\sigma_{1}\leq|F||A|+(N_{1}-|F|)(|Q|-|A|)=2|F||A|-|Q||F|+N_{1}|Q|-N_{1}|A|.$

Letting $n:=N_{1}/|F|$ and $q:=|Q|/|F|$, we obtain

$\displaystyle|A|^{2}$
$\displaystyle\leq\left\lfloor\frac{1}{2}\,k\right\rfloor|F|N_{0}+2|F||A|-|Q||F|+N_{1}|Q|-N_{1}|A|,$
$\displaystyle k^{2}|F|$
$\displaystyle\leq\left\lfloor\frac{1}{2}\,k\right\rfloor(N_{0}+N_{1})+2|A|-|Q|+\left(q-\left\lfloor\frac{1}{2}k\right\rfloor\right)N_{1}-kN_{1},$
$\displaystyle k^{2}$
$\displaystyle\leq\left(\left\lfloor\frac{1}{2}\,k\right\rfloor-1\right)\,q+2k+\left(q-k-\left\lfloor\frac{1}{2}\,k\right\rfloor\right)n.$

Since $q\leq\left\lceil\frac{5}{3}\,k\right\rceil-1$ by (4), we derive that

$k^{2}\leq\left(\left\lfloor\frac{1}{2}\,k\right\rfloor-1\right)\left(\left\lceil\frac{5}{3}\,k\right\rceil-1\right)+2k+\left(\left\lceil\frac{5}{3}\,k\right\rceil-1-k-\left\lfloor\frac{1}{2}\,k\right\rfloor\right)n.$

A routine analysis shows that for $k\geq 3$, the last inequality is false if
$n\leq k$. (Hint: exact computation for $3\leq k\leq 6$, substituting $n=k$
and using the crude estimates $\left\lfloor k/2\right\rfloor\leq k/2$ and
$\left\lceil 5k/3\right\rceil\leq(5k+2)/3$ for $k\geq 7$.) Therefore $n\geq
k+1$; that is,

$|Q_{1}|=N_{1}\geq|A|+|F|\quad(k\geq 3).$ (5)

From Lemma 6 and the definition of the set $Q_{1}$, we have
$g_{1}^{-1}g_{2}\in Q$ for any $g_{1},g_{2}\in Q_{1}$. Consequently,
$Q_{1}^{-1}Q_{1}\subseteq Q$ whence, by the choice of $A$, there is a finite
subgroup $H\leq G$ such that one of the following holds:

  * 1\.

$Q_{1}$ is contained in a left $H$-coset and $|Q_{1}|>\frac{3}{5}\,|H|$;

  * 2\.

$Q_{1}$ meets exactly two left $H$-cosets and $|Q_{1}|>\frac{9}{5}\,|H|$.

We investigate these two cases separately.

Case 1: There is a finite subgroup $H\leq G$ such that $Q_{1}$ is contained in
a left $H$-coset and $|Q_{1}|>\frac{3}{5}\,|H|$. We have

$|A|<|Q_{1}|\ \text{and}\ Q_{1}\subseteq H=Q_{1}^{-1}Q_{1}\subseteq Q;$ (6)

here the inequality follows from (5), the first inclusion from $1\in Q_{1}$,
the equality from $|Q_{1}|>\frac{3}{5}\,|H|$ and the box principle, and the
second inclusion from Lemma 6.

Consider the coset decomposition $A=A_{1}\cup\dotsb\cup A_{n}$ where
$n=|AH|/|H|$ and $A_{1},\dotsc,A_{n}$ are nonempty and reside in pairwise
distinct left $H$-cosets. We number the sets $A_{i}$ so that
$|A_{1}|=\min\\{|A_{i}|\colon 1\leq i\leq n\\}$. Fix $a_{1}\in A_{1}$. In view
of $a_{1}^{-1}A_{2}\cup\dotsb\cup a_{1}^{-1}A_{n}\subseteq Q\setminus H$ and
(6),

$|Q|\geq|H|+(|A_{2}|+\dotsb+|A_{n}|)>|A|+\left(1-\frac{1}{n}\right)\,|A|=\left(2-\frac{1}{n}\right)\,|A|.$

Since $|Q|<\frac{5}{3}\,|A|$, we conclude that $n=1$ or $n=2$. If $n=1$, then
$A$ resides in a single $H$-coset; moreover,
$|A|>\frac{3}{5}\,|Q|>\frac{3}{5}\,|H|$, showing that $A$ satisfies condition
(i).

Suppose now that $n=2$. Fix $a_{1}\in A_{1}$ and $a_{2}\in A_{2}$. Then

$\frac{5}{3}\,|A|>|Q|=|H|+|Q\setminus
H|\geq|H|+|a_{1}^{-1}A_{2}|=|H|+(|A|-|A_{1}|)$

whence $|A_{1}|>|H|-\frac{2}{3}\,|A|\geq\frac{1}{3}\,|A|$. Similarly, from

$\frac{5}{3}\,|A|>|Q|=|H|+|Q\setminus H|\geq|H|+|a_{1}^{-1}A_{2}|=|H|+|A_{2}|$

we obtain $|A_{2}|<\frac{2}{3}\,|A|$. Therefore,

$\frac{1}{3}\,|H|<|A_{1}|\leq\frac{1}{2}\,|A|\leq|A_{2}|<\frac{2}{3}\,|A|.$
(7)

Consider the set $S:=(A_{1}\times A_{2})\cup(A_{2}\times A_{1})$ and the
mapping $\varphi\colon S\to Q$ defined by $\varphi(a,b):=a^{-1}b$. Since the
image $\mathrm{Im}(\varphi)$ is disjoint from $H$, we have
$\mathrm{Im}(\varphi)\subseteq Q\setminus Q_{1}$. By the definition of the set
$Q_{1}$, every element of $\mathrm{Im}(\varphi)$ has at most
$\frac{1}{2}\,|A|$ inverse images in $S$. As a result,

$|Q|-|Q_{1}|\geq\mathrm{Im}(\varphi)\geq\frac{|S|}{|A|/2}=4\frac{|A_{1}||A_{2}|}{|A|}.$

Comparing this estimate with (5) and with the assumption
$|Q|<\frac{5}{3}\,|A|$, we obtain $|A|^{2}>6|A_{1}||A_{2}|$. This leads to
$|A_{2}|>(2+\sqrt{3})|A_{1}|$, contradicting (7).

Case 2: There is a finite subgroup $H\leq G$ such that $Q_{1}$ meets exactly
two left $H$-cosets and $|Q_{1}|>\frac{9}{5}\,|H|$. Since $1\in Q_{1}$, we can
write $Q_{1}=B_{0}\cup B_{1}$ where $B_{0}\subseteq H$ and $B_{1}\subseteq gH$
with some $g\in G\setminus H$. From $B_{1}=Q_{1}\setminus H$ and
$Q_{1}^{-1}=Q_{1}$ we get $B_{1}^{-1}=Q_{1}^{-1}\setminus H=Q_{1}\setminus
H=B_{1}\subseteq gH$. Thus, $B_{1}\subseteq gH\cap Hg^{-1}$. By Lemma 2, and
in view of

$|B_{1}|=|Q_{1}|-|B_{0}|\geq|Q_{1}|-|H|>\frac{4}{5}\,|H|$

we have $gH=Hg^{-1}$; that is, $H=gHg$, and it is easily seen that $H\cup gH$
is a subgroup. Moreover, $Q_{1}=B_{0}\cup B_{1}\subseteq H\cup gH$ and
$|Q_{1}|>\frac{9}{5}\,|H|>\frac{3}{5}\,|H\cup gH|$. This takes us back to the
Case 1 considered above.

##  5\. Concluding remarks

What is the structure of a finite set $A$ with
$\frac{5}{3}\,|A|<|A^{-1}A|<2|A|$? We make the following conjecture.

######  Conjecture 1.

Let $A$ be a finite subset of a group $G$, and let $n$ be a positive integer.
If

$|A^{-1}A|<\left(2-\frac{1}{n+1}\right)\,|A|,$

then there are a finite subgroup $H\leq G$ and a subset $A_{0}\subseteq A$ of
size $|A_{0}|\leq n$ contained in a single left $N(H)$-coset such that
$A\subseteq A_{0}H$, $|A_{0}H|=|A_{0}||H|$, and
$|A|>\left(2-\frac{1}{n+1}\right)^{-1}(2|A_{0}|-1)|H|$.

Moreover, $A^{-1}A=A_{0}^{-1}A_{0}H$ and $|A^{-1}A|=(2|A_{0}|-1)H$.

The inequality $|A|>\left(2-\frac{1}{n+1}\right)^{-1}(2|A_{0}|-1)|H|$ is worth
commenting on. It can be shown that, along with other conclusions of the
conjecture, it implies $|A|\leq|A_{0}H|<|A|+\frac{n}{2n+1}\,|H|$. Thus, this
inequality ensures that $A$ is a _dense_ subset of the set $A_{0}H$.

The particular case $n=1$ of the conjecture follows from Olson’s theorem,
while the case $n=2$ is the main result of this paper.

