2018_09 - Grhafite

QUANTUM THERMODYNAMICS:

WHAT IT MEANS AND WHAT IT STRIVES FOR

Gabriel T. Landi

Instituto de Física da Universidade de São Paulo www.fmt.if.usp.br/~gtlandi

Grupo Grhafite - IFUSP September 25th, 2018

S

o f I

Q u T

P N

B I f

spinoffqubit.info

IRREVERSIBILITY

Consider the simplest possible problem in thermodynamics:

A system S is put in contact with an environment E.

As time passes, the state of the system will eventually relax and achieve thermal equilibrium with the bath.

WHAT IS DIFFERENT IN QUANTUM SYSTEMS?

1. THERE ARE OTHER TYPES OF RESOURCES

In quantum systems there are also other resources, such as entanglement and coherence.

These Quantum resources can be used in information processing tasks to obtain quantum advantages:

Faster algorithms.

More precise measurements (metrology).

More secure communications.

More efficient quantum simulations.

1. THERE ARE OTHER TYPES OF RESOURCES

But these resources are affected by the contact of with the environment.

Aguilar, Valdés-Hernández, Davidovich, Walborn, Souto Ribeiro, Phys. Rev. Lett, 113, 240501 (2014)

1. THERE ARE OTHER TYPES OF RESOURCES

Coherence time.

Figure of merit for quantum information processing.

Amount of time that the system can retain its quantum properties.

Exponential growth in the last decades in some platforms.

We now have a handful of

“Controlled quantum platforms”.

1. THERE ARE OTHER TYPES OF RESOURCES

Consider a system of N qubits (i.e. spin 1/2 particles).

The most general state has the form:

| i = X

1,..., _N

1,..., _N

|

, . . . ,

i

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For instance, a state with an enormous amount of coherence is a GHZ state

This is a superposition of macroscopically distinct states. Such a state is a huge resource for metrology, for instance.

| i = 1

p 2

✓

| 1, . . . , 1 i + | 1, . . . , 1 i

◆

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1. THERE ARE OTHER TYPES OF RESOURCES

❖ Coherences and entanglement are usually washed away very quickly by the contact of a system with its environment.

❖ We start with a pure state:


❖ Then the contact with the environment will gradually degrade the coherences:


❖ If we wait long enough, we eventually get a classical state:

⇢(1) =

✓|a|² 0 0 |b|²

◆

| i = a|0i + b|1i =) ⇢ = | ih | =

✓|a|² ab^⇤ a^⇤b |b|²

◆

⇢(t) =

✓ |a|² e ^tab^⇤ e ^ta^⇤b |b|²

◆

1. THERE ARE OTHER TYPES OF RESOURCES

But such a state is also the most sensitive to decoherence.

If we put it in contact with a bath, the density matrix will change as

h | ⇢ |

i ! h | ⇢ |

i e

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⇤( ,

) = X

i

(

)

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The decoherence rate is stronger for states which are macroscopically more distinct.

M. A. Cipolla and GTL, “Processing quantum coherence using the spin-boson model”, arXiv 1808.01224

2. INFORMATION BECOMES ESSENTIAL

“The fragility of states makes quantum systems very difficult to isolate. Transfer of information (which has no effect on classical states) has marked consequences in the quantum realm. So, whereas fundamental problems of classical physics were always solved in isolation (it sufficed to prevent energy loss), this is not so in

quantum physics (leaks of information are much harder to plug).”

W. J. Zurek,

Nature Physics, 5, 181 (2009)

2. INFORMATION BECOMES ESSENTIAL

Objective reality: why different observers agree on what they are observing.

The environment in quantum mechanics plays an active role.

We access the system, by measuring small fractions of the environment.

e.g.: we measure the fraction of photons scattering from this screen.

And we all agree on what we are observing.

W. J. Zurek,

Nature Physics, 5, 181 (2009)

3. QUANTUM MECHANICS OFFERS MORE GENERAL BATHS

It is possible to work with engineered environments.

Example: squeezed thermal bath:

Klaers, Faelt, Imamoglu, Togan, Phys. Rev. X, 7, 031044 (2017)

Move beyond the standard paradigms of thermodynamics.

4. MEASUREMENTS PLAY A CENTRAL ROLE

Elouard, Herrera-Martí, Huard, Auffèves, Phys. Rev. Lett, 118, 260603 (2017)

Measurements can be directly implemented in thermodynamic engines.

Maxwell’s demons and information engines.

Xiong, et. al., Phys. Rev. Lett. 120, 010601 (2018)

Back-action (state collapse) affects how we extract thermodynamic information.

ENTROPY PRODUCTION

In thermodynamics the resources are heat and work, and irreversibility is quantified using the entropy production.

S Q

T ! ⌃ := S Q

T 0

(Clausius inequality)

(entropy production) (entropy flux)

We also express this in terms of rates:

⇧ = d⌃

dt

dS

dt = ⇧ = 1

T

dQ dt

(entropy production rate) (entropy flux rate)

EXAMPLE: RL CIRCUIT

⇧

=

= E

RT

��/��

Φ Π

0 1 2 3 4 5

0 2 4 6 8 10

t

ℰ²/RT

�

�_� ℰ

Steady-state

dS

dt = 0

EXAMPLE: TWO INDUCTIVELY COUPLED RL CIRCUITS

⇧_ss = E1²

R₁T₁ + E2²

R₂T₂ + m²R₁R₂

(L₁L₂ m²)(L₂R₁ + L₁R₂)

(T₁ T₂)² T₁T₂

GTL, T. Tomé and M. J. de Oliveira, J. Phys A. 46 (2013) 395001

CLASSICAL VS. QUANTUM MASTER EQUATIONS

Jader P. Santos, Lucas C. Céleri, Gabriel T. Landi and Mauro Paternostro The role of quantum coherence in non-equilibrium entropy production

arXiv 1707.08946 (submitted to Nature Quantum Information)

Consider a system with discrete energy levels and let pn denote de probability of being found in state n.

In a classical approach, the dynamics of the system in contact with a bath would be described by a Pauli master equation:

�

�

��

��

����

�

(�)

Contact with the bath generates a stochastic motion.

p^eq_n = e ^En Z Microscopically, the system will keep on jumping around, with an average probability given by the equilibrium distribution:

dp_n

dt = X

m

⇢

K(n|m)p_m K(m|n)p_n

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Schnakenberg [Rev. Mod. Phys., 48, 571 (1976)] found:

Thermal states satisfy detailed balance:

So that in this case we recover the thermal scenario:

= Q˙ T

See T. Tomé and M. J. de Oliveira, Dinâmica Estocástica, EDUSP.

⇧ = 1 2

X

n,m

⇢

K(n|m)p_m K(m|n)p_n ln K(n|m)p_m K(m|n)p_n

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= X

n,m

K(m|n)p_n ln K(m|n) K(n|m)

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K(n|m)

K(m|n) = p^eq_n p^eqm

= e ^{(En Em)}

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The entropy production may be written in a cooler way as

⇧ = dS(p(t)||p^eq) dt

S(p||p^eq) = X

n

p_n ln p_n/p^eq_n

Π due to system adapting to new population imposed by the bath.

Where

is the Relative entropy or Kullback-Leibler divergence.

It gives a type of “distance” between probability distributions.

OPEN QUANTUM SYSTEMS

When we first learn quantum mechanics, we introduce untiaries as the operators which take kets to kets, preserving probability.

|

i = U | i , U

U = 1

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But then we learn about density matrices.

We should then redo the question: what is the most general type of operation which takes density matrices into density matrices?

Answer: a quantum operation:

⇢

= X

k

M

⇢M

, X

k

M

M

= 1

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LINDBLAD MASTER EQUATIONS

The quantum operation is a map (like the unitary).

If we want a differential equation to generate the map, we get instead Lindblad’s equation:

d⇢

dt = i[H, ⇢] + X

k

k



L

⇢L

1

2 { L

L

, ⇢ }

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The L’s are jump operators: they describe how the environment induces jumps in the system.

EXAMPLE: QUBIT

D(⇢) = (1 f)



⇢ ₊ 1

2{ ⁺ , ⇢} + f



+⇢ 1

2{ ⁺, ⇢}

f = 1

e ^⌦ + 1 ⇢ =

✓p₀ q q^⇤ p₁

◆

dp₀

dt = f p₁ (1 f)p₀ dp₁

dt = (1 f)p₀ f p₁

(Pauli master equation)

dq

dt =

2 q

(Decoherence)

ENTROPY PRODUCTION

The entropy flux does not depend on the coherences:

But the entropy production, on the other hand, becomes.

⇧ = dS(⇢||⇢_eq) dt

Here we consider Thermal Operations (or Davies maps), which have simple thermal properties.

Thermalize correctly.

Populations evolve according to classical M Eq.

S(⇢||⇢_eq) = tr

⇢

⇢(ln ⇢ ln ⇢_eq)

= 1 T

dQ dt

GLOBAL UNITARY DYNAMICS

We can instead think about entropy production in terms of the global unitary dynamics of S+E. Then one may show that

1707.08946 and 1804.02970

see also: M. Esposito, K. Lindenberg, and C. Van Den Broeck, NJP12, 013013 (2010).