As the following proposition shows, in the appropriate range, Conjecture 1
gives a necessary and sufficient condition for $A$ to satisfy
$|A^{-1}A|<\left(2-\frac{1}{n+1}\right)\,|A|$.

######  Proposition 2.

Let $A$ be a finite subset of a group $G$, and let $n$ be a positive integer.
Suppose that there are a finite subgroup $H\leq G$ and a subset
$A_{0}\subseteq A$ of size $|A_{0}|\leq n$ contained in a single left
$N(H)$-coset such that $A\subseteq A_{0}H$, $|A_{0}H|=|A_{0}||H|$, and
$|A|>\left(2-\frac{1}{n+1}\right)^{-1}(2|A_{0}|-1)|H|$. If, in addition,
$|A^{-1}A|<2|A|$ then, indeed, $|A^{-1}A|<\left(2-\frac{1}{n+1}\right)\,|A|$.

We omit the proof since, anyway, Proposition 2 is not of much importance as
long as Conjecture 1 remains open.

## References

  * [F73] G. Freiman, Groups and the inverse problems of the additive number theory (Russian), _Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian)_ , 175–183, Kalinin. Gosudarstv. Univ., Moscow, 1973.
  * [H13] Y.O. Hamidoune, Two inverse results, _Combinatorica_ 33 (2) (2013), 217–230.
  * [Kn53] M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, _Math. Z._ 58 (1953), 459–484.
  * [Kn55] by same author, Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen, _Math. Z._ 61 (1955), 429–434.
  * [K56] J.H.B. Kemperman, on complexes in a semigroup, _Indag. Math._ 18 (1956), 247–254.
  * [O84] J.E. Olson, On the sum of two sets in a group, _Journal of Number Theory_ 18 (1984), 110–120.
  * [O86] by same author, On the symmetric difference of two sets in a group, European J. Combin. 7 (1) (1986), 43–54.
  * [SW] O. Serra and M. Wötzel, On a nonabelian Kneser theorem, _preprint_.
  * [T13] T. Tao, Noncommutative sets of small doubling, _European Journal of Combinatorics_ 34 (8) (2013), 1459–1465.
  * [Z94] G. Zemor, A generalization to noncommutative groups of a theorem of Mann, _Discrete Mathematics_ 126 (1994), 365–372.


# Quantum Advantage of Thermal Machines with Bose and Fermi Gases

Saikat Sur<EMAIL_ADDRESS>Department of Chemical and Biological
Physics, Weizmann Institute of Science, Rehovot 7610001, Israel  Arnab Ghosh
Department of Chemistry, Indian Institute of Technology, Kanpur 208016, India

(September 3, 2024)

###### Abstract

In this article, we show that a quantum gas, a collection of massive, non-
interacting, indistinguishable quantum particles can be realized as a
thermodynamic machine as an artifact of energy quantization and hence bears no
classical analog. Such a thermodynamic machine depends on the statistics of
the particles, the chemical potential, and the spatial dimension of the
system. Our detailed analysis demonstrates the fundamental features of quantum
Stirling cycles from the viewpoint of particle statistics and system
dimensions that helps us to realize desired quantum heat engines and
refrigerators by exploiting the role of quantum statistical mechanics. In
particular, a clear distinction between the behavior of a Fermi gas and a Bose
gas is observed in one dimension than in higher dimensions, solely due to the
innate differences in their particle statistics indicating the conspicuous
role of a quantum thermodynamic signature in lower dimensions.

##  I Introduction

The study of quantum thermodynamics comprises the basis for analyzing heat
engines and refrigerators at the microscopic level Scully (2002); Kosloff
(2013); Binder  _et al._ (2019); Deffner and Campbell (2019); Mukherjee and
Divakaran (2021); Vinjanampathy and Anders (2016); Bhattacharjee and Dutta
(2021); Opatrný  _et al._ (2021); Misra  _et al._ (2022). Although
thermodynamics is quite successful in the classical regime, its applications
towards quantum systems must be reconciled in view of energy quantization,
degeneracies, and most importantly particle statistics Zheng and Poletti
(2014); Gelbwaser-Klimovsky  _et al._ (2018a); Niedenzu  _et al._ (2019a);
Deffner (2019); Thomas  _et al._ (2019); Myers and Deffner (2020); Myers  _et
al._ (2021); Gupt  _et al._ (2021); Myers and Deffner (2021); Bouton  _et al._
(2021); Benjamin Yadin (2017). A fundamental departure from classical
mechanics is therefore imperative for analyzing the thermodynamic behavior of
quantum systems. As a result, one might expect new thermodynamic effects would
emerge in the quantum domain without having any classical correspondence. Most
of the cases that have been analyzed for understanding the generic
thermodynamic features in quantum thermal machines, do not exhibit any proper
evidence of quantum advantage over their classical counterparts Scully  _et
al._ (2003); Roßnagel  _et al._ (2014); Klaers  _et al._ (2017); Niedenzu  _et
al._ (2018); Ghosh  _et al._ (2019). In recent years, several proposals for
constructing quantum mechanical versions of thermal machines such as Otto,
Carnot, Stirling and Diesel Salamon  _et al._ (1980); Chen and Yan (1991);
Geva and Kosloff (1992a, b); Chen (1994); Chen and Yan (1998); Chen and
Schouten (1999); Arnaud  _et al._ (2002); Bhattacharyya and Mukhopadhyay
(2001); Abah  _et al._ (2012); Das and Mukherjee (2020); Mukherjee  _et al._
(2020) with different working media like a particle in a box, harmonic
oscillator and spin systems Geva and Kosloff (1992b); Feldmann and Kosloff
(2000); Chen  _et al._ (2002); Lin  _et al._ (2003); Henrich  _et al._ (2007);
Thomas and Johal (2014); Gelbwaser-Klimovsky  _et al._ (2018b); Levy and
Gelbwaser-Klimovsky (2018); Kosloff and Rezek (2017); Huang  _et al._ (2013,
2014); Abah and Lutz (2016); Stefanatos (2014, 2017) have been presented.
Among them only a few analyzes the quantum-enhanced machines specifically from
the viewpoint of a single particle within a potential well Gelbwaser-Klimovsky
_et al._ (2018a); Thomas  _et al._ (2019); Chatterjee  _et al._ (2021);
Chattopadhyay  _et al._ (2021).

The role of quantum statistics comes into play if the system comprises an
ensemble of identical particles in contact with a pair of low-temperature
baths. In this limit, the dynamics of the system are governed by the principle
of how a single particle state is occupied. This distinguishes the behavior of
two different types of particles, viz. fermions, and bosons. The chemical
potential that depends on the dimensionality of the system, in addition to its
statistics, can play a huge role in determining its behavior. Therefore, the
collective behavior of quantum particles is interlinked with the
dimensionality and hence the degeneracy of the energy levels Chowdhury and
Stauffer (2000); Pathria and Beale (2011a, b).

There have been several proposals to achieve quantum supremacy Jaramillo  _et
al._ (2016); Chen  _et al._ (2019); Watanabe  _et al._ (2020) by using many-
particle thermal machines over single particle machine. Many-particle quantum
thermal machines are shown to outperform the classical counterparts using
interacting Bose gases in a time-dependent harmonic trap and a tight waveguide
Jaramillo  _et al._ (2016); Chen  _et al._ (2019). In these studies, the
volume equivalent of the heat machines is taken to be the frequency of the
harmonic oscillator and the interparticle interaction strength, respectively.
Both these studies convey the quantum supremacy arising out of the interplay
between many-particle quantum effects. Moreover, multi-component engines have
been proposed, where quantum enhancement is shown exploiting the
indistinguishability of quantum components Watanabe  _et al._ (2020).

In this article, we elaborately, rigorously, and methodically study the
dependence of particle statistics and dimensionality of the system in the
context of a quantum Stirling cycle based on an infinite potential box. In the
present scenario, when the potential is distorted by introducing an infinite
barrier in the middle of the box, the odd levels shift upwards and overlap
with the immediately next even energy levels, creating a new energy level
structure with degeneracies (see e.g., Fig. 1(b)) Griffiths (2005); Belloni
and Robinett (2014). In quantum Stirling cycles, this distortion of the
potential may constitute a certain amount of work and heat exchange between
the baths exclusively based on quantized energy levels without having any
classical analog of volume Gurtin  _et al._ (2010); Paolucci (2016). Depending
on the behavior of the cycle, whether work is extracted or not, one can
construct a desired Stirling heat engine or refrigerator just by exploiting
the quantum nature of the identical particles. The approach to a many-particle
quantum system with symmetrizing/anti-symmetrizing wavefunctions is
ineffective, especially in the $N\rightarrow\infty$ limit, where distinct
statistical behavior is observed. Here, we show that one can bypass the
problem by using the tools of quantum statistical mechanics.