Thus, entropy production stems from:

1. Mutual information built up between S and E that is lost. 


2. The state of the environment being pushed away from equilibrium.

⇧ = dI^SE

dt + dS(⇢_E(t)||⇢^th_E ) dt

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I

= S (⇢

) + S (⇢

) S (⇢

)

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CONTRIBUTION FROM COHERENCES

As a result, we find that the entropy production can be divided in two parts:

But now we can separate:

S(⇢||⇢_eq) = S(p||p^eq) + C(⇢)

C(⇢) = S( _H(⇢)) S(⇢) (Entropy of coherence)

One part is the classical: entropy production due to population change.

But the other is genuinely quantum mechanical:

Entropy production due to loss of coherence.

⇧ = dS(p(t)||p^eq) dt

C(⇢) dt

QUANTUM PHASE SPACE FORMULATION

Jader P. Santos, Gabriel T. Landi and Mauro Paternostro The Wigner entropy production rate

Phys. Rev. Lett, 118, 220601 (2017)

We shall consider the relaxation of a bosonic mode in contact with a bath:

d⇢

dt = i[H, ⇢] + D(⇢) H = !(a^†a + 1/2)

D(⇢) = (¯n + 1)



a⇢a^† 1

2{a^†a, ⇢} + ¯n



a^†⇢a 1

2{aa^†, ⇢}

¯

n = 1

e ^! 1

�

�

��

��

����

�

(�⁾

Instead of working with density matrices, we work with the Wigner function:

W (↵, ↵^⇤) = 1

⇡² Z

d² e ^{⇤↵ ↵⇤}tr

⇢

⇢e ^{a† ⇤a}

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Re(↵) = q

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Im(↵) = p

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Introduces the notion of quantum phase space

Vacuum Thermal state Coherent state Squeezed state

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The Wigner function will satisfy a Quantum Fokker-Planck equation

J(W ) = 2



↵W + (¯n + 1/2)@_↵^⇤W

@_tW = @_↵J(W ) + @_↵^⇤J^⇤(W )

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This is a continuity equation for the quasi-probability.

J(W) is a probability current.

The current is zero if and only if the system is in thermal equilibrium.

J(W_eq) = 0 W_eq(↵, ↵^⇤) = 1

⇡(¯n + 1/2)e ^|↵|

2

¯

n+1/2

WIGNER ENTROPY

Instead of using the von Neumann entropy, we adopt instead the entropy of the Wigner function:

S =

Z

d²↵ W ln W

<latexit sha1_base64="giWgKob+yapItZlFIRnafYeRUF0=">AAACCHicbVBNS8NAEJ3Ur1q/qh49uFgEL5akCAoqFLx4rGibQhPLZrNtl242YXcjlNKjF/+KFw+KePUnePPfuG1z0NYHA4/3ZpiZFyScKW3b31ZuYXFpeSW/Wlhb39jcKm7vNFScSkLrJOaxbAZYUc4ErWumOW0mkuIo4NQN+ldj332gUrFY3OlBQv0IdwXrMIK1kdrF/Vt0iY6Rx4RG4X3FwzzpYeSdIxd5XCC3XSzZZXsCNE+cjJQgQ61d/PLCmKQRFZpwrFTLsRPtD7HUjHA6KnipogkmfdylLUMFjqjyh5NHRujQKCHqxNKUOWii/p4Y4kipQRSYzgjrnpr1xuJ/XivVnTN/yESSairIdFEn5UjHaJwKCpmkRPOBIZhIZm5FpIclJtpkVzAhOLMvz5NGpezYZefmpFS9yOLIwx4cwBE4cApVuIYa1IHAIzzDK7xZT9aL9W59TFtzVjazC39gff4AeViXEA==</latexit><latexit sha1_base64="giWgKob+yapItZlFIRnafYeRUF0=">AAACCHicbVBNS8NAEJ3Ur1q/qh49uFgEL5akCAoqFLx4rGibQhPLZrNtl242YXcjlNKjF/+KFw+KePUnePPfuG1z0NYHA4/3ZpiZFyScKW3b31ZuYXFpeSW/Wlhb39jcKm7vNFScSkLrJOaxbAZYUc4ErWumOW0mkuIo4NQN+ldj332gUrFY3OlBQv0IdwXrMIK1kdrF/Vt0iY6Rx4RG4X3FwzzpYeSdIxd5XCC3XSzZZXsCNE+cjJQgQ61d/PLCmKQRFZpwrFTLsRPtD7HUjHA6KnipogkmfdylLUMFjqjyh5NHRujQKCHqxNKUOWii/p4Y4kipQRSYzgjrnpr1xuJ/XivVnTN/yESSairIdFEn5UjHaJwKCpmkRPOBIZhIZm5FpIclJtpkVzAhOLMvz5NGpezYZefmpFS9yOLIwx4cwBE4cApVuIYa1IHAIzzDK7xZT9aL9W59TFtzVjazC39gff4AeViXEA==</latexit><latexit sha1_base64="giWgKob+yapItZlFIRnafYeRUF0=">AAACCHicbVBNS8NAEJ3Ur1q/qh49uFgEL5akCAoqFLx4rGibQhPLZrNtl242YXcjlNKjF/+KFw+KePUnePPfuG1z0NYHA4/3ZpiZFyScKW3b31ZuYXFpeSW/Wlhb39jcKm7vNFScSkLrJOaxbAZYUc4ErWumOW0mkuIo4NQN+ldj332gUrFY3OlBQv0IdwXrMIK1kdrF/Vt0iY6Rx4RG4X3FwzzpYeSdIxd5XCC3XSzZZXsCNE+cjJQgQ61d/PLCmKQRFZpwrFTLsRPtD7HUjHA6KnipogkmfdylLUMFjqjyh5NHRujQKCHqxNKUOWii/p4Y4kipQRSYzgjrnpr1xuJ/XivVnTN/yESSairIdFEn5UjHaJwKCpmkRPOBIZhIZm5FpIclJtpkVzAhOLMvz5NGpezYZefmpFS9yOLIwx4cwBE4cApVuIYa1IHAIzzDK7xZT9aL9W59TFtzVjazC39gff4AeViXEA==</latexit><latexit sha1_base64="giWgKob+yapItZlFIRnafYeRUF0=">AAACCHicbVBNS8NAEJ3Ur1q/qh49uFgEL5akCAoqFLx4rGibQhPLZrNtl242YXcjlNKjF/+KFw+KePUnePPfuG1z0NYHA4/3ZpiZFyScKW3b31ZuYXFpeSW/Wlhb39jcKm7vNFScSkLrJOaxbAZYUc4ErWumOW0mkuIo4NQN+ldj332gUrFY3OlBQv0IdwXrMIK1kdrF/Vt0iY6Rx4RG4X3FwzzpYeSdIxd5XCC3XSzZZXsCNE+cjJQgQ61d/PLCmKQRFZpwrFTLsRPtD7HUjHA6KnipogkmfdylLUMFjqjyh5NHRujQKCHqxNKUOWii/p4Y4kipQRSYzgjrnpr1xuJ/XivVnTN/yESSairIdFEn5UjHaJwKCpmkRPOBIZhIZm5FpIclJtpkVzAhOLMvz5NGpezYZefmpFS9yOLIwx4cwBE4cApVuIYa1IHAIzzDK7xZT9aL9W59TFtzVjazC39gff4AeViXEA==</latexit>

This will be real for Gaussian states (because then W > 0).