Our approach is interesting both from theoretical and practical viewpoints as
the recent advances in realizing heat machines with quantum components pave
the way for the realization of multi-particle energy-quantization assisted
machines in future Roßnagel  _et al._ (2014); Abah  _et al._ (2012); Roßnagel
_et al._ (2016); von Lindenfels  _et al._ (2019); Levy and Gelbwaser-Klimovsky
(2018); Deng  _et al._ (2018); Batalhão  _et al._ (2014); Josefsson  _et al._
(2018). Particularly, quantum dots (electrons or holes confined in three
spatial dimensions), wires (electrons or holes confined in two spatial
dimensions and free in the other direction) and wells (electrons or holes
confined in one spatial dimension but free in the other two directions)
Sattler (2010); Liu  _et al._ (2013); Du  _et al._ (2020); Stern  _et al._
(2014) and ion-trap systems Gangloff  _et al._ (2015); Bylinskii  _et al._
(2015); Karpa  _et al._ (2013) are potential candidates for realizing quantum
particles entrapped in a potential well.

This paper is organized as follows: In Sec. II, we present the basic
formulation with a brief overview of the quantum Stirling cycle, followed by
detailed characteristics of the working medium, and the fundamental principles
behind quantum statistics of identical particles. In Sec. III, we summarize
the main results and their interpretations together with the analytical
approach in the limiting case of a large number of particles and extremely low
temperature. Finally, we conclude in Sec. IV.

##  II  Formulation

###  II.1 Quantum Stirling cycle

Figure 1: (a) A schematic representation of a Stirling engine with $N$ massive
quantum particles confined in an infinite potential box. (b) Single particle
energy levels and wavefunctions of a particle in an infinite potential well
with and without the barrier.

A quantum Stirling cycle Thomas  _et al._ (2019) consists of two
configurations, governed by two different Hamiltonians $H$ and $H^{\prime}$
respectively, and each configuration is kept in equilibrium with two thermal
baths. Hence, the cycle has four stages, each connected to its preceding and
succeeding stages through two isothermal and two isochoric processes as shown
schematically in Fig 1(a). Initially, at stage $A$, the system is in
equilibrium with a hot bath at a temperature $T_{h}$. In the first step
$(A\rightarrow B)$, the potential is distorted quasi-statically and
isothermally to change the configurations of the energy levels and the
wavefunctions, while the system is still coupled to the hot bath [Fig 1(b)].
In the next step $(B\rightarrow C)$, the system is detached from the hot bath
and connected with a cold bath at a temperature $T_{c}$ via an isochoric
process. Consequently, the system fully thermalizes with the cold bath and the
temperature of the system falls from $T_{h}$ to $T_{c}$, without performing
any mechanical work. In the next step $(C\rightarrow D)$, the system is
brought back to its original configuration quasi-statically and isothermally
while keeping the system connected to the cold bath. In the final step
$(D\rightarrow A)$, the system is detached from the cold bath and connected to
the hot bath through an isochoric process without performing any mechanical
work. The system again fully thermalizes with the hot bath and the temperature
rises from $T_{c}$ to $T_{h}$.

The total heat transferred to the system from the bath in the isothermal
processes $(A\rightarrow B)$ and $(C\rightarrow D)$, while keeping the system
in equilibrium with the bath at temperature $T_{h}$ and $T_{c}$ respectively,
are combinations of the mechanical works performed due to deformation of the
potential and the change of the internal energies. Thus the heat transferred
to the bath from the system during two isothermal processes is given by

$\displaystyle Q_{AB}=-(U_{B}-U_{A}+k_{B}T_{h}\ln Z_{B}-k_{B}T_{h}\ln Z_{A}),$
(1)

and

$\displaystyle Q_{CD}=-(U_{D}-U_{C}+k_{B}T_{c}\ln Z_{D}-k_{B}T_{c}\ln Z_{C}),$
(2)

respectively. By definition, the internal energy at a temperature $T$ in terms
of the thermal partition function $Z_{T}$ is given as

$U_{T}=k_{B}T^{2}\frac{\partial}{\partial T}\ln Z_{T}.$ (3)

In what follows in Sec. II.3, we explicitly evaluate the thermal partition
functions of the Stirling cycle with noninteracting quantum gas of fermions
and bosons. On the contrary, the heat transferred to the system during the
isochoric processes $(B\rightarrow C)$ and $(D\rightarrow A)$ are only the
differences between the average energies of the initial and the final
configurations

$\displaystyle Q_{BC}=-(U_{C}-U_{B}),\;\;\text{and}\;\;Q_{DA}=-(U_{A}-U_{D}).$
(4)

Provided all the processes involved in the cycle are reversible and no leakage
takes place, the expressions for the net work done on the system $W$ and the
heat transferred to the system with the hot and cold baths, $Q_{h}$ and
$Q_{c}$ after completion of one cycle are given as follow:

$\displaystyle W$ $\displaystyle=$
$\displaystyle(Q_{AB}+Q_{BC}+Q_{CD}+Q_{DA})=-k_{B}\left(T_{h}\ln\frac{Z_{B}}{Z_{A}}-T_{c}\ln\frac{Z_{C}}{Z_{D}}\right),$
$\displaystyle Q_{h}$ $\displaystyle=$
$\displaystyle-(Q_{AB}+Q_{DA})=k_{B}T_{h}\ln\frac{Z_{B}}{Z_{A}}+U_{B}-U_{D},$
$\displaystyle Q_{c}$ $\displaystyle=$
$\displaystyle-(Q_{BC}+Q_{CD})=-k_{B}T_{c}\ln\frac{Z_{C}}{Z_{D}}-U_{B}+U_{D}.$
(5)

By the principle of conservation of energy or the first law of thermodynamics,
one can check that $Q_{h}+Q_{c}+W=0$. In our convention, if the quantities
$Q_{h}$ or $Q_{c}$ are positive, the heat is flowing into the system,
similarly, if work $W$ is positive, work is done on the system. In conformity
with the second law of thermodynamics, only four modes of operation are
possible. The four possible modes can be identified by the signs of $W$,
$Q_{h}$ and $Q_{c}$, as depicted in Table 1.

Modes of operation |  $Q_{h}$ |  $Q_{c}$ |  $W$
---|---|---|---
Engine |  $>0$ |  $<0$ |  $<0$
Refrigerator |  $<0$ |  $>0$ |  $>0$
Accelerator |  $>0$ |  $<0$ |  $>0$
Heater |  $<0$ |  $<0$ |  $>0$
Table 1: Different modes of operation of a quantum thermodynamic cycle.

Now, if heat is absorbed from the hot bath at a higher temperature by the
system, converts a fraction of it into work, and rejects the rest to the bath
at a lower temperature, we term it as a heat engine mode. The system works as
a heat engine if $W$ is negative, i.e., work can be extracted from the system.
The efficiency $\eta_{E}$ of the engine, defined as the ratio of the work
extracted to the total heat absorbed by the system is

$\displaystyle\eta_{E}=-\frac{W}{Q_{h}}.$ (6)

In contrast, if heat is absorbed from the cold bath and rejected into the hot
bath with external work done on the system, we term it refrigerator mode.
Here, $W$ is positive, i.e., work is done on the system. The coefficient of
performance of a refrigerator, the ratio of the heat extracted from the cold
bath to the work done on the system is given as,

$\displaystyle\eta_{R}=\frac{Q_{c}}{W}.$ (7)

Apart from the two modes discussed, there can be two other modes. In the
accelerator mode, there is a free flow of heat from the hot bath to the cold
bath, and work is done on the system. On the other hand in the heater mode, a
fraction of the work done on the system is rejected to the hot bath and the
rest to the cold bath.

The Carnot cycle, a theoretical proposition of Carnot, consists of only
reversible processes and therefore conserves the total entropy while the
system completes one cycle. However, it provides the maximum possible
efficiency for a given pair of thermal baths, which is solely dependent on the
bath temperatures Carnot (1897). In order to provide an estimate of its
performance, we will present the efficiency and coefficient of performance of
the quantum cycle scaled by the same to its Carnot equivalent. The efficiency
of a Carnot engine and the coefficient of performance of the Carnot
refrigerator are given respectively as

$\displaystyle\eta^{max}_{E}=1-\frac{T_{c}}{T_{h}},\mbox{\leavevmode\nobreak\
\leavevmode\nobreak\ and \leavevmode\nobreak\ \leavevmode\nobreak\
}\eta^{max}_{R}=\frac{T_{c}}{T_{h}-T_{c}}.$ (8)

The work done during a cycle is primarily dependent on the potential
deformation that changes the wavefunction of the working substance and shifts
the energy eigenvalues along with the bath temperatures. As discussed, quasi-
static deformation of the potential can cause extraction of work from the
system and hence serve as a quantum heat engine or a refrigerator depending on
whether heat is transferred from a hot bath to a cold bath and vice versa. As
a practical example of the aforementioned situations, we consider a quantum
thermal machine that uses a gas of quantum particles in a hard potential box
as its working medium Gelbwaser-Klimovsky  _et al._ (2018a); Thomas  _et al._
(2019) and discusses the details of its properties.

Although in this article, we demonstrate our results for a Stirling cycle, one
can study other thermodynamics cycles with Bose and Fermi gases case by case.
For example, an Otto cycle consists of a pair of isochoric and isentropic
(adiabatic) processes. In our context, the barriers are to be introduced
adiabatically and the rest of the steps are identical. With the approximation
of adiabaticity, that ensures that the level populations do not change upon
removal of the bath, one can have results similar to that of a Stirling engine
Gelbwaser-Klimovsky  _et al._ (2018a). One can generalize the results
replacing a single particle working medium with a quantum gas with fermions
and bosons. Whereas, an exact equivalence with a Carnot cycle is difficult to
construct in this setup since there should be a pair of expansion steps, an
isothermal and an adiabatic.

###  II.2  The working medium

We consider an ensemble of non-interacting, massive, indistinguishable
particles of mass $M$ confined in a $d$ dimensional potential box with length
$2L$ in each dimension as our working medium of the thermal machine. As the
particles are non-interacting, the full Hamiltonian of the system is the sum
of identical single-particle Hamiltonians. The single particle Hamiltonian is
given as

$H=\sum^{d}_{i=1}\frac{p^{2}_{x_{i}}}{2M},{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ }\mbox{for\leavevmode\nobreak\
}|x_{i}|<L.$ (9)

The potential is distorted by introducing $d$ impenetrable barriers at
$x_{i}=0$, one in each dimension, that changes the energy levels and distorts
the wavefunctions. In this context, to highlight the quantum advantage over
its classical counterpart, we note that introducing delta function barriers
does not change the volume of the classical system and thereby has zero
contribution to work. In addition to that, if the box becomes very large, gaps
between the energy levels go to zero forming free particles with a continuum
of energy levels.

The single-particle Hamiltonian of the system in the distorted configuration,
with $d$ impenetrable barriers, is given by

$H^{\prime}=\sum^{d}_{i=1}\frac{p^{2}_{x_{i}}}{2M}+\lambda\delta(x_{i}),{\leavevmode\nobreak\
\leavevmode\nobreak\ }\mbox{for\leavevmode\nobreak\ }|x_{i}|<L,$ (10)

where $\lambda$ is the strength of the delta function barrier. The single
particle eigen energies of the Hamiltonians $H$ and $H^{\prime}$, labeled by a
set of integer quantum numbers $n_{x_{i}}$ are given by the following
expressions

$\displaystyle
E(n_{x_{1}},n_{x_{2}},...,n_{x_{d}})=\frac{\pi^{2}\hbar^{2}}{8ML^{2}}\sum^{d}_{i=1}n^{2}_{x_{i}};\quad
n_{x_{i}}=1,2,3,...$ $\displaystyle
E^{\prime}(n_{x_{1}},n_{x_{2}},...,n_{x_{d}})=\frac{\pi^{2}\hbar^{2}}{8ML^{2}}\sum^{d}_{i=1}\left[n_{x_{i}}+\frac{\varepsilon(n_{x_{i}})}{2}({1-(-1)^{n_{x_{i}}}})\right]^{2};\quad$
$\displaystyle{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }n_{x_{i}}=1,2,3,...$ (11)

where $0\leq\varepsilon(n_{x_{i}})\leq 1$. The value of
$\varepsilon(n_{x_{i}})$ depends on the barrier strength $\lambda$ and can be
obtained from the graphical solution of the transcendental equation (see
Belloni and Robinett (2014); Griffiths (2005) for details). Since we consider
here an infinitely strong barrier, i.e., $\lambda\rightarrow\infty$, the
quantity $\varepsilon(n_{x_{i}})=1$. Inserting an impenetrable barrier shifts
each odd energy eigenstate towards its next even one and thus creates twice
degenerate energy levels. The single-particle energy levels and eigenfunctions
are schematically shown in Fig. 1(b). In a system with $d$ dimensions, one can
show that there will be $2^{d}$ degenerate energy levels Griffiths (2005). For
example, in two dimensions, the energy levels given by Eq. (11) are denoted by
a pair of quantum numbers $(n_{x_{1}},n_{x_{2}})$. Now, without the barriers
the pair $(n_{x_{1}},n_{x_{2}})$ takes values
$(1,1),(1,2),(2,1),(2,2),(1,3),(2,3),(1,4),(2,4),....$ and so on. After
introducing two impenetrable barriers at $x_{1}=0$ and $x_{2}=0$, the energies
of the same set change to the energy levels corresponding to the quantum
numbers $(2,2),(2,2),(2,2),(2,2),(2,4),(2,4),(2,4),(2,4),....$ of the
undeformed box and thus introduces a degeneracy of $4$.

The approach of dealing with a large number of quantum particles in the system
by symmetrizing/anti-symmetrizing their states is inefficient and intractable
even in computers, particularly in the limit where an average number of
particles $N\rightarrow\infty$. But the collective effects of bosons or
fermions are expected to be prominent in the said limit. At this juncture, the
problem can be bypassed using the tools of quantum statistical mechanics.

###  II.3 Statistics of quantum particles

Quantum statistical mechanics is the foundation of understanding the low-
temperature behavior of multi-particle physical systems consisting of
identical particles. In quantum mechanics, all identical particles are
classified into two categories; the class of particles with half-integer spin,
known as fermions, is described by Fermi-Dirac statistics, and the other class
with integer spins, known as bosons, is described by Bose-Einstein statistics.
The fundamental difference between these two categories arises entirely from
the principle of how a single particle state is occupied. Two fermions can
never occupy a single particle state; on the other hand, multiple bosons can
occupy the same state.

Quantum statistics dominate only when the interparticle spacing becomes
smaller than the thermal de Broglie wavelength under low temperature, which
can be termed as “quantum regime”. On the contrary, both statistics can be
approximated by Maxwell-Boltzmann distribution in the classical regime, where
quantum effects are negligible. The thermal grand partition functions
$Z^{+}_{T}$ and $Z^{-}_{T}$ for $N$ non-interacting fermions and bosons at a
temperature $T$ will respectively be Schroeder (2000); Arovas (2013)

$\displaystyle Z^{+}_{T}$ $\displaystyle=$
$\displaystyle\prod^{\infty}_{n=1}\left(\sum^{1}_{j=0}e^{-j(\tilde{E}_{n}-\tilde{\mu})/T}\right)^{g_{n}}=\prod^{\infty}_{n=1}\left(1+e^{-(\tilde{E}_{n}-\tilde{\mu})/T}\right)^{g_{n}},$
$\displaystyle Z^{-}_{T}$ $\displaystyle=$
$\displaystyle\prod^{\infty}_{n=1}\left(\sum^{\infty}_{j=0}e^{-j(\tilde{E}_{n}-\tilde{\mu})/T}\right)^{g_{n}}=\prod^{\infty}_{n=1}\left(\frac{1}{1-e^{-(\tilde{E}_{n}-\tilde{\mu})/T}}\right)^{g_{n}},$

where $g_{n}$ is the degeneracy of the single particle $n$-th energy level and
$\tilde{\mu}$ is the scaled chemical potential. Note that in the above
equation, the dependence on the particle number is hidden in the chemical
potential. For bosons, it is to be noted that the $N$ dependence on
$Z^{-}_{T}$ vanishes as the constraint on the particle number is relaxed in
the grand canonical ensemble. For the sake of simplicity, we express all the
energies and the chemical potentials in the units of $k_{B}$, i.e.
$\tilde{E}=E/k_{B}$ and $\tilde{\mu}=\mu/k_{B}$ in Eq.
(LABEL:partition_function).

As we will see in the next section, the chemical potential of the system plays
an essential role in determining the properties of the thermal machine. Now,
to estimate the value of the chemical potential for fermions in terms of an
average number of particles $N$, it is important to define the Fermi energy
$E_{F}$, the energy of the highest filled state at $T=0$, which is given as
$E_{F}=\frac{\hbar^{2}k_{F}^{2}}{2M}$ where $\vec{k}_{F}$ is the Fermi wave
vector. In $d$-dimension, the relation Chowdhury and Stauffer (2000); Pathria
and Beale (2011b, a); Acharyya (2010) $\frac{C_{d}k^{d}_{F}}{(2\pi/2L)^{d}}=N$
yields $k_{F}=2\pi(\frac{\sigma}{C_{d}})^{1/d}$. Hence the dimensionless Fermi
energy in terms of the volume of a $d$-dimensional unit sphere Blumenson
(1960); Chowdhury and Stauffer (2000) $C_{d}=\frac{\pi^{d/2}}{\Gamma(d/2+1)}$
and the particle density $\sigma=\frac{N}{(2L)^{d}}$, can be expressed as

$\displaystyle\tilde{E}_{F}=\frac{E_{F}}{k_{B}}=2\alpha
L^{2}\left(\frac{\sigma}{C_{d}}\right)^{2/d}=\frac{\alpha
N^{2/d}}{2C^{2/d}_{d}}.$ (13)

For example,

$\displaystyle\tilde{E}_{F}^{(d=1)}=\frac{\alpha
L^{2}\sigma^{2}}{2}=\frac{\alpha N^{2}}{8},$
$\displaystyle\tilde{E}_{F}^{(d=2)}=\frac{2\alpha
L^{2}\sigma}{\pi}=\frac{\alpha N}{2\pi},$
$\displaystyle\tilde{E}_{F}^{(d=3)}=\Big{(}\frac{6}{\pi}\Big{)}^{2/3}\frac{\alpha
L^{2}\sigma^{2/3}}{2}=\Big{(}\frac{6}{\pi}\Big{)}^{2/3}\frac{\alpha
N^{2/3}}{8}.$ (14)

At extremely low temperatures, for fermions, all the energy levels below the
Fermi energy are occupied, introducing barrier(s) in the system does not
change the Fermi energy appreciably. We, therefore, can consider the chemical
potentials in the two configurations, with and without the barrier(s) to be
equal, i.e., $\mu=\mu^{\prime}$.