Moreover, it coincides with the Rényi-2 entropy S<latexit sha1_base64="9TaMW2cQY8VSSAM/JjF6JEaMaSs=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBjSUpgi4UCm5cVrQPaGKZTCft0MkkzNyIJXTjxl9x40IRt/6DO//G6WOhrQcuHM65l3vvCRLBNTjOt5VbWFxaXsmvFtbWNza37O2duo5TRVmNxiJWzYBoJrhkNeAgWDNRjESBYI2gfznyG/dMaR7LWxgkzI9IV/KQUwJGatv7N+0yvsDH2BMSe8AeIAM1xJ7qxXfltl10Ss4YeJ64U1JEU1Tb9pfXiWkaMQlUEK1brpOAnxEFnAo2LHipZgmhfdJlLUMliZj2s/EXQ3xolA4OY2VKAh6rvycyEmk9iALTGRHo6VlvJP7ntVIIz/yMyyQFJulkUZgKDDEeRYI7XDEKYmAIoYqbWzHtEUUomOAKJgR39uV5Ui+XXKfkXp8UK+fTOPJoDx2gI+SiU1RBV6iKaoiiR/SMXtGb9WS9WO/Wx6Q1Z01ndtEfWJ8/K6eXFQ==</latexit><latexit sha1_base64="9TaMW2cQY8VSSAM/JjF6JEaMaSs=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBjSUpgi4UCm5cVrQPaGKZTCft0MkkzNyIJXTjxl9x40IRt/6DO//G6WOhrQcuHM65l3vvCRLBNTjOt5VbWFxaXsmvFtbWNza37O2duo5TRVmNxiJWzYBoJrhkNeAgWDNRjESBYI2gfznyG/dMaR7LWxgkzI9IV/KQUwJGatv7N+0yvsDH2BMSe8AeIAM1xJ7qxXfltl10Ss4YeJ64U1JEU1Tb9pfXiWkaMQlUEK1brpOAnxEFnAo2LHipZgmhfdJlLUMliZj2s/EXQ3xolA4OY2VKAh6rvycyEmk9iALTGRHo6VlvJP7ntVIIz/yMyyQFJulkUZgKDDEeRYI7XDEKYmAIoYqbWzHtEUUomOAKJgR39uV5Ui+XXKfkXp8UK+fTOPJoDx2gI+SiU1RBV6iKaoiiR/SMXtGb9WS9WO/Wx6Q1Z01ndtEfWJ8/K6eXFQ==</latexit><latexit sha1_base64="9TaMW2cQY8VSSAM/JjF6JEaMaSs=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBjSUpgi4UCm5cVrQPaGKZTCft0MkkzNyIJXTjxl9x40IRt/6DO//G6WOhrQcuHM65l3vvCRLBNTjOt5VbWFxaXsmvFtbWNza37O2duo5TRVmNxiJWzYBoJrhkNeAgWDNRjESBYI2gfznyG/dMaR7LWxgkzI9IV/KQUwJGatv7N+0yvsDH2BMSe8AeIAM1xJ7qxXfltl10Ss4YeJ64U1JEU1Tb9pfXiWkaMQlUEK1brpOAnxEFnAo2LHipZgmhfdJlLUMliZj2s/EXQ3xolA4OY2VKAh6rvycyEmk9iALTGRHo6VlvJP7ntVIIz/yMyyQFJulkUZgKDDEeRYI7XDEKYmAIoYqbWzHtEUUomOAKJgR39uV5Ui+XXKfkXp8UK+fTOPJoDx2gI+SiU1RBV6iKaoiiR/SMXtGb9WS9WO/Wx6Q1Z01ndtEfWJ8/K6eXFQ==</latexit><latexit sha1_base64="9TaMW2cQY8VSSAM/JjF6JEaMaSs=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBjSUpgi4UCm5cVrQPaGKZTCft0MkkzNyIJXTjxl9x40IRt/6DO//G6WOhrQcuHM65l3vvCRLBNTjOt5VbWFxaXsmvFtbWNza37O2duo5TRVmNxiJWzYBoJrhkNeAgWDNRjESBYI2gfznyG/dMaR7LWxgkzI9IV/KQUwJGatv7N+0yvsDH2BMSe8AeIAM1xJ7qxXfltl10Ss4YeJ64U1JEU1Tb9pfXiWkaMQlUEK1brpOAnxEFnAo2LHipZgmhfdJlLUMliZj2s/EXQ3xolA4OY2VKAh6rvycyEmk9iALTGRHo6VlvJP7ntVIIz/yMyyQFJulkUZgKDDEeRYI7XDEKYmAIoYqbWzHtEUUomOAKJgR39uV5Ui+XXKfkXp8UK+fTOPJoDx2gI+SiU1RBV6iKaoiiR/SMXtGb9WS9WO/Wx6Q1Z01ndtEfWJ8/K6eXFQ==</latexit> ₂ = ln tr⇢²

Adesso, Girolami, Serafini, PRL, 109, 190502 (2012)

With the Wigner entropy we can now separate

dS

dt = ⇧

As a result, we find

= n + 1/2



ha^†ai n = 1

!(n + 1/2)

dQ dt

⇧ = 4

(n + 1/2) Z

d²↵ |J(W )|²

W = dS(W ||W_eq) dt

At high temperatures which leads to ^{!(n + 1/2)} _' ^T _' ¹

T

dQ dt

NON-EQUILIBRIUM STEADY-STATES (NESS)

William B. Malouf, Jader P. Santos, Mauro Paternostro and Gabriel T. Landi Wigner entropy production in quantum non-equilibrium steady-states

In preparation (2018).

M. Brunelli, et. al., arXiv 1602.06958. Accepted in PRL.

OPTOMECHANICS

A thin membrane is allowed to vibrate in contact with radiation trapped in a cavity.

Aspelmeyer group Viena

H = !_ca^†a +

✓ p²

2m + 1

2m!_m² x²

◆

ga^†ax + ✏(a^†e ^i!pt + ae^i!pt)

d⇢

dt = i[H, ⇢] + D_c(⇢) + D_m(⇢)

Groeblacher, et. al., Nature Communications, 6, 7606 (2015)