For the convenience of our calculations, here, we define a parameter
$\alpha=\frac{\hbar^{2}\pi^{2}}{k_{B}ML^{2}}$ with the dimension of
temperature 111$[\alpha]=[\hbar]^{2}[k_{B}]^{-1}M^{-1}L^{-2}\\\
=(ML^{2}T^{-1})^{2}(ML^{2}T^{-2}\Theta^{-1})^{-1}M^{-1}L^{-2}=\Theta$ , such
that the low-temperature condition is scaled w.r.t $\alpha$ as
$T_{h,c}/\alpha\lesssim 1$. We re-emphasize here that as the bath temperatures
$T_{h},T_{c}$ become greater than $\alpha$, the energy levels become
continuous and the quantum advantage is gradually lost; we term this limit as
“classical regime”. In order to get an estimate of the different parameters
related to the system, let us consider a system of electrons entrapped in a
potential well. In terms of the fundamental constants, $\alpha=1$ corresponds
to a box length $L\sim 100\leavevmode\nobreak\ $nm. Since $\alpha$ is the
characteristic temperature associated with the system, the low-temperature
limit, in this case, reduces to $T_{h},T_{c}\lesssim 1K$. From the expression
of $\alpha$, it is evident with the decrease of the size of the box, i.e., the
more confined the quantum system becomes, the constraint on the two bath
temperatures becomes more relaxed.

Finally, we note that in the low-temperature limit, the chemical potential for
fermions can be taken to be equal to the Fermi energy; as the difference is
extremely small even at room temperature Cowan (2019); Chowdhury and Stauffer
(2000).

In the case of bosons, one can easily observe from the Bose-Einstein
distribution that the chemical potential must be smaller than the ground state
of the system in order to ensure a positive value for the occupation number.
As the ground state energies of a system with $d$ dimensions are $E_{1}=\alpha
d/8$ and $E^{\prime}_{1}=\alpha d/2$ for the configurations without and with
the barriers respectively, the corresponding chemical potentials must satisfy
the conditions $\mu\leq\alpha d/8$ and $\mu^{\prime}\leq\alpha d/2$. Hence,
the relation between the chemical potential and the average particle number
becomes involved in this case. Starting from the boundary condition that the
sum over the occupation numbers $N_{n}$ of all energy levels equals the total
number of particles, i.e., $N=\sum_{n}N_{n}$, one can write

$\displaystyle N$
$\displaystyle=\sum^{\infty}_{n=1}\frac{g_{n}}{e^{(\tilde{E}_{n}-\tilde{\mu})/T}-1},$
(15)

In the extremely low-temperature limit, all the terms except the one
corresponding to the ground state are negligible. With this approximation, the
$N$ dependence of the chemical potential for the bosons is given by the
following form,

$\displaystyle\tilde{\mu}\approx\tilde{E}_{1}-T\ln(1+\frac{g_{1}}{N}).$ (16)

For a system of $d$ dimensions, the values of the chemical potential will be
given as

$\displaystyle\tilde{\mu}\approx\frac{\alpha d}{8}-T\ln(1+\frac{1}{N}),$
$\displaystyle\tilde{\mu}^{\prime}\approx\frac{\alpha
d}{2}-T\ln(1+\frac{2^{d}}{N}).$ (17)

Further note that, in the limit $N\rightarrow\infty$, the values of
$\tilde{\mu}$ limit the respective ground state energies from below. Unlike
fermions, the ground state occupation of bosons contributes to the chemical
potential predominantly in the low-temperature limit. Since inserting the
barrier(s) significantly changes the ground state, the chemical potentials
with and without the barrier(s) are different.

The chemical potential for a Bose gas is slightly less than the ground state
energy at a very low temperature and the difference between them is of the
order of inverse volume of the system Cowan (2019); Chowdhury and Stauffer
(2000). It goes to zero at a finite temperature only if the volume of the
system becomes large enough so that the energy level spacing becomes
continuous, i.e., $\alpha\rightarrow 0$, forming free particles and hence
Bose-Einstein condensate (BEC) is formed at a finite temperature for $d\geq
3$. In the limit $\alpha\rightarrow 0$, as explained before, the quantum
advantage is lost implying that quantum thermal machines based on quantized
energy levels and BEC cannot be achieved in the same limit. A few proposals
for quantum heat engines or refrigerators that use the BEC phase of the
working medium or the bath have been made Myers  _et al._ (2022); Niedenzu
_et al._ (2019b), but their thermodynamic advantage is not related to
quantized energy levels as they are in this case.

##  III Results

It is often useful to express the work done in terms of relative partition
functions, i.e., the ratios of the partition functions for the adjacent
stages, i.e., $\zeta(T_{h})={Z_{B}}/{Z_{A}}$ and
$\zeta(T_{c})={Z_{C}}/{Z_{D}}$. Then the work done given by Eq. (5) can be
rewritten as

$\displaystyle W=-(T_{h}\ln\zeta(T_{h})-T_{c}\ln\zeta(T_{c})).$ (18)

The relative partition function is the ratio between the partition functions
of the system with and without the barrier(s) at the same temperature and thus
implies the work done due to inserting the barrier(s) to deform the
wavefunctions to create a new set of degenerate energy levels Thomas  _et al._
(2019). This insertion of the barrier(s) accounts for an extra amount of work
$-T\ln\zeta(T)$. Given a pair of bath temperatures $(T_{h},T_{c})$, one can
compute the total work done in a cycle using Eq. (18) in terms of the relative
partition functions. For $\zeta(T)=1$, the insertion of the barrier(s) does
not contribute to any extra work and therefore no advantage is obtained as we
expect in the classical limit. In what follows, before we discuss the quantum
regime, we briefly review the well-known results of the classical limit for
the sake of completeness.

In the classical regime, i.e., $T\gg\alpha$, the thermodynamics of the
particles are governed by Maxwell-Boltzmann statistics. The single particle
thermal partition function in this regime, in $d$ dimensions, is given by

$\displaystyle
Z_{T}=\sum^{\infty}_{n_{x_{1}}=1}...\sum^{\infty}_{n_{x_{d}}=1}e^{-\tilde{E}(n_{x_{1}},...,n_{x_{d}})/T}.$
(19)

In terms of the eigenenergies given in Eq. (11), one readily finds the
expression for the relative partition function

$\displaystyle\zeta(T)=$
$\displaystyle\frac{\sum^{\infty}_{n_{x_{1}}=1}...\sum^{\infty}_{n_{x_{d}}=1}e^{-{\alpha(n^{2}_{x_{1}}+...+n^{2}_{x_{d}})}/8T}}{2^{d}\sum^{\infty}_{n_{x_{1}}=1}...\sum^{\infty}_{n_{x_{d}}=1}e^{-{\alpha(n_{x_{1}}^{2}+...+n_{x_{d}}^{2})}/2T}}$
(20)
$\displaystyle\approx{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{\left(\frac{1}{2}\sqrt{\frac{8\pi
T}{\alpha}}\right)^{d}}{2^{d}\times\left(\frac{1}{2}\sqrt{\frac{2\pi
T}{\alpha}}\right)^{d}}}=1.$

This implies that introducing barrier(s) at $x=0$ does not cost any extra work
in the classical limit. In view of Eqs. (18) and (20), one concludes that the
work done (extracted) by (from) the system is identically zero in the
classical limit. Therefore, one can easily see that the counterpart of such a
quantum cycle operating between two thermal baths, is an incompressible
classical engine/refrigerator with zero efficiency Gelbwaser-Klimovsky  _et
al._ (2018a); Paolucci (2016); Gurtin  _et al._ (2010). This paradigm breaks
down in the quantum domain, where the discreteness in the energy levels and
the inhomogeneous shift of the population distribution can lead to efficient
quantum thermal machines with no classical analogue Gelbwaser-Klimovsky  _et
al._ (2018a); Thomas  _et al._ (2019).

First, let us consider the case with particles in a one-dimensional potential
well. The relative partition functions from Eq. (LABEL:partition_function) for
fermions and bosons are found to be

$\displaystyle\zeta^{+}(T)=\prod^{\infty}_{n=1}\frac{\left(1+e^{-(\alpha(2n)^{2}/8-\tilde{\mu}^{\prime})/T}\right)}{\left(1+e^{-(\alpha(2n-1)^{2}/8-\tilde{\mu})/T}\right)},$
(21)

and

$\displaystyle\zeta^{-}(T)=\prod^{\infty}_{n=1}\frac{\left(1-e^{-(\alpha(2n-1)^{2}/8-\tilde{\mu})/T}\right)}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(1-e^{-(\alpha(2n)^{2}/8-\tilde{\mu}^{\prime})/T}\right)}}.$
(22)

The expressions for the internal energies at $B$ and $D$ for fermions and
bosons are respectively given by

$\displaystyle U^{\pm}_{B}=\sum^{\infty}_{n=1}2\frac{\alpha
n^{2}/2-\tilde{\mu}^{\prime}}{e^{(\alpha
n^{2}/2-\tilde{\mu}^{\prime})/T_{h}}\pm 1};$ $\displaystyle
U^{\pm}_{D}=\sum^{\infty}_{n=1}\frac{\alpha n^{2}/8-\tilde{\mu}}{e^{(\alpha
n^{2}/8-\tilde{\mu})/T_{c}}\pm 1}.$ (23)

We now generalize the result given in the previous section for a quantum gas
in a $d$-dimensional box, with a barrier in each dimension. As discussed in
Sec. II.2, this will render an energy level structure with a degeneracy of
$2^{d}$. The expressions for the relative partition functions of the particles
are modified according to Eq. (11) as

$\displaystyle\zeta^{+}(T)=\prod^{\infty}_{n_{x_{i}}=1}\prod^{1}_{j_{x_{i}}=0}\frac{\left(1+e^{-(\alpha\sum_{i}((2n_{x_{i}})^{2}/8-\tilde{\mu}^{\prime})/T}\right)}{\left(1+e^{-(\alpha\sum_{i}((2n_{x_{i}}-j_{x_{i}})^{2}/8-\tilde{\mu})/T}\right)},$
(24)

for fermions and

$\displaystyle\zeta^{-}(T)=\prod^{\infty}_{n_{x_{i}}=1}\prod^{1}_{j_{x_{i}}=0}\frac{\left(1-e^{-(\alpha\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2}/8-\tilde{\mu})/T}\right)}{\left(1-e^{-(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu}^{\prime})/T}\right)},$
(25)

for bosons, respectively.

The extension for the $d$-dimensional internal energies at $B$ and $D$, for a
system of fermions and bosons are respectively given by

$\displaystyle
U^{\pm}_{B}=\sum^{\infty}_{n_{x_{i}}=1}2^{d}\frac{\alpha(\sum_{i}(2n_{x_{i}})^{2})/8-\tilde{\mu}^{\prime}}{e^{(\alpha(\sum_{i}(2n_{x_{i}})^{2})/8-\tilde{\mu}^{\prime})/T_{h}}\pm
1},$ $\displaystyle
U^{\pm}_{D}=\sum^{\infty}_{n_{x_{i}}=1}\frac{\alpha(\sum_{i}n^{2}_{x_{i}})/8-\tilde{\mu}}{e^{(\alpha(\sum_{i}n^{2}_{x_{i}})/8-\tilde{\mu})/T_{c}}\pm
1}.$ (26)

Figure 2: Relative Partition function $\zeta^{\pm}(T)$ [Cf. Eqs. (24) and
(25)] as a function of $T$ for (a) fermions and (b) bosons with $N=20$.

To see the contrasting role of the chemical potential on the relative
partition functions for fermions and bosons, the temperature dependence of
$\zeta^{\pm}(T)$ have been plotted in Fig. 2 following Eqs. (24) and (25) for
$d=1,2$ and $3$. It is evident from Fig. 2(a) that the dimension of the system
of fermions decides the qualitative behavior of $\zeta^{+}(T)$. The value of
$\zeta^{+}(T)$ is zero at $T$ tends to zero irrespective of $d$ but shows
different behavior depending on the $d$ owing to the particle number
dependence of $\tilde{\mu}$ ($\sim\tilde{E}_{F}$). In $T/\alpha\lesssim 1$
regime, the value of $\zeta^{+}(T)$ does not change appreciably from zero for
$d=1$ ($\zeta^{+}(T)\sim 10^{-10}$ at $T=1$), but increases for $d=2$ and
$d=3$. However, the value of $\zeta^{+}(T)$ for $d=3$ is much smaller compared
to $d=2$. As we have explained earlier, the relative partition function is
related to the amount of work $-T\ln\zeta(T)$ to change the wavefunctions
while inserting the barrier. This work done is always non-zero irrespective of
$d$. On the other hand, for bosons, the qualitative behavior of $\zeta^{-}(T)$
is similar irrespective of the dimension as seen in Fig. 2(b). The value of
$\zeta^{-}(T)$ is the maximum at $T=0$ and then decays to zero quickly
irrespective of the dimension as the lowest energy level is filled with all
the particles but the value at $T=0$ increases with $d$ mainly owing to the
values of dimension dependent chemical potentials given in Eq. 17.

It is further evident from Fig. 2 that low-temperature behavior of fermions
and bosons would be significantly contrasting as the relative partition
functions for fermions and bosons tend to zero and finite nonzero values
respectively as $T\rightarrow 0$ and their slopes are positive and negative
respectively. It is also expected that the behavior of both fermions and
bosons will strongly depend on the dimension of the system. These corollaries
solely follow particle statistics and are independent of the specific form of
the cycle. Therefore, it captures the generic nature of the low-temperature
behavior of any quantum thermal machines with quantum gases. We show in the
next subsection in detail that our results with the Stirling cycle are
consistent with these observations.

Given the two baths with two different temperatures and a system with
arbitrary dimensions and a number of particles, it is not straightforward to
simplify the above expressions to predict the behavior of the cycle. One can
easily plot the expressions to observe the characteristics of the system for a
given pair of baths set at two arbitrary temperatures. However, in the next
subsection, we elaborate that it is actually possible to look at the extremely
low temperature of the system in $N\rightarrow\infty$ limit analytically and
show the contrasting behavior of the system with fermions and bosons in
different dimensions.

###  III.1  Analytical approach for extremely low temperature &
$N\rightarrow\infty$ limit of the cycle

We are specifically interested in the behavior of the Stirling cycle in the
quantum regime, i.e., in the extremely low temperatures, given by
$T_{h}\rightarrow 0,T_{c}\rightarrow 0$ with $\Delta T=(T_{h}-T_{c})>0$. In
this limit the expression for work [Eq. (18)] reduces to

$W=\lim_{T\rightarrow 0,\Delta T\rightarrow 0}-((T+\Delta T)\ln\zeta((T+\Delta
T))+T\ln\zeta(T).$ (27)

Now, let us define the following function of temperature,

$\omega(T)=T\ln\zeta(T).$ (28)

The slope of this function solely determines, whether, at low temperatures,
work can be extracted from the system or not. Work can be extracted from the
system only if the slope of the function, i.e., $\lim_{T\rightarrow
0}\frac{d}{dT}\omega(T)$ as $T\rightarrow 0$ is positive and work is done on
the system if the same is negative. Now, in the thermodynamic limit, i.e.,
$N\rightarrow\infty$, one can analytically evaluate the sign of the above
quantity and thereby decide the nature of the system. With reference to Fig.
2, one can find that the slope of the relative function for fermions and
bosons are opposite in the low-temperature regime and predict their
distinctive behaviors.

Figure 3: Different modes of operation of a Stirling engine with a gas of
fermions. The modes of operation as functions of $T_{h}$ and $T_{c}$ for: (a)
$d=1$, $N=20$; (b) $d=1$, $N=40$; (c) $d=2$, $N=20$; (d) $d=2$, $N=40$; (e)
$d=3$, $N=20$; (f) $d=3$, $N=40$. The regimes corresponding to Engine,
Refrigerator, Accelerator, and Heater are marked with red, blue, grey, and
black respectively.

Figure 4: Different modes of operation of a Stirling engine with a gas of
bosons. The modes of operation as functions of $T_{h}$ and $T_{c}$ for: (a)
$d=1$, $N=20$; (b) $d=1$, $N=40$; (c) $d=2$, $N=20$; (d) $d=2$, $N=40$; (e)
$d=3$, $N=20$; (f) $d=3$, $N=40$. The regimes corresponding to Engine,
Refrigerator, Accelerator, and Heater are marked with red, blue, grey, and
black respectively.

Figure 5: Dependence on the particle number $N$ on engine efficiency. Engine
efficiency scaled w.r.t. Carnot efficiency $\eta_{E}/\eta^{max}_{E}$ for a
fixed $T_{c}/T_{h}=0.50$ for (a) fermions in $d=1$ and (b) bosons in $d=2,3$.

####  III.1.1 Fermions

Let us first consider the case of non-interacting fermions in a one-
dimensional potential well. The function $\omega(T)$ takes the form of [Cf.
Eq. (21)]

$\displaystyle\omega(T)=T\ln\prod^{\infty}_{n=1}\frac{\left(1+e^{-\alpha((2n)^{2}-N^{2}/)/8T}\right)}{\left(1+e^{-\alpha((2n-1)^{2}-N^{2})/8T}\right)}.$
(29)

In the aforementioned limiting cases, the quantities $\lim_{T\rightarrow
0}\lim_{N\rightarrow\infty}e^{\frac{\alpha}{8T}(4n^{2}-N^{2})}$ and
$\lim_{T\rightarrow
0}\lim_{N\rightarrow\infty}e^{\frac{\alpha}{8T}((2n-1)^{2}-N^{2})}$ both are
very small positive numbers and

$e^{\frac{\alpha}{8T}(4n^{2}-N^{2})}>e^{\frac{\alpha}{8T}((2n-1)^{2}-N^{2})}{\leavevmode\nobreak\
}\quad\quad\forall{\leavevmode\nobreak\ }n.$

Hence, expanding $\frac{d}{dT}\omega(T)$ series of $n$ we get

$\displaystyle\frac{d}{dT}\omega(T)=\frac{\alpha}{8T}\sum^{\infty}_{n=1}\left(N^{2}-4n^{2}+1\right)\left[e^{\frac{\alpha}{8T}(4n^{2}-N^{2})}\right.$
$\displaystyle\left.-e^{\frac{\alpha}{8T}((2n-1)^{2}-N^{2})}\right]+(1-4n)e^{\frac{\alpha}{8T}((2n-1)^{2}-N^{2})}.$
(30)

Clearly, for $N\rightarrow\infty$, the quantity $\frac{d}{dT}\omega(T)$ is
positive. This implies that work can be extracted from a system with a large
number of non-interacting fermions in low-temperature limits. Therefore from
Table 1 one concludes that the system behaves exclusively as a heat engine.

The efficiency of the heat engine is then given as

$\displaystyle\eta_{E}=-\frac{W}{Q_{h}}=\frac{1-\frac{T_{c}\ln\zeta^{+}(T_{c})}{T_{h}\ln\zeta^{+}(T_{h})}}{1+\frac{U^{+}_{B}-U^{+}_{D}}{T_{h}\ln\zeta^{+}(T_{h})}}.$
(31)

In the aforementioned limit, the quantities
$T_{h}\ln\zeta^{+}(T_{h})\rightarrow-\infty$ and
$\frac{T_{c}\ln\zeta^{+}(T_{c})}{T_{h}\ln\zeta^{+}(T_{h})}\rightarrow\frac{T_{c}}{T_{h}}$.
Therefore, efficiency $\eta_{E}\rightarrow 1-\frac{T_{c}}{T_{h}}$, i.e., the
Carnot limit. The above equation connotes that a large number of non-
interacting fermions entrapped in a one-dimensional potential well behave like
a heat engine and its efficiency tends to Carnot limit as the bath
temperatures approach absolute zero. Here we restate that it is purely a
quantum effect with no classical correspondence, yet the machine still works
with the highest possible efficiency and abides by the second law of
thermodynamics. This is the first important result of our analysis exhibiting
true quantum signature at a macroscopic scale.

But for fermions at higher dimensions, the behavior of the system is entirely
different because of the different number dependence of the chemical
potential. The function $\omega(T)$, in this case, is given by [Cf. Eq. (24)]

$\displaystyle\omega(T)=T\ln\prod^{\infty}_{n_{x_{i}}=1}\prod^{1}_{j_{x_{i}}=0}\frac{\left(1+e^{-(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu})/T}\right)}{\left(1+e^{-(\alpha\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2}/8-\tilde{\mu})/T}\right)}.$

We know that in the low-temperature limit, the chemical potential
$\tilde{\mu}$ can be taken equal to be the Fermi energy $\tilde{E}_{F}$. It
can be seen from Eq. (13) that the chemical potential in higher dimensions
varies in a sub-quadratic fashion with $N$ as $\tilde{\mu}\propto N^{\gamma}$,
$\gamma<2$, whereas the energy dispersion is still quadratic as given in Eq.
(11). As a result, in the macroscopic thermodynamic limit
$N\rightarrow\infty$, the energy always outgrows the chemical potential. Now,
in the limit $N\rightarrow\infty$ and $T\rightarrow 0$, the expression
$\frac{1}{T}(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu})$ tends to $\infty$
or $-\infty$ depending on the values of $\tilde{\mu}$. If
$\frac{1}{T}(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu})$ tends to $\infty$,
then
$e^{-\frac{1}{T}(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu})}\rightarrow 0$,
on the other hand if
$\frac{1}{T}(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu})$ tends to
$-\infty$, then
$e^{\frac{1}{T}(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu})}\rightarrow 0$.
So, we split up the sum over $n_{x_{i}},j_{x_{i}}$ in two parts and denote
them by $\sum_{1}$ and $\sum_{2}$ for these two cases respectively. Hence,

$\displaystyle\frac{d}{dT}\omega(T)=\sum_{1}\left[e^{-F_{1}(n_{x_{i}},j_{x_{i}})}-e^{-F_{2}(n_{x_{i}},j_{x_{i}})}\right]$
$\displaystyle+\sum_{1}\left[F_{1}(n_{x_{i}},j_{x_{i}})e^{-F_{1}(n_{x_{i}},j_{x_{i}})}-F_{2}(n_{x_{i}},j_{x_{i}})e^{-F_{2}(n_{x_{i}},j_{x_{i}})}\right]$
$\displaystyle-\sum_{2}\left[F_{1}(n_{x_{i}},j_{x_{i}})e^{F_{1}(n_{x_{i}},j_{x_{i}})}-F_{2}(n_{x_{i}},j_{x_{i}})e^{F_{2}(n_{x_{i}},j_{x_{i}})}\right],$
(33)

where,
$F_{1}(n_{x_{i}},j_{x_{i}})=\frac{1}{T}(\frac{\alpha}{8}\sum_{i}(2n_{x_{i}})^{2}-\tilde{\mu})$
and
$F_{2}(n_{x_{i}},j_{x_{i}})=\frac{1}{T}(\frac{\alpha}{8}\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2}-\tilde{\mu})$.

Using the following inequalities for each set of integers
$(n_{x_{1}},n_{x_{2}},...)$ and $j_{x_{i}}=0,1$,

$\displaystyle
e^{\frac{1}{T}(\frac{\alpha}{8}\sum_{i}(2n_{x_{i}})^{2}-\tilde{\mu})}\geq
e^{\frac{1}{T}(\frac{\alpha}{8}\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2}-\tilde{\mu})},$
$\displaystyle
e^{-\frac{1}{T}(\frac{\alpha}{8}\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2}-\tilde{\mu})}\geq
e^{-\frac{1}{T}(\frac{\alpha}{8}\sum_{i}(2n_{x_{i}})^{2}-\tilde{\mu})},$ (34)

one finds that the first and the third terms are negative and positive
respectively and the second term can be both positive and negative. Therefore,
$\frac{d}{dT}\omega(T)$ can be both positive and negative depending on the
actual functional forms of the chemical potential and dimension. Therefore,
the non-interacting fermions at low temperatures in more than one dimension
can behave as a heat engine or a refrigerator, accelerator, and heater
depending on the sign of $Q_{h}$ and $Q_{c}$. The interplay between the number
dependence on the chemical potential and the energy levels in $d>2$ dictates
the behavior of fermions, which we explicitly show in numerical results later.
This is the second important observation of our analysis.

####  III.1.2 Bosons

Now, we consider the case of non-interacting bosons in a one-dimensional box.
The function $\omega(T)$ takes the form of [Eq. (22)]

$\displaystyle\omega(T)=T\ln\prod^{\infty}_{n=1}\frac{\left(1-e^{-(\alpha(2n-1)^{2}/8-\tilde{\mu})/T}\right)}{\left(1-e^{-(\alpha(2n)^{2}/8-\tilde{\mu}^{\prime})/T}\right)},$
(35)

Expanding $\frac{d}{dT}\omega(T)$ in the $T\rightarrow 0$ limit we get

$\displaystyle\frac{d}{dT}\omega(T)=\sum^{\infty}_{n=1}\ln\frac{(1-e^{-(\alpha(2n-1)^{2}/8-\tilde{\mu})/T})}{(1-e^{-(\alpha(2n)^{2}/8-\tilde{\mu}^{\prime})/T})}$
$\displaystyle+\frac{1}{T}\sum^{\infty}_{n=1}((\frac{\alpha}{8}(2n)^{2}-\tilde{\mu}^{\prime})e^{-(\alpha(2n)^{2}/8-\tilde{\mu}^{\prime})/T}$
$\displaystyle-(\frac{\alpha}{8}(2n-1)^{2}-\tilde{\mu})e^{-(\alpha(2n-1)^{2}/8-\tilde{\mu})/T}).$

In the $N\rightarrow\infty$ limit, using the inequality

$\displaystyle\frac{\alpha}{8}(2n)^{2}-\tilde{\mu}^{\prime}>\frac{\alpha}{8}(2n-1)^{2}-\tilde{\mu}{\leavevmode\nobreak\
\leavevmode\nobreak\ }\forall n\geq 1$ (37)

with $\mu=\alpha/8$ and $\mu^{\prime}=\alpha/2$ (Cf. Eq. 17), one readily
shows the quantity $d\omega(T)/dT$ to be negative and therefore, implying that
a system with large number of non-interacting bosons in $d=1$, behaves as a
refrigerator or an accelerator or a heater depending on the signs of $Q_{h}$
and $Q_{c}$.

But for bosons at higher dimensions, the function $\omega(T)$, in this case,
is given by [Cf. Eq. (25)]

$\displaystyle\omega(T)=T\ln\prod^{\infty}_{n_{x_{i}}=1}\prod^{1}_{j_{x_{i}}=0}\frac{\left(1-e^{-(\alpha\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2}/8-\tilde{\mu})/T}\right)}{\left(1-e^{-(\alpha\sum_{i}(2n_{x_{i}})^{2}/8-\tilde{\mu}^{\prime})/T}\right)}.$

In the limit $T\rightarrow 0$, unlike fermions, only the ground state is
occupied by the bosons and consequently, the occupation at the excited states
is negligible. In the $N\rightarrow\infty$ limit, we use the following
inequality for the set of integers $n_{x_{i}}=1$ and $j_{x_{i}}=0,1$,

$\displaystyle\frac{\alpha}{8}(\sum_{i}(2n_{x_{i}})^{2})-\tilde{\mu}^{\prime}\leq\frac{\alpha}{8}(\sum_{i}(2n_{x_{i}}-j_{x_{i}})^{2})-\tilde{\mu},{\leavevmode\nobreak\
\leavevmode\nobreak\ }\forall d\geq 2,$ (39)

with $\tilde{\mu}=\alpha d/8$ and $\tilde{\mu}^{\prime}=\alpha d/2$, one finds
that $d\omega/dT$ is positive (note that the inequalities in Eqs. (37) and 39
have reversed signs). This implies that the system with non-interacting bosons
in $d\geq 2$ will behave as a heat engine in the low-temperature limit. The
efficiency of the heat engine is then given as

$\displaystyle\eta_{E}=-\frac{W}{Q_{h}}=\frac{1-\frac{T_{c}\ln\zeta^{-}(T_{c})}{T_{h}\ln\zeta^{-}(T_{h})}}{1+\frac{U^{-}_{B}-U^{-}_{D}}{T_{h}\ln\zeta^{-}(T_{h})}}.$
(40)

In the limit $T_{h},T_{c}\rightarrow 0$, both the quantities
$\zeta^{-}(T_{h})$ and $\zeta^{-}(T_{c})$ are nonzero,
$\ln\zeta^{-}(T_{c})>\ln\zeta^{-}(T_{h})$ [Fig. 2(b)] and
$U^{-}_{B}>U^{-}_{D}$ ensure an efficiency less than the Carnot bound, unlike
the fermions in $d=1$. Later, we plot the efficiency as a function of $N$ in
Fig. 5(b) for $d=2$ and $d=3$.

In Fig. 3 we present different modes of operations for fermions in one, two,
and three dimensions for $N=20$ and $40$ from numerical calculations. Fermions
in a one-dimensional box behave strictly as a heat engine in the $T_{h}-T_{c}$
parameter space [Fig. 3(a),(b)] as predicted from the analytical approach. The
efficiency is almost constant and equal to the Carnot bound in the low-
temperature regime $T_{h}\lesssim\alpha$. On the other hand, in $d=2$ and
$d=3$ we predicted from Eq. (33) that all the four modes can coexist in the
$T_{h}-T_{c}$ plane in the low-temperature limit. In $d=2$, from numerical
calculations also we find that the three modes (refrigerator, accelerator, and
heater) with $W>0$ for $N=20$ and all the four modes (engine, refrigerator,
accelerator, and heater) with $N=40$ coexist in the $T_{h}-T_{c}$ plane [Fig.
3(c), (d)]. In $d=3$, we find that all the four modes (refrigerator, engine,
heater, and accelerator) for $N=20$ and the three modes (refrigerator, heater,
and accelerator) with $W>0$ for $N=40$ coexist [Fig. 3(e), (f)]. In all the
cases above, when $T_{c}\lesssim T_{h}$, the system works as a refrigerator
but as one increases $T_{h}$ while keeping $T_{c}$ fixed, the region of the
accelerator is reached. A narrow region of the heater demarcates the boundary
between the regions of the refrigerator and accelerator where $Q_{h}$ flips
its sign. The numerical results conclude that the low-temperature behavior of
fermions for $d>1$ depends both on the dimensions and the number of particles
in the system owing to the interplay between the number dependence of the
chemical potential and energy levels, as explained through our analytical
approach.

In contrast to fermions, as predicted from our analytical approach, the system
with bosons in $d=1$ predominantly behaves as an accelerator with small
regions of refrigerator and heater [Figs. 4(a)-(b)]. But it shows
qualitatively different behavior in $d=2,3$ [Figs. 4(c)-(f)], where there
exist all four distinct regions but near $T_{h},T_{c}\rightarrow 0$, the
system behaves as a heat engine as predicted from the analysis. We have shown
the results with two different values of $N$; Figs. 4(a)-(e) for $N=20$ and
Figs. 4(b)-(f) for $N=40$, but no qualitative difference depending on $N$ is
observed.

Some common remarks can be made from the results shown in Fig. 3 and Fig. 4.
(a) The fermi gas in one dimension is the most desired system to construct a
heat engine with Carnot efficiency. However, a boson gas is also a candidate
for heat engines provided the dimension of the system is greater than one. (b)
The region of refrigerator mode is located near the boundary $T_{h}=T_{c}$ and
the region of the accelerator on the other boundary $T_{c}\sim 0$ for bosons
and fermions in $d=2,3$. A small region of the heater demarcates the boundary
of transition between the refrigerator and the accelerator. Therefore, the
refrigerator is not a suitable mode for quantum gases, in general.

###  III.2 Dependence on average particle number $N$

We have already shown that a Stirling cycle with fermions in one dimension
exclusively works as an engine whereas with bosons it both behaves as a
refrigerator and a heater. To explore the $N$ dependence for fermions, we plot
the heat engine efficiency scaled w.r.t the Carnot efficiency,
$\eta_{E}/\eta^{max}_{E}$ in Fig. 5(a), with the number of particles $N$ and
$T_{h}$, in one-dimensional box keeping the ratio $T_{c}/T_{h}=0.5$ fixed. The
efficiency reaches the Carnot bound as the bath temperatures tend to zero. It
is interesting to see that for a given pair of baths with non-zero
temperatures, a system with larger $N$ yields better efficiency. Owing to a
prominent number of dependence on fermions, the engine efficiency can be
boosted by adding more fermions to the system.

In Fig. 5(b), we plot the engine efficiency for bosons scaled w.r.t the Carnot
efficiency, with the number of particles $N$, and $T_{h}$, for $d=2,3$ keeping
the ratio $T_{c}/T_{h}=0.5$ fixed. The scaled efficiency tends to values much
less than 1 as the bath temperatures tend to zero irrespective of $N$ and $d$
as predicted from our analysis and it also goes to zero when the system
switches to the accelerator mode as shown already in Figs. 4(c)-(f). The
maximum coefficient of performance increases slightly with $N$, as seen from
Fig. 5(b), reveals that bosons do not show any prominent dependence on $N$,
unlike fermions.

##  IV Conclusions

To conclude, we have demonstrated some fundamental features of quantum
particles in the context of the quantum Stirling cycle. Given a thermodynamics
cycle, one can anticipate that the behaviour of the working medium in the
quantum regime is fundamentally determined only by the quantum statistics of
the particles. We analyze the role of quantum statistics and system dimensions
in determining the collective thermodynamic behaviour of particles. It is
worth mentioning that the efficiency considered here exhibits a pure quantum
signature with no classical analogue and manifests only in the quantum regime.
Though our analysis is focused on a specific working medium, viz., quantum gas
trapped in an infinite potential well or a square well, one can trivially
generalize our analysis for any system. A similar analysis with other trapping
potentials would connote a qualitatively similar result as a consequence of
the statistical nature of identical quantum particles.

We find that bosons behave completely opposite to fermions as a manifestation
of the fundamental difference between the particle statistics in the quantum
domain. Therefore, fermions and bosons are useful in a different way from the
viewpoint of constructing a desirable thermodynamic machine. A Stirling cycle
with fermions confined in a one-dimensional potential well, when connected to
two low-temperature baths, behaves exclusively like a heat engine ($W<0$)
while a Stirling cycle with bosons behaves like an accelerator, refrigerator
and a heater ($W>0$) depending on the bath temperatures. On the other hand,
the behaviour of bosons and fermions is qualitatively similar in two and
three-dimensional systems. In both cases, unlike in one dimension, a mixture
of four different modes can be observed depending on the bath temperatures and
the number of particles.

We have also shown that the particle number or the dimension does not
appreciably affect the performance of bosons. Unlike bosons, the number
dependence of the chemical potential for fermions decides the behaviour of the
system. As the dimension of the system decides the number dependence of the
chemical potential, it determines the overall thermodynamic behaviour of the
system. The behaviour of fermions and bosons in a one-dimensional well is
completely different compared to that of two or higher-dimensional boxes. We
have found that increasing the number of particles in a system can boost its
performance in spite of the engine efficiency/coefficient of performance being
bounded by the Carnot bound.

###### Acknowledgements.

S.S. thanks Dr. Amit Kumar Jash (WIS) for fruitful discussions regarding
recent experiments on quantum wires and wells. A.G. acknowledges ICTS for
participating in the program - Physics with Trapped Atoms, Molecules and Ions
(ICTS/TAMIONs-2022/5).

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